Numerical modelling in large strain plasticity with application to tube collapse analysis
271
0
0
Texto completo
(2) 2. ABSTRACT. Numerical. methods are. proposed f o r t h e a n a l y s i s. of 2 or. dimensional l a r g e s t r a i n p l a s t i c i t y problems. A F i n i t e. 3-. Difference. program, w i t h 2 - d i m e n s i o n a l continuum elements and e x p l i c i t. time. i n t e g r a t i o n , has been developed and applied to model the axisymmetric crumpling of c i r c u l a r tubes. New types of mi xed el ements ( T r i a n g l e s - Q u a d r i l a t e r a l s for 2-D, Tetrahedra-Bricks. for. 3-D). are. proposed. for. the. spatial. d i s c r e t i z a t i o n . These elements model accurately incompressible p l a s t i c f l o w , without unwanted " z e r o - e n e r g y " d e f o r m a t i o n modes or t a n g l i n g over of the mesh. E l a s t i c - p l a s t i c , r a t e dependent laws are modelled w i t h a "radial by p l a s t i c. return" algorithm. The transmission of heat generated. work and m a t e r i a l. dependence on t e m p e r a t u r e are. included, enabling a f u l l y coupled thermo-mechanical A 2-D and a x i s y m m e t r i c implementing the numerical. also. analysis.. computer program has been developed, techniques. described.. Computational. e f f i c i e n c y was e s s e n t i a l , as l a r g e s c a l e , c o s t l y a p p l i c a t i o n s were intended. An important part of the program was the contact a l g o r i t h m , enabling the modelling of i n t e r a c t i o n between surfaces. The a x i s y m m e t r i c c r u m p l i n g of tubes under a x i a l ("concertina". mode) has been analyzed N u m e r i c a l l y .. compression Quasi-static. experiments on Aluminium tubes were modelled, using v e l o c i t y s c a l i n g . Mery l a r g e s t r a i n s are developed i n the c r u m p l i n g p r o c e s s ; w i t h the help of tension t e s t s , material laws v a l i d f o r such s t r a i n ranges were developed. Good agreement was obtained between numerical and experimental. predictions. r e s u l t s . Modelling choices such as mesh refinement,. element type and v e l o c i t y scaling were studied, and found to have an i m p o r t a n t i n f l u e n c e on the numerical p r e d i c t i o n s . F i n a l l y , a l a r g e scale impact a n a l y s i s of a s t e e l tube at 176m/s was p e r f o r m e d . The r e s u l t s compared well w i t h experiment, i n d i c a t i n g differences w i t h the behaviour of low v e l o c i t y crumpling mechanisms. To c o n c l u d e , F i n i t e D i f f e r e n c e procedures w i t h e x p i i c i t. time-.
(3) 3. marching techniques are proposed for large s t r a i n p l a s t i c i t y problems, at low or medium impact v e l o c i t i e s . A f a i r l y. r o b u s t code has been. developed and applied successfully to a range of large s t r a i n and tube crumpling problems..
(4) 4. ACKNOWLEDGEMENTS. I wish to express my sincere gratitude to the f o l l o w i n g : Dr. G.L. England who s u p e r v i s e d t h i s work, f o r h i s c o n s t a n t support and guidance throughout the period of research; Dr. J. M a r t i theoretical. from P r i n c i p i a , f o r h i s i n v a l u a b l e e x p e r t advice on and n u m e r i c a l. matters,. and h i s f r i e n d l y. support. and. suggestions; Or. £. A l a r c o n , f o r h i s help and m o t i v a t i o n i n the i n i t i a l. stages of. work; The s t a f f. and t e c h n i c i a n s of the C i v i l. E n g i n e e r i n g Department. at. King's College, f o r t h e i r help and cooperation; My w i f e , Tereca Marin, for her help in the preparation of f i g u r e s , and her unending patience and encouragement during the research period.. I would also l i k e to thank the f o l l o w i n g bodies for sponsoring my research at t h e v a r i o u s s t a g e s : The B r i t i s h C o u n c i l , The Spanish Government ( M i n i s t r y of Education), P r i n c i p i a Mechanica Ltd. (UK), and the Committee of Vicechancellors and Principals. (UK)..
(5) TABLE OF CONTENTS. Pa TITLE. ABSTRACT. ACKNOWLEDGEMENTS. TABLE OF CONTENTS. BASIC NOTATION. 10. CHAPTER 1 - INTRODUCTION. 13. 1.1 Objectives. 14. 1.2 Non-linear modelling. 16. 1.3 Layout. 17. CHAPTER 2 - CONTINUUM MECHANICS DESCRIPTIONS. 18. 2.1 Introduction. 19. 2.2 Kinematics. 20. 2.2.1 Configurations. 21. 2.2.2 Deformation tensors. 21. 2.2.3 Deformation and spin rates. 23. 2.2.4 Strains. 23. 2.2.5 Transformations. 24. 2.3 Stress. 25. 2.3.1 Cauchy. 25. 2.3.2 Piola-Kirchhoff. 25. 2.4 Balance laws 2.4.1 Balance of momentum. 26 26.
(6) Pa 2.4.2 Balance of mass. 26. 2.4.3 Balance of energy. 27. 2.5 Constitutive relations. 28. 2.5.1 Rate equations. 29. 2.5.2 Elasticity. 30. 2.5.2.1 Hyperelastic materials. 31. 2.5.2.2 Hypoelastic materials. 31. 2.5.3 Plasticity. 32. 2.5.3.1 Von Mises model. 34. 2.5.3.2 Other plasticity models. 35. CHAPTER 3 - NON-LINEAR NUMERICAL MODELS FOR SOLID MECHANICS. 37. 3.1 Introduction. 38. 3.2 Finite Difference methods. 39. 3.3 Finite Element methods. 41. 3.4 Mesh descriptions. 44. 3.4.1 Lagrangian. 44. 3.4.2 Eulerian. 45. 3.4.3 Arbitrary Lagrangian-Eulerian. 45. 3.5 Large displacement formulations. 46. 3.5.1 Total Lagrangian. 46. 3.5.2 Cauchy stress - velocity strain. 47. 3.5.3 Updated Lagrangian. 49. 3.6 Time integration. 49. 3.6.1 Central Difference (explicit). 50. 3.6.2 Trapezoidal rule (implicit). 52. 3.6.3 Operator split methods. 54. 3.7 Practical considerations for discrete mehes. 55. 3.7.1 "Locking-up" for incompressible flow. 55. 3.7.2 "Hourglassing". 58. 3.8 Conclusions. 59. CHAPTER 4 - EXPLICIT FINITE DIFFERENCE NUMERICAL MODEL. 61. 4.1 Introduction. 63.
(7) Pa 4.1.1 General methodology 4.2 Spatial semidiscretization. 63 66. 4.2.1 Constant Strain Triangles and Tetrahedra (CST elements) 67 4.2.2 Mixed Discretization (MTQ, MTB elements). 72. 4.2.3 Prevention of negative volumes (MTQC, MTBC elements). 75. 4.2.4 Mass lumping procedure. 78. 4.3 Momentum balance. 78. 4.4 Central difference time integration. 79. 4.5 Constitutive models. 79. 4.5.1 Hypoelasticity. 80. 4.5.2 Plasticity; radial return algorithm. 80. 4.5.3 Hardening and uniaxial stress-strain laws. 84. 4.5.4 Objective stress rates. 87. 4.6 Damping. 88. 4.7 Stability of time integration. 90. 4.8 Modelling of contacts. 91. 4.8.1 Contact interface laws. 92. 4.8.2 Contact detection algorithm. 94. 4.9 Heat conduction. 96. 4.10 Energy computations. 100. 4.11 Implementation into Fortran program. 102. CHAPTER 5 - BENCHMARKS AND VALIDATION EXAMPLES. 106. 5.1 Introduction. 107. 5.2 Wave propagation. 107. 5.2.1 E l a s t i c waves in bars. 107. 5.2.2 E l a s t i c - p l a s t i c waves. 110. 5.2.3 Elastic waves in cone. 112. 5.3 Vibration of a c a n t i l e v e r. 114. 5.4 Static e l a s t i c - p l a s t i c problems. 117. 5.4.1 Punch t e s t. 117. 5.4.2 E l a s t i c - p l a s t i c sphere under i n t e r n a l pressure. 121. 5.5 Heat conduction. 123. 5.5.1 Coupled thermomechanical analysis. 123. 5.5.2 Temperature r e d i s t r i b u t i o n in a slab. 124. 5.6 Large s t r a i n s and r o t a t i o n s. 124.
(8) Pa 5.7 Impact of c y l i n d e r. 126. 5.8 Conclusions. 128. CHAPTER 6 - TENSION TESTS ON ALUMINUM BARS. 129. 6.1 Introduction. 130. 6.1.1 Constitutive idealization. 132. 6.1.2 Tension tests - a review. 133. 6.2 Theoretical interpretation of tension tests. 136. 6.2.1 Strain distribution at minimum neck section. 137. 6.2.2 Stress distribution. 138. 6.3 Tension tests. 140. 6.3.1 Specimens and material. 140. 6.3.2 Programme. 141. 6.3.3 Procedure. 142. 6.3.4 Results. 142. 6.3.5 Microhardness tests. 144. 6.4 Material hardening law. 147. 6.5 Numerical calculations for tension tests. 151. 6.5.1 Model. 151. 6.5.2 Analysis. 153. 6.5.3 Results. 154. 6.6 Conclusions. 156. CHAPTER 7 - CONCERTINA TUBE COLLAPSE ANALYSIS. 164. 7.1 I n t r o d u c t i o n. 166. 7 . 1 . 1 Scope 7.2 Overview of energy d i s s i p a t i n g devices. 166 167. 7.2.1 D e f i n i t i o n and c r i t e r i a. 167. 7.2.2 Types of energy d i s s i p a t i n g devices. 168. 7.2.3 Tubes as energy absorbers. 169. 7 . 2 . 3 . 1 Lateral compression. 169. 7.2.3.2 Axisymmetric axial crumpling. 170. 7.2.3.3 Diamond-fold axial crumpling. 171. 7.2.3.4 Tube inversion. 171.
