TWO ENABLING DEVELOPMENTS FOR COMPUTATIONAL CONTACT MECHANICS By Gaurav Chauda A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering โ€“ Doctor of Philosophy 2021 ABSTRACT TWO ENABLING DEVELOPMENTS FOR COMPUTATIONAL CONTACT MECHANICS By Gaurav Chauda Two approaches for computational contact mechanics are explored to provide an accurate and robust numerical simulation of frictional contact. First, a novel quasi-static contact algorithm (Method of First Violation) is developed, which provides a solution over the load history that is admissible in the sense that all equality and inequality constraints are satisfied at each load step and every point between load steps. The central idea of this algorithm is to calculate each load increment to be exactly which is necessary to cause the first nodal inequality constraint to be violated. At the end of that load increment, the status of the associated node is switched to a different status (such as from contact to non-contact) where the inequality constraint is replaced with an equality constraint, and a new inequality constraint is introduced. Importantly, the constraints of the previous and the new node status are simultaneously satisfied at that point in the loading cycle. The Method of First Violation (MFV) algorithm is illustrated using a compliance formulation for degrees of freedom on the surface of a half-plane loaded by a rigid indenter with friction. The compliance formulation permits considering only the degrees of freedom in the vicinity of contact, therefore extremely fine surface discretization is possible using only a modest number of degrees of freedom. Because this contact problem has been well studied using analytic techniques and because of the extremely fine mesh resolution, it is possible to perform a critical comparison with analytical solutions. An added benefit of this algorithm is its greatly improved robustness. This indenter/half-plane discretization was also used to test MFV where friction was represented by a two-parameter (static and kinematic) model. Though such problems are notoriously ill-conditioned, the MFV remained robust. It was also used to explore a problem involving the Dahl friction model. Second, a compliance matrix formulation is derived for a two-dimensional elastic disk and for a hole in an infinite elastic plane. The circular disk/hole surface is discretized finely with a uniform distribution of nodes and a set of basis functions for traction are defined so that each basis function has a value of 1 in the vicinity of the corresponding node and a value of zero elsewhere. The surface displacement field associated with each traction basis function is obtained through a laborious derivation involving Fourier series and terms of the Michellโ€™s Airy stress function. The compliance matrix is constructed using the calculated displacement fields evaluated at each node. Fortunately, one may restrict attention to only the tractions and displacements in the vicinity of contact because all surface tractions outside that region are zero so the resulting compliance matrix is of tractable size. This semi-analytical formulation provides a direct construction of a compliance matrix with an extremely fine surface discretization, easy implementation of friction models, and accommodation of elastic coupling. The Method of First Violation and the compliance formulation of two-dimensional circular bodies are implemented together to numerically simulate multiple contact problems. Five different geometries are designed using cylinders, holes, and half-plane, where two contacting bodies are pressed against each other and then sheared cyclically for four cycles. Surface tractions, stick ratios, steady-state dissipation, and error in surface tractions are calculated using Goodman decoupling and full decoupling approximations. The results are compared with calculations employing full elastic coupling to assess the qualitative and quantitative ramifications of the decoupling assumptions. The surface tractions calculated using Goodman decoupling show qualitative similarity with those of full elastic coupling but showed some severe errors in predicting the dissipation. The full decou- pling approximation simulations predictions are significantly different from the predictions of full coupling calculations. However, they are no worse than the Goodman decoupling approximations in predicting dissipation. This ability to introduce extremely fine mesh resolution in a contact region with a robust contact algorithm allows us to explore many contact characteristics through numerical simulation, which were previously intractable. Copyright by GAURAV CHAUDA 2021 This thesis is dedicated to my parents. v ACKNOWLEDGEMENTS Your acknowledgements here. I express my sincere gratitude to my advisor Prof. Daniel J Segalman for mentoring me throughout my doctoral studies. Your motivation kept me working even in the gloomy days of my studies. I learned a lot from your stories and experiences. My sincere thanks to my Dissertation committee Prof. Brian Feeny, Prof. Thomas Pence, Changyong Cao, and Prof. James Barber (from University of Michigan) for guiding me during my studies. I want to thank Prof. Matt Allen (from University of Wisconsin) and his research group for providing helpful suggestions during this research. I would like to thank Prof. Matthew Brake and Prof. Christoph Schwingshackl for letting me participate in the NDCSI summer programs. I thank my parents Kailash and Anita, my sister Damini, and brother-in-law Nitin for their support in my studies. I would like thank lab colleagues Fatemeh, Ayse, Aakash, Jun, and Jamal for making these studies fun and enjoyable. I appreciate all the help and support provided by friends Pratap, Anuj, Suhas, Dhrubajit, Anshu, HP, Vivek, Jhalak, Hannah, and Prithvi specially for the meaningful discussions and suggestions for PhD life and academic studies. I would like to thank my friends Chelsea, Paige, Andya, Marisol, Thang, and my MSU salsa club family for making my stay at MSU enjoyable. vi TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Friction-Free Contact Algorithms . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Contact Algorithms with Friction . . . . . . . . . . . . . . . . . . . . . . 7 1.3.3 Path Dependence in Contact Algorithms . . . . . . . . . . . . . . . . . . . 8 1.4 Thesis Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 I The Method of First Violation (Contact Algorithm) 11 CHAPTER 2 CONTACT ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Compliance Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Taxonomy of Contact Status . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Compliance Matrix with Contact Constraints . . . . . . . . . . . . . . . . 14 2.2 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Algorithm II: Method of First Violation . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Admissible Contact Histories . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Contact Status Violations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2.1 Geometric Violation . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2.2 Normal Pressure Violation . . . . . . . . . . . . . . . . . . . . . 23 2.3.2.3 Violation in Stick Constraints . . . . . . . . . . . . . . . . . . . 24 2.3.2.4 Violation in Slip Constraints . . . . . . . . . . . . . . . . . . . . 24 2.3.3 Single-Node Illustrations of Computing ๐›ฟ๐ฟ . . . . . . . . . . . . . . . . . 24 CHAPTER 3 VERIFICATION OF CONTACT ALGORITHMS . . . . . . . . . . . . . . 28 3.1 Frictionless Normal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Rigid Cylinder Pressed against an Elastic Half-plane . . . . . . . . . . . . 28 3.1.2 Rigid Flat Indenter Pressed against an Elastic Half-plane . . . . . . . . . . 29 3.2 Normal Contact with the Coulomb Friction . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 Comparison with Spence (1973) Stick Ratio Results . . . . . . . . . . . . . 32 3.2.2 Verification of Numerical Lateral Traction with an Analytical Expression . 35 3.2.3 Verification of Stick-Ratio in the Mindlin Problem. . . . . . . . . . . . . . 35 3.3 Coulomb Friction Model in Mindlin Problem . . . . . . . . . . . . . . . . . . . . 37 3.4 Two-parameter Coulomb Friction Model in Mindlin Problem . . . . . . . . . . . . 38 vii 3.5 High-Coefficient of Friction with Full Elastic Coupling . . . . . . . . . . . . . . . 40 3.6 Implementation of Dahl Friction Model . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Two-parameter Coulomb Friction for a Cylinder Pressed against an Elastic Half-plane 44 3.7.1 Cyclic Shear Loading with Two-Parameter Coulomb Friction . . . . . . . . 46 3.8 Algorithm Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.9 Summary of the Algorithm Verification and Comparison . . . . . . . . . . . . . . 49 II An Extremely Fine Mesh Strategy Using Compliance Matrices 52 CHAPTER 4 COMPLIANCE FORMULATION FOR AN ELASTIC DISK AND FOR A HOLE IN AN INFINITE ELASTIC PLANE . . . . . . . . . . . . . . . . 53 4.1 Computational Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 2D-Cylinder Airy Stress Function (Michell Solution) . . . . . . . . . . . . . . . . 57 4.2.1 General Discretization Strategy . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.2 A Solid Elastic Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.2.1 Creating the Surface Displacement Field in an Elastic Disk . . . . 61 4.2.3 A Hole in an Infinite Elastic Plane . . . . . . . . . . . . . . . . . . . . . . 64 4.2.3.1 Creating Surface Displacement Field for a Hole in an Infinite Plane 64 4.3 Creating a Compliance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Summary of Compliance Matrix Formulation . . . . . . . . . . . . . . . . . . . . 71 CHAPTER 5 VERIFICATION OF COMPLIANCE FORMULATION AND OTHER CONTACT PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 Verification Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.1 An Elastic Cylinder Pressed against a Rigid Half-plane . . . . . . . . . . . 73 5.1.2 Rigid Cylinder Pressed inside of a Hole in an Infinite Elastic Plane . . . . . 76 5.1.3 Normal Contact with Friction for an Elastic Cylinder Pressed against a Rigid Half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.4 von-Mises Stress Distribution in an Elastic Cylinder in Contact . . . . . . . 78 5.2 Two Identical Elastic Cylinders Pressed and Sheared . . . . . . . . . . . . . . . . . 80 5.2.1 Comparing Surface Tractions and Displacement . . . . . . . . . . . . . . . 81 5.2.2 Comparing Stress Distributions inside the Cylinders . . . . . . . . . . . . . 83 5.3 Summary of the 2D Compliance Verification . . . . . . . . . . . . . . . . . . . . . 85 IIIComparison of Decoupling Approximations and their Ramifications 86 CHAPTER 6 ELASTIC DECOUPLING APPROXIMATIONS . . . . . . . . . . . . . . . 87 6.1 Decoupling Approximations in Elastic Contact . . . . . . . . . . . . . . . . . . . . 87 6.2 Analytical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2.1 Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.3 Surface Tractions for a Cyclic Lateral Force . . . . . . . . . . . . . . . . . . . . . 91 6.4 Variation with Lateral Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5 Variation with Coefficients of Friction . . . . . . . . . . . . . . . . . . . . . . . . 99 viii 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.7 Summary of Effects of Elastic Decoupling Approximations . . . . . . . . . . . . . 103 CHAPTER 7 CONCLUSION AND FUTURE RESEARCH . . . . . . . . . . . . . . . . . 105 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 APPENDIX A COMPLIANCE MATRIX FORMULATION FOR ELASTIC HALF- PLANE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 APPENDIX B IMPLEMENTAION OF DAHL FRICTION MODEL WITH ALGORITHM- II (METHOD OF FIRST VIOLATION) . . . . . . . . . . . . . . . . 114 APPENDIX C MICHELL COEFFICIENTS FOR A SOLID ELASTIC DISK WITH NO HOLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 APPENDIX D MICHELL COEFFICIENTS FOR AN INFINITE ELASTIC PLANE WITH A CIRCULAR HOLE . . . . . . . . . . . . . . . . . . . . . . 123 APPENDIX E THE MICHELL SOLUTION- STRESS AND DISPLACEMENT COMPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 ix LIST OF TABLES Table 2.1: Equality and inequality constraints for each node status type. . . . . . . . . . . 14 Table 4.1: Maximum contact arc for compressing an elastic disk onto a flat rigid surface above which the disk material exceeds its elastic limit. . . . . . . . . . . . . . . 55 Table 4.2: Michell coefficients for applied radial and tangential unit tractions over ฮ”๐œƒ centered at ๐œƒ = 0 and ๐‘Ÿ = ๐‘… on the elastic solid disk . . . . . . . . . . . . . . . 63 Table 4.3: Michell coefficients for applied radial and tangential tractions over ฮ”๐œƒ centered at ๐œƒ = 0 and ๐‘Ÿ = ๐‘… for the hole in the elastic plane. . . . . . . . . . . . . . . . . 65 Table B.1: Equality and inequality constraints for Dahl friction model. . . . . . . . . . . . 115 Table E.1: The terms of Michell Solution (Airy Stress Function) and their corresponding stresses and displacements. Terms shown in the cyan are the self-equilibrating part of the solid disk stress function, terms shown in white are the self- equilibrating part of the Hole in a elastic plane, and terms shown in gray are modified secular terms removing their multi-value through linear combina- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Table E.2: The terms of Michell Solution (Airy Stress Function) and their corresponding stresses and displacements. Terms shown in the cyan are the self-equilibrating part of the solid disk stress function and terms shown in white are the self- equilibrating part of the Hole in a elastic plane. . . . . . . . . . . . . . . . . . . 132 x LIST OF FIGURES Figure 2.1: Taxonomy of nodal status. The node status types encountered in contact analysis are indicated by thick borders. . . . . . . . . . . . . . . . . . . . . . . 14 Figure 2.2: Algorithm I: A contact algorithm similar to Ahn & Barber flow chart . . . . . . 17 Figure 2.3: Algorithm II: The Method of First Violation contact algorithm with adaptive load step flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Figure 2.4: Comparison between a conventional contact algorithm and the Method of First Violation contact algorithm. With conventional contact algorithms, a load step is selected and an iterative procedure is used to find an admissible set of loads and displacements consistent with that load step. With the Method of First Violation, the load step is computed to be exactly that necessary to bring the first node to change status (contact, non-contact, stick, and slip) . . . . 22 Figure 2.5: Single node illustration for contact and non-contact violations. The thick (blue) solid dashed lines represent admissible configurations and the thin (orange) solid and dashed lines represent inadmisible configurations. For instance if a node begins not-in-contact (thick dashed line on the left), as the load step ๐›ฟ๐ฟ increases, the gap closes linearly until contact occurs at ๐›ฟ๐ฟ ๐‘ค , as the load further increases the gap stays at zero and the contact pressure increases linearly (thick solid line on the right.) . . . . . . . . . . . . . . . . . . 25 Figure 2.6: Single node illustration for stick and slip violations. The dark (blue) solid and thick lines represent admissible configurations and the lighter (orange) solid and dashed lines represent inadmissible configurations. For an initially stuck node (dashed thick (blue) line), calculation of shear traction exploits linearity of that traction to obtain the new ๐›ฟ๐ฟ ๐œ of stick violation. For an initially slipping node (solid thick (blue) line), calculation of change in ๐‘ ๐‘”๐‘›( ๐‘ข) ยค will require the method of bisection to obtain ๐›ฟ๐ฟ ๐‘ขยค of slip violation. . . . . . . . . . 25 Figure 3.1: Verifying the half-plane compliance matrix formulation with the analytical solution. The two numerical algorithms yielded identical solutions. The โˆ— and  represent numerically calculated traction at the centers of intervals. The continuous curves are analytic . . . . . . . . . . . . . . . . . . . . . . . . 29 Figure 3.2: Verifying the half-plane compliance matrix formulation with the analytical solution. The two numerical algorithms yielded identical solutions. The โˆ— and  represent numerically calculated traction at the centers of intervals. The continuous curves are analytic . . . . . . . . . . . . . . . . . . . . . . . . 30 xi Figure 3.3: Verifying numerical normal and lateral traction distribution with analytical expression for rigid cylinder pressing against an elastic half-plane with coef- ficient of friction (๐œ‡ ๐‘  = 0.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 3.4: Verifying numerical normal and lateral traction distribution with analytical expression for rigid flat indenter pressing against an elastic half-plane with coefficient of friction (๐œ‡ ๐‘  = 0.3). . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 3.5: Stick ratio for cylindrical indenter with and without decoupling assumption . . . 33 Figure 3.6: Stick ratio for flat indenter with and without decoupling assumption . . . . . . . 33 Figure 3.7: Stick ratio vs friction with full elastic coupling for cylindrical indenter . . . . . 34 Figure 3.8: Stick ratio vs friction with full elastic coupling for flat indenter . . . . . . . . . 34 Figure 3.9: Mapping of two elastic cylinder contact to the rigid cylinder over the half-plane 36 Figure 3.10: Verifying numerical stick ratio with analytical expression for Mindlin solution. . 37 Figure 3.11: Comparing numerical tractions from Algorithm-I, Algorithm-II, and analyt- ical expression for one-parameter friction model (๐œ‡ = 0.7) after application of lateral force: normal pressure distribution (left) and lateral traction distri- bution (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Figure 3.12: Comparing numerical tractions from Algorithm-I and Algorithm-II for two- parameter friction model (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.42) after application of lateral force: normal pressure distribution (left) and lateral traction distribution (right). For Algorithm-I, the region of stick is indicated by a solid black line, and the region of slip is indicated by a black line connecting solid green circles (โ€ข). For Algorithm-II, the region of stick is indicated by a dashed blue line, and the region of slip is indicated by a dashed blue line connecting red dots(โ€ข). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 3.13: Normal and lateral traction distribution after a normal and lateral pressing of the cylinder against an elastic half-plane with coefficient of friction ๐œ‡ ๐‘  = 3. Comparing the Algorithm-I and Algorithm-II with 85 contact nodes in both cases. 41 Figure 3.14: Normal and lateral traction distribution after a normal and lateral pressing of the cylinder against an elastic half-plane with ๐œ‡ ๐‘  = 3. Comparing the tractions predicted by Algorithm-II using a fine mesh (FM) having 85 nodes in the contact patch and those predicted by Algorithm-II using a coarse mesh (CM) having only 42 nodes in the contact patch. . . . . . . . . . . . . . . . . . 42 xii Figure 3.15: Implementing of the Method of First Violation algorithm with the Dahl fric- tion model: (๐œŽ = 1000, ๐›พ = 1, ๐œ‡ ๐‘  = 0.3). Algorithm-I would not converge to a solution for Dahl friction model. . . . . . . . . . . . . . . . . . . . . . . . . 43 Figure 3.16: Normal and lateral traction distribution calculated for two-parameter friction (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.49) through Algorithm-I and Algorithm-II after a rigid cylinder is pressed against an elastic half-plane with a normal force (๐‘ƒ = 0.001). 44 Figure 3.17: Normal and lateral traction distribution calculated for two-parameter friction (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.49) through Algorithm-I and Algorithm-II after a rigid cylinder is pressed against an elastic half-plane with a normal and lateral forces (๐‘ƒ = 0.001, ๐‘„ = 0.000185). Slipping nodes are represented through โ€™red oโ€™ and โ€™Green *โ€™ for A-I and A-II respectively. . . . . . . . . . . . . . . . . 45 Figure 3.18: Evolution of lateral traction with lateral cyclic loading (Arrow shows the direction of lateral loading.) All tractions shown are physical traction nor- malized by ๐บ. [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Figure 3.19: Number of iterations (๐‘๐‘–๐‘ก๐‘’ ) vs load-step (๐›ฟ๐ฟ 0 ) using Algorithm-I for coarse to fine contact mesh (red marks shows the bad converged solution). Optimal load increments for this method on this problem are suggested by the dashed blue curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Figure 3.20: Number of iterations (๐‘๐‘–๐‘ก๐‘’ ) vs load-step (๐›ฟ๐ฟ 0 ) using Algorithm-II(MFV) for coarse to fine contact mesh (Black marks shows the optimal load-step for faster and accurate solution). The blue dashed line is copied from Figure 3.19. . 50 Figure 4.1: A contact problem with normal (๐‘ƒ) and lateral (๐‘„) forces are applied pressing an elastic disk against a rigid surface with 33 boundary nodes in contact (11 nodes in each 4๐‘œ contact region). . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 4.2: Flow chart using Michell expansion to create a compliance matrix. . . . . . . . 58 Figure 4.3: A solid disk with discretizaton from โˆ’๐œ‹ to ๐œ‹ . . . . . . . . . . . . . . . . . . . 61 Figure 4.4: The displacement field resulting from the unit pressure over ฮ”๐œƒ centered at ๐œƒ = 0 is obtained by summing a Fourier series. Though there would be Gibbs phenomena if the pressure field were recovered from Fourier series, there is no Gibbs phenomena associated with the displacement field. . . . . . . . . . . 62 Figure 4.5: An infinite plane with a hole discretized from โˆ’๐œ‹ to ๐œ‹ . . . . . . . . . . . . . . 65 Figure 4.6: On moving the unit normal pressure the corresponding displacement field (dashed) is also moving that is one of the columns of compliance matrix . . . . 68 xiii Figure 4.7: Shifting of the displacement basis to create compliance matrix . . . . . . . . . 70 Figure 5.1: Two concentric elastic cylinders pressed against each other . . . . . . . . . . . 73 Figure 5.2: A rigid cylinder pressed inside a hole in an elastic half-plane . . . . . . . . . . 74 Figure 5.3: Comparison of analytical and numerical normal pressure distribution for a elastic cylinder pressed against a rigid half-plane with a coarse mesh (on top) and with a fine mesh (on bottom). . . . . . . . . . . . . . . . . . . . . . . . . . 75 Figure 5.4: Rigid cylinder indenter pressed against a hole inside an infinite elastic plane. . . 76 Figure 5.5: Elastic cylinder pressed against a rigid half-plane with friction. The stick ratios for multiple friction coefficients are compared with the analytical ex- pression provided by Spence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Figure 5.6: Mindlin contact mapping is created using elastic cylinder pressing on a rigid half-plane with friction. Stick ratio for multiple friction coefficients are compared with analytical expression provided by Mindlin. . . . . . . . . . . . . 78 Figure 5.7: The Mindlin two-cylinder contact problem is mapped into that of an elastic cylinder pressed against a rigid half-plane with friction. The numerically calculated lateral traction distribution (connected lines) for multiple friction coefficients are compared with analytical values (dashed lines). . . . . . . . . . 79 Figure 5.8: Contour plot for normalized von-Misses stress inside the elastic disk on pressing against rigid plane on friction free surface with contact angle 0.64โ—ฆ . The positions are non-dimensionalize with the contact length parameter (๐‘Ž). . . 80 Figure 5.9: Surface traction and displacement distributions for three cases: normal force ๐‘ƒ, only (left); Shear force, ๐‘„ applied after normal force ๐‘ƒ and employing a Coulomb friction model (center); Shear force ๐‘„ applied after normal force ๐‘ƒ and employing a two-parameter friction model (right).[17] . . . . . . . . . . . . 82 Figure 5.10: Stress distributions for three cases: normal force, ๐‘ƒ, only (left); Shear force, ๐‘„ applied after normal force and employing a Coulomb friction model (center); shear force, ๐‘„ applied after normal force and employing a two-parameter friction model (right). Horizontal and vertical coordinates are normalized by ๐‘Ž. [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 6.1: Surface traction distributions for three cases: fully coupled (FC) (in left); Goodman decoupled approximation (GD) (in center); fully decoupled ap- proximation (FD) (in right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 xiv Figure 6.2: Normalized dissipation hysteresis and non-linear part of hysteresis: fully coupled (in left); Goodman decoupled (in center); fully decoupled (in right). Non-linear part is generated by removing the linear . . . . . . . . . . . . . . . 94 Figure 6.3: For the five contact geometries, contact parameters such as Dissipation, Stick ratio, and errors in surface tractions are plotted against over a range of lateral หœ All calculation involved ๐œ‡ = 0.7 friction coefficient. . . . . . . . . . 96 force (๐‘„). Figure 6.4: For the five contact geometries, contact parameters such as Dissipation, Stick ratio, and errors in surface tractions are plotted against over a range of friction coefficients (๐œ‡). All calculation involved ๐‘„หœ = ๐‘„/(๐œ‡๐‘ƒ) = 0.99 lateral force. . . . 100 Figure 6.5: For a rigid cylinder pressed inside a circular hole of twice radius, contact parameters such as Dissipation, Stick ratio, and errors in surface tractions are plotted against over a range of friction coefficients (๐œ‡) and five different หœ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 lateral ratio ๐‘„. Figure A.1: Coordinates ๐‘ฅ, ๐‘ฆ; tractions ๐‘, ๐œ; and displacements ๐‘ข, ๐‘ค on the surface of an elastic half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure A.2: Unit normal pressure and corresponding displacement on shifting by โˆ’ฮ”๐‘ฅ and ฮ”๐‘ฅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Figure B.1: Comparing Dahl friction model with Coulomb friction model. . . . . . . . . . . 114 Figure C.1: Stress distribution on the solid disk is equlibrated with forces and moment near the center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Figure D.1: Stress distribution on the hole should be equilibrated with forces and moment applied on the indenter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 xv KEY TO ABBREVIATIONS ๐›ฝ Material Mismatch parameters (Dundurโ€™s Parameter) ๐œƒ angle at the contacting node form center ๐œ‡ coefficient of friction for one parameter friction ๐œ‡ ๐‘  Static coefficient of friction for two parameter friction ๐œ‡ ๐‘˜ Dynamic coefficient of friction for two parameter friction ๐œˆ Poissonโ€™s ratio ๐œŽ๐‘Ÿ๐‘Ÿ Radial Stress field ๐œŽ๐‘Ÿ๐œƒ Tangential Stress field ๐œŽ๐œƒ๐œƒ Circumferential Stress field ๐œŽ๐‘ฃ๐‘€ von-Mises Stress field ๐›ฟ๐ฟ ๐‘œ Load-step ฮ”๐œƒ angle of discretized interval in the contact region ฮ”๐‘ฅ length of discretized interval in the contact region ฮฆ๐‘œ Michellโ€™s Airy stress function a Half of the contact length b Half of the stick length c Stick ratio (๐‘ = ๐‘/๐‘Ž) ๐‘ Normal surface traction ๐‘ = ๐œ ๐‘ฆ ๐‘๐‘Ÿ Radial surface traction ๐‘ ๐œƒ Tangential surface traction ๐‘ 0 Normalizing pressure = 2๐‘ƒ/๐œ‹๐‘Ž ๐œ Horizontal surface traction ๐œ = ๐œ ๐‘ฅ ๐‘ข Horizontal displacement ๐‘ข ๐œƒ Tangential displacement ๐‘ค Normal displacement field xvi ๐‘ค ๐‘Ÿ Radial displacement ๐ถCart Compliance matrix in Cartesian coordinate ๐ถPolar Compliance matrix in polar coordinate ๐ถ๐‘ค ๐‘ Compliance sub-matrix correlates normal displacements to normal pressure ๐ถ๐‘ค๐œ Compliance sub-matrix correlates normal displacements to lateral traction ๐ถ๐‘ข ๐‘ Compliance sub-matrix correlates lateral displacements to normal pressure ๐ถ๐‘ข๐œ Compliance sub-matrix correlates lateral displacements to lateral traction ๐บ Elasticity shear modulus ๐พ Twice the mean curvature of contacting bodies ๐‘ƒ Normal force ๐‘„ Lateral force ๐‘„หœ Non-dimensional lateral force ratio ๐‘„หœ = ๐‘„/(๐œ‡๐‘ƒ) ๐‘๐‘–๐‘ก๐‘’ Number of iterations A-I Algorithm-I, Conventional algorithm A-II Algorithm-II, Method of First Violation ๐ด๐‘š๐‘› Coefficient of Airy stress function ๐ต๐‘š๐‘› Coefficient of Airy stress function FD Full decoupling approximation FC Full elastic coupling GD Goodman decoupling approximation MFV Method of First Violation Contact Algorithm xvii CHAPTER 1 INTRODUCTION 1.1 Overview Two approaches for computational contact mechanics are explored to provide an accurate and robust numerical simulation of frictional contact. First, a novel contact algorithm - the Method of First Violation (MFV) - is developed and verified with available analytical solutions. The algorithm is compared with a more conventional algorithm on standard Coulomb friction indenter problems and then tested on problems involving two other friction models. Second, because of the computational advantages of compliance matrices in contact analysis, a compliance matrix formulation is derived for two-dimensional elastic disks and for circular holes in an infinite elastic plane. 1.2 Motivation Structures involving bolted, riveted, and other types of joints for which frictional interfaces are fundamental, are ubiquitous in our industrial environment and the dynamics of such structures are greatly complicated by their interfaces, where these contacting surfaces suffer wear and tear and dissipate energy [2, 14, 61, 33, 73]. These contacting effects are good or bad depending on the structureโ€™s requirements, such as linearity, energy conservation, or higher damping near resonance. It is not easy to understand the actual dynamics due to the inaccessibility of the interfaces during the dynamic loading. Researchers have worked on the contact problems analytically and numerically in several ways, such as considering or ignoring friction, 2D or 3D elastic contact, and with full elastic coupling or with various decoupling approximations. To be tractable, these problem formulations still require substantial idealization. The simplest problem that contains the interface mechanics associated with joint structures is the 2D contact between two co-axial elastic cylinders pressed together or a rigid cylinder pressing on an elastic half-plane. The frictionless case of such problems is commonly known as a 2D 1 Hertzian problem. Introducing Coulomb friction to the problem brings complications because of the inequality conditions for stick or slip constraints and an elastic coupling between normal and shear tractions due to Poissonโ€™s effects. Though coupling is not necessarily an issue with numerical solutions, it makes analytical analysis difficult and is usually mitigated using various decoupling approximations. The most common decoupling approximation is attributed to Goodman [31] as- serting that โ€œnormal displacements resulting from lateral traction that are negligible in comparison to normal displacements resulting from normal traction." This decoupling approximation has been exploited regularly by researchers such as Mindlin [52], and Spence [63, 64] to provide analytical expressions for the width of the contact patch, stick-slip regions in the contact patch, lateral trac- tion, displacement distributions, and relations between the lateral force and stick-slip ratio in terms of applied loads or displacements. Another, more severe, decoupling approximation ignores any elastic coupling between normal and lateral loads so that โ€œnormal and lateral displacements result only from normal force and lateral force respectively" [30, 69]. This decoupling approximation (complete decoupling) reduces the complexity of the problem significantly and results in faster though not-so-accurate predictions for contact parameters. It is mostly implemented to understand cyclic loading and uneven contact. A serious limitation of most contact analyses is that only Coulomb friction is considered. However, Coulomb friction is the ๐‘ง๐‘’๐‘Ÿ๐‘œ๐‘กโ„Ž order approximation of frictional contact, and it performs poorly in cases involving the calculation of partial slip, such as is the case in structures with joint [67]. In order to develop better predictive joints models, it is necessary to explore other friction models, particularly with respect to their performance in partial slip. The underlying goal of this research effort is to develop a test-bed for the exploration of various friction models (such as identified by [11] ) that, when used in contact models, will yield predictions that can be compared to experiments reported for mechanical joints [67]. These friction models bring their own complexity due to the introduction of multiple friction parameters and possibly more inequality constraints that must be incorporated in the implementation. 