ABSTRACT ANALYSIS OF A DYNAMIC SAINT-VENANT REGION m A SEMI-INFINITE CIRCULAR CYLINDER By Joseph F. Binkowski This investigation examines the response of an elastic, homogeneous, semi-infinite cylinder to dynamic end loadings. These loads consist of harmonic, axisymmetric normal and shear stresses applied at the finite end, Z = 0. The analytical formulation is based on Cauchy's equations of motion and the dynamic Beltrami- Michell equations of compatibility. The solution of this formulation is expressed in the form of a four vector biorthogonal eigenfunction expansion. The solutions for three different stress distributions at one frequency are examined in detail, revealing non-decaying modes. Eigenvalues are tabulated for six frequencies between 100 and 50,000 cycles per second. ANALYSIS OF A DYNAMIC SAINT-VENANT REGION IN A SEMI-INFINITE CIRCULAR CYLINDER By "} 5‘4 ‘ Joseph FfoBinkowski, Jr. A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DU: TOR OF PHILOSOPHY Department of Matallurgy, Mechanics and Material Science 1974 I‘O CONNIE J. STEWART AND MY MOTHER, MARY B INKOWS KI ii ACKNOWLEDGEMENTS The author wishes to thank Dr. Robert W. Little for his suggestion of the problem, counsel and encouragement. A special note of appreciation to fellow graduate students Jack Webb, Dave Butler, Fred Mendenhall, Sandy Carr, M. Fahmy and C.B. Ullagaddi for their friendship and encouragement. A special note of graditude to Beverely Anderson and Dr. William Bradley for their assistance and thoughtfulness. I can only inadequately express my deepest graditude to Connie J. Stewart for her continual encouragement, understanding and assistance throughout the course of this research. Thanks are also extended to Dr. R. Summitt, Chairman, Department of Metalluragy, Mechanics, and Materials Science, and Mr. J.W. Hoffman, Director, Division of Engineering Research, for financial assistance. Thanks are due to Mrs. Noralee Burkhardt for her excellent typing. iii Chapter II III TABLE OF CONTENTS List of Tables List of Figures List of Principal Symbols INTRODUCTION 1.1 General Discussion 1.2 Progress in the Development of Analytic Solutions of a Dynamically End Loaded Cylinder 1.3 Problem Statement 1.4 Method of Analysis FORMULATION OF THE FIRST MIXED BOUNDARY VALUE PROBLEM 2.1 Problem Statement 2.2 Equations of Elasticity 2.3 Defining Equations for the First Mixed Case 2.4 Solutions of the Defining Equations for the NNN NO‘U‘ First Mixed Case Boundary Conditions Definition of the Biorthogonality Operator Definition and Solution of the Adjoint Equation THE SECOND MIXED BOUNDARY VALUE PROBLEM WWW WNH b.3009) Chm-b Problem Statement Definition of the Auxiliary Variable Q Defining Equations of the Second Mixed Boundary Value Problem Solutions of the Second Mixed Case Boundary Conditions Definition and Solution of the Adjoint Equation Development of the Biorthogonality Operator iv Page vii ix 11 13 17 20 22 22 22 24 26 27 29 31 Chapter IV VI VII PURE STRESS CASE 4.1 Four-Vector Formulation of the Pure Stress Solution 4.2 Four4Vector Form of the Biorthogonality Operator for the Pure Stress Case DETERMINATION OF THE EIGENFUNCTION CONSTANTS 5.1 Outline of Procedure 5.2 Evaluation of the Adjoint Eigenfunction * * Constants BL - C L 5.3 Evaluation of Terms Represented in Equation (5.1-2) DETERMINATION OF THE EIGENVALUES 6.1 General Discussion 6.2 Evaluation of Pochhammer's Frequency Equation STRESS SOLUTIONS OF SPECIFIC BOUNDARY VALUE PROBLEMS 7.1 Specification of Applied Boundary Stresses 7.2 Results 7.3 Conclusions BIBLIOGRAPHY APPENDICES A - Reduction of Boundary Condition Operators From that of Equation (2.4-4) to Equation (2.4-5) B - Tabulated Eigenvalues of.Pochhammer's Frequency Equation, Equation (2.4-15) Page 34 34 36 39 39 40 41 46 46 48 57 57 59 6O 9O 92 97 Table VI-l VI-2 VI-3 VII-1 VII—2 VII-3 VII-4 B«3 B-4 B-5 LIST OF TABLES Phase Velocities of the First Three Longitudinal Modes Phase Velocities of the First Four Longitudinal Modes Comparison of Low Frequency Complex Eigen- values with Static Problem Eigenvalues Axisymmetric Trial Problem No. 1: Com- parison of Eigenfunction Expansion Solutions with Prescribed Boundary Values Axisymmetric Trial Problem No. 2: Com- parison of Eigenfunction Expansion Solution with Prescribed Boundary Values Axisymmetric Trial Problem No. 3: Com— parison of Eigenfunction Expansion Solution with Prescribed Boundary Values Convergence of Eigenfunction Expansions for Problem 2 Boundary Functions at Frequencies of 100, 2000 and 6380 c.p.s. Roots (QN) of Transcendental Equation (2.4-15), Frequency = 100 c.p.s. Roots (an) of Transcendental Equation (2.4-15), Frequency = 1000 c.p.s. Roots (qN) of Transcendental Equation (2.4-15), Frequency = 2000 c.p.s. Roots (0N) of Transcendental Equation (2.4—15), Frequency = 6380 c.p.s. Roots (0N) of Transcendental Equation (2.4-15), Frequency = 20,000 c.p.s. Roots (0N) of Transcendental Equation (2.4-15), Frequency = 50,000 c.p.s. vi Page 49 50 51 63 64 65 66 98 99 100 101 102 103 Figure VI-l VI-2 VI-3 VI-4 VI-S VII-1 VII-2 VII-3 VII-4 VII-5 VII-6 VII-7 VII-8 LIST OF FIGURES Plot of Pochhammer's Frequency Equation for Lowest Three Branches of Frequency Spectrum Plot of Pochhammer's Frequency Equation Showing Linkage of Real, Imaginary and Complex Branches at High Frequencies Phase Velocity Variation of Extensional Waves in Cylindrical Bars; v = .29 Phase Velocity Variation of Extensional waves in Cylindrical Bars; v = .33 Eigenvalues in First Quadrant of Complex Plane for Frequencies of 100 c.p.s. and 20,000 c.p.s.; a'x+1Y Comparison of Eigenfunction Expansions with the Prescribed Boundary Functions for Problem 2 at Frequencies of 100, 2000, 6380 c.p.s. Applied Boundary Stress 022 and its Non- Decaying Stress Modes at Z - 0; Problem 1 Resultant Stress Orr and its Non-Decaying Stress wave Modes at Z = 0; Problem 1 Resultant Stress 099 and its Non-Decaying Stress wave Modes at Z = 0; Problem 1 Resultant Stress and its Non-Decaying T rZ Stress Wave Modes at Z = 0; Problem 1 Decaying Modes of 022 at R = l and Tr at R = .5; Problem 1 Z Non-Decaying Modes of 022 at R = 1; Problem 1 Non-Decaying Modes of TrZ at R = .5; Problem 1 vii Page 52 53 54 55 56 67 69 7O 71 72 73 74 v V Figure VII-9 VII-IO VII-11 VII-12 VII-13 VII—14 VII-15 VII-16 VII-17 VII-18 VII-19 VII-20 VII-21 VII-22 VII-23 Decaying Modes of o and O at R = .1, Problem 1 rr 99 Non-Decaying Modes of O at R = .1; Problem 1 rr Non-Decaying Modes of g at R = .1; Problem 1 99 Applied Stress T and its Non-Decaying Stress rZ Wave Modes at Z = 0; Problem 2 and its Non-Decaying Z = 0; Problem 2 Resultant Stress Orr Stress Wave Modes at and its Non-Decaying = 0; Problem 2 Resultant Stress 099 Stress Wave Modes at Resultant Stress and its Non-Decaying = 0; Problem 2 022 Stress Wave Modes at Z Decaying Modes of GZZ and Problem 2 TrZ at R = .5; a at R ' .5; Non-Decaying Modes of 22 Problem 2 Non-Decaying Modes of T Z at R = .5; Problem 2 r Decaying Stress wave Modes of on and a at R - .1; Problem 2 99 Applied Stress and its Non-Decaying °zz Stress Wave Modes at = 0, Problem 3 Resultant Stress Orr and its Non-Decaying Stress wave Modes at Z = 0; Problem 3 Resultant Stress and its Non-Decaying Z = 0; Problem 3 °ee Stress Wave Modes at Resultant Stress T and its Non-Decaying Z = 0; Problem 3 r2 Stress Wave Modes at viii Page 75 76 77 78 79 8O 81 82 83 84 85 86 87 88 89 LIST OF SYMBOLS adjoint eigenfunction constants = 5', phase velocity = /E/p , phase velocity of a wave having infinite wavelength Young's modulus of elasticity 2 2 = [a - %%j;;]/[a + ——fl——] = “(1263 2 (012 + am + 53;) J2(e) (1 + wazxo J1 (1 + vKO)e2 J1“) (l-v) =crzil - fi- 2v8 2 (a +8) J (5) H II rfi Q xv + | F C u; \ Q 03 II II f” i” - + \y4 ng‘ stress variables defined by R = c + 0' 3 rr 90 LQ = (l-v)§% - vafi ix {S} () ()- , eigenfunctions of the 4-vector formulation displacements in polar cylindrical coordinates propagation constant in Z-direction 2 =P—;°—<1+v> 2 g -2 z (0' + 6(1'V; (a2 + 26) (a2 +'B) 1 n , eigenvector of the second 1.