(9) Page 7.3 Quasi-static concertina tube collapse mechanisms 7.3.1 Related experimental work. 172 172. 7.3.1.1 Experimental programme and method. 173. 7.3.1.2 Typical experimental results. 174. 7.3.1.3 Microhardness tests. 177. 7.3.1.3.1 Equipment and procedure. 177. 7.3.1.3.2 Derivation of material strength, Y. 178. 7.3.2 Numerical model. 182. 7.3.2.1 Discretization and material. 183. 7.3.2.2 Velocity scaling. 184. 7.3.2.3 Interpretation of output. 185. 7.3.3 Results form numerical calculations and experiment. 187. 7.3.3.1 Tube geometry A: ID=19.05mm, t=1.64mm, L=50.8mm. 188. 7.3.3.2 Tube geometry B: ID=19.05mm, t=1.17mm, L=50.8mm. 193. 7.3.3.3 Tube geometry C: 0D=38.10mm, t=1.65mm, L=50.8mm. 197. 7.3.3.4 Tube geometry D: 0D=25.40mm, t=0.95mm, L=25.4mm. 203. 7.3.4 Parametric studies in numerical analyses. 208. 7.3.4.1 Influence of friction. 208. 7.3.4.2 Influence of velocity scaling. 213. 7.3.4.3 Influence of mesh refinement. 216. 7.3.4.4 Influence of element type. 223. 7.3.5 Discussion 7.4 Medium velocity (176m/s) tube impact analysis. 228 233. 7.4.1 Description of problem. 233. 7.4.2 Numerical idealization. 235. 7.4.3 Numerical results. 237. 7.4.4 Discussion. 243. 7.5 Conclusions. 249. CHAPTER 8 - CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH. 251. 8.1 Conclusions. 252. 8.2 Suggestions for further research. 253. 8.2.1 Theoretical and numerical developments. 254. 8.2.2 Additional applications. 255. REFERENCES. 256.
(10) 10. BASIC NOTATION. A. Area; Mass damping c o e f f i c i e n t. B. Stiffness. (eqn. 4.56). damping c o e f f i c i e n t. (eqn. 4.56);. strain-rate. parameter (eqn. 4.50) B (B.j-). Left Cauchy-Green deformation tensor (eqn. 2.12). B.:j. Gradient operator for F i n i t e Elements (eqn. 3.6). c. Stress wave v e l o c i t y. C (Cj-jki) C o n s t i t u t i v e tensor f o r Jaumann rate of Cauchy stress (eqn. 3.23) C (CJJ). Right Cauchy-Green tensor (eqn. 2.12). C. Damping matrix (eqn. 3.30). (^ (Cj-jki ) C o n s t i t u t i v e t e n s o r f o r T r u e s d e l l. r a t e of Cauchy. stress. (sect. 3.5.2) CST. Constant s t r a i n elements. d (d.j-j). Rate of deformation tensor (eqn. 2.15). d (d.j ). Penetration in contact (eqn. 4.68). D (DJJKL) C o n s t i t u t i v e tensor ( t o t a l Lagrangian) (eqns. 2.47, 3.21) D. Diameter. E. Young's modulus of E l a s t i c i t y. E (Ejj ). Green's s t r a i n tensor (eqn. 2.17). F. Force; Yield function (eqn. 2.54). F (F j ). Deformation gradient tensor (eqn. 2.7). FD. F i n i t e Difference. FE. F i n i t e Element. G. Elastic. g (g-j-j). Metric tensor. h,ha,hY,h'Height;. (method). (method). shear modulus (eqn.. Plastic. (eqn.. 2.45). 2.2). hardening moduli (eqn.. (eqns. 4 . 4 4 ,. h (hi ). Heat f l o w r a t e. 2.32). I. I d e n t i t y tensor. ID. Inner diameter. J. Jacobian of motion (eqn. 2.10). K,K. S t i f f n e s s , s t i f f n e s s matrix (eqns. 3.10, 3.11). L. Length. 1 Oi j>. Velocity gradients (eqn. 2.14). m , M , mi,M. Mass, mass matrices (eqn. 3.8). MD. Mixed D i s c r e t i z a t i o n (sect. 4.2.2). MTB(C). Mixed Tetrahedra-Brick. (Corrected) elements. 4.45).
(11) 11. MTQ(C). Mixed Triangles-Quadrilateral. (Corrected) elements. N, n. Normal vectors. n. Time i n s t a n t corresponding to n. N. Shape functions f o r FE (eqn. 3.5). OCi. Outer Diameter. P. Internal. q r, R. Body heat supply. t. forces. Radius; External force (eqn. 3.8). R (Rij) s, s. Rotation tensor (eqn. 2.11). s. 2nd P i o l a - K i r c h h o f f stress tensor (eqn. 2.25). Surface; Distance along a curve. (SIJ) s (sid). Cauchy deviatoric stresses (eqn. 2.59). t. Time; Thickness. T. Temperature. u. Internal energy. u. (U-j). Displacements Right stretch tensor (eqn. 2.11). U (UIJ) V. Volume. V (Vi-j). Left stretch tensor (eqn. 2.11). v,v (v.j ) Velocity (eqn. 2.6) w (w-j ^) W. Spin tensor (eqn. 2.15) Work, Energy. x (xn- ), (x,y,z) Spatial coordinates (eqn. 2.5) X. Particle (sect. 2.2.1). X (Xj). Lagrangian coordinates (eqn. 2.4). X. Vector product (eqn. 2.30). V. Yield stress. a. Thermal. expansion c o e f f i c i e n t ;. Back s t r e s s. Mixed D i s c r e t i z a t i o n c o r r e c t i o n c o e f f i c i e n t (3. P r o p o r t i o n of c r i t i c a l (eqn.. (eqn.. 2.61);. (eqn. 4.19). damping; Radial return c o e f f i c i e n t. 4.32). T. P l a s t i c flow a r b i t r a r y m u l t i p l i e r. (eqn. 2.55). A. Increment. 5-j -;. Kronecker delta. eP. E f f e c t i v e p l a s t i c s t r a i n (eqn. 2.60b). e(e-j-j). Small s t r a i n tensor (eqn. 2.43). X. Lame's Elastic constant. \i. C o e f f i c i e n t of Coulomb f r i c t i o n (eqn. 7.11).
(12) 12. v. Poisson's. ratio. P 9. Mass d e n s i t y ; Radius of curvature. a (cr i:j ). Cauchy stress tensor (eqn. 2.24). d. Partial. Angular coordinate derivative. Pull-back, push-forward of tensors (eqns. 2 . 2 1 , 2.22^. x. Mesh coordinates (sect. 3.4). 0). Angular frequency.
(13) 13. CHAPTER 1. INTRODUCTION. 1.1 OBJECTIVES 1.2 NON-LINEAR MODELLING 1.3 LAYOUT.
(14) 14. 1.1 OBJECTIVES This. work c o n s i s t s. of a t h e o r e t i c a l. part. (mathematical. numerical models, c h a p t e r s 2 - 4 ) , and a p r a c t i c a l p a r t and numerical. and. (applications. s i m u l a t i o n of l a r g e s t r a i n tube c o l l a p s e. analysis,. chapters 5-7). The motivation f o r the t h e o r e t i c a l. part of the work l i e s in the. author's i n t e r e s t in non-linear s o l i d mechanics modelling,. understood. broadly as encompassing the f o l l o w i n g phenomena: - large s t r a i n s and large displacements (geometric - p l a s t i c and v i s c o p l a s t i c. behaviour (material. nonlinearities);. nonlinearities);. - contacts and impact (nonlinear boundary c o n d i t i o n s ) ; - thermomechanical coupling. On the p r a c t i c a l. side, the source of motivation was the research. program on tube c o l l a p s e mechanisms being c a r r i e d out at the C i v i l Engineering. department. of. King's. College,. University. (Andrews, England and Ghani, 1983). Such mechanisms are. of. London. efficient. energy d i s s i p a t i n g systems (Johnson and Reid, 1978), for use in impact s i t u a t i o n s . A d d i t i o n a l l y , t u b e s a r e frequent s t r u c t u r a l components f o r aerospace vehicles and other equipment or components which may suffer accidental c o l l i s i o n s . The o b j e c t i v e of t h i s work was the development of. numerical. methods of s i m u l a t i o n f o r n o n l i n e a r a n a l y s i s , capable of m o d e l l i n g tube c o l l a p s e restricted. mechanisms.. to collapse. (Concertina mode).. More s p e c i f i c a l l y ,. through. Numerical. axi symmetric. predictions. the a t t e n t i o n sequential. was. folding. for tube collapse should be. obtained and compared to experimental r e s u l t s , available from previous work on a l u m i n i u m tubes by Ghani (1982). This o b j e c t i v e posed some important challenges, such as the development of a numerical model for large s t r a i n s and large displacements, w i t h e l a s t i c - p l a s t i c. behaviour,. capable of m o d e l l i n g a r b i t r a r y t u b e - t u b e and t u b e - p l a t e n c o n t a c t s (chapters 2, 3, 4). R e l i a b l e data would have t o be o b t a i n e d f o r the.
(15) 15. TIME = 0.00 ms. TIME = 0.77 ms. TIME = 1.37 ms. TIME = 2.09 ms. TIME = 2.93 ms. TIME = 3.62 ms. 3=. igur. 1, TYPICAL RESULTS FOR AXISYMMETRIC TUBE COLLAPSE ANALYSIS 0D = 38..1mm.t.= 1.22mm.L=88.9 mm - TUBE9, GEOMETRY F (SEE TABLE 7.4).
(16) 16. c o n s t i t u t i v e behaviour of t h e A l u m i n i u m at l a r g e s t r a i n s. With t h i s. objective. in mind, a F i n i t e Difference. (chapter. 6).. method. with. e x p l i c i t t i m e i n t e g r a t i o n was chosen. The method uses a F i n i t e topology,. and. arbitrary. can. therefore. continua.. be. An e f f i c i e n t. success of t h e s i m u l a t i o n s .. applied. contact. A typical. to. irregular. logic. Element. meshes. was e s s e n t i a l. example of t h e r e s u l t s. for. in the. obtained. i s shown i n f i gu r e 1 . 1 .. 1.2 NON-LINEAR MODELLING. W i t h i n t h e p a s t decade t h e r e has been c o n s i d e r a b l e i n t e r e s t nonlinear. solid. mechanics. simulations,. due t o t h e g r e a t. problem-. s o l v i n g power a v a i l a b l e f r o m t h e new g e n e r a t i o n s of d i g i t a l The p o s s i b i l i t y of d e t a i l e d s o l u t i o n s problems. has occassioned. renewed. for. interest. m a t h e m a t i c a l d e s c r i p t i o n s and n u m e r i c a l them. highly. complex. and pressure. techniques. computers. non-linear. for. which. powerful implement. efficiently.. In. many aspects. available:. explicit. of. non-linear. or. implicit. E u l e r i a n meshes, t o t a l. numerical time. modelling,. integration,. Lagrangian or Cauchy s t r e s s. choices. Lagrangian. formulations.. o f t h e s e c h o i c e s has i t s own a d v a n t a g e s and d r a w b a c k s . On t h e hand,. non-linear. advantages. of. different. techniques. method, A r b i t r a r y Lagrangian E u l e r i a n. The procedure. chosen. for. this. (e.g.. for. (wave p r o p a g a t i o n. impact scenarios,. Winfrith. reported. work. Difference techniques. Barr. (1983a). are a l s o u s e f u l. through t h e use of v e l o c i t y 7).. By. for. Finite. and. Element. Difference). short duration A natural. impact t e s t s. (sect.. order t o t a k e advantage of t h e n o n - l i n e a r. 6,. other. efficiently. Element. (Explicit. type problems).. e.g. t h e m i s s i l e. by. or. descriptions).. is i d e a l l y suited for steeply non-linear, phenomena. are. Each. mechanics i s a f i e l d under c o n s t a n t development,. new approaches are being e x p l o r e d which a t t e m p t t o combine the. in. slow. 7.4).. transient. application. is. done a t UKAEA. Explicit. Finite. l o a d i n g phenomena,. in. robustness and c a p a b i l i t i e s ,. s c a l i n g or dynamic r e l a x a t i o n. (chapters. 4,.