2 There is a very long history of calculating interface mechanics in the context of classical elasticity, but the intrinsic difficulty of the problem has restricted the number and nature of analytic solutions available, as well as the robustness and fidelity of numerical simulations. The numerical contact algorithms that have been developed over the decades appear to address most of the problems of practical importance, but they still suffer from issues of robustness and fidelity to the underlying equations. These limitations must be overcome to provide a test-bed for the study of the interface mechanics of joints. A traditional complaint about numerical contact algorithms is that they often do not adequately take into account or correctly reproduce the path dependence found in analytic solutions [12, 47, 45]. The core notion motivating the new approach is that a valid numerical contact algorithm must satisfy all elasticity equations and all equality and inequality constraints of contact not only at the beginning and end of each load or displacement increment, but at all points in between. In this study, contact histories that conform to the above conditions are referred to as admissible. An algorithm, referred to as the Method of First Violation, developed to achieve admissible contact histories, appears to provide more physically reasonable predictions โ€“ particularly with respect to path dependence โ€“ than standard contact algorithms as well as greater robustness. In order to get insight into interface mechanics through computation, an extraordinary fine mesh on the interface is required. The more prominent methods of surface discretization are: assembly of stiffness matrices obtained from finite elements, obtaining mixed system matrices from the boundary integral method, and, where possible, the development of a compliance matrix via semi- analytic methods. The stiffness matrix of the finite element formulation for contact problems tends to be very large because accuracy requires not only many nodes on the surface, but at least a gradual transition of element sizes away from the surface. A process of static reduction [23] is then used to replace the (often huge) stiffness matrix with one that involves only surface degrees of freedom. This process of resolving out internal degrees of freedom is routinely done, but is computationally challenging [25, 69]. The boundary integral method involves only nodes on the surface and requires a gradual 3 transition of element size away from the contact patch. The results is two system matrices much smaller than those obtained via finite element method: one acting on a column of surface tractions and the other acting on a column of displacements [48]. These two matrices are resolved to yield a stiffness or compliance matrix, but a compliance matrix is preferable for the reasons discussed next. There are a few geometries (such as the surface of a half plane) where it is possible to construct a compliance matrix using semi-analytic methods. The only rows and columns of such a compliance matrix that need to be considered are those associated with nodes in the vicinity of the contact patch, all surfaces outside that contact region are traction-free. This facilitates the creation of an extremely fine mesh without necessarily requiring the solution of an unreasonably large system of equations. For numerical experiments on contact algorithms, compliance matrix for the half-plane is constructed using analytic expressions for the displacement field resulting from uniform tractions over finite intervals on the surface [35, 53]. The advantage of constructing a compliance matrix is that it avoids having to resolve down the system of equations resulting from finite element or boundary integral method. Additionally, the compliance formulation accommodates the complexity of frictional contact with various levels of elastic coupling, which can be turned on and off on demand. In order to have these same advantages in contact calculations of curved 2D elastic bodies, a method for direct calculation of compliance matrices for planar elastic disks and holes in an infinite elastic plane was developed. 1.3 Literature Review Solving a contact problem depends on a robust and efficient algorithm (static, dynamic, friction, or no-friction). There are multiple kinds of algorithms in literature such as contact detection algo- rithms, mortar-based algorithms, Lagrangian-based contact algorithms with or without Coulomb friction, contact optimization algorithms, mesh-based and meshless contact algorithms, and Quasi- static contact algorithms [13]. In this paper, we focus on the Quasi-static contact algorithms for elastic contact and discuss the various algorithms to capture the quasi-static contact process with 4 and without friction. 1.3.1 Friction-Free Contact Algorithms Conry and Seireg[22], and Kalker and van Randen[43] are among the earlier examples of algorithms for friction-free contact analysis. Conry and Seiberg [22] used a simplex algorithm for solving the contact problem through linear complementarity conditions. On the other hand, Kalker and van Randen [43] solved the normal contact between two dissimilar elastic materials by employing two energy functions (contact elastic energy and contact enthalpy). Minimization of these energy functions was used to estimate the pressure distribution between two contacting surfaces through a quadratic programming approach because of their convex behavior. This approach became the baseline for developing new contact methods and approaches with advancement in computational capability. Polonski and Keer [54] used the conjugate gradient method (CGM) to minimize the contact energy functions to find the pressure distributions on a coarser mesh and then used the multi-level multi-summation (MLMS) technique to get the results for a finer mesh in a normal contact analysis. This approach resulted in a faster estimation of results in a very finely discretized domain. Johnson [38] used the Fourier representation of pressure and displacement to develop a rela- tionship for estimating normal pressure distribution through surface displacements and vice-versa. Ju and Farris [39] further investigated this convolution and developed an error estimation method. They observed that the loss/error of high-frequency content in pressure FFT does not cause large errors in displacement and load, while the loss/error in the low frequency does lead to large errors in displacement and load. Therefore the contact length needs to be significantly small compared to the length of the FFT window. They also suggested various other ways to reduce errors for generalized two surface contact. Liu and Wang [50] further explained this convolution domain for an elastic punch on a rigid half-plane. Because of the implicit periodicity associated with the use of Fourier methods, there can be artifacts of interference between tractions fields in adjacent periods and Liu and Wang showed how this is mitigated if the spatial periods are chosen to be large relative 5 to the contact patch. They further introduced a Discrete Convolution Fast Fourier Transformation (DC-FFT) technique, where influence coefficients from finite element analysis were mapped onto the spatial frequency domain. They explored five different approaches to reduce the numerical errors when solving the discretized normal contact problem. They identified DC-FFT as the fastest method to solve the line contact problem. Stanely and Kato [66] used the FFT relations provided by Johnson [38] with minimization of total complementary energy to solve for the contact between two surfaces, demonstrating fast convergence for two-dimensional normal contact. Allwood [3] provided a brief survey on contemporary (2005) methods of tackling the contact problem. He employed Kalkerโ€™s approach of obtaining a stiffness matrix through FEM analysis and then used that stiffness matrix to compare three different contact algorithms. In the first technique he examined, a reduced stiffness matrix is developed using Duncan/Gunyan [25] reduction, and that reduced matrix is solved subject to the constraints associated with the currently assumed status of the nodes. An iterative process is established where one node status is changed at a time, and the elasticity problem is resolved. The second method was based on Kalkerโ€™s [43] approach of first formulating the energy function and later minimizing it with the conjugate gradient method. After minimization, if any contact nodes do not satisfy their inequality constraints, then their status is updated, and the minimization is done again. This iterative procedure is continued until all nodes satisfy all contact constraints and inequalities. This method is slower than the other two because the energy function must be minimized in each iteration. Polonsky and Keerโ€™s [54] approach of FFT-based MLMS to calculate the contact was the third method explored by Allwood. This last method is the fastest but least accurate of the three methods. Allwood and Ciftci [4] provided a new algorithm based on incremental load again using a reduced stiffness matrix. The normal load is increased slowly to solve for the contact area and pressure iteratively until the required load is reached. In the case of any inequality condition that is not satisfied, a new load increment gets selected through a linear correlation such that the gap or penetration reduces to zero. This incremental approach resulted in a faster solution with a low number of iterations. This method by Allwood and Ciftci can be seen as an antecedent to the 6 Method of First Violation algorithm. 1.3.2 Contact Algorithms with Friction Friction brings complexity to the contact problem due to nonlinear interaction between shear loads and normal loads at each contacting point, requiring a solution for lateral traction fields and satisfaction of stick or slip inequalities. Kalker [40, 41] performed analysis on elastic cylinders in rolling using dry Coulomb friction. He used known Hertzian analytic expressions for the normal pressure and then calculated the lateral traction. He posed the traction problem as one of minimization subject to linear inequality constraints that could be solved using a simplex algorithm. He later derived a more general approach [42] by separately solving normal and lateral contact as a convex minimization problem. He devised algorithms NORM, TANG, and PANA to solve only normal contact, lateral contact, and coupled normal and lateral contact, respectively. In PANA, TANG and NORM were solved in turn one by one, where the result of one was input of the next. This process of sequential solving was repeated until the solution converged. Klarbring et al. [46] provided another approach for the quasi-static contact problem with dry friction and linear elasticity using variational inequalities. They observed that small increments in the loads could result in faster convergence. In other work, Klarbring et al. [45] discussed the non-uniqueness and non- existence of contact solutions for single node contact in several cases, when the friction coefficient is larger than the ratio of stiffness for normal contact (๐‘† ๐‘ ) to stiffness coupling between normal displacement and tangential load (๐‘† ๐‘๐‘‡ ), i.e. ๐œ‡ > ๐‘† ๐‘ /๐‘† ๐‘๐‘‡ . Cho and Barber [20] extended this by incorporating inertia of contacting interfaces to consider the dynamic effects of the problem. This non-uniqueness is also shown for a two node system by Andersson et al. [5] which can extend to multi-node systems. Cho and Barber [21] also addressed the issue of stability in three-dimensional contact as defining the direction of slip and friction force is difficult and can result in an inconsistent solution. Ahn and Barber [1] devised a contact algorithm for two-dimensional cyclic frictional contact that can distinguish the stick-slip boundary and capture the contact history. They solved normal 7 and lateral contact simultaneously, and used a reduced-order stiffness matrix with a Guass-Seidel procedure while updating each node to satisfy all the contact inequalities within some tolerance. Note that, this is a relaxation scheme requiring iteration at each load increment. All the contact equality and inequality constraints are satisfied at the end of each load step, but multiple node statuses are permitted to change in a single iteration. Chen and Wang [19] extended the FFT- based normal contact analysis of Liu and wang [50] to lateral contact analysis with Coulomb friction. They created discrete influence coefficients to obtain explicit relations between pressures and displacement distributions. They also solved normal and lateral contact separately using CGM and later combined those to satisfy the contact inequalities. Moreover, they changed constraints status in each iteration and verified the results with an analytical expression provided by Hills et al. [35]. Spinu [65] presented an approach based on Allwoodโ€™s method [4], first solving the normal contact and lateral contact separately and later coupling them to approximate a better contact solution. The method appears to provide fast convergence to solutions in three-dimensional elastic contact. 1.3.3 Path Dependence in Contact Algorithms Several authors [12, 45, 46, 49, 59] discuss the path dependence of contact problems with friction. Klarbring [46] demonstrated that the dissipative nature of friction leads to path dependence, so โ€œthe final configuration depends on the way loads reach their final state, not only on the state. For an incremental problem to capture exactly this phenomenon, the increments must be infinitesimal, which is impossible from the point of view of practical numerical calculations. Within one increment the path dependence is neglected, although when more increments are considered the path dependence is preserved". It is generally presumed โ€“ that for the discretized problem โ€“ the time steps may be finite, but they must be sufficiently small in a manner determined by the discretization. Bertocchi [12] showed that for a problem involving a single node, convergence is achieved, and path dependence is respected when the load increments are chosen precisely to bring the node to its 8 next status change. In this way, he addressed the change in node status resulting from increments in normal or lateral load or increments in both. He described the geometric nature of admissible status for various loading conditions, which can be used for multiple nodes wherein only one node changes its status at a time. As mentioned above, Klarbring et al. [45, 46] discussed the issue of existence and uniqueness of solutions of frictional contact problems and the role of load history with respect to the current traction/displacement state. Another view of the observation of Klarbring et al. [45, 46] is that two different load histories resulting in the same net force can have different interface traction distributions, differing by some self-equilibrating shear traction distribution. Or equivalently, the load history is captured in the traction distribution at the interface. A concern with methods that involve iterating through solutions without consistent states and status (non-physical configurations) is that they scramble the histories stored as interface traction. The above is not an issue with friction- free problems, where the solution is independent of history and can be solved by minimization of energy functions. 1.4 Thesis Synopsis This thesis presented here is divided into three parts. Each part is detailed and provides a proper explanation on its own. The reader may skip a part without loosing the thread of the discussion. In the first part, the development of a new contact algorithm (Method of First Violation) is described. To understand the need and motivation, a conventional contact algorithm is also discussed before delving into the Method of First Violation. Both contact algorithms are employed in a contact problem โ€“ a rigid cylinder pressing against an elastic half-plane โ€“ using a compliance matrix formulation. Both algorithms are initially verified with analytical expressions available in the literature and later compared with each other. Multiple friction models are employed to ensure the robustness of the Method of First Violation (MFV). The MFV contact algorithm also appears to capture path dependence associated with contact mechanics with good fidelity. In the second part, an extremely fine discretization method is engineered to create a compliance 9 matrix for the two-dimensional circular elastic cylinders, disks, and holes in an infinite elastic plane. The Michellโ€™s Airy stress function is used with the Fourier series expansion to derive a displacement basis. These displacement bases are assembled together to form a compliance matrix. The compliance matrices for elastic cylinders and holes are verified with analytical expressions where available. The test-bed of compliance matrix and Method of First Violation is tested for a contact problem of two identical elastic cylinders pressed together and sheared. This compliance formulation can analyze the contacting interface using a very fine mesh in the surface requiring the minimum number of degrees of freedom. In the third part, Goodman decoupling and full decoupling approximations are compared with full elastic coupling using five contact problem geometries. Contact problems, where contacting bodies are pressed against each other under a cyclic shear loading, are numerically simulated using the compliance matrix formulation and the MFV contact algorithm. Contact parameters such as surface tractions, stick ratios, dissipation, and L1 error in surface tractions are measured over a range of friction coefficients and lateral force ratios. Both qualitative and quantitative differences are identified and discussed. Lastly, a summary of the work undertaken in this dissertation is reported, highlighting the final conclusions and remarks. A few recommendations are provided for the continuation of this research. 10 PART I The Method of First Violation (Contact Algorithm) 11 CHAPTER 2 CONTACT ALGORITHMS Though the algorithm - Method of First Violation - can be employed for the contact of surfaces of two curved (and not necessarily convex) elastic bodies, for simplicity this method is illustrated here for the case of contact of a rigid indenter on an elastic half-plane. Further, though there is no barrier to employing this new algorithm on problems where discretization is developed in a stiffness formulation (as is most common in finite element method) or in a mixed formulation (such as usually results from boundary integral method), or as illustrated below where the discretization results in a compliance formulation. 2.1 Compliance Formulation In contact calculations discussed below, discretization is achieved using the equations for displace- ment resulting from a uniform normal or shear traction on a finite interval of a half-plane ([35, 72]). Each column of the compliance matrix is the collection of displacements due to the traction on a corresponding interval evaluated at the centers of the array of intervals. Because there are arbitrary horizontal and vertical displacements in the elasticity equations for an elastic half-plane, these arbitrary displacements are absorbed in rigid body displacements ๐‘Š๐‘ก๐‘œ๐‘ก and ๐‘ˆ๐‘ก๐‘œ๐‘ก in the compliance formulation of Eq. 2.1. ๏ฃฑ .. ๏ฃผ ๏ฃด ๏ฃฎ .. .. ๏ฃน ๏ฃฑ .. ๏ฃผ ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃฏ . . ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด 1 0 ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐ถ๐‘ค ๐‘ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐ถ๐‘ค๐œ ๏ฃบ ๏ฃด ๏ฃด ๐‘๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ .. .. ๏ฃบ ๏ฃด ๏ฃบ ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ . . ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๏ฃด ๏ฃบ๏ฃด๏ฃฒ ๏ฃฝ ๏ฃด ๐‘ข ๐‘› = ๏ฃฏ ๐ถ๐‘ข ๐‘ (2.1) ๏ฃฏ ๐ถ๐‘ข๐œ 0 1 ๏ฃบ ๐œ๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃด ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃฏ .. .. ๏ฃบ๏ฃบ ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ . . ๏ฃบ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃบ๏ฃด ๏ฃด ๏ฃฏ ฮ”๐‘ฅ . . . 0 0 0 ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ƒ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘Š๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘„ ๏ฃด ๏ฃฏ 0 . . . ฮ”๐‘ฅ ๏ฃด ๏ฃด 0 0 ๏ฃบ๏ฃป ๏ฃด ๏ฃด ๐‘ˆ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃณ ๏ฃพ ๏ฃฐ ๏ฃณ ๏ฃพ 12 where ๐‘ ๐‘› is the uniform pressure applied in interval ๐‘›; ๐œ๐‘› is the uniform shear traction applied in interval ๐‘›; ๐‘ค ๐‘› is the downward motion at the center of the interval ๐‘›; ๐‘ข ๐‘› is the horizontal displacement at the center of the interval ๐‘›; ๐‘ƒ is the net downward force applied by the indenter; and ๐‘„ is the net horizontal force applied by the indenter. The full mathematical formulation details are presented in appendix A. In this equation, the compliance matrix itself is divided into four parts (๐ถ๐‘ค ๐‘ , ๐ถ๐‘ค๐œ , ๐ถ๐‘ข ๐‘ , and ๐ถ๐‘ข๐œ ), where ๐ถ๐‘ค ๐‘ correlates normal displacements to normal pressure, ๐ถ๐‘ข๐œ correlates lateral displacement to lateral traction, ๐ถ๐‘ข ๐‘ correlates lateral displacement to normal pressure, and ๐ถ๐‘ค๐œ correlates normal displacement to lateral traction. Also matrices ๐ถ๐‘ค ๐‘ and ๐ถ๐‘ข๐œ are identical, i.e. ๐ถ๐‘ค ๐‘ = ๐ถ๐‘ข๐œ , and matrices ๐ถ๐‘ข ๐‘ = โˆ’๐ถ๐‘ค๐œ . The Goodman decoupling approximation is achieved by setting ๐ถ๐‘ค๐œ = 0. The numerical efficiency of the surface compliance discretization approach shown here is worth discussing. Not only are there no nodes internal to the half-plane, but one needs to discretize only the interval of interest (โˆ’๐‘…, ๐‘…) containing the anticipated contact. Admittedly, this particular compliance formulation applies only to the half-plane, but this provides an excellent test-bed for preliminary investigation of the contact algorithm discussed in this chapter. The sort of discretization shown here, is discussed in [35] and more recently in [72], where the compliance matrix is presented as a step to obtaining a stiffness matrix used in deducing tractions associated with an imposed indentation. Sayles discusses this approach to discretization broadly in [60]. Webster and Sayles [71] employed this approach to address rough surface contact. The first to have used this approach (using chapeau functions) appears to have been Bentall and Johnson [10]. 2.1.1 Taxonomy of Contact Status With reference to figure 2.1, we observe that when one considers all possible node statuses in the context of Coulomb friction (and several other models for friction), nodes may be 1) Not in Contact, 2) In Contact and Stuck, 3) In Contact and Slipping Left, or 4) In Contact and Slipping Right. These four node statuses and the corresponding equality constraints and inequality constraints for Coulomb friction are shown in Table 2.1. (In this table, for Coulomb friction ๐‘“ ( ๐‘) = ๐œ‡ ๐‘.) 13 Nodal Statuses Contact Non-Contact Stick Slip Left Slip Right Slip Figure 2.1: Taxonomy of nodal status. The node status types encountered in contact analysis are indicated by thick borders. Nodal Statuses Equality Constr. Inequality Constr. Contact and Stick ๐‘ขยค = 0, ๐‘ค = ๐‘” |๐œ| โ‰ค ๐‘“ , ๐‘ โ‰ฅ 0 Contact and Slip Right ๐œ = โˆ’ ๐‘“ ( ๐‘) , ๐‘ค = ๐‘” ๐‘ขยค โ‰ฅ 0, ๐‘ โ‰ฅ 0 Contact and Slip Left ๐œ = ๐‘“ ( ๐‘) , ๐‘ค = ๐‘” ๐‘ขยค โ‰ค 0, ๐‘ โ‰ฅ 0 Not-in-Contact ๐‘ = 0, ๐œ = 0 ๐‘คโ‰ค๐‘” Table 2.1: Equality and inequality constraints for each node status type. ๏ฃฑ 0 ๏ฃผ ๐ถ๐‘ค๐œ 1 0 ๏ฃด ๐‘๐‘› ๏ฃฎ ๏ฃฎ ๐ถ๐‘ค ๐‘ ๏ฃน ๏ฃฑ ๏ฃผ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃบ ๏ฃฎ โˆ’1 ๏ฃน ๏ฃน๏ฃบ ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด 1 0 . ๏ฃฏ ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃบ ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃฏ โˆ’1 ๏ฃบ ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃฏ .. .. ๏ฃบ ๏ฃฏ ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ . . ๏ฃบ ๏ฃฏ .. ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๐œ๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃฏ . ๏ฃบ ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 1 ๏ฃบ๏ฃบ ๏ฃฏ . ๏ฃฏ ๐ถ๐‘ข ๐‘ ๐ถ๐‘ข๐œ ๏ฃฏ ๏ฃบ ๏ฃบ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด โˆ’1 ๏ฃฏ ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 ๏ฃฏ ๏ฃฏ ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 1 ๏ฃบ๏ฃบ ๏ฃฏ ๏ฃฏ ๏ฃฏ ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃด ๐‘Š๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃฏ ... โˆ’1 0 ๏ฃบ๏ฃบ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ฮ”๐‘ฅ ... 0 0 0 ๏ฃบ๏ฃบ ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๐‘ˆ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃฏ 0 ๏ฃฏ ... 0 โˆ’1 ๏ฃป ๏ฃด๏ฃบ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฐ ... ฮ”๐‘ฅ 0 0 ๏ฃบ๏ฃป ๏ฃฐ ๏ฃบ๏ฃด๏ฃฒ . .. ๏ฃฝ ๏ฃด = ๏ฃฒ ๏ฃด ๐‘๐‘ ๏ฃฝ ๏ฃด (2.2) ๏ฃฏ ๏ฃบ ๏ฃฏ ๏ฃฎ ๐ฟ๐‘ 0 0 ๏ฃน๏ฃบ ๏ฃฎ ๐ฟ๐‘ค 0 0 ๏ฃน ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค๐‘ ๏ฃด ๏ฃด ๏ฃด ๐‘ค๐‘› .. ๏ฃฏ ๏ฃฏ ๏ฃฏ ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. .. ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 0 ๏ฃบ 0 0 ๏ฃบ ๏ฃบ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃบ ๏ฃฏ . ๏ฃฏ ๏ฃฏ . ๏ฃฏ . ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ .. .. ๏ฃบ๏ฃบ ๏ฃฏ . . ๏ฃบ ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐œ๐‘ ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๐ฟ ๐ฟ๐œ . . ๏ฃบ ๏ฃฏ ๐ฟ . . . . ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๐œ๐‘ ๏ฃฏ ๐‘ข ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๐‘ข๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ข๐‘ ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ .. ๏ฃบ ๏ฃฏ .. ๏ฃบ ๏ฃบ ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด . 0 0 ๏ฃบ๏ฃบ . 0 0 ๏ฃบ ๏ฃบ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃฏ ๏ฃบ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃฏ ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ 0 0 ... 0 0 ๏ฃบ๏ฃบ ๏ฃฏ 0 0 ... 1 0 ๏ฃบ ๏ฃบ ๏ฃด ๏ฃบ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃฏ ๏ฃด ๐‘ƒ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ƒ ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ 0 0 ... 0 0 ๏ฃบ๏ฃป ๏ฃฏ 0 0 . . . 0 1 ๏ฃบ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃณ ๐‘„ ๏ฃป ๏ฃป๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฐ ๏ฃฐ ๏ฃฐ ๏ฃพ ๏ฃด ๏ฃณ ๐‘„ ๏ฃด ๏ฃพ 2.1.2 Compliance Matrix with Contact Constraints Say at an instant, we know the correct status of each node. Then our inequality constraints are all satisfied and equality constraints are incorporated as additional linear equations. The larger matrix 14 is shown in Eq. (2.2), where the upper part of the matrix captures elasticity equations and the lower part captures the contact equality constraints through ๐ฟ ๐‘ , ๐ฟ ๐‘ข , ๐ฟ ๐œ ๐‘ , ๐ฟ ๐œ and ๐ฟ ๐‘ข . Normal pressure constraint matrix ๐ฟ ๐‘ fixes the pressure of non-contact nodes to be zero by setting constraint pressure ๐‘ ๐‘ = 0 and its corresponding value in matrix ๐ฟ ๐‘ ๐‘ = 1. Geometric constraint matrix ๐ฟ ๐‘ค fixes normal displacements of nodes in-contact by setting normal displacement ๐‘ค ๐‘ = ๐‘”๐‘› and its corresponding value in matrix ๐ฟ ๐‘ค ๐‘ = 1. Similarly, stick constraint matrix ๐ฟ ๐‘ข sets the lateral displacements of the stick status node through ๐‘ข ๐‘ = ๐‘ข ๐‘› and ๐ฟ ๐‘ข ๐‘ = 1. For the case of slip constraints, we use a relation between lateral and normal pressure |๐œ๐‘› | = ๐œ‡ ๐‘ ๐‘› for the Coulomb friction model through matrices ๐ฟ ๐œ ๐‘ and ๐ฟ ๐œ . (The sign of ๐œ๐‘› depends on the direction of travel and is reflected in ๐ฟ ๐œ ๐‘ .) We set ๐ฟ ๐œ๐‘ = 1 and ๐ฟ ๐œ ๐‘ ๐‘ = โˆ’๐œ‡ to the corresponding slip status node. The other four rows in the matrix allow us to fix the imposed load conditions through normal ๐‘ƒ and lateral ๐‘„ force or normal ๐‘Š๐‘ก๐‘œ๐‘ก and lateral ๐‘ˆ๐‘ก๐‘œ๐‘ก displacement constants. With contact status and imposed boundary conditions, the matrix (2.2) is used to solve for new displacements and pressure resulting from increments in load, which are then compared with the inequality contact constraints to identify any violations. Changing the contact status of nodes requires a robust contact algorithm to perform all these comparisons and provide solutions. We will discuss this in the next sections. This approach โ€“ implementing equality constraint equations in the elasticity matrix to solve and later verifying the inequality constraints โ€“ is similar to the linear complementarity problem (LCP) formulation. The size of the matrix (2.2) would appear to be large, but its dimension grows roughly as four times the number of nodes in the contact patch, so the computational requirements are still modest. One may either employ a fixed number of nodes and their degrees of freedom which are sufficient to capture the contact mechanics for the calculation of interest, or one may continuously change the set of nodes under consideration to include all those currently in the contact plus one more node on the either side. The resulting overall matrix size becomes (4๐‘› + 12) ร— (4๐‘› + 12) for ๐‘› nodes in contact. Almost 70% of the matrix elements are zero, facilitating the use of sparse matrix solvers. When the number of nodes in contact is O (๐‘›), the time to solve the elasticity matrix will be O (๐‘›2 ). The purpose of introducing the surface compliance formulation above is solely to define the 15 discretization on which the Method of First Violation is illustrated below. This contact algorithm is not restricted to such discretizations. 