\2 mixed case Poisson's Ratio density stress components in polar cylindrical coordinates l = Y , adjoint eigenvector of the first Y2 mixed case 1 = T , eigenvector of the first mixed 2 case T I..l = , adjoint eigenvector of the 1..2 second mixed case fourdvector form of the biorthogonality Operator angular frequency complex conjugate transpose complex conjugate at CHAPTER I INTRODUCTION Section 1.1. General Discussion. Saint-Venant's principle is one of the frequently applied principles in elasticity today. In essence, this principle allows a mathematically difficult set of tractions Specified on a finite boundary to be replaced by a statically equivalent, mathematically simple set of tractions. This replacement is possible only when the difference problem, a self-equilibrated loading, can be shown to have only local effects. The influence of these self-equilibrated forces, as Saint-Venant postulated, "extend a very short distance beyond the parts of the body upon which they act." A more popular version of this principle, published by A.E.H. Love, stated, "the strains that are produced in a body by the application, to a small part of its surface, of a system of forces statically equivalent to zero force and zero couple (self- equilibrating), are of negligible magnitude at distances which are large compared with the linear dimensions on the part" [18]. The validity of Love's version of Saint-Venant's principle has been confirmed for the semi-infinite geometries, of the strip [10], the cylinder [16], the wedge and cone [17]. The above papers have been solely concerned with the existence of a Saint-Venant region for static self-equilibrated 1 loadings. The question as to the existence of a dynamic Saint- Venant region remains unanswered. Several papers examine, in whole or in part, the existence of a dynamic Saint-Venant region. Boley [2] in 1954 deve10ped an analytical model of a three element thick bar, showing that stresses are restricted to the loading end of the bar if the dynamic loads are applied slowly. These stresses extend to greater distances from the loading surface if the application of the loading increases in frequency. A report by O.E. Jones and Norwood [12], on the dynamic loading of a semi-infinite cylinder, suggests that stress differences between two normalized loadings, a pressure step load and a velocity impact load with a radially constrained end, is negligible at propaga- tion distances greater than fifteen to twenty bar diameters. Stress differences existing within the above region were attributed to radial inertia being neglected in one of the above loadings. The authors comment that these modes should constitute an upper bound for the dynamic Saint-Venant region. Jones and Kennedy [11] investigated the reSponse of a semi- infinite cylinder to statically equivalent, but different radial end stress distributions. Stress differences were found to be negligible at distances greater than five bar diameters from the loading surface. Recently, Grandin and Little [8] developed an analytical formulation for stress wave propagation in the semi-infinite strip. This investigation shows that for self-equilibrating tractions applied on the finite end, a dynamic Saint-Venant region does not exist. Also, pr0pagating stress modes of much greater magnitude than the applied boundary loads can exist. Section 1.2. Progress in the Development of Analytic Solutions of a Dynamically End Loaded Cylinder. The first analytical solution to the problem of elastic wave propagation in a homogeneous, isotropic cylinder of infinite length was presented by Pochhammer in 1876. Pochhammer's solution of the equations of motion for the cases of compressional, flexural and torsional waves in an infinite rod, had the form of traveling harmonic wave trains. The requirement of a traction free surface generated a frequency equation for each of these three cases. Initial work with Pochhammer's frequency equation was con- cerned with numerical evaluation of the lower modes of wave trans- mission. Through the efforts of Davies [4], it was established that the faster traveling low frequency-long waves from the lowest mode could account for the important features of transient disturbance at rod stations remote from the source. A revival of interest in wave prepagation of bars and plates in the 1950's initiated investigations using multi-integral trans- forms and residue theory as a means of solution. One of the first solutions for the problem of a semi-infinite cylindrical rod sub- jected to a step axial velocity was given by Skalak [25]. Skalak's approach towards a solution, was to take appropriate combinations of Fourier sine and cosine transforms of the equations of motion. Inversion of the displacement variables resulted in solutions which have the form of simple integrals of the Airy function, representing a non-decaying with time, dispersive disturbance. Evaluating these solutions, it was shown that long waves from the lowest mode formed the main contribution to the strains for large time, in agreement with Davies work. Using a transform technique similar to Skalak's, Folks, Fox, Shook, and Curtis [7] presented a solution for the elastic strain produced by the sudden application of pressure to the end of a semi-infinite cylinder with a radially constrained end. Results similar to Skalak's were noted, that is, low frequency waves from the lowest mode form the main contributions for large time. An extension of the above transformation technique for non-axisymmetric mixed end conditions on a semi-infinite cylinder is given by Devault and Curtis [5]. Miklowitz [19] considered the case of a semi-infinite rod with a stress-free lateral surface subjected to a pressure shock at the end. The method of solution chosen is the Laplace transform technique. Solutions were based on Mindlin-Herrmann theory [20], which takes into account radial inertia in conjunction with longitudinal motion. Results for end conditions of a pressure step load or a velocity impact source excitation show identical results for large distances from the finite end. Axial strain caused by either of the above loadings is initially of a diSpersive nature, but for large time, its magnitude approaches a constant value. Mindlin and MCNiven [21] proposed a system of approximate, one dimensional equations for axially symmetric motions in an elastic rod of circular cross section. The approximate equations of motion take into account the coupling between the longitudinal, axial shear and radial shear modes. These modes, representing the first three general branches of the Pochhammer frequency equation, were presented in graphical form, showing the imaginary and complex portions in- Volvod. The authors point out that the complex branches correSponded to wave modes experimentally identified by Oliver [23] as edge waves on the finite end. Despite all of the work cited above for the semi-infinite rod, it is important to emphasize that exact analytical solutions have been obtained only in cases when the end conditions are given in mixed form.* The solutions for these cases are expressions involving inverses of multiple transforms, which in their evaluation resulted in solutions for large times and large distances. The more realistic case of end conditions in pure form are more complicated and there- fore have received very little treatment. There have been only a few papers which have approached this problem. Folk [6] in dis- cussing an approach to the problem of a semi-infinite cylinder, using transform theory as a means of solution, states that for mixed end conditions, one is able to choose the proper combination of sine and cosine transforms so that in taking the transform of the equations of motion, the information asked for by the transform, was given by the end conditions. This is not true of pure stress boundary con- ditions. Recently Grandin and Little [8] considered the case of axi- symmetric loading on the finite end of a semi-infinite strip. The solution represented the steady-state response of the strip to self— End conditions are in "mixed form" if they are specified either in terms of a normal stress and tangential diSplacements or in terms of the normal displacement and the tangential stresses. equilibrated loads that were in the form of pure end conditions. Solutions of the exact equations of elasticity indicate that waves of a non-diSpersive nature exist, implying that a dynamic Saint Venant region does not exist for the strip. It is the purpose of this thesis to present an analytic solution of an elastic, homogeneous, semi-infinite cylinder, sub- jected to dynamically applied self-equilibrated end loads. The underlying question to be answered is, "Does a dynamic Saint-Venant region exist for the semi-infinite cylinder"? Section 1-3. Problem Statement: A semi-infinite cylinder with stress free lateral boundaries is loaded on the finite end with tractions perpendicular and parallel to the plane, Z = 0. These end loads are time harmonic, self-equi- librated normal and shear stresses. The stress wave propagation characteristics are analytically determined within short distances of the finite loading surface. Section 1-4. Method of Analysis: Thetechnique of solving a pure stress formulation will follow the technique of Little and Childs [l6], Klemm and Little [l3], and Grandin and Little [8], in its use of biorthogonal eigenfunction expansions. The above method is extended for the case of time- dependent loadings on a solid semi-infinite cylinder. * End conditions are in "pure form" if they are Specified in terms of normal and tangential stresses or in terms of normal and tangential displacements. The method involves solving two mixed boundary value problems. The first mixed problem is the case of specifying a normal stress and radial diSplacement at the finite end of the rod, Z = 0, while the second mixed problem considers Specification of a normal dis- placement and radial Shear Stresses. The purpose in developing the intermediate mixed cases, is to aid in formulating the pure stress problem. These intermediate mixed cases introduce stress related variables which carry the information of the displacement boundary conditions, thus making a pure stress formulation feasible. The governing equations for the above mixed cases are derived from the dynamic Beltrami-Michell equations and Cauchy's equations of motion. These defining, second order, vectorial partial differential equations are reduced to ordinary matrix differential equations by separation of variables. The separation of variables solution, an eigenfunction expansion, represents a non-self adjoint system and an adjoint operator for