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THESIS Date 0-7639 August 11, 1981 :7 IIUW||H||WI||HI|7 ‘3‘ 3 1293 106993 "1,. wwvrwgfif 2' Wm?“ v» em sin-ante. wmfifi {an‘gf EV This is to certify that the thesis entitled THE NUMERICAL TREATMENT OF BODY FORCES, DISLOCATION FIELDS, AND ARRAYS OF CRACKS IN PLANE ELASTOSTATICS PROBLEMS presented by Gary J. Burgess has been accepted towards fulfillment of the requirements for Ph'D- degree in Mechanics Major pr fess r 0mm: nuts: 2“ per «In! par 1t- BETUIIIIIE IBM MTERIALS: P110. in book return to mow diam from circulatton neon THE NUMERICAL TREATMENT OF BODY FORCES, DISLOCATION FIELDS, AND ARRAYS 0F CRACKS IN PLANE ELASTOSTATICS PROBLEMS By Gary J. Burgess A DISSERTATION Submitted to Michigan State University in partiai fuifiIIment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of MetaTTurgy, Mechanics and MateriaTs Science 1981 I; ABSTRACT The numerical solution of plane elasticity problems in which the body is of arbitrary shape and is subjected to various loading conditions is first considered. The method employed is based on embedding the body of interest in the infinite plane and satisfying prescribed boundary conditions using a collection of concentrated loads acting outside the body. The procedure developed is then extended to include internal sources of stress such as body forces, fields of edge dislocations and their dipoles, isolated cracks, and finally dense arrays of cracks. In the treatment of crack problems, the actual crack boundary is replaced by a dislocation dipole which acts as a passive radiator of stress in the presence of an applied external stress field. Since each crack is given this status and since the strength of any given dipole depends on the stresses produced by its neighbors and by boundary tractions, there is a high degree of interaction between dipoles. The approach yields a system of matrix equations to be solved for the dipole strengths which are subsequently used to determine the state of stress anywhere within the body. Comparison of numerical results to known solutions indicates that the dipole model of a crack is an accurate one for most applications and that the numerical methods developed perform satisfactorally. This work should therefore be useful in the study of damage due to progressive crack growth in engineering materials such as metals and rock. Copyright by GARY J. BURGESS 198l ACKNOWLEDGEMENTS I would like to thank the members of my doctoral committee, Professors Nicholas J. Altiero, George E. Mase, Robert Wm. Little, and Albert N. Andry for contributing to the academic background necessary for the preparation of this dissertation and for the accommodating atmosphere they provided during the course of my research. I would also like to thank Ms. Jan Swift for her haste in preparing this manuscript on such short notice. Welcomed financial assistance was provided for by the Department of Metallurgy, Mechanics, and Materials Science, the Division of Engineering Research, and by the National Science Foundation (NSF Grant # ENG—76-l8355). ii TABLE OF CONTENTS LIST OF TABLES ........................................... v LIST OF FIGURES .......................................... vi LIST OF SYMBOLS .......................................... viii CHAPTER 1 ................................................ l INTRODUCTION ........................................ l CHAPTER 2 ................................................ 6 THE NUMERICAL SOLUTION OF ELASTICITY PROBLEMS BY THE SUPERPOSITION METHOD ......................... 6 2.l Background ................................ 6 2.2 The Superposition Method in the Absence of Body Forces .................... 15 2.3 The Superposition Method in the Presence of Body Forces and Other Sources of Internal Stress ................ 21 2.4 Fields of Edge Dislocations and Their Dipoles ............................. 29 CHAPTER 3 ................................................ 42 THE NUMERICAL SOLUTION 0F CRACK PROBLEMS ............ 42 3.l The Modelling of Cracks by Dipoles ........ 42 3.2 Solutions to Crack Problems ............... 54 CHAPTER 4 ................................................ 65 APPLICATIONS AND NUMERICAL RESULTS .................. 65 4.l The Basic Computer Program ................ 65 4.2 The Inclusion of Body Forces .............. 72 4.3 The Inclusion of Edge Dislocations and their Dipoles ......................... 80 4.4 Analytic Solutions to Crack Problems ...... 8T 4.5 Numerical Solutions to Crack Problems ..... 90 iii CHAPTER 5 ................................................ 94 CONCLUSIONS ......................................... 94 APPENDICES ............................................... 97 APPENDIX A - THE EVALUATION OF FOUR BASIC AREA INTEGRALS ...................................... 97 APPENDIX B - DIPOLE TRANSFORMATIONS ................. 109 APPENDIX C - THE EVALUATION OF FOUR BASIC LINE INTEGRALS ...................................... 116 APPENDIX D - THE COMPUTER PROGRAM FOR THE SUPERPOSITION METHOD ................................ 123 APPENDIX E - THE COMPUTER PROGRAM FOR BODY FORCES...124 APPENDIX F - THE EDGE DISLOCATION SUBROUTINE ........ 126 APPENDIX G - THE COMPUTER PROGRAM FOR CRACK PROBLEMS ............................................ 127 LIST OF REFERENCES ....................................... 129 iv LIST OF TABLES 4.1 Numerical results for the beam problem ..... ........ 71 4.2 Numerical results for the rotating disk problem ....79 4.3 The results for the problem of Figure 4.4 ........... 93 2.l LIST OF FIGURES Concentrated loads in the infinite plane ............ 9 Edge dislocations in the infinite plane ............. ll Transformation Properties of edge dislocations ...... 13 Embedding the body in the infinite plane ............ 16 Embedding the body in a rectangle in preparation for its Fourier series representation ............... 24 Normal and shear dipoles in the infinite plane ...... 34 A slit crack and its dislocation equivalent ......... 43 The self-stress field at point P .................... 46 A crack and its dipole model ........................ 51 A rotated crack and its dipole model ................ 52 A beam loaded on all faces .......................... 69 The rotating disk body force problem ................ 76 An infinite array of cracks ......................... 82 A square block containing five identical cracks at symmetrical locations in a state of uniform stress .............................................. 92 The region of integration broken up into four parts ............................................... 98 The normal-shear dipole and its equivalent .......... 110 The rotated normal dipole and its equivalent normal—shear dipole pair ............................ lll Transformation properties of the normal dipole ...... 113 Transformation properties of the shear dipole ....... 114 Integration in the complex plane .................... 