2.2 Algorithm I The new algorithm - Method of First Violation - is a major improvement on conventional algorithms. First, a conventional algorithm is presented for comparison. Contact algorithms are complex and have multiple parameters, so before delving into specific algorithms, let us assign a term โ€œnodal parameters (๐ป)", which contains these four contact surface parameters normal pressure, lateral traction, normal displacement, and lateral displacement (๐‘, ๐œ, ๐‘ค, ๐‘ข). We also refer to the four kinds of contact status associated with nodes provided in the Tab. 2.1. Each node will have only one contact status and corresponding equality and inequality constraints (๐‘†). The collection of node statuses and nodal parameters (๐‘†, ๐ป) defines the contact configuration. An equilibrium configuration is one for which the parameters for each node satisfy the equality and inequality constraints of each nodeโ€™s assigned status, and the collection of displacements and tractions on each side of the interface are consistent with the elasticity equations for the corresponding body. The first contact algorithm presented here is very similar to that of Ahn and Barber [1]. It differs from [1] in that to make the process more robust, at the beginning of each new load step all the nodes that were in contact at the end of the previous load step are assumed to be stuck. On each iteration, the contact inequality constraints are checked and just one node status change is allowed at each iteration until a converged solution is found for the given load increment. This contact algorithm is briefly explained with a flow chart shown in Fig 2.2. The algorithm starts with an initial equilibrium contact configuration (๐ป0 & ๐‘†0 ) and total final imposed load (๐ฟ ๐‘ก๐‘œ๐‘ก ), such as total normal force or displacement. To reach the final imposed load (๐ฟ ๐‘ก๐‘œ๐‘ก ), a load step (๐›ฟ๐ฟ 0 ) is selected according to the number of steps anticipated to be needed to cover the contact history. Even a single load (๐›ฟ๐ฟ 0 = ๐ฟ ๐‘ก๐‘œ๐‘ก โˆ’ ๐ฟ 0 ) step may be enough to reach the final state, but a small load step enhances the chances of convergence. After choosing a load step, matrix (Eq. 2.2) is solved with all contact nodes status to stick. The resulting new contact parameters (๐‘ ๐‘› , ๐œ๐‘› , ๐‘ค ๐‘› , ๐‘ข ๐‘› ) are 16 Initial Equilibrium Configur- tion (๐ป0 = ( ๐‘, ๐œ, ๐‘ค, ๐‘ข) and ๐‘†0 = Only contact node status) no is ๐ฟ 0 โ‰ค ๐ฟ๐‘‡ ๐‘œ๐‘ก Quit Yes Set ๐›ฟ๐ฟ = ๐›ฟ๐ฟ 0 , Set all contact nodes to stick status Solving elasticity matrix Check contact inequalities in the Yes Change one node No contact violation preference order of status at a time violation correction Figure 2.2: Algorithm I: A contact algorithm similar to Ahn & Barber flow chart compared to their respective inequality constraints to check for the contact violation. Only one violation gets fixed at a time in the preference order of โ€œgeometric inter-penetration", โ€œnegative normal pressure", and โ€œslipping". An elasticity matrix (Eq. 2.2) is solved again using the new contact status. Multiple iterations may be needed to get a converged contact solution without any contact violation. This converged solution is used for finding a new solution with a new load step, where it again starts with all contact nodes to stick status. This algorithm usually works well with the Coulomb friction, showing a few issues, especially 17 in an elastically fully coupled case. For instance, there are situations where the algorithm identifies a violation at the end of a time step, changes node status to avoid that violation, but then finds a different violation at the end of the time step. This can be an unending cycle. This issue can be resolved by changing the multiple statuses manually or with a small load step, or implementing both simultaneously. This algorithm can also have issues with other friction models such as the two-parameter friction model (๐œ‡ ๐‘  and ๐œ‡ ๐‘˜ ) and the Dahl friction model where the constraint inequality (๐œ โ‰ค ๐œ‡ ๐‘  ๐‘) is different from the slipping traction (๐œ = ๐œ‡ ๐‘˜ ๐‘). This problem can arise even with extremely small time steps. To resolve these issues, we developed a new algorithm with an adaptive load step that shows substantially more robustness. 2.3 Algorithm II: Method of First Violation As discussed at the end of section 1.3.3, a concern with methods that involve iterating through solutions without consistent nodal parameters and nodal statuses (non-physical states) is that they scramble the contact histories stored as interface traction. The contact algorithm method that follows is designed to avoid iterating through non-physical configurations, so as to preserve consistent evolution of traction distribution with load history, and to provide unique and consistent solutions. The second algorithm is a modification of the more standard algorithms, such as Algorithm I, and follows the linear complementarity problem (LCP) approach. It is best understood through the following observations: 1. the contact problem is piece-wise linear. Consider the following thought experiment. โ€ข Say at some instant the status of every node is known and reflected in Eq. 2.2. The horizontal load at that time is ๐‘„. โ€ข Say that the horizontal load is incremented by ๐›ฟ๐ฟ and that all of the inequality conditions consistent with the initial node statuses remain satisfied. โ€ข Say that instead of using load increment ๐›ฟ๐ฟ, we used ๐›ผ๐›ฟ๐ฟ where 0 โ‰ค ๐›ผ โ‰ค 1. Because 18 Initial Equilibrium Configurtion (๐ป0 = ( ๐‘, ๐œ, ๐‘ค, ๐‘ข) and ๐‘†0 = all node status (contact, slip and stick) ) no is ๐ฟ 0 โ‰ค ๐ฟ๐‘‡ ๐‘œ๐‘ก Quit yes Set ๐›ฟ๐ฟ = ๐›ฟ๐ฟ 0 Set ฮ”๐‘† = 0 ๐‘†0 , ๐›ฟ๐ฟ ๐‘†0 , ๐›ฟ๐ฟ Solving elasticity matrix ๐ท โˆ— , ๐‘†0 , ๐›ฟ๐ฟ Check consistency and contact status violation ๐‘‰๐‘›โˆ— (๐›ฟ๐ฟ) yes Find ๐›ฟ๐ฟ so that Set ๐‘†0 = ๐‘†0 +ฮ”๐‘†0 , ๐ท 0 = is ๐‘‰๐‘›โˆ— โ‰ค๐œ– ๐‘‰๐‘›โˆ— (๐›ฟ๐ฟ) = 0, Record ๐ท โˆ— , ๐ฟ 0 = ๐ฟ 0 + ๐›ฟ๐ฟ no status Change ฮ”๐‘† Figure 2.3: Algorithm II: The Method of First Violation contact algorithm with adaptive load step flow chart 19 matrix of Eq. 2.2 remains unchanged and the right hand side of Eq. 2.2 changes only through the value of ๐‘„ + ๐›ผ๐›ฟ ๐ฟ, then all the changes in the contact parameters ๐‘ ๐‘› , ๐œ๐‘› , ๐‘ค ๐‘› , and ๐‘ข ๐‘› will be linear in ๐›ผ. โ€ข Examination of the inequalities of Table 2.1, shows that they all remain satisfied for all 0 โ‰ค ๐›ผ โ‰ค 1. We conclude that if all the node statuses at the beginning of the load step also prevail at the end of the load step, then the nodes will not change status at the loads in between. The version of the compliance matrix (Eq. 2.2) that reflects the node status at the beginning of the load step will consistently reflect the contact values and status throughout the load step, and consequently the discretized contact problem is piece-wise linear. 2. Except for the case of macro-slip, there exists a minimum load increment that will cause any violation of inequality constraints associated with the current node status. This greatest lower bound is a consequence of the completeness of the real number line [55](p.7). We can define ๐›ฟ๐ฟ such that ๐ฟ + ๐›ฟ๐ฟ is that greatest lower bound. 3. The value of ๐›ฟ๐ฟ can be calculated through a finite number of steps as follows: a) Choose a load increment ๐›ฟ๐ฟ 0 sufficient to cause one or more violations. Let us call this set of violations ๐‘‰0 . b) Select one of those violated inequalities of ๐‘‰0 and calculate the load increment ๐›ฟ๐ฟ 1 that exactly turns that inequality constraint into an equality constraint. c) Substitute ๐ฟ + ๐›ฟ๐ฟ 1 for ๐‘„ in Eq. 2.2 to solve for all tractions and displacements. Test those states against the inequality conditions to see if any violations remain. If there are no other violations, then define ๐›ฟ๐ฟ = ๐›ฟ๐ฟ 1 . d) If there are remaining violations, note that because of the linearity of Eq. 2.2, they are a subset of ๐‘‰0 . Choose one and find the corresponding ๐›ฟ๐ฟ 2 . By construction, ๐›ฟ๐ฟ 2 โ‰ค ๐›ฟ๐ฟ 1 . 20 Also, note that each recalculation of ๐›ฟ๐ฟ ๐‘› reduces the number or remaining inequality violations by at least one. e) Continue this process until inequalities of ๐‘‰0 are resolved, and ๐›ฟ๐ฟ is obtained. Note that the above process involves a maximum number of steps if one were at every opportunity to make an incorrect choice of which violation to fix. In fact, there are strategies to make good choices, such as identifying all violations within each class โ€“ say interpenetrations โ€“ and starting with the worst of these violations. 4. At the end of that load step ๐›ฟ๐ฟ, the inequalities of the beginning of the load step are still satisfied. Immediately beyond that load increment, some node status will change, and by definition, the new inequality and equality constraints will also be satisfied at the special ๐›ฟ๐ฟ. The distinction between the conventional approach and the Method of First Violation is il- lustrated in figure 2.4. The conventional algorithm (Algorithm-I) solves the contact violation by iterating on the same load step ๐›ฟ๐ฟ 0 . In the case of a large load-step ๐›ฟ๐ฟ 0 , solving multiple con- tact violations through an iterative approach may lead to a non-converging or a wrong solution. Conversely, the Method of First Violation (Algorithm-II) finds the new ๐›ฟ๐ฟ 1 (independent of initial load-step ๐›ฟ๐ฟ 0 ) satisfying all contact equality and inequality constraints. 2.3.1 Admissible Contact Histories The issue of convergence of conventional numerical contact algorithms has been a continuing issue. A reasonable hypothesis is that iterative processes to find consistent tractions and displacements at the end of each load step may wipe out the information on the interface contact/slip history - as recorded in self-equilibrating traction fields. This loss of information is suggested by occasional node-to-node discontinuities in traction or jumps in nodal traction from one load step to the next, even during monotonic loading, when conventional contact algorithms are employed. The algorithm outlined in Steps 3a through 3e above is presented as flow chart in figure 2.3. By construction, all equality and inequality constraints associated with the node statuses of each 21 Conventional Algorithm Method of first Violation P P P =L P =L P = L + ฮดL0 Iteration-1 P = L + ฮดL1 Adding nodes with zero Pressure P = L + ฮดL0 Iteration-2 P = L + ฮดL0 Iteration-3 P = L + ฮดL1 + ฮดL2 Adding nodes with zero Pressure Half-plane Indenter Figure 2.4: Comparison between a conventional contact algorithm and the Method of First Violation contact algorithm. With conventional contact algorithms, a load step is selected and an iterative procedure is used to find an admissible set of loads and displacements consistent with that load step. With the Method of First Violation, the load step is computed to be exactly that necessary to bring the first node to change status (contact, non-contact, stick, and slip) load step are satisfied at the beginning of the load step, and the end of the load step, and at all loads in between. When one considers one load step after another, one sees that these calculations result in a consistent set of loads and interface tractions and displacements that satisfy all elasticity equations, all equality constraints, and all inequality constraints thorough-out the loading history. In this way, at any point in the loading history, all node statuses, all tractions, and all displacements are consistent with all that has happened prior to that point. One would hope that this fidelity of contact history would eliminate โ€“ or at least mitigate โ€“ the issues of tractions oscillating from one node to the next, as well as the problems with mesh convergence. Finally, though the above explication used increments in shear load ๐‘„ for the purpose of discussion, one could easily employ increments in normal load ๐‘ƒ, increments in lateral displacement ๐‘ˆ๐‘ก๐‘œ๐‘ก , or increments in normal displacement ๐‘Š๐‘ก๐‘œ๐‘ก . Of course, any desired history of imposed loading or imposed displacements could be constructed from the sequential incremental imposition of these 22 four quantities. 2.3.2 Contact Status Violations Finding a Contact Status Violation (CSV) and resolving it is key to solving contact problems through this new algorithm (2.3). We define four kinds of CSV: 1) Geometric Violation, 2) Normal Pressure Violation, 3) Slip Violation, and 4) Stick Violation. As indicated above, in the case of multiple CSVs, we can choose any of them to start with to calculate the new ๐›ฟ๐ฟ. If there remain other CSVs at this load increment, they will be used to calculate new and smaller ๐›ฟ๐ฟs. We continue to shrink ๐›ฟ๐ฟ in this process until we have the load increment that is just enough to trigger just one CSV. The calculation of the new ๐›ฟ๐ฟ associated with a violation depends on the type of violation, as discussed below. 2.3.2.1 Geometric Violation A geometric violation occurs when the contact configuration suggests an interpenetration between two contacting surfaces. Say a node is assumed to be non-contacting, the corresponding geometric condition is ๐‘ค ๐‘› โ‰ค ๐‘”๐‘› [39] (where ๐‘”๐‘› is the indenter node displacement), and if it transpires that a load increment ๐›ฟ๐ฟ 0 results in nodal displacement ๐‘ค ๐‘› > ๐‘”๐‘› , then inter-penetration has taken place. In this case, a new ๐›ฟ๐ฟ is calculated to bring the interpenetrated node exactly to the contacting surface (๐‘ค ๐‘› = ๐‘”๐‘› ) through Eq. (2.3), where ๐‘ค โˆ— , ๐‘ค 0 , and ๐‘” are the current violating/penetrating displacement, previous equilibrium displacement, and indenter displacement. ๐›ฟ๐ฟ ๐‘ค ๐‘” โˆ’ ๐‘ค0 = โˆ— (2.3) ๐›ฟ๐ฟ 0 ๐‘ค โˆ’ ๐‘ค 0 2.3.2.2 Normal Pressure Violation A normal pressure violation occurs when load increment ๐›ฟ๐ฟ 0 causes a node that is nominally in contact to have negative normal pressure. The new load step ๐›ฟ๐ฟ is calculated to make the normal pressure exactly zero: 23 ๐›ฟ๐ฟ ๐‘ ๐‘0 = (2.4) ๐›ฟ๐ฟ 0 ๐‘0 โˆ’ ๐‘โˆ— where ๐‘ โˆ— and ๐‘ 0 are the normal pressure associated with ๐›ฟ๐ฟ 0 and the normal pressure of previous equilibrium state, respectively. 2.3.2.3 Violation in Stick Constraints Violation in a stick constraint occurs when the load step ๐›ฟ๐ฟ 0 causes a node with that contact status to have a new lateral traction sufficient to overcome the friction force (|๐œ| โ‰ฅ ๐‘“ ). The load step that resolves this violation is the one which is just enough to bring the shear traction just to the limit of slip (|๐œ| = ๐‘“ ), i.e. ๐›ฟ๐ฟ ๐œ (|๐œ0 | โˆ’ ๐œ‡๐‘ 0 ) = (2.5) ๐›ฟ๐ฟ 0 (|๐œ0 | โˆ’ ๐œ‡๐‘ 0 ) โˆ’ (|๐œ โˆ— | โˆ’ ๐œ‡๐‘ โˆ— ) 2.3.2.4 Violation in Slip Constraints Violation in a slip constraint occurs when a node that nominally slips in one direction, is slipping in the opposite direction, or has stopped slipping. If it were known, the rate of change in the lateral displacement speed would be used to find the ๐›ฟ๐ฟ which was just enough to change the node state, but because of the quasi-static nature of the constitutive model and the algorithm, this information is unavailable. Instead, if a violation of the slipping state occurs in the load step ๐›ฟ๐ฟ 0 , that load step is repeatedly bisected to find a smaller interval in which resides the ๐›ฟ๐ฟ that is just enough to cause that violation. It occasionally happens โ€“ such on a load reversal โ€“ that more than one sliding node becomes stuck at the same time. On such occasions, all of those sliding nodes that have a direction reversal are set to stick. 2.3.3 Single-Node Illustrations of Computing ๐›ฟ๐ฟ At this stage it is helpful to walk through the calculation of ๐›ฟ๐ฟ for an individual node for a couple of types of node status: 24 (g โˆ’ w0 > 0) Gap (p > 0) Contact 0 P ressure(p) Pressure (g โˆ’ w) ฮดLw ฮดLp L  L Interpenetration (g โˆ’ wโˆ— < 0) Negative Pressure (pโˆ— < 0) Node Status Transition Load Figure 2.5: Single node illustration for contact and non-contact violations. The thick (blue) solid dashed lines represent admissible configurations and the thin (orange) solid and dashed lines represent inadmisible configurations. For instance if a node begins not-in-contact (thick dashed line on the left), as the load step ๐›ฟ๐ฟ increases, the gap closes linearly until contact occurs at ๐›ฟ๐ฟ ๐‘ค , as the load further increases the gap stays at zero and the contact pressure increases linearly (thick solid line on the right.) (|๐œ โˆ— | โˆ’ ๐œ‡๐‘ โˆ— ) Slipping to the right Stuck 1 ๐‘ ๐‘”๐‘›( ๐‘ขยค0 ) = 1 |๐œ| โˆ’ ๐œ‡๐‘ > 0 ๐›ฟ๐ฟ ๐œ (|๐œ| โˆ’ ๐œ‡๐‘) ๐ฟ ยค |๐œ| โˆ’ ๐œ‡๐‘ = 0 ยค =0 ๐‘ ๐‘”๐‘›( ๐‘ข) ๐ฟ ๐‘ ๐‘”๐‘›( ๐‘ข) 0 ยค =0 ๐‘ ๐‘”๐‘›( ๐‘ข) |๐œ| โˆ’ ๐œ‡๐‘ = 0 |๐œ| โˆ’ ๐œ‡๐‘ < 0 ๐›ฟ๐ฟ ๐‘ขยค (|๐œ0 | โˆ’ ๐œ‡๐‘ 0 ) โˆ’1 ๐‘ ๐‘”๐‘›( ๐‘ขยคโˆ— ) = โˆ’1 ๐ฟ๐‘œ๐‘Ž๐‘‘ Figure 2.6: Single node illustration for stick and slip violations. The dark (blue) solid and thick lines represent admissible configurations and the lighter (orange) solid and dashed lines represent inadmissible configurations. For an initially stuck node (dashed thick (blue) line), calculation of shear traction exploits linearity of that traction to obtain the new ๐›ฟ๐ฟ ๐œ of stick violation. For an ยค will require the initially slipping node (solid thick (blue) line), calculation of change in ๐‘ ๐‘”๐‘›( ๐‘ข) method of bisection to obtain ๐›ฟ๐ฟ ๐‘ขยค of slip violation. 25 1. Non-contact and Contact Problems: In Fig 2.5, at load ๐ฟ a node has not-in-contact status but at load ๐ฟ = ๐ฟ + ๐›ฟ๐ฟ 0 , there would be interpenetration. Because of the linearity of the problem, we can calculate a load increment ๐›ฟ๐ฟ ๐‘ค such that at load ๐ฟ + ๐›ฟ๐ฟ ๐‘ค the node just touches the surface (๐‘ค โˆ’ ๐‘” โ‰ˆ 0) with zero nodal traction (๐‘ โ‰ˆ 0). At this point, the node status is switched from non-contact to being in contact. Note that at load ๐ฟ + ๐›ฟ๐ฟ ๐‘ค , the nodal constraints associated with non-contact (๐‘” โˆ’ ๐‘ค โ‰ฅ 0) and with in-contact (๐‘ โ‰ฅ 0) are satisfied simultaneously. On adding a node in contact, it can have three possible contact status stick, left slip, or right slip. There are two possible approaches to identify the contact status. โ€ข First stick status is assigned to the new contact node. Then, the elasticity matrix is solved with a tiny load step. Then, calculated shear traction is used to decide the direction of the slip. After deciding the contact status, the elasticity matrix is solved again with a nominal load step. โ€ข A slip status is assigned to the new contact node, where slip direction is assigned based on the previous motion of the node. For example, if a non-contacting node moves toward the left, then on coming into the contact, its contact status would be contact left slip. Both approaches provide the same solution. We have used the second approach which is a little quicker than the first one. Conversely, if the load is initially ๐ฟ, where there is contact and the value of load is reduced to ๐ฟ, there would be a pressure violation (๐‘ < 0) at ๐ฟ, so an incremental load of ๐›ฟ๐ฟ ๐‘ (negative) is found so that at ๐ฟ + ๐›ฟ๐ฟ ๐‘ , ๐‘ = 0 and ๐‘ค โˆ’ ๐‘” = 0. Again, this is a state where the constraints of both the status at the beginning of the load step and the status that holds just after that load step are satisfied. Also, note that even though these are linear calculations, the values of load at which node status transition takes will not be exactly identical in the two calculations 26 discussed above, there will be some numerical error, and computational tests against zero must accommodate that numerical uncertainty. 2. Sticking and Slipping Problems: If initially the load ๐ฟ corresponds to a node sliding in one direction (which we choose as positive for the sake of discussion), constraints being ๐œ = โˆ’๐œ‡๐‘ and ยค =1 sgn( ๐‘ข) and then the load is changed to ๐ฟ corresponding to ๐œ = ๐œ‡๐‘ and ยค = โˆ’1 sgn( ๐‘ข) then ๐‘ขยค must go through zero for some ๐ฟ โˆ— = ๐ฟ + ๐›ฟ๐ฟ ๐‘ขยค for some ๐›ฟ๐ฟ ๐‘ขยค . At that load, causing the velocity to be approximately zero, the node becomes stuck and the node status must be changed appropriately (See Fig. 2.6.) Since the constraint on sgn( ๐‘ข) ยค is nonlinear, there is no linear equation to solve for ๐›ฟ๐ฟ ๐‘ขยค ; instead ๐›ฟ๐ฟ ๐‘ขยค must be found through a method of bisections. When that load is found, the constraints holding at the previous step (๐‘ขยค โ‰ฅ 0 and |๐œ| = ๐œ‡๐‘) as well as the constraints for the new state (๐‘ขยค = 0 and |๐œ| โ‰ค ๐œ‡๐‘) are satisfied simultaneously. If initially under load ๐ฟ on the body, a node is stuck, with that node status the nodal constraints are ๐‘ขยค = 0 and |๐œ| โ‰ค ๐œ‡๐‘. Say that the load is changed to ๐ฟ and at that load, Eq. 2.2 yields |๐œ| > ๐œ‡๐‘, violating a constraint of the previous node status (for the sake of discussion, let us assume that ๐œ > 0). There exists a load ๐ฟ โˆ’ ๐›ฟ๐ฟ ๐œ between ๐ฟ and ๐ฟ at which |๐œ| = ๐œ‡๐‘ (see Fig. 2.6). From Equation 2.2 where the constraints are those of the initial node status, the change in node traction changes linearly with change in applied load. We obtain a linear equation for ๐›ฟ๐ฟ ๐œ (Eq. 2.5) satisfying |๐œ| = ๐œ‡๐‘. Note that at load ๐ฟ โˆ’ ๐›ฟ๐ฟ ๐œ , the stick conditions (๐‘ขยค = 0 and |๐œ| โ‰ค ๐œ‡๐‘) are satisfied as are the slip conditions at inception of slip (๐‘ขยค โ‰ฅ 0 and |๐œ| = ๐œ‡๐‘). The algorithms discussed here are not fixed to particular mesh construction techniques such as finite element method, boundary integral method, or compliance matrix method. They work with almost all discretization techniques if contact surface nodes should follow the proper equality and inequality constraints from Eq. 2.1. 27 CHAPTER 3 VERIFICATION OF CONTACT ALGORITHMS The coupled and complex nature of contact problems makes them difficult to solve analytically and usually requires simplifying assumptions โ€“ most often the Goodman decoupling approximation [31] โ€“ to reduce the complexity of the problem. In this chapter, we implemented and compared algorithms of the previous chapter with available analytical results; imposing the Goodman ap- proximation in those problems where it is invoked in the corresponding analytic solution. The contact length, stick ratio, normal pressure, and lateral traction distributions are the most natural choices for comparing numerical and analytical solutions for a rigid cylindrical indenter pressing against an elastic half-plane. The closed-form expressions invoked here can be found in [6, 35]. We also compared the numerical solutions using both algorithms and explained the significant improvements through multiple examples. 3.1 Frictionless Normal Contact 3.1.1 Rigid Cylinder Pressed against an Elastic Half-plane For the case of two identical elastic cylinders pressing against each other, we have analytic expres- sions for the length of the contact patch and pressure distribution [6] which can be mapped onto the problem of a rigid cylinder pressing against a frictionless elastic half-plane: r ๐‘ƒ(๐œ… + 1)๐‘… ๐‘Ž= , (3.1) 2๐œ‹๐บ โˆš 2๐‘ƒ ๐‘Ž 2 โˆ’ ๐‘ฅ 2 ๐‘(๐‘ฅ) = (3.2) ๐œ‹๐‘Ž 2 where 2๐‘Ž is the contact length, ๐‘ƒ is the total normal force, R is the radius of the rigid cylinder, ๐บ is the shear modulus of the half-plane, ๐œ… = 3 โˆ’ 4๐œˆ (for plane strain), and ๐œˆ is the Poissonโ€™s ratio. In the calculations that follow, ๐บ = 1 and ๐œˆ = 0.3, ๐‘… = 0.25, and ๐‘ƒ = 0.015 are used. The contact length 28 ยท10โˆ’2 Numerical Press. Analytical Press. Numerical Displ. Indenter 1.5 0.2 1 Displacement Pressure 0.1 0.5 0 Contact Length(2a) 0 โˆ’0.1 โˆ’0.05 โˆ’0.025 0 0.025 0.05 Contact Region (x) Figure 3.1: Verifying the half-plane compliance matrix formulation with the analytical solution. The two numerical algorithms yielded identical solutions. The โˆ— and  represent numerically calculated traction at the centers of intervals. The continuous curves are analytic (with 21 nodes in-contact) calculated using Eq. 3.1 is used for calculating the pressure distribution using Eq. 3.2. Figure 3.1 shows the numerical pressure distribution calculated from the compliance formulation, which matches almost identically with analytical results for a given total normal force (๐‘ƒ). 3.1.2 Rigid Flat Indenter Pressed against an Elastic Half-plane Furthermore, another analytical solution is available for a rigid flat indenter pressing against the elastic half-plane. In this case, the contact length between the indenter and half-plane remains unchanged and is equal to the rigid indenterโ€™s length. The pressure distribution for a normal force 29 ยท10โˆ’3 ยท10โˆ’4 7 Numerical Press. Analytical Press. 8 Numerical Displ. Indenter 6 5 6 4 Displacement Pressure 4 3 2 2 1 Contact Length(2a) 0 0 โˆ’2 โˆ’1 โˆ’0.4 โˆ’0.2 0 0.2 0.4 Contact Region (x) Figure 3.2: Verifying the half-plane compliance matrix formulation with the analytical solution. The two numerical algorithms yielded identical solutions. The โˆ— and  represent numerically calculated traction at the centers of intervals. The continuous curves are analytic ๐‘ƒ is shown in the Eq. 3.3. ๐‘ƒ ๐‘(๐‘ฅ) = โˆš . (3.3) ๐œ‹ ๐‘Ž2 โˆ’ ๐‘ฅ 2 The numerical expression for pressure distribution is calculated using the same material properties of elastic half-plane with normal force ๐‘ƒ = 0.001. The contact region has 103 nodes in-contact for the contact length of 0.8, equivalent to the length of the flat rigid punch. Figure 3.2 compares the analytical and numerical pressure distributions for the rigid flat indenter, which are the same in most of the contact region. There are minor discrepancies at both endpoints due to the singular nature of the pressure distribution at the ends for the flat rigid indenter. 30 ยท10โˆ’2 8 Normal Press. Num. Normal Press. Ana. Lateral Trac. Num. Lateral Trac. Ana. 6 Normal/Lateral Traction 4 2 0 Contact Length(2a) Stick Length(2b) โˆ’2 โˆ’0.01 โˆ’0.005 0 0.005 0.01 Contact Region (x) Figure 3.3: Verifying numerical normal and lateral traction distribution with analytical expression for rigid cylinder pressing against an elastic half-plane with coefficient of friction (๐œ‡ ๐‘  = 0.3). 3.2 Normal Contact with the Coulomb Friction Next, the simplest friction model, Coulomb friction, is employed with contact algorithm-II. The rigid cylinder is pressed against an elastic half-plane with normal force (๐‘ƒ = 0.001), friction coefficient (๐œ‡ ๐‘  = 0.3), and contact problem has same material and geometric properties from previous section 3.1. Figure 3.3 shows the normal and lateral traction distribution due to the normal pressing of the cylinder. Lateral traction is anti-symmetric with respect to the center of the contact and divides the contact length(2๐‘Ž) into following three regions; 1. slipping to the left at the left end of the contact. 2. stick region with length (2๐‘) in the center. 3. slipping to the right at the right end of the contact. 31 ยท10โˆ’2 1 Normal Press. Num. Normal Press. Ana. Lateral Trac. Num. Lateral Trac. Ana. 0.8 Normal/Lateral Traction 0.6 0.4 0.2 Contact Length(2a) 0 Stick Length(2b) โˆ’0.2 โˆ’0.4 โˆ’0.3 โˆ’0.2 โˆ’0.1 0 0.1 0.2 0.3 0.4 Contact Region (x) Figure 3.4: Verifying numerical normal and lateral traction distribution with analytical expression for rigid flat indenter pressing against an elastic half-plane with coefficient of friction (๐œ‡ ๐‘  = 0.3). Figure 3.4 shows similar characteristics such as anti-symmetric lateral traction and stick-slip region for the rigid flat indenter. 3.2.1 Comparison with Spence (1973) Stick Ratio Results The stick ratio is one of the relevant contact parameters for the comparison and verification during the frictional contact. Spence [63] derived analytical solutions for a rigid indenters normally pressed against a rough elastic half-planes. Relying on the Goodman approximation, he developed the expression for the stick ratio as shown in Eq. (3.4). ๐›ฝ๐พ (๐‘) = ๐œ‡๐พ 0 (๐‘) (3.4) where ๐›ฝ is the Dundurโ€™s parameter (๐›ฝ = (1 โˆ’ 2๐œˆ)/(2 โˆ’ 2๐œˆ))[26], ๐œˆ is the Possionโ€™s ratio, ๐‘ = ๐‘/๐‘Ž p  is the stick ratio, ๐พ (๐‘) is the complete elliptic integral of the first kind and ๐พ 0 (๐‘) = ๐พ (1 โˆ’ ๐‘2 ) . 32 1 Decoupled Analytical Decoupled Numerical Coupled ยต = 0.10 0.8 Coupled ยต = 0.15 Coupled ยต = 0.20 Stick ratio (c) Coupled ยต = 0.25 0.6 Coupled ยต = 0.30 Coupled ยต = 0.35 Coupled ยต = 0.40 0.4 Coupled ยต = 0.45 Coupled ยต = 0.50 Coupled ยต = 0.55 Coupled ยต = 0.60 0.2 0 0 1 2 3 4 5 6 ( ฮฒยต ) Figure 3.5: Stick ratio for cylindrical indenter with and without decoupling assumption 1 Decoupled Analytical Decoupled Numerical Coupled ยต = 0.10 0.8 Coupled ยต = 0.15 Coupled ยต = 0.20 Stick ratio (c) Coupled ยต = 0.25 0.6 Coupled ยต = 0.30 Coupled ยต = 0.35 Coupled ยต = 0.40 0.4 Coupled ยต = 0.45 Coupled ยต = 0.50 Coupled ยต = 0.55 Coupled ยต = 0.60 0.2 0 0 1 2 3 4 5 6 ( ฮฒยต ) Figure 3.6: Stick ratio for flat indenter with and without decoupling assumption Remarkably, this relationship holds for both flat and round indenters. Figures. 3.5 and 3.6 show that numerical results obtained by setting ๐ถ๐‘ค๐œ = 0 are in good agreement with the analytical results for both cylindrical and flat indenters. At a higher value of ๐›ฝ/๐œ‡, there is a little discrepancy between analytical and numerical results due to discretization in 33 1 0.8 Stick ratio (c) 0.6 0.4 Numerical ฮฝ = 0 0.2 Spence ฮฝ = 0 Numerical ฮฝ = 0.3 Spence ฮฝ = 0.3 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Friction (ยต) Figure 3.7: Stick ratio vs friction with full elastic coupling for cylindrical indenter 1 0.8 Stick ratio (c) 0.6 0.4 Numerical ฮฝ = 0 0.2 Spence ฮฝ = 0 Numerical ฮฝ = 0.3 Spence ฮฝ = 0.3 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Friction (ยต) Figure 3.8: Stick ratio vs friction with full elastic coupling for flat indenter the contact region. If stick length is less than the discretization length, then the discretization length is taken as the stick length resulting in a higher value of slip ratio. Also shown in these figures are contact algorithm results where the Goodman approximation was not performed. The predictions of the coupled calculations lie slightly below those associated with the uncoupled system. 34 Spence [63] further solved cases of coupled systems for rigid indenters pressing against rough elastic half-planes. He solved the system of coupled integral equations numerically. Figures 3.7 and 3.8 show stick-slip ratios for both Spenceโ€™s numerical results and those of the contact algorithm. For the case of the cylindrical indenter, the agreement between the predictions of the contact algorithm and the analytic results are again extremely good. In the case of a flat punch, the contact solutions predict slightly higher stick ratios, but the discrepancy is likely due to the singularity of the tractions at the edges of the indenter, and it is difficult to get accurate solutions. 3.2.2 Verification of Numerical Lateral Traction with an Analytical Expression Using Spenceโ€™s [63] results for stick ratio, an analytical expression (Eq. 3.5) was developed [35] for the lateral traction distribution for frictional cylindrical contact. Using a similar approach, an analytical expression (Eq. 3.6) for the lateral traction distribution for the frictional contact of the rigid flat indenter was developed. hp n โˆš 2 2 oi 2๐œ‡๐‘ƒ ๏ฃฑ ๏ฃฒ (๐œ‹๐‘Ž)๐พ (๐‘) 1 โˆ’ (๐‘ฅ/๐‘Ž) 2 ฮจ(๐œ™, ๐‘) โˆ’ 2๐‘ ๐‘ฅ ln ๐‘โˆ’โˆš ๐‘2 โˆ’๐‘ฅ2 ๐‘ ๐‘”๐‘›(๐›ฝ๐‘ฅ) |๐‘ฅ| โ‰ค ๐‘ ๏ฃด ๏ฃด ๏ฃด ๐‘ž ๐‘๐‘ฆ๐‘™ (๐‘ฅ) = ๐‘+ ๐‘ โˆ’๐‘ฅ (3.5) p  2๐œ‡๐‘ƒ 2 1 โˆ’ (๐‘ฅ/๐‘Ž) โ‰ค |๐‘ฅ| โ‰ค ๏ฃด ๏ฃด ๏ฃด (๐œ‹๐‘Ž) ๐‘ ๐‘”๐‘›(๐›ฝ๐‘ฅ) ๐‘ ๐‘Ž ๏ฃณ ๏ฃฑ  n โˆš 2 2 o ๐œ‡๐‘ƒ 1 ๐‘โˆ’ ๐‘ โˆ’๐‘ฅ ๏ฃฒ (๐œ‹๐‘Ž)๐พ (๐‘) โˆš1โˆ’(๐‘ฅ/๐‘Ž) 2 ฮจ(๐œ™, ๐‘) โˆ’ 2๐‘ ln ๐‘+โˆš๐‘2 โˆ’๐‘ฅ 2 ๐‘ ๐‘”๐‘›(๐›ฝ๐‘ฅ) |๐‘ฅ| โ‰ค ๐‘ ๏ฃด ๏ฃด ๐‘ฅ ๏ฃด ๏ฃด ๐‘ž ๐‘“ ๐‘™๐‘Ž๐‘ก (๐‘ฅ) =   (3.6) ๐œ‡๐‘ƒ โˆš 1 ๐‘ โ‰ค |๐‘ฅ| โ‰ค ๐‘Ž ๏ฃด ๏ฃด ๏ฃด (๐œ‹๐‘Ž) ๐‘ ๐‘”๐‘›(๐›ฝ๐‘ฅ) ๏ฃด ๏ฃณ 1โˆ’(๐‘ฅ/๐‘Ž) 2 where sin(๐œ™) = ๐‘ฅ/๐‘ and ฮจ(๐œ™, ๐‘) is an incomplete integral of the first kind, and ๐พ (๐‘) is the complete integral of first kind. Figures 3.3 and 3.4 show that the lateral traction distribution calculated numerically are in line with analytical expression for both cylinder indenter and flat indenter contact. 3.2.3 Verification of Stick-Ratio in the Mindlin Problem. The frictionless Hertz problem is equivalent to pressing two identical rough cylinders against each other. In that case, the full contact path results in stick status. Next, if the cylinders are each subject to equal and opposite lateral forces, then the slip will occur on the contact patchโ€™s outer edges. This 35 Elastic Cyl.-1 Rigid Cylinder Rigid Cylinder P P P Q Q x=0 x=0 Friction free Elastic Half- Friction Elastic Half-plane plane Q P Elastic Cyl.