this system is derived by the use of a generalized biorthogonality condition. The general solution for the pure stress boundary conditions will be a four component vector formulation, derived by stacking the two mixed case solutions. The biorthogonality operator for the above four component vector form is developed in similar manner. Numerical work, for pure stress end conditions, are performed by: a) substituting boundary data into the four-vector solution, b) applying the biorthogonality operator, c) solving the finite system of equations obtained from the infinite system of equations by truncation. Sec tion 2.2. CHAPTER II FORMULATION OF THE FIRST MIXED BOUNDARY VALUE PROBLEM Section 2.1. Problem Statement: The problem studied is the stress distribution due to dynamic, axisymmetric, self-equilibrated end loads on a semi-infinite, elastic, homogeneous, solid cylinder. This problem will be formulated in polar cylindrical coordinates, where u, v, and w denote dis- placements in the r, e, and Z directions respectively. The radius of the cylinder is taken to be one, and the Z-axis, represents an axis of symmetry. The first mixed boundary value problem denotes the following mixed boundary value problem: Dynamic, time harmonic end loads con- 8 isting of a normal stress, a Z’ and a radial displacement, u, are Z S pecified on the finite end, Z = 0. The lateral surface, r = 1, remains stress free. Equations of Elasticity: The equations of motion of an axisymmetric, elastic body, in PC) 1a 1- cylindrical coordinates are: Cauchy's Equations of Motion: (2 - O aTrZ .. 2‘1) L°rr"§&+gz‘“=pu (2 - 2 8022 .. 2) aZ + L TrZ - pw 8 ‘ H Vhi where 1 2 L=§~r—+; "=5—2- at Dynamic Beltrami-Mitchell Equations The form of the general dynamic compatibility equations are: (22-3) {7’ 7=+ 1 65R . .Vg (1+V) =9. 21+ 7"- V =°' E I ( v)o (1_v) I K] where 6:2, a—+z=§ la-+’é 3— r 5r 9 r 89 Z 52 (2.2-4) ..=a2__ atz K=c1~r+°eae+dzz Expanding equation (2.2-3) yields: 2 2 1 2K 22 - - ——. _ + ___.h__ C 2 5) V Crr+ 2 (099 Orr) l+u 2 r 5r 9 2 = 2- -2 oo - u u E(1-v) [( V V )orr v(oee + 022)] (2 -2_ 2 _;_ _ +le5 6) V 699 r2 (099 Orr) 1+v r 5r _ -- - 0. a. STI:_T [(2 v 2V )oee v(orr + 022)] (2 2 1 2K - 12 _. ___.a__ 7) Vc,ZZ-'-1+\: 2 Q 2 .. .. .. E(l-v) [(Z'V’ZV )ozz - v(orr + 099)] 10 2 2 .1__ _l_ a._1< = 2(1+v)%-¥rz DiSplacement - Stress Relations: For a linear elastic material, the displacements are related to the stresses by the following equations - as = = l - (2.2 9) at an, E [orrr Woes + 022)] 2.: _ l _ (2°2-10) r 699 _ E [099 “(°ir + OZZ)] _ aw. = = l - (2'2 11) az ezz E [°22 VRJ 2 l+v 2. _ 32 + am = 2 = —L—l. ( 2 12) az 3r 6rz E Trz where (2.2—13) R = Orr +’Oee Addition of equations (2.2-9) and (2.2-10) results in 1 (2.2-14) Lu = E'[(l - v)R - 2vozz] . The above equation shows that a mixed formulation specified by 022 and u at the bars end, Z = 0, is equivalent to the Specification of R and 022, up to a rigid body diSplacement. The defining equations are expressed in terms of the variables R and OZZ’ where R is a displacement variable in a stress form. 11 Section 2.3. Defining Equations for the First Mixed Case. The governing equations in sec. 2.1 can be reduced to two hdefining equations in terms of R and Adding equations 022' (2.2-5) - (2.2-7) yields the first defining equation =ig1+v)11—2v) Q 2 (2.3-1) V (R + cm) (1-\,) E (R + 522) The second defining equation is the third compatibility equation, equation (2.2-7) 2 2 1 a 2. - _ ( 3 2) v °zz + 1+v 522 (022 + R) 2 u . _ E(1-\)) [(2-v-2v )OZZ - VR] . Solutions of equations (2.3-1) and (2.3-2) may be readily found if they are represented in matrix notation as a single vector partial differential equation. 2 2 - a. ,a__ - a. .a__ a _ (2.3 3) (L at + [A1] 2 E [A2] 2){¢) _ o 52 at where -1 1 1 V 2.3-4 = a_.+ — = -— ( ) L at 1' [A1] 1+V 1 2+V 2 {<9} = [A2] = 2 .22 (v-D ,, 2., +.-2 Section 2.4. Solutions of the Defining Equations for the First Mixed Case The matrix form of the defining system (2.3-3) is solved by separation of variables. Since the end boundary stresses for 12 the final pure stress formulation is time harmonic, the separation of variables solution will include a time harmonic component. ASSuming a vector solution of the form (2.4—1) {$) = $(r)¢(Z)eiwt one obtains -o m —o -0, z ' (2.4-2) {:9} = 2 ¢,(r)e 3 9.1th = j=0 J 1 q3(r) -ajZ iwt a) 53 2 J=0 q3(r) where j is an integer and aj is an undetermined parameter. {$1,} satisfies 2 2 2. - a—+ +9-59— 7 = ( 43) (L at ojIAl] E [A2]){cpj} 0 When the above system of equations is rearranged, the operators acting on {EU} become - a—«+ 2 a— 2 F = (2.4 4) [L a, 6 JJEL a, + e ljwj} 0 where 2 2 M ijg-zg _ 2 u, bj = (11+ E (l-v) ”j + (at? 2 2 ”,2 2 2 e =oz.+-P——(1+\)) =01 +<‘-’-°—-> j J E 3 CL with CL and CT representing the characteristic longitudinal and transverse wave Speeds. The solution of equation (2.4-4) has the following form: 1 cpj(r) =[B1 JO(6r) + B2 J0(er)]j (2.4-5) 2 .pjm = [133 J0(6r) + 34 J0(er)]j 13 1 2 d . . where m, and q3 are components of the vector fmj]. Substituting J ‘ solutions (2.4-5) into either of the equations represented by equa- tion (2.4-3), defines two of the four constants. 1 (2.4-6) 2 = .. + . ij [K0 Bl JO(6r) 32 Jo(er)]j The remaining constants are determined by use of the boundary conditions at r = l and those at Z = 0. Section 2.5. Boundary Conditions. At the surface of the cylinder, r = l, the two stress com- ponents, Orr and Trz’ vanish. When these stress components vanish, two boundary conditions arise in terms of the stress components, R and 022. Because the defining matrix equation (2.3-3) does not explicitly contain a relationship between TrZ’ R, and 1022 or Grr’ R and OZZ , two relations shall be derived. First Boundary condition: (2.5-1) TrZ\r=1 = 0 . An explicit relationship between T R and o is rZ’ Z2 2 derived by differentiating equation (2.2-2) with reSpect to 2:32 yielding 2 an 3"'22 . -2 3---L = - - -—-—- . <25 > 2a,. 221 gin... .22 1 Z is obtained from equation (2.2-8). Substituting this into equation (2.5-2) An equivalent representation for :;'L Tr gives 14 2 3T _ a. 2 Q " _ _l_lil$______£§. (2'5 3) 32 [ (1+v) E Trz 1+v araZ 822 3 2 =a_[pa§.i_""’zz1 at AZ 22 Equivalent representations of (BE + 53) and flu. in terms ar 52 82 of stresses are obtained by differentiating equations (2.2—11) and 2 (2.2-12) with respect to 9-5 . Substituting these into equation at (2.5-3) yields: a. 211.21 a: a: (2'5'4) 52 [ E p 2 ' 23Trz at 62 2 a o 2 =a_.[9_(d 'Vfi)- ZZ+_J:_B._I_<.] 2 5r E 22 322 1+v az * The above equation is equivalent to the expression a c 2 _ a_. e. _ ZZ.+ _l_.a_§ (2.5-5) Trz ar [E (022 vR) 2 l+v 2] AZ 32 The boundary condition TrZ\r=1 = 0 is assured of being satisfied if a o 2 B_ D- " _ .. _ 42. .i- L— = (2.5-6) at IE (ozz vR) 22 + 1+v az§]\r=1 0 Substituting solutions for R and 0 represented by ZZ equation (2.4-2) into equation (2.5-6), results in 2 2 2 2 a .. + :0 (2-5 7) (BIG 6 6 J1(6) B28 (a + 1_v)e J1(e))j Second boundary condition: - O . (2.5-8) .9 orr‘r=l _ Once again the defining matrix equation (2.3-5) does not contain a relationship between Orr’ R and 022. Developing this Discussed in Appendix A. 15 relationship begins by differentiating equation (2.2-1) w.r.t. the Operator L. aOI’I‘ TrZ 1 a— ' 2.5-9 L ——_—'+ L —-—'+ '— - = ” . ( ) Br 62 [r or (Orr 096)] pLu Subtracting (2.5-9) from equation (2.2-S) and simplifying, results in 2 2 R + 2 522 r rr 99 1+v 3r2 i a. aTrZ - r at (o - 069) L 82 _v_ -- Bi 2 if .- = pt“, (Lu + 32) + 33;] - pLu Differentiating equation (2.2-2) w.r.t. a- and multiplying Z 2 B aTrZ equation (2.2-1) by ;' allows substitutions to be made for L :2 2 and -§ 089 in the previous expression. Simplifying gives r 2 2 2 1 5014-022) 135 30;; aOrr (2.5-10) 1+ 2 +. r + 2 + 2 V at r a 32 AZ 2 aTrZ Q S1‘2V2 -' .. + r 52 ' E (l-v) (R + °zz) 2 1 a (R‘+ 022) An alternate representation of 1+v 2 is obtained ar ‘ from the defining equation for the first mixed case, equation (2.3-1). Substituting this into equation (2.5-10) produces: (2.5-11) rr+-——+—L-l"a—[vR-o 16 * Equation (2.5-11) is equivalent to the expression: d|ho 0) 01 N a '1 b4 2 1 B | 1 l a - = _ - — _ - - The boundary condition a = O is satisfied if rr‘r=l 2 a__ 2 [vozz - R]‘ 52 II 0 (2.5-13) a:[vR - 0221‘ + B r=1 r=l Substituting solutions for R and 022 in the above equa- tion re3ults in: 2 (2.5-14) 2 0 {EN [J1(6) + n J (5)13 , fl 2 0 1 La +B(1-v)] + [e J1(e) + a2J0(e)]B2 j = 0 Boundary conditions (2.5-1) and (2.5-8) give two linear homogeneous equations containing the constants B1 and 32’ equa- tions (2.5-7) and (2.5-14). A necessary and sufficient condition for these two equations to have a non-trivial solution is that their determinant of coefficients should vanish. Simplifying this, yields the transcendental equation for the eigenvalues, qj 2 (2 5-15) (a e a JO(e)J1(6) + 68 J1(6)J1(e) - 32J0(5)J1(e))j = o where 2 2 2 B = EfiL-(l + v) s = a + a 2 2 1-2 2 2 6=a+61(—1—-% €=a+28. * 2 The operator 5.7 is shown in Appendix A to correSpond to solutions 52 which yield zero stresses for axisymmetric vibration and are therefore dropped. 17 Development of boundary condition (2.5-6) provided a means of relating the constants Blj and sz contained in the series . 1 2 . solution of qfi and “B in equation (2.4-6). The general solution of equation (2.4-3) is _. (p1 1 K1 J (6r) (2.5-16) {q3} = i = 131 0 . -K K .J(a) = l O l . 