120 vi LIST OF SYMBOLS Area Burger's vector in the i-direction Body force in the i-direction Crack half-length Matrix with m rows and n columns Fourier coefficients Dislocation dipole strengths Modulus of elasticity Resultant forces at point i Shear modulus Dipole spacings Unit vectors in the x, y, 2 directions Integral with parameter k Auxilliary integral Constants related to dislocations and dipoles Mode I and II stress intensity factors Lengths in the x and y directions Indices in Fourier series Concentrated load in the i-direction Terms used in the influences due to Fourier dis- tributions of internal stress Relative distance between source and field points vii Olj Olj 1>ss Shape factor for grains Traction vector in the i-direction Displacements in the x and y directions Displacements or integral substitutions Column vector containing n rows Rectangular coordinates Coordinates in a rotated system Constants Dislocation density along a line Dipole density in an area Gradient and Laplace operators Crack density in a granular medium Engineering strains Angle between x and R axes Poisson's ratio The ij stress component Dislocation dipole stresses Summation symbol Shear stress Airy stress function or special integral Rigid body rotation or angular velocity Integral over area A viii dislocation Dislocation dipoles 516952 Superposition of states 1 and 2 ix CHAPTER I INTRODUCTION The linear and nonlinear theories of elasticity, plasticity and fracture mechanics are cemposed of two parts: the develop— ment of constitutive equations describing the material response to applied loads and the determination of the stress fields resulting from an application of these equations to boundary value problems of interest. This dissertation deals with the latter. Beginning with Chapter 2 the numerical solution to linear elasticity problems with emphasis on efficiency is addressed and the method developed is then extended to include body forces, fields of dislocations, dislocation dipoles, isolated cracks and finally a dense array of microcracks. Common to all solution methods in problems of plasticity, nonlinear elasticity and frac- ture mechanics is the need for an accurate assessment of the stress fields arising from distributed internal sources of stress, such as a body force field. Also common to these methods is the incre- mental nature of the approach. Consequently, the basic process of solving a linear elasticity problem must be performed repeatedly, requiring the numerical procedure to be both accurate and efficient. In classical plasticity theory, a rearrangement of the elas- tic—plastic Navier equation shows that the plasticity problem can be regarded as an elasticity problem with a suitably modified body force [1,2]. The body force carries the effects of plasticity for a given set of constitutive relations and is typically not known apriori since it is expressed in terms of the displacement field. The same is true of nonlinear elasticity problems [3]. The nonlinear terms occupy the same position in the Navier equation as does the body force term. In both plasticity and non-linear elasticity, the determination of the body force distributions for a given loading system is typically done iteratively. The load is then given an increment and the procedure is repeated until some predetermined loading program is completed. It is therefore necessary to solve a series of elasticity problems with body forces for each increment of the load. Attempts to obtain the stress field due to body forces by replacing the distribution with a collection of point loads [1], gives poor results near the points of application of these loads due to the singular nature of the point load solution [4]. The same is true of quadrature rules used in numerical integration techniques [2] which are aimed at smearing the point load solution over polygon-shaped areas. The finite element approach [5] of dividing the body into elements and assigning some simple distri— bution to the body force over each element is cumbersome and numerically inefficient. In Chapter 2 it is shown that the problem can be treated very effectively by characterizing the body force field by a set of Fourier coefficients which are then used to determine the stress field at any point from simple expressions. Equally important is that the integrals associated with the smearing of point loads over an area to obtain a body force distri- bution are exactly the same as those required to smear dislocations and dislocation dipoles over an area. In problems where the dis- location distribution is known, the methods of Chapter 2 allow for the calculation of the associated stress field with a minimum of effort. A known distribution however is likely to be the exception rather than the rule. As with plasticity and nonlinear elasticity, the distribution is usually determined iteratively by satisfying some kind of constitutive criterion [6]. Nye [7] and Eshelby [8] have independently developed continuum theories of dislocations but both are described by a set of partial differential equations which are unlikely to be solved in even the most elementary problems. In fracture mechanics [9], the prediction of stress levels at which crack growth takes place is of primary importance. The deter- mination of stress intensity factors [5, 9, 10, 11, 12] is aimed at accomplishing this. The subsequent problem of following the pro— gression of crack growth is an area of current research [13, 14, 15, 16, 17, 18, 19]. Fundamental to the success of this research is the ability to characterize the stress field of a collection of cracks with a minimum of effort. It has been customary to focus attention on one or two cracks in simple geometries under conditions of uniform stress [9, 10, ll, 12]. The solution to the problem of a single crack in uniaxial tension in the infinite plane is no simple matter and therefore points to the need to develop unconventional methods of solution. In Chapter 3 a dislocation dipole model for a crack [6] is proposed which simplifies matters considerably while preserving the general features of a crack and eventually allows for the solution of general boundary value problems in which the body contains an arbitrary array of cracks with varying sizes, orientations and locations within the body. The application of this model to problems with known solutions indicates that it compares favorably with more refined models and requires very little effort to apply numerically. Ultimately, this model is used in conjunction with the results of Chapter 2 to treat the limiting case of a dense array of microcracks characterized by a 'microcrack density'. It is hoped that, when used with consti- tutive models of crack growth (yet to be developed), these results will prove to be an integral part of the problem solution process. No attempt has been made to develop or incorporate already existing crack growth models into the material presented here. Efforts to classify materials according to their behavior in the realm of progressive crack growth [13, 14, 15, l6, 17, 18, 19, 20] indicate that this stage of the problem needs further development. Dragon and Mroz [14] come closest to defining the state of the art approach to crack problems. In their approach, they use an abstract quantity intro- duced by Vakulenko and Kachanov [21] called the crack tensor and pursue a potential theory development similar to that used in plasticity. Their development, however, is largely phenomenological and lacks justification in many places. The crack tensor idea is nevertheless an advantageous contribution and is very similar to the'dipole state' used here to represent a crack field. In view of the successes met with in the example problems presented in Chapter 4, this type of approach looks promising for future use in the incremental treatment of crack growth problems. For the sake of clarity, the methods presented in the following chapters deal with the treatment of body forces, dislocations and cracks individually. The versatility of the numerical approach must however be emphasized; there is no res— triction preventing these methods from being used in concert to treat the general problem of elastic-plastic-dislocation-fracture mechanics. CHAPTER 2 THE NUMERICAL SOLUTION OF ELASTICITY PROBLEMS BY THE SUPERPOSITION METHOD 2.1 Background The governing equations