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Thermo-plastic constitutive relations and a variational solution technique using finite elements Charlwood, Robin Gurney 1970

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THERMO - PLASTIC CONSTITUTIVE RELATIONS AND VARIATIONAL SOLUTION TECHNIQUE USING FINITE ELEMENTS by ROBIN GURNEY CHARLWOOD Diploma i n C i v i l Engineering, Brighton College of Technology, 1 9 6 l M.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1 9 6 8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1 9 7 0 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . R. G. Charlwood Department o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada May, 1970. THERMO - PLASTIC CONSTITUTIVE RELATIONS AND A VARIATIONAL SOLUTION TECHNIQUE  USING FINITE ELEMENTS ABSTRACT This study i s concerned with a thermo-elasto-p l a s t i c continuum i n which the thermodynamic co u p l i n g between m a t e r i a l deformations, heat generation and flow are included. The e l a s t o - p l a s t i c behavior i s represented by a l i n e a r " r a t e -type" theory and r e s t r i c t e d to i n f i n i t e s i m a l deformations. The heat flow equations are posed i n incremental form f o r the l i n e a r theory of heat conduction. The theory i s such t h a t the conservation laws of thermodynamics are s a t i s f i e d . A displacement f o r m u l a t i o n i s used and the f i e l d equations are shown to be l i n e a r operator equations. The operators are t e s t e d f o r symmetry and p o s i t i v e - d e f i n i t e n e s s i n order to t e s t t h e i r s u i t a b i l i t y f o r s o l u t i o n by a v a r i a t i o n a l method. I t i s shown that the s p e c i a l case of t h e r m o - e l a s t i c i t y may be solved by m i n i m i s a t i o n of a f u n c t i o n a l and t h a t convergence of approximate s o l u t i o n s may be p r e d i c t e d . In the general case of t h e r m o - e l a s t o - p l a s t i c i t y , the operator i s shown to be unsymmetric. Therefore an i t e r a t i v e f u n c t i o n a l i s introduced i n order to o b t a i n a s o l u t i o n by a v a r i a t i o n a l method. Approximate s o l u t i o n s to some i l l u s t r a t i v e problems are found using a f i n i t e - e l e m e n t - f o r m u l a t i o n , and i t i s shown t h a t the r e s u l t s are c o n s i s t e n t w i t h expected p h y s i c a l behavior. TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOTATION ACKNOWLEDGEMENTS CHAPTER 1 INTRODUCTION 2 STATEMENT OF THE PROBLEM 2.1 I n t r o d u c t i o n 2.2 Domain of the Problem 3 CONSTITUTIVE LAWS 3.1 I n t r o d u c t i o n 3.2 E l a s t i c - P l a s t i c C o n s t i t u t i v e R e l a t i o n s 3.3 Thermodynamic C o n s t i t u t i v e R e l a t i o n s 3 . 4 Comments on Existence of C o n s t i t u t i v e R e l a t i o n s 4 FIELD EQUATIONS 4 . 1 I n t r o d u c t i o n 4 . 2 Review of F i e l d Equations 4 . 3 Rate Form of F i e l d Equations ( i i i ) Page CHAPTER 5 THE VARIATIONAL METHOD 5.1 I n t r o d u c t i o n 37 5.2 Basic D e f i n i t i o n s 38 5 . 3 The Minimal F u n c t i o n a l Method 40 5 . 4 A p p l i c a t i o n to Thermo-Elastic 45 Equations 5.5 A p p l i c a t i o n to Thermo-Elastic- 55 P l a s t i c Equations 5 . 6 An I t e r a t i v e Method f o r 58 Unsymmetric Operators 5.7 A p p l i c a t i o n of I t e r a t i v e 62 Method to Thermo-Elastic-P l a s t i c Equations 6 A FINITE ELEMENT FORMULATION 6.1 I n t r o d u c t i o n 67 6.2 Choice of C o n s t i t u t i v e R e l a t i o n s 68 6.3 D e r i v a t i o n of Equations f o r 73 ! F i n i t e Element 6 . 