0 qd J J J where 2 liL... 2 a ‘ a-..) PAL KO " 2 + a B " E (1 + V) a (l-v) 2 -a be J1(6) K = 1 2 2 ._Ji__ [a + B][oz + (1_v)]~11(e) Section 2.6. Definition of the Biorthogonality Operator. The biorthogonality operator for the first mixed case is derived from equation (2.4-3) and its adjoint, equation (2.6—1). 3.2 _a_ ~— (2.4-3) (L at + aj [A1] + 1+v [A2]){wj} - 0 _ 5.. -2 + _B_ + - = (2.6 1) (at L1 + UL [A1] + 1+v [A2] >9“) 0 where + = complex conjugate transpose - = complex conjugate L = a_., l . 1 5r r The biorthogonality operator is derived by: a + l.) Pre-multiplying equation (2.4-3) by the vector {Yb} 2.) Post-multiply (the complex conjugate transpose of equation l8 2.6-1) by the vector {3%}. 3.) Subtract the result of step 2 from the result of step 1. The above operations give (2.6-2) {Y }L s. {cm} - (:3; L10? DWJ-l 2 2 -v+ -o + , - Y A , = O . Integrating the above equation (the first two terms by parts) yields: _ 4+5_.1__a_*t“ - (2.6 3) [{YL} at +(r ar){Yt}]{q3}\r=1 + IJL — 0 where: I <2 2)1?"'}£A{“} d = -0 0 1L 0’1 L JAN, 1] ‘93 \r=1r Expanding the first two terms of equation (2.6-3) gives _ 1 a_.+ _. h_ -1 1 (2.6 4) [ML at (1 “arWLkpj +[\y a;+(_-§_)q;2 =0. Substituting solution (2.4-2) into boundary conditions (2.5-6), (2.5-13), yields, after simplifying, acpz [2 + v8] . at _.J... i 1 2 (2.6-5> at - 2 [c9 - vcp 1,4,4 (v - 1 2 - __ia‘P=[“"+B]i 1_ 2 (2.6 6) at (V2 _ 1) [cp vcp jj\r_1 Substituting equations (2.6-5) and (2.6-6) into equation (2.6-4), produces l9 -1 —2 (2.6-7) ‘TL—_"- [02 v-+ B] '+"L———— [a2 + v8]j + ("' 3‘9§1 q&\ _ (v2 -1) j (v2 -1) -‘ L J “1 -1[v02+ B] V §2 [02 + v8]. +(L_l- 3r r)YL (pj‘r=1 + IjL = 0 . The adjoint boundary conditions are defined as, ~71 2 q?- .1.— _L__ .1. a. -1 _ (2.6-8) [.1 v + a] + [a + v8] ( )v — o (vZ-l) L (vZ-l) L r er L tr=1 (2-6-9) —-4‘—— w + a] + -—L— [a + 63] + ( -)\y = 0 (vz-l) L (vZ-l) L 5r r L \r=l The above adjoint boundary conditions are defined in this - 2 manner because their form contains the terms (a: Y:.a 0L Ti). When these adjoint boundary conditions are substituted into equation 2 2 (2.6-4), the term (aj - UL) is removed, thus showing the -*f a biorthogonality of the vectors {Yb} and {mj). The Substitution of adjoint boundary conditions (2.6-8) and (2.6-9) into equation (2.6-7) gives 1 2 2 —-'+ ~0 0 - - d (2 6 10) (aj aL) A {YL}[A1]QG r 1 —1+—2 1- 2 ___ + (v-l) [VYL YL][¢ vm 1j‘r=l 0 where v -1 A = t 1] [;. 2+w;] 20 Expression (2.6-10) represents the two vector form of the biorthogonality condition for the first mixed boundary value problem. Section 2.7. Definition and Solution of the Adjoint Equation for the First Mixed Case. The adjoint equation correSponding to equation (2.3-3) is - a. -2 + ‘+ e _ (2.7 1) [Br L1 + 0L [A1] + 3[A2] ]{YL} - 0 where + = complex conjugate transpose - = complex conjugate = a__i L1 r The adjoint operator in equation (2.7-l), when rearranged, takes the form (2.7-2) Lat L1 + 6L][5r L1 + 6L]{YL} - o 2 2 gl-sz = + where 6L 0L (l-v) B 2 2 = + 2 , cc qL 8 Solutions of the system (2.7-2) yields (2 7-3) f1 = [3* 6r J (Jr) +-B* er J ( r)] ' L 1 0 2 0 e L 2 4 _2 - [3* + * J ( .7- ) Yb - 3 6r JO(5r) 84 er 0(a)]L ~ Substitution of solutions (2.7-3) and (2.7-4) into either of the defining equations represented by equation (2.7-1) gives (2 7 5 ‘1 — * J ( + 3* . - ) YL — [Bl 6r 0 6r) 2 er JO(€r)]L 21 -—2— * *EE (2.7-6) Yb - [Bl 6r JO(6r) +B2 K0 JO(€r)]L . Using adjoint boundary conditions (2.6-8) or (2.6-9) inter- * * relates the two constants B and B yielding u. 24. 1 + a 1+ Y; _ m * F1 ‘ Kz-l 6r JO(5r) (2.7-7) {YL} = 2 — 2 BL Y L=l K2 er JO(er) 1 — 4. K0 L. .4 L 2 (l+v)6 J1(6) where K2 = - ;g_ 2 (wxon J1(e) Substituting solutions (2.7-5) and (2.7-6) into the adjoint boundary conditions (2.6-8) and (2.6-9) yields two linear homogeneous * * equations containing the constants B1 and B2. Proceeding as before the transcendental equation determining the a '5, equation L (2.5-15),is again obtained. CHAPTER III THE SECOND MIXED BOUNDARY VALUE PROBLEM Section 3.1. Problem Statement. The following case represents the second mixed boundary value problem: the diSplacement w along with the shearing trac- tion TrZ is Specified on the finite boundary, Z = O. The lateral surface, r = l, is again free of stress. To permit a stress-related formulation of the problem equi- valent to the mixed formulation, a new stress related variable Q is introduced. The following section shows that Specifying T rZ and Q on the boundary, Z = 0, is equivalent to Specifying Trz and w up to a rigid body displacement. Section 3.2. Definition of the Auxiliary Variable Q. The variable Q is derived by examination of the first two diSplacement-stress relations, equations (2.2-9) and (2.2-10), which yields (3.2-1) ' Lu = Pip—l [(l-v)R - 2v 022] By differentiating equation (3.2-1) w.r.t. :23 an equivalent ad 22 can be found in equation (2.1-2), Cauchy's representation for second equation of motion. This yields - figs}- - 53-- "- (3.2 2) L 82 E [(1 v) 82 2v(pw L Trz)] . 22 23 Let (3-2‘3) LQ = (l-v §%’- 2v pw . Substituting equation (3.2-3) into equation (3.2-2) gives _ 33 = (3.2 4) L 82 Fill—4 [LQ + ZvL Trz] Equation (3.2-4) can be written in the form (3.2-5) L f = o where L=a"‘+l’o 5r r Solving equation (3.2-5) yields Since f must be bounded as r a 0, g(Z,t) must be identically zero. EQuation (3.2-4) becomes - 53=-1- +2 (3.2 6) 52 E [Q v Trz] . From equation (2.1-12), the term 2%' is also equivalent to (3.2-7) BE.=.ZLLtEL T _ 33,. 32 E rZ 3r Equating equations (3.2-6) and (3.2-7) yields (i=1- (3.2-8) at E [ZTrZ Q] Equation (3.2-8) reveals that Specification of T and Q rZ on the finite boundary 2 = O is equivalent to Specifying .Trz and w. 24 Section 3.3. Defining Equations of the Second Mixed Boundary Value Problem. The defining equations for this case are based on the general governing equations in section 2.2 and the definition of Q, equa- tion (3.2-3). The above equations are reduced to two equations in terms of T and Q. rZ The first defining equation is based on compatibility equa- tion (2.2-8), 2 2 (3.3_1) [V _ l.— _ 2(1+\)) % a._2..JTrz +—_1__ LIL. = 0 r2 Eat +1+v araZ Adding equations (2.1-9) - (2.1-11) gives .. __L_ A‘i (3.3-2) K “' (1'2V) [LU + 82] The end result of differentiating equation (3.3-2) w.r.t. 2:32, can be substituted into equation (3.3-1) giving 2 1 20+) 3: __ v (3.3-3) [v - r2 - E p atzlwrz 2w E a_. a_.+ a_.(L + (l+v)(1—2v) [Br L 52 52 arZZ>1 = 0 ° w Substituting equivalent expressions for L 32- and B___ 52 araZ from equations (3.2-4) and (3.2-7) into equation (3.3-3) yields 2 + (3.3_4) [V —2-1-._L]‘__\_)LDA—]Trz r at 2 1 +(l+v) (1-2v) aQ_ [EL'LQ + 2v :—-LT+ 2 +2 2 - 22] — o . After simplifying and grouping terms, the first governing equation is, 25 (3 3_5) [a__L+ 3+2v a__._ 2(1+V)2(1"231.Q.EZ_.¢ ' r 1+2v 822 (l-v)(1+2v) E 5:2 r2 1 a_L _ a__ _ + (1-v)(1+2v) [a 23Q ' The second defining equation is derived from the definition of Q, equation (3.2-3), ag_ .. (3.3-6) LQ = (l-v) AZ - vaw . Differentiating equation (3.3—6) w.r.t. §;- gives 2 Bo _ a_ = - .a_E_ _ _ ZZ - 3i (3.3 7) Br LQ (1 v) araz (1 v) araz va ar . Equivalent expressions for the first two terms on the right hand side of equation (3.3-5) are derived by differentiating the 2 addition of equations (2.2-9) - (2.2-11) w.r.t. 2:32" and dif- ferentiating Cauchy's third equation of motion, (2.2-2) w.r.t. 3;. 2 Substituting the above equivalent expressions for §—§-' and 2 _ r32 a C22 . 3 3 6 . , BFAZ into equation ( . - ) gives. 2 a. = £l_21_.g a_.L a_.+ h— (3.3-8) at LQ (1_2V) EL at LaZ AZ (araz)] _ afi. _ a. (1+v)p Br +‘(1 v) at L Trz . The term contained in brackets above is the same expression found in equation (3.3-3). Use of this and equation (3.2-8) for ER. an equivalent representation of in equation (3.3-8) gives _ a_. ilixl.a__ _ a.£li:&.L (3.3 9) [ar L + 322 2 E(1'V)a %} 1 a_ _ e. a__. _ - v(1-v) [ L E (1+v a::]Q _ 26 The defining equations (3.3-5) and (3.3-9) are written in vector form as follows: 2 2 (3.3-10) (g; L + [v1] 37 + [v2] 94;“) a QW} = 0 52 at where 1 (l-v)(2+v) -1 [V1] = 2 2 l-v (l-v) v(1-V) 1 2(v2+V-1) 1 [V2] = 2 1-v 2v(1-v) (1+2v)(v-1) {ii} = Ea] . Q Section 3.4. Solutions of the Second Mixed Boundary Value Problem. The method of solution for the matrix form of the defining system, equation (3.3-10), is the separation of variables technique. Since the end boundary stresses are time harmonic, the separation of variables solution will include a time harmonic component. Assuming a vector solution of the'form ,., .4 —02 1 ‘02. (3.4_1) {n} = {nj(r)} e j eiwt = n e j elwt {fij(r)} satisfies ‘ ' 2 . 