(17) 17. 1.3 LAYOUT In chapter 2 a number of essential. s o l i d mechanics concepts are. i n t r o d u c e d and discussed b r i e f l y . N o n - l i n e a r numerical models and techniques. t o implement those concepts i n t o n u m e r i c a l. reviewed i n c h a p t e r 3. The E x p l i c i t. Finite. codes are. Difference. model. and. computer code developed here are described in chapter 4, while chapter 5 contains some v a l i d a t i o n examples which t e s t the main aspects of the formulation.. Chapter. 6 concerns. the. derivation. of. a. material. c o n s t i t u t i v e law f o r Aluminium a l l o y through t e n s i l e t e s t s , w i t h some applications. to. the. numerical. simulation. of. the. tensile. tests. themselves. Chapter 7 contains applications to tube collapse analysis, comparing the results w i t h experimental data f o r q u a s i - s t a t i c collapse of Aluminium t u b e s . The c o n s t i t u t i v e law from c h a p t e r 6 i s used f o r the numerical predictions. An analysis f o r a medium v e l o c i t y (176 m/s) tube. impact. is. also. described.. Finally,. conclusions. suggestions for f u r t h e r work are given in chapter 8.. and. some.
(18) CHAPTER 2. CONTINUUM MECHANICS DESCRIPTIONS. 2.1 INTRODUCTION. 2.2 KINEMATICS. 2.2.1 Configurations 2.2.2 Deformation tensors 2.2.3 Deformation and spin rates 2.2.4 Strains 2.2.5 Transformations 2.3 STRESS. 2.3.1 Cauchy 2.3.2 Piola-Kirchhoff. 2.4 BALANCE LAWS. 2.4.1 Balance of mass 2.4.2 Balance of momentum 2.4.3 Balance of energy. 2.5 CONSTITUTIVE RELATIONS. 2.5.1 Rate equations 2.5.2 Elasticity 2.5.2.1 Hyperelastic materials 2.5.2.2 Hypoelastic materials 2.5.3 Plasticity 2.5.3.1 Von Mises model 2.5.3.2 Other plasticity models.
(19) 19. 2.1 INTRODUCTION The advances made in d i g i t a l. computing w i t h i n the l a s t decades. have opened up new f i e l d s f o r engineers and s c i e n t i s t s .. Problems. previously regarded as unsolvable, only approached through experiments and s i m p l i f i e d empirical formulae, can now be analyzed numerically in great d e t a i l . In the f i e l d of continuum mechanics t h i s has g r e a t l y increased the i n t e r e s t in d e t a i l e d mathematical d e s c r i p t i o n s , amenable t o be used i n n u m e r i c a l models w i t h d i s c r e t i z a t i o n t e c h n i q u e s (e.g. F i n i t e Element or F i n i t e Difference methods). Having confusion. said in. the. mathematicians,. this,. there. specialist. still. exists. a certain. literature.. On t h e. degree. part. rigorous mechanical descriptions are often. i n ways d i f f i c u l t. the. presented. t o be grasped by engineers and implemented. numerical production codes. As a r e s u l t , many engineers s t i l l t o outdated and much l e s s p o w e r f u l theoretical. of. presentations. of. in. c l i n g on. n o t a t i o n s . On the o t h e r hand,. are not u n i q u e , c a u s i n g some degree of. confusion to researchers f i r s t approaching seriously these t o p i c s . An e f f o r t. has been made i n t h i s. chapter. t o present. overview of c e r t a i n continuum mechanics concepts,. a brief. indispensable in a. rigorous treatment, without unnecessary mathematical fuss. The purpose of t h i s exposition i s : - t o i n t r o d u c e the nomenclature and d e f i n i t i o n s of concepts used i n l a t e r chapters; - t o discuss the s i g n i f i c a n c e of and i n t e r p r e t some concepts w i t h a view to numerical modelling (basis of t h i s work); - t o ensure c e r t a i n completeness f o r t h e ideas presented i n. this. thesis. It. must be s t r e s s e d , however, t h a t t h i s. e x p o s i t i o n does not. pretend to be complete. Only the concepts which are relevant for. the. r e s t of t h i s t h e s i s w i l l be d w e l t upon. In p a r t i c u l a r , emphasis i s l a i d on s o l i d mechanics and el a s t i c - p i a s t i c behaviour. A number of results. will. be presented. without. proof.. For. a more. complete.
(20) 20. discussion of these t o p i c s , the interested reader is referred to Fung (1965), Malvern (1969), and B i l l i n g t o n and Tate (1981) for the general concepts,. and t o. Marsden and Hughes (1983) f o r. a more. detailed. and up-to-date mathematical d e s c r i p t i o n . In the f o l l o w i n g presentation, the ambient space is assumed to be an Euclidean p o i n t space ( i . e . i n t e r i o r product d e f i n e d ) , and where necessary t h i s w i l l. be p a r t i c u l a r i z e d to R3. The coordinate bases may. be c u r v i l i n e a r and a r b i t r a r y , a l t h o u g h when equations are given i n component form, often orthonormal bases (not n e c e s s a r i l y. cartesian). are assumed for s i m p l i c i t y . The usual conventions for tensor notation are employed: repeated i n d i c e s i n d i c a t e summation over t h e i r range unless e x p l i c i t l y. stated,. and commas indicate covariant. derivatives.. Vectors and tensors are represented by boldface characters. Superposed dots indicate material time d e r i v a t i v e s .. Given two t e n s o r s A and B the product AB i s understood t o be contracting the near indices with opposite variance: (AB)ij = AikBkj. (2.1). I f the i n d i c e s have the same v a r i a n c e , e.g. both are c o n t r a v a r i a n t , the metric tensor g is necessary to lower one: (AB) 1J - A l k g k l B 1 j = A l k 8 k j. (2.2). When the t e n s o r components are r e f e r r e d t o orthonormal v e r t i c a l position of the indices is i r r e l e v a n t ,. bases the. as the metric. tensor. is u n i t y . A colon indicates doubly contracted product: A:B = A ^ B ^. (2.3). 2.2 KINEMATICS A body (or continuum) i s a set whose e l e m e n t s , c a l l e d m a t e r i a l p a r t i c l e s , have a o n e - t o - o n e correspondence w i t h a r e g i o n V of the.
(21) 21. Euclidean point space. The following kinematics concepts are intended to provide a description of the motion of deformable bodies.. 2.2.1 CONFIGURATIONS Each particle X of the body B may be identified by its position X in the original configuration, V 0 , which is taken as reference: X = k(X). (2.4). X (components Xi) are c a l l e d Material or lagrangian coordinates of the p a r t i c l e . The motion of the body at a l a t e r time is given by the t i m e dependentpositions x of the p a r t i c l e s in the current c o n f i g u r a t i o n , V:. x = x(X,t). x (components x i ) are the spatial. (2.5). or Eulerian coordinates.. Hereafter. upper case i n d i c e s s h a l l r e f e r t o Lagrangian c o o r d i n a t e s , and lower case to Eulerian. The v e l o c i t i e s are defined as :2.6' where the dot s i g n i f i e s a material time d e r i v a t i v e , i.e. f o l l o w i n g the p a r t i c l e X.. 2.2.2 DEFORMATION TENSORS. Central to deformation measurements is the deformation gradient tensor:. F = dx/dX. (2.7). with components F n j = x 1 j . The t e n s o r F i s used as the base f o r a number of s t r a i n and d e f o r m a t i o n measures. An element of a curve dX i s t r a n s f o r m e d by.
(22) 22. dx = FdX. The inverse of F gives the spatial gradients of the material coordi nates:. F"1 = dX/dx (2.9) 1. 1. ( (F" ) ! = X. ! si. ). F c o n s t i t u t e s a two-point tensor. Another i n t e r p r e t a t i o n that. relates. F to transformations between configurations i s given in section 2.2.5. The Jacobian of the motion is J = det(F). (2.10). The polar decomposition of F gives F = RU = VR where R i s an o r t h o g o n a l. (rotation) tensor,. (2.11) RRT = I .. p o s i t i v e d e f i n i t e , and are c a l l e d the r i g h t and l e f t. U and V are. stretch tensors. r e s p e c t i v e l y . Equations (2.11) r e p r e s e n t two ways t o v i s u a l i z e the deformation: f i r s t. s t r e t c h i n g (U) and then r o t a t i n g (R), or. first. r o t a t i n g (R) and then s t r e t c h i n g (V) Other deformation measures are the Cauchy-Green tensors: C = F'F. ( r i g h t Cauchy-Green) (2.12). B = FF1. (left. Cauchy-Green). The l e n g t h of an element of curve i s given by ds2 = dxdx i n current c o n f i g u r a t i o n ,. and dS2 = dXdX in the o r i g i n a l. the. configuration.. The significance of C and B is given by the r e l a t i o n s ds2 = dXCdX (2.13) dS2 = dxB-ldx.
(23) 23. 2.2.3 DEFORMATION AND SPIN RATES The spatial velocity gradient tensor is defined as: 1 = dv/dx. (2.14). which can be decomposed i n t o symmetric and skew-symmetric p a r t s : d = (l+lT)/2 (2.15) w = (l-lT)/2 These are c a l l e d the rate of deformation (or v e l o c i t y s t r a i n ) and spin rate tensors respectively. The rate of change of length of an element of curve is given by ds = (dxddx)/ds. (2.16). 2.2.4 STRAINS A measure of the t o t a l. strain. is. given by the Green. strain. tensor, defined as E = (C - I ) / 2 where I is the I d e n t i t y tensor. I t is t r i v i a l ds2-dS2 = 2dXEdX. (2.17) to see that (2.18). and that the rate of E is given by E = FTdF. (2.19).