4 Comments on Computer Program 85 f o r S o l u t i o n 7 SOLUTION OF SOME ILLUSTRATIVE PROBLEMS 7.1 I n t r o d u c t i o n 90 7.2 P r e s e n t a t i o n of Sol u t i o n s 95 8 SUMMARY 107 BIBLIOGRAPHY 109 ( i v ) LIST OF TABLES Page I Table of Comparative S o l u t i o n s 92 I I M a t e r i a l P r o p e r t i e s f o r Computer 94 Sol u t i o n s I I I Comparison of Pressures f o r Y i e l d i n g 104 (v) LIST OF FIGURES Page 1 Domain of the Problem 5 2 Generalised Co-ordinates 75 3 Surface T r a c t i o n s and Temperatures 8 l k Computer Program Flow Chart 86 5 Averaging Technique f o r Stresses and 93 Temperatures 6 Hollow C y l i n d e r f o r Cases 1 and 3 97 7 F i n i t e Element I d e a l i s a t i o n of Hollow 97 C y l i n d e r f o r Cases 1 and 3 8 Load Diagram f o r Case 1 - I n t e r n a l Pressure 98 9 Graph of Stresses and Displacement f o r Case 1 99 10 Graph of Temperatures f o r Case 1 98 11 S o l i d C y l i n d e r f o r Case 2 101 12 F i n i t e Element I d e a l i s a t i o n of S o l i d 101 C y l i n d e r f o r Case 2 13 Load Diagram f o r Case 2 - Transient 102 Temperature lk Graph of Temperature f o r Case 2 102 15 Load Diagram f o r Case 3 - I n t e r n a l Pressure and Temperature 16 Graph of Stresses and Displacement f o r Case 3 17 Graph of Temperature f o r Case 3 ( v i i ) NOTATION The s p e c i f i c usage and meaning of symbols i s defined i n the t e x t where they are introduced. The summation convention holds f o r s u b s c r i p t e d v a r i a b l e s with repeated i n d i c e s except where s p e c i f i c a l l y noted. The range of summation i s from 1 to 3, except i n parts of Chapter 7 where i t i s reduced to be from 1 to 2 where noted. P a r t i a l d i f f e r e n t i a t i o n w i t h respect to s p a t i a l (X^) co-ordinates i s denoted by a comma. A dot denotes a time (t) d e r i v a t i v e . Egs ',3 lxt2>t denotes an array w i t h f r e e v a r i a b l e t = /, 3 ( v i ACKNOWLEDGEMENTS The author wishes to express h i s g r a t i t u d e to h i s p r i n c i p a l a d v i s o r , Dr. D. L. Anderson, and a l s o to Dr. N. D. Nathan f o r t h e i r i n v a l u a b l e guidance during the research and pr e p a r a t i o n of t h i s t h e s i s . The f i n a n c i a l support of the N a t i o n a l Research Co u n c i l of Canada i n the form of a Research A s s i s t a n t s h i p i s g r a t e f u l l y acknowledged. G r a t i t u d e i s a l s o expressed to the U n i v e r s i t y Computing Centre f o r the f r e e use of t h e i r f a c i l i t i e s . The author deeply appreciates the support and co n s i d e r a t i o n given to him by h i s wife during the peri o d of h i s s t u d i e s . May, 1970 Vancouver, B r i t i s h Columbia 1 CHAPTER 1  INTRODUCTION This study i s concerned with a thermo-elasto-p l a s t i c continuum i n which the thermodynamic co u p l i n g between m a t e r i a l deformations, heat generation and flow are included. The e l a s t o - p l a s t i c behavior i s represented by a l i n e a r " r a t e -type" theory and r e s t r i c t e d to i n f i n i t e s i m a l deformations. The heat flow equations are posed i n incremental form f o r the l i n e a r theory of heat conduction. The theory i s such t h a t the conservation laws of thermo-dynamics are s a t i s f i e d . A displacement f o r m u l a t i o n i s used and the f i e l d equations are shown to be l i n e a r operator equations. The operators are tes t e d f o r symmetry and p o s i t i v e - d e f i n i t e n e s s i n order to t e s t t h e i r s u i t a b i l i t y f o r s o l u t i o n by a v a r i a t i o n a l