boundary element programming in mechanics_104207485

7/29/2019 Boundary Element Programming in Mechanics_104207485

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B O U N D A R Y E L E M E N TP R O G R A M M I N G

I N M E C H A N I C S

X I A O - W E I G A OA r iz o n a S ta t e U n i v e r s it y

T R E V O R G . D A V I E SG l a s g o w U n i v e r s i t y

S i* 1*ftpCAMBRIDGEUNIVERSITY PRESS


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C o n t e n t s

Preface , page xiiiLegal Matters . xvP A R T I. L IN E A R P R O B L E M S

1 Introduction 31.1. Intro duc tion 31.2. A No te on Program ming 31.3. M athem atical Preliminaries 51.4. His torical Ske tch 121.4.1. Ap pro xim ate M ethod s 12

1.4.2. BE M in Solid Mechanics 131.4.3. BE M in Elasticity 141.4.4. BE M in Elasto-Plasticity 141.5. Closu re 15

2 Theory of Elasticity 1 62.1. Introduction 162.2. Displacements 162.3. Stresses 172.4. Stress-S train Relationsh ips 182.5. Nav ier-Cauc hy Equ ations of Equilibrium 192.6. Redu ced Forms in Two Dimensions 20

2.6.1. Plane Strain 212.6.2. Plan e Stress 222.6.3. Axisymmetry 222.7. An isotropic M aterials ; 232.8. Closure 24

3 Boundary Integral Equations fo r Elasticity 2 53.1. Introduction 253.2. The Kelvin Fu nd am enta l Solution 253.3. Betti 's Reciprocal Work Theorem 273.4. Somigliana Iden tity 283.5. Boundary Integral Equ ations 30

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3.6. Intern al Stresses 313.7. Closure 334 Numerical Implementation 34

4 . 1 . Introduction 344.2. Bo und ary Discretization 344.3. Interp olatio n of Field Qu antities 374.4. Discretized Bo und ary Integral Eq uations 384.5. Ad aptive Integration 394.6. Singular Integ ration 47

4.6.1. Weakly Singular Integrals in Three Dim ensions 474.6.2. Weakly Singular Integrals in Two Dim ension s 494.6.3. Strongly Singular Integrals 514.7. Eva luation of Bou nda ry Stresses 534.8. Symmetry 574.9. Co rners and Edg es 604.10. M ultiple Re gions 634.11. System Eq uation Solution 644.12. Closure 66

5 The Elastic Program Code 675.1. Introduction 675.2. Scope of the Prog ram 675.3. Program Structure 675.4. Glo bal Variables 68

5.4.1. Glob al Variables in M odule Program Units 685.4.2. Glo bal Variables Passed throu gh A rgu m en t Lists 755.5. Main Program: BE M EC H 755.6. Sub routine INPU T_C TR 765.7. Subroutine BLO CKJD ATA 775.8. Subroutine INP UT -EL 785.9. Subrou tine TR EA T_T 795.10. Subroutine AX ES .CO S 805.11. Subrout ine SH AP EF 815.12. Subroutines D SH A PE and DS HA P3D 825.13. Subroutine EL_CO EFS 845.14. Subroutine AD AP TIN T 855.15. Subroutine CH O SE GP 875.16. Subroutines SET GA S and GA USSV 895.17. Subroutines M IND IST and IVSNR 123 905.18. Sub routine INT_HG 915.19. Subroutine EV A L.H G 925.20. Subroutine FO R M -H G ' 935.21. Subrout ine SIN GU HG 945.22. Subrou tine SIN 2D H G 955.23. Subroutine SETD SUB 965.24. Sub routine BD STR S 975.25. Subroutine BS CO EF 985.26. Subrou tine SYM TRY 995.27. Subroutine H G TO EQ S 100


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C O N T E N T S ix5.28. Subrou tine INN ER PS 1015.29. Subroutine EL-SO LV E 1025.30. Subroutine INVSO LVR 1035.31. Subroutines OU TP UT and SIGTITL 1045.32. Closure 106

6 Linear Applications 1076.1. Introduction 1076.2. Thick-Walled Cylinder un der Inte rna l Pressure 1076.3. Circular Rigid Fo und ation on a Semi-Infinite M edium 1116.4. A Three-Dim ensional M achine Co mp onen t 1126.5. Closure 117

PART I I . NONLINEAR PROBLEMS7 Rate-Independent Plas ticity Theory 1 2 1

7.1. Introduction 1217.2. Isotropic Yield Criteria 1217.2.1. Stress Inva riants and Principal Stresses 1227.2.2. Th e Tresca Crite rion 1237.2.3. Th e Von Mises Criterion 1247.2.4. The M ohr-C oulo m b Criterion 1247.2.5. The Druc ker-Prage r Criterion 125

7.3. Principles of Elas to-Pla stic Flow 1267.4. Co nstitutive Re lationships 1287.5. Isotropic Ha rden ing M aterials 1307.5.1. Eq uiva lent Un iaxial Yield Criteria 1317.5.2. Eq uiva lent Plastic Strain 131