(24) 24. 2 . 2 . 5 TRANSFORMATIONS. For the t e n s o r s d e f i n e d above, some of the i n d i c e s r e f e r t o the original configuration. (upper case),. while others are related to the. current c o n f i g u r a t i o n (lower case). Here some transformation laws are given. to. find. the. corresponding. tensor. in. the. alternative. confi g u r a t i o n . Although the transformed tensors w i l l. be considered as d i f f e r e n t. t e n s o r i a l e n t i t i e s , one way t o v i s u a l i z e the t r a n s f o r m a t i o n i s as a mere change of base. Imagine a base ( O . e ^ f i x e d i n space t h r o u g h o u t the m o t i o n , and another base. ( 0 ' ( t ),e ' i U ) ). which. deforms. and. t r a n s l a t e s w i t h the body. In t h i s convected c u r v i l i n e a r base, the coordinates of a material point remain constant throughout the motion, and equal to the material coordinates, XI. The spatial components of F p r o v i d e t h e m a t r i x f o r the change of c o o r d i n a t e s between the two bases. Given a 2nd order c o n t r a v a r i a n t. t e n s o r a by i t s. convected. material components, a_U, the spatial components are a i j = OxVdxMOxJ/ax^aU. (2.20). Hence F provides a means for transforming between spatial and material coordinates, a"iJ and a_U are the components in d i f f e r e n t same t e n s o r ,. bases of the. a. I f now one assumes components a_U t o apply t o the. spatial basis, a new tensor is obtained: A = a_IJ ejBej. (2.21a). where s s i g n i f i e s a t e n s o r i a l product. A is c a l l e d the pull-back of a, and may be obtained as A = F _1 aF" T = (d£(a). (2.21b). while the push-forward is defined by the inverse r e l a t i o n : a = 0 t * ( A ) = FAFT. (2.22). These relations may be t r i v i a l l y generalized to tensors of any rank..
(25) 25. Elements. of. area. and. volume. in. reference. and. current. configurations are transformed by the f o l l o w i n g transport formulae: nda = JF"TNdA. (2.23a). dv - JdV. (2.23b). These r e l a t i o n s may be used t o express i n t e g r a l balance laws ( s e c t . 2.4) i n e i t h e r c o n f i g u r a t i o n . Eqn. (2.23a) c o n d i t i o n s t h e form of the Piola transformations. for the stress tensor (eqn. 2.25). 2.3 STRESS. 2.3.1 CAUCHY The concept of s t r e s s r e s t s upon the Cauchy p o s t u l a t e t h a t the action of the rest of the material upon any volume element of i t is of the same form as d i s t r i b u t e d s u r f a c e f o r c e s . A t r a c t i o n v e c t o r t ( n ) may be d e f i n e d at each p o i n t , infinitesimal. as t h e f o r c e. exerted. per. unit. area, for each o r i e n t a t i o n n.. A p p l y i n g e q u i l i b r i u m c o n s i d e r a t i o n s , i t may be deduced t h a t a stress tensor a. must e x i s t , such that for every o r i e n t a t i o n n t(n) = na. a i s c a l l e d the Cauchy or true stress tensor, and i t. (2.24) is. related. to the current c o n f i g u r a t i o n .. 2.3.2 PIOLA-KIRCHHOFF If. both the f o r c e and the area components of the concept. of. s t r e s s are t r a n s f o r m e d back i n t o the o r i g i n a l c o n f i g u r a t i o n , a new stress tensor is obtained: S = JF -1 (7 F _ t = J 0 t * ( c r ). (2.25).
(26) 26. This r e l a t i o n is c a l l e d the backward Piola t r a n s f o r m a t i o n . I t. defines. the 2nd P i o l a - K i r c h h o f f s t r e s s t e n s o r S, which i s a s t r e s s measure referred to the o r i g i n a l. configuration.. 2 . 4 BALANCE LAWS. Balance laws (mass, momentum, angular momentum, and energy) may be s t a t e d a l t e r n a t i v e l y Integral. in i n t e g r a l. form or as f i e l d. forms provide "weaker" expressions. equations.. for the same p r i n c i p l e s .. This w i l l be commented f u r t h e r in section 4.3.. 2 . 4 . 1 BALANCE OF MASS. Conservation of mass i m p l i e s. that. the mass of the. material. ocupying a c e r t a i n r e g i o n V of the body remains c o n s t a n t t h r o u g h o u t the motion:. (d/dt)fpdV = 0. (2.26). where p is the mass density. As a field. equation,. balance. of mass. is expressed. by the. continuity equation:. p + pdiv(v) = 0. (2.27). 2 . 4 . 2 BALANCE OF MOMENTUM. For a region V of the body w i t h boundary S, the i n t e g r a l form of the equation of l i n e a r momentum balance is ( d / d t ) f pvdV = I pfdV + f V where. f. is. the. V body. force. nadS. (2.28). S per u n i t. mass.. The. corresponding.
(27) 27. differential. expression is Cauchy's equation of motion, p v = d i v ( a). + pf. (2.29). The i n t e g r a l s i n eqn. (2.28) i n v o l v e v e c t o r s , and as p o i n t e d out by Marsden and Hughes (1978), may not provide a covariant statement of the momentum balance p r i n c i p l e in a general manifold. However, for the Euclidean space to which t h i s exposition r e f e r s , the objection is not relevant.. The i n t e g r a l. expression. is. better. suited. for. finite. difference numerical models (section 4.3), f o r which weak v a r i a t i o n a l global expressions (as employed in F i n i t e Elements) are not obtained. Taking moments i n eqn. (2.28) w i t h respect t o the o r i g i n , the balance of angular momentum is expressed by ( d / d t ) /p(xXv)dV = fp(xXf)dV + /xX(nc7)dS. (2.30). where xXv denotes vector product of x and v. The corresponding field equation states simply the symmetry of O": U= GJ (2.31) Symmetry of S may be deduced from eqns. (2.25) and (2.31).. 2.4.3 BALANCE OF ENERGY In a continuum, the first law of thermodynamics may be expressed as (d/dt) JpudV = / ( pq+a:d)dV + fhndS. (2.32;. V where u is the internal energy per unit mass q is the rate of body heat supply per unit mass h is the heat flux vector; for an oriented infinitesimal area the heat flow rate is given by H = hndS The corresponding field equation is.
(28) 28. p u = aid The term time,. aid. + pq + div(h). (2.33). r e p r e s e n t s the s t r e s s work per u n i t volume and. a and d are said to be conjugate stress and s t r a i n measures. An. a l t e r n a t i v e representation of the energy balance p r i n c i p l e. involves. the use of S and E, also conjugate:. p 0 u = S:E + p 0 q + &- d i v ( h ). (2.34). 2.5 CONSTITUTIVE RELATIONS The balance. laws. provide. a set. of. equations. which. are. not. s u f f i c i e n t to determine the behaviour of a material body. Some f u r t h e r equations are necessary, s t a t i n g the r e l a t i o n between kinematic and dynamic variables. (constitutive. equations).. Constitutive equations are based on judgement, a - p r i o r i. knowledge. of how the material behaves. However, c e r t a i n general p r i n c i p l e s must be s a t i s f i e d in t h e i r f o r m u l a t i o n . For our purpose, the most important p r i n c i p l e i s t h a t of o b j e c t i v i t y ,. which s t a t e s t h a t. constitutive. equations must be i n v a r i a n t under changes of reference frame, in order to represent the material behaviour o b j e c t i v e l y . For a homogeneous material 1981), t h a t. an o b j e c t i v e. it. may be seen ( B i l l i n g t o n and Tate,. relation. b e t w e e n Cauchy. stress. and. deformation takes the form: a = R p ( C t ( s ) , T t ( s ) ) RT. (2.35). where R is the r o t a t i o n tensor (eqn. 2.11) and T the temperature. The n o t a t i o n Ct(s) s i g n i f i e s the h i s t o r y of C (eqn. 2.12) from - o ° < s < t . Note that in general the complete h i s t o r y of the deformation C (or of E equi val e n t l y ,. eqn.. (2.17)). are r e q u i r e d ,. while. for. R only. the. instantaneous current value is used, f o r r o t a t i n g the stresses. The second Piola-Kirchhoff. stress S is objective as such (being. r e l a t e d to a f i x e d reference c o n f i g u r a t i o n ) .. In terms of. it. eqn..
(29) 29. (2.35) may be rephrased as S -. B(Ct(s),Tt(s)). (2.36). The advantage of using the 2nd P i o l a - K i r c h h o f f stress tensor for t o t a l formulations is evident (see also sect.. 3.5.1).. Some types of E l a s t i c and P l a s t i c r a t e e q u a t i o n s are discussed below. For s i m p l i c i t y , a t t e n t i o n is centred on the isothermal case.. 2.5.1 RATE EQUATIONS Materials without memory or w i t h smooth memory may be described with rate equations, e . g . a= g ( d , a , F ). (2.37). where a is a stress rate which is objective f o r r i g i d body r o t a t i o n s . The choice of o b j e c t i v e r a t e i s not u n i q u e . A v a r i e t y of o p t i o n s are a v a i l a b l e , the two most widely used being the Jaumann rate a = a + a w + wTa and the Truesdell. (2.38). rate, G = <7 - ( j l T - 1(7+ a t r ( l ). In e q u a t i o n. (2.37). a. (2.39). p r o v i d e s the c o n s t i t u t i v e. part. of. the. s t r e s s i n c r e m e n t . Eqns. (2.38) or (2.39) d e f i n e the r e m a i n i n g t e r m s t h a t must be added f o r o b j e c t i v i t y . objective. rate may be made equivalent. Formulations. based on e i t h e r. by adjusting the. constitutive. law, g. However, i f g i s p o s t u l a t e d a - p r i o r i , i n d e p e n d e n t l y of the choice of o b j e c t i v e r a t e , both f o r m u l a t i o n s give r i s e t o. different. c o n s t i t u t i v e behaviour. The need for special objective rates is avoided i f the equations are formulated in a material s e t t i n g , e . g ..
(30) 30. S = g.(E,E,S). (2.40). S is the material time rate of a tensor on the current (2nd P i o l a - K i r c h h o f f ) ,. configuration. which is already o b j e c t i v e .. 2.5.2 ELASTICITY E l a s t i c m a t e r i a l s are those f o r which a n a t u r a l ,. stress-free. state e x i s t s , to which the body returns upon removal of a l l. external. f o r c e s . The stress depends on the deformation from t h i s natural. state:. S = f(C,t). (2.41). A perfect memory of the natural s t a t e , with no memory of. intermediate. states, is e x h i b i t e d . For. linear. elasticity. and small. strains. the. relation. i s as. fol1ows: o = c:e. (2.42). er1J = c^-j e*'). (in component form. c is termed the elasticity tensor, and e is the small strain tensor:. e. iij. =. (ui,j. +u. j,i>/2. (2'43>. where u are displacements. For i s o t r o p i c m a t e r i a l s , and provided a and e are both symmetric, c must take the form c. where. ijkl. "X«ij. 5. kl. +. 2 G. 5ikSjl. (2.44). X and G are c a l l e d Lame's c o n s t a n t s . This gives r i s e t o the. classic generalized Hooke's l a w :. ff. ij. =. ^kk. 5. ij. +. 2G. e. ij. (2-45).