of linear elastostatics applied to a homogeneous isotropic body in a state of plane stress are [4] the equilibrium equations, 1.. + 7?: + 03 II C U (2.1) II C) ox + 0y) = -(l + v) V -B} (2.2) _ l EX —'E_ (OX ' VOy), _ l_ _ 6y — E (0y vo ), (2.3) .1, ny G xy’ and the strain-displacement relations, 6.8;: x ax’ av = __, 2.4 8y 3y () —3_U§l ny — 3y + 8x’ where ox, Oy’ Ixy are the stresses; Bx’ By are the body forces; ex, 6y, ny are the strains; u, v are the displacements; E, G, v are the usual material properties. In what follows, the body will always be assumed to be in a state of plane stress. The interested reader can easily convert any results to satisfy plane strain conditions. Any stress field, (OX, 0y: Txy compatibility is a solution to some boundary value problem in ), which satisfies both the equilibrium equations and elastostatics. Satisfaction of compatibility insures the inte- grability of Equations (2.4) to obtain a displacement field, (u, v). Of particular importance in the theory of elasticity are the principle of superposition, which follows from the linearity of the governing equations, and the uniqueness theorem due to Kirchhoff. The principle of superposition allows one to construct new solutions from linear combinations of old solutions. The solution to a typical problem in elasticity involves the satis— faction of the governing equations at every point inside the body and the matching of boundary conditions at every point on the boundary of the body. The theorem of Kirchhoff guarantees unique- ness of the solution provided that the boundary conditions are of the form usually encountered, (2.5) where Bt and Bu are those parts of the boundary on which tractions, t:, and displacements, u:, are prescribed, respectively. Conse- quently, a solution obtained by any means is the only solution. Two basic solutions, the concentrated load and the edge dislocation solutions, will be used in what follows. Figures 2.1a and 2.lb show concentrated loads of magnitudes PX and P acting at the origin in the infinite plane in the absence of body forces. The resultant stress and displacement fields shown in matrix form are u Lube ln r + L2] x G l+v r2 u _ L . 5.x .v 46 F2 P (l+v) 2 _ X X X Ox ‘ ‘ T' "2‘ (B + ‘2‘) (2-66) Y‘ Y‘ x y2 __ _ + __ 0y l"2 ( 8 r2) y X2 Txy ”2—(B+—’2‘) a. A concentrated force acting in the x direction. b. A concentrated force acting in the y direction. Figure 2.1. Concentrated loads in the infinite plane. l0 and I ll ux ' 4G'r2 2 l 3-v x uy fig [(TTU7 1" F + ;§ P (1+v) 2 = __l____._. y _ X 0x 2n ‘ ‘2” 8 + '7) (Mb) r r y yZ o —(B+—-) y r2 r2 x y2 Txy ;§‘(B + :f) 9 respectively, where (2.7) An edge dislocation is given the symbol,J.. Operationally, the dislocation is formed by slicing the infinite plane along a line and inserting a slab of material of width b into the cut. The stem of the.J_refers to this slab of material. Figure 2.2 shows dislocations of various orientations situated at the origin in the infinite plane. The accompanying 'Burger's vectors', 9, represents the net discontinuity in the elastic displacement taken counterclockwise around a closed circuit enclosing the leading edge of the inserted slab [6]: in Figure 2.2, the leading a. An edge dislocation with Burger's vector, bx' b. An edge dislocation with Burger's vector, by. Figure 2.2. Edge dislocations in the infinite plane. 12 edge lies along the z axis. An excellent discussion of edge dis- locations and their stress and displacement fields is given in [6]. The stress fields corresponding to Figures 2.2a and 2.2b are 2 2 = , y(3x +y) 0x Kx 4 ’ r y<) , xy y r4 where be Kx = 4F_ ’ (2.8c) E K = —1. y 4n It can easily be shown that the so—called Burger's vector is in fact a vector. Referring to Figure 2.3a, the proof is effected by writing out the stress field for this rotated dislocation l3 a. A rotated edge dislocation with Burger's vector, b. (b) (b-cose) (bosine) b. The vector nature of the 'Burger's vector'. Figure 2.3. Transformation properties of edge dislocations. l4 referred to the Ely system, —2 —2 _ ,y(3x +y) O__K _4 9 Y‘ _: K,§(T2 -Y2) (29a) 0 __4 ’ . Y‘ ——2 —2 T—=K'X( 'X) xy F4 ’ where K = 45" This state of stress may now be transformed back to the x—y system. For example, with sin e = s and cos a = c, _ 2 2 ox — o;c + oys — 21;; sc. (2.9b) Substituting Equations (2.9a) into (2.9b) and rearranging gives K [—3 X2 _ — 22 ——2—2 oX — F7? y + y (3c —s ) + 2x(x —y ) sc]. (2.9c) Finally, using the coordinate transformations, >< ll xc + ys, ‘ Actual Body Congruent Boundary Figure 2.4. Embedding the body in the infinite plane. l7 3) Satisfy boundary conditions by adjusting load magnitudes so that the superposed effect of all 2n loads acting in concert gives the desired conditions. Since there are 2n unknown magnitudes, one has the capability of satisfying boundary conditions at n points on the boundary. For simplicity, the concentrated loads are arranged so as to act on a contour which is congruent to the actual boundary at a distance 5 from it. The parameter s is arbitrary and can be used to prevent C_from becoming too ill-conditioned. Let w be the column vector containing the magnitudes of the 2n concentrated loads and let Crbe the square matrix whose element, jj,lrepresents the contribution of the jth unit concentrated load to the ith boundary condition. Finally, let b_be the column C vector containing the prescribed values of the Zn boundary condi- tions. The superposition of the Zn concentrated loads to satisfy the boundary conditions is represented by b =_C w, (2.10a) and if C'is nonsingular, b. (2.10b) The stresses and displacements at any point in the body can be calculated by summing the contributions due to each of concen- trated loads, the magnitudes of which are now known. 18 The numerical procedure just described solves an elasto- statics problem in the sense that equilibrium and compatibility are satisfied at every point inside the body and boundary condi- tions are satisfied at a finite number of points on the boundary. The approximation therefore lies in the neglect of boundary con- ditions not directly accounted for in the matching technique. Refinements in the solution may therefore be made by considering more points at which conditions are to be satisfied. This of course necessitates more concentrated loads so that the size of the matrix, C, and hence the amount of computation increase considerably. An alternative procedure allows for the satisfaction of an arbitrary number of boundary conditions in a least squares sense while maintaining a constant matrix size. The problem is still defined as yet, where C_now has 2N rows and 2h columns with N > n. Since C>is nonsquare, the inverse of C_does not exist and there is no way in general to choose the Zn unknowns in w_so as to satisfy the 2N conditions in 9, Therefore, for a given w, one may define the l'lOYlZEY‘O EY‘Y‘OY‘ VECtOY‘, that 6(E2) = 3T6__+ 6_Tg_= 29T62_= 0. However, 6§_= - §_§w, (2.13) so that which must hold for arbitrary 5w, Therefore, uT—E§m=0, or, transposing, ££E=JE- can In retrospect, this merely amounts to premultiplying the original' equation, 20 Equation (2.14) has the property that §:§_is square so that w’is uniquely determined by E: <29) 22- (2.15) It should be remarked that the least squares approach may not be desirable in certain cases. If, for example, the boundary conditions vary rapidly over small distances, this approach will tend to smooth out these fluctuations giving conservative values in cases where accurate values are desired. As a final note, one may generalize the type of boundary condition prescribed to the form auX + buy + coX + doy + 8Txy = f, (2.16) where the constants a, b, c, d, e, and f are specified. The matrix element, ij, is then the influence given by the left hand side of Equation (2.16) at the boundary point i due to a unit concentrated load applied at point j outside the body. The right hand side, f, appears in the ith row of the column vector, b, Of course this type of boundary condition forces one to forfeit cer- tainty about the uniqueness of the solution, but is nevertheless useful for contact problems and problems in which the boundary is subject to friction and spring loading. 