-2 Figure 3.9: Mapping of two elastic cylinder contact to the rigid cylinder over the half-plane contact problem โ€“ Mindlin Problem [52] โ€“ can be solved with use of the Goodman decoupling approximations (see Fig. ??). Analytical equations for stick-slip ratio (c) vs lateral tractions (๐‘ž(๐‘ฅ)) are shown in Eqn. (3.7) and (3.8) [56]. s ๐‘„ ๐‘ ๐‘= 1โˆ’ = (3.7) ๐œ‡๐‘ƒ ๐‘Ž "r r # ๐œ‡(2๐‘ƒ) ๐‘ฅ2 ๐‘2 ๐‘ž(๐‘ฅ) = 1โˆ’ 2 โˆ’๐‘ 1โˆ’ 2 ๐น๐‘œ๐‘Ÿ |๐‘ฅ| < ๐‘ ๐œ‹๐‘Ž ๐‘Ž ๐‘Ž r (3.8) ๐œ‡(2๐‘ƒ) ๐‘ฅ2 ๐‘ž(๐‘ฅ) = 1โˆ’ 2 ๐น๐‘œ๐‘Ÿ ๐‘ โ‰ค |๐‘ฅ| โ‰ค ๐‘Ž ๐œ‹๐‘Ž ๐‘Ž The above two-cylinder Mindlin problem can be mapped to a similar problem of a rigid cylindrical indenter pressed against an elastic half-plane (see Fig. ??) in the following manner: โ€ข In the absence of friction, the indenter is pressed against with force ๐‘ƒ over an elastic half- plane. โ€ข Friction is turned on, but by construction, there is no shear traction on the interface, and there is no slip. โ€ข A lateral force ๐‘„ is introduced to the indenter. (See Eqs. from 2.20 to 2.33 of ref [35] for the mapping). Also, this strategy of turning friction on and off in a calculation has a precedent [15]. 36 1 Mindlin Analytical Numerical mapping 0.8 Stick Ratio (c) 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( fQP ) Figure 3.10: Verifying numerical stick ratio with analytical expression for Mindlin solution. As the force ๐‘„ is increased, slipping on each edge of the contact path will evolve. When the contact algorithm is employed with Goodman decoupling approximation (๐ถ๐‘ข๐‘ค = 0), the resulting predicted stick-slip ratios are in good agreement with those of the Mindlin solution (see fig. 3.10). 3.3 Coulomb Friction Model in Mindlin Problem We implemented the Coulomb friction model (๐œ‡ ๐‘  ) with both contact algorithms. We used the well-known Mindlin solution for comparison. Note the Mindlin solution is employing Goodman decoupling approximation. The mapping of the Mindlin problem to the rigid cylinder over an elastic half-plane is shown in section 5. In the following analysis, the geometric and material properties of the problem from Section 3.1 are used again (radius of cylinder ๐‘… = 0.25, shear modulus G=1, and Poissonโ€™s ratio ๐œˆ = 0.3). The elastic half-plane is more finely discretized in this problem than the previous, and 87 nodes are found to be in-contact with the normal load (๐‘ƒ = 0.001). This finer mesh was chosen to obtain an adequate resolution of the stick-slip boundary and the shear stress at that boundary for comparison with the analytic solutions. The lateral load is set at ๐‘„ = 2.8 ร— 10โˆ’4 . Figure 3.11 compares the normal pressure and lateral traction distributions of the analytic 37 Analytical Algorithm-I Algorithm-II (MFV) ยท10โˆ’2 ยท10โˆ’2 3 6 Normal Pressure p(x) Lateral Tractions ฯ„ (x) 2 4 1 2 0 0 -0.01 0 0.01 -0.01 0 0.01 Contact Region (x) Contact Region (x) Figure 3.11: Comparing numerical tractions from Algorithm-I, Algorithm-II, and analytical ex- pression for one-parameter friction model (๐œ‡ = 0.7) after application of lateral force: normal pressure distribution (left) and lateral traction distribution (right) expression and of the two numerical algorithms. All three seem to be in line with each other for a one-parameter Coulomb friction model with friction coefficient ๐œ‡ = 0.7. For the problem with these parameters, we are well within the limit calculated by Klarbring for existence and uniqueness of a single node (or rigid body) contact. Both Algorithm-I and Algorithm-II are verified with the given analytical expressions. These verification established that the numerical results using the contact algorithms are in-line with the analytical results. In the upcoming sections, we illustrated the issues with Algorithm-I and improvements due to Algorithm-II. 3.4 Two-parameter Coulomb Friction Model in Mindlin Problem Next, We demonstrate the robustness of the Method of First Violation (Algorithm-II) on a more complicated friction model โ€“ the two-parameter friction model โ€“ involving a coefficient of static friction, ๐œ‡ ๐‘  , and a coefficient of kinetic friction, ๐œ‡ ๐‘˜ . In this numerical problem ๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.42, but the loading sequence and geometry of the previous problem are employed again. For the sake of consistency, the Goodman decoupling approximation is used in this analysis as well. Figure 3.12 38 Algorithm-I Algorithm-II(MFV) Slip Nodes Algor.-I Slip Nodes Algor.-II(MFV) ยท10โˆ’2 ยท10โˆ’2 3 6 Normal Pressure p(x) Lateral Tractions ฯ„ (x) 2 4 1 2 0 0 -0.01 0 0.01 -0.01 0 0.01 Contact Region (x) Contact Region (x) Figure 3.12: Comparing numerical tractions from Algorithm-I and Algorithm-II for two-parameter friction model (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.42) after application of lateral force: normal pressure distribution (left) and lateral traction distribution (right). For Algorithm-I, the region of stick is indicated by a solid black line, and the region of slip is indicated by a black line connecting solid green circles (โ€ข). For Algorithm-II, the region of stick is indicated by a dashed blue line, and the region of slip is indicated by a dashed blue line connecting red dots(โ€ข). shows the normal pressure and lateral traction predictions of the two algorithms. (An analytical expression is not available in the literature). As expected, the two algorithms yield identical normal traction fields. However, there are serious discrepancies between the lateral tractions predicted by the two algorithms. Algorithm-II (MFV) predicts a smooth and symmetric distribution of shear traction with a clear separation between stick and slip boundaries (Fig. 3.12b). Algorithm-I shows a much wider stick region than does Algorithm-II. Most significantly, the stick region is not smooth; this is an artifact of nodes fluctuating back and forth between stick and slip status in the iterative process. In contrast, Algorithm-II assures that all constraints are satisfied at the beginning and end of each load step, and all points in between. It should be noted that in order to get convergence of Algorithm-I in this problem, it is necessary to set all nodes to stuck status after each load increment and then to release nodes according to how much they violate the constraining inequality (๐œ โ‰ค ๐œ‡ ๐‘  ๐‘). The above example demonstrates a situation where the more conventional contact algorithm 39 (Algorithm I) has difficulty converging to a unique solution; the two-parameter friction model is notorious for causing convergence problems [44]. Yet the Method of First Violation appears to address this problem with little difficulty. A major source of interest in elastic frictional contact is predicting and understanding the contact mechanics associated with cyclic loading. Some preliminary results with the cyclic loading are presented in the later this chapter. 3.5 High-Coefficient of Friction with Full Elastic Coupling We implemented both algorithms with a high coefficient of friction (๐œ‡ ๐‘  = 3.0) and a full elastic coupling. In this instance, the rigid cylinder is first pressed against a frictional elastic half-plane with normal load(๐‘ƒ = 10โˆ’3 ), and later pressed with a lateral load(๐‘„ = 2 ร— 10โˆ’4). The geometric and material properties of the rigid cylinder and elastic half-plane remain unchanged. The Klarbringโ€™s results [45] apply to bodies in contact at a single point. His formula is suggestive that higher coefficients of friction might cause issues of existence, uniqueness, and numerical stability to manifest even in problems of distributed contact. Consequently, one should not be surprised that distributed contact problems become more challenging as the coefficient of friction increases. Figure 3.13 shows that the lateral traction distribution predicted by Algorithm-I jumps alter- nately between positive and negative from node to node in the slip region and is irregular in stick region. Also, the discontinuous direction of friction in the slip regions obscures the boundary between stick and slip regions. Because of the full elastic coupling of this calculation, the jumps in lateral traction also influence the normal pressure, which also oscillates from node to node in the vicinity of the slip region. On the other hand, the lateral traction distribution predicted using Algorithm-II is smooth with a proper boundary between the slip and the stick regions. It should be pointed out that Algorithm-I often fails to converge for high values of friction coefficient and very fine meshes, yet this does not seem to be an issue with Algorithm-II. A quick exploration of mesh convergence using Algorithm-II is summarized in Figure 3.14. In 40 ยท10โˆ’2 Normal Press. A-II Lateral Trac. A-II Normal Press. A-I Lateral Trac. A-I Normal and Lateral Tractions 6 3 0 โˆ’3 โˆ’0.01 โˆ’0.005 0 0.005 0.01 Contact Region (x) Figure 3.13: Normal and lateral traction distribution after a normal and lateral pressing of the cylinder against an elastic half-plane with coefficient of friction ๐œ‡ ๐‘  = 3. Comparing the Algorithm- I and Algorithm-II with 85 contact nodes in both cases. the figure, the Algorithm-II calculations of 3.13 are re-plotted and labeled as FM, corresponding to the fine mesh of 85 nodes in the contact patch. Also shown in that figure are the predictions of Algorithm-II when a coarser mesh (CM) having only 42 nodes is employed. The two sets of traction predictions align closely except near the edges of the slip zones, where the coarse mesh is incapable of capturing fine detail. Though not definitive, this calculation is at least consistent with mesh convergence of Algorithm-II. Of course, the finer meshes will be associated with smaller load steps to reach the first violations. 3.6 Implementation of Dahl Friction Model The Method of First Violation (Algorithm-II) was also tested on a problem involving the Dahl friction model [24]. The Dahl model is a rate-independent, quasi-static continuous slip model, 41 ยท10โˆ’2 8 Normal Press. FM Lateral Trac. FM Normal Press. CM Lateral Trac. CM 6 Normal and Lateral Tractions 4 2 0 โˆ’0.01 โˆ’0.005 0 0.005 0.01 Contact Region (x) Figure 3.14: Normal and lateral traction distribution after a normal and lateral pressing of the cylinder against an elastic half-plane with ๐œ‡ ๐‘  = 3. Comparing the tractions predicted by Algorithm- II using a fine mesh (FM) having 85 nodes in the contact patch and those predicted by Algorithm-II using a coarse mesh (CM) having only 42 nodes in the contact patch. based on an idea of bristle stiffness (Eq. 3.9). ๐›พ   ๐‘‘๐‘“ ๐‘“ ๐‘“ = ๐œŽ 1 โˆ’ sgn( ๐‘ข) ยค sgn 1 โˆ’ sgn( ๐‘ข) ยค (3.9) ๐‘‘๐‘ข ๐‘“๐‘ ๐‘“๐‘ where, ๐‘“ , ๐‘“๐‘ , and ๐‘ข are the friction force, maximum/cut-off friction force and the lateral displacement of contacting node respectively. ๐œŽ and ๐›พ are the contact stiffness and exponents (Dahl parameters), respectively. In this model, lateral displacement of a node changes even while the friction force is less than cut-off friction ( ๐‘“ โ‰ค ๐‘“๐‘ ) - unlike Coulomb friction. But, as friction force reaches the maximum force ( ๐‘“๐‘ ) the Dahl model behaves similar to Coulomb friction at macro-slip. This similarity with Coulomb friction is exploited in the numerical implementation of this friction model, which is briefly explained in the appendix B. A contact problem is considered where the rigid indenter of the previous illustrations is pressed 42 ยท10โˆ’2 ยท10โˆ’4 8 6 Normal Press. Lateral Trac. Normal Disp. Lateral Disp. 6 4 Displacement 4 Pressure 2 2 0 0 โˆ’2 โˆ’0.01 โˆ’0.005 0 0.005 0.01 Contact Region (x) Figure 3.15: Implementing of the Method of First Violation algorithm with the Dahl friction model: (๐œŽ = 1000, ๐›พ = 1, ๐œ‡ ๐‘  = 0.3). Algorithm-I would not converge to a solution for Dahl friction model. with a normal force (๐‘ƒ = 0.001) against an elastic half-plane where the frictional interface is represented by a Dahl model having parameters ๐œŽ = 1000 and ๐›พ = 1, and the geometric and material properties are those of the from the previous problems. The maximum friction force at each node is ( ๐‘“๐‘ ๐‘› = ๐œ‡๐‘ ๐‘› ) with ๐œ‡ = 0.3. Figure 3.15 shows the predicted normal and lateral traction and displacement distributions when the MFV algorithm is employed. Those predicted tractions and displacements are smooth with clear boundaries between stick and slip regions. This demonstrates that the MFV algorithm can provide stable results in problems beyond Coulomb friction. The authors attempted to apply Algorithm-I to this problems, but could not obtain a convergent solution. 43 0.08 Normal Press. A-I Lateral Trac. A-I Normal Press. A-II Lateral Trac. A-II Pressure/ Lateral Traction 0.06 0.04 0.02 0 โˆ’0.02 โˆ’0.01 โˆ’0.005 0 0.005 0.01 Contact Region Figure 3.16: Normal and lateral traction distribution calculated for two-parameter friction (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.49) through Algorithm-I and Algorithm-II after a rigid cylinder is pressed against an elastic half-plane with a normal force (๐‘ƒ = 0.001). 3.7 Two-parameter Coulomb Friction for a Cylinder Pressed against an Elastic Half-plane A two-parameter Coulomb friction model is used between the indenter and the half-plane with static coefficient ๐œ‡ ๐‘  = 0.7 and dynamic coefficient ๐œ‡ ๐‘˜ = 0.49. To be able to compare numerical results with analytic ones, the Goodman decoupling approximation [31] is imposed by setting ๐ถ๐‘ข ๐‘ = 0. Initially, the rigid cylinder is pressed against the elastic half-plane with a normal force ๐‘ƒ = 0.001, bringing 43 nodes in the contact region. This results in symmetric normal pressure distribution and anti-symmetric lateral traction between two surfaces. In this case, the predictions calculated through Algorithm-I and Algorithm-II are identical, both shown on the left-hand side of Figure 3.16. Because of the imposition of the Goodman decoupling approximation, the normal pressure field is independent of the shear tractions and identical to the analytical expression, provided by Mindlin [52, 56]. There is no analytical expression available for lateral traction distribution for two-parameter Coulomb friction models. 44 0.08 Normal Press. A-I Lateral Trac. A-I Normal Press. A-II Lateral Trac. A-II Pressure/ Lateral Traction 0.06 0.04 0.02 0 โˆ’0.02 โˆ’0.01 โˆ’0.005 0 0.005 0.01 Contact Region Figure 3.17: Normal and lateral traction distribution calculated for two-parameter friction (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.49) through Algorithm-I and Algorithm-II after a rigid cylinder is pressed against an elastic half-plane with a normal and lateral forces (๐‘ƒ = 0.001, ๐‘„ = 0.000185). Slipping nodes are represented through โ€™red oโ€™ and โ€™Green *โ€™ for A-I and A-II respectively. Next, a lateral load of ๐‘„ = 0.000185 is applied gradually on the indenter in the x-direction. The plot on the right-hand side of Fig 3.17 shows the normal and lateral traction distribution as calculated by Algorithms I and II. The normal pressure distribution remains unchanged because of Goodmanโ€™s decoupling approximation; however, the lateral traction distributions calculated by the two methods show a fair difference. It is worth noting that the right-hand side of the shear traction field predicted by Algorithm-I follows a similar pattern to that predicted with Algorithm-II but shows significant spatial oscillation between stick and slip states. This is an indication of a failure of the algorithm to converge to a valid solution. Algorithm-II shows a proper separation between the stick and the slip regions. Both Algorithms involve small load increments, both are converged to the same loads, and the sums of the lateral tractions are also equal. Because the stress and displacement fields evolve from those of the previous load step, this small discrepancy can grow into large error as the load also evolves. 45 3.7.1 Cyclic Shear Loading with Two-Parameter Coulomb Friction Having provided evidence that this formulation using the method of first violation appears to provide a proper solution with the two-parameter Coulomb friction, it seems reasonable to explore the path dependence of traction fields during cyclic loading [16]. We consider the same configuration as before (see 3.7), impose a vertical compressive load of ๐‘ƒ = 0.001, and then apply a cyclic lateral loading with amplitude ๐‘„ max = 0.000185. The resulting lateral traction fields at multiple steps in the process are shown in Fig. 3.18. The following discussion is with reference to that figure. Sub-figure(1) has the same traction distribution as of Fig. 3.17: that which exists immediately after the imposition of the vertical load. As soon as a rightward lateral load is applied, the left side node that had been slipping leftward switches to being stuck. Sub-figure(2) with lateral load 0.3 ๐‘„ max shows a continuous slipping region on the right resulting from the rightward load adding to the initially rightward tractions from the previous load step. Sub-figure(3) with lateral load 0.7 ๐‘„ max shows a growth in the slipping region on the right and the beginning of slip on the left, where the rightward load has become sufficient to overcome the initial leftward traction and to bring that node into sliding to the right. Sub-figure(4) with lateral load ๐‘„ max shows further growth in slipping regions on both ends. There is a clear separation between sticking and slipping regions. This traction distribution provides a pattern for the residual stresses in cyclic loading that we see below. Sub-figure (5) shows the traction distribution just after the change in the direction of lateral load. All the nodes that had been slipping towards the right, now suddenly stick because the traction at each point has been reduced from that necessary for slip to just below that. These nodes remain stuck in sub-figure(6); the reduced load is again accommodated by a redistribution of traction among the stuck nodes without any actual sliding. In sub-figure (7), we see that as the applied load continues to decrease, the residual stress due to previous rightward sliding requires two nodes to start slipping leftward to maintain the force balance. Sub-figures (8) and (9) are in the same state and repeated to preserve periodicity. There is zero lateral load, but again some node must go into 46 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 4 4 4 4 FL = 0 1 FL = 0.3Fmax 2 FL = 0.7Fmax 3 FL = Fmax 4 Traction Value 2 2 2 2 0 0 0 0 โˆ’2 โˆ’2 โˆ’2 โˆ’2 โˆ’4 โˆ’4 โˆ’4 โˆ’4 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 4 4 4 4 FL = Fmax 5 FL = 0.7Fmax 6 FL = 0.3Fmax 7 FL = 0 8 Traction Value 2 2 2 2 0 0 0 0 โˆ’2 โˆ’2 โˆ’2 โˆ’2 โˆ’4 โˆ’4 โˆ’4 โˆ’4 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 4 4 4 4 FL = 0 9 FL = โˆ’0.3Fmax 10 FL = โˆ’0.7Fmax 11 FL = โˆ’Fmax 12 Traction Value 2 2 2 2 0 0 0 0 โˆ’2 โˆ’2 โˆ’2 โˆ’2 โˆ’4 โˆ’4 โˆ’4 โˆ’4 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 4 4 4 4 FL = โˆ’Fmax 13 FL = โˆ’0.7Fmax 14 FL = โˆ’0.3Fmax 15 FL = 0 16 Traction Value 2 2 2 2 0 0 0 0 โˆ’2 โˆ’2 โˆ’2 โˆ’2 โˆ’4 โˆ’4 โˆ’4 โˆ’4 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 4 4 4 4 FL = 0 17 FL = 0.3Fmax 18 FL = 0.7Fmax 19 FL = Fmax 20 Traction Value 2 2 2 2 0 0 0 0 โˆ’2 โˆ’2 โˆ’2 โˆ’2 โˆ’4 โˆ’4 โˆ’4 โˆ’4 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 โˆ’0.01 0 0.01 Contact Region Contact Region Contact Region Contact Region Lateral Traction Slipping Right Slipping Left Figure 3.18: Evolution of lateral traction with lateral cyclic loading (Arrow shows the direction of FIGURE 3: Evolution of lateral traction with lateral cyclic loading (Arrow shows the direction of lateral loading.) All tractions shown lateral loading.) All tractions shown are physical traction normalized by ๐บ. [16] are physical traction normalized by G. 4 Copyright c 2020 by ASME 47 slip in order for the integrated traction to be zero. Sub-figures (10), (11), and (12) show that the increase in load leftward increases the extent of the slip regions. Sub-figure (12) with lateral load โˆ’๐‘„ max shows a new negative peak of traction, which separates the stick and slip regions. The region between these two peaks will remain stuck throughout the cycling process. The traction fields of sub-figures 13-20 show the similar pattern of evolution as discussed for sub-figures 1-12, but the important observation is that steady-state has been reached by sub-figure 20: sub-figures 12 and 20 are right-left/up-down reflections of each other. These traction fields illustrated in Fig. 3.18 provide an insight on how the traction distribution evolves, how slip zones form, grow, and disappear, and generally how each traction field unfolds from that of the previous load. This logical progression argues that the correct path dependence has been adequately captured by the "Method of First Violation" algorithm. 3.8 Algorithm Computational Efficiency The relative efficiencies of the two algorithmic approaches is explored on our paradigm problem. A rigid cylinder is pressed normally over the elastic half-plane (same geometry and material properties from section 3.1 ) with normal load (๐‘ƒ = 0.004) and friction (๐œ‡ = 0.3) for multiple load steps (๐›ฟ๐ฟ 0 ). The efficiency of algorithms can be compared by counting the number (๐‘๐‘–๐‘ก๐‘’ ) of linear solves of Eq. 2.2, the most time-consuming part of the procedure. Figure 3.19 shows that the number of iterations using Algorithm-1 reduces with an increase in load step and that the number of iterations increases with an increase in contact nodes (mesh density). Though the increasing load step results in a faster solution, eventually the converged solution manifests visible anomalies. Red marks in Fig. 3.19 indicate non-physical solutions due to the attempt to resolve multiple contact violations associated with large load steps(๐›ฟ๐ฟ 0 ). The optimal load step for this algorithm would appear to be a bit smaller than those which cause visible anomalies in the solution; this is suggested by the dashed blue line. Figure 3.20 shows a concave up relation between the number of iterations on increasing the load step using Algorithm-II(MFV). On each curve in Fig. 3.20, all points represent an identical 48 4 Nodes 11 Nodes 21 3.5 Nodes 43 Nodes 85 Nodes 170 3 log10 (Nite ) 2.5 2 1.5 1 โˆ’5.5 โˆ’5 โˆ’4.5 โˆ’4 โˆ’3.5 โˆ’3 log10 (ฮดL0 ) Figure 3.19: Number of iterations (๐‘๐‘–๐‘ก๐‘’ ) vs load-step (๐›ฟ๐ฟ 0 ) using Algorithm-I for coarse to fine contact mesh (red marks shows the bad converged solution). Optimal load increments for this method on this problem are suggested by the dashed blue curve. solution; the choice of nominal load step impacts only the compute time to achieve that solution. These solutions capture the same load history and are admissible solutions throughout that history. The black star in each curve in Fig. 3.20, indicates the optimal load step size for maximum efficiency (faster convergence). The blue curve of optimal load increments associated with Figure 3.19 is reproduced in Figure 3.20 showing that both methods have about the same optimal load steps and the compute times when those optimal load steps are employed. 3.9 Summary of the Algorithm Verification and Comparison The contact algorithm - Method of First Violation (MFV) - based on the principle of first contact status violation is presented and its predictions are compared to those of a more conventional algorithm and with analytic expressions when available. The important role of contact history is emphasized and the capability of the MFV to capture and to accommodate that history dependence 49 Nodes 11 5 Nodes 21 Nodes 43 Nodes 85 Nodes 170 4 log10 (Nite ) 3 2 1 โˆ’5.5 โˆ’5 โˆ’4.5 โˆ’4 โˆ’3.5 โˆ’3 log10 (ฮดL0 ) Figure 3.20: Number of iterations (๐‘๐‘–๐‘ก๐‘’ ) vs load-step (๐›ฟ๐ฟ 0 ) using Algorithm-II(MFV) for coarse to fine contact mesh (Black marks shows the optimal load-step for faster and accurate solution). The blue dashed line is copied from Figure 3.19. is discussed. The fidelity to load-displacement history derives from this algorithm providing node states that are consistent with all contact constraints at the beginning and end of every load step as well as all points in between. The MFV algorithm predictions compared well with analytic solutions where they were available and also well with the predictions of a conventional algorithm when then that conventional algorithm provided numerically stable results. The robustness of the Method of First Violation was also tested on three categories of contact problem know to be numerically challenging 1. A contact problem involving two-parameter friction. Here the MFV provided robust and physically plausible solutions while the conventional algorithm anomalies at the stick-slip interfaces. 50 2. A contact problem of Coulomb but with full coupling and a high coefficient of friction. The analogy with Klarbringโ€™s results for single point contact would suggest numerical difficulties for problems of coupled contact of elastic bodies with high coefficient of friction. Indeed the conventional algorithm resulted in very discontinuous shear tractions in the vicinity of the slip zone and corresponding jaggedness in the normal traction. The predictions of the MFV shows none of these difficulties. 3. The MFV was applied to contact involving an entirely different form of contact model, the Dahl equation. Here the MFV again performs robustly and makes plausible predictions of traction and displacement fields, though there do not appear to be alternate estimates to compare to. The authors were unable to get the conventional algorithm to converge on this problem. 4. The MFV is shown to be stable when used with the two-parameter friction model and to provide a reasonable evolution of the traction fields from each load step to the next. 5. In the problems studied, when the Method of First Violation is used with optimal nominal load steps, its efficiency is comparable to that of the conventional algorithm. 51 PART II An Extremely Fine Mesh Strategy Using Compliance Matrices 52 CHAPTER 4 COMPLIANCE FORMULATION FOR AN ELASTIC DISK AND FOR A HOLE IN AN INFINITE ELASTIC PLANE The fundamental mechanics of contact problems is often elusive because of the intrinsic experimen- tal difficulties (the physics takes place exactly where it cannot be seen). This intrinsic experimental difficulty is matched by similarly difficult computational impediments [37, 51] that requires a very fine mesh near the contacting interfaces. The finite element method is the most general approach to create very fine mesh/discretization on the interfaces. However, with a large number of the nodes on the interface also creates many more nodes inside the body, which means more degrees of freedom to solve with a big stiffness matrix, making the contact problem hard to analyze. The reduction methods are fairly common to reduce the stiffness matrix to only contacting nodes, but they are challenging due to the required inversion of the big matrix. 4.1 Computational Issues The scope of the problem to be addressed is suggested in Figure 4.1. On the left is a problem of multiple contacts on an elastic disk; there are shear and normal tractions applied at the contact patches. It is only in the regions of the three contact patches that tractions and displacements are ๐‘ƒ ๐‘ƒ ๐‘„ ๐‘„ (a) Contact problem expected to (b) Boundary mesh of the (c) A coarse finite element have 33 nodes in contact among the disk with 147 nodes on the mesh with 1903 nodes inside three contact regions. boundary. the elastic disk Figure 4.1: A contact problem with normal (๐‘ƒ) and lateral (๐‘„) forces are applied pressing an elastic disk against a rigid surface with 33 boundary nodes in contact (11 nodes in each 4๐‘œ contact region). 53 essential to the problem in the sense that if the tractions in the contact patch were known, that would be sufficient to calculate stresses and displacements throughout the body. The figure in the center is a surface discretization such as could be used for a boundary integral element [9] analysis of the problem; note that extraordinarily fine meshes are required in the regions of contact in order to have enough nodes to solve the contact problem with any fidelity. A gradual mesh transition away from contact reduces the number of degrees of freedom of the corresponding numerical problem. However, the surface mesh must be fine enough overall to achieve accuracy [68, 34, 70]. One would like to have several tens of nodes over each contact patch. Contact patches are usually substantially less than 2๐‘œ of the osculating circle; for visibility, the surface mesh shown in Fig. 4.1b has 11 nodes over each contact patch of 4๐‘œ and the mesh transition is relatively aggressive (15%). In reality, one would expect each contact patch to be on the order of 1๐‘œ of the radius for the stresses to remain below yield, to have on the order of 50 โˆ’ 200 nodes per contact patch to obtain a reasonable resolution of the traction field, and several thousand nodes around the surface. Further, the nature of the boundary element method is such that the corresponding matrices are dense (not sparse), and the systems of equations that must be solved at each iteration of the contact algorithm becomes onerous. One can use the analytic expressions [35, 57] for deformation of frictionless contact for a cylinder pressed against a rigid surface (or of two identical cylinders pressed against each other) to evaluate the width of the contact patch and the maximum von Mises stress, each as functions of geometry, elastic constants, and net compression load. Table 4.1 provides the contact angle achieved just as the peak stress within the cylinder reaches yield stress for each of six common metals. Note that most of the metals exceed their elastic limit within a degree of contact arc. In the case of friction and applied lateral loads, the contact angles would be yet smaller. To understand the interface physics and to have a conclusive test model, extremely fine discretizations are needed for each contact region. The right-most mesh (Fig. 4.1c) is such as might be used for a finite element calculation, though for visibility the surface mesh is the same as the crude mesh of the center figure and the mesh transition away from the surface is also aggressive. Even so, there are approximately 1903 54 Metals Elasticityโ€™s Modu- Yield Strength Poissonโ€™s Contact arc an- lus (GPa) (MPa) Ratio gle (von-Mises) Copper 117 70 0.36 0.223๐‘œ Brass 102 117 0.35 0.428๐‘œ Mild Steel ASTM 302 200 250 0.3 0.4670 Aluminum 69 95 0.33 0.514๐‘œ Stainless Steel AISI 302 180 502 0.3 1.041๐‘œ Titanium 105 730 0.34 2.595๐‘œ Table 4.1: Maximum contact arc for compressing an elastic disk onto a flat rigid surface above which the disk material exceeds its elastic limit. nodes in the system shown in Fig. 4.1c. In an actual contact problem, one would expect to have to employ several tens of thousands of nodes. The corresponding equations โ€“ though banded โ€“ are still formidable and must be solved, either directly or indirectly, with the help of a static condensation, with each iteration of the contact algorithm. In general, solving contact problems with high resolution often becomes intractable. A strategy that can mitigate the difficulty of computational contact analysis greatly is to pose the problem using compliance matrices. For the sake of discussion, consider a frictionless contact problem such that each node has one normal displacement, one tangential displacement, and poten- tially one normal pressure. Because the problem is frictionless, it is only the normal displacement (๐‘ค ๐‘˜ ) and normal pressure (๐‘ ๐‘˜ ) that are necessary to address elastic contact. The usual finite element formulation where elasticity is expressed in terms of a stiffness matrix appears as: ๏ฃฑ ๏ฃด ๏ฃผ ๏ฃฑ ๏ฃผ ๏ฃด ๏ฃด ๏ฃด ๐‘1 ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค1 ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๐‘2 ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃฒ ๐‘ค2 ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด .. ๏ฃด = [๐พ] .. ๏ฃด (4.1) ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘๐‘ ๏ฃด ๏ฃด ๐‘ค๐‘ ๏ฃด ๏ฃด ๏ฃณ ๏ฃพ ๏ฃณ ๏ฃพ In this formulation, as the tractions change on just a few nodes in contact, the whole large system of equations must be solved. If we can formulate the elasticity in terms of a compliance matrix: 55 ๏ฃฑ ๏ฃด ๏ฃผ ๏ฃฑ ๏ฃผ ๏ฃด ๏ฃด ๏ฃด ๐‘ค1 ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘1 ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๐‘ค2 ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃฒ ๐‘2 ๏ฃด ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด .. ๏ฃด = [๐ถ] .. ๏ฃด (4.2) ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค๐‘ ๏ฃด ๏ฃด ๐‘๐‘ ๏ฃด ๏ฃด ๏ฃณ ๏ฃพ ๏ฃณ ๏ฃพ then we observe that all the nodes not in contact are subject to zero pressure, and we can dispense with all the columns in ๐ถ corresponding to those nodes. Further, the displacements outside the contact regions are not necessary to solve the contact problem, so the corresponding rows of ๐ถ can also be deleted. The resulting system can be orders of magnitude smaller than the original. A compliance representation of the elasticity is highly favorable to a solution of contact prob- lems, but it does not naturally come out of boundary integral methods or stiffness-based finite element implementations. In fact, there are only a few shapes that lend themselves to a compliance formulation. One is the an elastic half-plane [35, 72]. A compliance formulation for the surface of a disk or for a hole in an elastic plane is developed in the body of this chapter. Though the methods discussed below are restricted to narrow categories of two-dimensional geometries, they should provide the foundation for a computational test-bed sufficient to address some of the outstanding questions on interface mechanics, such as the significance of different frictional constitutive models in resulting contact stress distributions or elastic coupling between shear and normal displacement in conformal and non-conformal contact. 