(31) 31. 2.5.2.1 HYPERELASTIC MATERIALS The concept of H y p e r e l a s t i c i t y was introduced by Green and given i t s present name by T r u e s d e l l. (e.g. T r u e s d e l l and Toupin, 1960). I t. postulates the existence of a strain-energy. function from which the. stresses may be derived as S = ft, ( d W / d E ). (2.46). Assuming the necessary d i f f e r e n t i a b i 1 i t y , t h e e l a s t i c i t y t e n s o r i s defined as. D = P0 (d^/d^). (2.47). and a rate equation may be written as S = D:E (in component form. S. (2.48) = D\\ E. ). For a c o n s t a n t value of D, a l i n e a r h y p e r e l a s t i c t o t a l. equation. is obtained: S = D:E. (2.49). 2.5.2.2 HYPOELASTIC MATERIALS The term Hypoelastic, also introduced by Truesdell (Truesdell and Toupin, 1960), c h a r a c t e r i z e s a m a t e r i a l f o r which the behaviour i s defined. i n the c u r r e n t. configuration. by an i n c r e m e n t a l l y. linear. r e l a t i o n s h i p of the form:. 0= c:d (in component form. & 1J = c 1 J k -]d. (2.50) ). An o b j e c t i v e s t r e s s r a t e must be used f o r eqn. (2.50) (see s e c t i o n.
(32) 32. 2.5.1). Hypoelastic. behaviour is very convenient for descriptions. based. on the c u r r e n t c o n f i g u r a t i o n . M a t e r i a l data based on t r u e s t r e s s natural. strain relationships. hypoelastic. (see section 6.1) give rise n a t u r a l l y. to. interpretations.. For i s o t r o p i c materials eqn. (2.50) takes the form a. = \ t r ( d ) I + 2Gd. (in orthonormal components. (2.51). o^- = A d ^ 8^. + 2Gd^ •). 2.5.3 PLASTICITY For most s o l i d s , behaviour may be assumed e l a s t i c only w i t h i n a certain. stress. range. Beyond t h e. elastic. range y i e l d. occurs,. d e f o r m a t i o n s being c h a r a c t e r i z e d by permanent changes occasioned by s l i p or dislocations at the atomic level After y i e l d ,. flow).. E l a s t i c and P l a s t i c d e f o r m a t i o n s are assumed t o. happen c o n c u r r e n t l y ( E l a s t i c - P l a s t i c idealizations. (Plastic. m a t e r i a l s ) . More. are p r o v i d e d by r i g i d - p l a s t i c. restrictive. models (only. plastic. deformations). An a d d i t i v e decomposition of the rate of deformation is assumed here: d = d e + dP. (2.52). where s u p e r i n d i c e s e and p i n d i c a t e e l a s t i c and p l a s t i c components r e s p e c t i v e l y . A d d i t i v e d e c o m p o s i t i o n of s t r a i n s in t h i s fashion was proposed by H i l l. (1950). Lee (1969) has proposed a m u l t i p l i c a t i v e. decomposition of deformation gradients instead, F = Fepp, while Green and Naghdi (1965) have advocated an a d d i t i v e d e c o m p o s i t i o n of t o t a l s t r a i n , E = Ee + EP. Classical configuration. plasticity (Hill,. is. formulated. 1950). Hence the. i n terms of the. popularity. decomposition of the rates of deformation,. of. an. current additive. eqn. (2.52), coupled w i t h.
(33) 33. hypoelastic behaviour, f o r E l a s t i c - P l a s t i c. material descriptions. (e.g.. Wi 1 kins (1964), H i b b i t , Marcal and Rice (1970)). In t h i s case <7= c : d e = c : ( d - d P ). (2.53). The y i e l d c r i t e r i o n determines the l i m i t of the e l a s t i c range: F( a , Q ) = 0. (2.54). where Q is a set of p l a s t i c hardening parameters. For F<0 the material behaves e l a s t i c a l l y . Two additional sets of r e l a t i o n s must be provided to determine f u l l y the s t r e s s - s t r a i n. behaviour:. - Flow rule. dP = 7R( a,Q). (2.55a). - Hardening rule. Q=. (2.55b). 7H( a ,Q). where 7 is an a r b i t r a r y m u l t i p l i e r ,. whose value is determined from o. the simultaneous s o l u t i o n of eqns. (2.54), (2.55). An objective rate Q must be used in eqn. (2.55b). Drucker (1951) p o s t u l a t e d a c r i t e r i o n f o r stable work-hardening materials. This involves the work done by a set of. self-equilibrating. forces, r e q u i r i n g : a:d > 0. (2.56a). p. (2.56b). a:d ^ 0. The equal sign i n (2.56b) holds f o r p e r f e c t l y p l a s t i c m a t e r i a l s hardening). A consequence of Drucker's postulate is the. (no. associativity. of p l a s t i c flow: for a smooth part of the y i e l d surface, dp = 7 ( d F / d a ). (2.57). which i n a n i n e - d i m e n s i o n a l s t r e s s space may be i n t e r p r e t e d as the normality of dP to the surface F( c , Q ) ..
(34) 34. 2.5.3.1. VON MISES MODEL. P a r t i c u l a r l y usefu] and simple models are derived from the y i e l d c r i t e r i o n of Von Mises (1913). This may be w r i t t e n as F = ( 3 / 2 ) s : s - Y2 = 0. (2.58). where s are the deviatoric Cauchy stresses, s = a - (l/3)tr(a)I. (2.59a). (in orthonormal components, s ^ = a^-. - (1/3) o - ^ 5 . .. (2.59b). Y i s the y i e l d s t r e n g t h of the m a t e r i a l , which c o i n c i d e s w i t h the y i e l d stress in uniaxial. t e n s i o n (see s e c t i o n 6.1). The Von Mises. y i e l d c o n d i t i o n i s independent of v o l u m e t r i c s t r e s s e s , which are assumed to behave e l a s t i c a l l y . An i s o t r o p i c. hardening model is obtained by making Y a function. of the e f f e c t i v e p l a s t i c s t r a i n ,. eP:. Y = Y(e p ) with. eP. (2.60a). = J 6 e p = J A 2/3) dp: dP dt. (2.60b). A more general hardening model, incorporating Bauschinger e f f e c t , may be obtained by combining i s o t r o p i c. hardening w i t h the kinematic. hardening proposed by Prager (1956) and Z i e g l e r (1959), g i v i n g the y i e l d condition. F = ( 3 / 2 ) ( s - a ) : ( s - a ) - Y2 o: i s c a l l e d the b a c k - s t r e s s and r e p r e s e n t s a k i n e m a t i c. (2.61) hardening. parameter ( t r a n s l a t i o n of the Von Mises c i r c l e ) . The associative flow ru 1 e i s dP = 7 ( s - a ). (2.62).
(35) 35. and the hardening laws, a=. (2/3)hadP. (2.63a). Y = h y eP Imposing the c o n s i s t e n c y combining. eqns.. (2.51),. (2.63b). condition (2.53),. (F = 0) d u r i n g. (2.61)-(2.63),. the. loading,. and. stress-strain. r e l a t i o n is found to be. a = c:Ld - ( s - a ) ( 3 / 2 ) d : ( s - a ) / Y 2 ( i + h ' / 3 G ) J where h' = h y + n a. (2.64). ( p l a s t i c modulus). Purely i s o t r o p i c hardening i s. obtained with h a =0, and purely kinematic with h Y =o. In a u n i a x i a l. test,. law (2.64) w i l l. p r o v i d e an. Elastoplastic. hardening modulus of h = 1/L1/E + 1/h'J. (2.65). where E - G(3 X+2G)/( X+G) (Young's modulus of elasticity).. 2.5.3.2 OTHER PLASTICITY MODELS Plasticity. i n s o i l s i s g e n e r a l l y considerably more complicated. than the above Von Mises model. Pressure dependent y i e l d , d i l a t a t i o n and n o n - a s s o c i a t i v i t y ,. anisotropy,. h y s t e r e t i c c y c l i c behaviour, pore. p r e s s u r e , are i m p o r t a n t f e a t u r e s f o r s o i l p l a s t i c i t y . An e x c e l l e n t review of t h i s topic has been given by Marti and Cundall (1980). The mathematical t h e o r y of p l a s t i c i t y development. (e.g.. Mroz. Very refined phenomenological (1967),. multiple-surface. Prevost. (1978)).. is a f i e l d s t i l l. under. models have been proposed. These. models. are. based. on. i d e a l i z a t i o n s . They provide elaborate s t r e s s - s t r a i n. laws r e q u i r i n g c o n s i d e r a b l e c o m p u t a t i o n a l. cost. for. numerical. m o d e l l i n g , thus i n p r a c t i c e they are h a r d l y used. This f a c t has been acknowledged by O r t i z and Popov (1983), who propose s i m p l e r , surface models for metal. plasticity.. one-.
(36) 36. Finally, plasticity. a principle. model,. sophisticated. is. as t h e. that. to it. be h e l d p r e s e n t can o n l y. experimental. be as. when c h o o s i n g a reliable. information. and as. on w h i c h. the. determination of the model parameters is based. For example, there is little. point. i n u s i n g a n y t h i n g o t h e r than a Von Mises. isotropic. hardening model i n a m e t a l , i f a l l the i n f o r m a t i o n a v a i l a b l e i s a uniaxial of. s t r e s s - s t r a i n law. On the other hand, the added complication. some models. monotonic.. may not. be necessary. if. the. loading. is. mainly.
(37) CHAPTER 3 NONLINEAR NUMERICAL MODELS FOR SOLID MECHANICS. 3.1 INTRODUCTION 3.2 FINITE DIFFERENCE METHODS. 3.3 FINITE ELEMENT METHODS 3.4 MESH DESCRIPTIONS 3.4.1 Lagrangian 3.4.2 Eulerian 3.4.3 Arbitrary Lagrangian-Eulerian 3.5 LARGE DISPLACEMENT FORMULATIONS 3.5.1 Total Lagrangian 3.5.2 Cauchy stress - velocity strain 3.5.3 Updated Lagrangian 3.6 TIME INTEGRATION. 3.6.1 Central difference (explicit) 3.6.2 Trapezoidal rule (implicit) 3.6.3 Operator split methods 3.7 PRACTICAL CONSIDERATIONS FOR DISCRETE MESHES 3.7.1 "Locking up" for incompressible flow 3.7.2. "Hourglassing". 3.8 CONCLUSIONS.