2l 2.3 The Superposition Method in the Presence of Body Forces and Other Sources of Internal Stress The presence of body forces does not seriously complicate the numerical solution to an elastostatics problem. The procedure is as follows. 1) Embed the body in the infinite plane of the same material and let the body forces act. 2) Determine the stresses and displacements on the boundary of the embedded body caused by these body forces. 3) The difference between these boundary values and the pres— cribed boundary conditions is then eliminated by adjusting the magnitudes of the concentrated loads outside the body. This procedure gives a unique solution in the sense that the equations of elasticity with body forces are satisfied at every point inside the body while boundary conditions of the type men— tioned in the uniqueness theorem are partially satisfied on the boundary. The stresses and displacements at any point inside the body can be calculated by superposing the effects of the body force field and the concentrated loads, the magnitudes of which are known at the conclusion of step 3. It is important to note that the solution inside the embedded body is unique. The difference referred to in step 3 is certainly dependent upon the body forces acting outside the body but this difference is compensated for by the selection of the concentrated load magnitudes. 22 The only difficult part of this procedure is the calculation of the stress field produced by an arbitrary distribution of body forces (step 2). One may approximate the continuous body force field by a set of concentrated loads acting at various locations throughout the body, but due to the singular nature of the con— centrated force solution, the stresses and displacements near the points of application of these loads are highly exaggerated and even diverge at the points themselves. For the sake of accuracy, the only alternative is to continuously distribute infinitesimal point forces over the body. For example, using Equations (2.6), the stress, ox, at the point (a,b), caused by the body force, Bx(x,y), acting inside of an area, A, embedded in the infinite plane is, by superposition, ox(a,b) = f _ %%X_(1+V)(a-x)(l—v . l + (a—x) A r r where and dPX = BX(x,y) dA. With BX known, (2.17) As mentioned earlier, the field, Bx(x,y), may be allowed to act over the entire infinite plane without affecting the solu- tion to the elastostatics problem, provided that the stress and displacement fields caused by this do not diverge as a result. To insure convergence, one may assign to Bx the desired value inside the body and any value whatsoever outside which does not accumulate influences to the point of divergence. Such a dis- tribution may, for example, be the Fourier series representation for B , X 1] L L B (x,y) = Z Z c.. cos 115-cos All . (2.18) i J' x y Here, the body is embedded in a rectangle as shown in Figure 2.5 and the Fourier coefficients, Cij’ are evaluated by point matching or by Ly Lx - - c1. = L f f f Bx(x,y) cos %15 cos %11 . (2-19) 3 x y o 0 X y It is not necessary to place the rectangle in the first quadrant but it is imperative that it be placed in one of the four quadrants to the exclusion of all others since this distribution is an even function in both x and y and consequently forces BX(x,y) to be even Figure 2.5. Embedding the body in a rectangle in preparation for its Fourier series representation. The completion of the stress field produced by Bx and By follows in the same manner, altogether requiring the evaluation of four basic integrals, Ik -mf a-x 3;: b' )k cos mx - cos ny dA, (2.21) where k = 0, l, 2, 3 61nd r2 = (ax-x)2 + (my)? Note that m and n need not be integers. These integrals are difficult to evaluate and appear in Appendix A. From Appendix A, 26 Io- (g—+l)a, I - (1-1)b, l 77 (2.22a) 12 = (7' ])a: I3=(-721+1)b, for m=n=O. Otherwise, 2 2 I0 = ImAg—igflél-sin ma - cos nb, (m +n) _ nn(n2-m2) . I1 — ——§——§—§—-cos ma - Sin nb, (2.22b) (m Hi) 2 2 I = EEKEL431—A sin ma ~ cos nb, 2 (m2+n2)2 2 2 I3 = IDI%_i%E§l.COS ma - sin nb. (m +n) The stress, ox(a,b), due to constant body forces, Bx = c00 Eilnd By = doo’ acting over the entire infinite plane is, from Equa- t;ion (2.20), _ l—v l Ox(a’b) _ -‘__— C [1+v ' 2'(Io+12) + Io:l (2.23) 1 2h 00 [' ‘13'2‘(11+I3) + 117’ where the values for the four integrals are taken from the m=n=0 case of Equation (2.22a). After simplification, this becomes Performing similar operations for the remaining stresses and making the interchanges, a+x and b+y, the stress field at the point (x,y) due to constant body forces is _ l+v l+v Ox " ' 2[Coo(1+ TV + doo (V ' TUJ’ o=—ltc (v-flwd (Ml—3m (224) y 2 00 n 00 n ’ ' _ l l+v Txy - ' 2'( 7"F_) (Cooy + doox) 1’1: is a simple matter to verify that both equilibrium and com- F>£a.tibility are satisfied. The extension of these results to body forces of the form B = 5 mx - cos n x Cmn CO y (2.25) By = dmn cos mx - cos ny acting over the entire infinite plane follows in exactly the same manner. The complete stress field at the point (x,y) generated by these body forces for m and n not both zero is 28 _ 2 2 2 2 . Ox - R[m(m + (2 + v)n )cmn - P + n(vn —m )dmn Q] o = R[m(va-n2)c - p + n(n2 + (2 + v)m2)d . Q] (2 26) y mn mn ’ ' _ 2 2 2 2 . Txy — R[n(n -vm )cmn Q + m(m - vn )dmn P], where P = sin mx - cos ny, Q = cos mx - sin ny, l R — — ——————-——. (m2+n2)2 lngain, it is easy to show that both equilibrium and compatibility £3 r‘e satisfied. The body forces are now completely characterized tDJV' their Fourier coefficients, cm and dmn’ and the state of n 55 1: ress associated with these is given by Equations (2.24) and ( 23 .26). The displacement field can be obtained by inserting the SS‘tLresses into Equation (2.3) to obtain the strains and then i ritegrating the strain-displacement relations of Equation (2.4) 'Fc3r the displacements, u = — R(;+V) [(m2(l-v) + 2n2>cmni + nmdmn51 - my + d, (2.27) v _ _ R(E+V) [mn(i+v)cmns + (n2(l-v) + 2m2)dmnT] IF; 29 where R is defined as in Equation (2.26), and S = sin mx - sin ny, T = cos mx . cos ny, w, a, B are arbitrary constants. One may choose w so that the rigid body rotation is zero and a and £3 so that the displacements at the origin are zero, 0. E n a = R(1E+v) [m2(l-v) + 2n2]cmn, (2.28) B = R(E+V) [n2(l-V) + 2m2]dmn. 23 - 4 Fields of Edge Dislocations and Their Dipoles Fields of dislocations can be handled in a manner similar to Tltiat used for body forces. Ascribe to the field a Burger's \Iector density given by = s - os n dA cmn co mx c y, (2.29) I_l_= . dA dmn cos mx cos ny. 3O Recalling the edge dislocation solution given by Equations (2.8), the stress, ox, at the point (a,b) due to this field acting over the infinite plane is E_ 4n (b-y)(3(a-x = ) . ox(a,b) ob r cmn cos mx cos ny dA . cos mx - cos ny dA, 4n°° r4 mn or ox(a,b) = E? [cmn(311+13) + dmn (IE-10)]. (2.30) For a constant density, m=n=O, and this reduces to E ox(a,b) = g} [(V'I)Coo - dooa]. (2.31) (3therwise, 2 ox(a,b) = ——%fl—§—§ [ncmn cos ma - sin nb - mdmn sin ma - cos nb]. (m +n ) (2 32) The remaining stresses, oy and Txy’ are obtained in a similar