2 a (3.4-2) (3171. + ozj [v1] - %— [v2]){nj(r)} = o . The above system of equations can be rearranged such that the Operators acting on {fij(r)} become 2 a_ 2 _ (3.4-3) EggL + 5 Mar L + e ij’jun — o 27 where 5 2 1-2 2 2 j j (I‘V) j T 2 2 2 m 2 ej 01 8 dj + (CL) with CL and CT representing the characteristic longitudinal and transverse wave speed. The solution of equation (3.4-3) has the following form: 1 - (3.4-4) nj — [C1 J1(6r) +C2 J1(er)]j 2 (3.4-5) “j = [C3 J1(6r) + C4 J1(er)]j 1 2 4 where TB and Tb are components of the vector {nj}. Substituting solutions (3.4-4) and (3.4-5) into either of the equations represented by equation (3.4-2) interrelates two of the four constants. (3.4-6) n; = [Cl J1(6r) +c2 J1(er)]j 2 (3.4-7) “j = [K3 c1 J1(6r) +1<4 c2 J1(er)]j . The remaining two constants are determined by use of the boundary conditions at r = l and those at Z = 0. Section 3.5. Boundary Conditions. The long side of the cylinder is stress free, therefore, (3.5-1) TrZ\r=l 0’ (3.5-2) crr‘r=l The second boundary condition cannot be used directly for this second mixed case, until Orr is related to the stress components 28 Tr and Q. Deriving this interrelationship is accomplished by determining how R and 022 are related to Trz and Q. Then boundary condition (2.4-13) may be used, which relates Orr to the functions R and 022. The definition of Q, equation (3.2-3) and equation (2.2-2), Cauchy's second equation of motion represent two expressions in which the variables R, 022, TIZ and Q are directly related. Sub- stituting solutions (2.3-2) and (3.4-1) into equations (2.2-2) and (3.2—3) results in 2 2 (3.5-3) [a (l-v)6 R1j = [-2vaa L Trz - a(a2 + emuj (3.5-4) [02(1-V)520223j = ['vae LQ + (a2(1-v> - 2vze>L Terj 2 Multiplying boundary condition (2.5-13) by a2(l—v)6 and substituting solution (2.3-2), allows equations (3.5—3) and (3.5-4) to be substituted into equation (2.5-13). This leaves the boundary condition Orr\r=l = O, solely in terms of TrZ and Q. (3.5-5) crr‘r=1 = O 2 571—2 2 1 2 59 =[(1-v)e ar + s Q + v(01 + (1-v)B) 5ri -- 0 Substituting solutions (3.4-6) and (3.4-7) into boundary conditions (3.5-1) and (3.5-5), provides two linear equations in the coefficients Clj and C2j' When the determinant formed by the coefficients in the above two linear equations vanishes, this provides a transcendental equation for the determination of the eigen- values aj, for which a non-trivial solution exists. The resulting equation for is the same transcendental equation as shown in 03 equat ion (2 .5 -15) . 29 Boundary condition (3.5-1), Tr = 0, provides a means Z‘r=1 by which the constants C1i and C21 (contained in series solution of TrZ and Q) can be interrelated. Doing so, the solution of equation (3.4-3) becomes 4 1 m 1 J (5r) (3.5-6) {nj} = “j = 2 Cj K5 1 ll; J=1 K3 KQKS J1(€r) j 2 1 2 where K3 = (l-v) K4 = a (Z-V)- v3 (0 +'B) = - J1(6) K5 J1(e) Section 3.6. Definition and Solution of the Adjoint Equation. The adjoint equation correSponding to equation (3.4-1) is [ a. '2 +' + -+ (3.6-1) L1 at + 0L [v1] - BEVZ] ]{rL} = o . The adjoint operator in the above equation, when re- arranged, takes the form _ a_, 2 EL. 2 + = 2 _ 2 agl-sz where 6L - 0L + (l-v) 2 2 Solving the system (3.6-1) yields —1 * * (3.6-3) rt = [c1 6r J1(6r) +c2 er J1(er)]L ._2 * * (3.6-4) FL = [C3 6r Jl(6r) +c4 er J1(er)]L . 30 Substitution of solutions (3.6-3) and (3.6-4) into either of the equations of (3.6-1), defines two of the constants, giving _ ~k * (3.4-5) r‘1 = [C1 or Jl(5r) +C2 er J1(er>]L ._ 3&- 6r 9: r 3.6-6 = - c ——-J + -5- ( > 5.2 [ 1 K4 1(61') 02 K3 5(a)], Use of adjoint boundary equation (3.7-8), interrelates the * * constants C and CL , yielding 1 2 P .1 F F" “l + 1 K or .1 (5r) 4 + FL m * 6 1 (3.6—7) {FL} = 1 = 2 CL 1 K6 1" L=1 —— -— er J (er) ‘2 K4 K3 1 where K3 = (l-v) 2 = or (l-v) - 2V8 K 4 32 2 3 'J (5) aéKK K5=._L—. K =-—————3——S- 4 2 2 S=a+8. Substituting the form of the adjoint functions from equations (3.6-5) and (3.6-6) into equations (3.7-8) and (3.7-9), generates * * two linear equations in the knowns C1 and C2' Taking the deter- * minant of these equations containing C1 which is the same transcendental equation as that represented by * and C2, gives an expression equation (2.5-15). 31 Section 3.7. Development of the Biorthogonality Operator. The biorthogonality operator for the second mixed boundary value problem can be derived from equation (3.4-2) and its adjoint equation (3.6-1). Development of this operator proceeds in the same manner as shown in section 2.6. The biorthogonality condition re- duces to all -+ _1 l A. = (3.7'1) [PK at + (r Bijf: 131317;]. Ij’c 0 2 1* where IjL= (a? ‘ Oi) g FL [V1]“j dr . Expanding the first two terms of equation (3.7-1) gives an1 5712 551 552 "1 __l "’2 __i- __L 1 .Jc. 2 3. -2 + 1 2 _. n "l 1 _i = +F¢r+jr4r1r=1+ljt 0. Note that because of boundary conditions (3.5—1), Trz(1,Z,t) = O, the third and fifth terms in the above expression are zero. Equation (3.7-2) now is of the form 1 2 an an. - 1-——1-+"2-—-1-+-1--3—"22 +I =0. From boundary condition (3.5-5), (3.7-4) ((;1 :31- = -G,_ 211-1" G3T\2 )j where: Glj = (l-v)e§ czj =1; (0132+ 8(1-v)) GBj = a; , 32 Multiplying equation (3.7-3) by G allows an equivalent 1 . 11 representation of G1j igL (obtained from equation (3.7-4)) to be substituted into equation (3.7-3). This yields 2 2 _ _" aIL. 2 ‘2 afl. (3.7 5) [ FL(G2 + G3n )j + FL(GI at )j at l _ a_.‘2 2 = + (r ar)rL(G1“ )j]r=1 + Glle o . Grouping and simplifying, equation (3.7-5) reduces to -2 3.. (3.7-6) (GljFL- sz r4)x lr= _ a. .l"2+ = o . (Gij(ar' r)I‘L+ G31F:)n§ \F +6leM The adjoint boundary conditions are taken as follows: _2 _1 - (3.7-8) [GIF - 62? 1L - - EL._ l.-€ ‘1 = (3.7 9) [61(3r r)? + 63? 1L O Substitution of the adjoint boundary conditions, (3.7-8) and (3.7-9), into equation (3.7-6), yields 2 22 _-2'Ii_231 (3.7-10) (aj - at) [(1 v)? - v 3t at \r=1 - [<1-v)(§-;- -)r2 + r 1:. “fir- _1 2 2 l _ + (1-v)[a + 2 £9—-(1+v)] j F+ Lv 3fi dr = o . J E 0 4 1 1 Note that from the defining equation for the second mixed case, equation (3.4-2), that 33 2 a; £v11fi.=} <9“- mm l-§;L>{T\}- Making the above substitution in equation (3.7-10), the final form of the biorthogonality relation is 2 1 2 2 w -w+ q a+ 3— (3.7—11) (aj - 01,) LE {FL}[F]{nj}dr + (v-1) guy 51' L {'n'jwr Cfi—wh‘ l -2 - +[(1-v)(; ' 2;)? - P1]Qj‘r=l + 2 1 $2 :1- all - o [( 'V) VJL 4‘r=13r — 2 - where ‘ [F] = 1 2(l-v) -(1-v) CHAPTER IV PURE STRESS CASE Section 4.1. Four Vector Formulation of the Pure Stress Solution. The purpose of obtaining eigenfunction expansions for the previous two mixed boundary cases was to obtain a means of formulat- ing a solution for a solid cylinder with pure stress end conditions Specified on the finite boundary Z = O. The eigenfunction expansion for this pure stress case is obtained from the previous two mixed boundary value problems. This is accomplished by stacking the two mixed eigenfunction expansions into a four vector expansion. Con- stants of each of the two 2-vector problems are interrelated by the following differential relationship: 3R - - __i- z = (4.1 1) (1 v) 52 2v 52 j LQ + 2"“: . Z. j J The above expression is derived by differentiating equation (3.2-l) w.r.t. 3; and then equating the result to equation (3.2-4). 1 a The eigenfunction expansions for the stress variables shown in equation (4.1-l) are 34 35 co -a2 = J Rj jEIBJEJO(5r) + K1J0(er)]je e iwt m -a'z J - K J K J j jil Bj[ 0 0(6r) +' 1 0(er)]je e iwt (4.1-2) -a.Z m J j jEI Cj[J1(5r) + K5J1(er)]je e iwt q '1 N II (b ca 2 _ 3 Q]. — 1:1 chK3J1(6r) + K4K5J1(er)]je e iwt Substitution of the above expansions into equation (4.1-l) yields the following relationship between the constants Bj and C1: 2 “[02 + lul— m E (l-oh (4.1-3) B1 = a5 cj = lqjcj The desired eigenfunction expansion has the form _, .. F 1 FR 51 4 1 4 - 32 ( . ') O'zz " -ajz 1111!: c e T j 83 r2 4 Q S where 31- K7(J0(5r) + K1J0(er)) (4.1-5) 52 = "8(K0Jo‘5r) + K1J0(er)) s4 - * K3J1(6r) + K4K5J1(er) 36 Section 4.2. Four-Vector Form of the Biorthogonality Operator for the Pure Stress Case. The biorthogonality condition for the first mixed boundary value problem, equation (2.5-10), 1 42-1) (2'2)1-‘F+[A 4d ( ° aj dab g { L} 1]{(‘Pj} r l —1 -2 l 2 + (v-l) [VY + Y 1LE¢ ' Vw Jj‘ra = 0 where [A1] = v -1 l (2+v) The eigenfunctions {2;} of the four vector form of the biorthogonality condition are equal to the eigenfunctions {Eé} of the two vector form multiplied by a constant. The biorthogonality condition for the second mixed boundary value problem, (3.7-ll), is 2 1 1 (4.2-2) (0': - 01:) Q%_£ {T'L}+[F]{flj}dr + (v-l) 2E {11’} §;L{'fij}dr + [(1-v)(-' §-)1‘2 -r‘ 2], T1“ r_ _1 ‘1] + [(1-v)52 - 1L at where [F] = 2 I-l 2(1 -\)) -(1-\J) . The eigenfunctions {2;} of the four vector form of the biorthogonality condition are again equal to the eigenfunctions {fig} of the two vector form of the biorthogonality condition, divided by a constant. 37 Stacking biorthogonality conditions (4.2-l) and (4.2-2), the four vector form of the biorthogonality condition becomes 1‘_ __ _. r' V '1 (4°2-3) I£Y13Y23E13r2}!’ V '1 0 0 T X1 o 1 (2+v) o o x2 dr 0 0 23 -B x3 o o 26(1-v) -B(1-v) x“ L. .2 L. 4.3 1 _ r- F' +(v-1)&{Y,~y¥2,,rl 1‘2}, 0 o o .1 xfl 0 0 0 x2 dr 0 o B—1. 0 x3 4 o o 23--L at D L. _J .J J r- . '1 r’ j +{l’1.?2,i‘1.(;1:' :1???)L 0 0 0 0 x1 o o o o x2 0 0 O -l x3 o o 0 (1m) x4 \ L. L. ‘4 j r81 2 _2 _. r- F- 1 H171, mks}, 3%, x: o o 7 x7 __1__ .2. 2 v-l l-v O 0 X o o o _la. x3 v ar 4. o o o 1- a. b ( war 3 Lx .. j\,=1 The four vector form of the biorthogonality operator is 38 0 0 where matrices H1, H2, H3, H4, H5, and H6 are defined as F r v -1 ‘1 992.1 2 -l .1 H = ’ H = 1 1 (2+4) 2 E 2(1-v) -(l-v) _a L .J r- r- Ei—L o 7 o -1 1 ar H = H a: 3 a— 9 4 - 3 L 0 Br I; g L0 (1 v: P ,, -vfl r'0 -1. a. 1 .. _L v 3r Hs ‘- v-1 1 _ ’ H6 g 0 1 a. L v L ( «Dar and d CHAPTER V DETERMINATIW OF THE EIGENFUNCTION CONSTANTS Section 5.1. Outline of Procedure. Eigenfunction expansions for each of the following stress and stress-like variables, R, 022’ TrZ’ and Q are resolved in their final form as shown in equation (4.1-4). Biorthogonality operator (4.2-4) is used to evaluate the constants found in expansion (4.1-4) and those constants involved in the general eigenfunction expansion for R and Q at the finite boundary, 2 = 0. On the boundary 2 = O, the above stress components have the following eigenfunction expansion form: t... '°° 1' FR 7 2 cjsjJ SJ 1'1 S2 “ZZ 0221: iwt iwt = e 3 Cj 3 e TrZ a Ter ' S 4 4 Q ‘ 2 C S s L .J 2’0 1:1 .1 J j .