(38) 38. 3.1 INTRODUCTION The Governing e q u a t i o n s. in s o l i d. mechanics. are the e q u a t i o n s. of. m o t i o n , which can be w r i t t e n i n component form as. aij5j. + p(fi-iii). = 0. (3.1). where cr^j - j s the Cauchy s t r e s s t e n s o r , p t h e mass d e n s i t y , f-j are body f o r c e s per u n i t equations. mass ( t y p i c a l l y. originate. from. the. gravity),. balance. of. and u i. displacements.. momentum p r i n c i p l e. 2 . 4 . 2 ) . T h i s p r i n c i p l e may be s t a t e d a l t e r n a t i v e l y (eqn.. (section. in i n t e g r a l. form. 2.28).. The p a r t i a l. differential. space and 1 t i m e. eqns. of. variables.. are. discretized. differential. in. motion. The n u m e r i c a l. p e r f o r m independent s e m i d i s c r e t i z a t i o n s (3.1). These. space,. (3.1) depend upon 3 described. here. i n space and t i m e . F i r s t. eqns.. yielding. models. a. system. of. ordinary. e q n s . i n t i m e . These a r e t h e n i n t e g r a t e d w i t h a t i m e -. stepping procedure.. The d i s c r e t i z a t i o n. of. t h e continuum. F i n i t e E l e m e n t (FE) or F i n i t e D i f f e r e n c e have had s e p a r a t e h i s t o r i c a l convergence Belytschko, different: However,. has. local. equivalent. proposed. for. Considerable simplification BEM lose. Boundary solid. in. the. principles for. literature. need. Element. mechanics. advantage. FD, g l o b a l. error. can. only. Methods by. Rizzo. be. gained. a. often. not be reviewed here.. produce. discretization. (BEM). (1967). of t h e continuum. for. 1981).. by. appeal. Volume i n t e g r a l s. discretization. are. norms f o r FE.. These and. the. in. were. Cruse. the first. (1969).. reduction. of t h e d i s c r e t i z a t i o n . For n o n l i n e a r p r o b l e m s ,. reason BEM w i l l. (e.g.. f o r b o t h methods. FE and FD f o r m u l a t i o n s. methods. much of t h e i r. an a d d i t i o n a l this. lately. (e.g. Kunar and Minowa,. numerical the. (FD) m e t h o d s . Both methods. a l s o based on independent shape f u n c t i o n s. As a r e s u l t ,. algorithms. Other. reached. truncation errors. FE methods are. with. d e v e l o p m e n t s , a l t h o u g h some d e g r e e o f. 1 9 8 3 ) . The t h e o r e t i c a l. each e l e m e n t .. boundary:. been. may be achieved e i t h e r. appear which. (e.g. G a r c i a ,. and. however, require. 1981).. For.
(39) 39. Time i n t e g r a t i o n methods or by d i r e c t. may be performed e i t h e r integration. by modal. analysis. ( t i m e - m a r c h i n g ) . Modal. analysis. r e q u i r e s t r a n s f o r m a t i o n s i n t o the frequency domain which are o n l y v a l i d in a l i n e a r regime, for which reason they must be ruled out for nonlinear models. As to time-marching procedures two main a l t e r n a t i v e s exist,. explicit. or. implicit. disadvantages, which w i l l. methods.. Both. have advantages. and. be reviewed b r i e f l y in t h i s chapter. Recent. a l t e r n a t i v e procedures based on operator s p l i t t i n g methods w i l l. also. be considered.. 3.2 FINITE DIFFERENCE METHODS F i n i t e D i f f e r e n c e methods have been used f o r a long t i m e by engineers w i t h i n relaxation procedures (e.g. Southwell,. 1940).. Finite. D i f f e r e n c e o p e r a t o r s p r o v i d e l o c a l a p p r o x i m a t i o n s f o r a system of coupled d i f f e r e n t i a l e q u a t i o n s . Due t o t h i s f a c t , a one-step g l o b a l solution is not possible and recourse must be made to relaxation and i t e r a t i v e techniques. A d d i t i o n a l l y , FD methods have been a s s o c i a t e d n o r m a l l y w i t h r e g u l a r zoning (at l e a s t t o p o l o g i c a l l y r e g u l a r ) . For these two reasons, FD methods were eclipsed by the F i n i t e Element boom in the 1960's f o r s t r u c t u r a l. and s o l i d mechanics a p p l i c a t i o n s . FD has. always been p o p u l a r , however, i n o t h e r areas such as E u l e r i a n mechanics (e.g. Nichols, H i r t , For a r e g u l a r. and Hotchkiss,. mesh w i t h. coordinate d i r e c t i o n s ,. 1980).. " I " and " J " l i n e s. standard f i n i t e. fluid. difference. along the. approximations. two for. the gradient of a vector u are given by: uI+l/2,J+l/2a. u. J_(uI+l,J+l/2_uI,J+l/2). | : i / 2 , J + l / 2 = 1_. (u. I+l/2,J+l_uj+l/2,J). (3.2). Ax.£ Eqns. (3.2) r e q u i r e the mesh t o be t o p o l o g i c a l l y and g e o m e t r i c a l l y regular. The use of contour i n t e g r a l. formulas. (Wilkins,. 1964) allows the. application of FD approximations t o t o p o l ogi cal l y and g e o m e t r i c a l l y i r r e g u l a r meshes. The basic idea is to employ Gauss' theorem in order.
(40) t o express the g r a d i e n t of a f i e l d i n a c e l l. i n terms of a c o n t o u r. i n t e g r a l . Considering c e l l VE enclosed by contour SE, and the gradient of the displacement vector u,. J. u i5J dV = [ u^jdS. (3.3). If the gradient is assumed constant in the cell, then. 1. f. ui ,j = - I u^jdS. (3.4). V EJ S E. The contour i n t e g r a l may be evaluated assuming a l i n e a r v a r i a t i o n of u along the edge of the c e l l . Contour. integrals. may. be used f o r. any. 2-0. polygon. or. 3-D. polyhedron, t o i n t e r p o l a t e a value f o r the g r a d i e n t at the c e n t r e o the c e l l , knowing the values at the c o r n e r nodes. For the p a r t i c u l a r cases of t r i a n g l e s available (e.g. M a r t i ,. and t e t r a h e d r a ,. an a l t e r n a t i v e. technique. is. 1981), in which the gradients are i n t e r p o l a t e d. d i r e c t l y by i n v e r t i n g t h e s p a t i a l f i n i t e d i f f e r e n c e e q u a t i o n s . This technique has been followed in the present work, and w i l l. be detailed. i n s e c t i on 4 . 2 . 1 . For e x p l i c i t time-marching models the semi-discrete equations of motion become uncoupled. This means that only local approximations the p a r t i a l. differential. to. e q u a t i o n s (3.1) are performed w i t h i n each. t i m e - s t e p , no i t e r a t i o n s being needed f o r a FD o p e r a t o r . Such a f a c t was exploited in the development of the f i r s t. FD "Hydrocodes" at the. U.S. n a t i o n a l l a b o r a t o r i e s i n the 1950's. These were o r i e n t e d m a i n l y towards sensitive nuclear and defence a p p l i c a t i o n s . was given u n t i l Noh (1964)).. the 1960's (Wilkins. At t h i s. time. Finite. Little. publicity. (1964), Maenchen and Sack (1964), Element. Methods. had j u s t. been. i n t r o d u c e d f o r s o l i d mechanics (Clough, 1960), and techniques were being developed for l i n e a r analysis. Not much a t t e n t i o n was given to FD f o r s o l i d mechanics by the e n g i n e e r i n g community, as indeed FE methods seemed much more powerful and indeed advantageous for. linear. systems, being able t o provide a one-step global s o l u t i o n .. Interest. i n the n o n l i n e a r. and w a v e - p r o p a g a t i o n. regimes. for. s p e c i a l i z e d e n g i n e e r i n g a p p l i c a t i o n s i n t h e l a t e 1960's and 1970's.
(41) 41. c r e a t e d a resurgence of the Hydrocodes (Bertholf and Benzley (1968), Wilkins (1975)), and a c e r t a i n degree of convergence between FD and FE literature. (Belytschko. (1978), K r i e g and Key (1976), Goudreau and. Hallquist (1982)). E x p l i c i t f i n i t e - d i f f e r e n c e methods were popularized t o wider s e c t o r s of the e n g i n e e r i n g community, and new codes were c r e a t e d such as PISCES (Hancock, 1976), t h e rock mechanics codes of Cundall and Marti (1979), and PR3D for s o l i d mechanics impact by Marti (1981).. 3 . 3 FINITE ELEMENT METHODS. The f i r s t. a p p l i c a t i o n of F i n i t e Element techniques for. continua. was by Clough (196U), a l t h o u g h the t h e o r e t i c a l bases f o r t h e method had already been set by Courant (1943) and applications to analysis had been proposed e a r l i e r. (Argyris and Kelsey,. F i n i t e Element d i s c r e t i z a t i o n s rely on two essential a variational. or weak form of the eqns.. of. motion. structural. 1954). ingredients: (3.1),. and a. c o n s t r u c t i o n of approximate s o l u t i o n s based on g e n e r a l i z e d nodal coordinates and independent element shape functions. The domain V i s s u b d i v i d e d i n t o elements VE, i n t e r c o n n e c t e d by nodes. An approximate s o l u t i o n is constructed w i t h i n an element E as a product of shape functions N^x) and the nodal displacements u f ( t ) : u(x,t). = u^(t)Nj(x). (3.5). where I is summed over the nodes of the element. The shape functions are chosen so t h a t u i s c o n t i n u o u s. over t h e element. boundaries,. a l t h o u g h i t s g r a d i e n t need not be c o n t i n u o u s (CO c o n t i n u i t y ) . The shape f u n c t i o n s Ni. so. constitutes. fact. in. d e f i n e d are a. local. independent of t i m e ; eqn. (3.5) separation. of. variables. (semi d i s c r e t i z a t i on ). The discrete form of the gradient operator may be w r i t t e n as. u. i'j. ". B. jluil. (3.6).
(42) 42. where B. Ji dx. i. Let the s o l i d continuum be V w i t h boundary S, c o n s i s t i n g of Sy where. and S j '. u = u* a n = T*. on Su. on. sT. A weak form of the eqns. of motion (3.1) may be o b t a i n e d by u s i n g e i t h e r Galerkin weighed residuals or the v i r t u a l work p r i n c i p l e ,. both. of which y i e l d the same r e s u l t : f. Vi,jCTijdV. +rpviuidV. V. = J p V i f i d V + [v-jT^dS. V. V. (3.7). ST. where v is the t e s t function (or v a r i a t i o n ) and u the t r i a l Eqns. (3.7) r e q u i r e only C° c o n t i n u i t y f o r both t r i a l. function. and t e s t. f u n c t i o n s , as opposed t o (3.1), f o r which Cl c o n t i n u i t y i s needed. I f the approximations defined in (3.5) are used for u and v, and because (3.7) must hold f o r a r b i t r a r y v, the g l o b a l d i s c r e t e equations. are. deduced: Mu + P(u) = R. (3.8). where the g l o b a l c o e f f i c i e n t m a t r i c e s M, P, R are assembled from individual element matrices that take the form: ME= / pNjNj&jjdV. (mass matrix;. E. V E. P = / Bjj^jdV J VE RE= /. f-j NjdV v£. ( i n t e r n a l forces) +f J. s$. N I T i dS. (external. forces). (3.9;.