manner. The complete state of stress at the location (x,y) generated by the dislocation field described by Equation (2.29) acting over the entire infinite plane is l" for m=n=0, and _ 2 o - Rn [ncmnP — mdan], x _ 2 oy — Rm [ncmnP - mdan], (2.33b) Txy = Rmn[-ncan + mdmnP], 1=c>r'm and n not both zero, where E R = —_—— , (m2+n2)2 P = cos mx - sin ny, (2.33c) Q = sin mx - cos ny. Equilibrium in the absence of body forces is satisfied at every point whereas compatibility is satisfied nowhere since V (0 +0 ) = -E(ncmnP - mdan) (2.34) It is not surprising that compatibility is violated in view of the fact that the hallmark of a dislocation is its discontinuous elastic displacement field. Nonetheless, the significance of Equation (2.34) is not understood. Of course, violation of com— patibility destroys the integrability of the strain-displacement relations so that the stresses are the only influences obtainable, —y except in the case where D is irrotational, V'x D = O, (2.35) for which V2(ox+oy) = O For this case, a displacement field exists. Closely related to the edge dislocation is the dislocation dipole, which is useful in the modelling of cracks. There are two basic dipoles, the normal and the shear dipoles. Both are the limiting cases of two edge dislocations of equal and opposite Burger's 33 vectors approaching each other in such a way that the product of their separation and Burger's vector remains constant. Figure 2.6 shows normal and shear dipoles at various orientations. In Figure 2.6a, let o(x,y) be the state of stress at the point (x,y) generated by the edge dislocation with the positive Burger's vector, bx’ situated at the origin. Let o*(x,y) be the combined effect of this dislocation and its opposite, located at a distance, hy, above it. Then 0*(X9y) = 0(X9Y) ' o(x,y-h ) h U(X,Y), (2'36) y yW as hy+0. The stress field, o*(x,y), is defined to be that generated by the normal dipole of Figure 2.6a in the limiting case, hy»0 and bx+m, in such a way that the product, dx=bxhy’ remains constant. This product is termed the strength of the dipole. The important result here is that the stress field, 0*, of the dipole can be obtained from that of the edge dislocation by replacing the Burger's vector, bx’ occuring in the stress field, 0, by an operator, dX 333 and per- forming the indicated operation. Symbolically, (b +d 8—) » (o+o*), (2.37) x X By which is read: the replacement of bX by d generates the stress x57 field 0* from 0. Similar operations, listed symbolically in Figure 2.6, hold for shear dipoles. Upon performing the operations indicated in Figure 2.6, the following stress fields are obtained: 34 (b »d 8—)+(o+o*> a. A normal dipole With dX = bxhy' h (b 'P-dy—X)—>(0—+O*) b. A l d' l 'th d = b h . norma ipo e Wl y y x .___. _> .+ * xy 8x) (0 O ) d. A shear dipole with d = b h . XX y y Figure 2.6. Normal and shear dipoles in the infinite plane. for the dipole shown in Figure 2.6a: _ 4_ 22 4 a;—Ky(x 6xy HI). 4 2 2 4 *= + _ 0y Ky(X 6xy 3y), _ . 2_ 2 T;y - Ky 2xy (x 3y ), Ed where K = _._SXE y 4hr for the dipole shown in Figure 2.6b: 2 2 x ny - 2xy (y - 3x ) Q FI- II o; = K - 2xy (X2-3y2), Xy 1* = K - (x4 - 6x2y2 + y4), (2.38b) (2.38c) de Ed x where KX = — ———%— or - __XE y 4nr 4hr for the dipoles shown in Figures 2.6c or 2.6d. If a dipole is not located at the origin of coordinates, then the x and y coordinates appearing in these equations are relative coordinates, x = xP — XD’ y = yp ' yD: where (xP,yP) and (x0, yD) are the coordinates of the point P, at which the stress field is desired, and of the point D, at which the dipole is located, both referred to some global coordinate system. Evidently, 2.2 EX _ Bxp’ and 2.22 By " Byp’ so that there is no ambiguity in the operation, dX gy, if the relative y coordinate is replaced by the global y coordinate of the point P, irrespective of the location of the dipole. It is for this reason that the stress field due to the dipole density, X = cos mx - cos ny, (2.39a) 37 acting over the infinite plane can be calculated easily from Equations (2.33) by replacing the scalar, Cmn’ by the operator, 3 Cmn 5y , with dmn = 0. Having done this, cmn is associated with the dipole density of Equation (2.39a). The smearing of dipoles over the infinite plane is equivalent to the smearing of edge dislocations with their Burger's vectors replaced by the operators described in Figure 2.6. Therefore, this trick merely takes advantage of the interchangeability of the order of integration involved in the smearing process and partial differentiation involved in the process of generating dipole stress fields from dislocation stress fields. For the distribution of normal dipoles whose density is given by Equation (2.39a), the corresponding state of stress is 0x = 2E-'Ecoo’ o =Lc (2 39b) y 2h 00’ ' T = 0, XY for m=n=O, whereas for m and n not both zero, _ 2 0X Rn cmn cos mx cos ny, 2 : I 2. oy Rm cmn cos mx cos ny, ( 39c) : 7n o . Txy Rmncmn Sl mx Sln ny, 38 2 En where R = —-——-——-. (m2+n2)2 The normal and shear dipoles of Figures 2.6b and 2.6c can also be continuously distribued over the infinite plane with densities similar to that of Equation (2.39a), with dX replaced by dy and dxy respectively. The corresponding stress fields are obtained by replacing cm and dmn in Equations (2.33) by the operators n 3 _ dmn + 'dmn 5;- and cmn - O, (2.40) and c + -c §—-and d = 0 mn mn 3x mn ’ thereby obtaining for the distribution, d(d ) _dfix_': dmn cos mx - cos ny, (2.4la) E 0x _ 2F'doo’ O = 3:1.Ed (2-4lb) for m=n=0. For m and n not both zero, With R = 7—237. m+n For the shear dipole distribution with density J: C COS "1X ‘ COS ny, =E_ Txy 2n Coo’ for m=n=0, and for m and n not both zero, _ 2 . . ox Rn cmn Sln mx Sln ny, o = Rmzc sin mx - sin n y mn y, Txy — RmnCmn cos mx - cos ny, with R = Em“ (m2+n2)2 ' (2.42a) (2.42b) (2.42c) 40 Of course, for the stress fields of Equations (2.39c), (2.4lc), and (2.42c), equilibrium is satisfied at every point in the infinite plane while compatibility is satisfied nowhere. Both the normal and shear dipole may be rotated through an arbitrary angle and the resulting stress fields calculated. The topic of dipole transfor- mations is taken up in Appendix B and is later referred to in the discussion of crack fields. The numerical procedure for the solution of elasticity problems described earlier can now be extended to include sources of internal stress other than body forces, such as edge dislocation fields and their dipoles. As with body forces, the field is first charac- terized by a set of Fourier coefficients chosen to represent the field inside the body without regard to what it represents outside. These coefficients are then used to ascertain boundary conditions already present as a result of the field, and the concentrated loads acting outside the body are used to adjust these boundary conditions to those desired. The resulting interior stress field is obtained by superposing the separate effects of the concentrated loads and of all sources of internal stress. The only restriction placed on this method when used with dislocations and dipoles is that dis- placement boundary conditions may not be prescribed. Violation of compatibility prohibits the integration of the strain—displacement relations so that displacement fields for distributions of dislocations and dipoles are unobtainable. This inadequacy can be attributed to the fact that a continuum of dislocations is physically unattainable. The initial continuum into which the body is embedded is altered by 4l the introduction of dislocations which add and subtract material at every point, thereby creating a new continuum. The preceding results