56 4.2 2D-Cylinder Airy Stress Function (Michell Solution) The Michell solution is an expansion of the Airy stress function in polar coordinates and is often used in studying the elasticity of two-dimensional circular bodies [6, 56, 62]. ฮฆ0 (๐‘Ÿ, ๐œƒ) =๐ด01๐‘Ÿ 2 + ๐ด02๐‘Ÿ 2 ln(๐‘Ÿ) + ๐ด03 ln(๐‘Ÿ) + ๐ด04 ๐œƒ   + ๐ด11๐‘Ÿ 3 + ๐ด12๐‘Ÿ ln(๐‘Ÿ) + ๐ด14๐‘Ÿ โˆ’1 cos ๐œƒ + ๐ด13๐‘Ÿ๐œƒ sin ๐œƒ   + ๐ต11๐‘Ÿ 3 + ๐ต12๐‘Ÿ ln(๐‘Ÿ) + ๐ต14๐‘Ÿ โˆ’1 sin ๐œƒ + ๐ต13๐‘Ÿ๐œƒ cos ๐œƒ (4.3) ร•โˆž   + ๐ด๐‘›1๐‘Ÿ ๐‘›+2 + ๐ด๐‘›2๐‘Ÿ โˆ’๐‘›+2 + ๐ด๐‘›3๐‘Ÿ ๐‘› + ๐ด๐‘›4๐‘Ÿ โˆ’๐‘› cos(๐‘›๐œƒ) ๐‘›=2 ร•โˆž   + ๐ต๐‘›1๐‘Ÿ ๐‘›+2 + ๐ต๐‘›2๐‘Ÿ โˆ’๐‘›+2 + ๐ต๐‘›3๐‘Ÿ ๐‘› + ๐ต๐‘›4๐‘Ÿ โˆ’๐‘› sin(๐‘›๐œƒ) ๐‘›=2 where the ๐ด๐‘š๐‘› and ๐ต๐‘š๐‘› are unknown coefficients. By definition, an Airy stress function satisfies the biharmonic equation (โˆ‡4 ฮฆ0 = 0)), and indeed every term of the Michell expansion does so. The stress and displacement fields associated with an Airy stress function can all be expressed in terms of partial derivatives of ฮฆ, and those fields for the Michell expansion have been tabulated in many texts, but are combined in Appendix Table ?? and E.2 in consistent notation for the convenience of the reader. The Michell expansion has been used to find analytic expressions for traction and displacement distributions for many problems involving two-dimensional elastic disks or holes in an infinite elastic plane. The following is used in a somewhat different manner; it will be used to discretize the surfaces of disks or holes and to obtain a very efficient compliance formulation for elastic, frictional contact analysis. Which terms in Equation 4.3 are actually employed depends on the geometry of the problem. Terms that generate non-integrable singular stress fields as ๐‘Ÿ โ†’ 0 are not used in problems involving a solid disk. Terms associated with stress fields that grow without bound as ๐‘Ÿ โ†’ โˆž are not used in problems involving unbounded boundaries. Some terms generate multi-valued displacement fields (secular terms not periodic in ๐œƒ), and these must be combined to yield displacement fields that are periodic in ๐œƒ. 57 4.2.1 General Discretization Strategy A set of evenly spaced nodes are defined on the free surface(s) of the body of interest. This may be a very fine mesh because only the nodes in the vicinity of contact will contribute to the size of the numeric problem to be solved. A compliance matrix associated with that mesh is constructed using the strategy indicated Fig. 4.2. Traction basis Michell expansion, function ฮ ๐‘› (๐œƒ) unknown coef. centered at ๐œƒ ๐‘› Fourier Se- Michell Traction ries of ฮ ๐‘› Fourier Series Deduce Michell Coefficients Displacements ๐‘ข๐‘Ÿ๐‘Ÿ , ๐‘ข๐‘Ÿ๐œƒ at each node Column of compliance matrix Figure 4.2: Flow chart using Michell expansion to create a compliance matrix. The strategy of exploiting the Michell expansion to construct a compliance matrix is outlined in Figure 4.2. 1. A set of 2๐‘ basis functions ฮ ๐‘› for traction (one for each of radial traction and shear traction at each node) on the surface(s) of the region of interest is defined. This set of basis functions must be a partition of unity, have only local support, and must achieve their maximum amplitude at their associated node. The simplest such functions are the rectangular or box function ๐œ™๐‘› (๐‘ฅ) = ๐ป (๐‘ฅ โˆ’ (๐‘ฅ ๐‘› โˆ’ ฮ”๐‘ฅ/2)) โˆ’ ๐ป (๐‘ฅ โˆ’ (๐‘ฅ ๐‘› + ฮ”๐‘ฅ/2)) (4.4) 58 which has a value of 1.0 between (๐‘ฅ ๐‘› โˆ’ ฮ”๐‘ฅ/2) and (๐‘ฅ ๐‘› + ฮ”๐‘ฅ/2) and is zero elsewhere and the chapeau function ๐ถ๐‘› (๐‘ฅ) = ๐œ™๐‘› (๐‘ฅ) (๐‘ฅ ๐‘› โˆ’ |๐‘ฅ|) /(ฮ”๐‘ฅ/2) (4.5) which has a triangular shape, taking the value of 1.0 at ๐‘ฅ ๐‘› , zero at ๐‘ฅ ๐‘› ยฑ ฮ”๐‘ฅ/2, and zero outside of (๐‘ฅ ๐‘› โˆ’ ฮ”๐‘ฅ/2, ๐‘ฅ ๐‘› + ฮ”๐‘ฅ/2) [10]. These basis functions are each decomposed into their Fourier components. 2. The contributions of each term of the Michell expansion to surface tractions are all trigono- metric functions. Coefficients of each trig function are grouped. 3. Matching terms of both Fourier series yields a linear system of simultaneous equations for coefficients of the Michell expansion. In the problems considered below, these equations can be decoupled. 4. Given the Michell expansion coefficients, the radial and tangential displacements on the surface are evaluated at every node. 5. These displacements become the ๐‘›th column of the compliance matrix. 4.2.2 A Solid Elastic Disk For an elastic disk with no holes (Fig. 4.3), all terms of the Michell expansion associated with non- integrable stress singularities at ๐‘Ÿ = 0 are discarded ({๐ด02 , ๐ด03 , ๐ด14 , ๐ต14 , ๐ด๐‘›2 , ๐ด๐‘›4 , ๐ต๐‘›2 , ๐ด๐‘›4 , ๐ด04 } = 0). Similarly, terms associated with multi-values of displacement should also be removed to provide a meaningful solution ({ ๐ด12 , ๐ต12 , ๐ด13 , ๐ต13 } = 0). These restrictions result in a single valued displacement stress function expression with non-secular terms (Eq. 4.6) at each value of (๐‘Ÿ, ๐œƒ). ฮฆ0 (๐‘Ÿ, ๐œƒ) = ๐ด01๐‘Ÿ 2 + ๐ด11๐‘Ÿ 3 cos(๐œƒ) + ๐ต11๐‘Ÿ 3 sin(๐œƒ) ร•โˆž   (4.6) + ( ๐ด๐‘›1๐‘Ÿ ๐‘›+2 + ๐ด๐‘›3๐‘Ÿ ๐‘› ) cos(๐‘›๐œƒ) + (๐ต๐‘›1๐‘Ÿ ๐‘›+2 + ๐ต๐‘›3๐‘Ÿ ๐‘› ) sin(๐‘›๐œƒ) ๐‘›=2 59 The expression (4.6) is suitable for solid disks with no net forces or moments. When there are net forces and moments, some of the discarded terms can be brought in with the assurance that all singularities are integrable and that all displacements are single-valued. The ฮฆ0 term ๐ด04 ๐œƒ has stresses values as ๐œŽ๐‘Ÿ๐‘Ÿ = 0,๐œŽ๐œƒ๐œƒ = 0 and, ๐œŽ๐‘Ÿ๐œƒ = 1/๐‘Ÿ 2 (see the table ??). The ๐œŽ๐‘Ÿ๐œƒ = 1/๐‘Ÿ 2 has a singularity at r=0, but this is an integrable singularity at the center of the disk, corresponding to a finite torque. If one wishes to incorporate non-self equilibrating tractions to the boundary of a disk, one must balance them with some internal traction that will oppose the net forces and moments for the external distribution. Such terms are available from the Michell solution, but the cost is accommodating singularities at the center of the disk since our focus is on the surface of the disk, singularities in stress or displacement are acceptable. The ฮฆ0 terms (๐ด12๐‘Ÿ ln(๐‘Ÿ) cos(๐œƒ), ๐ด13๐‘Ÿ๐œƒ sin(๐œƒ), ๐ต12๐‘Ÿ ln(๐‘Ÿ) sin(๐œƒ), and ๐ต13๐‘Ÿ๐œƒ cos(๐œƒ)) have multi- valued displacement terms and contain displacement elements that are proportional to ๐œƒ. This issue can be resolved by taking linear combinations of these terms for which the multi-valued secular terms disappear (see Eq. 4.7). ๐œ…+1 ๐œ…+1 ๐ด13 = โˆ’๐ด12 and ๐ต13 = ๐ต12 (4.7) ๐œ…โˆ’1 ๐œ…โˆ’1 where, ๐œ… is a Kolosovโ€™s constant (Eq. 3.20 of [6]): ๏ฃฑ ๏ฃฒ 3 โˆ’ 4๐œˆ for plane strain ๏ฃด ๏ฃด ๏ฃด ๐œ… = 3โˆ’๐œˆ (4.8) ๏ฃด ๏ฃด for plane stress ๏ฃณ 1+๐œˆ ๏ฃด As these terms are non-secular now and represent the net forces and moments. The new stress function will be the following: ฮฆ0 (๐‘Ÿ, ๐œƒ) = ๐ด01๐‘Ÿ 2 + ๐ด04 ๐œƒ   3 ๐œ…+1 + ๐ด11๐‘Ÿ cos(๐œƒ) + ๐ด12 ๐‘Ÿ log(๐‘Ÿ) cos(๐œƒ) โˆ’ ๐‘Ÿ๐œƒ sin(๐œƒ) ๐œ…โˆ’1 (4.9)   3 ๐œ…+1 + ๐ต11๐‘Ÿ sin(๐œƒ) + ๐ต12 ๐‘Ÿ log(๐‘Ÿ) sin(๐œƒ) + ๐‘Ÿ๐œƒ cos(๐œƒ) ๐œ…โˆ’1 ร•โˆž   + ( ๐ด๐‘›1๐‘Ÿ ๐‘›+2 + ๐ด๐‘›3๐‘Ÿ ๐‘› ) cos(๐‘›๐œƒ) + (๐ต๐‘›1๐‘Ÿ ๐‘›+2 + ๐ต๐‘›3๐‘Ÿ ๐‘› ) sin(๐‘›๐œƒ) ๐‘›=2 60 โˆ’ฯ€ ฯ€ Elastic Disk M P โˆ’ฯ€/2 Q ฯ€/2 โˆ†ฮธ 0 ฮธ ฯ†0 (ฮธ) Figure 4.3: A solid disk with discretizaton from โˆ’๐œ‹ to ๐œ‹ Stresses and displacements were calculated for the above stress function, which are shown in Appendix C.1. 4.2.2.1 Creating the Surface Displacement Field in an Elastic Disk A very fine discretization is created on the elastic disk shown in Fig. 4.3 to formulate a numerical solution. The circular disk discretized into ๐‘ intervals with the equal length of ฮ”๐œƒ = 2๐œ‹/๐‘ from โˆ’๐œ‹ to ๐œ‹ as shown in Fig. 4.3. Unit tractions (normal and then shear) are imposed, one at a time in each interval, and the corresponding displacement field is evaluated at nodes placed at the centers of that and of every other interval. For instance, unit pressure load ๐œ™0 (๐œƒ) is applied radially on the disk at ๐œƒ = 0; taking a value of one in the region of (โˆ’ฮ”๐œƒ/2, ฮ”๐œƒ/2), and zero in at any other region of (โˆ’๐œ‹, ๐œ‹). The coefficients of the Fourier expansion for ๐œ™0 (๐œƒ) can be calculated in closed form, as shown in Eq. 4.10 and those coefficients are used to deduce the coefficients of the terms of the Michell solution for that traction field and in turn the corresponding displacement at each node. (This process is outlined in C.2. ) 61 0.5 1.2 k= N k= 4N 1 k= 16N 0.4 k= 32N 0.8 Unit Basis 0.3 Displacement 0.6 Pressure 0.4 0.2 0.2 0 0.1 โˆ’0.2 0 โˆ’0.4 โˆ’0.1 โˆ’0.05 0 0.05 0.1 Contact Length (x) Figure 4.4: The displacement field resulting from the unit pressure over ฮ”๐œƒ centered at ๐œƒ = 0 is obtained by summing a Fourier series. Though there would be Gibbs phenomena if the pressure field were recovered from Fourier series, there is no Gibbs phenomena associated with the displacement field. ๐ป๐‘Ž๐‘Ÿ๐‘š ฮ”๐œƒ 2 ร• 1 ๐œ™0 (๐œƒ) = + cos(๐‘˜๐œƒ) sin(๐‘˜ฮ”๐œƒ) (4.10) 2๐œ‹ ๐œ‹ ๐‘˜=1 ๐‘˜ The displacement field corresponding to basis traction ๐œ™0 (๐œƒ) is also expressed in the Fourier series. It should be noted that though there would be a Gibbs phenomenon if it were necessary to construct ๐œ™0 (๐œƒ) from its Fourier series, there are no Gibbs phenomena in the construction of the corresponding displacement field (Fig. 4.4). The convergence of the displacement field with the number of harmonics (๐ป๐‘Ž๐‘Ÿ๐‘š) depends on the width of ฮ”๐œƒ and as one sees from Fig. 4.4 it is more 2๐œ‹ than sufficient to use ๐ป๐‘Ž๐‘Ÿ๐‘š = 4 ฮ”๐œƒ = 4๐‘ harmonics. The Michell terms (Aโ€™s and Bโ€™s) for unit normal and shear tractions over ฮ”๐œƒ centered at ๐œƒ = 0 for the solid elastic disk are listed in the Tab. 4.2. These Aโ€™s and Bโ€™s values are substituted in formulae (Eqs.C.6) and the corresponding radial and tangential displacements are evaluated. Eqs. 62 Terms Radial Traction Tangential Traction ๐ด01 ฮ”๐œƒ/(4๐œ‹) 0 ๐ด04 0 2 ฮ”๐œƒ๐‘… /(2๐œ‹) (๐œ… โˆ’ 1) sin( ฮ”๐œƒ 2 ) ๐ด11 0 2(1 + ๐œ…)๐œ‹๐‘… (1 โˆ’ ๐œ…)๐‘… sin( ฮ”๐œƒ2 ) ๐ด12 0 (1 + ๐œ…)๐œ‹ โˆ’(๐œ… + 3) sin( ฮ”๐œƒ 2 ) ๐ต11 0 2(1 + ๐œ…)๐‘…๐œ‹ โˆ’(๐œ… โˆ’ 1)๐‘… sin( ฮ”๐œƒ 2 ) ๐ต12 0 (1 + ๐œ…)๐œ‹ ๐‘… โˆ’๐‘› sin( ๐‘›ฮ”๐œƒ 2 ) ๐ด๐‘›1 0 ๐‘›(๐‘› + 1)๐œ‹ 2โˆ’๐‘› ๐‘… sin( ๐‘›ฮ”๐œƒ 2 ) ๐ด๐‘›3 0 ๐‘›(๐‘› โˆ’ 1)๐œ‹ ๐‘… โˆ’๐‘› sin( ๐‘›ฮ”๐œƒ 2 ) ๐ต๐‘›1 0 ๐‘›(1 + ๐‘›)๐œ‹ ๐‘… 2โˆ’๐‘› (๐‘› โˆ’ 2) sin( ๐‘›ฮ”๐œƒ 2 ) ๐ต๐‘›3 0 ๐‘›2 (๐‘› โˆ’ 1)๐œ‹ Table 4.2: Michell coefficients for applied radial and tangential unit tractions over ฮ”๐œƒ centered at ๐œƒ = 0 and ๐‘Ÿ = ๐‘… on the elastic solid disk 4.11 are displacements due to unit radial pressure, Equations 4.12 are displacements due to unit tangential tractions, and the subscript ๐‘†๐ท refers to โ€™Solid Diskโ€™. ๐‘…(๐œ… โˆ’ 1)ฮ”๐œƒ (๐œ… โˆ’ 3 + 4 ln(๐‘…))๐œ…๐‘… sin(ฮ”๐œƒ/2) cos(๐œƒ) ๐‘ค ๐‘Ÿ0๐‘†๐ท (๐œƒ) = + 8๐œ‹๐œ‡ 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(1 + ๐œ…(๐‘› โˆ’ 1) + ๐‘›) sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) + 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ 2 (4.11) (๐œ… + ๐œ… โˆ’ 4 + 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐‘ข๐‘Ÿ0๐‘†๐ท (๐œƒ) = 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(๐œ…(๐‘› โˆ’ 1) โˆ’ 1 โˆ’ ๐‘›) sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) + 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ (๐œ… 2 + ๐œ… โˆ’ 4 โˆ’ 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐‘ค 0๐œƒ ๐‘†๐ท (๐œƒ) = โˆ’ 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(1 โˆ’ ๐œ…(๐‘› โˆ’ 1) + ๐‘›) sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) + 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ (4.12) ๐‘…ฮ”๐œƒ (๐œ… 2 + 5๐œ… + 8 + 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) cos(๐œƒ) ๐‘ข 0๐œƒ ๐‘†๐ท (๐œƒ) = + 4๐œ‹๐œ‡ 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(๐œ…(๐‘› โˆ’ 1) + 1 + ๐‘›) sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) + 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ 63 We used these displacement fields (๐‘ค ๐‘Ÿ0๐‘†๐ท (๐œƒ), ๐‘ข๐‘Ÿ0๐‘†๐ท (๐œƒ), ๐‘ค 0๐œƒ ๐‘†๐ท (๐œƒ), and ๐‘ข 0๐œƒ ๐‘†๐ท (๐œƒ)) to construct a compliance matrix in the later section 4.3. 4.2.3 A Hole in an Infinite Elastic Plane Next consider another 2D circular geometry โ€“ a hole in an elastic infinite plane (see Fig. 4.5). Its stress and displacement fields can also be expressed in terms of the Michell Airy stress function (Eq. 4.3). Terms that are unbounded for large ๐‘Ÿ (especially at ๐‘Ÿ โ†’ โˆž) (those associated with {๐ด01 ,๐ด03 ,๐ด11 , ๐ต11 ,๐ด๐‘›1 ,๐ด๐‘›3 ,๐ต๐‘›1 , ๐ต๐‘›3 }) must be dropped. Terms that are multi-value must be dropped or combined so as to remove that feature (Eq.4.7). This results in the following stress function: ฮฆ0 (๐‘Ÿ, ๐œƒ) = ๐ด01๐‘Ÿ 2 + ๐ด03 ln(๐‘Ÿ) + ๐ด04 ๐œƒ + ๐ด23๐‘Ÿ 2 cos(๐œƒ) + ๐ต23๐‘Ÿ 2 sin(๐œƒ)   ๐œ…+1 + ๐ด12 ๐‘Ÿ log(๐‘Ÿ) cos(๐œƒ) โˆ’ ๐‘Ÿ๐œƒ sin(๐œƒ) + ๐ด13๐‘Ÿ๐œƒ cos(๐œƒ) ๐œ…โˆ’1 (4.13)   ๐œ…+1 + ๐ต12 ๐‘Ÿ log(๐‘Ÿ) sin(๐œƒ) + ๐‘Ÿ๐œƒ cos(๐œƒ) + ๐ต13๐‘Ÿ๐œƒ sin(๐œƒ) ๐œ…โˆ’1 ร• โˆž ( ๐ด๐‘›2๐‘Ÿ 2โˆ’๐‘› + ๐ด๐‘›4๐‘Ÿ โˆ’๐‘› ) cos(๐‘›๐œƒ) + (๐ต๐‘›2๐‘Ÿ 2โˆ’๐‘› + ๐ต๐‘›4๐‘Ÿ โˆ’๐‘› ) sin(๐‘›๐œƒ)   + ๐‘›=2 Stresses and displacements were calculated for this stress function using Michell tables ?? and E.2 are shown in the D.1. 4.2.3.1 Creating Surface Displacement Field for a Hole in an Infinite Plane The hole of radius R in the infinite plane is very finely discretized from โˆ’๐œ‹ to ๐œ‹ with ๐‘ small nodes, as shown in the Fig. 4.5. A unit pressure ๐œ™0 (๐œƒ) (Eq. 4.10) is applied over an interval ฮ”๐œƒ centered at ๐œƒ = 0. This pressure distribution is expanded in Fourier series (Eq. 4.10) and coefficients are matched to the trigonometric polynomials of stresses (Eqs. D.2), yielding Michell coefficients (Appendix D.2). A similar process is performed with respect to a unit shear traction over ฮ”๐œƒ. The Michell terms (Aโ€™s and Bโ€™s) are listed in Tab. 4.3 for both unit radial and tangential tractions. These A and B values are substituted in the displacement (Eqs. D.7) and yield corresponding radial and 64 Infinite Elastic Plane โˆ’ฯ€ ฯ€ โˆ’ฯ€/2 ฯ€/2 โˆ†ฮธ 0 ฮธ ฯ†0 (ฮธ) Figure 4.5: An infinite plane with a hole discretized from โˆ’๐œ‹ to ๐œ‹ Terms Radial Traction Tangential Traction ฮ”๐œƒ๐‘… 2 ๐ด03 2๐œ‹ 0 ฮ”๐œƒ๐‘… 2 ๐ด04 0 2๐œ‹ โˆ’(๐œ… โˆ’ 1)๐‘… sin( ฮ”๐œƒ 2 ) ๐ด12 0 (1 + ๐œ…)๐œ‹ โˆ’(๐œ… โˆ’ 1)๐‘… 3 sin( ฮ”๐œƒ 2 ) ๐ด14 0 2(1 + ๐œ…)๐œ‹ (๐œ… โˆ’ 1)๐‘… sin( ฮ”๐œƒ 2 ) ๐ต12 0 (1 + ๐œ…)๐œ‹ (๐œ… + 3)๐‘… 3 sin( ฮ”๐œƒ2 ) ๐ต14 0 2(1 + ๐œ…)๐œ‹ ๐‘… ๐‘› sin( ๐‘›ฮ”๐œƒ 2 ) ๐ด๐‘›2 โˆ’ 0 ๐‘›(๐‘› โˆ’ 1)๐œ‹ ๐‘… ๐‘›+2 sin( ๐‘›ฮ”๐œƒ 2 )) ๐ด๐‘›4 0 ๐‘›(๐‘› + 1)๐œ‹ ๐‘… ๐‘› sin( ๐‘›ฮ”๐œƒ 2 ) ๐ต๐‘›2 0 โˆ’ ๐‘›(๐‘› โˆ’ 1)๐œ‹ ๐‘… ๐‘›+2 (2 + ๐‘›) sin( ๐‘›ฮ”๐œƒ2 ) ๐ต๐‘›4 0 ๐‘›2 (๐‘› + 1)๐œ‹ Table 4.3: Michell coefficients for applied radial and tangential tractions over ฮ”๐œƒ centered at ๐œƒ = 0 and ๐‘Ÿ = ๐‘… for the hole in the elastic plane. 65 tangential displacement fields (Eqs. 4.14 and 4.15). Equations 4.14 are the displacements due to application of unit radial pressure and equations 4.15 are displacements due to unit tangential traction. ๐‘…ฮ”๐œƒ (๐œ… + 1 โˆ’ 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) cos(๐œƒ) ๐‘ค ๐‘Ÿ0๐ป ๐‘œ๐‘™๐‘’ (๐œƒ) = โˆ’ 4๐œ‹๐œ‡ 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(1 + ๐œ…(๐‘› + 1) + ๐‘›) sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) โˆ’ 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ (4.14) (๐œ… + 1 + 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐‘ข๐‘Ÿ0๐ป ๐‘œ๐‘™๐‘’ (๐œƒ) = โˆ’ 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(๐œ…(๐‘› + 1) + 1 โˆ’ ๐‘›) sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) + 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ (๐œ… + 1 + 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐‘ค 0๐œƒ ๐ป ๐‘œ๐‘™๐‘’ (๐œƒ) = 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(1 + ๐œ…(๐‘› + 1) โˆ’ ๐‘›) sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) โˆ’ 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ (4.15) ๐‘…ฮ”๐œƒ (๐œ… + 1 โˆ’ 4๐œ… ln(๐‘…))๐‘… sin(ฮ”๐œƒ/2) cos(๐œƒ) ๐‘ข 0๐œƒ ๐ป ๐‘œ๐‘™๐‘’ (๐œƒ) = โˆ’ โˆ’ 4๐œ‹๐œ‡ 4๐œ‹๐œ‡(๐œ… + 1) ๐‘…(๐œ…(๐‘› โˆ’ 1) โˆ’ 1 + ๐‘›) sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) โˆ’ 2๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐œ‡ These displacement fields (๐‘ค ๐‘Ÿ0๐ป ๐‘œ๐‘™๐‘’ (๐œƒ), ๐‘ข๐‘Ÿ0๐ป ๐‘œ๐‘™๐‘’ (๐œƒ), ๐‘ค 0๐œƒ ๐ป ๐‘œ๐‘™๐‘’ (๐œƒ), and ๐‘ข 0๐œƒ ๐ป ๐‘œ๐‘™๐‘’ (๐œƒ)) are the displacements on the surface of a hole in an infinite elastic plane for an unit radial and tangential force. These displacement fields can be used to construct a compliance matrix. 4.3 Creating a Compliance Matrix In previous sections, the pressure-displacement relationships for a solid elastic disk and for a hole in an infinite elastic plane were developed for cases of unit tractions over intervals ฮ”๐œƒ centered at ๐œƒ = 0. In the following, these results are expanded to construct the full compliance matrix. As before, the surface of the elastic disk is discretized with N nodes evenly distributed over the surface (Fig. 4.3). The inter-nodal distance is ฮ”๐œƒ = 2๐œ‹/๐‘. Each node corresponds to radial and tangential displacements to their tractions centered on them. Applied traction fields are represented 66 using 2๐‘ basis functions is shown in Eq. 4.16. ๐‘€โˆ’1 ร• ๐‘€โˆ’1 ร• ๐‘๐‘Ÿ (๐œƒ) = ๐‘๐‘Ÿ๐‘˜ ๐œ“ ๐‘˜ (๐œƒ) ๐‘ ๐œƒ (๐œƒ) = ๐‘ ๐œƒ๐‘˜ ๐œ“ ๐‘˜ (๐œƒ) (4.16) ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ where ๐‘€ is half the number of Nodes ๐‘, ๐‘๐‘Ÿ๐‘˜ and ๐‘ ๐œƒ๐‘˜ are the radial and tangential tractions at the corresponding ๐œƒ ๐‘˜ = ๐‘˜ฮ”๐œƒ, and ๐œ“ ๐‘˜ (๐œƒ) is a basis function centered at ๐œƒ ๐‘˜ . At this stage, the basis functions are assumed to form a partition of unity and by symmetry, each basis function is assumed to be obtained from the previous one by a shift of ฮ”๐œƒ. It would be natural here to use either box functions (Eq 4.4) or chapeau functions (4.5); box functions were chosen because of the simpler expressions that result, but chapeau functions would be only slightly more difficult to implement. The box function ๐œ“ ๐‘˜ is defined by ๐œ“ ๐‘˜ (๐œƒ) = ๐œ™0 (๐œƒ โˆ’ ๐œƒ ๐‘˜ ) (4.17) Similarly, the radial and tangential displacement fields can be expressed ๐‘˜=๐‘€โˆ’1 ร• ๐‘˜=๐‘€โˆ’1 ร• ๐‘ค ๐‘Ÿ (๐œƒ) = ๐‘ค ๐‘Ÿ๐‘˜ (๐œƒ) ๐‘๐‘Ÿ๐‘˜ + ๐‘ค ๐œƒ๐‘˜ (๐œƒ) ๐‘ ๐œƒ๐‘˜ + ๐‘Š๐‘ก๐‘œ๐‘ก ๐‘Ÿ ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ (4.18) ๐‘˜=๐‘€โˆ’1 ร• ๐‘˜=๐‘€โˆ’1 ร• ๐‘ข ๐œƒ (๐œƒ) = ๐‘ข๐‘Ÿ๐‘˜ (๐œƒ) ๐‘๐‘Ÿ๐‘˜ + ๐‘ข ๐œƒ๐‘˜ (๐œƒ) ๐‘ ๐œƒ๐‘˜ + ๐‘ˆ๐‘ก๐‘œ๐‘ก ๐œƒ ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ where ๐‘Š๐‘ก๐‘œ๐‘ก๐‘Ÿ and ๐‘ˆ ๐œƒ are unknown rigid body constant for radial and tangential displacements, ๐‘ก๐‘œ๐‘ก respectively; and ๐‘ค ๐‘Ÿ๐‘˜ (๐œƒ) = ๐‘ค ๐‘Ÿ0๐‘†๐ท (๐œƒ โˆ’ ๐œƒ ๐‘˜ ), ๐‘ข๐‘Ÿ๐‘˜ (๐œƒ) = ๐‘ข๐‘Ÿ0๐‘†๐ท (๐œƒ โˆ’ ๐œƒ ๐‘˜ ), (4.19) ๐‘ค ๐œƒ๐‘˜ (๐œƒ) = ๐‘ค 0๐œƒ ๐‘†๐ท (๐œƒ โˆ’ ๐œƒ ๐‘˜ ), ๐‘ข ๐œƒ๐‘˜ (๐‘ฅ) = ๐‘ข 0๐œƒ ๐‘†๐ท (๐œƒ โˆ’ ๐œƒ๐‘˜ ) These relationships are illustrated for ๐œ“ ๐‘˜ (๐œƒ) and ๐‘ค ๐‘Ÿ๐‘˜ (๐œƒ) for the ๐‘˜ = โˆ’1, 0, 1 in Fig. 4.6. The displacement corresponding to the pressure distribution would be calculated through Eq. 4.18. These linear equations can be employed directly to create a compliance matrix in polar coordinate involving radial and tangential components of tractions and displacements as shown in Eq. 4.21. Furthermore, the net forces and moments applied to the cylinder must equilibrate the tractions of the contact region as shown in Fig. 4.7. These are introduced in the compliance matrix through the Eq. 4.20. 67 0.5 1.2 Pโˆ’1 P0 P1 N odes 1 0.4 0.8 0.3 Displacement 0.6 Pressure 0.4 0.2 0.2 0 0.1 โˆ’0.2 0 โˆ’0.4 โˆ’0.1 โˆ’0.05 0 0.05 0.1 Contact Region (x) Figure 4.6: On moving the unit normal pressure the corresponding displacement field (dashed) is also moving that is one of the columns of compliance matrix ๐‘˜=๐‘€โˆ’1 ร• ๐‘ƒ= (โˆ’๐‘๐‘Ÿ๐‘˜ cos ๐œƒ ๐‘˜ + ๐‘ ๐œƒ๐‘˜ sin ๐œƒ ๐‘˜ )๐‘…ฮ”๐œƒ, ๐‘˜=โˆ’๐‘€ ๐‘˜=๐‘€โˆ’1 ร• ๐‘„= ( ๐‘๐‘Ÿ๐‘˜ sin ๐œƒ ๐‘˜ + ๐‘ ๐œƒ๐‘˜ cos ๐œƒ ๐‘˜ )๐‘…ฮ”๐œƒ, (4.20) ๐‘˜=โˆ’๐‘€ ๐‘˜=๐‘€โˆ’1 ร• and ๐‘€= ๐‘ ๐œƒ๐‘˜ ๐‘… 2 ฮ”๐œƒ ๐‘˜=โˆ’๐‘€ The external forces and moments applied to the disk can be imposed either through Michel compo- nents that are singular at the center of the disk (such as ๐‘Ÿ ln(๐‘Ÿ) cos(๐œƒ)) or more smoothly by tractions imposed on the disk surface. If one is interested only in the traction and displacement fields near contact, St. Venantโ€™s principle permits one to use either approach. In fact, if one wants to analyze the stress and displacement fields arising from forces conveyed through a rod going through the center of the disk, the singular forces and moment applied at the center might be more appropriate. 68 ๏ฃฑ .. ๏ฃผ ๏ฃฎ .. .. .. ๏ฃน๏ฃบ ๏ฃฑ .. ๏ฃผ ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ . . . ๏ฃบ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค ๐‘Ÿ๐‘› ๐ถ๐œƒ๐‘Ÿ โˆ’ cos ๐œƒ ๐‘› sin ๐œƒ ๐‘› 0 ๏ฃบ๏ฃบ ๏ฃด ๐‘๐‘Ÿ๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐ถ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐‘Ÿ๐‘Ÿ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ .. .. .. ๏ฃบ ๏ฃด๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃฏ . . . ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๐‘ข ๐‘›๐œƒ sin ๐œƒ ๐‘› cos ๐œƒ ๐‘› 1 ๏ฃบ๏ฃด ๐‘ ๐‘›๐œƒ ๏ฃด ๏ฃด ๏ฃฏ ๐ถ๐‘Ÿ๐œƒ ๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด ๐ถ๐œƒ๐œƒ ๏ฃบ ๏ฃฒ ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด = (4.21) ๏ฃฏ .. ๏ฃฏ .. .. ๏ฃบ .. ๏ฃบ ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃฏ ๏ฃด . . . ๏ฃบ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ƒ/๐‘…ฮ”๐œƒ ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ โˆ’ cos ๐œƒ ๐‘› . . . sin ๐œƒ ๐‘› 0 0 0 ๏ฃบ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๐‘Š๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘„/๐‘…ฮ”๐œƒ ๏ฃด ๏ฃฏ sin ๐œƒ ๐‘› . . . cos ๐œƒ ๐‘› 0 0 0 ๏ฃบ๏ฃด ๐‘ˆ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘€/๐‘… 2 ฮ”๐œƒ ๏ฃด ๏ฃบ๏ฃด ๏ฃด 0 1 0 0 0 ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃพ ๏ฃฐ ... ๐œƒ ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃณ ๏ฃป๏ฃณ ๏ฃพ where ๐‘Š๐‘ก๐‘œ๐‘ก , ๐‘ˆ๐‘ก๐‘œ๐‘ก and ๐œƒ ๐‘ก๐‘œ๐‘ก are the rigid body displacements in normal, horizontal and angular directions. ๐‘ƒ, ๐‘„, and ๐‘€ are normal force, lateral force, and moment applied on the disk. The four different parts of compliance matrix ๐ถ๐‘Ÿ๐‘Ÿ , ๐ถ๐œƒ๐‘Ÿ , ๐ถ๐‘Ÿ๐œƒ , and ๐ถ๐œƒ๐œƒ correlate radial displacement to radial traction, tangential displacement to radial traction, radial displacement to tangential traction, and tangential displacement to tangential traction, respectively. Together, these compliance matrices comprise to form the compliance matrix ๐ถPolar in polar coordinate. The Goodman decoupling approximation [31] can be implemented by setting ๐ถ๐œƒ๐‘Ÿ = 0. However, in most of the contact analyses, we are dealing with problems expressed in terms of rectangular coordinates. This is accommodated by transforming each nodal traction and nodal displacement from polar to rectangular coordinates in the usual manner. Eq. 4.22 shows a single node transform at ๐œƒ ๐‘˜ . ๐‘ฆ ๐‘ ๐‘˜ = โˆ’๐‘๐‘Ÿ๐‘˜ cos(๐œƒ ๐‘˜ ) + ๐‘ ๐œƒ๐‘˜ sin(๐œƒ ๐‘˜ ), ๐‘ ๐‘ฅ๐‘˜ = ๐‘๐‘Ÿ๐‘˜ sin(๐œƒ) + ๐‘ ๐œƒ๐‘˜ cos(๐œƒ), (4.22) ๐‘ค๐‘˜ = โˆ’๐‘ค ๐‘Ÿ๐‘˜ cos(๐œƒ ๐‘˜ ) + ๐‘ข ๐œƒ๐‘˜ sin(๐œƒ ๐‘˜ ), ๐‘ข๐‘˜ = ๐‘ค ๐‘Ÿ๐‘˜ sin(๐œƒ) + ๐‘ข ๐œƒ๐‘˜ cos(๐œƒ), Formally, ๐ถRect = ๐‘‡๐‘ค๐‘‡ ๐ถPolar๐‘‡๐‘ (4.23) where ๐‘‡๐‘ค๐‘‡ and ๐‘‡๐‘ are the rotational transformation matrices for the displacement and pressure, ๐ถpolar and ๐ถRect are the compliance matrices in polar and rectilinear coordinates. Because ๐ถRect involve 69 โˆ’๐œ‹ ๐œ‹ Elastic Disk ๐‘ƒ ๐‘€ โˆ’๐œ‹/2 ๐‘„ ๐œ‹/2 ๐‘ฆ ๐‘๐‘˜ ๐‘ ๐‘˜๐œƒ ๐œƒ๐‘˜ ๐‘ ๐‘˜๐‘ฅ 0 ๐‘ ๐‘Ÿ๐‘˜ Pressure distribution ๐‘(๐œƒ), ๐œ(๐œƒ) Figure 4.7: Shifting of the displacement basis to create compliance matrix only nodes in the contact region, it is a fairly small matrix, and the above matrix multiplications would not be onerous; in practice, the transformations are done analytically as the compliance matrix in the rectangular frame is assembled. The final compliance matrix for a solid disk is shown in Eq. 4.24. .. ๏ฃด๏ฃผ ๏ฃฎ๏ฃฏ .. .. .. ๏ฃน๏ฃฑ . ๏ฃผ ๏ฃฑ ๏ฃด . ๏ฃด . . . ๏ฃบ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ฆ 1 0 sin(๐œƒ ) ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐ถ ๐‘ฆ๐‘ฆ ๐ถ ๐‘ฆ๐‘ฅ ๐‘› ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๐‘ ๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ . . . . . . ๏ฃบ ๏ฃด ๏ฃด . . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ . . . ๏ฃบ ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด ๏ฃด 0 1 cos(๐œƒ ๐‘› ) ๏ฃบ ๏ฃด ๐‘ฅ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๏ฃด ๐‘ข๐‘› ๏ฃด ๏ฃด ๏ฃฝ ๏ฃฏ ๏ฃฏ ๐ถ๐‘ฅ๐‘ฆ ๐ถ๐‘ฅ๐‘ฅ ๏ฃบ ๏ฃฒ ๐‘๐‘› ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด .. ๏ฃด = ๏ฃฏ .. .. .. ๏ฃบ .. ๏ฃด (4.24) ๏ฃด ๏ฃด . ๏ฃด ๏ฃฏ . . . ๏ฃบ๏ฃด๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ƒ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ ๐‘…ฮ”๐œƒ . . . 0 0 0 0 ๏ฃบ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘Š ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘„ ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ 0 . . . ๐‘…ฮ”๐œƒ 0 0 0 ๏ฃบ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๐‘ˆ ๏ฃด ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด 2 2 ๏ฃด ๏ฃด sin ฮ”๐œƒ cos ฮ”๐œƒ 0 0 0 ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃณ ๐‘€ ๏ฃด ๏ฃพ ๏ฃฏ๏ฃฐ ๐‘… ๐œƒ ๐‘› . . . ๐‘… ๐œƒ ๐‘› ๏ฃด ๐œƒ ๏ฃบ ๏ฃณ ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃพ ๏ฃป ๐‘ฆ where ๐‘ ๐‘› and ๐‘ ๐‘ฅ๐‘› are nodal normal and horizontal tractions; ๐‘ค ๐‘› and ๐‘ข ๐‘› are nodal normal and 70 horizontal displacements. The four different parts of compliance matrix ๐ถ๐‘ฅ๐‘ฆ , ๐ถ๐‘ฆ๐‘ฅ , ๐ถ๐‘ฅ๐‘ฆ , and ๐ถ๐‘ฅ๐‘ฅ correlate normal displacement to normal pressure, normal displacement to horizontal traction, horizontal displacement to normal pressure, and horizontal displacement to horizontal traction, respectively. Together these four matrices comprise the compliance matrix in rectilinear coordinate ๐ถRect . A similar procedure, but using Eqs. 4.14 and 4.15, can be used to calculate as a compliance matrix for the surface of a hole in an infinite plane. Further, to actually perform analysis of contact between two surfaces, one needs both the discretization of elasticity and a contact algorithm. There are multiple contact algorithms available, such as [1, 4, 18] that could be employed, and the Method of First Violation of [18] was selected because of its robustness. In the next chapter, compliance matrix formulation is employed with the MFV contact algorithm, where it is first verified with available analytical expressions and later used for simulating other contact problems. 4.4 Summary of Compliance Matrix Formulation An extremely fine discretization strategy for two-dimensional circular surfaces such as disks, long cylinders, and holes in an infinite plane is described in detail. The closed-form analytical expressions of displacement basis are provided to create a compliance matrix. The discretization strategy has several visible benefits with respect to contact analysis. 1. Only degrees of freedom on the surface are involved, resulting in a smaller system of equations than is the case with finite element analysis. 2. The process generates a compliance rather than stiffness matrix, permitting the retention of only the degrees associated with the contact region and associated equations. 3. The strategy facilitates extremely fine meshing over the contact patch, as is necessary to obtain clear resolution of interface tractions and displacments. 