method. I t i s shown th a t the s p e c i a l case of t h e r m o - e l a s t i c i t y may be solved by minimisation of a f u n c t i o n a l and th a t convergence of approximate s o l u t i o n s may be p r e d i c t e d . In the general case of t h e r m o - e l a s t o - p l a s t i c i t y , the operator i s shown to be unsymmetric. Therefore an i t e r a t i v e f u n c t i o n a l i s introduced i n order to o b t a i n a s o l u t i o n by a v a r i a t i o n a l method. Approximate s o l u t i o n s to some i l l u s t r a t i v e problems are found using a f i n i t e - e l e m e n t - f o r m u l a t i o n , and i t i s shown that the r e s u l t s are c o n s i s t e n t with expected p h y s i c a l behavior. A theory of the e l a s t i c - p l a s t i c continuum has been 2 presented by Green and Naghdi [ 8 } and [ 9 ] who, i n a d d i t i o n to f o r m u l a t i n g the c o n s t i t u t i v e p o s t u l a t e s of the l i n e a r " r a t e -type" theory i n general form, have shown the r o l e of the laws of thermodynamics. The thermodynamical c o n s i d e r a t i o n s i n [ 8 ] are i n terms of the Hel;miholz f r e e energy f u n c t i o n , the form of which i s not s p e c i f i e d . In t h i s study the works of Naghdi [ l 8 ] , K e s t i n [ l l ] and Lee f l 4 j are used to p o s t u l a t e e x p l i c i t forms f o r the entropy, i n t e r n a l energy and f r e e energy f u n c t i o n s . An important assumption i s that entropy and i n t e r n a l energy are s t a t e v a r i a b l e s f o r the i r r e v e r s i b l e processes of p l a s t i c deformation and heat flow. Using the postula t e d theory, the r e s t r i c t i o n s which the laws of thermo-dynamics place on the p l a s t i c flow law are i n v e s t i g a t e d . The work of B i o t [ 1] i s used as the b a s i s f o r for m u l a t i n g the f i e l d equations. These equations are w r i t t e n i n incremental form i n terms of the m a t e r i a l displacement v e c t o r and the entropy displacement v e c t o r introduced i n [ l ] . To o b t a i n s o l u t i o n s to the f i e l d equations the f i n i t e element method was u t i l i z e d . To do t h i s , advantage was taken of the work of Melosh [ l 5 ] , Key [ l 2 ] and O l i v e i r a [19J i n the f i e l d of c l a s s i c a l e l a s t i c i t y i n which they showed that the f i n i t e element method may be used as a d i r e c t v a r i a t i o n a l method, s i m i l a r to the R i t z method. The advantage of t h i s f o r m u l a t i o n i s t h a t bounds and convergence c h a r a c t e r i s t i c s of the s o l u t i o n s may be i d e n t i f i e d . The mathematical foundation f o r t h i s a p p l i c a t i o n of the v a r i a t i o n a l technique i s given by M i k h l i n [l6~\ and [17]. 3 M i k h l i n shows tha t the s u f f i c i e n t c o n d i t i o n s on the operator of the f i e l d equations f o r the v a r i a t i o n a l method to apply are that the operator i s symmetric and p o s i t i v e d e f i n i t e . T o n t i [ 2 1 ] proves t h a t formal symmetry i s a necessary c o n d i t i o n . In t h i s study i t i s found t h a t i n the case of the thermo-elastic equations the necessary and s u f f i c i e n t c o n d i t i o n s are s a t i s f i e d and the d i r e c t v a r i a t i o n a l method may "be a p p l i e d . In the general case of the t h e r m o - e l a s t o - p l a s t i c equations i t i s shown that the necessary c o n d i t i o n s of formal symmetry cannot be s a t i s f i e d . Consequently the formal method could not be a p p l i e d and an a l t e r n a t i v