are nevertheless useful, for if a large but finite number of dis- locations are known to exist in a given area, the stress field caused by these is much more easily and accurately approximated by proceeding to the limit and assigning a density to this collection than by treating each dislocation individually. Furthermore, since the displacement field will oscillate rapidly inside this area and since uncertainties in the displacement field due to the approxi- mations used in the linear theory of elasticity are on the order of magnitude of the displacements themselves [6], the value of such results is questionable. CHAPTER 3 THE NUMERICAL SOLUTION OF CRACK PROBLEMS 3.l The Modelling of Cracks by Dipoles Edge dislocations can be used advantageously to model cracks in both tension and shear. Figure 3.la shows a slit crack of length 2c situated in an infinite plate in uniform tension, 0, applied at infinity. Figure 3.lb shows the same plate with the crack removed and a line distribution of infinitesimal edge dislocations in its place. The distribution can be made to model the crack by requiring that the total stress field satisfy traction free boundary conditions along the portion of the x axis, |xl 2*= 9 (4.15) The expected form of the resolved stresses is 0*T = (0,0*, 0. 0. 0*, 0, 0. 0*.0...), (4.l6) which indicates that only the normal dipoles of the type shown in Figure 2.6b are needed to model these cracks. Now from Equations (2.38b) the stress produced by such a dipole located at the origin at the point (x,0) is O._.Ed_yi_ y 40 X2 The same dipole located at the point (x2,0) produces a stress at the point (x],0), Ed Oy:——4_i/T.—12—’ (4.17) X 12 where x12 = x2—x]. Taking advantage of symmetry and using the labelling of Figure 4.3a, the 0 matrix used in Equation (4.l3) has the form, 0,0,0 0,2x;§,0 0,2 {5,0 . Q=-E— 000 000 02x'20 __ 4W 9 9 9 9 a 239 - 0,0,0 0,0,0 0,0,0 2 and g: - 27E” g, (4.l8) where J is the m x m diagonal matrix whose diagonal elements are 0 O O 011 = O l O , O O l and whose off diagonal elements are the 3 x 3 zero matrix. This form for P_was obtained from Equations (3.l6). In a tedious but straightforward manner it can be shown that 2 _ c 4 SIB—7' ("E—“‘9- Substitution of this result into Equation (4.l5) and inspection of the result shows that this system of linear equations can be reduced to a single equation, ] 0* = o. (4.l9) 86 But I n2 n2 = 2 o _— . = 2. (4-20) 12b Substituting this result into Equation (4.l9) gives 2 2 0* = 0/[1 - 24:2] (4.21) which should be compared to the exact result of Equation (4.l2). Before doing so it should be pointed out that this result could have been obtained easily (in this case) by applying Equation (4.l7) repeatedly to the crack at the origin. That is, for x1=0, Ed w nEd 0*(0,0) = 0 + (- 40x9 2 z —%—-= 0 — g, (4.22) i=2 x]. 48b 1 and using the dipole model, 2 d = - ZIC 0*, (4.23) y E the result of Equation (4.2l) is obtained. 87 Now, defining _ ha 2 - 23, (4.24) and expanding Equation (4.l2) in a Taylor series in z, 2 _ z l9 4 O*-O(I+—6—+§-6'—OZ ...), (4.25) whereas the expansion of Equation (4.2l) is ° 22 Z4 0* = 0(1 + T6'+ ———...). (4.26) 3 Evidently, the dipole model produces acceptable results for this problem. In fact, the error in Equation (4.2l) does not exceed l0 percent for ratios of (c/b) up to .76 and any error is due of course to treating the crack as a point source, ignoring its spatial extent. These results can easily by extended to cover the more general case shown in Figure 4.3b. At the origin, the resolved stresses are, from Equations (2.38), Ed Ed = __1_ _ __X O; 01 + 40 3R 40 R, Ed Edy *: __..__. _._._ oy 02 40 R 4W R, (4.27) d * =~ EXYR, Substituting Equations (4.28) into (4.27) and rearranging gives a system of linear equations to be solved for the resolved stresses, O;(l+3pa) - 0§P(l-a) + T;y(2p8) = 0], -0;(pa) + o;(l+p(l-a)) - I;y(2pB) = 02, (4.29) -O;(pB) - 0;(p8) + T;y(l-p) = T12, “ZCZ where p =-——7?, 24b 0 = sinze, B = -sin 0 cos 0. 89 For 0:0, Equation (4.2l) is recovered and O P , 2 0* ‘ 01 + 175’ O _ 2 O; — T_—p , (4.30) T* = :l— , xy -D For 0=90°, Equations (4.29) produce C’1 *: Ox l+3p ’ o; = 02 + 01p, (4.3l) T _ l2 Tiy—m. The reduced resolved stress, 0;, of Equation (4.3l) is attributed to the shielding effect of the cracks in front of and behind the crack at the origin. Evidently, the straining mechanisms find it difficult to 'reach behind' the two cracks on either side of the origin to produce the resolved stress, 0], which would appear if these cracks were not present. A similar situation arises in quantum mechanics when first order perturbation theory is applied to orbital energies in atomic arrangements. The electrons in outer orbitals are shielded from the electrostatic field of the 9O nucleus by electrons in inner orbitals resulting in a reduced field as seen by the outer electrons. 4.5 Numerical Solutions to Crack Problems A computer program which handles bodies of arbitrary shape in a state of plane stress/plane strain, subjected to pure trac— tion boundary conditions and containing an arbitrary number of cracks of any (reasonable) size, orientation and location within the body is shown in Appendix G. The program uses the direct method described in Section 3.2. The labelling used in the COMMENT statements is the same as that used in Equations (3.l8) through (3.24) with the additional abbreviations, "...stored in RH of C, NBB + ...”, which indicates that a particular matrix is being stored in the right half of the general purpose matrix, C) beyond column NBB, for reasons of economy. The additional variables not already accounted for are described below. NC The number of cracks. XC(NC x 2) The coordinates of the crack sites. D(NC x 2) The lengths and inclinations (in degrees) of the cracks. P(3NC x 3) The E_matrix of Equations (3.l6) for each crack. C(a x b) General purpose matrix where a = max (2NB, 3NC), b = a + 3NC + 2. B(3NC x 2N8) The B_matrix in Equation (3.22). 9l The program prints out the determinant of (I-QP), the determinant of (C + HPB), the crack coordinates and the corresponding resolved stresses. The particular problem chosen for solution appears in Figure 4.4 and the results are shown in Table 4.3. The results for the resolved stresses in the x—direction for the two cracks located at the section, x=.5, are consistent with the crude ob— servation that since the amount of material left to carry the load at this section after the introduction of the cracks is reduced by an amount equal to the sum of the crack lengths, the average stress (an approximation to the resolved stress) is lx2 = 2-2 .1 =1.111... Moreover, it is easily demonstrated that the imposition of a uniform stress, 0y, on the top and bottom faces of the block in Figure 4.4 alters only the resolved stresses in the y—direction. 