(43) 43. (3.8) is a system of ordinary d i f f e r e n t i a l in t i m e ;. integration. equations of second order. of these i s discussed i n s e c t i o n 3.6. For a. l i n e a r model (small deformations and e l a s t i c behaviour), (3.8) becomes Mii + Ku = R. (3.10). The s t i f f n e s s matrix K is assembled from element matrices of the type KE = /. 6jjCijklB1KdV. (3.11). J vE In s t a t i c analysis the i n e r t i a term may be dropped from eqns. ( 3 . 8 ) : P(u) = R. (3.12). which for the linear case becomes Ku = R. (3.13). For l i n e a r a n a l y s i s , a s o l u t i o n i s o b t a i n e d merely by i n v e r t i n g the stiffness. m a t r i x K i n eqns. (3.10) or (3.13). In a n o n l i n e a r case,. eqns. (3.8) or (3.12) must be solved in a number of steps using NewtonRaphson or i t e r a t i v e techniques. I t is i n t e r e s t i n g to note the a b i l i t y of FE t o give a one-step s o l u t i o n t o the l i n e a r p r o b l e m , which F0 methods. lack,. relaxation. having. to. and i t e r a t i o n .. approach. the. global. Hence t h e p o p u l a r i t y. solution of FE f o r. through linear. problems. For nonlinear behaviour, however, t h i s advantage disappears'* as both FE and FD have to perform some sort of i t e r a t i o n s .. For large systems, the assemblage of matrix K i s undesirable, as core memory l i m i t s may be exceeded and recourse must be made to slow, c o s t l y disk Input/Output. This f a c t accounts f o r the popularization of relaxation. techniques. for. equation. solving. (e.g.. Flanagan. Belytschko,. 1981a), which avoid the assemblage of global. and. coefficient. m a t r i c e s . The FE o p e r a t o r s are used only at a l o c a l l e v e l . In t h i s case FE become conceptually very s i m i l a r to FD methods with general topology, s p e c i a l l y as they o f t e n p r o v i d e e q u i v a l e n t a l g o r i t h m s the local approximations (Kunar and Minowa, 1981).. for.
(44) 44. 3.4 MESH DESCRIPTIONS. Let a p a r t i c l e X of body B be d e f i n e d by i t s p o s i t i o n at t=0 ( r e f e r e n c e c o n f i g u r a t i o n ) , X . At t i m e t ( c u r r e n t c o n f i g u r a t i o n ) the p o s i t i o n of the p a r t i c l e w i l l be x = x(X,t). (3.14). x are called the spatial coordinates, and X the material coordinates of X. Eqns. (3.14) describe the motion of 8. For the discretization of B three types of meshes may be used, depending on the motion of the nodes of the mesh. The position of a point of the mesh, initially coincident with particle X, will be given by. X = X(X,t).. 3.4.1 LAGRANGIAN In a lagrangian description the mesh follows the motion of the body,. X(X,t) = x(X,t) A given node remains c o i n c i d e n t. (3.15). w i t h the same m a t e r i a l. throughout the motion. Each element w i l l. particle. contain the same domain of. material throughout the d e f o r m a t i o n , thus e n f o r c i n g i m p l i c i t l y. the. c o n t i n u i t y equation. Motion of the boundary does not present d i f f i c u l t i e s ,. as. it. always c o i n c i d e s w i t h t h e mesh boundary. For a s c a l a r f i e l d g ( X , t ) , the material. time d e r i v a t i v e. (i.e. f o l l o w i n g the p a r t i c l e ). coincides. w i t h the p a r t i a l time d e r i v a t i v e : dg g = -. (3.16). dt The only disadvantage of t h i s description comes from the fact that the mesh can become e x c e s s i v e l y. d i s t o r t e d f o r c e r t a i n problems. (e.g.. f l u i d s , high v e l o c i t y i m p a c t ) . In some cases, " r e z o n i n g " techniques.
(45) 45. may be used to circumvent this problem (e.g. Kalsi and Marti, 1985).. 3.4.2 EULERIAN In an Eulerian description the mesh is fixed in space, i.e. X(X,t) = X Nodes are no longer coincident. (3.17). w i t h material. p a r t i c l e s through t i m e ,. and the material flows through the c e l l s . Continuity must be enforced e x p l i c i t l y . The m a t e r i a l t i m e d e r i v a t i v e of g ( X > t ) i n c l u d e s a f l u x term: 69 39 dx dg d9 g = _ + = _ + _ v. dt. dx dt. dt. (3.18). dx. Numerical computations f o r the f l u x of s c a l a r f i e l d s tend t o smear t h e i r values, which w i l l not be defined as sharply as f o r a Lagrangian mesh. M a t e r i a l. boundaries are d i f f i c u l t t o d e s c r i b e , as they move. r e l a t i v e to the mesh. On the c r e d i t side, d i s t o r t i o n is not a problem, making Eulerian meshes preferrable to Lagrangian meshes for very large deformations.. 3 . 4 . 3 ARBITRARY LAGRANGIAN-EULERIAN. Arbitrary. Lagrangian-Eulerian. (ALE) d e s c r i p t i o n s. attempt. to. combine the advantages of Lagrangian and E u l e r i a n meshes. The mesh moves w i t h an a r b i t r a r i l y d e f i n e d m o t i o n ,. X ( X , t ) . d X / d t = 0 f o r an. Eulerian mesh, d X / d t = v for a Lagrangian mesh. X can be defined so as to f o l l o w the material in the boundary, but without causing excessive d i s t o r t i o n i n the i n t e r i o r . ALE f o r m u l a t i o n s have been developed by Noh (1964) and H i r t et a l . (1974) i n FO f o r m a t s , and by Donea et a l . (1977) and Belytschko and Kennedy (1978) in FE. Material derivatives are given by. d9 g = _. dt. dx 59 +. (3.19). dt dx.
(46) 46. With an ALE d e s c r i p t i o n p r o p e r t i e s s t i l l. need t o be f l u x e d t h r o u g h. c e l l s , and some smearing may occur as a r e s u l t . A c r u c i a l aspect i n ALE d e s c r i p t i o n s i s the d e f i n i t i o n of the a r b i t r a r y motion of t h e mesh. X. for internal. points. Generally a. complex rezoning algorithm is necessary for optimizing the new mesh p o s i t i o n s at each s t e p . Such a general r e z o n i n g a l g o r i t h m has been proposed f o r 2-D by G i u l i a n i. (1982). Schreurs (1983) has proposed a. mesh o p t i m i z i n g a l g o r i t h m based on the d e f o r m a t i o n of a f i c t i t i o u s material. from an " i d e a l " mesh.. In f a c t. equivalent to Lagrangian descriptions at every s t e p . Some a p p l i c a t i o n s. ALE techniques. would be. in which rezoning is. performed. (e.g. metal f o r m i n g ) may not need. such f r e q u e n t r e z o n i n g , and Lagrangian t e c h n i q u e s w i t h r e z o n i n g at wider i n t e r v a l s could be p r e f e r r a b l e .. 3 . 5 LARGE DISPLACEMENT FORMULATIONS. Several. formulations. are. possible. depending. on. which. c o n f i g u r a t i o n s the s t r e s s and d e f o r m a t i o n t e n s o r s are r e f e r r e d t o . Three a l t e r n a t i v e s widely used in s o l i d mechanics are presented below.. 3.5.1 TOTAL LAGRANGIAN The 2nd P i o l a - K i r c h h o f f. s t r e s s t e n s o r S and the Green. strain. t e n s o r E, both of which r e l a t e t o the r e f e r e n c e c o n f i g u r a t i o n , used t o d e s c r i b e the m a t e r i a l. behaviour. H i b b i t ,. (1970) proposed t h i s description. in the f i r s t. are. Marcal and Rice. published. large-strain,. large-displacement nonlinear formulation f o r general purpose FE codes. A c o n s t i t u t i v e r e l a t i o n is given by S = S(E). (3.20).
(47) 47. and in rate form by S = D:E ( S IJ. = D. (3.21). IJKLEKL components). S and E being m a t e r i a l t e n s o r s , t h e i r m a t e r i a l rates are o b j e c t i v e . This formulation is advantageous f o r H y p e r e l a s t i c m a t e r i a l s , whose behaviour is described on the reference c o n f i g u r a t i o n .. In t h i s case,. c a l l i n g W the s t r a i n energy functional per u n i t mass, d2W dE2. (3.22). aw s = — d£ Elastic-piastic. material. behaviour. i s best d e s c r i b e d on the. c u r r e n t c o n f i g u r a t i o n , x ( H i l l , 1950). I t i s p o s s i b l e t o t r a n s f o r m such a law i n t o one of the type (3.21) (e.g. H i b b i t et a l . (1970), Krieg and Key (1976)), but complex and c o m p u t a t i o n a l l y. expensive. transformations are necessary. However, Simo and O r t i z (1985) suggest that t o t a l. Lagrangian,. Hyperelastic-type formulations. provide a more. r i g o r o u s approach f o r i n c r e m e n t a l , n o n - l i n e a r c a l c u l a t i o n s . Such r i g o u r i s not j u s t i f i e d i n e x p l i c i t c a l c u l a t i o n s w i t h very. small. steps.. 3.5.2 CAUCHY STRESS-VELOCITY STRAIN A description based on the current c o n f i g u r a t i o n may be used to model the behaviour of materials w i t h smooth memory. In the simplest case,. the. Jaumann. rate. of. Cauchy. stress. ( a ) and the. rate. of. deformation tensor ( v e l o c i t y s t r a i n , d) are related by ff= C:d ( ajj. = Cj^idki. (3.23) in component form).