e method was th e r e f o r e derived u s i n g an i t e r a t i o n scheme. This method i s developed using a modified v a r i a t i o n a l method by s p l i t t i n g the operator i n t o symmetric and anti-symmetric p a r t s . The method i s a p p l i e d to the f i n i t e element method and a minimal f u n c t i o n a l i s given. To o b t a i n s o l u t i o n s to the f i e l d equations f o r some problems a f i n i t e element was developed f o r the case of plane s t r a i n and two dimensional heat flow. The methods of F e l l i p p a [ 6 ] were used to produce the sim p l e s t p o s s i b l e two dimensional element c o n t a i n i n g l i n e a r l y v a r y i n g displacements and constant s t r e s s e s and temperature. A computer program was w r i t t e n and s o l u t i o n s were obtained f o r some i l l u s t r a t i v e problems. Owing to the s c a r c i t y of closed form s o l u t i o n s to e l a s t i c - p l a s t i c work-hardening problems, and the l a c k of any s o l u t i o n s to f u l l y coupled thermo-plastic problems, 4 the numerical r e s u l t s could not be compared d i r e c t l y with e x i s t i n g s o l u t i o n s . In order to a l l o w maximum use of e x i s t i n g s o l u t i o n s f o r comparison and to i l l u s t r a t e the features of t h i s study the f o l l o w i n g cases were analysed: 1. E l a s t i c - p l a s t i c work-hardening with an i n i t i a l l y uniform temperature f i e l d . 2. Transient heat f l o w with no mechanical c o u p l i n g . 3. E l a s t i c - p l a s t i c work-hardening with a non-uniform temperature f i e l d . I t i s shown that the s o l u t i o n s to the above cases are c o n s i s t e n t with the expected behavior of a thermo-plastic m a t e r i a l . 5 CHAPTER 2 STATEMENT OF THE PROBLEM 2.1 I n t r o d u c t i o n The purpose of t h i s chapter i s to define the domain of the problem and i t s boundaries. The basic f i e l d v a r i a b l e s , displacements e t c . , are a l s o presented. 2.2 Domain of the Problem The domain of the problem considered i s defined below with reference to Figure 1. Figure 1 - Domain of the Problem Sl i s the region of 3-dimensional Euclidean space occupied by the volume of the body, not i n c l u d i n g the boundary S . The 6 c l o s e d r e g i o n SI i s defined asJZ + S . A point P = P ( x W l t } i s defined a t time t through i t s three r e c t a n g u l a r c a r t e s i a n co-ordinates X w , 10=1,3-The m a t e r i a l displacements u.j = u-i (x*», t ) and entropy displacements = ^ (Xm^l) are assumed to be defined throughout Jl . In t h i s study displacements are r e s t r i c t e d to be i n f i n i t e s i m a l . Stresses <TVj w i l l be defined i n JZ , surface t r a c t i o n s T; on S and temperatures Q i n Jl . The temperature 0 i s defined as the d i f f e r e n c e 9*T-~T0 , whereT i s the current absolute temperature and To i s the absolute temperature i n the unstrained n a t u r a l s t a t e . The n a t u r a l s t a t e i s assumed to be i n thermodynamic e q u i l i b r i u m and consequently T0 w i l l be uniform throughout SL . Body fo r c e s Ft" may be s p e c i f i e d i n Jl . The boundary S i s subdivided i n t o f o u r regions, S^, 5^, ST and Sg>. i s the regi o n where ut- are s p e c i f i e d , where ^ are s p e c i f i e d , Sj where are s p e c i f i e d and S& where 0 i s s p e c i f i e d . The s u b d i v i s i o n i s subject to the f o l l o w i n g r e s t r i c t i o n s . 