71 CHAPTER 5 VERIFICATION OF COMPLIANCE FORMULATION AND OTHER CONTACT PROBLEMS Contact problems are notoriously difficult, and most do not have analytical solutions, these that do usually have solutions obtained by virtue of some simplifying assumptions. A particularly important analytical solution is the two-dimensional Mindlin problem of two identical elastic cylinders pressed together by a normal force ๐‘ƒ and then subject to a shear force ๐‘„ [52] which is discussed in the previous chapter . These expressions for contact length(2๐‘Ž) and normal pressure distribution (๐‘(๐‘ฅ)) can be used for the preliminary comparisons and verification, and they are available in [6, 56]. Figure 5.1 shows two frictionless concentric disks/cylinders pressed against each other, and Figure 5.2 shows a cylinder pressed against the inside of a hole in a plane. If both bodies are elastic, it is necessary to create commensurate meshes on each body in the vicinity of contact. For a small contact region, analytic expressions for contact length and pressure distribution are shown in Eq. 5.1. r โˆš 2๐‘ƒ ๐ด 2๐‘ƒ ๐‘Ž 2 โˆ’ ๐‘ฅ 2 ๐‘Ž= , ๐‘(๐‘ฅ) = โˆ’ ๐œ‹๐พ ๐œ‹๐‘Ž 2 (5.1) ๐œ…1 + 1 ๐œ…2 + 1 1 1 where, ๐ด= + , ๐พ= + ๐บ1 ๐บ2 ๐‘…1 ๐‘…2 where, ๐œ… 1 , ๐บ 1 , and ๐‘…1 are the Kolosovโ€™s Constant, shear modulus, and radius of the first cylinder. Similarly, ๐œ…2 , ๐บ 2 , and ๐‘…2 are properties of the second cylinder. ๐‘ƒ is the total force applied through the center of the cylinder. By varying these geometric and material properties, we can formulate multiple contact problems and use these expressions to verify the above compliance formulation. Numerical and analytical results are compared below. Because analytical solutions are usually expressed in terms of normal pressure ๐‘ and shear traction ๐œ, results are reconciled by observing that for the configurations shown, ๐‘ = ๐‘ ๐‘ฆ and ๐œ = ๐‘ ๐‘ฅ . Expressions are normalized and non- dimensionalized using Eq. 5.2. ๐‘หœ and ๐œหœ are the dimensionless normal and lateral tractions. ๐‘คหœ and ๐‘ขหœ are the dimensionless normal and lateral displacements. ๐‘ฅหœ is the dimensionless coordinate along 72 Elastic Cylinder-1 P โˆ†ฮธ1 R1 R2 โˆ†ฮธ2 P Elastic Cylinder-2 Figure 5.1: Two concentric elastic cylinders pressed against each other the contact region. ๐‘ƒหœ and ๐‘„หœ are dimensionless normal and lateral forces. ๐‘หœ = ๐‘ ๐ด, ๐œหœ = ๐œ ๐ด, ๐‘คหœ = ๐‘ค๐พ, ๐‘ขหœ = ๐‘ข๐พ (5.2) ๐œƒ 1 ๐‘…1 ๐พ ๐œƒ 2 ๐‘…2 ๐พ หœ ๐‘ฅหœ = = ๐‘ƒ = 4๐œ‹๐‘ƒ ๐ด๐พ, ๐‘„หœ = 4๐œ‹๐‘„ ๐ด๐พ 4๐œ‹ 4๐œ‹ 5.1 Verification Problems 5.1.1 An Elastic Cylinder Pressed against a Rigid Half-plane The contact problem of an elastic cylinder pressed against a rigid half-plane, can be obtained from the two-dimensional Mindlin solution by setting the radius of the second cylinder ๐‘…2 to infinity and setting its shear modulus to infinity as well (consider Fig. 5.1). For this verification case, the geometric and material properties for the first elastic cylinder are radius ๐‘…1 = 2๐‘š, Poisson 73 โˆ†ฮธ1 R1 Rigid Cylinder-1 R2 โˆ†ฮธ2 P Hole in an infinite elastic plane Figure 5.2: A rigid cylinder pressed inside a hole in an elastic half-plane ratio ๐œˆ1 = 0.3, and shear modulus ๐บ 1 = 77๐บ๐‘ƒ๐‘Ž, causing Eq. 5.1 to yield contact parameters ๐ด = 9.1 ร— 10โˆ’12 ๐‘š 2 /๐‘ƒ๐‘Ž and ๐พ = 0.5๐‘š โˆ’1 . The first cylinder is nominally discretized with 2049 nodes on itโ€™s surface from โˆ’๐œ‹ to ๐œ‹ with ฮ”๐œƒ = 0.18โ—ฆ , but in fact only a very small compliance matrix is constructed โ€“ involving only degrees of freedom of nodes anticipated to be in the vicinity of contact. The cylinder is pressed with a normal force ๐‘ƒ = 1.09 ร— 109 ๐‘/๐‘š against the rigid surface, resulting in a contact arc of 6.67โ—ฆ and a contact length of 2๐‘Ž = 0.233๐‘๐‘š (Eq. 5.1). Figure 5.3 shows that the normalized analytical and numerical pressure fields appear identical. In this calculation, there are 37 nodes in the contact patch, and each load step involved solving an elasticity problem of roughly double that number of equations and degrees of freedom. The same contact problem is again analyzed using an extremely fine discretization in the anticipated contact region. The cylinder is discretized with 513 nodes from โˆ’5โ—ฆ to 5โ—ฆ with ฮ”๐œƒ = 0.0195โ—ฆ . The normal force ๐‘ƒ is again 1.09 ร— 109 ๐‘/๐‘š is resulting in 331 nodes (6.47โ—ฆ contact angle) in contact as shown in the fig 5.3b. 74 ยท10โˆ’2 ยท10โˆ’3 8 8 Numerical Pressure Analytical Pressure Deformed cylinder Undeformed cylinder 6 6 Dimensionless Displacement ๐‘คหœ Dimensionless Pressure ๐‘หœ 4 4 2 2 0 0 Contact Length โˆ’2 โˆ’2 โˆ’0.006 โˆ’0.003 0 0.003 0.006 Dimensionless Contact Region ๐‘ฅหœ (a) Elastic cylinder pressed against the rigid half-plane: coarse mesh ยท10โˆ’2 ยท10โˆ’3 8 8 Numerical Pressure Analytical Pressure Deformed cylinder Undeformed cylinder 6 6 Dimensionless Displacement ๐‘ฅหœ Dimensionless Pressure ๐‘ฅหœ 4 4 2 2 0 0 Contact Length โˆ’2 โˆ’2 โˆ’0.006 โˆ’0.003 0 0.003 0.006 Dimensionless Contact Region ๐‘ฅหœ (b) Elastic cylinder pressed against the rigid half-plane:very fine mesh Figure 5.3: Comparison of analytical and numerical normal pressure distribution for a elastic cylinder pressed against a rigid half-plane with a coarse mesh (on top) and with a fine mesh (on bottom). 75 5.1.2 Rigid Cylinder Pressed inside of a Hole in an Infinite Elastic Plane A rigid cylinder pressed against the inside of a hole in an infinite elastic plane can be constructed by making the first cylinder rigid (๐บ 1 = โˆž) and changing the radius of the second cylinder to a negative value (as shown in Fig. 5.2). The first cylinder has radius ๐‘…1 = 1๐‘š and the hole has radius ๐‘…2 = โˆ’4๐‘š, Poissonโ€™s ratio ๐œˆ = 0.3 and shear modulus ๐บ 2 = 77๐บ๐‘ƒ๐‘Ž. The hole is discretized to have 513 nodes in the estimated contact region from โˆ’1.5โ—ฆ to 1.5โ—ฆ , providing an extremely fine mesh for analysis with ฮ”๐œƒ 2 = 0.0058โ—ฆ . The rigid cylinder is subject to a normal force ๐‘ƒ = 0.104 resulting in the contact length of 2๐‘Ž = 0.50๐‘๐‘š and contact angle of 0.5โ—ฆ on the hole. In the numerical solution, 81 nodes on the surface of the hole are in contact with the indenter. Figure 5.4 shows that analytical and numerical pressure distributions appear to be identical. ยท10โˆ’2 ยท10โˆ’4 2 8 Numerical Pressure Analytical Pressure Indenter before load Indenter after load 1.5 Hole before load Hole after load 6 Dimensionless Displacement ๐‘คหœ Dimensionless Pressure ๐‘หœ 1 4 0.5 2 0 0 Contact Length(2๐‘Ž) โˆ’0.5 โˆ’2 โˆ’0.002 โˆ’0.001 0 0.001 0.002 Contact Region ๐‘ฅหœ Figure 5.4: Rigid cylinder indenter pressed against a hole inside an infinite elastic plane. 76 5.1.3 Normal Contact with Friction for an Elastic Cylinder Pressed against a Rigid Half- plane Consider the case of an elastic cylinder pressed against a frictional rigid half-plane. In this case, the elastic cylinder is discretized with 1025 nodes in the anticipated contact region from โˆ’1.5โ—ฆ to 1.5โ—ฆ . The cylinder has the same material and geometric properties as those employed in previous Section 5.1.1. The numerical solution for stick ratio is compared with the same Spence expression discussed in chapter 3 section 3.2.1. Figure 5.5 shows that the stick ratio from the Spence expression appears identical to the stick ratio calculated from numerical simulation with Goodman decoupling is also imposed. The stick ratio curve with full elastic coupling is only slightly different due to a small Dundur coupling parameter [26, 35]. 1 Spence-Goodman Numerical-Goodman Numerical-Coupled 0.8 0.6 stick-ratio 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 ๐›ฝ /f Figure 5.5: Elastic cylinder pressed against a rigid half-plane with friction. The stick ratios for multiple friction coefficients are compared with the analytical expression provided by Spence. Another contact problem โ€“ that of two identical cylinders pressed against each other with normal and lateral forces (the two-dimensional Mindlin problem) โ€“ is mapped to the problem of an elastic cylinder pressed against a rigid half-plane in the same manner as the problem of a rigid cylinder pressed on an elastic half-plane, which is discussed in the chapter 3 section 5 with corresponding 77 1 Analytical Numerical 0.8 stick-ratio (๐‘) 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ๐‘„/( ๐‘“ ๐‘ƒ) Figure 5.6: Mindlin contact mapping is created using elastic cylinder pressing on a rigid half-plane with friction. Stick ratio for multiple friction coefficients are compared with analytical expression provided by Mindlin. analytical expressions. In this verification problem, the elastic cylinder is pressed against the rigid half-plane with normal force ๐‘ƒ = 2.2 ร— 107 ๐‘/๐‘š and then subjected to lateral force ๐‘„ = 4.4 ร— 106 ๐‘/๐‘š. The stick ratios are calculated using multiple friction coefficients and compared with analytical results: Fig. 5.6. The distributed lateral tractions fields ๐œ( หœ ๐‘ฅ) หœ are calculated numerically for four friction coefficients (1, 0.8,0.6, 0.2) and compared with analytical values as shown in Fig. 5.7. In each of these figures, the numerical results appear to be almost indistinguishable from the analytic values. 5.1.4 von-Mises Stress Distribution in an Elastic Cylinder in Contact In previous sections, very fine meshes made possible by the compliance formulation introduced here were employed to perform contact calculations with very high resolution of displacements and tractions in the contact patch. Observing that knowing the traction distribution over the surface of a cylinder is sufficient to uniquely determine the Michell coefficients (Tab. 4.2), one may use those coefficients to construct the stress distribution throughout the cylinder explicitly (Eqs. C.2.1). This 78 ยท10โˆ’3 4 Fric: 1 Fric: 0.6 Dimensionless Lateral Traction (๐œ) หœ 3 Fric: 0.3 Fric: 0.2 2 1 0 โˆ’0.0008 โˆ’0.0004 0 0.0004 0.0008 Dimensionless Contact Region (๐‘ฅ) หœ Figure 5.7: The Mindlin two-cylinder contact problem is mapped into that of an elastic cylinder pressed against a rigid half-plane with friction. The numerically calculated lateral traction distribu- tion (connected lines) for multiple friction coefficients are compared with analytical values (dashed lines). is true also for the case of a hole in the elastic plane (Tab. 4.3 & Eqs. D.2.1). This construction of the stress field from the results of contact analysis is illustrated on the following problem. An elastic cylinder with the same geometric and material properties as in Section 5.1.1 is pressed against the friction-free rigid plane with normal force ๐‘ƒ = 1.09 ร— 108 , resulting in 33 nodes in contact. Normal and lateral tractions are calculated during the contact analysis that is used with Eqs. C.9 and C.10 to calculate stresses (๐œŽ๐‘Ÿ๐‘Ÿ , ๐œŽ๐‘Ÿ๐œƒ , and ๐œŽ ๐œƒ๐œƒ ) throughout the interior of the body and these are used to evaluate von-Mises stress. The von-Mises stress is normalized by ๐‘ 0 and plotted in Fig. 5.8 where the circumferential and radial coordinates are normalized by ๐‘Ž, the half-width of the contact patch. The maximum von-Misses stress does not occur on the surface but at a point (0, 0.71๐‘Ž) inside the cylinder. This stress contour plot is almost identical to Figure 3.5(a) of [56]. 79 0.5 1.5 0.4 Position (๐‘ฆ/๐‘Ž) 1 0.3 (0, 0.71) 0.2 0.5 0.1 0 โˆ’2 โˆ’1.5 โˆ’1 โˆ’0.5 0 0.5 1 1.5 2 Position (๐‘ฅ/๐‘Ž) Figure 5.8: Contour plot for normalized von-Misses stress inside the elastic disk on pressing against rigid plane on friction free surface with contact angle 0.64โ—ฆ . The positions are non-dimensionalize with the contact length parameter (๐‘Ž). 5.2 Two Identical Elastic Cylinders Pressed and Sheared Two identical concentric cylinders are pressed against each other with equal and opposite normal forces ๐‘ƒ = 8.24 ร— 106 ๐‘. Each cylinder has radius ๐‘… = 3๐‘š, modulus of elasticity ๐ธ = 200๐บ๐‘ƒ๐‘Ž, and Poissonโ€™s ratio ๐œˆ = 0.3. Normal compression of cylinders results in the contact length of 2๐‘Ž = 2.4๐‘๐‘š and contact angle of 0.47โ—ฆ with 79 nodes in contact. After the normal force has been applied, equal and opposite lateral forces ๐‘„ = 2.74 ร— 106 ๐‘ are applied on the two cylinders. In these numerical experiments, two friction models were employed: the simple Coulomb friction model (๐œ‡ ๐‘  = 0.7), and the two-parameter Coulomb friction model (๐œ‡ ๐‘  = 0.7, ๐œ‡ ๐‘˜ = 0.49). Three load cases of contact between identical cylinders are considered: 1. The cylinders are pressed together by a normal force ๐‘ƒ. 2. Normal force ๐‘ƒ and then lateral force ๐‘„ are applied; friction is represented by a one-parameter Coulomb friction model. 3. Normal force ๐‘ƒ and then lateral force ๐‘„ are applied; friction is represented by a two-parameter 80 Coulomb friction model. The surface tractions and displacements of each of the three cases are shown in Fig. 5.9, and the stress fields near the surface are shown in 5.10. 5.2.1 Comparing Surface Tractions and Displacement 2๐‘ƒ The numerical tractions and stresses are normalized using ๐‘ 0 = , and the horizontal and vertical ๐œ‹๐‘Ž coordinates are each normalized by contact length ๐‘Ž. The cases are presented column-wise: the first column is associated with normal loading and, by symmetry, all shear tractions on the interface are zero; the center column shows the effect of a lateral force applied after the normal force and where friction is modeled by the one-parameter Coulomb model; and the right column shows the effect of a lateral force applied after the normal force and where friction is modeled by a two-parameter Coulomb friction model (static and dynamic friction). In the first column of Fig. 5.9, the normal pressure distribution from numerical simulation agrees almost exactly with the analytical expressions. The surfaces of the two cylinders are flat in the contact patch. By symmetry, there is no slipping in the contact patch, independent of the friction model. There is an equal lateral displacement on each interface, towards the interface. In the second column of Fig. 5.9, applied lateral force ๐‘„ results in lateral traction that is indistinguishable from the analytical distribution. The contact region divides into two slip regions near the edges, with 22 nodes slipping, and a stuck region in the middle with 57 nodes sticking. Lateral displacement differs between the two bodies in the regions of slip. In the third column of Fig. 5.9, the same lateral force ๐‘„ is applied, but a two-parameter Coulomb friction model (๐œ‡ ๐‘  = 0.7 and ๐œ‡ ๐‘˜ = 0.49) employed. Due to the complexity of the friction model, there are no available analytic expressions to compare against these numerical solutions. The discontinuous nature of lateral traction clearly shows the boundary between the stick region and the slip regions, with 30 nodes slipping and 49 nodes sticking. The stick region reduces by 0.2๐‘Ž, and relative lateral displacement in the slip region increases compared to the one-parameter 81 ๐‘ƒ ๐‘ƒ ๐‘ƒ ๐‘„ ๐‘„ ๐‘„ ๐‘„ ๐‘ƒ ๐‘ƒ ๐‘ƒ (a) Identical cylinders pressed together by normal force ๐‘ƒ (left); normal force ๐‘ƒ applied first followed by application of shear force ๐‘„ where Coulomb friction ๐œ‡ ๐‘  is employed (center); But employing a two-parameter Coulomb friction ๐œ‡ ๐‘˜ , ๐œ‡ ๐‘  (right) Normal Pressure Lateral Traction Analytical Normal Analytical Lateral Trac. 1.5 1.5 1.5 Tractions ( ๐‘(๐‘ฅ)/๐‘ 0 ) 1 1 1 0.5 0.5 0.5 0 0 0 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (b) Non-dimensionalized normal and lateral tractions (๐‘(๐‘ฅ)/๐‘ 0 and ๐œ(๐‘ฅ)/๐‘ 0 ) Normal Disp. C1 Normal Disp. C2 Lateral Disp. C1 Lateral Disp. C2 ยท10โˆ’2 ยท10โˆ’2 ยท10โˆ’2 0.5 0.5 0.5 Displacements (๐‘ค(๐‘ฅ) โˆ— ๐‘˜) 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 โˆ’1 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (c) Non-dimensionalized normal and lateral displacement fields (๐‘ค(๐‘ฅ)๐พ and ๐‘ข(๐‘ฅ)๐พ) for cylinder 1 (C1) and cylinder 2 (C2) Figure 5.9: Surface traction and displacement distributions for three cases: normal force ๐‘ƒ, only (left); Shear force, ๐‘„ applied after normal force ๐‘ƒ and employing a Coulomb friction model (center); Shear force ๐‘„ applied after normal force ๐‘ƒ and employing a two-parameter friction model (right).[17] 82 Coulomb friction case. The normal pressure and normal displacement remain unchanged in all three cases because of the Goodman decoupling approximation. 5.2.2 Comparing Stress Distributions inside the Cylinders The numerical contact analysis provides the surface tractions at the contact path and those are used, via the Michell expansion, to calculate stresses inside the cylinder. A region from โˆ’2๐‘Ž to 2๐‘Ž on the surface and extending a distance of 2๐‘Ž into the cylinder is chosen to illustrate the stress contours. Stresses due to normal compression of the disks are shown in the first column of Fig. 5.10. Radial stress (๐œŽ๐‘Ÿ๐‘Ÿ ), circumferential stress (๐œŽ๐œƒ๐œƒ ), and von-Mises stress (๐œŽ๐‘ฃ๐‘€ ) are symmetric with respect to y-axis, but shear stress ๐œŽ๐‘Ÿ๐œƒ is anti-symmetric and is zero on the surface. The maximum von-Mises stress occurs on the y axis at a distance of 0.7๐‘Ž from the contact plane, and is consistent with values calculated from the analytic solutions [56, 58]. The application of the lateral force ๐‘„ leaves radial stress ๐œŽ๐‘Ÿ๐‘Ÿ contours almost unchanged. Small changes are seen in the ๐œŽ๐œƒ๐œƒ as its symmetry breaks down and it slightly increases near the contacting interface. Both friction models provide almost the same ๐œŽ๐‘Ÿ๐‘Ÿ and ๐œŽ๐œƒ๐œƒ . The tangential stresses ๐œŽ๐‘Ÿ๐œƒ and von-Mises stress ๐œŽ๐‘ฃ๐‘€ differ dramatically from the case of ๐‘„ = 0 due to the non-zero surface shear stresses ๐œŽ๐‘Ÿ๐œƒ . When the one-parameter Coulomb friction model is employed, the maximum von-Mises stress ๐œŽ๐‘ฃ๐‘€ occurs either on the contact patch within the slip region or slightly above that, depending on the friction coefficient [58]. In the problem shown, the maximum von-Mises stress occurs near the trailing edge of the contact surface at 0.82๐‘Ž. When the two-parameter Coulomb friction model is used, the maximum von-Mises stress ๐œŽ๐‘ฃ๐‘€ occurs exactly at the stick-slip boundary (0.62๐‘Ž). The stresses fields inside the body are not significantly affected by the choice of using the simple Coulomb friction model or using the two-parameter Coulomb friction model. The differences are visible near the contact interface, and those differences might have significance with respect to material fatigue and failure. The maximum von-Mises stress associated with one-parameter 83 1 1.5 1.5 1.5 0.8 0.6 (๐‘ฆ/๐‘Ž) 1 1 1 0.4 0.5 0.5 0.5 0.2 0 0 0 0 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 (a) Normalized radial Stress (๐œŽ๐‘Ÿ๐‘Ÿ /๐‘ 0 ). 1.5 1.5 1.5 0.5 (๐‘ฆ/๐‘Ž) 1 1 1 0 0.5 0.5 0.5 โˆ’0.5 0 0 0 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 (b) Normalized circumferential stress (๐œŽ๐œƒ ๐œƒ /๐‘ 0 ) 0.4 1.5 1.5 1.5 0.2 (๐‘ฆ/๐‘Ž) 1 1 1 0 0.5 0.5 0.5 โˆ’0.2 0 0 0 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 (c) Normalized shear stress (๐œŽ๐‘Ÿ ๐œƒ /๐‘ 0 ) 1 1.5 1.5 1.5 0.8 0.6 (๐‘ฆ/๐‘Ž) 1 1 1 0.4 0.5 0.5 0.5 0.2 0 0 0 0 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 (๐‘ฅ/๐‘Ž) (๐‘ฅ/๐‘Ž) (๐‘ฅ/๐‘Ž) (d) Normalized von-Mises stress (๐œŽ๐‘ฃ ๐‘€ /๐‘ 0 ) Figure 5.10: Stress distributions for three cases: normal force, ๐‘ƒ, only (left); Shear force, ๐‘„ applied after normal force and employing a Coulomb friction model (center); shear force, ๐‘„ applied after normal force and employing a two-parameter friction model (right). Horizontal and vertical coordinates are normalized by ๐‘Ž. [17] 84 Coulomb friction is located in the trailing slip region, but for the case of two-parameter Coulomb friction, that maximum is located at the trailing boundary of the stick-slip region. 5.3 Summary of the 2D Compliance Verification The compliance matrix formulation is verified with available analytical expressions for normal pressure, normal displacements, contact length, lateral tractions, and stick ratio. Numerical calcu- lations agree extremely well with analytic results. The compliance formulation has been shown to increase the discretization resolution possible in elastic, frictional contact mechanics. The contact analysis of two identical cylinders pressed against each other with normal and lateral forces provides a similar distribution of stress fields inside the cylinders for the one-parameter and the two-parameter Coulomb friction models. On the other hand, the tractions and displacements on the contact patch differ between the two friction models, and this difference might have ramifications with respect to friction and failure. 85 PART III Comparison of Decoupling Approximations and their Ramifications 86 CHAPTER 6 ELASTIC DECOUPLING APPROXIMATIONS Frictional contact is a major part of the tribology universe, yet analytical and numerical tools to provide detailed insight have been elusive. What analytic solutions exist are generally reliant on simplifying assumptions โ€“ such as the Goodman decoupling assumption โ€“ and where tribological analysis and design is based on those solutions, the decoupling assumptions have to be taken on faith. It has been difficult to assess the validity of these assumptions numerically because of the very fine spatial resolution required in the contact regime and the resulting huge system of equations to be solved at each contact iteration. This numerical limitation has been mitigated recently by the development of a compliance formulation for the case of contact among two dimensional round surfaces such as disks or holes and the validity of the decoupling assumptions has been investigated in the context of such geometries. 6.1 Decoupling Approximations in Elastic Contact Solving contact problems analytically is extremely complex and only possible for a small number of surface geometries, usually with some level of elastic decoupling approximation. The most common decoupling is the Goodman decoupling (GD) approximation [31] asserting that โ€œnormal displacements resulting from lateral traction are negligible compared to the normal displacements resulting from normal traction." This decoupling approximation has been exploited regularly by researchers such as Mindlin [52] and Spence [63, 64] to provide analytical expressions for the width of the contact patch, stick-slip regions in the contact patch, lateral traction, displacement distributions, and relations between the lateral force and stick-slip ratio in terms of the applied loads or displacements. Another, more severe, decoupling approximation ignores any elastic coupling between normal and lateral loads so that โ€œnormal displacement and lateral displacement result only from normal force and lateral force respectively." [30, 69]. This full decoupling (FD) approximation reduces the complexity of the problem significantly and results in faster though generally not as 87 accurate predictions. It is most often implemented for understanding cyclic loading and uneven contact where the contact calculations of a coupled system over a very large number of steps would be prohibitive. Researchers [7, 8, 29, 30] have explored the impact of elastic coupling to understand the process of frictional shakedown behavior (frictional shakedown happens as initial residual tractions and displacements reduces the amount of partial slip during the steady state). The elastic coupling is required to have a shakedown in the contact problem [29, 69]. Because of this, we narrowed our focus to understand the impact of decoupling approximations on contact parameters. Flicek et al. [30] explored the impact of elastic coupling on the contact problem with cyclic shear loading. They measured level of coupling by a norm of matrices and analyzed the effect of initial transient conditions on steady state contact parameters over the range of geometries, material combinations, and friction coefficients. They showed that the coupling effects are higher for the material mismatch than geometric mismatch for a contact problem. The Goodman approximation is helpful in deriving analytic contact solutions, but itโ€™s assistance in numerical contact calculations is minimal. On the other hand, complete decoupling facilitates both development of approximate analytic solutions and substantially speeds up numerical calcu- lations - presumably at the cost of accuracy. In this study, effects of Goodman decoupling and full decoupling are assessed by comparison to similar calculations done with full coupling; examining contact parameters such as surface traction distributions, stick ratios, and steady-state dissipation. This numerical investigation is now tractable because of a recently derived compliance formula- tion for elastic two-dimensional circular bodies, dropping degrees of freedom outside the contact regions from the system of equations to be solved and making extremely fine meshes at the contact tractable. For this study, five different contact configurations are formulated using circular elastic cylinders and holes. Multiple simulations are conducted by varying the lateral load and friction coefficient for each contact problem. 88 6.2 Analytical Expressions Most of the analytical expressions have been discussed in the verification process in previous chapters. Some of these expressions represents the level of elastic coupling through material properties and geometries. For a quick refresher, we have reintroduced them here. ๐œ… 1 โˆ’1 ๐œ… 2 โˆ’1 ๐œ…1 + 1 ๐œ…2 + 1 ๐บ1 โˆ’ ๐บ2 1 1 ๐ด= + , ๐›ฝ= ๐œ… 1 +1 ๐œ… 2 +1 and ๐พ= + (6.1) ๐บ1 ๐บ2 ๐บ1 + ๐บ2 ๐‘…1 ๐‘…2 A is the measure of the mutual compliance of the two bodies, ๐›ฝ is a material mismatch parameter also known as Dundurโ€™s parameter, and K is the mean curvature of the contacting bodies. G represents the shear modulus, R denotes the radius of curvature of the body, and ๐œ… is the Kolosovโ€™s constant (Eq. 3.20 of [6]); with subscripts 1 and 2 refer to each of the two cylinders, ๏ฃฑ ๏ฃฒ 3 โˆ’ 4๐œˆ for plane strain, ๏ฃด ๏ฃด ๏ฃด ๐œ… = 3โˆ’๐œˆ (6.2) ๏ฃด ๏ฃด for plane stress ๏ฃณ 1+๐œˆ ๏ฃด where ๐œˆ is the Poissonโ€™s ratio. Parameter ๐›ฝ captures the elastic mismatch between the two bodies and is strongly associated with elastic coupling seen in compliance or stiffness matrices. If the material properties (G and ๐œˆ) are the same for both elastic bodies (๐›ฝ = 0) that results in equal amount of transverse yielding in lateral displacement. This corresponds to entire contact region with zero traction for a normal force which is same as employing a full decoupling approxi- mation [17, 69]. If one of the materials is rigid and other one is isotropic elastic, then this choice of materials brings the maximum level of elastic coupling for the analysis [35] because the material mismatch parameter ๐›ฝ becomes maximum and is independent of shear modulus. ๐œ…1 โˆ’ 1 ๐œ…1 + 1 ๐›ฝ= and ๐ด= (6.3) ๐œ…2 + 1 ๐บ1 Energy dissipation due to frictional work is one of the contact parameters of interest during cyclic shear loading. It connects vibration-damping analysis with the contact mechanics. During a cyclic shear loading, two contacting surfaces move relative to each other in the slip region. The 89 friction force in the slip region generates non-conservative and dissipative work which can be calculated numerically by first summing up the frictional work at each interval for a load step and later summing up load steps for a cyclic interval. The analytical expressions for the dissipation due to cyclic shear loading are limited. For two-identical cylinders pressed together and cyclically sheared, Fleury et al. [27, 28] extended the Mindlin solution to provide a closed-form integral expression for the dissipation per unit length: s ! โˆซ ๐‘Ž r  ๐‘ฅ  2 ๏ฃฎ๏ฃฏ  ๐‘ฅ   ๐‘ฅ  2  ๐‘   ๐‘  2 ๐‘ฅ r ๐‘ฅ  ๏ฃน๏ฃบ ๐ท= 2๐‘Ž ๐ด๐œ‡2 ๐‘ 20 1โˆ’ ๏ฃฏ โˆ’ โˆ’ log + ) 2 โˆ’ 1 ๏ฃบ๏ฃบ ๐‘‘๐‘ฅ (6.4) โˆ’๐‘Ž ๐‘Ž ๏ฃฏ๏ฃฏ ๐‘Ž ๐‘Ž ๐‘Ž ๐‘Ž ๐‘ ๐‘ ๏ฃบ ๏ฃฐ ๏ฃป The dimensionless dissipation per cycle is defined " r  !# ๐ท 8๐œ‡2 ๐‘Ž โˆซ q 2 2 p ๐‘  ๐‘  D= 2 1 โˆ’ (๐‘ ) ๐‘  ๐‘ 2 โˆ’ ๐‘ โˆ’ ๐‘2 log + โˆ’ 1 ๐‘‘๐‘  ๐‘ƒ ๐ด ๐œ‹ 2 โˆ’๐‘Ž ๐‘ ๐‘ (6.5) 8๐œ‡2 = 2 G(๐‘) ๐œ‹ ๐‘ฅ ๐‘ where, ๐‘  = ๐‘Ž and ๐‘ = ๐‘Ž is a stick ratio during cyclic shear loading at maximum shear force. Note that because the above expression is for identical cylinders pressed together, therefore ๐›ฝ = 0, and it does not provide insight into coupling, but it will be shown to be a useful nondimen- sionalization. 6.2.1 Contact Problems Simulations are performed for fully coupled, partly coupled (Goodman), and fully decoupled analyses for multiple load combinations and following five different geometries: 1. A rigid indenter of radius 0.97๐‘… is pressed inside a elastic circular hole of radius ๐‘… (๐พ ๐‘… = 0.03). 2. A rigid indenter of radius 0.5๐‘… is pressed inside a elastic circular hole of radius ๐‘… (๐พ ๐‘… = 1). 3. A elastic cylinder of radius ๐‘… is pressed over a rigid half-plane (๐พ ๐‘… = 1). 90 4. A elastic cylinder of radius ๐‘… is pressed over a rigid cylinder of radius 0.5๐‘… (๐พ ๐‘… = 3/2). 5. A elastic cylinder of radius ๐‘… is pressed over a rigid cylinder of radius ๐‘… (๐พ ๐‘… = 2). Note that the radius ๐‘… is the larger of the radii of each configuration and the dimensionless quantity ๐พ ๐‘… could range from zero to infinity, but in these analyses the range of ๐พ ๐‘… is from 0.03 to 2.0 (In the numerical calculations performed for this study ๐‘… had a value of 4.0 meters, but the particular value of ๐‘… becomes irrelevant after normalization and non-dimensionalization). In all calculations, shear modulus of the larger cylinder (or hole) ๐บ = 77 GPa, Poissonโ€™s ratio ๐œˆ = 0.3 and coefficient of friction ๐œ‡ = 0.7 is assumed. In each case, the larger cylinder (or hole) is elastic and the smaller one is rigid. 6.3 Surface Tractions for a Cyclic Lateral Force In this case, a rigid cylinder is pressed against the inside of an elastic hole of twice its radius. The indenter is pressed with sufficient force ๐‘ƒ = 8.0 ร— 107 ๐‘ to cause a contact area of 2๐‘Ž = 0.022๐‘…; this corresponds to 1.26โ—ฆ of the surface of the hole. After the normal load is imposed, a cyclic lateral force ๐‘„ = 0.65๐œ‡๐‘ƒ is applied to the rigid cylinder. The numerical experiments are conducted with three levels of elastic coupling: full elastic coupling (FC), Goodman decoupling (GD), full elastic decoupling (FD). The resulting surface tractions are shown in figure 6.1. There are three columns: the first column with FC, the second with GD, and the third is associated with the FD. In the first row of Fig. 6.1, the indenter is pressed against the hole with normal force ๐‘ƒ. The resulting normal traction is symmetric about the y-axis, but the contact patch and the pressure distribution are a little narrower for the fully coupled case than for the GD and FD cases because of the retarding effects of surface shear traction. The lateral traction is self-equilibrating and anti- symmetric about the y-axis for FC and GD. For the full decoupling case, there is no lateral traction caused by the application of the normal load. The second row of Figure 6.1 corresponds to a lateral force ๐‘„ applied while the indenter remains pressed against the surface of the hole. For the GD and FD analyses, the normal pressure distribution remains unchanged from that calculated in the previous step (recall that for GD and FD 91 Normal Traction Lateral Traction 1 1 1 Tractions ๐‘(๐‘ฅ) หœ or ๐œ(๐‘ฅ) หœ 0.5 0.5 0.5 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (a) Indenter is pressed with normal load P 1 1 1 Tractions ๐‘(๐‘ฅ) หœ or ๐œ(๐‘ฅ) หœ 0.5 0.5 0.5 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (b) Indenter is pressed with normal load P and lateral load Q 1 1 1 Tractions ๐‘(๐‘ฅ) หœ or ๐œ(๐‘ฅ) หœ 0.5 0.5 0.5 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (c) Indenter is pressed with normal load P and lateral load -Q 1 1 1 Tractions ๐‘(๐‘ฅ) หœ or ๐œ(๐‘ฅ) หœ 0.5 0.5 0.5 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (d) Indenter is pressed with normal load P and lateral load Q Figure 6.1: Surface traction distributions for three cases: fully coupled (FC) (in left); Goodman decoupled approximation (GD) (in center); fully decoupled approximation (FD) (in right). 