-.. 1 L .4 j L .1 or r' n 0° m 1 F" 1‘} z c s 5 1:1 11 2 (5 1 1) or22b S . - = C 3 Ter j S 2: (3133‘ s“ H 1 = 1 L. J 1 L— .J 39 40 Operating on equation (5.1-1) with the biorthogonality operator Q which is represented by equation (4.2-4), yields the following L, infinite system of equations: . m * * 5.1.-2 T + M C = B N +C N C ( ) E1 111 (411 1494 L j where TL - involves the known boundary stresses OZZb and Ter° sz - represents a summed term created by the eigenfunction expansion of R and Q on the boundary Z'= O. * * BLNLl’CfNLZ - represents the biorthonormalized eigenfunctions of the first and second mixed boundary cases. The infinite system of equations represented by equation (5.1-2) is solved by truncating the series, then utilizing the Gaussian- Elimination method as a means of solving for the j-number of constants. Section 5.2. Evaluation of the Adjoint Eigenfunction Constants - * * BL - CL . A closer examination of equation (5.1-2) shows that the * * constants BL and C need to be evaluated before this infinite L system of equations can be solved. These constants are contained within the series solution for the adjoint eigenvectors of the first and second mixed boundary value problems, equations (2.6-7) and (3.6-7). Since no individual defining condition exists for the de- termination of these constants, the constants B: and c: are chosen such that the eigenvectors for each of the mixed cases become a biorthonormal set. That is ‘k * B 3.1. C a N .1. L 2 N L (“H 41 Section 5.3. Evaluation of Terms Represented in Equation (5.1-2). °° N1 5. -2 + C = B +C: N2 = 00. m o ( 1 ) TL 121’“ij [:N 4. 3C4, " 1 The above representation of an infinite system of equations is obtained by operating with biorthogonality operator (4.2-4) on the series expansion (5.1-1), representative of the eigenfunction expansion for the stress components R, 022, TrZ and Q. A repre- sentation of the terms shown in equation (5,1-2) is as follows: 1 = -2 -1 (5.3-1) TL. JQUZWW - 1' 3L ozzbdr 1 + 25 gin-v)? + 5111’ 'rerdr 1"1 3_, v ‘1 -2 + (v-1)& PL at L Terdr - 3tl (v? + Y )LOZZb‘r=1 1 1 _ 3 -1 —2 1 _ -1 _ -2 a (5.3 2) MU £1ij + y 1szdr 5&[1‘ + (1 v)? j‘bsjdr + (v-1)§LF2 g—'L der + [<1-v) é - sir-)5: PL]Sj\r-1 —1 -2 [WL + YL]Sj‘r=1 534 __SJ. JB\ 1 (v-l) + 1- ‘2-.1..p1 [(WerFL r=l (5 .3-3) 3* N1‘= 1 [ $1 + g2]81 dr ° chvt L4, 1 —2 _ -1 .2 d +3 “2+th YLJSL r 1 -1 -2 1 2 (v-l) [Wt + VLHSL ' “Stun-=1 42 1 * 2 -2 -1 3 _ .2 1- s d (5.3 4) CLNL 39;“ v)? +FJL L r 1 -B£L(1-v)I‘2 4511‘s: dr 1 + (v-1) 5?: §;Ldr + [(1-v)(‘:" - gpr r11, Sit-=1 4 + [(1-\:)'f2 - _JLBI' —-3‘ \r=1 The term T in expression (5.3-l) can be evaluated if the L boundary stresses and T are specified. CT is evaluated or22b 1% L for Specific boundary stresses in chapter seven.) The adjoint eigen- , + 4- functions {T;} and {fl} involved in the definition of TL’ are defined in equations (2.7-7) and (3.6-7) respectively. The evaluation of sz, expression (5.3-2), is entirely de- pendent upon the eigenvalues of the frequency equation, 535 2 J + J(6) s2 ()J(6 =0 ( - - ) a £6 0(e)J1(6) 86¢ J1(s) 1 J1 e 0 ) - An analysis of the above frequency equation for roots, yields the following: 1.) real roots 2.). pure imaginary roots 3.) complex roots. Due to the nature of the above roots, M must be evaluated Lj independently for the following cases: 43 Case 1: at # aj Case 2: a = a L Case 3: a = -a 4. 3 Evaluation of MLJ - Case 1.) JO(5L)JQ(6’1) (5.3-6) M3. = B: K71 (vi-D6 {gro J (a, r)JO (aj r)dr + (w _1) ' J Q(6 MN (6 ) + (v+1)6 HK1 {:gro J (ej r)J0 (6 Lr)dr + (v 1) J (e )J (6 ) 0 4' 0 i + K2L(1+VK0L)3L {it “10(f’jm'o("w.r)dr + > + K4K5(eJo(e) - 5(0)} 4. . Evaluation of MLj ' Case 3.) — 3*K c*[ (5.3-s) MM 4. n' L 1- The brackets above represent the identical terms found in expression (5.3-7) for MLj. 45 The normalization constants, N: and NL2 evaluated, are: 1 1 2 2 (5.3-9) NL = Kn {(1-K0) {OWN gr J0(6r)dr + Kze(1-Ko)£r Jo(er)J0(6r)dr} 1 _ 2 l - e(1+\))l(2l(1 gr J0(¢r)dr + (v-l) (v+l)6Jo(6) + K2(1+vK0) eJo(e) ' {(IWKO)JO(6) '1' (1+v)K1JO(e)}} L K 1 ' 1 2 (5.3-10) NL = 5(1 — {in {gangs J:(5r)dr + (2-K4)K5‘gr J1(5r)J1(er)dr} K1 K1 3 3 2 3 2 } +(1-v) 5(1-—-)rJ(5r)+KeK(l-—&)rJ(er)dr { K4; 1 . 15 K3; 1 K K 2 2 .. El + E1)6J1(6) + 2¢K6J1(e) - E1 5 J0(6) "' K66 J0(s)} 4 4 -J1 1+5.” (a>+9—+—"9—w(> - 6(v K4 1 v 6 l‘ {g(bJOM) - J1(6)) + K4K5(eJo(e) - J1(e))} L CHAPTER VI DETERMINATION OF THE EIGENVALUES Section 6.1. General Discussion. The eigenvalues aj’ which correSpond to possible modes of wave propagation in the solid cylinder, are roots of Pochhammer's frequency equation. This is 2 2 (6.1-1) 0 a. J1<5)Jo(e) - s J0(6)J1(e) + be J1(6)J1(e) = o where 52 = a2 +'B %%E%¥l» e2 = a2 + 23 2 2 2 s =a+e 5 3%(14'0. Roots of equation (6.1-l), correspond to modes of axisymmetric vibration inla solid circular cylinder. The specific mode of axi- symmetric vibration is dependent upon the form of the eigenvalue, which may be real, imaginary or complex, depending upon the fre- quency of wave motion. Axisymmetric vibration consists of three primary motions, an axial shear, a radial shear and a longitudinal motion. Figure VI-Z graphically illustrates several modes of each of the primary motions, represented by Specific branches over a wide range of the frequency Spectrum. Note that for low frequencies, a < 1 (Table VI-l): (a) Mode (l) which represents longitudinal vibration, has eigenvalues or propagation constants which are pure imaginary. 46 47 (b) Mode (2), the axial shear mode, has propagation constants that are imaginary or complex, depending upon the frequency of the wave motion. (c) Mode (3), which is the radial shear mode, has roots which may be real, pure imaginary or complex, again depending upon the frequency of the wave motion. Researchers who have contributed to the investigation of these primary modes are as follows: Davies [4] investigated the lower modes of longitudinal vibration. Holden [8] examined the next five higher longitudinal modes (correSponding to real propagation constants.) Onoe [24] first published the imaginary propagation constants; Adem [1] investigated the complex propagation constants. Both Adem and Onoe related the complex propagation constants to either an axial or radial shear mode. A composite of the previous work is graphically illustrated in Figure VI-2, as analyzed by Onoe,'Mindlin and MCNiven [24]. This plot represents Pochhammer's frequency equation, over a wide range of frequencies, showing both real and complex branches of the axisymmetric modes of vibration. Further observation shows that the longitudinal modes will always have the largest pure imaginary prepagation constants (eigenvalues) for any given frequency. The next higher mode of longitudinal vibration is identified as having the second largest root. This would imply that the order of the lowest to the highest mode of longitudinal wave propagation can be identified by the magnitude of the pure imaginary propagation con- stants. 48 Davies [4],in 1948, correlated phase velocity with wave- length for the first three longitudinal modes. Figure VI-3 illustrates these results. Figure VI-4 shows the same correlation for data presented in Table VI—l and Table VI-2. This last section represents a qualitative view with respect to correlating root types with correSponding modes of wave propagation. Section 6.2. Evaluation of Pochhammer's Frequency Equation. Roots, a , of Pochhammer's frequency equation, 3 2 2 (6.1-1) P(aj) = (a be J1(6)Jo(e) ' S JO(6)J1(s) + 68(1+v)J1(5)J1(s))j = 0 were determined only within the first quadrant of the complex plane, since P(a) is symmetric about the real and imaginary axis. The above frequency equation is evaluated using Muelleris method [21]. Plots of these eigenvalues in the first quadrant of the complex plane are shown in Figure VI-S for frequencies of 100 and 20000 c.p.s. . It is interesting to note that eigenvalues computed for the low frequency of w = 100 c.p.s. compare very closely with those obtained by Little and Childs [14] for the static solution of this problem. This former comparison is made in Table VI-3. Roots for six frequencies ranging from 100 c.p.s. to 50000 c.p.s. are listed in appendix B. The calculations were made for a material density of 15.1 me./ft3., a Young's modulus of 30 X 106 LBf/in2., and a Poisson's ratio v of .33. All dimensions are in units of feet. 49 Table VI-l. Phase Velocities of the First Three Longitudinal Modes C0 = E-- phase velocity of waves with infinite wavelength Velocity Frequency Eigenvalue Mode wavelength Phase Velocity Ratio .1. 1r. . 2.. . 9. FT. 9.. w ’ C°P°3' 9," (FT. 1 2“ C a " (sac.) c0 100 .03714 1 .0059 16917. 1.002 1000 .3729 1 .0593 16846. .9960 2000 .7558 1 .12 16575. .98 6380 3.670. ‘ 1 .58 10994. .65 1.600 2 .25 25032. 1.48 8000 4.953 1 .7883 10147. .60 2.566 2 .40 19589. 1.16_ .9216 3 .15 54541. 3.22 3.582 2 .5701 17540. 1.03 2.147 3 .3417 29264 1.73 Frequency w - Capos. 15000 20000 50000 50 Phase Velocities of the First Four Longitudinal Modes Eigenvalue .1. 0’ (FL) 9. 7. 4. 