(48) 48. where C i s the c o n s t i t u t i v e t e n s o r . For an e l a s t i c - p l a s t i c. materia'. (hypoelastic with associated p l a s t i c i t y ) C takes the form C. ijkl. =. ^ij5ki. + 2G(5 i k 5 J 1. (3.24). -^ijnki). where rj >0 f o r p l a s t i c loading, =0 otherwise n is the u n i t normal to the y i e l d surface The Jaumann d e r i v a t i v e. used i n. eqn.. (3.23) p r o v i d e s. c o n s t i t u t i v e part of the stress rate. To obtain the t o t a l. stress. the rate. the r o t a t i o n a l components must be added: a• • = o- • + Ww• CTn • + Ww- a • 1J 1J 1P PJ JP pi. (3.25). I f the m a t e r i a l behaviour i s a n i s o t r o p i c , C must be updated w i t h a s i m i l a r objective r a t e : c. ijkl. _. c. (3.26). ijkl+wipcpjkl+wjpcipkl+wkpcijpl+wlpcijkp. An a l t e r n a t i v e formulation results from the use of the Truesdell rate in eqn. (3.23): :3.27. (J= C:d. (3.23) and (3.27) are equivalent i f one sets. Cijkl =. C. ijkl. +. CT 5. ij kl-^ik5jl+oril5jk+(7jk5il. +. CT 5. il ik)/2. The T r u e s d e l l s t r e s s r a t e i s the f o r w a r d P i o l a (eqns.. 2.22,. (3-28). transformation. 2.25) of the r a t e of the 2nd P i o l a - K i r c h h o f f. stress. tensor: O = 0t*(J-1S). [3.29). Pinsky, Ortiz and Pister (1983) have suggested that the Truesdell rate. formulation. is. the natural. configuration) for hyperelasticity. tensor i s obtained d i r e c t l y ,. one t o. use. (in. the. current. In t h i s case the c o n s t i t u t i v e. from the t o t a l Lagrangian tensor D, as.
(49) 49. £= Classical (Hill,. 0t*(j-lD).. plasticity. i s d e s c r i b e d on the current c o n f i g u r a t i o n. 1950). Jaumann Cauchy stress formulations have been widely and. successfully Maenchen. used f o r. and Sack. formulations. elastic-plastic. (1964),. hyperelastic. Krieg. behaviour. and Key. behaviour. (Wilkins. (1976)).. (1964),. With. ( r e l a t e d t o the. such. original. c o n f i g u r a t i o n ) may also be described, a l b e i t in a less convenient way, as the c o n s t i t u t i v e. relations. need t o be pushed f o r w a r d i n t o the. current c o n f i g u r a t i o n . Finally, integrable. one problem w i t h t h i s. (i.e. i t. f o r m u l a t i o n is t h a t d i s not. i s not the r a t e of any v a l i d s t r a i n. tensor).. Additional s t r a i n computations must be done i f a t o t a l s t r a i n measure is requi r e d .. 3.5.3 UPDATED LAGRANGIAN In t h i s. formulation. the. model. is. described. on a r e f e r e n c e. c o n f i g u r a t i o n , which is updated at each increment to coincide w i t h the c u r r e n t c o n f i g u r a t i o n . From t h i s updated reference, the incremental c o n f i g u r a t i o n is described w i t h a t o t a l. Lagrangian f o r m u l a t i o n .. This. method was f i r s t proposed by Yaghmai and Popov (1971), and has been w i d e l y used since f o r. incremental. nonlinear. analysis:. Osias and. Swedlow (1974), Bathe et a l . (1975), Nagtegaal and de Jong (1981). For t h i s d e s c r i p t i o n , F = I ( i d e n t i t y ) and J = 1. Hence, eqn. (3.29) i m p l i e s S = a. I t i s a l s o easy t o see from eqn. (2.19) t h a t E = d. In f a c t t h i s f o r m u l a t i o n r e v e r t s t o t h e T r u e s d e l l Cauchy s t r e s s r a t e f o r m u l a t i o n , eqn. (3.27). This means t h a t the t e n s o r t o be used f o r the tangential s t i f f n e s s. is £.. 3.6 TIME INTEGRATION. Using e i t h e r F i n i t e Difference or F i n i t e Element Methods for the spatial (3.1). semidiscretization,. the p a r t i a l. differential. eqns. of motion. may be t r a n s f o r m e d i n t o a system of o r d i n a r y. differential.
(50) 50. equations in time: Mii + Cu + P(u) = R These eqns. can be s o l v e d e i t h e r. (3.30) by modal a n a l y s i s. or by. direct. i n t e g r a t i o n . Modal a n a l y s i s methods (e.g. Bathe and Wilson (1976), c h p t . 8) p e r f o r m t r a n s f o r m a t i o n s of eqns. (3.30) which are only v a l i d for l i n e a r or q u a s i - l i n e a r systems (i.e. P(u) = Ku) For n o n l i n e a r. analysis,. direct. integration. (time-marching). methods must be used. For these the time domain is divided i n t o t i m e steps {At),. and an incremental analysis is performed for each step.. Time i n t e g r a t i o n procedures may be c l a s s i f i e d i n t o e x p l i c i t. and. i m p l i c i t . E x p l i c i t schemes compute the incremental displacements u from the e q u i l i b r i u m c o n d i t i o n s at t i m e t . I m p l i c i t schemes, on the c o n t r a r y , solve the eqns. of motion (3.30) at t+hAt > t , producing an i m p l i c i t system of eqns. for ut+At. Two of the most common and r e p r e s e n t a t i v e. time-integration. schemes, one in e i t h e r c l a s s , are presented below.. 3.6.1 CENTRAL DIFFERENCE (EXPLICIT) Central. d i f f e r e n c e methods are the most w i d e l y used e x p l i c i t. schemes for s o l i d mechanics, being the optimal from a very wide class (Key,. 1978). The F i n i t e Difference expressions used for v e l o c i t y and. acceleration are an+l/2=. (l/i+l_ji. ) M t. (3>31a). ii n = ( u n + 1 / 2 - u n " 1 / 2 ) / A t Note t h a t. each d e r i v a t i v e. lags. the. value. (3.31b) by h a l f. a. time-step.. P a r t i c u l a r i z i n g the equations of motion (3.30) at time n, Mun + Cun + P(u n ) = Rn. (3.32).
(51) 51. Using eqn. (3.31b) and l e t t i n g. On = ( u n - l / 2 + u n + l / 2 ) / 2 ,. eqn.. (3.32) may. be s o l v e d , y i e l d i n g :. u " + l / 2 = (MMt-C/2)- 1 [MMt-C/2)u n - 1 / 2 +R n -P(u n )]. (3.33). The new d i s p l a c e m e n t s un + 1 a r e t h e n f o u n d f r o m e q n . ( 3 . 3 1 a ) . I f. the. system has no damping (C = 0) and t h e mass m a t r i x M i s d i a g o n a l , eqns. (3.33) become u n c o u p l e d :. •n+l/2 = - n - l / 2 + These e q n s . freedom. can t h e n. M-l[Rn_p(u%t. be s o l v e d. (3.34). independently. for. each. degree. of. I: •n+l/2=. Gn-l/2+A(Rn. _. p. n)/lt)i. (3>35). The equations a l s o become uncoupled i f be of t h e R a y l e i g h t y p e ,. t h e damping i s assumed t o. as shown i n s e c t i o n 4.6.. This u n c o u p l i n g of the e q u a t i o n s of motion i s the major advantage of e x p l i c i t. i n t e g r a t i o n procedures. No mass or s t i f f n e s s. matrices. need. be i n v e r t e d or even assembled, as a l l t h e i n c r e m e n t a l c a l c u l a t i o n s each degree of. freedom can be done i n d e p e n d e n t l y. at. the. local. for. level.. This not o n l y a l l o w s f o r a s i m p l e r a r c h i t e c t u r e i n computer codes, but i t enables t h e t r e a t m e n t of n o n - l i n e a r i t i e s. (be i t. Geometric. no added c o s t f r o m. or Boundary t y p e ) w i t h v i r t u a l l y. of. Constitutive, the. l i n e a r c a s e . The number o f o p e r a t i o n s p e r t i m e - s t e p i s much s m a l l e r than f o r i m p l i c i t. methods ( s e c t i o n 3 . 6 . 2 ) , and s t o r a g e. grow only l i n e a r l y. w i t h t h e s i z e of t h e p r o b l e m .. The explicit. main. disadvantage. of. the. methods i s t h a t c o m p u t a t i o n s. central. requirements. difference. and. are only c o n d i t i o n a l l y. other stable. depending on the t i m e - s t e p s i z e . The t i m e - s t e p must be s m a l l e r than a certain c r i t i c a l This. constitutes. value f o r numerical a major. obstacle. e r r o r s n o t t o grow u n b o u n d e d . for. certain. problems. excessive number of t i m e - s t e p s makes t h e a n a l y s i s t o o. The s t a b i l i t y. of. the c e n t r a l. difference. where. an. considered. in. costly.. method i s. s e c t i o n 4.7. The t i m e - s t e p i s l i m i t e d by t h e C o u r a n t c r i t e r i o n ,. i.e..
(52) 52. the time i t takes the stress waves to travel across one element. This l i m i t a t i o n is consistent w i t h the l o c a l , uncoupled i n t e g r a t i o n of the equations of m o t i o n . I f the t i m e - s t e p was l a r g e r than the Courant c r i t i c a l value, stress waves would t r a v e l across an element w i t h i n one time-step,. affecting. the. surrounding. elements.. The. incremental. behaviour of that element would no longer be independent from the rest of the model. C e n t r a l d i f f e r e n c e schemes have been w i d e l y used i n n o n l i n e a r numerical codes, from the e a r l y FD hydrocodes of Wi1kins(1964) and Maenchen and Sack(1964), to the FE codes of H a l l q u i s t. (1982a, 1982c),. Key (1974), and B e l y t s c h k o and Tsay(1982). The accuracy and s t a b i l i t y of central difference methods has been studied and discussed by v a r i o u s. authors. Belytschko(1978),. (e.g.. Belytschko,. Holmes. and M u l l e n. Krieg and Key(1973)). The central. (1975),. difference. method. is considered as the most convenient w i t h i n the e x p l i c i t class.. 3.6.2. TRAPEZOIDAL RULE (IMPLICIT) The. so-called. trapezoidal. i n t e g r a t i o n methods. In f a c t Newmark f a m i l y ,. rule. is. an example. of. implicit. i t c o n s t i t u t e s a p a r t i c u l a r case of the. probably the most popular of the i m p l i c i t. schemes. A. c o n s t a n t average a c c e l e r a t i o n i s assumed f o r each i n c r e m e n t At. The difference equations are: u n + h = (ii n + u n + 1 )/2 n+1. n. n. (0 s< h < 1). n+1. u = u + (ii + i i )At/Z u n + l = u n + a n 4 t + ( .jjn + yn+1. (3.36) )At. 2/4. For obtaining un+l the equations of motion are enforced for time t+At.. In an undamped case, Mii n+1 + P(u n + 1 ) = R n+1. Eqn.. (3.37). difference obtained:. is. an i m p l i c i t. expressions. (3.37). relation. (3.36). in. for. un+l.. (3.37) the. Substituting. following. system. the is.
Documento similar