5 T •+ Su. = 3 The problem i s posed i n a displacement f o r m u l a t i o n . Hence the s o l u t i o n w i l l be to f i n d the displacements u,- and $A as a f u n c t i o n of time o v e r J I . The s o l u t i o n w i l l be req u i r e d to s a t i s f y the equations of continuum mechanics and the postulate d c o n s t i t u t i v e laws. Since the e l a s t i c - p l a s t i c c o n s t i t u t i v e law i s non-. . . . ( 2 . 2 . 1 ) 7 l i n e a r i n terms of t o t a l deformations, the problem i s formu-l a t e d as a " r a t e " theory and solved i n c r e m e n t a l l y w i t h time steps dt . Consequently the boundary c o n d i t i o n s and body fo r c e s w i l l , need to be p r e s c r i b e d f o r each increment. The incremental procedure a l s o allows s o l u t i o n of the time dependent heat flow problem once the r e l a t i o n s h i p between the f i e l d v a r i a b l e s and time i s s p e c i f i e d . The equations w i l l be developed i n the general three dimensional case. However, i n the s o l u t i o n of problems i n Chapter 7 the s p e c i a l case of plane s t r e s s , i . e . two dimensions, w i l l be considered. 8 CHAPTER 3  CONSTITUTIVE LAWS 3.1 I n t r o d u c t i o n The e l a s t i c - p l a s t i c c o n s t i t u t i v e p o s t u l a t e s considered are those of the "rate-type" theory. Green and Naghdi [ 8 ] give a general theory f o r an e l a s t i c - p l a s t i c continuum f o r non-isothermal deformations. In t h e i r work they g i v e a f o r m u l a t i o n which may be made c o n s i s t e n t with the f i r s t and second laws of thermodynamics. P r e v i o u s l y , there had been some confusion between the e l a s t i c - p l a s t i c c o n s t i t -u t i v e p o s t u l a t e s and the requirements of thermodynamics. Drucker's [ 5 ] n o r m a l i t y hypothesis, which was introduced on the b a s i s of a pseudo-thermodynamic requirement and a d e s i r e f o r uniqueness of s o l u t i o n s , i s seen i n Green and Naghdi's work as an e l a s t i c - p l a s t i c c o n s t i t u t i v e p o s t u l a t e and i s not a necessary c o n d i t i o n . The c o n s t i t u t i v e p o s t u l a t e s are considered i n t h i s chapter and the requirements of thermodynamics are considered w i t h the f i e l d equations i n Chapter 3.2 E l a s t i c - P l a s t i c C o n s t i t u t i v e R e l a t i o n s The c o n s t i t u t i v e r e l a t i o n s used i n t h i s study are based on "A General Theory of an E l a s t i c - P l a s t i c Continuum" 9 by A. E. Green and P. M. Naghdi [ 8 ]. In t h i s study only i n f i n i t e s i m a l displacements are considered. A general c o n s t i t u t i v e r e l a t i o n i s used which a p p l i e s to i n i t i a l l y a n i s o t r o p i c m a t e r i a l s with a n i s o t r o p i c y i e l d f u n c t i o n s and hardening r u l e s . The r e l a t i o n s are w r i t t e n as a "rate-type" theory as i n [ 8 ] . This i s d i r e c t l y analogous to an "incremental" theory. The two t h e o r i e s are interchangeable by m u l t i p l y i n g by an increment of time d t . For i n f i n i t e s i m a l displacements the t o t a l s t r a i n r ate tensor e;j i s defined as * i -* "vi** ) • •••• ( 3 . 2 . . 1) In the i n f i n i t e s i m a l theory of p l a s t i c i t y a fundamental assumption i s that the s t r a i n r a t e tensor can be separated i n t o e l a s t i c and p l a s t i c p a r t s . That i s eij =z. &ij •+ ejj . . . . ( 3 . 2 . 