3 13 11 32 28 26 24 757 511 931 .841 .046 .06 .62 .22 .59 .17 .48 .44 Radius wavelength r = q_ 1 211 1.553 1.195 .785 .611 2.08 1.76 1.37 .98 5.18 4.48 4.21 3.88 Phase Velocity =m~ 9659. 12547. 19113 24533 9632. 11332. 14546. 20127 9640. 11163 11834 12854 Velocity Ratio C/C0 .571 .7418 1.13 1.45 .57 .67 .86 1.19 .57 .66 .70 .76 51 Table VI-3. Comparison of Low Frequency Complex Eigenvalues with Static Problem Eigenvalues* n Static Problem Eigenvalues Low Frequency Eigenvalues 1 2.722176 +.1 1.362197 2.721465 + 1 1.362190 2 6.060083 1.637624 6.060082 1.637622 3 9.266835 1.828256 9.266835 1.828256 4 12.442529' 1.967241 12.442528 1.967241 5 15.605440 2.076284 15.605440 2.076284 6 18.761738 2.165933 18.761738 2.165933 7 21.914138 2.24023 21.914137 2.242023 8 25.064033 2.308104 25.064033 ‘ 2.308105 9 28.212220 2.366500 28.212220 2.366502 10 31.359185 2.418808 31.359186 2.418809 From Little and Childs_[16]. .Eacuooam xocozaocm we mogucmcn omega umwzop so» :o_um:ao zucoaoogw m.cossagsuoa we pay; 1- p1~> mmawmm .mmcupn mumcpvgoou any co mucmsuom xopnsoo any we mcovuoofioca as“ men mmcvp cognac 52 moves Lumen Payout saw: wouuwoomma mucmumcoo cowaaooaocm moves camgm Fmpxa sow: umuawoommo magnumcoo cowouaoaoca 66665 46:.6asvacop :81: umuapuommm mucoamcoo cowaammaoca penumcoo covpammaocg mmw—COPmCOEwO fl >w+x fl 3| 3m dl¢o hocmaamc$ mmmpcowmcmswo u u no 1 «u r- 19111 Hu Nu a mu 1% >1 I '/ . \ ‘\ av ‘- \ \‘R‘Q‘ \\ v1 )1 5“)“. was. ‘4. 7 1X1 .1" 7 V fl,“ is ‘1 f \ \, l,4,7,lZ,lS - Branches representative of longitudinal modes. 2,5,8,lO.lB - Branches representative of axial shear modes. 3,6,9,ll,l4 - Branches representative of radial shear modes. 9 = Dimensionless frequency C = X+ iY t Dimensionless propagation constant. ale 8 IE Dashed lines are complex segments. Order of branches are given by numbers next to curves. Bar over branch indicates reflection of branch about v=0 plane FIGURE VI-2 -- Plot of Pochhammer's frequency equation showing linkage of *real. imaginary and complex branches at high frequencies. .9 Phase Velocity - Q U! l. / ( ’(oo.cs) .571 6 P __ —- —11 t—o- .‘0 1- Modes 1,2,3 - Longitudinal modes .2 ’ l L 1 l 1 1 r .9 .8 ‘ 1.2 1.6 2.0 L ___ 0: _ ( Radius ) A 24 Wavelength CO = {E' - ’Phase velocity of waves having infinite wavelength 0 FIGURE VI-3 -- Phase velocity variation of extensional waves in cylindrical bars for v = .29. (Davies[4] ) c '6 Phase velocity - O 1.1 ,- 1.4 F 3 1.2 f \2 ‘0 R l .8 i- .6 P Co 21 Modes l.2,3 - Longitudinal modes .4 r- .Zr 0 L l _l 4:» .4 .8 . 1.2 1.6 2.0 3_ = a - ( Radius ) A 28 ave ength Co =‘ {E = Phase velocity of waves having infinite wavelength 0 FIGURE VI-4 -- Phase velocity variation of extensional waves in cylindrical bars for v = .33 56 52.3.85..de 80.8 can .mdd oo— »o mmwocmzamcw go$ ocmpq xmaasoo on» we acmgvcmsc was?» as» c? .Acv mmapm>cmmvm u- m-H> mmawHu ov mm mm em .m.a.o ooo.oN 1 omoaoU 00—. I om op q NF unwl ii (11 U o.p o.~ o.m o.¢ o.m cg o.w 3.. 3: 9: of 0.2 CHAPTER VII STRESS SOLUTIONS OF SPECIFIC BOUNDARY VALUE PROBLEMS Section 7.1. Specification of Applied Boundary Stresses. The general solution for the pure stress formulation is given by equation (4.1-4), where FR '7 9° PSI-1 o S2 zz -ojz iwt = 3 'r Cj S e e rz 4 L—Q .4 Lns .J J’ = 1 j The solution above is complete, if the coefficients Cj are known. The constants Cj are determined by solving the infinite system of equations represented by equation (5.1-2), where *1 - + MC=BN (5.12) T 2:. [LL +C*N2 L 6J3 l—l...a. C L L] L Equation (5.1-2) is obtained by operating on equation (4.1-4) with the biorthogonality operator 6 after boundary stresses for K) T and o are Specified at Z = 0 in equation (4.1-4). There- rZ ZZ _ 's is dependent on the fore, solving equation (5.1-2) for the Cj boundary stresses specified at Z = 0. These Specified stresses are contained only in the term T where L! 57 58 1 (7.1-1) TL g [(2+v)v - y JLOZZbdr 1 -2 1 + 23 g [(1-0)r + r ]LTerdr 1 _1 a. + (0-1); FL Br L Terdr .2. ‘1 '2 - v-l (vY + Y ) t O2Zblr=1 The adjoint eigenvectors 7:, Pi, E1 and f: are defined in equations (2.6-7) and (3.6-7). The term TL, represented in equation (7.1-1) will be evaluated for the following Specific pure stress end conditions Problem OZZb Ter l 1 - 2r2 0 2 0 2.4r - 2.6r3 + .2r5 3 [1 - 1.5r2]4 - .137499 0 Section 7.2. Evaluation of the Boundary Term TL . Problem 0Z2) TrZ Loading 1 1 - 27:2 O self-equilibrated Substitution of the above stresses into equation (7.1-1) yields = ' L _ ' L , ‘2. TL [(1+V)[62 1]J1(6) + (1+v)[(v_1) 6170(6) + [£2- - 11K2[2+v - KOJJ1(8) C v 4; + K2[(V'1) (1 + vK0)e - 6 (2+V - K0>1J0(e)]l, . 59 Problem OZZb Ter 2 0 2.4: - 2.6r3 +1.2r5 Proceeding as before K3 22 .4 76.8 4.4 38.4 T = 2 1 --— -+-——- --——- - -———- -—- 1 M K :65, L 0:116 56 62 ]J1(6) [65 63JJ0(6)]L * l§+ 38. 4 22.4 26.8 + (v-l)6LCL115+ --30]J (60-1 62 + 67. 111(6)]L __ 22.4 76.8 + (v-1)K6e C :111 + —:-§-]J 0(8) - 1““: + €41J1(e) 1L Problem CZZb Ter Loading 3 [1 - 1.5r2]4 - .137499 0 Self-equilibrated Proceeding as before * - 1 1 3 1 5 = . 2 - . TL BL(I+V)64[ 86 5 gr J0(5r) 6£r JO(6r) + 13 5g: J0(6r) 1 7 1 9 - 13.53r Jo(5r) + 5.06gr J0(6r)]Ldr K0 1 7 1 9 - 13.5 it Jo(er) + 5.06 St Jo(gr)]Ldr * . K e 1 1 1 + B [(2+0 - )-2-j [ 8625 r J ( r) - 6 r3J ( r) +.13 5 jrSJ ( r) 4 Kb L ' g 0 e g 0 e ' 0 0 e K e 2 (130 [awe J 0(6) + (2+v- «0) ig—Jo 0091!; - B :(. 075) Section 7.2. Results. This investigation entails a study of the stress modes which exist on or near the end of a semi-infinite solid circular cylinder. These stress modes were initiated by dynamic, self- 60 equilibrated loads being applied on the finite surface, Z = 0. The analysis was carried out for three Specific problems, listed in section 7.2. The accuracy of the pure stress solution (equation 4.1-4), is determined by how closely this eigenfunction expansion matches the known applied boundary stresses at the loading surface, Z = 0. Con- vergence of the eigenfunction expansion for problems 1, 2 and 3 are listed in Tables VII-l through VII-3. The eigenvalues used in calculation of the above eigenfunction expansions are listed in Appendix B, Table B-4 (w = 6380 c.p.s.). The reSponse of the solid cylinder to self-equilibrated loads consists of decaying and non-decaying stress modes. These stress modes for problems 1, 2, and 3 are listed in Figure VII-1 through VII-23. Section 7.3. Conclusions. A dynamic Saint-Venant region does not exist for the solid circular semi-infinite cylinder. Examination of Figures VII-1 through VII-23 shows that non-decaying modes exist for all stresses (Orr’ a , azz, Trz), in each of the three problems examined. These 00 prOpagating modes have propagations constants which are pure imaginary. In section 6.1, these pure imaginary eigenvalues were shown to correspond to longitudinal modes. This was exhibited by the fact that phase velocities of modes corresponding to the imaginary propagation constants (illustrated in Figure VI-3) were identical to the phase velocities of the first 3 longitudinal modes calculated by Davies [4] and illustrated in Figure VI-4. 61 Figure VI-2 is a plot of Pochhammer's frequency equation over a wide range of frequencies, showing both real and complex branches of the axisymmetric modes of vibration. This figure illustrates the fact that longitudinal modes, correSponding to pure imaginary propagation constants, exist for all frequencies of axisymmetric vibration. Therefore propagating stress modes will occur for any frequency of axisymmetric vibration. This figure also shows that as the frequency of vibration increases, higher modes of longitudinal vibration become prevalent, with each mode correlat- ing on a one to one basis with a pure imaginary propagation constant. An examination of eigenvalues for several frequencies, located in Appendix B, shows that as the frequency increases,the number of pure imaginary propagation constants (oj), increase. Therefore at higher frequencies, more stress modes of the non—decaying type will exist. The applied self-equilibrated loads cause non-decaying stress modes of much greater magnitude than the applied loads themselves. Figure VII-2 shows the applied boundary stress 022 at Z = 0. Figure VII-3 through VII-5 displays the resultant stresses at Z = 0 due to the applied boundary stress. Figures VII-7, VII-8, VII-10, VII-11, exhibits in what manner the non-decaying modes propagate away from the end Z = 0. Note that in Figure VII-ll a non-decaying mode, is propagated down the bar of magnitude 71.5 times the O' 00 maximum applied boundary stress. The non-decaying stress modes are very sensitive to the dis- tribution of the self-equilibrated applied boundary loads. Problems 1 and 3 both involve applied normal stresses OZZ of comparable 62 magnitude with zero shear stresses. The non-decaying stress modes for Orr’ 006’ and TrZ caused by the above applied boundary stresses, differ radically as illustrated in Figures VII-3 through VII-5 for problem 1 and Figures VII-21 through VII-23 for problem 3. An analysis of the stress modes which exist at lower or higher frequencies could have been studied, had the convergence of the eigen- function expansion been better. Figure VII-l shows convergence of the eigenfunction expansion for frequencies of 100, 2000, and 6380 c.p.s. 0.0 l r .0 Table VII-1 63 . Axisymmetric Trial Problem No l: 2 022 - 1 - 2r w = 6380 c.p.s. Trz=0 Number of eigenvalues used prescribed 1.0 .98 .92 .82 .68 .50 .28 .02 -.28 -.62 -l.0 .85455 .82412 .73463 .5914 .4029 .1806 .0623 .3113 -.5514 .7685 -.9499 .9739 .9609 .9169 .8296 .6886 .4934 .2561 -.0056 -.284 -3935 -1.0044 15 .9997 .9798 .9201 .8201 .6797 .5000 .2801 .0198 -.2800 -.6198 -1.000 21 .9999 .9799 .9201 .8198 .6801 .4998 .2801 .0199 -.2801 -.6l98 -.9999 0.0 1.0 Table VII-2 TrZ 64 Axisymmetric Trial Problem No. 2: = 2.4r - 2.61:3 + .Zr Number of eigenvalues used prescribed 0.0 .2374 .4592 1.6502 .7956 .8812 .8939 .8218 1.6543 .3826 .00 0.0 .3337 .6369 .8817 1.044 1.109 1.069 .9233 .6829 .3666 .0000 0.0 .2589 .4924 .6808 .8127 .8832 .8877 .8167 .6547 .3846 .0000 5 °22 15 0.0 .2369 .4592 .6504 .7955 .8811 .8940 .8219 .6542 .3827 .0000 = 0 21 0.0 .2372 .4593 .6502 .7956 .8812 .8938 .8219 .6542 .3827 .0000 r 0.0 1.0 Table VII-3. prescribed oz 65 2 4 z = [1 - 1.5: 3 — .137499 Number of eigenvalues used .8625 .8038 .6432 .4223 .1961 .0151 .0927 .1325 .1375 .1354 .075 .223 .204 .1511 .0689 30327 .1426 .2485 .3393 .4056 .4413 .4439 .5769 .6174 .6841 .6507 .4147 -.0086 '04530 —.6643 -.4447 .1884 .919 Axisymmetric Trial Problem No. 3: T 15 .8755 .8195 .6415 .4133 .2108 .0138 .1103 .1300 .1282 .1249 .1041 r2 21 .8629 .8034 .6458 .4214 .2001 .0146 .0893 .1317 .1388 .1333 .0663 66 Table VII-4. Convergence of Eigenfunction Expansions for Problem 2 Boundary Functions at Three Frequencies r prescribed w = 100 c.p.s. w = 2000 c.p.s. w = 6380 c.p.s. 0.0 0.0 0.0 0.0 0.0 .1 .2374 .2181 .2385 .2374 .2 .4592 .4230 .4518 .4593 .3 .6502 .6014 .6337 .6502 .4 .7956 .7400 .7905 .7956 .5 .8812 .8258 .8577 .8812 .6 .8939 .8455 .8717 .8938 .7 .8218 .7859 .8044 .8219 .8 .6543 .6338 .6119 .6542 .9 .3826 .3762 .3608 .3827 1.0 0.0- , 0.0 0.0 0.0 .NL .m.a.u owm.e .ooo.~ .oo. 46 ”maucmaomat 46 N Empaoaa so» meowuocze atmucaon nonwgommea we» ;u_3 meowmcmaxo cowuocawmcmwm mo comwaoqeoo .. FuHH> mmame 06F me Q0 Q. N. \ l N. O "NND .7 mLN. + mas.~ - t¢.~ "N44 .1 a. 6 meme: :o_»o==w acaecaoa emnwgommta .11..11:- 0. @- .m.a.o omm.m u 3 .III.|11 .m.a.u ooo.m u 3 ....... .m.a.o oo— 8 3 -11---- NLP a Em_noea .ouN “a moses mamas” measauau-=6= as? acaNNa wastes seaeesoa umapaa< -- ~-HH> manual Eu 6 6 . ol 0 I O0”! N.F ‘ m a o m N P 31.1 .44 A . . a . . / — /. 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LN: 85 \ \ \ I \ \ \ l \ \ \ \ I. \ \ . \ \ 1 \ \ ‘\\\ I 3: P...‘ \sm\\\ .2: .maN .2: _IIIII. . . . . l ............ 3o / .Co / 86 .NN m Em_noga .onN um waves m>m3 mmmgpm mcwmmumuuco: new acaNNo mmmgum umwpaa< -- omuHH> umaon 0 mo mmuoe m>mz mmmgum mewaaumuucoz ura..:|. . u . u - NN mNmF efimgm _ NV . o .NNo mmmgpm vmwpaq< -uununu 87 .— Lo mo mmuoe m>m3 mmeum mcwxaumvicoz III..|ou NNo op mat Lgo mmmgpm “cup—ammm ..... .- m Ew—QOLA .OHN um meOE m>63 mmeum mcwhmuwblcoc mar bcm o¢b mmmLHm Hcmupzmmm II NNIHH> mmzwmm $0 0.7 ON. 3”. y 88 «2...: 333 3383-52 mob $0 mmUoE § \ o._1_ s NNo on can Nan mmogum pamppzmmm ------- ft: 89 .m Empaoga .ouN um mmuoe m>m3 mmmgum a:_xmumuucoc mu? uca ng mmmgum acmapammm .. mN-HH> “mama; ng p. o F.. N.. mum. .9 fl _ _ . J! o / N. _ / e. -n // _ F muoz o _ \‘ N «no: ng yo mucos m>oz mmmgum ch»mumuucoz .||.alt. NNo oa «av Nsp mmmgum ucmupammm u-------- o; 3: .. B IBLImRAPHY Ff. '-' - '.—£ «7.2 w 10. ll. 12. BIBLIOGRAPHY Adem, J., On the axially symmetric steady wave propagation in elastic circular rods, Quart. Appl. Math. 12, pp. 261-275, 1954. Boley, B.A., The application of Saint-Venant's principle in dynamical problems, J. Appl. Mech., vol. 22, no. 2, pp. 204- rvq 206, 1955. Curtis, C.W., Propagation of an elastic strain pulse in a semi- infinite bar, Stress wave Propagation in Materials, Davids, N., £-“ ed., Interscience Publishers, New York, pp. 15-43, 1960. ' Davies, R.M., Phil. Trans. Roy. Soc., A240, pp. 375, 1948. Devault, G.P., and Curtis, C.W., Elastic cylinder with free lateral surface and mixed time—dependent end conditions, J. Acoust. Soc. of Am., vol. 34, pp. 421-432, 1962. Folk, R., Time dependent boundary value problems in elasticity, thesis, Lehigh university, 1958, (University Microfilms, Ann Arbor, Michigan). ‘ Folk, R., Fox, 6., Shook, C.Am, and Curtis, c.w., Elastic strain produced by sudden application of pressure to one end of a cylindrical bar, 1. Theory, J. Acoust. Soc. Amer. 30, pp. 552- 558, 1958. Grandin, HmT., and Little, R.W., Dynamic Saint-Venant region in a semi-infinite elastic strip, Journal of Elasticity, vol. 4, no. 2, pp. 131-146, 1974. Holden, A.N., Longitudinal modes of elastic waves in isotrOpic cylinders and slabs, Bell System Technical Journal, vol. 30, pp. 956-969, 1951. Johnson, M;W., Jr. and Little, RQW., The semi-infinite elastic strip, Quart. Appl. Math., vol. 22, pp. 335-344, 1965. Jones, O.E. and Kannedy, L.W., Longitudinal wave propagation in a circular bar loaded suddenly by a radially distributed end stress, J. Appl. Mach., vol. 36, no. 1, pp. 470-478, 1968. Jones, 0.E. and Norwood, F.R., Axially symmetric cross-sectional strain and stress distributions in suddenly loaded cylindrical elastic bars, J. Appl. Mech., vol. 34, no. 1, pp. 718-724, 1967. 90 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 240 25. 26. 91 Klemm, J.L. and Little, R.W., The semi-infinite elastic cylinder under self-equilibrated end loading, Siam J. Appl. Math., vol. 19, pp. 712-729, 1970. Kolsky, R., Stress waves in Solids, Oxford, Clarendon Press, pp. 54, 1953. Langer, R.E., A problem in diffusion in the flow of heat for a solid in contact with a fluid, Toboku Math. Jour., vol. 35, 1932. Little, R.W. and Childs, S.B., Elastostatic boundary region in solid cylinders, Quart. of Appl. Math., vol 25, no. 3, pp. 261- 274, October 1967. Little, R.W. and Thompson, T.R., End effects in a truncated semi-infinite cone, Quart. J. Mech. Appl. Math., vol. 23, no. 2, pp. 185-196, 1970. Hm v2.1.7 Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, Inc., New York, 1944, 4th ed. (1926), pp. 131-132. ' Miklowitz, J., The propagation of compressional waves in a dispersive elastic rod, J. of Applied. Mech., pp. 231-239, 1957. Mindlin, R.D. and Herrmann, G., A one dimensional theory of compressional waves in an elastic rod, Proc. First 0.8. National Congress of.Applied Mechanics, ASHE, New York, 1952, pp. 187-191. Mindlin, R.D. anndNiven, R.D., Axially symmetric waves in elastic rods, J. Appl. Mech., vol. 27, pp. 145-151, 1960. Mu11er, D.E., A method for solving algebraic eQuations using an automatic computer, Mathematical Tables and other Aids to Computation, vol. 9-10, pp. 208-215, 1955-56. Oliver, J., Elastic wave diSpersion in a cylindrical rod by a wide-band short-duration pulse technique, Journal of the Acoustical Society of America, vol. 29, pp. 189-194, 1957. Onoe, O.E., McNiven, H.D. and Mindlin, R.D., Dispersion of axially symmetric waves in elastic rods, J. of Appl. Mech. , vol. 29, pp. 724-734, 1962. Skalak, R., Longitudinal impact of a semi-infinite circular elastic, J. Appl. Mech., Trans. ASME, vol. 70, pp. 59-64, 1957. Todhunter, I. and Pearson, R., A history of the theory of elasticity and of the strength of materials, vol. 2, pt. 1, Cambridge Univ. Press, London, Chapter 10, pg. 12, 1893. APPENDICES APPENDIX A REDUCTION OF BOUNDARY CONDITION OPERATORS FROM THAT OF EQUATION (2.4-4) TO Boundary Condition Equation (2.4-4) Pd- 2 2 a2°22 1. 2K B-1 BEEM‘TB'E‘VZ =a:[1%(°zz -VR>- 2 +IIGL§3° Boundary Condition Equation (2.4-5) B.2 . a o g n . - " ZZ+1 Trz 9- [ (d 2 v3) - #1 322 +1+v 322 The above boundary conditions have equivalent right hand sides, therefore substitution of solutions for R and 022 (represented by equation 2.3-7) yields the same functions for the right hand sides in both equations. Equations 3.1 and 8.2 are now of the form 2 2 '“z i t B.3 5.. [‘0' L5 - a—z-Trrz = F(r)e e u) at -a . _ z lmt 3.4 Trz F(r)e e where F(r) = F(J0(6r): J0(gr)) Solving equation (B.3) for TrZ gives 'a Z = + TrZ TrZ(homogeneous) C1F(r)e e 3.5) lwt 92 93 where homogeneous solutions of are dependent on the eigen- TrZ (e = 0)- values a = 0 and a = fu/D Note that solutions B.4 and 3.5 are equivalent, when T is zero. An examination of the diSplacements rZ(homogeneous) f th d or e mo es of wave propagation corresponding to TrZ(homogeneous) show diSplacements of magnitude zero. These modes in particular correspond to eigenvalues of a = O and a ='—J£- (s = O). fu/p The displacement formulation for the axisymmetric loading of a solid Cylinder provides a means of demonstrating displacements are zero correSponding to eigenvalues a = 0 and a ='J£—- (e = 0). /E The governing equations for axisymmetric vibration of a solid cylinder are B.6 pfi-(i+2p)gf+2u§3 3.7 pei=(x+2u):%-§*§;(rie) where 3.8 A =§§f§l+g§ A - dilitation B.9 . 69 = %‘[§%'- 3%] me - rotation about the e-axis The stress components which vanish at the surface of the cylinder, Orr and Trz’ related to the diSplacements are = + 2 32. 8.10 Orr AA u Br = as 23$ 94 The stress variables for the stress formulation of this problem are harmonic w.r.t. time and decay exponentially w.r.t. Z, therefore the displacements are taken as 'dZ iwt 8.12 u U 8 e u -aZ . t 8.13 w = W e elm w Substituting solutions 8.12 and 8.13 U(r) W(r) into equations 8.6 and 8.7 gives 7- 3A - o - = “2 B 14 pm U (l‘+ 2“) at u a we 8.15 -w2W=-a(x+2)A-z“‘a-(rt5). Using equations (8.8) and (8.9), 89 or A are eliminated from equations (8.14) and (8.15). This leads to the two equations, 2 1 2 r 5r at 32' as ’ . w m g 8.17 —79-+l“—e'--a+ezw =0 r at 2 9 5r r where 3 18 57- 2+2_