92 Figure 6.1 (contโ€™d): Normal Traction Lateral Traction 1 1 1 Tractions ๐‘(๐‘ฅ) หœ or ๐œ(๐‘ฅ) หœ 0.5 0.5 0.5 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 0 1 โˆ’1 0 1 โˆ’1 0 1 Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) Contact Region (๐‘ฅ/๐‘Ž) (e) Indenter is pressed with normal load P and lateral load -Q analyses the shear tractions do not influence normal displacement). However, for the FC analysis, normal traction does change slightly from the previous case as symmetry is lost due to coupling with the shear traction. Also, the contact region is shifted to the left by 0.1๐‘Ž. The lateral traction distributions for the FC and GD cases are similar and both distributions loose symmetry on the application of the lateral force. In the case of FD, lateral traction is symmetric. In the third row of Fig. 6.1, the applied lateral force is slowly reversed to โˆ’๐‘„. The normal traction for FC is changed again; the shape is altered to have one more inflection and the contact region shifts toward the right by 0.2๐‘Ž. One can identify the stick and slip region boundaries through the inflections in normal traction distribution. The lateral traction reflects the shift in the contact patch as well. For the GD case, the normal traction distribution is unchanged and the shear traction is similar to, though distinguishable, from that of the FC case. For the FD analysis, the lateral traction distribution is reflected vertically. The fourth row of the Fig. 6.1 reflects the applied lateral force slowly reverting to ๐‘„ to complete a cycle. For the FC case, the normal traction continues to evolve and acquires another inflection and the lateral traction also changes, acquiring one more local maximum. For the GD analysis, the lateral traction distribution continues to show qualitative similarity to that of the FC case. For the FD case, the lateral traction is the same as that of step 2 where ๐‘„ had the same value. The fifth row of the Fig. 6.1 has to do with reversing the shear force again to โˆ’๐‘„. Of course, the tractions in FD case are identical to those of step three when the shear and normal forces were 93 1 1 1 Dissipation Hysteresis Dissipation Hysteresis Dissipation Hysteresis 0.5 0.5 0.5 Force ๐‘„หœ 0 0 0 โˆ’0.5 โˆ’0.5 โˆ’0.5 โˆ’1 โˆ’1 โˆ’1 โˆ’1 โˆ’0.5 0 0.5 1 โˆ’1 โˆ’0.5 0 0.5 1 โˆ’1 โˆ’0.5 0 0.5 1 Displacement ๐‘ˆหœ Displacement ๐‘ˆหœ Displacement ๐‘ˆหœ (a) Normalized dissipation hysteresis where ๐‘„หœ and ๐‘ˆหœ are normalized with force ๐‘„ and maximum ๐‘ˆ๐‘ก๐‘œ๐‘ก from FD respectively NL Hysteresis NL Hysteresis NL Hysteresis 1 1 1 Non-linear Force ๐‘„ ๐‘›๐‘™ 0 0 0 โˆ’1 โˆ’1 โˆ’1 โˆ’1 โˆ’0.5 0 0.5 1 โˆ’1 โˆ’0.5 0 0.5 1 โˆ’1 โˆ’0.5 0 0.5 1 Displacement ๐‘ˆหœ Displacement ๐‘ˆหœ Displacement ๐‘ˆหœ (b) Non-linear part of Hysteresis Figure 6.2: Normalized dissipation hysteresis and non-linear part of hysteresis: fully coupled (in left); Goodman decoupled (in center); fully decoupled (in right). Non-linear part is generated by removing the linear the same. Particularly interesting are the tractions of the FC and GD analyses, where the normal and shear tractions appear to have reverted to what they were in Step 3. From this point on, the traction distributions at the end of each shear loading are exactly what they were in the previous step having that shear load. The same surface traction behavior is observed for the other four contact geometries. From the above, one can observe that though the normal traction distribution associated with the GD analysis can be remarkably different from that of the FC analysis, but the shear tractions are quantitatively very similar. The FD analyses show extremely different shear tractions profiles than either the FC or GD cases. Frictional dissipation is associated with wear and is also important as a source of vibration damping, but in addition it provides some insight into frictional contact. The contact formulation 94 employed here provides net force on each body and net relative displacement at each point in the cycle. These histories are used to construct the hysteresis curves shown in Fig. 6.2a. Once the curves become closed, a steady-state has been achieved and the area within the curve is the dissipation per cycle. Because these frictional contact are almost elastic, these curves result in almost linear and difficult to differentiate. This limitation is addressed by subtracting its linear part from each curve (figure 6.2b). For the FC and the GD, the first three-quarters of the first cycle reflects the effect of the initial shear traction distribution. Once a steady-state is achieved, the dissipation per cycle is the area enclosed by the loop. For the FD case, the steady-state curve results after a quarter cycle. If scaled in the same way (removing the same linear part), steady-state curves for GD and FD are nearly identical and result in the same dissipation. 6.4 Variation with Lateral Force The five contact geometries discussed in the section 6.2.1 are simulated over a range of shear load amplitudes. For each case, four load cycles are simulated resulting in at least three steady-state cycles. Dissipation and stick ratio for all three coupling choices are calculated and compared. Also, the ๐ฟ 1 norm in each case is calculated for the differences between the FC tractions and those of the GD and FD approximations. In each case the larger-diameter body is treated as elastic with the parameters employed in Section 6.3 and the other body is treated as rigid. The applied normal force ๐‘ƒ is adjusted for geometry such that the contact patch will have a width of 0.088m with a contact angle of 1.26โ—ฆ relative to the larger diameter body. The elastic parameters ๐ด and ๐›ฝ are the same for all geometries though the curvature ๐พ (defined Eq. 6.1) will differ for each geometry. For each contact geometry, steady-state dissipation is calculated and plotted against the normal- ized lateral force ratio (๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ) for FC as shown in Fig 6.3a. As expected, dissipation increases with increase in lateral load. Each contact geometry (๐พ ๐‘…) is associated with a different dissipation curve. On the other hand, when properly normalized and nondimensionalized, all the dissipation curves converge to a single curve (Fig. 6.3b ) โ€“ this is anticipated from the nondimensionalization of the relevant integral equations. The log-log plot of dimensionless dissipation vs dimensionless 95 10โˆ’5 ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 Dissipation ๐ท ๐พ๐‘… = 2 10โˆ’10 10โˆ’15 10โˆ’1 100 Normalized Lateral Force ( ๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ) (a) Dissipation values with full elastic coupling and ๐œ‡ = 0.7 Normalized Dissipation ๐ท๐บ/๐‘ƒ2 100 ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 ๐พ๐‘… = 2 10โˆ’3 10โˆ’6 10โˆ’1 100 Normalized Lateral Force ( ๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ) (b) Non-dimensional Dissipation with full elastic coupling and ๐œ‡ = 0.7 ยท10โˆ’12 6 FC Dissipation ๐ท 4 GD FD 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Lateral Force ( ๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ) (c) Dissipation values for ๐พ ๐‘… = 1 with ๐œ‡ = 0.7 friction coefficient Dissipation ratio ๐ท ๐บ๐ท /๐ท ๐น๐ถ 1.8 ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 1.6 ๐พ๐‘… = 2 1.4 1.2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 หœ Normalized Lateral Force ( ๐‘„ = ๐‘„/๐œ‡๐‘ƒ) (d) Ratio of Goodman dissipation to fully coupled dissipation with ๐œ‡ = 0.7 Figure 6.3: For the five contact geometries, contact parameters such as Dissipation, Stick ratio, หœ All calculation and errors in surface tractions are plotted against over a range of lateral force (๐‘„). involved ๐œ‡ = 0.7 friction coefficient. 96 Figure 6.3 (contโ€™d): 1 Stick Ratio (Forward Load) FC 0.8 GD FD 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Lateral Force ( ๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ) (e) Stick ratio after a forward lateral load with ๐œ‡ = 0.7 1 Stick Ratio (Backward Load) FC 0.8 GD FD 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 หœ Normalized Lateral Force ( ๐‘„ = ๐‘„/๐œ‡๐‘ƒ) (f) Stick ratio after a backward lateral load with ๐œ‡ = 0.7 0.4 ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 0.3 ๐พ๐‘… = 2 ๐‘ ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Lateral Force ( ๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ) (g) Errors in Normal traction ๐‘ ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ with ๐œ‡ = 0.7 GD FD ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 ๐พ๐‘… = 2 ๐œ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 หœ Normalized Lateral Force (๐‘„ = ๐‘„/๐œ‡๐‘ƒ) (h) Errors in Lateral traction ๐œ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ with ๐œ‡ = 0.7 97 shear force is nearly linear over the range of loads applied. At low values of load, the slope of the curve for this FC problem is roughly 3.15. For each decoupling approximation, dissipation is calculated with varying lateral force as shown in figure 6.3c. When plotted on a log-log graph (not shown here), the slope is close to 3.0 โ€“ as suggested in [32]. Dissipation values are nearly equal for GD and FD, which are higher than those of the FC analysis at each level of lateral force. The dissipation values for GD are divided by dissipation values of FC, and the resulting values are plotted in Fig. 6.3d. This plot indicates that the decoupling approximations over-predict dissipation by 20% to 50% as compared to the full coupling solution. These ratios are independent to the mean curvature (K). Figure 6.3e and 6.3f show the stick ratios for forward shear load and the reverse direction load respectively. The stick-ratios for all three couplings are independent of the mean curvature (K). In the case of backward lateral loading (โˆ’๐‘„), stick ratios are nearly the same for all three coupling cases. It is in application of the initial shearing load that the cases differ. Stick ratios are lowest for FC contact, highest for FD contact, and intermediate for the GD case. For the case of FD contact, the stick ratios are the same for both loading and unloading. Errors in the surface tractions due to the GD and FD approximations are calculated with aid of the ๐ฟ 1 norm using the following relation: โˆซ | ๐‘(๐‘ฅ) โˆ’ ๐‘(๐‘ฅ) FC |๐‘‘๐‘ฅ FC ร๐‘ ๐‘–=1 | ๐‘ ๐‘– โˆ’ ๐‘ ๐‘– |ฮ”๐‘ฅ ๐‘ error = โˆซ = ร๐‘ FC , | ๐‘(๐‘ฅ) FC |๐‘‘๐‘ฅ ๐‘–=1 | ๐‘ ๐‘– |ฮ”๐‘ฅ โˆซ (6.6) |(๐œ(๐‘ฅ) โˆ’ ๐œ(๐‘ฅ) FC )|๐‘‘๐‘ฅ FC ร๐‘ ๐‘–=1 |๐œ๐‘– โˆ’ ๐œ๐‘– |ฮ”๐‘ฅ ๐œerror = โˆซ = ร๐‘ FC , |๐œ(๐‘ฅ) FC |๐‘‘๐‘ฅ ๐‘–=1 |๐œ๐‘– |ฮ”๐‘ฅ where ๐‘๐‘– , and ๐œ๐‘– are nodal normal and lateral tractions respectively, while superscript ๐น๐ถ represents the tractions for full elastic coupling; ๐‘ is the total number of nodes in the contact region and ฮ”๐‘ฅ is the length of the interval. By assumption, the normal traction distributions for GD and FD do not change with change in the lateral load though those of the FC cases do, so it is sufficient to show the error in normal traction calculated for GD as shown in Fig. 6.3f. The error increases from 0% to 23% with increase in the lateral force. Error in lateral traction is calculated for both GD and FD and shown in Fig. 6.3g. For 98 the GD case, error in lateral traction first increases to a maximum of 35% and later reduces to 20% with increase in lateral force. For the FD case, error in lateral traction reduces with an increase in lateral force from 100% to 20%. For small lateral load, the lateral traction distribution is similar for GD and FC. This is understood by observing that at zero lateral load the shear traction of the FD case is zero โ€“ representing 100% error with respect to the FC case. At very high lateral load, nearly all of the the contacting region starts to slip, resulting in 20% error for GD and FD because the contact region shifts by 0.1๐‘Ž for FC. 6.5 Variation with Coefficients of Friction Again, the same five contact geometries are simulated over the range of friction coefficients (from 0.05 to 1) , but with a fixed lateral force ratio (๐‘„หœ = ๐‘„/๐œ‡๐‘ƒ = 0.99). The applied normal forces (๐‘ƒ) and the elastic properties (๐ด and ๐›ฝ) are the same as in the previous case such that the contact patch has a width of 0.088m for all five geometries. Steady-state dissipation and errors in the surface tractions are compared for the five contact geometries as shown in Fig. 6.4. Numerical simulations (not shown here) indicate that in either forward or reverse loading, so long as the force ratio ๐‘„หœ is constant, there is little dependence on coefficient of friction ๐œ‡. The steady-state dissipation is calculated and plotted against the friction coefficients. The dissipation increases with the increase in friction coefficient and also with the increase in ๐พ ๐‘…, as shown in Fig. 6.4a. Again on non-dimesionalizing the dissipation, all dissipation curves converge to a single curve, as shown in Fig. 6.4b and become independent of mean curvature values, also expected by the dissipation Eq. 6.5. The log-log plot of dissipation against friction coefficients is linear in nature. The slope of the curve is 2 for all three cases (FC, GD, and FD), which represents a quadratic relation also anticipated from the dimensionaless dissipation equation 6.5. Errors in surface traction are calculated over the range of coefficients of friction using Eq. 6.6 that result independent of the mean curvature value ๐พ. Figure 6.4 shows that both errors in normal and lateral traction increase with the increase in the coefficient of friction and are independent of the decoupling approximation due to tiny stick region at high lateral force ratio (๐‘„หœ = 0.99). The 99 10โˆ’4 ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 Dissipation ๐ท ๐พ๐‘… = 2 10โˆ’9 10โˆ’14 10โˆ’1 100 Friction Coefficient (๐œ‡) (a) Dissipation with full elastic coupling and ๐‘„หœ = 0.99 100 Normalized Dissipation D ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 10โˆ’1 ๐พ๐‘… = 2 10โˆ’2 10โˆ’3 10โˆ’1 100 Friction Coefficient (๐œ‡) (b) Non-dimensional dissipation values with full elastic coupling and ๐‘„หœ = 0.99 0.4 ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 0.3 ๐พ๐‘… = 2 ๐‘ ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Friction Coefficient (๐œ‡) (c) Errors in Normal traction ๐‘ ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ with ๐‘„หœ = 0.99 0.4 GD FD ๐พ ๐‘… = 0.03 ๐พ๐‘… = 1 0.3 ๐พ๐‘… = 1 ๐พ ๐‘… = 1.5 ๐œ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ ๐พ๐‘… = 2 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Friction Coefficient (๐œ‡) (d) Error in Lateral traction ๐œ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ with ๐‘„หœ = 0.99 Figure 6.4: For the five contact geometries, contact parameters such as Dissipation, Stick ratio, and errors in surface tractions are plotted against over a range of friction coefficients (๐œ‡). All calculation involved ๐‘„หœ = ๐‘„/(๐œ‡๐‘ƒ) = 0.99 lateral force. 100 Dissipation ratio ๐ท ๐บ๐ท /๐ท ๐น๐ถ 1.4 ๐‘„หœ = 0.99 ๐‘„หœ = 0.93 ๐‘„หœ = 0.65 ๐‘„หœ = 0.46 ๐‘„หœ = 0.32 1.2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Friction Coefficient (๐œ‡) (a) Dissipation with full elastic coupling for ๐พ ๐‘… = 1 0.4 ๐‘„หœ = 0.99 ๐‘„หœ = 0.93 0.3 ๐‘„หœ = 0.65 ๐‘„หœ = 0.46 ๐‘„หœ = 0.32 ๐‘ ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Friction Coefficient (๐œ‡) (b) Errors in Normal traction ๐‘ ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ for ๐พ ๐‘… = 1 GD FD ๐‘„หœ = 0.99 ๐‘„หœ = 0.93 ๐‘„หœ = 0.65 ๐‘„หœ = 0.46 ๐‘„หœ = 0.32 1 ๐œ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Friction Coefficient (๐œ‡) (c) Errors in Lateral traction ๐œ๐‘’๐‘Ÿ๐‘Ÿ ๐‘œ๐‘Ÿ for ๐พ ๐‘… = 1 Figure 6.5: For a rigid cylinder pressed inside a circular hole of twice radius, contact parameters such as Dissipation, Stick ratio, and errors in surface tractions are plotted against over a range of หœ friction coefficients (๐œ‡) and five different lateral ratio ๐‘„. 101 non-dimensionalized contact parameters are independent of the mean curvature even with variation in friction coefficients. Next, a contact problem โ€“ a rigid cylinder pressed inside the elastic hole twice its radius โ€“ is simulated with five different lateral load ratios (๐‘„) หœ over a range of friction coefficients. The dissipation value for GD is divided by dissipation value for FC to calculate a dissipation ratio. The ratio is plotted against the friction coefficients as shown in Fig. 6.5. The dissipation ratio increases with increasing friction coefficients for high lateral load (๐‘„หœ = 0.99, 0.93). But this ratio results in a concave up curve for low lateral load, as shown in Fig. 6.5. The dissipation ratio varies from 1.14 to 1.30 for the five lateral forces, concluding at least a 14% overestimation of dissipation calculated with decoupled approximation. Errors in normal tractions increase with increasing friction coefficients and lateral load, as shown in Fig. 6.5b. Errors in lateral traction depend significantly on the decoupling approximations. For the GD case, error increases with increase in the coefficient of friction. The maximum value of error observed is 35%. But for the FD case, error in lateral tractions reduces for lower lateral ratios and increases for higher lateral ratios, as shown in Fig. 6.5c. 6.6 Discussion Both qualitative and quantitative ramifications of the decoupling assumptions can be assessed on the basis of the simulations reported here. Shear tractions and stick ratios calculated using the Goodman decoupling approximation are consistently in good qualitative agreement with the fully coupled (correct) traction distributions. Dissipations and normal tractions remain the same for the Goodman decoupling and full decoupling approximations. As expected, the shear traction distributions associated with full decoupling bear little resemblance to the traction distributions predicted by the full coupling model or the Goodman decoupling model. The fully coupled analysis shows the contact patch shifting from side to side as lateral forces are applied. By construction, neither the Goodmanโ€™s model or fully-decoupled analyses show any motion of the contact patch. As expected from analyses of the underlying integral equations, the appropriately nondimension- 102 alized energy dissipation plots are independent of the geometrical parameter ๐พ ๐‘…. The decoupling approximations almost always lead to over-predicting the dissipation. For the cases of Goodman decoupling and full decoupling, the log-log plots of dissipation vs shear load show slopes close to 3.0, consistent with Goodmanโ€™s observation [32]. On the other hand, the fully coupled model yields slightly higher slopes, close to 3.15. For all three coupling treatments, the dissipation in- creases proportionally to the square of friction coefficient. Other nondimensional parameters such as stick-ratios, errors in the surface tractions also happen to be independent of ๐พ ๐‘…. The normalized normal traction is the same for GD and FD, but visibly distinguishable from FC. The errors in surface traction are sensitive to lateral force ratio and friction coefficients. Because increase in lateral force ratio or the friction coefficients causes high lateral traction in the contact region, the impact of the normal tractions for the fully coupled system and results in the higher normal traction error. The errors in lateral traction are significantly different for Goodman decoupling and full de- coupling. For the Goodman decoupling, errors in lateral traction increase with increase in lateral traction due to its qualitative similarity with fully coupled case. For full decoupling, small lateral force leads to small lateral traction which is different than lateral traction calculated for the fully coupled case. In some cases, this difference results in almost 100% error. 6.7 Summary of Effects of Elastic Decoupling Approximations The research presented here assess the seriousness of compromises in accuracy that results from exploitation of the simplifying approximations of Goodman and of full decoupling. A few notable points are: โ€ข Goodman decoupling approximation showed surprisingly good results in capturing the shape of lateral tractions and stick ratio. However, Goodman decoupling shows some serious errors in measuring dissipation and surface tractions. The Goodman decoupling does not predict the shift of contact region which is important for tribological analysis. โ€ข Full decoupling approximation did not predict a proper lateral traction. However, it is no 103 worse than Goodman decoupling in estimating the dissipation and normal traction. It shows serious errors both quantitatively and qualitatively compared to the fully coupled case. 104 CHAPTER 7 CONCLUSION AND FUTURE RESEARCH In this chapter, a brief summary of the dissertation is presented, highlighting the main achievements. Brief concluding remarks are provided with directions to continue this research work in the future. 7.1 Summary The aim of this research was to develop a numerical test-bed to solve and analyze contact problems. A robust contact algorithm and an extremely fine discretization method are required to create numerical test-bed. Contact Algorithm: To address this, a contact problem โ€“ a rigid indenter pressed against an elastic half-plane is analyzed where an elastic half-plane is discretized to create a compliance matrix. Applying the conventional algorithm showed issues with convergence, so a new contact algorithm โ€“ the Method of First Violation (MFV) is developed that showed robustness and fidelity with contact path. The conventional algorithm and MFV are compared and verified using analytical expressions for surface tractions, stick-ratio, and displacements. The MFVโ€™s robustness was shown by employing a two-parameter Coulomb friction model, Dahl friction, and one-parameter Coulomb friction with a high coefficient of friction. For the same cases, a conventional algorithm would not converge or provide correct solutions. The path-dependent integrity of the MFV is shown through surface traction evolution during cyclic shear loading when the two-parameter friction model is employed. The computational efficiency of MFV is of the similar order to the conventional algorithm if optimal nominal load step is used. Fine Mesh Strategy: A compliance matrix formulation is derived for two-dimensional circular elastic cylinders, disks, and holes in an infinite plane. Michellโ€™s Airy stress function is used with the Fourier series to calculate stresses and displacements inside and on the surface of an elastic body. First, radial and tangential displacements over the contacting surface are calculated by applying a unit radial and tangential force on a discretized interval for a circular disk or hole. This calculated 105 displacement basis is used as a column of compliance matrix, and other columns are assembled by the circular shifting of the displacement basis. The resulting polar compliance matrix is converted to Cartesian form through rotational transformations. Contact problems of an elastic cylinder pressed against a rigid half-plane and a rigid cylinder pressed inside a circular hole are analyzed using compliance matrices. These compliance matrices with the MFV contact algorithm are verified with available analytical expressions of surface traction, stick ratio, stresses, and surface displacements. Another contact problem โ€“ Two identical elastic cylinders pressed against each other and sheared โ€“ is analyzed for two friction models (one-parameter Coulomb and two-parameter Coulomb) while employing Goodman decoupling approximation. Contact parameters such as surface traction, surface displacements, and stresses in and over the contacting surfaces are compared. The change in friction model caused a slight change in stress fields inside the contacting surfaces, and most of the differences are noticed in surface tractions, slip, and the location of maximum von-Mises stress. Comparing Decoupling Approximations: Multiple contact problems are numerically simu- lated using MFV and the compliance matrix strategy. Five different contact geometries are designed using elastic cylinders, holes, and half-plane, where Goodman decoupling approximation and full decoupling approximations are employed along with full elastic coupling in contact problems. Numerical simulations are conducted over a range of lateral force ratios and friction coefficients. Contact parameters calculated from decoupling approximations are compared with corresponding full elastic coupling. Both qualitative and quantitative ramifications of the decoupling approxima- tions are assessed, showing some serious differences between contact parameters. 7.2 Concluding Remarks The frictional contact problems are notoriously complex and require a robust contact algorithm with an extremely fine discretization in the contact region for its analysis. These concerns are addressed by developing a robust contact algorithm โ€“ Method of first Violation โ€“ and deriving a compliance formulation for the two-dimensional disks, cylinders, and holes. These two developments provide 106 a robust test-bed to analyze the contact problems. This test-bed is used to compare the Goodman and full decoupling approximations with full coupling and to assess the level of accuracy and compromises due to these decoupling approximations. Now, this test-bed offers opportunities to explore other complex contact problems that were previously difficult or intractable. 7.3 Future Research In the future, this research can be expanded in multiple directions. Some of these have been planned to be explored in the near future. โ€ข Develop a similar compliance matrix formulation for other geometries such as annular disks and cams. A compliance formulation for annular disks or cylinders may require a similar approach with a little more complicated math. โ€ข Employ other friction models on this test-bed and compare and correlate the numerical results with the experimental results of joints. โ€ข Develop a formulation that can connect the finite element approach with compliance formu- lation of elastic bodies with circular contact profiles such as cams and gears. One would hope to explore the contact analysis through compliance formulation and predict the other important parameters outside the contact region through finite element analysis. โ€ข Implement this extremely fine mesh for elastohydrodynamic lubricant contact and conduct contact analysis for contact problems such as rotating shafts and roller bearings. 107 APPENDICES 108 APPENDIX A COMPLIANCE MATRIX FORMULATION FOR ELASTIC HALF-PLANE Consider a rigid cylinder pressing an elastic half-plane with consequent normal pressure ๐‘(๐‘ฅ) and a lateral traction ๐œ(๐‘ฅ) as shown in Fig A.1. These distributed tractions result in the normal and lateral displacements ๐‘ค(๐‘ฅ) and ๐‘ข(๐‘ฅ) respectively, which can be calculated using Eqns. (A.1) and (A.2) [6, 35]. These expressions involve singular kernels and constants ๐ถ1 and ๐ถ2 representing arbitrary rigid body displacements. ๐‘ฆ๐‘ฆ ๐‘ฅ๐‘ฅ ๐‘๐‘ ๐‘ค๐‘ค ฯ„ ๐‘ข๐‘ข Figure A.1: Coordinates ๐‘ฅ, ๐‘ฆ; tractions ๐‘, ๐œ; and displacements ๐‘ข, ๐‘ค on the surface of an elastic half-plane โˆซ โˆซ ๐‘(๐œ)(๐œ… + 1) log |๐‘ฅ โˆ’ ๐œ | ๐œ(๐œ)(๐œ… โˆ’ 1)sgn(๐‘ฅ โˆ’ ๐œ) ๐‘ค(๐‘ฅ) = โˆ’ ๐‘‘๐œ โˆ’ ๐‘‘๐œ + ๐ถ1 (A.1) 4๐œ‹๐œ‡ 8๐œ‡ โˆซ โˆซ ๐œ(๐œ)(๐œ… + 1) log |๐‘ฅ โˆ’ ๐œ | ๐‘(๐œ)(๐œ… โˆ’ 1)sgn(๐‘ฅ โˆ’ ๐œ) ๐‘ข(๐‘ฅ) = โˆ’ ๐‘‘๐œ + ๐‘‘๐œ + ๐ถ2 (A.2) 4๐œ‹๐œ‡ 8๐œ‡ These equations are the basis for creating the discretization method discussed below. A.1 Discretization In order to use numerical simulation to obtain insight into the mechanics taking place in a contact patch, it is necessary to have a very fine mesh in that region. In most general applications, this is done using finite element analysis where a fine mesh in the contact region requires meshing the full substrate structure with a grid that gradually increases in coarseness away from the contact patch. This involves a huge number of degrees of freedom that must be solved at each iteration of each load step. Though there are various flavors of efficiencies available, such as static reduction 109 of the degrees of freedom not in the region of the contact patch [23, 36], implementation of those efficiencies also involve tremendous computer resources and substantial compute time. In the particular case where the elastic component is a half-plane, it is possible to formulate the problem so that the only degrees of freedom are the displacement and tractions on the surface of the half-plane in the vicinity of the region of contact. With reference to Equations A.1 and A.2, we consider some set of ๐‘ basis functions ๐œ™ ๐‘˜ (๐‘ฅ) such as those used in one-dimensional finite element analysis so that in the interval (โˆ’๐‘…, ๐‘…) containing the region of contact, we express our tractions in terms of those basis functions. ๐‘˜=๐‘€ ร• ๐‘˜=๐‘€ ร• ๐‘(๐‘ฅ) = ๐œ™ ๐‘˜ (๐‘ฅ) ๐‘ ๐‘˜ and ๐œ(๐‘ฅ) = ๐œ™ ๐‘˜ (๐‘ฅ) ๐œ๐‘˜ (A.3) ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ where ๐‘ ๐‘˜ and ๐œ๐‘˜ are the values of pressure and shear stress at at location ๐‘ฅ ๐‘˜ , respectively. In the development that follows, the shape functions used are disjoint step-functions constructed so as to serve as a partition of unity in (โˆ’๐‘…, ๐‘…). Implicit in the above definition is an assignment of ๐‘ = 2๐‘€ + 1 nodal locations: ๐‘ฅ๐‘˜ = 2 ๐‘Ž ๐‘˜ where ๐‘Ž = ๐‘…/๐‘ (A.4) The step basis functions ๐œ™ ๐‘˜ (๐‘ฅ) are now defined as ๏ฃฑ 0 if ๐‘ฅ โ‰ค ๐‘ฅ๐‘˜ โˆ’ ๐‘Ž ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๐œ™ ๐‘˜ (๐‘ฅ) = 1 if ๐‘ฅ๐‘˜ โˆ’ ๐‘Ž < ๐‘ฅ < ๐‘ฅ๐‘˜ + ๐‘Ž (A.5) ๏ฃด ๏ฃด ๏ฃด ๏ฃด 0 if ๐‘ฅ โ‰ฅ ๐‘ฅ๐‘˜ + ๐‘Ž ๏ฃด ๏ฃด ๏ฃณ ๐‘ ๐‘ ๐‘ The displacements ๐‘ค ๐‘˜ (๐‘ฅ), ๐‘ข ๐‘˜ (๐‘ฅ), ๐‘ค ๐œ๐‘˜ (๐‘ฅ), and ๐‘ข ๐‘˜ (๐‘ฅ) associated with these pressure basis func- tions are provided in [35] for the case of ๐‘˜ = 0 and are presented in the below A.3. The remaining ๐‘ ๐‘ ๐‘ basis functions expressed in terms of ๐‘ค 0 (๐‘ฅ), ๐‘ข 0 (๐‘ฅ), ๐‘ค 0๐œ (๐‘ฅ), and ๐‘ข 0 (๐‘ฅ): ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ค ๐‘˜ (๐‘ฅ) = ๐‘ค 0 (๐‘ฅ โˆ’ ๐‘ฅ ๐‘˜ ), ๐‘ข ๐‘˜ (๐‘ฅ) = ๐‘ข 0 (๐‘ฅ โˆ’ ๐‘ฅ ๐‘˜ ), (A.6) ๐‘ค ๐œ๐‘˜ (๐‘ฅ) = ๐‘ค 0๐œ (๐‘ฅ โˆ’ ๐‘ฅ ๐‘˜ ), ๐‘ข ๐œ๐‘˜ (๐‘ฅ) = ๐‘ข 0๐œ (๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ ) ๐‘ These relationships are illustrated for ๐œ™ ๐‘˜ (๐‘ฅ) and ๐‘ค ๐‘˜ (๐‘ฅ) for ๐‘˜ = โˆ’1, 0, 1 in Figure A.2, where normal displacement distribution also shifted by +ฮ”๐‘ฅ or โˆ’ฮ”๐‘ฅ on shifting the normal unit pressure. 110 ยท10โˆ’2 1.5 1.2 Pโˆ’1 P0 P1 N odes 1 1 0.8 0.5 Displacement 0.6 Pressure 0 0.4 0.2 โˆ’0.5 0 โˆ’1 โˆ’0.2 โˆ’0.4 โˆ’1.5 โˆ’3 โˆ’2 โˆ’1 0 1 2 3 Contact Region (x) ยท10โˆ’2 Figure A.2: Unit normal pressure and corresponding displacement on shifting by โˆ’ฮ”๐‘ฅ and ฮ”๐‘ฅ The displacements corresponding to Equation A.3 are ๐‘˜=๐‘€ ร• ๐‘˜=๐‘€ ร• ๐‘ ๐‘ค(๐‘ฅ) = ๐‘ค ๐‘˜ (๐‘ฅ) ๐‘๐‘˜ + ๐‘ค ๐œ๐‘˜ (๐‘ฅ) ๐œ๐‘˜ + ๐‘Š๐‘ก๐‘œ๐‘ก ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ (A.7) ๐‘˜=๐‘€ ร• ๐‘˜=๐‘€ ร• ๐‘ ๐‘ข(๐‘ฅ) = ๐‘ข ๐‘˜ (๐‘ฅ) ๐‘๐‘˜ + ๐‘ข ๐œ๐‘˜ (๐‘ฅ) ๐œ๐‘˜ + ๐‘ˆ๐‘ก๐‘œ๐‘ก ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ and ๐‘Š๐‘ก๐‘œ๐‘ก and ๐‘ˆ๐‘ก๐‘œ๐‘ก are yet unknown rigid body vertical and horizontal displacements, respectively. A.2 Compliance Matrix Evaluating Equation A.6 at the nodal locations ({๐‘ฅ ๐‘˜ }) yields a compliance matrix for displacements in terms of tractions. In order to accommodate the unknown rigid body displacements ๐‘Š๐‘ก๐‘œ๐‘ก and ๐‘ˆ๐‘ก๐‘œ๐‘ก , it is necessary to add two force equilibrium equations ๐‘˜=๐‘€ ร• ๐‘˜=๐‘€ ร• ๐น๐‘ = ๐‘ ๐‘˜ ฮ”๐‘ฅ and ๐น๐ฟ = ๐œ๐‘˜ ฮ”๐‘ฅ (A.8) ๐‘˜=โˆ’๐‘€ ๐‘˜=โˆ’๐‘€ where ฮ”๐‘ฅ = 2๐‘Ž and ๐น๐‘ and ๐น๐ฟ are the net normal and lateral forces imposed on the half plane, respectively. All these relationships are shown in Equation A.9. It is assumed that at every step two of ๐‘Š๐‘ก๐‘œ๐‘ก , ๐‘ˆ๐‘ก๐‘œ๐‘ก , ๐น๐‘ , and ๐น๐ฟ are known. 111 ๏ฃฑ ๏ฃด .. ๏ฃผ ๏ฃด ๏ฃฎ .. .. ๏ฃน ๏ฃฑ .. ๏ฃผ ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃฏ . . ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด 1 0 ๏ฃบ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐ถ๐‘ค ๐‘ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ค๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๐ถ๐‘ค๐œ ๏ฃด ๏ฃด ๐‘๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃฏ .. .. ๏ฃบ ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ . . ๏ฃบ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฒ ๏ฃด ๏ฃด ๏ฃฝ ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃบ๏ฃด๏ฃฒ ๏ฃฝ ๏ฃด ๐‘ข ๐‘› = ๏ฃฏ ๐ถ๐‘ข ๐‘ (A.9) ๏ฃฏ ๐ถ๐‘ข๐œ 0 1 ๏ฃบ ๐œ๐‘› ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃฏ .. .. ๏ฃบ๏ฃบ ๏ฃด ๏ฃด .. ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃฏ . . ๏ฃบ๏ฃด ๏ฃด ๏ฃด . ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃบ๏ฃด ๏ฃด ๏ฃฏ ฮ”๐‘ฅ . . . 