2 ) • i . . il where cSij = e l a s t i c s t r a i n r a t e tensor, and <Sij = p l a s t i c s t r a i n r a t e tensor, then the c o n s t i t u t i v e r e l a t i o n s are s p e c i f i e d s e p a r a t e l y f o r the e l a s t i c and p l a s t i c s t r a i n r a t e s as f u n c t i o n s of the s t r e s s r a t e tensor (T-j and temperature r a t e T . The e l a s t i c law f o r a l i n e a r theory i s defined i n terms of r a t e s as S t | = C.iju -4- c<-ij T . . . . ( 3 * 2 . 3 ) r ' _ l where L-ijia i s the inverse of WjK* , the f o u r t h order tensor i of e l a s t i c constants, <*<j i s a second order tensor of the c o e f f i c i e n t s of l i n e a r thermal expansion. The inverse of Cijkl i s defined by the r u l e t h a t i f then <Hj = Cijki €LKI . . . . . ( 3 . 2 . 5 ) The same r u l e w i l l apply to other f o u r t h order c o n s t i t u t i v e tensors used l a t e r i n t h i s study. From the theory of e l a s t i c i t y i t i s known t h a t L\'JK£ i s p o s i t i v e d e f i n i t e . A lso, the f o l l o w i n g symmetrises e x i s t Cijki =c'Uij = c\iM ~ C-iJL* ( 3 - 2 . 6 ) = ^ • ( 3 - 2 . 7 ) I t should a l s o be noted that the symmetrises and p o s i t i v e i i -) d e f i n i t e character of Cijict a l s o apply to i t s inverse dj**. The c o n s t i t u t i v e equations f o r the p l a s t i c s t r a i n of a work-hardening m a t e r i a l are now presented. A fundamental p o s t u l a t e i s the existence of a y i e l d f u n c t i o n f which i s r e l a t e d to a hardening f u n c t i o n f- . Here £ i s considered to be a r e g u l a r (continuously d i f f e r e n t i a b l e ) f u n c t i o n of i t s v a r i a b l e s and of the form h f ( ^ , e J * , T ) . . . . . ( 3 . 2 . 8 ) The hardening f u n c t i o n -f i s s c a l a r which depends on the 11 h i s t o r y of deformation. I t s r a t e of change j£ i s defined, f o r t h i s study, to be of the form f = ^ ( C , e'mn J ) e!L . . . . . ( 3 . 2 . 9 ) Here hk£ i s a tensor f u n c t i o n of i t s v a r i a b l e s . The above form forj- ensures t h a t j^-O when e-i|= O. For a l i n e a r r a t e theory, the most general c o n s t i t -u t i v e equation f o r the p l a s t i c s t r a i n r a t e tensor &^ i s shown by Green and Naghdi to take the form 4 = *M (Vr ^ ^ 4 ^ ' •••• ( 3 , 2 , 1 0 ) during l o a d i n g , when P T < „*d t ^ ^ j T f > 0 ' ^ 0 ) (3 .2 .11) Here >^,j i s a symmetric tensor f u n c t i o n and ?) a s c a l a r f u n c t i o n of , -elm and T . . if I t i s a l s o p o s t u l a t e d t h a t &ij-0 during n e u t r a l l o a d i n g , when W ^ + = 0 , ( / ^ ) . . . . ( 3 . 2 . 1 2 ) during unloading, when a ~ * ? o i ^ - f - f T f A O , ^ " 0 ) . . . . ( 3 . 2 . 1 3 ) or when the m a t e r i a l i s i n the e l a s t i c s t a t e , when J < / and | i ^ ^ J T ^ J ^ ^ , , , , (3.2.HH Without l o s s of g e n e r a l i t y ?) i s chosen to be p o s i t i v e . 12 ?l > o . . . . ( 3 . 2 . 1 5 ) Then since . f -> J ^ IT -+ - T > 0 during l o a d i n g , i t can be shown from equations ( 3 . 2 . 9 ) and ( 3 . 2 . 1 0 ) t h a t • • • • ( 3 - 2 - l 6 > J from which )) may be determined i f ^ , ^ij and J are pre s c r i b e d . As mentioned p r e v i o u s l y , the above c o n s t i t u t i v e equations apply to work-hardening p l a s t i c i t y . The case of p e r f e c t p l a s t i c i t y (non-work hardening) r e q u i r e s s p e c i a l treatment. P e r f e c t p l a s t i c i t y i s c h a r a c t e r i s e d by the equation ft ^  ^ ( 3 . 2 . 1 7 ) where k i s a m a t e r i a l constant. In t h i s case j- = 7( = ^  ^^' f^V T = = 0 ( 3 . 2 . 1 8 ) The flow r u l e ( 3 . 2 . 1 0 ) w i l l be indeterminate s i n c e ?i w i l l tend to i n f i n i t y . Therefore f o r t h i s case the flow r u l e i s w r i t t e n i n the form tiif - A f>i} ( 3 . 2 . 1 9 ) where A ^ O i f t*k and ?1 4 + - ^~0 ( 3 . 