7.5.3. Explicit De rivations 1327.6. Kinematic Ha rden ing Ma terials 1347.7. Mixed H arde nin g M aterials 1357.8. Closure 136

8 Boundary Integra l Equations in ElastoPlasticity 1388.1. Introduction 1388.2. Bou ndary Integral Eq uation s 1388.3. Internal Stress Integral Eq uation s 1408.4. Integ ration of Strongly Singular D om ain Integrals 1438.5. Closu re 145

9 Numerical Imp lementation 1469.1. Introduction 1469.2. Do ma in Discretization 1469.3. Weakly Singular Dom ain Integrals ', 1489.4. Strongly Singular D om ain Integrals 1499.5. Bou ndary Stresses - Traction-Recovery M ethod 1509.6. System Eq uatio ns 153

9.6.1. Initial Stress Re pre sen tation 1539.6.2. Plastic M ultiplier Re pre sen tation 1549.7. System Eq ua tion Solution 1569.7.1. New ton-Raphson Method 156


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9.7.2. Transition from Elastic to Ela sto- Pla stic States 1579.7.3. Autom atic Increm entation of Bou ndary Loading 1589.7.4. Sum mary 1599.8. Closu re 160

10 The Elasto -Plastic Program Code 16210.1. Introduction 16210.2. Scope of the Program 16210.3. Program Structure 16210.4. Glo bal Variables 16410.5. Subrout ine INP UT .NL 16510.6. Subroutine NL .COE FS 16610.7. Subroutine INT_CELL 16710.8. Subrou tine SESLCELL 16810.9. Subroutine CE LL-B OU ND 17010.10. Subrou tine EV AL JCE 17110.11. Subroutine INTSU BC 17310.12. Subroutine PBS CO EF 17310.13. Subroutine NL-SOLV E 17410.14. Subroutine PL-FLOW 17610.15. Subroutine DF -DS IG 17710.16. Subroutine P -M JT E R 17810.17. Subrou tine M AT RIC ES 18110.18. Subroutine SIG-SC ALE 18210.19. Sub routine SIG CR OS S 18310.20. Sub routine DF JvlA TR X 18410.21. Subroutine UP DA TE V 18510.22. Closure 186

11 Nonlinear Applications 18711.1. Introduction 18711.2. A Cu be Subjected to Uniax ial Tension 18711.3. A Thick-Walled Cylinder Subjected to Interna l Pressure 19111.4. A Rigid Pun ch un der Plane Strain - 19311.5. A Flexible Square Footing 19611.6. M ultiplanar Tubular DX -Joint 19911.7. Closure 201

12 Epilogue 20312.1. Review 20312.2. The Way Forward 203

12.2.1. Autom atic Integration 20412.2.2. Com putation of Bou ndary Stresses 20412.2.3. Stress-Return Algorithm 20412.2.4. System Eq ua tion Solver 20512.2.5. Local Boun dary Conditions 20512.2.6. No nlinear H arde nin g 20612.2.7. Adv anced Yield Functions 20612.2.8. Finite Strain Elasto-Plasticity 20612.2.9. Infinite Bo und ary Elem ents 20712.2.10. M ultiple Reg ions 207


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CONTENTS xiAppendix A. D erivation of Kernel F unctions 2 0 9

A.I. Derivation of the Strain Kerne l 209A.2. Derivation of the Stress Ke rnel 210A.3. Derivation of the Traction Kern el 211A.4. Ke rnel Fun ctions for Plane Strain and Plane Stress 211

Appendix B. Shape Functions 2 1 3B.I. One -Dime nsional Shape Functions 213B.2. Two-Dimensional Shape Functions 214B.3. Three-Dim ensional Shape Functions 215

Appendix C. D egenerate E lements: S ingular M apping 2 1 7Appendix D. Elasto-Plastic Flow Theory 2 2 1D.I. Derivation of the Plastic Flow Rule and Plastic Loading Ru le 221

D.2. Derivations for Kinem atic Ha rden ing Ma terials 224D.3. Derivations for Mixed Ha rdening Materials 225D.4. Derivation of the Deform ation State Function F 226

Appendix E. D omain Integra l Formulations 2 2 8E.I. Bou ndary Integral Eq uation s: Initial Strain Form ulation 228E.2. Ana lytical Integ ration of the Strongly Singular Volum e Integ ral 228E.3. Interio r Stress Eq ua tion: Initial Strain Form ulation 230E.4. Analytical Integration of E,-y# in Two Dim ensions 230E.5. Ana lytical Integ ration : Initial Strain Form ulation 231

Appendix F. S olution of the Nonlinear S ystem Equations 2 3 2F.I. The New ton-Rap hson I terat ive Algorithm 232F.2. System Eq uation Solution Strategies 233

F.2.1. The Initial Stress Iteratio n Tech nique 234F.2.2. The Implicit Solution Techn ique 235F.2.3. The Variable Stiffness Techn ique 236F.2.4. The Mixed Re presen tation Technique 237

Appendix G. Elements of Elasto-Plasticity 2 3 9Appendix H. Description of Input Data 2 4 2References 247Index 253

boundary element programming in mechanics_104207485

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