92 Figure 4.4. A square block containing five identical cracks at symmetrical locations in a state of uniform stress. CP = .4 seconds Table 4.3. The results for the problem of Figure 4.4. CHAPTER 5 CONCLUSIONS A new point of view regarding the numerical treatment of distributed sources of internal stress has been presented in Chapter 2. The Fourier series approach to smearing point load type singularities over the infinite plane was emphasized because of its effectiveness in the numerical treatment of linear elasticity problems with body forces. Since a large class of nonlinear problems can be rearranged to look like linear problems with pseudo-body forces carrying the nonlinear effects, the numerical method is applicable to nonlinear problems as well. In addition, any source of stress whose influence on the surrounding medium is of the same functional form as that of a point load qualifies for a similar treatment. Dipoles of these sources are also admissable. Hence, the numerical procedure is also applicable to distributed crack problems which use dislocation dipoles in their treatment. Although the dislocation model for a crack has been in use for quite some time [6], there appears to have been no attempts made to use it in fracture mechanics problems involving more than one crack. This is surprising since the model is easily incorporated into a numerical scheme (see Section 3.2 and Appendix G) which produces stress intensity factors for crack configurations which 94 95 are not easily treated by conventional methods [ll, l2]. The dipole model is however not without limitations. There are two possible sources of error in the model. The first occurs in treating the crack as a point effect and the second in ignoring the variation in the resolved stresses over the length of the crack. The first source of error can easily be removed by using the results of Equations (3.7) and (3.8) in conjunction with Appendix C to refine the dipole stress fields of Equations (2.38a, b, c) taking into account the spatial extent of the crack. Similar to Equations (3.l8), (3.l9), and (3.20), the problem would be defined as where the H and Q_matrices are a bit more involved computationally but are nevertheless straightforward. The removal of the second source of error, however, presents an almost insurmountable problem since the distribution cited in Equation (3.3b) is valid only for a straight crack in uniform tension. If the stress field due to external sources (the resolved stress) varies appreciably over the length of the crack, as would be the case if the crack were near the boundary or another crack, this result is seriously in error. The true distribution is determined by satisfying Equation (3.3a) with 0 replaced by 0(x). The solution to this integral equation for a given 0(x) is no simple matter and is further 96 complicated by the fact that for finite bodies, 0(x) is not known until the end of the problem. Consequently, the removal of these errors involves as much work as would be required in solving the problem by conventional methods [ll, l2]. An inherent source of error related to the practical aspects of crack modelling deserves attention. The treatment of cracks as straight slits is obviously an oversimplification. Naturally occuring cracks in, for example, metals and rock are expected to violate these conditions regularly. Fortunately, the straight line model is justifiable insofar as crack propagation is con— cerned since the Griffith criterion utilizes only gross properties of the crack, the crack surface area and the strain energy change in an infinite medium associated with the crack's presence, both of which are relatively insensitive to local variations in crack geometry. That is, it is reasonable to expect that there are a large number of crack configurations possible, all of which have the same surface area and produce the same strain energy change. The modelling of a continuum of microcracks by a field of dipoles (latter part of Section 3.2) is an area which needs development. The dipole density concept was introduced in Chapter 2 primarily to facilitate the transition from a small collection of cracks, which would be handled by the method in Appendix G, to a dense array of microcracks, where the individual handling of these cracks would be prohibitively complicated. The utility of this approach remains to be seen and will undoubtedly find its application in the study of progressive damage due to crack growth. APPENDICES APPENDIX A THE EVALUATION OF FOUR BASIC AREA INTEGRALS The four basic integrals to be evaluated here are I :1 J - ~ 2?— 8V 3m3 [§(:VT e ] 4(2+mv)e Consequently, Isl = % sin ma f-R+b (2+mv)emV cos n(v+b) dV, —R—b which vanishes in the limit since the integrand behaves like Re over a finite range of width 2b. —R+2a d R 152 = 7 x(a-x)3 which can be written as —R+a 3 152: f -du-u cos m(u+a) —R—a COS mx lOO cos n (v+b) - J, The integral over STRIP2 is [R cos ny 4 -R+2b r f -mR dy, R—b cosnv-cosnb - sinnv.sinnb dv -R+b (u2+v2) lOl or I = —2 cos nb f'R+a du - u3 cos m (u+a) - J, 52 —R-a where cos nv = w —-——————— dv. J f0 (u2+v2)2 Now, 2 _ l_.§_. n -n(-u) = ;1_ _ “U J 2u Bu [2 —u e ] 4u (nu l)e . Consequently, Is2 = ' E cos nb I'R+a (nu—l)enuCOS m(u+a) 9”, -R—a which vanishes in the limit since the integrand behaves like -nR Re over a finite range of width 2a. It appears then that only the integral over the PLANE contributes to IO: 4 cos mx cos ny dy dx, -R+2a —R+2b r 2y,udnégneed only extract from the expanded fOrm 0f O that part which is both odd in u and even in v. This results in I = 4 sin ma - cos nb f0° du - u3 . sin mu - J, o o where w cosnv-dv J = f -——-———-—. o (u2+v2)2 As before, _ l a J - - 2U'EU'Q (n,v), and since u is positive in 10, l 3 n -nu 703.4209 1 J = = —E§-(l+nu)e'nu. 4u Now, Hr l03 IO = n sin ma cos nb [m (l+nu)e'nu sin nu du. 0 Introducing K = f00 e—nu sin mu du = 2m 2, o m +n IO becomes _ . 8K I0 — n s1n ma cos nb (K-n 53), giving nm(m2+3n2) I = sin ma - cos nb. (A.4) 0 2 2 2 (m +n) Consider next, I]: R R (a-x)2(b-y) I1 = f fl————~7f—————-cos mx cos ny dx dy. The integrals over the PATCH, STRIPl, and STRIP2 are all zero for the same order of magnitude reasons used previously in the evaluation of Io. The only contribution to I1 then, is that due to the integral over the PLANE, = I = fR fR Iglllfiélll I1 P 4 cos mx cos ny dy dz, -R+2a -R+2b r which can be written as 12;. é I1 = 4 cos ma sin nb I" duouz-cos mu-J, o where w v sin nv J = I -——-—-———— dv. o (u2+v2)2 By inspection, 133 fifi'gfi'gfi' @(n,v), C.- II giving Finally, . 00 - I1 = nn cos ma Sin nb f ue nu cos mu du, 0 which, upon using the previously defined K, becomes _ o I 0 8K I1 — nn cos ma Sin nb am’ giving nn(n2-m2) . I = ——§——§—§—-cos ma . s1n nb. (A.5) ‘ (m+n) The remaining integrals, I2 and I3, can be obtained from 11 and I0 respectively by interchanging a with b and m with n. That this is true can be seen by making these exchanges in the original definitions of the integrals along with the dummy variable exchange of x with y. That is, Ik(a,b,m,n) = I3_k(b,a,n,m). (A 6) It should not be assumed that these results are valid when either m or n are zero, the reason being that the order of magnitude arguments used in the STRIP integrals break down in this case. How— ever, independent evaluation of these integrals reveals that the above results are applicable if either m or n is zero, but not both zero at the same time. That the validity of Equation (A.5), for example, is suspect for the case, m=n=0, is made apparent by the fact that different results are obtained for the limit depending on the order of the limiting process, lim lim I1 2 lim lim 1,. n»O m+0 m+0 n+0 lO6 Consider Io(a,b,0,0). The PATCH integral is zero by the usual order of magnitude analysis. The STRIPl and PLANE integrals are zero because their integrands are odd functions of u, inte- grated over symmetric limits. Therefore, or, 10 = -2 7"R+a du 03 0, -R-a where R-b dv J = f . o (u2+v2)2 The upper limit on this last integral can be replaced by R without affecting 10 since the net result is to add to J an amount not exceeding b/R4, which when coupled with the integral over u produces a result of order 4ab/R which vanishes in the limit. Hence, R dv l -l R Ru J ‘ f = ——— (tan —-+ ) o (u2+v2)2 2u3 R2+u2 I R n Note that tan— U-cannot be replaced by é-at this point because u assumes unbounded values. Now, It which, in the limit as R becomes large approaches —l R Ru 0 I - -2a[tan —-+ ] = a(—+l). o u R2+u2 U=_R 2 Finally, consider