0 0 0 ๏ฃบ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘ƒ ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘Š๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃด ๏ฃฏ ๏ฃบ๏ฃด๏ฃด ๏ฃด ๏ฃด ๏ฃด ๐‘„ ๏ฃด ๏ฃฏ 0 . . . ฮ”๐‘ฅ ๏ฃด ๏ฃด 0 0 ๏ฃบ๏ฃป ๏ฃด ๏ฃด ๐‘ˆ๐‘ก๐‘œ๐‘ก ๏ฃด ๏ฃด ๏ฃณ ๏ฃพ ๏ฃฐ ๏ฃณ ๏ฃพ A.3 Displacements Associated with Unit Normal Pressure and Unit Shear Traction Functions for ๐‘˜ = 0 The displacements associated with the traction basis functions for pressure and shear are provided in [35]. The normal displacement due to unit normal pressure in (โˆ’๐‘Ž, ๐‘Ž) is   ๐‘Ž + ๐‘ฅ 2  ๐‘Ž โˆ’ ๐‘ฅ 2 ๐‘ โˆ’๐ด ๐‘ค 0 (๐‘ฅ) = (๐‘Ž + ๐‘ฅ) ln + (๐‘Ž โˆ’ ๐‘ฅ) ln (A.10) 2๐œ‹ ๐‘Ž ๐‘Ž for ๐‘ฅ โˆˆ (โˆ’โˆž, โˆž) The lateral displacement due to unit pressure in (โˆ’๐‘Ž, ๐‘Ž) is1 . ๏ฃฒ โˆ’๐ด๐›ฝ๐‘ฅ for ๐‘ฅ โˆˆ (โˆ’๐‘Ž, ๐‘Ž) ๏ฃฑ ๏ฃด ๐‘ ๏ฃด ๐‘ข 0 (๐‘ฅ) = (A.11) ๏ฃด โˆ’๐ด๐›ฝ๐‘Ž sgn (๐‘ฅ) otherwise ๏ฃด ๏ฃณ The lateral displacement due to unit shear traction in the interval (โˆ’๐‘Ž, ๐‘Ž) is:   ๐‘Ž + ๐‘ฅ 2  ๐‘Ž โˆ’ ๐‘ฅ 2 ๐œ ๐ด ๐‘ข 0 (๐‘ฅ) = (๐‘Ž + ๐‘ฅ) ln + (๐‘Ž โˆ’ ๐‘ฅ) ln (A.12) 2๐œ‹ ๐‘Ž ๐‘Ž for ๐‘ฅ โˆˆ (โˆ’โˆž, โˆž) The normal displacement due to unit shear traction in (โˆ’๐‘Ž, ๐‘Ž) is1. ๏ฃฒ โˆ’๐ด๐›ฝ๐‘ฅ for ๐‘ฅ โˆˆ (โˆ’๐‘Ž, ๐‘Ž) ๏ฃฑ ๏ฃด ๐œ ๏ฃด ๐‘ค 0 (๐‘ฅ) = (A.13) ๏ฃด โˆ’๐ด๐›ฝ๐‘Ž sgn (๐‘ฅ) otherwise ๏ฃด ๏ฃณ 1 Equations of lateral displacement due to normal pressure A.11 and normal displacement due to lateral traction A.13, differ by factor of two from the corresponding equations in "Mechanics of Elastic Contacts" [35] in chapter 14 at page 427. However, substituting these traction and displacement expressions into the integral equations of Chapter 2 of that book and of Chapter 12 of [6] indicate that the expressions used here are correct. 112 Above, 1 โˆ’ ๐œˆ2 1 โˆ’ 2๐‘ฃ ๐ด=2 and ๐›ฝ= (A.14) ๐ธ 2(1 โˆ’ ๐‘ฃ) For plane strain ๐œ… = 3 โˆ’ 4๐œˆ, and ๐ธ = 2๐บ (1 + ๐œˆ) (๐œ… โˆ’ 1) ๐ด๐›ฝ = (A.15) 4๐บ 113 APPENDIX B IMPLEMENTAION OF DAHL FRICTION MODEL WITH ALGORITHM-II (METHOD OF FIRST VIOLATION) The rate-independent Dahl model can be compared to Coulomb friction model and has similarities to it as shown in the Fig. B.1. The Dahl model and Coulomb friction differ in the regime where force is less than ๐‘“๐‘ , the force necessary to cause sliding. In that regime the Dahl model provides a force evolution equation (Eq. 3.9) while Coulomb friction imposes a no-slip condition. The parameter ๐œŽ is a contact stiffness associated with interface mechanics before slip. This differential relationship can be implemented numerically using a finite difference method with explicit integration. In the partial slip regime ( ๐‘“ < ๐‘“๐‘ ) the Dahl model becomes (Eq. B.1), which can be expressed in the simpler form of Eq. B.2. ๐›พ   ๐‘“ โˆ’ ๐‘“0 ๐‘“0 ๐‘“ = ๐œŽ 1โˆ’ sgn( ๐‘ขยค0 ) sgn 1 โˆ’ sgn( ๐‘ขยค0 ) ๐‘ข โˆ’ ๐‘ข0 ๐œ‡๐‘ 0 ๐œ‡๐‘ 0 ๐›พ   (B.1) ๐‘“0 ๐‘“ using, ๐พ = ๐œŽ 1 โˆ’ sgn( ๐‘ขยค0 ) sgn 1 โˆ’ sgn( ๐‘ขยค0 ) ๐œ‡๐‘ 0 ๐œ‡๐‘ 0 ๐‘“ โˆ’ ๐พ๐‘ข = ๐‘“0 โˆ’ ๐พ๐‘ข 0 (B.2) where, ๐‘“0 , ๐‘ข 0 and ๐‘ 0 are the known friction force, lateral displacements, and normal pressure values of the last equilibrium configuration. ๐œŽ and ๐›พ are the Dahl parameters. Quantities ๐‘“ and ๐‘ข are the f slope=ฯƒ1 fc = ยตp slope=ฯƒ2 Coulomb Friction U Dahl Model โˆ’fc Figure B.1: Comparing Dahl friction model with Coulomb friction model. 114 friction force and lateral displacement, to be calculated for the end of the load step. Nodal Statuses Equality Constr. Inequality Constr. Contact and Partial slip ๐œ โˆ’ ๐พ๐‘ข = ๐œ0 โˆ’ ๐พ๐‘ข 0 , ๐‘ค = ๐‘” |๐œ| โ‰ค ๐‘“ , ๐‘ โ‰ฅ 0 Contact and Slip Right ๐œ= ๐‘“ ,๐‘ค=๐‘” ๐‘ขยค โ‰ฅ 0, ๐‘ โ‰ฅ 0 Contact and Slip Left ๐œ = โˆ’๐‘“ , ๐‘ค = ๐‘” ๐‘ขยค โ‰ค 0, ๐‘ โ‰ฅ 0 Not-in-Contact ๐‘ = 0, ๐œ = 0 ๐‘คโ‰ค๐‘” Table B.1: Equality and inequality constraints for Dahl friction model. The equality and inequality constraints are similar to those of Coulomb friction except in the region of partial slip (which corresponds to the stick region for Coulomb friction). The equality constraints for each node (B.1) can be incorporated into a system matrix similar to Eq. 2.2 and Algorithm-II can be applied with few other changes. Among those changes is resetting the shear traction of sliding nodes when normal traction decreases, also, because of the explicit formulation of the constitutive equation, accuracy and stability require use of small times steps as well calculation of those that cause transition of node status. 115 APPENDIX C MICHELL COEFFICIENTS FOR A SOLID ELASTIC DISK WITH NO HOLES The Michell Airy stress function for a solid elastic disk after discarding the secular and multi-value terms is following:   2 3 ๐œ…+1 ฮฆ0 (๐‘Ÿ, ๐œƒ) = ๐ด01๐‘Ÿ + ๐ด04 ๐œƒ + ๐ด11๐‘Ÿ cos(๐œƒ) + ๐ด12 ๐‘Ÿ log(๐‘Ÿ) cos(๐œƒ) โˆ’ ๐‘Ÿ๐œƒ sin(๐œƒ) ๐œ…โˆ’1   3 ๐œ…+1 + ๐ต11๐‘Ÿ sin(๐œƒ) + ๐ต12 ๐‘Ÿ log(๐‘Ÿ) sin(๐œƒ) + ๐‘Ÿ๐œƒ cos(๐œƒ) (C.1) ๐œ…โˆ’1 ร•โˆž   + ( ๐ด๐‘›1๐‘Ÿ ๐‘›+2 + ๐ด๐‘›3๐‘Ÿ ๐‘› ) cos(๐‘›๐œƒ) + (๐ต๐‘›1๐‘Ÿ ๐‘›+2 + ๐ต๐‘›3๐‘Ÿ ๐‘› ) sin(๐‘›๐œƒ) ๐‘›=2 C.1 Stresses and Displacements The stresses and displacements are calculated for the stress function Eq. C.1 using Michell Table ??. Radial stress ๐œŽ๐‘Ÿ๐‘Ÿ and tangential stress ๐œŽ๐‘Ÿ๐œƒ at ๐‘Ÿ = ๐‘… results in the radial and tangential tractions applied on the surface of the disk and are sufficient to formulate a compliance matrix. Thus, circumferential stress ๐œŽ๐œƒ๐œƒ is not required for calculation.       1 (๐œ… + 1) 2 1 (๐œ… + 1) (โˆ’2) ๐œŽ๐‘Ÿ๐‘Ÿ = 2๐ด01 + 2๐‘Ÿ ๐ด11 + ๐ด12 โˆ’ cos(๐œƒ) + 2๐‘Ÿ ๐ต11 + ๐ต12 + sin(๐œƒ) ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ร•โˆž   + ( ๐ด๐‘›1 (โˆ’(๐‘› + 1)(๐‘› โˆ’ 2)๐‘Ÿ ๐‘› ) + ๐ด๐‘›3 (โˆ’๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) cos(๐‘›๐œƒ) ๐‘›=2 ร•โˆž   + (๐ต๐‘›1 (โˆ’(๐‘› + 1)(๐‘› โˆ’ 2)๐‘Ÿ ๐‘› ) + ๐ต๐‘›3 (โˆ’๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) sin(๐‘›๐œƒ) ๐‘›=2     ๐ด04 1 1 ๐œŽ๐‘Ÿ๐œƒ = 2 + 2๐‘Ÿ ๐ด11 + ๐ด12 sin(๐œƒ) + 2๐‘Ÿ ๐ต11 + ๐ต12 cos(๐œƒ) ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ร•โˆž   + ( ๐ด๐‘›1 (๐‘›(๐‘› + 1)๐‘Ÿ ๐‘› ) + ๐ด๐‘›3 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) sin(๐‘›๐œƒ) ๐‘›=2 ร•โˆž   + (๐ต๐‘›1 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ ๐‘› ) + ๐ต๐‘›3 (โˆ’๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) cos(๐‘›๐œƒ) ๐‘›=2 (C.2) 116 โˆ’๐œ‹ ๐œ‹ Elastic Disk ๐‘€ ๐‘ƒ โˆ’๐œ‹/2 ๐‘„ ๐œ‹/2 ๐œŽ๐‘Ÿ ๐œƒ 0 ๐œƒ ๐œŽ๐‘Ÿ๐‘Ÿ Figure C.1: Stress distribution on the solid disk is equlibrated with forces and moment near the center. Stresses distributed on the surface of the disk due to contact should be equilibrated by applying the forces on the disk away from the contact or by applying the forces and moment near the center of the disk as shown in Fig. C.1. A normal force per unit length (๐‘ƒ) is equilibrated with components of radial (๐œŽ๐‘Ÿ๐‘Ÿ ) and tangential (๐œŽ๐‘Ÿ๐œƒ ) stress as shown in Eq. C.3. Expressions of stresses from Eq. C.2 are substituted in Eq. C.3 and solved. The expression results with only Michell coefficient (๐ด12 ) to equilibrate the normal force (๐‘ƒ). โˆซ ๐œ‹ ๐‘ƒ= (๐œŽ๐‘Ÿ๐‘Ÿ cos ๐œƒ โˆ’ ๐œŽ๐‘Ÿ๐œƒ sin ๐œƒ)๐‘…๐‘‘๐œƒ โˆ’๐œ‹    (๐œ… + 1)2 2 ๐œ‹ โˆ’ 2๐‘… 2 ๐ด11 + ๐ด12 ๐œ‹   ๐‘ƒ = 2๐‘… ๐ด11 + ๐ด12 1 โˆ’ (C.3) (๐œ… โˆ’ 1)   (๐œ… + 1) ๐‘ƒ = โˆ’2๐ด12 ๐œ‹ (๐œ… โˆ’ 1) Similarly, lateral force per unit length (๐‘„) is equilibrated with components of radial and tangential stress as shown in C.4. The expression results with only Michell coefficient (๐ต12 ) to equilibrate the lateral force (๐‘„). โˆซ ๐œ‹ ๐‘„= (๐œŽ๐‘Ÿ๐‘Ÿ sin ๐œƒ + ๐œŽ๐‘Ÿ๐œƒ cos ๐œƒ)๐‘…๐‘‘๐œƒ โˆ’๐œ‹    2 (๐œ… + 1)2 ๐œ‹ + 2๐‘… 2 ๐ต11 + ๐ต12 ๐œ‹   ๐‘„ = 2๐‘… ๐ต11 + ๐ต12 1 โˆ’ (C.4) (๐œ… โˆ’ 1)   (๐œ… + 1) ๐‘„ = โˆ’2๐ต12 ๐œ‹ (๐œ… โˆ’ 1) 117 A moment at the center of disk results because of the tangential stress (๐œŽ๐‘Ÿ๐œƒ ) at the contact surface which is equilibrated using Eq. C.5. The moment expression results with Michell coefficient (๐ด04 ). โˆซ ๐œ‹ ๐‘€= (๐œŽ๐‘Ÿ๐œƒ )๐‘… 2 ๐‘‘๐œƒ โˆ’๐œ‹ (C.5) ๐‘€ = 2๐œ‹ ๐ด04 Three Michelle coefficients ๐ด12 ,๐ต12 , and ๐ด04 are sufficient to equilibrate the forces and moment applied on an elastic circular disk. To calculate displacement at the surface, first strains at the contact surface are calculated using Hookโ€™s law and stresses shown in Eq. C.2. Later, strains are integrated over the length of the contact surface that results in displacements and some constant of integration. The procedure is explained in Chapter 9 (especially section 9.2) of Elasticity book [6]. As pointed out in the procedure, three constants of integration result from derivation. Two constants represent the rigid body translation and one constant represents rigid body rotation. Displacements expressions (Eq. C.6) are without these rigid body constants. One would need to consider the rigid body displacements during the calculation as shown in the section 4.3. 118 2๐œ‡๐‘ˆ๐‘Ÿ = ๐ด01 (๐œ… โˆ’ 1)๐‘Ÿ    2 1 (๐œ… + 1) + ๐ด11 (๐œ… โˆ’ 2)๐‘Ÿ + ๐ด12 ((๐œ… โˆ’ 1) ln(๐‘Ÿ) โˆ’ 1) โˆ’ ((๐œ… + 1) ln(๐‘Ÿ) โˆ’ 1) cos(๐œƒ) 2 (๐œ… โˆ’ 1)    2 1 (๐œ… + 1) + ๐ต11 (๐œ… โˆ’ 2)๐‘Ÿ + ๐ต12 ((๐œ… โˆ’ 1) ln(๐‘Ÿ) โˆ’ 1) + (1 โˆ’ (๐œ… + 1) ln(๐‘Ÿ)) sin(๐œƒ) 2 (๐œ… โˆ’ 1) ร• โˆž   + ( ๐ด๐‘›1 ((๐œ… โˆ’ ๐‘› โˆ’ 1)๐‘Ÿ ๐‘›+1 ) + ๐ด๐‘›3 ((โˆ’๐‘›)๐‘Ÿ ๐‘›โˆ’1 )) cos(๐‘›๐œƒ) ๐‘›=2 ร• โˆž   + (๐ต๐‘›1 ((๐œ… โˆ’ ๐‘› โˆ’ 1)๐‘Ÿ ๐‘›+1 ) + ๐ต๐‘›3 ((โˆ’๐‘›)๐‘Ÿ ๐‘›โˆ’1 )) sin(๐‘›๐œƒ) ๐‘›=2 (C.6) 1 2๐œ‡๐‘ˆ๐œƒ = โˆ’๐ด04  ๐‘Ÿ   2 1 (๐œ… + 1) + ๐ด11 (๐œ… + 2)๐‘Ÿ + ๐ด12 (โˆ’1 + (๐œ… + 1) ln(๐‘Ÿ)) โˆ’ (โˆ’1 โˆ’ (๐œ… + 1) ln(๐‘Ÿ)) sin(๐œƒ) 2 (๐œ… โˆ’ 1)    2 1 (๐œ… + 1) + โˆ’๐ต11 (๐œ… + 2)๐‘Ÿ + ๐ต12 (1 + (๐œ… โˆ’ 1) ln(๐‘Ÿ)) + (โˆ’1 โˆ’ (๐œ… + 1) ln(๐‘Ÿ)) cos(๐œƒ) 2 (๐œ… โˆ’ 1) ร• โˆž   + ( ๐ด๐‘›1 ((๐œ… + ๐‘› + 1)๐‘Ÿ ๐‘›+1 ) + ๐ด๐‘›3 (๐‘›๐‘Ÿ ๐‘›โˆ’1 )) sin(๐‘›๐œƒ) ๐‘›=2 ร• โˆž   + (๐ต๐‘›1 (โˆ’(๐œ… + ๐‘› + 1)๐‘Ÿ ๐‘›+1 ) + ๐ต๐‘›3 (โˆ’๐‘›๐‘Ÿ ๐‘›โˆ’1 )) cos(๐‘›๐œƒ) ๐‘›=2 C.2 Michell Terms for Unit Radial/Tangential Load The unit pressure ๐œ™0 is applied radially in the interval centered at node ๐œƒ = 0 generating a non-zero radial stress and a zero tangential stress. Trigonometric terms from the Eqs.C.2 and 4.10 are equated and result in the linear equations in Aโ€™s and Bโ€™s shown in C.7. 119 ๐œŽ๐‘Ÿ๐‘Ÿ = ๐œ™0 (๐œƒ) ๐œŽ๐‘Ÿ๐œƒ = 0 ฮ”๐œƒ 2๐ด01 = ๐ด04 /๐‘Ÿ 2 = 0 2๐œ‹ 1 (๐œ… + 1) 2  2 ฮ”๐œƒ 1 2๐‘Ÿ ๐ด11 + ๐ด12 โˆ’ = sin( ) 2๐‘Ÿ ๐ด11 + ๐ด12 =0 ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐œ‹ 2 ๐‘Ÿ 1 (๐œ… + 1) 2  1 2๐‘Ÿ ๐ต11 + ๐ต12 โˆ’ =0 2๐‘Ÿ ๐ต11 + ๐ต12 = 0 ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ (C.7) ( ๐ด๐‘›1 (โˆ’(๐‘› + 1)(๐‘› โˆ’ 2)๐‘Ÿ ๐‘› ) โˆ’ ๐ด๐‘›3 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 ) = ( ๐ด๐‘›1 (๐‘›(๐‘› + 1)๐‘Ÿ ๐‘› ) + ๐ด๐‘›3 (๐‘›(๐‘› โˆ’ 2 ๐‘›ฮ”๐œƒ sin( ) 1)๐‘Ÿ ๐‘›โˆ’2 )) = 0 ๐‘›๐œ‹ 2 (๐ต๐‘›1 (โˆ’(๐‘› +1)(๐‘› โˆ’2)๐‘Ÿ ๐‘› ) โˆ’ ๐ต๐‘›3 (๐‘›(๐‘› โˆ’1)๐‘Ÿ ๐‘›โˆ’2 ) = 0 (๐ต๐‘›1 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ ๐‘› ) โˆ’ ๐ต๐‘›3 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) = 0 Similarly, the unit traction ๐œ™0 is applied tangentially at same interval creating a non-zero tangential stress and a zero radial stress. This results in the linear equations shown in C.8. ๐œŽ๐‘Ÿ๐‘Ÿ = 0 ๐œŽ๐‘Ÿ๐œƒ = ๐œ™0 (๐œƒ) ฮ”๐œƒ 2๐ด01 = 0 ๐ด04 /๐‘Ÿ 2 = 2๐œ‹ 1 (๐œ… + 1) 2  1 2๐‘Ÿ ๐ด11 + ๐ด12 โˆ’ =0 2๐‘Ÿ ๐ด11 + ๐ด12 = 0 ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ 1 (๐œ… + 1) 2  1 2 ฮ”๐œƒ 2๐‘Ÿ ๐ต11 + ๐ต12 โˆ’ =0 2๐‘Ÿ ๐ต11 + ๐ต12 = sin( ) ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ ๐œ‹ 2 ( ๐ด๐‘›1 (โˆ’(๐‘› + 1)(๐‘› โˆ’ 2)๐‘Ÿ ๐‘› ) โˆ’ ๐ด๐‘›3 (๐‘›(๐‘› โˆ’ ( ๐ด๐‘›1 (๐‘›(๐‘› + 1)๐‘Ÿ ๐‘› ) + ๐ด๐‘›3 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) = 0 1)๐‘Ÿ ๐‘›โˆ’2 ) = 0 (๐ต๐‘›1 (โˆ’(๐‘› + 1)(๐‘› โˆ’ 2)๐‘Ÿ ๐‘› ) โˆ’ ๐ต๐‘›3 (๐‘›(๐‘› โˆ’ (๐ต๐‘›1 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ ๐‘› ) โˆ’ ๐ต๐‘›3 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ ๐‘›โˆ’2 )) = 2 ๐‘›ฮ”๐œƒ 1)๐‘Ÿ ๐‘›โˆ’2 ) = 0 sin( ) ๐‘›๐œ‹ 2 (C.8) These linear Eqs. C.7 and C.8 are solved separately for ๐ดโ€™s and ๐ตโ€™s terms of radial and tangential load respectively (at the surface ๐‘Ÿ = ๐‘…), and the resulting values of ๐ดโ€™s and ๐ตโ€™s are shown in the Tab. 4.2. 120 C.2.1 Stresses and Displacements inside the Circular Disk For a unit radial stress on the interval about ๐œƒ = 0, stresses and displacement inside the disk are the following: ฮ”๐œƒ (๐œ… โˆ’ 1)๐‘Ÿ 2 + (๐œ… + 3)๐‘… 2 ๐œŽ๐‘Ÿ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = + sin(ฮ”๐œƒ/2) cos(๐œƒ) 2๐œ‹ ๐œ‹(๐œ… + 1)๐‘…๐‘Ÿ ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’2 (๐‘›๐‘… 2 + 2๐‘Ÿ 2 โˆ’ ๐‘›๐‘Ÿ 2 ) + sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘… ๐‘› ฮ”๐œƒ (๐œ… โˆ’ 1)(3๐‘Ÿ 2 โˆ’ ๐‘… 2 ) ๐œŽ๐œƒ๐œƒ (๐‘Ÿ, ๐œƒ) = + sin(ฮ”๐œƒ/2) cos(๐œƒ) 2๐œ‹ ๐œ‹(๐œ… + 1)๐‘…๐‘Ÿ ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’2 ((๐‘› + 2)๐‘Ÿ 2 โˆ’ ๐‘›๐‘… 2 ) + ๐‘› sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘… (๐œ… โˆ’ 1)(๐‘Ÿ 2 โˆ’ ๐‘… 2 ) ๐œŽ๐‘Ÿ๐œƒ (๐‘Ÿ, ๐œƒ) = sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐œ‹(๐œ… + 1)๐‘…๐‘Ÿ ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’2 (๐‘Ÿ 2 โˆ’ ๐‘… 2 ) (C.9) + sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘… ๐‘› ๐‘Ÿ (๐œ… โˆ’ 1)ฮ”๐œƒ ((๐œ… 2 โˆ’ 3๐œ… + 2)๐‘Ÿ 2 + 4๐œ…๐‘… 2 ln(๐‘Ÿ)) ฮ”๐œƒ 2๐œ‡๐‘ˆ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = + sin( ) cos(๐œƒ) 4๐œ‹ 2๐œ‹(๐œ… + 1)๐‘… 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’1 ((๐œ… โˆ’ ๐‘› โˆ’ 1)(๐‘› โˆ’ 1)๐‘Ÿ 2 + (๐‘›2 + ๐‘›)๐‘… 2 ) ๐‘›ฮ”๐œƒ + 2 sin( ) cos(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› โˆ’ 1)๐œ‹๐‘… ๐‘› 2 (๐œ… 2 + ๐œ… โˆ’ 2)๐‘Ÿ 2 โˆ’ 4(1 + ๐œ… ln(๐‘…))๐‘… 2 ฮ”๐œƒ 2๐œ‡๐‘ˆ๐œƒ (๐‘Ÿ, ๐œƒ) = sin( ) sin(๐œƒ) 2๐œ‹(๐œ… + 1)๐‘… 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ((๐œ… + ๐‘› + 1)(๐‘› โˆ’ 1)๐‘Ÿ โˆ’ (๐‘› + ๐‘›)๐‘… 2 ) ๐‘›โˆ’1 2 2 ๐‘›ฮ”๐œƒ + 2 sin( ) sin(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› โˆ’ 1)๐œ‹๐‘… ๐‘› 2 121 For unit tangential stress at ๐œƒ = 0, stresses and displacements inside the disk are the following: (๐œ… + 3)(๐‘Ÿ 2 โˆ’ ๐‘… 2 ) ๐œŽ๐‘Ÿ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = โˆ’ sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐œ‹(๐œ… + 1)๐‘…๐‘Ÿ ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’2 (๐‘› โˆ’ 2)(๐‘Ÿ 2 โˆ’ ๐‘… 2 ) + sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘… ๐‘› (3(๐œ… + 3)๐‘Ÿ 2 + (๐‘˜ โˆ’ 1)๐‘… 2 ) ๐œŽ๐œƒ๐œƒ (๐‘Ÿ, ๐œƒ) = โˆ’ sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐œ‹(๐œ… + 1)๐‘…๐‘Ÿ ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’2 ((๐‘› โˆ’ 2)๐‘… 2 โˆ’ (๐‘› + 2)๐‘Ÿ 2 ) + ๐‘› sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘… ฮ”๐œƒ (๐œ… + 3)๐‘Ÿ 2 + (๐œ… โˆ’ 1)๐‘… 2 ) ๐œŽ๐‘Ÿ๐œƒ (๐‘Ÿ, ๐œƒ) = + sin(ฮ”๐œƒ/2) cos(๐œƒ) 2๐œ‹ ๐œ‹(๐œ… + 1)๐‘…๐‘Ÿ ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’2 (๐‘›๐‘Ÿ 2 + 2๐‘… 2 โˆ’ ๐‘›๐‘… 2 ) (C.10) + sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘… ๐‘› ((๐œ… 2 โˆ’ ๐œ… + 6)๐‘Ÿ 2 + 4๐œ…๐‘… 2 ln(๐‘Ÿ)) ฮ”๐œƒ 2๐œ‡๐‘ˆ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = sin( ) sin(๐œƒ) 2๐œ‹(๐œ… + 1)๐‘… 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’1 ((๐‘›2 + ๐œ… โˆ’ ๐‘›๐œ… โˆ’ 1)๐‘Ÿ 2 + (2 + ๐‘› โˆ’ ๐‘›2 )๐‘… 2 ) ๐‘›ฮ”๐œƒ + sin( ) sin(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘›2 โˆ’ 1)๐œ‹๐‘… ๐‘› 2 ๐‘Ÿฮ”๐œƒ (๐œ… 2 + 5๐œ… + 6)๐‘Ÿ 2 + 4(1 + ๐œ… ln(๐‘…))๐‘… 2 ฮ”๐œƒ 2๐œ‡๐‘ˆ๐œƒ (๐‘Ÿ, ๐œƒ) = + sin( ) cos(๐œƒ) 2๐œ‹ 2๐œ‹(๐œ… + 1)๐‘… 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘Ÿ ๐‘›โˆ’1 ((๐œ… + ๐‘› + 1)(๐‘› โˆ’ 1)๐‘Ÿ 2 + (2 โˆ’ ๐‘›2 + ๐‘›)๐‘… 2 ) ๐‘›ฮ”๐œƒ + 2 sin( ) cos(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› โˆ’ 1)๐œ‹๐‘… ๐‘› 2 Due to the elastic nature of the problem, the stress calculated using each contacting node can be summed to get the final stresses or displacements. 122 APPENDIX D MICHELL COEFFICIENTS FOR AN INFINITE ELASTIC PLANE WITH A CIRCULAR HOLE The Michell Airy stress function for a hole in an infinite elastic plane after discarding the secular and multi-value terms is following: ฮฆ0 (๐‘Ÿ, ๐œƒ) = ๐ด01๐‘Ÿ 2 + ๐ด03 ln(๐‘Ÿ) + ๐ด04 ๐œƒ + ๐ด23๐‘Ÿ 2 cos(๐œƒ) + ๐ต23๐‘Ÿ 2 sin(๐œƒ)   ๐œ…+1 + ๐ด12 ๐‘Ÿ log(๐‘Ÿ) cos(๐œƒ) โˆ’ ๐‘Ÿ๐œƒ sin(๐œƒ) + ๐ด13๐‘Ÿ๐œƒ cos(๐œƒ) ๐œ…โˆ’1 (D.1)   ๐œ…+1 + ๐ต12 ๐‘Ÿ log(๐‘Ÿ) sin(๐œƒ) + ๐‘Ÿ๐œƒ cos(๐œƒ) + ๐ต13๐‘Ÿ๐œƒ sin(๐œƒ) ๐œ…โˆ’1 ร•โˆž ( ๐ด๐‘›2๐‘Ÿ 2โˆ’๐‘› + ๐ด๐‘›4๐‘Ÿ โˆ’๐‘› ) cos(๐‘›๐œƒ) + (๐ต๐‘›2๐‘Ÿ 2โˆ’๐‘› + ๐ต๐‘›4๐‘Ÿ โˆ’๐‘› ) sin(๐‘›๐œƒ)   + ๐‘›=2 D.1 Stresses and Displacements The stresses and displacements are calculated for the stress function Eq. D.1 using Table ??.         ๐ด03 1 (๐œ… + 1) 2 โˆ’2 1 (๐œ… + 1) โˆ’2 โˆ’2 ๐œŽ๐‘Ÿ๐‘Ÿ = 2๐ด01 + 2 + ๐ด12 โˆ’ + ๐ด14 3 cos(๐œƒ) + ๐ต12 + + ๐ต14 3 sin(๐œƒ) ๐‘… ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ ร• โˆž ( ๐ด๐‘›2 (โˆ’(๐‘› โˆ’ 1)(๐‘› + 2)๐‘Ÿ โˆ’๐‘› ) + ๐ด๐‘›4 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ โˆ’๐‘›โˆ’2 )) cos(๐‘›๐œƒ)   + ๐ด23 (โˆ’2) cos(2๐œƒ) + ๐‘›=2 ร•โˆž (๐ต๐‘›2 (โˆ’(๐‘› โˆ’ 1)(๐‘› + 2)๐‘Ÿ โˆ’๐‘› ) + ๐ต๐‘›4 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ โˆ’๐‘›โˆ’2 )) sin(๐‘›๐œƒ)   + ๐ต23 (โˆ’2) sin(2๐œƒ) + ๐‘›=2     ๐ด04 1 โˆ’2 1 โˆ’2 ๐œŽ๐‘Ÿ๐œƒ = 2 + ๐ด12 + ๐ด14 3 sin(๐œƒ) + ๐ต12 + ๐ต14 3 cos(๐œƒ) ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ โˆž ร• ( ๐ด๐‘›2 (โˆ’๐‘›(๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘› ) + ๐ด๐‘›4 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ โˆ’๐‘›โˆ’2 )) sin(๐‘›๐œƒ)  + ๐ด23 2 sin(2๐œƒ) + ๐‘›=2 ร• โˆž (๐ต๐‘›2 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘› ) + ๐ต๐‘›3 (๐‘›(๐‘› + 1)๐‘Ÿ ๐‘›โˆ’2 )) cos(๐‘›๐œƒ)   + ๐ต23 (โˆ’2) cos(2๐œƒ) + ๐‘›=2 (D.2) 123 From the above Eq. D.2, stresses can be non-zero at ๐‘Ÿ = โˆž and can have some initial stresses in the elastic infinite plane (see Eq. D.3). For the known values of initial stress at ๐‘Ÿ = โˆž, ๐ด01 ,๐ด23 and ๐ต23 can be calculated through Eq. D.3 and used for further calculation. For this case, no initial stress in infinite plane is assumed which results in ({๐ด01 , ๐ด23 ๐ต23 } = 0). There is no loss of generality in setting stress to zero at infinity. As any displacements and stresses in the infinite plane can be superposed with the solution due to the elastic nature of the infinite plane. ๐œŽ๐‘Ÿ๐‘Ÿ (โˆž, ๐œƒ) = 2๐ด01 โˆ’ 2๐ด23 cos(๐œƒ) โˆ’ 2๐ต23 sin(๐œƒ) ๐œŽ๐‘Ÿ๐œƒ (โˆž, ๐œƒ) = 2๐ด23 sin(๐œƒ) โˆ’ 2๐ต23 cos(๐œƒ) (D.3) ๐œŽ๐œƒ๐œƒ (โˆž, ๐œƒ) = 2๐ด01 + 2๐ด23 cos(๐œƒ) + 2๐ต23 sin(๐œƒ) Infinite Elastic Plane โˆ’๐œ‹ ๐œ‹ โˆ’๐œ‹/2 ๐œ‹/2 ๐‘€ ๐‘„ ๐‘ƒ ๐œŽ๐‘Ÿ ๐œƒ 0 ๐œƒ ๐œŽ๐‘Ÿ๐‘Ÿ Figure D.1: Stress distribution on the hole should be equilibrated with forces and moment applied on the indenter. Stresses distributed on the surface of the hole due to contact (shown in Fig. D.1) should be equilibrated by the forces and moment applied on the indenter that is pressing inside the hole. A normal force per unit length (๐‘ƒ) is equilibrated with components of radial (๐œŽ๐‘Ÿ๐‘Ÿ ) and tangential (๐œŽ๐‘Ÿ๐œƒ ) stress as shown in Eq. D.4. The expression results with only Michell coefficient (๐ด12 ) to equilibrate the normal force (๐‘ƒ). 124 โˆซ ๐œ‹ ๐‘ƒ= (๐œŽ๐‘Ÿ๐‘Ÿ cos ๐œƒ โˆ’ ๐œŽ๐‘Ÿ๐œƒ sin ๐œƒ)๐‘…๐‘‘๐œƒ โˆ’๐œ‹      1 (๐œ… + 1)2 1 ๐‘ƒ = โˆ’2๐ด14 2 + ๐ด12 1 โˆ’ ๐œ‹ โˆ’ โˆ’2๐ด14 2 + ๐ด12 ๐œ‹ (D.4) ๐‘… (๐œ… โˆ’ 1) ๐‘…   (๐œ… + 1) ๐‘ƒ = โˆ’2๐ด12 ๐œ‹ (๐œ… โˆ’ 1) Similarly, lateral force per unit length (๐‘„) is equilibrated with components of radial and tangential stress as shown in D.5. The expression results with only Michell coefficient (๐ต12 ) to equilibrate the lateral force (๐‘„). โˆซ ๐œ‹ ๐‘„= (๐œŽ๐‘Ÿ๐‘Ÿ sin ๐œƒ + ๐œŽ๐‘Ÿ๐œƒ cos ๐œƒ)๐‘…๐‘‘๐œƒ โˆ’๐œ‹      1 (๐œ… + 1)2 1 ๐‘„ = 2๐ต14 2 + ๐ต12 1 โˆ’ ๐œ‹ + 2๐ต14 2 + ๐ต12 ๐œ‹ (D.5) ๐‘… (๐œ… โˆ’ 1) ๐‘…   (๐œ… + 1) ๐‘„ = โˆ’2๐ต12 ๐œ‹ (๐œ… โˆ’ 1) A moment at the center of the indenter results due to the tangential stress (๐œŽ๐‘Ÿ๐œƒ ) at the contact surface which is equilibrated using Eq. D.6. The moment expression results with Michell coefficient (๐ด04 ). โˆซ ๐œ‹ ๐‘€= (๐œŽ๐‘Ÿ๐œƒ )๐‘… 2 ๐‘‘๐œƒ โˆ’๐œ‹ (D.6) ๐‘€ = 2๐œ‹ ๐ด04 Three Michelle coefficients ๐ด12 ,๐ต12 , and ๐ด04 are sufficient to equilibrate the contact stresses on the circular hole similar to the case of solid elastic disk. Note that these Michelle coefficients are common coefficients both for a solid elastic disk and for a circular hole in an infinite elastic plane. Other Michell coefficients do not affect the forces and moment applied on the rigid or elastic indenter body. Similar to the previous appendix C.1, the displacements are calculated by first calculating strains from stresses and later integrating strains over the length of the contact surface that result in displacements expression shown in Eq. D.7 and three constant of integration. The procedure is explained in Chapter 9 (especially section 9.2) of Elasticity [6]. Two constants represent the rigid body translation and one constant represents rigid body rotation. One should consider these rigid body displacements during the calculations. 125 โˆž โˆ’1 ร•  ( ๐ด๐‘›2 ((๐œ… + ๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘›+1 ) + ๐ด๐‘›4 ((๐‘›)๐‘Ÿ โˆ’๐‘›โˆ’1 )) cos(๐‘›๐œƒ)  2๐œ‡๐‘ˆ๐‘Ÿ = ๐ด03 + ๐‘Ÿ ๐‘›=2 ร• โˆž (๐ต๐‘›2 ((๐œ… + ๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘›+1 ) + ๐ต๐‘›4 ((๐‘›)๐‘Ÿ โˆ’๐‘›โˆ’1 )) sin(๐‘›๐œƒ)   + ๐‘›=2    2 1 (๐œ… + 1) + ๐ด14 (1/๐‘Ÿ ) + ๐ด12 ((๐œ… โˆ’ 1) ln(๐‘Ÿ) โˆ’ 1) โˆ’ ((๐œ… + 1) ln(๐‘Ÿ) โˆ’ 1) cos(๐œƒ) 2 (๐œ… โˆ’ 1)    2 1 (๐œ… + 1) + ๐ต14 (1/๐‘Ÿ ) + ๐ต12 ((๐œ… โˆ’ 1) ln(๐‘Ÿ) โˆ’ 1) + (1 โˆ’ (๐œ… + 1) ln(๐‘Ÿ)) sin(๐œƒ) 2 (๐œ… โˆ’ 1) โˆž (D.7) 1 ร• ( ๐ด๐‘›2 (โˆ’(๐œ… โˆ’ ๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘›+1 ) + ๐ด๐‘›4 (๐‘›๐‘Ÿ โˆ’๐‘›โˆ’1 )) sin(๐‘›๐œƒ)  2๐œ‡๐‘ˆ๐œƒ = โˆ’๐ด04 + ๐‘Ÿ ๐‘›=2 ร• โˆž (๐ต๐‘›2 ((๐œ… โˆ’ ๐‘› + 1)๐‘Ÿ ๐‘›+1 ) + ๐ต๐‘›4 (โˆ’๐‘›๐‘Ÿ โˆ’๐‘›โˆ’1 )) cos(๐‘›๐œƒ)   + ๐‘›=2    2 1 (๐œ… + 1) + ๐ด14 (1/๐‘Ÿ ) + ๐ด12 (โˆ’1 + (๐œ… + 1) ln(๐‘Ÿ)) โˆ’ (โˆ’1 โˆ’ (๐œ… + 1) ln(๐‘Ÿ)) sin(๐œƒ) 2 (๐œ… โˆ’ 1)    2 1 (๐œ… + 1) + โˆ’๐ต14 (1/๐‘Ÿ ) + ๐ต12 (1 + (๐œ… โˆ’ 1) ln(๐‘Ÿ)) + (โˆ’1 โˆ’ (๐œ… + 1) ln(๐‘Ÿ)) cos(๐œƒ) 2 (๐œ… โˆ’ 1) D.2 Michell Terms for Unit Radial/Tangential load The unit pressure ๐œ™0 is applied radially on the interval centered at node ๐œƒ = 0 generating non-zero radial stress and zero tangential stress. Trigonometric terms from the Eqs.D.2 and 4.10 are equated and result in the linear equations in Aโ€™s and Bโ€™s shown in D.8. ๐œŽ๐‘Ÿ๐‘Ÿ = ๐œ™0 (๐œƒ) ๐œŽ๐‘Ÿ๐œƒ = 0 1 1 ๐ด03 2 = ฮ”๐œƒ/(2๐œ‹) ๐ด04 2 = 0 ๐‘Ÿ  ๐‘Ÿ 1 (๐œ… + 1) 2 2 2 ฮ”๐œƒ 1 2 ๐ด12 โˆ’ โˆ’ ๐ด14 3 = sin( ) ๐ด12 โˆ’ ๐ด14 3 = 0  ๐‘Ÿ (๐œ… โˆ’ 1)  2 ๐‘Ÿ ๐œ‹ ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ 1 (๐œ… + 1) 2 โˆ’2 1 2 ๐ต12 โˆ’ + ๐ต14 3 = 0 ๐ต12 โˆ’ ๐ต14 3 = 0 ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ (D.8) ๐ด๐‘›2 (โˆ’(๐‘›โˆ’1)(๐‘›+2)๐‘Ÿ โˆ’๐‘› ) + ๐ด ๐‘›4 (โˆ’๐‘›(๐‘›+1)๐‘Ÿ โˆ’๐‘›โˆ’2 ) = ( ๐ด๐‘›2 (โˆ’๐‘›(๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘› ) + ๐ด๐‘›4 (โˆ’๐‘›(๐‘› + 2 sin(๐‘›ฮ”๐œƒ/2) 1)๐‘Ÿ โˆ’๐‘›โˆ’2 )) = 0 ๐‘›๐œ‹ ๐ต๐‘›2 (โˆ’(๐‘›โˆ’1)(๐‘›+2)๐‘Ÿ โˆ’๐‘› ) +๐ต๐‘›4 (โˆ’๐‘›(๐‘›+1)๐‘Ÿ โˆ’๐‘›โˆ’2 ) = (๐ต๐‘›2 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘› ) + ๐ต๐‘›4 (๐‘›(๐‘› + 0 1)๐‘Ÿ ๐‘›โˆ’2 )) = 0 126 Similarly, the unit traction ๐œ™0 is applied tangentially on the interval about node ๐œƒ = 0 creating non-zero tangential stress and zero radial stress. This results in the linear equations shown in D.9. ๐œŽ๐‘Ÿ๐‘Ÿ = 0 ๐œŽ๐‘Ÿ๐œƒ = ๐œ™0 (๐œƒ) 1 1 ๐ด03 2 = 0 ๐ด04 2 = ฮ”๐œƒ/(2๐œ‹) ๐‘Ÿ  ๐‘Ÿ 1 (๐œ… + 1) 2 โˆ’2 1 โˆ’2 ๐ด12 โˆ’ + ๐ด14 3 = 0 ๐ด12 + ๐ด14 3 = 0  ๐‘Ÿ (๐œ… โˆ’ 1)  ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ 1 (๐œ… + 1) โˆ’2 โˆ’2 1 โˆ’2 2 ๐ต12 + + ๐ต14 3 = 0 ๐ต12 + ๐ต14 3 = sin(ฮ”๐œƒ/2) ๐‘Ÿ (๐œ… โˆ’ 1) ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐œ‹ ๐ด๐‘›2 (โˆ’(๐‘› โˆ’ 1)(๐‘› + 2)๐‘Ÿ โˆ’๐‘› ) + ๐ด๐‘›4 (โˆ’๐‘›(๐‘› + ( ๐ด๐‘›2 (โˆ’๐‘›(๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘› ) + ๐ด๐‘›4 (โˆ’๐‘›(๐‘› + 1)๐‘Ÿ โˆ’๐‘›โˆ’2 ) = 0 1)๐‘Ÿ โˆ’๐‘›โˆ’2 )) = 0 ๐ต๐‘›2 (โˆ’(๐‘› โˆ’ 1)(๐‘› + 2)๐‘Ÿ โˆ’๐‘› ) + ๐ต๐‘›4 (โˆ’๐‘›(๐‘› + (๐ต๐‘›2 (๐‘›(๐‘› โˆ’ 1)๐‘Ÿ โˆ’๐‘› ) + ๐ต๐‘›4 (๐‘›(๐‘› + 1)๐‘Ÿ ๐‘›โˆ’2 )) = 2 1)๐‘Ÿ โˆ’๐‘›โˆ’2 ) = 0 sin(๐‘›ฮ”๐œƒ/2) ๐‘›๐œ‹ (D.9) These linear Eqs. D.8 and D.9 are solved separately for ๐ดโ€™s and ๐ตโ€™s terms of radial and tangential load respectively (at the inner surface ๐‘Ÿ = ๐‘…), and the resulting values of ๐ดโ€™s and ๐ตโ€™s are shown in the Tab. 4.3. 127 D.2.1 Stresses and Displacements inside the Circular Hole For a unit radial stress on ๐œƒ = 0, stresses and displacement inside the infinite plane of the hole are following: ฮ”๐œƒ ๐‘…((๐œ… + 3)๐‘Ÿ 2 + (๐œ… โˆ’ 1)๐‘… 2 ) ๐œŽ๐‘Ÿ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = + sin(ฮ”๐œƒ/2) cos(๐œƒ) 2๐œ‹ ๐œ‹(๐œ… + 1)๐‘Ÿ 3 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› (๐‘›๐‘Ÿ 2 + 2๐‘Ÿ 2 โˆ’ ๐‘›๐‘… 2 ) + sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘Ÿ ๐‘›+2 ฮ”๐œƒ (๐œ… โˆ’ 1)(๐‘Ÿ 2 + ๐‘… 2 )๐‘… ๐œŽ๐œƒ๐œƒ (๐‘Ÿ, ๐œƒ) = โˆ’ โˆ’ sin(ฮ”๐œƒ/2) cos(๐œƒ) 2๐œ‹ ๐œ‹(๐œ… + 1)๐‘Ÿ 3 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› ((๐‘› โˆ’ 2)๐‘Ÿ 2 โˆ’ ๐‘›๐‘… 2 ) โˆ’ sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘Ÿ ๐‘›+2 (๐œ… โˆ’ 1)(๐‘Ÿ 2 โˆ’ ๐‘… 2 )๐‘… ๐œŽ๐‘Ÿ๐œƒ (๐‘Ÿ, ๐œƒ) = โˆ’ sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐œ‹(๐œ… + 1)๐‘Ÿ 3 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› (๐‘Ÿ 2 โˆ’ ๐‘… 2 ) (D.10) + ๐‘›+2 sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘Ÿ ๐‘Ÿฮ”๐œƒ ๐‘…(2๐‘Ÿ 2 + (๐œ… โˆ’ 1)๐‘… 2 โˆ’ 4๐œ…๐‘… 2 ln(๐‘Ÿ)) ฮ”๐œƒ 2๐œ‡๐‘ˆ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = โˆ’ โˆ’ sin( ) cos(๐œƒ) 2๐œ‹ 2๐œ‹(๐œ… + 1)๐‘Ÿ 2 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› ((๐œ… + ๐‘› โˆ’ 1)(๐‘› + 1)๐‘Ÿ 2 โˆ’ (๐‘›2 โˆ’ ๐‘›)๐‘… 2 ) ๐‘›ฮ”๐œƒ โˆ’ 2 sin( ) cos(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› โˆ’ 1)๐œ‹๐‘Ÿ ๐‘›+1 2 (๐œ… โˆ’ 1)๐‘… 3 + 2(1 + 2๐œ… ln(๐‘…))๐‘…๐‘Ÿ 2 ฮ”๐œƒ 2๐œ‡๐‘ˆ๐œƒ (๐‘Ÿ, ๐œƒ) = โˆ’ 2 sin( ) sin(๐œƒ) 2๐œ‹(๐œ… + 1)๐‘Ÿ 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› ((๐œ… โˆ’ ๐‘› + 1)(๐‘› + 1)๐‘Ÿ 2 + (๐‘›2 โˆ’ ๐‘›)๐‘… 2 ) ๐‘›ฮ”๐œƒ + 2 sin( ) sin(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› โˆ’ 1)๐œ‹๐‘Ÿ ๐‘›+1 2 For unit tangential stress at ๐œƒ = 0, stresses and displacements inside the infinite plane of the 128 hole are following: (๐œ… + 3)(๐‘Ÿ 2 โˆ’ ๐‘… 2 )๐‘… ๐œŽ๐‘Ÿ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐œ‹(๐œ… + 1)๐‘Ÿ 3 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› (๐‘› + 2)(๐‘Ÿ 2 โˆ’ ๐‘… 2 ) + ๐‘›+2 sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘Ÿ ๐‘…((๐œ… โˆ’ 1)๐‘Ÿ 2 + (๐‘˜ + 3)๐‘… 2 ) ๐œŽ๐œƒ๐œƒ (๐‘Ÿ, ๐œƒ) = โˆ’ sin(ฮ”๐œƒ/2) sin(๐œƒ) ๐œ‹(๐œ… + 1)๐‘Ÿ 3 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› ((๐‘› โˆ’ 2)๐‘Ÿ 2 โˆ’ (๐‘› + 2)๐‘… 2 ) โˆ’ sin(๐‘›ฮ”๐œƒ/2) sin(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘›๐‘Ÿ ๐‘›+2 ฮ”๐œƒ ๐‘…((๐œ… โˆ’ 1)๐‘Ÿ 2 + (๐œ… + 3)๐‘… 2 )) ๐œŽ๐‘Ÿ๐œƒ (๐‘Ÿ, ๐œƒ) = + sin(ฮ”๐œƒ/2) cos(๐œƒ) 2๐œ‹ ๐œ‹(๐œ… + 1)๐‘Ÿ 3 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› (๐‘›๐‘… 2 + 2๐‘… 2 โˆ’ ๐‘›๐‘Ÿ 2 ) (D.11) + ๐‘›+2 sin(๐‘›ฮ”๐œƒ/2) cos(๐‘›๐œƒ) ๐‘›=1 ๐œ‹๐‘Ÿ ๐‘…((๐œ… + 3)๐‘… 2 โˆ’ 2(1 โˆ’ 2๐œ… ln(๐‘Ÿ))๐‘Ÿ 2 ) ฮ”๐œƒ 2๐œ‡๐‘ˆ๐‘Ÿ (๐‘Ÿ, ๐œƒ) = 2 sin( ) sin(๐œƒ) 2๐œ‹(๐œ… + 1)๐‘Ÿ 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› ((๐œ… + ๐‘› โˆ’ 1)(๐‘› + 1)๐‘Ÿ 2 โˆ’ (๐‘›2 + ๐‘› โˆ’ 2)๐‘… 2 ) ๐‘›ฮ”๐œƒ โˆ’ 2 sin( ) sin(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› โˆ’ 1)๐œ‹๐‘Ÿ ๐‘›+1 2 ๐‘Ÿฮ”๐œƒ ๐‘…(2๐‘Ÿ 2 โˆ’ (3 + ๐œ…)๐‘… 2 4๐œ…๐‘Ÿ 2 ln(๐‘Ÿ)) ฮ”๐œƒ 2๐œ‡๐‘ˆ๐œƒ (๐‘Ÿ, ๐œƒ) = โˆ’ โˆ’ 2 sin( ) cos(๐œƒ) 2๐œ‹ 2๐œ‹(๐œ… + 1)๐‘Ÿ 2 ๐ป๐‘Ž๐‘Ÿ๐‘š ร• ๐‘… ๐‘› ((๐œ… โˆ’ ๐‘› + 1)(๐‘› + 1)๐‘Ÿ 2 + (๐‘›2 + ๐‘› โˆ’ 2)๐‘… 2 ) ๐‘›ฮ”๐œƒ โˆ’ 2 โˆ’ 1)๐œ‹๐‘Ÿ ๐‘›+1 sin( ) cos(๐‘›๐œƒ) ๐‘›=1 ๐‘›(๐‘› 2 129 APPENDIX E THE MICHELL SOLUTION- STRESS AND DISPLACEMENT COMPONENTS ๐œ… is a Kolosovโ€™s constant, can be defined as following: ๏ฃฑ ๏ฃฒ 3 โˆ’ 4๐œˆ for plane strain ๏ฃด ๏ฃด ๏ฃด ๐œ… = 3โˆ’๐œˆ (E.1) ๏ฃด ๏ฃด for plane stress ๏ฃณ 1+๐œˆ ๏ฃด 130 Coef. ๐œ™ ๐œŽ๐‘Ÿ๐‘Ÿ ๐œŽ๐‘Ÿ๐œƒ ๐œŽ๐œƒ๐œƒ 2๐œ‡๐‘ข๐‘Ÿ 2๐œ‡๐‘ข ๐œƒ ๐ด01 ๐‘Ÿ2 2 0 2 (๐œ… โˆ’ 1)๐‘Ÿ 0 ๐ด02 2 ๐‘Ÿ ln(๐‘Ÿ) 2 ln(๐‘Ÿ) + 1 0 2 ln(๐‘Ÿ) + 3 (๐œ… โˆ’ 1)๐‘Ÿ ln(๐‘Ÿ) โˆ’ ๐‘Ÿ (๐œ… + 1)๐‘Ÿ๐œƒ ๐ด03 ln(๐‘Ÿ) 1/๐‘Ÿ 2 0 โˆ’1/๐‘Ÿ 2 โˆ’1/๐‘Ÿ 0 ๐ด04 ๐œƒ 0 1/๐‘Ÿ 2 0 0 โˆ’1/๐‘Ÿ ๐ด11 3 ๐‘Ÿ cos ๐œƒ 2๐‘Ÿ cos ๐œƒ 2๐‘Ÿ sin ๐œƒ 6๐‘Ÿ cos ๐œƒ (๐œ… โˆ’ 2)๐‘Ÿ 2 cos ๐œƒ (๐œ… + 2)๐‘Ÿ 2 sin ๐œƒ 1 1 ๐ด12 ๐‘Ÿ ln(๐‘Ÿ) cos ๐œƒ cos ๐œƒ/๐‘Ÿ sin ๐œƒ/๐‘Ÿ cos ๐œƒ/๐‘Ÿ (๐œ… + 1)๐œƒ sin ๐œƒ โˆ’ cos ๐œƒ (๐œ… + 1)๐œƒ cos ๐œƒ โˆ’ sin ๐œƒ 2  2  +(๐œ… โˆ’ 1) ln(๐‘Ÿ) cos ๐œƒ โˆ’(๐œ… โˆ’ 1) ln(๐‘Ÿ) sin ๐œƒ 1  1  ๐ด13 ๐‘Ÿ๐œƒ sin ๐œƒ 2 cos ๐œƒ/๐‘Ÿ 0 0 (๐œ… โˆ’ 1)๐œƒ sin ๐œƒ โˆ’ cos ๐œƒ (๐œ… โˆ’ 1)๐œƒ cos ๐œƒ โˆ’ sin ๐œƒ 2  2  +(๐œ… + 1) ln(๐‘Ÿ) cos ๐œƒ โˆ’(๐œ… + 1) ln(๐‘Ÿ) sin ๐œƒ ๐ด14 cos ๐œƒ/๐‘Ÿ โˆ’2 cos ๐œƒ/๐‘Ÿ 3 โˆ’2 sin ๐œƒ/๐‘Ÿ 3 2 cos ๐œƒ/๐‘Ÿ 3 cos ๐œƒ/๐‘Ÿ 2 sin ๐œƒ/๐‘Ÿ 2 ๐ต11 ๐‘Ÿ 3 sin ๐œƒ 2๐‘Ÿ sin ๐œƒ โˆ’2๐‘Ÿ cos ๐œƒ 6๐‘Ÿ sin ๐œƒ (๐œ… โˆ’ 2)๐‘Ÿ 2 sin ๐œƒ โˆ’(๐œ… + 2)๐‘Ÿ 2 cos ๐œƒ 1 1 ๐ต12 ๐‘Ÿ ln(๐‘Ÿ) sin ๐œƒ sin ๐œƒ/๐‘Ÿ โˆ’ cos ๐œƒ/๐‘Ÿ sin ๐œƒ/๐‘Ÿ โˆ’(๐œ… + 1)๐œƒ cos ๐œƒ โˆ’ sin ๐œƒ (๐œ… + 1)๐œƒ sin ๐œƒ + cos ๐œƒ 2  2  +(๐œ… โˆ’ 1) ln(๐‘Ÿ) sin ๐œƒ +(๐œ… โˆ’ 1) ln(๐‘Ÿ) cos ๐œƒ 1 1 ๐ต13 ๐‘Ÿ๐œƒ cos ๐œƒ โˆ’2 sin ๐œƒ/๐‘Ÿ 0 0 (๐œ… โˆ’ 1)๐œƒ cos ๐œƒ + sin ๐œƒ โˆ’ (๐œ… โˆ’ 1)๐œƒ sin ๐œƒ โˆ’ cos ๐œƒ 2  2  โˆ’(๐œ… + 1) ln(๐‘Ÿ) sin ๐œƒ โˆ’(๐œ… + 1) ln(๐‘Ÿ) cos ๐œƒ ๐ต14 sin ๐œƒ/๐‘Ÿ โˆ’2 sin ๐œƒ/๐‘Ÿ 3 2 cos ๐œƒ/๐‘Ÿ 3 2 sin ๐œƒ/๐‘Ÿ 3 sin ๐œƒ/๐‘Ÿ 2 โˆ’ cos ๐œƒ/๐‘Ÿ 2 Table E.1: The terms of Michell Solution (Airy Stress Function) and their corresponding stresses and displacements. Terms shown in the cyan are the self-equilibrating part of the solid disk stress function, terms shown in white are the self-equilibrating part of the Hole in a elastic plane, and terms shown in gray are modified secular terms removing their multi-value through linear combinations. 131 Coef. ๐œ™ ๐œŽ๐‘Ÿ๐‘Ÿ ๐œŽ๐‘Ÿ๐œƒ ๐œŽ๐œƒ๐œƒ 2๐œ‡๐‘ข๐‘Ÿ 2๐œ‡๐‘ข ๐œƒ ๐ด๐‘›1 ๐‘Ÿ ๐‘›+2 cos ๐‘›๐œƒ โˆ’(๐‘› + 1) (๐‘› โˆ’ 2) ๐‘›(๐‘› + 1) (๐‘› + 1) (๐‘› + 2) (๐œ… โˆ’ ๐‘› โˆ’ 1) (๐œ… + ๐‘› + 1) ร— ๐‘Ÿ ๐‘› cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘› sin ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘› cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›+1 cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›+1 sin ๐‘›๐œƒ ๐ด๐‘›2 ๐‘Ÿ โˆ’๐‘›+2 cos ๐‘›๐œƒ โˆ’(๐‘› + 2) (๐‘› โˆ’ 1) โˆ’๐‘›(๐‘› โˆ’ 1) (๐‘› โˆ’ 1) (๐‘› โˆ’ 2) (๐œ… + ๐‘› โˆ’ 1) โˆ’(๐œ… โˆ’ ๐‘› + 1) ร— ๐‘Ÿ โˆ’๐‘› cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘› sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘› cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›+1 cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›+1 sin ๐‘›๐œƒ ๐ด๐‘›3 ๐‘Ÿ ๐‘› cos ๐‘›๐œƒ โˆ’๐‘›(๐‘› โˆ’ 1) ๐‘›(๐‘› โˆ’ 1) ๐‘›(๐‘› โˆ’ 1) โˆ’๐‘›๐‘Ÿ ๐‘›โˆ’1 cos ๐‘›๐œƒ ๐‘›๐‘Ÿ ๐‘›โˆ’1 sin ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›โˆ’2 cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›โˆ’2 sin ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›โˆ’2 cos ๐‘›๐œƒ ๐ด๐‘›4 ๐‘Ÿ โˆ’๐‘› cos ๐‘›๐œƒ โˆ’๐‘›(๐‘› + 1) โˆ’๐‘›(๐‘› + 1) ๐‘›(๐‘› + 1) ๐‘›๐‘Ÿ โˆ’๐‘›โˆ’1 cos ๐‘›๐œƒ ๐‘›๐‘Ÿ โˆ’๐‘›โˆ’1 sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›โˆ’2 cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›โˆ’2 sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›โˆ’2 cos ๐‘›๐œƒ ๐ต๐‘›1 ๐‘Ÿ ๐‘›+2 sin ๐‘›๐œƒ โˆ’(๐‘› + 1) (๐‘› โˆ’ 2) โˆ’๐‘›(๐‘› + 1) (๐‘› + 1) (๐‘› + 2) (๐œ… โˆ’ ๐‘› โˆ’ 1) โˆ’(๐œ… + ๐‘› + 1) ร— ๐‘Ÿ ๐‘› sin ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘› cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘› sin ๐‘›๐œƒ ๐‘›+1 ร— ๐‘Ÿ sin ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›+1 cos ๐‘›๐œƒ ๐ต๐‘›2 ๐‘Ÿ โˆ’๐‘›+2 sin ๐‘›๐œƒ โˆ’(๐‘› + 2) (๐‘› โˆ’ 1) ๐‘›(๐‘› โˆ’ 1) (๐‘› โˆ’ 1) (๐‘› โˆ’ 2) (๐œ… + ๐‘› โˆ’ 1) (๐œ… โˆ’ ๐‘› + 1) ร— ๐‘Ÿ โˆ’๐‘› sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘› cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘› sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›+1 sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›+1 cos ๐‘›๐œƒ ๐ต๐‘›3 ๐‘Ÿ ๐‘› sin ๐‘›๐œƒ โˆ’๐‘›(๐‘› โˆ’ 1) โˆ’๐‘›(๐‘› โˆ’ 1) ๐‘›(๐‘› โˆ’ 1) โˆ’๐‘›๐‘Ÿ ๐‘›โˆ’1 sin ๐‘›๐œƒ โˆ’๐‘›๐‘Ÿ ๐‘›โˆ’1 cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›โˆ’2 sin ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›โˆ’2 cos ๐‘›๐œƒ ร— ๐‘Ÿ ๐‘›โˆ’2 sin ๐‘›๐œƒ ๐ต๐‘›4 ๐‘Ÿ โˆ’๐‘› sin ๐‘›๐œƒ โˆ’๐‘›(๐‘› + 1) ๐‘›(๐‘› + 1) ๐‘›(๐‘› + 1) ๐‘›๐‘Ÿ โˆ’๐‘›โˆ’1 sin ๐‘›๐œƒ โˆ’๐‘›๐‘Ÿ โˆ’๐‘›โˆ’1 cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›โˆ’2 sin ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›โˆ’2 cos ๐‘›๐œƒ ร— ๐‘Ÿ โˆ’๐‘›โˆ’2 sin ๐‘›๐œƒ Table E.2: The terms of Michell Solution (Airy Stress Function) and their corresponding stresses and displacements. 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