I](a,b,0,0). As before, the PATCH integral is zero as are the STRIP2 and PLANE integrals due to integrands which are odd in v integrated over symmetric limits. Consequently, -R+2b R (a—x)2(b—y) d I = I = f f x dy, 1 S] -R —R+2a r2 or, 11 — —2 I‘R+b dv-v-J, —R—b where _ R—a u2du J—f 222. o (u +v ) Replacing the upper limit on the last integral by R adds to J an amount, a/4R2 at most, which couples with the integral over v to produce a term of order ab/R which vanishes in the limit. Hence, R u2du l (u2+v2)2 2v J = f [tan-1 5-— RV ], 0 and in the limit, 11 = -2b[tan'1 %- -§-"—2-] = b(l} - 1). The remaining integrals follow directly from Equation (A.6), which applies for all values of m and n. APPENDIX B DIPOLE TRANSFORMATIONS In the generalized normal—shear dipole of Figure B.la, let 0 be the stress field generated by the edge dislocation located at the origin and let 0* be the state of stress generated by the dipole shown. Then 0*(X.y) = G(x.y) - (x—h. y-k), (8 l) which for small h and k becomes or, reverting back to the difference form, 0* = [0(x,y) - 0(x-h.y)l + [0(x.y) - 0(x,y—k),l. (B 3) Figure B.lb interprets this result symbolically. Now examine the rotated normal dipole of Figure B.2a. Using the vector nature of b and adopting the notation, lO9 llO a. A generalized normal-shear dipole. b. A normal—shear dipole decomposed into a normal and a shear dipole. Figure B.l. The normal-shear dipole and its equivalent. lll a. A rotated normal dipole of strength bh. b. A normal and a shear dipole superposed. Figure 8.2. The rotated normal dipole and its equivalent normal—shear dipole pair. ll2 b - s = b sin 0 b - c = b cos 0 h - s = h sin 0 h - c = h cos 0, the normal dipole can be reduced to the pair of normal-shear dipoles shown in Figure B.2b. Finally, with the aid of Figure B.lb, each of these normal-shear dipoles can be reduced to a normal and a shear dipole, yielding the four dipoles shown in Figure B.3a, which, when taken altogether, are equivalent to the rotated normal dipole of Figure B.2a. This quartet can be reduced to a trio by recalling that the first and fourth configurations generate the same stress field. Figure B.3b depicts this equivalence symbolically; the dipole strengths are written in parenthesis below each symbol. The rotated shear dipole of Figure B.4a is treated in the same manner resulting in the equivalence depicted in Figure B.4b. That these transformation properties of dipoles bear a resemblance to the transformation properties of second order tensors is not surprising since the dipole strength is itself like a tensor in that it is the outer product of two vectors, the Burger's vector and the spacing. These properties can now be used to transform a general dipole state, (dx’ d , d ), to a rotated coordinate Y XY system. That is, (0;, d—, d——) s (d , d , d ), (13.5) aI-Is ll3 Y Y J4. X x T _.,, —> bC bc Y Y a. Two normal and two shear dipoles superposed. >4: (bh) TEFD—II—EFDiT (bhcos2 0) (bhsin 20) (2bhsinecose) b. The transformation of a rotated dipole. Figure 8.3. Transformation properties of the normal dipole. ll4 a. A rotated shear dipole of strength bh. >42 “6041-094 (- bhsinecose) (bhsinecose) (bhcos20) b. The transformation of a rotated shear dipole. Figure 8.4. Transformation properties of the shear dipole. ll5 where the latter dipole state is given, provided that d cosze + d sinze + d sine cosO , x y XY o. I 11 _ - 2 2 . d—-— dX s1n 6 + dy cos 0 - dXy $10 6 cos 0, (B.6) _ . 2 . 2 diy-- 2(dy—dx) $10 0 cos 0 + dxy (cos 0 - Sin 0). It is worthwhile to note that the above transformations mgy be cast into a form which is consistent with second—order tensor trans- formations by redefining dXy in Figure 2.6c to be 6 h. (8.7) This amounts to replacing dxy in Equations (B.6) by dey' The same type of situation is encountered in the transformation of strain; there are two definitions of shear strain in use, 6 X7 and ny' The choice of I 8xy 2'ny results in tensor transformation rules for strain. ) I Under the substitution, x = -c . cos 0, the four basic integrals quoted in Equation (3.6) assume the form ”3-n n I = _fn b (a+cosg) cosg dg (C.l) " o [(a+cos¢)2+b2]2 ’ where n = 0, l, 2, 3. The workload can be reduced to the evaluation of two simpler integrals, J = f“ cos -d , o o (a+cos¢)2+b2 (0.2) J = In (a+cosg) cosg d9 0 (a+cos¢)2 + b2 , ll6 It will be assumed hereafter, that a>0 and b>0. Now, 2 (c050 -0)(cos¢ —0*), (a+cos¢)2 + b where a = -a + bi, and 0* is the complex conjugate of a. It is easy to verify that J =11;Im(K), (0.4) C: ..a I - n + Re(K), complex quantity, K, respectively. z = tan %, K becomes K = ' TEE 5: 22:2, where B= (%1/2=u+iV. with u = (4.41/2. and Under the substitution, ll9 where F = ((1—a2—02)2 + 402)1/2, e = ((1—a)2+bz). and H = 1-a2-b2 + F. Now, I - dz K = _OL_ (2+8) where the circuit integral is taken around the closed path shown in Figure C.l. The integral over the circular part of this path vanishes in the limit as R+w. Also, since 8 lies in the region enclosed by the path and since the function, l/(Z+B), is analytic everywhere inside this region, by Cauchy's integral theorem, . l niq K =-Jl—' 201 —— = . 1+0 8 B(I+q) Substituting this result into Equation (C 4) and simplifying yields Real Axis pole Figure 0.1. Integration in the complex plane. l2l J _ _ g_ 2b + a-a —b )H 0 b H2011)”2 (C.6) _ b(G+F) J 'TTU -—'——), 1 H2014)”2 which can now be used in Equations (C.3) to produce the desired integrals. An independent reevaluation of the four basic integrals for the special case, b=0, gives I0 = I1 ‘ 12 = O, (C 7) whereas I3 = 0, for a < l (C.8a) and I = n(————§————-- l), for a > l. (C 8b) 3 (62_1)1/2 In what follows, the crack tip coordinates, (r,e), pre— viously referred to in Figure 3.2 and defined by a = l + r cos 0, b = r sin 0, will be used. Of particular importance is the limiting case, r+0. For small r, it is a simple matter to show that, asymptotically, (r, - - ,«3‘414‘5.» [Mg (7M): b ‘1 5;) 3' 1 .3. = . cose 3 3b _ Sine 3 5F'+ r 36’ the first of the relations in Equation (C.3) becomes 2 . 2 SJ SJ _ r Sin 0 . o cose o J I _ ___2__(51ne VIP ___r_._a_e_) . After tedious but straightforward manipulations, this and the remaining integrals are found to be, for r+0, . 3 0 I = 2 Sin 2'(] + 4 cos2 6) 9 I1 = -2 sin2 g-cos g-(l - 4 cos2 g), (C.ll) I2 = 2 sin g—(l + cos2 g(l - 4 sin2 9)), I3 = 2 cos g-(Z + sin2 g-(l - 4 cos2 %)), where - 2(2r)1/2' APPENDIX D THE COMPUTER PROGRAM FOR THE SUPERPOSITION METHOD A N O 1 \— W AX “I an HA 8—H VN D¢ H O A no IH N FD av o D¢ vm 7 0V WX VN NU O” W. N. U> XI MA II x lA no (a a a» (N U > Nfi Fv A § .> OH 0 ”H x HO FIW O 9 H “X DR! 0 A? A u’ 0. 9A \ A Av 3 Xx 2am A k Hm .5 “v HNZ 0 Z VX HQ >-\ ll 9' H 0 HI H O A HOH v fl IA vm ”H mm” 2 ”AQH U2 H QVH H Hmv. fl A9 (LL 0 U) OZZH ”N Hm PX!" m W9 QOVAVHHvHWXCih X3 FommHIImH 1X4 I? pmxzxuaxowwnuz Am OVOOQVV DA >-A 00-12 OWAfiHMZAHH1‘1JO'Um FXHHAOHHO¢+>14HHUl DooHHuwommm+odmvl> aAH.vu«szszunm>« ZNVAHWWVCHH¥V OXO> HimHZO+mNH1xUWHHA+ voyvuAAXOHdv QXHU ZNvH HNVH \wm 4" kvfixfiooovwfin CWWmG ammo HHmU 41> 2 «vamvvzvmmwaommqn a 2mm é4¢X>ZZN FZAADKXAAOO>'H ) XF(I’1)9XF(IQZ),PQGQS O'OQLKK‘ O‘fi—HHI¥#Hfi+X'W%WT#¥' swoonnnoouuw dmv—lfiHHNHHH—JUhj—qju .g * v mzvv oovv #1. axxm OMODHHHDQNNXOH¢WX++H Cst ~VQ< VHH H N 123 vat/)HD iHmLCWWMMCCHVVOOHHHQZ QDEKGXXKIQDKUUQQKQWJD 7")? HA #2 Z. 022 Hy. OOH XH+ ”PH 7U“ C 9 ( I S PRINT*.DET K S)/C(IoI) IOICUZ MZHZAVM ZOvoUH HrfiUo—q One FHHHHH D£A¥V+Z C j UZX mfion+oD a H WHP DOVOHVN WOUOWUK d NW 0 H o D o G o O N o N o o #0 O 0 -o N o 0 D so N o o N0 c D m o I N H o to w O HO 0 H N o o om ¢ G N0 0 I N N o o m G No a o 9 ON G o o O I D N0 F. 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