2 . 2 0 ) and-A = 0 i f (<K or when f= fr and f < 0 ( 3 . 2 . 2 1 ) 13 In general JL i s non-unique, however, i n c e r t a i n cases by ensuring c o m p a t a b i l i t y of the p l a s t i c r e g i o n w i t h surround-ing e l a s t i c regions or boundaries a unique s o l u t i o n may be obtained. The previous r e s u l t s may be extended to s i n g u l a r y i e l d f u n c t i o n s using K o i t e r ' s [ 13 ] g e n e r a l i s a t i o n . He r ( x ) /*> proposed the use of a set of y i e l d f u n c t i o n s j- and 7C However, i n t h i s study only r e g u l a r y i e l d f u n c t i o n s w i l l be considered which have unique normals at every p o i n t . I t should be noted that the above f o r m u l a t i o n i s i n i t s most general form f o r a l i n e a r r a t e theory. The funda-mental p o s t u l a t e of p l a s t i c i t y , t h a t i s the existence of a y i e l d f u n c t i o n and the hardening f u n c t i o n , have y i e l d e d the above flow r u l e s . In these flow r u l e s there e x i s t s a choice of the f u n c t i o n s J , hki and ^ i j f o r p a r t i c u l a r m a t e r i a l s . However, there are thermodynamical r e s t r i c t i o n s to be imposed on these choices. These are considered l a t e r i n t h i s study. A combined e l a s t i c - p l a s t i c c o n s t i t u t i v e law may be w r i t t e n i n terms of the t o t a l s t r a i n r a t e s &ij as £ij = Cij*± <T£4 H-^J-T .... ( 3 . 2 . 2 2 ) where the e l a s t i c - p l a s t i c moduli / ^ J i J ^ < r * j L J ( 3 . 2 . 2 3 ) and the e l a s t i c - p l a s t i c thermal expansion c o e f f i c i e n t s o<ij =- <^ ij (3.2.24) 14 The existence of the inverse Cijkt i s discussed i n s e c t i o n (3«4) of t h i s chapter. The f o l l o w i n g symmetry p r o p e r t i e s e x i s t C^KI — Cjjik — cCj-i kw!. • • • • (3*2.25) fa: = o<p . . . . (3.2.26) However, Ciii(i4 CK^- unless 0*1= — 15 3.3 Thermodynamic C o n s t i t u t i v e R e l a t i o n s In the f o l l o w i n g development the f i r s t and second laws of thermodynamics are used as b a s i c axioms. By post-u l a t i n g a form of Gibb's equation f o r entropy and assuming that entropy i s a s t a t e f u n c t i o n f o r the i r r e v e r s i b l e processes of heat flow and p l a s t i c i t y the form of the entropy f u n c t i o n i s determined. Then using the c o n s t i t u t i v e post-u l a t e s l i n k i n g (Tjj , Bjj and T » an e x p l i c i t equation f o r entropy i s w r i t t e n . Using t h i s expression f o r entropy the i n t e r n a l energy and f r e e energy f u n c t i o n s are defined. Using the derived form of the f r e e energy f u n c t i o n the r e s t r i c t i o n s which the second law of thermodynamics (regarding entropy production) places on the p l a s t i c i t y c o n s t i t u t i v e assumptions are examined. An entropy flow v e c t o r i s introduced to describe the heat flow. R e l a t i o n s l i n k i n g ^ to temperature and s t r a i n are derived. Also i n t h i s s e c t i o n the equations of heat flow are introduced f o r the l i n e a r case and r e s t r i c t i o n s are placed on the c o n d u c t i v i t y c o e f f i c i e n t s i n order to s a t i s f y the second law of thermodynamics. The f i n a l p a r t of t h i s s e c t i o n i s concerned with using the e l a s t i c - p l a s t i c c o n s t i t u t i v e p o s t u l a t e s of the previous s e c t i o n to develop a combined t h e r m o - e l a s t i c - p l a s t i c theory. The f i r s t law of thermodynamics i s p o s t u l a t e d i n the form 16 j " f> iw nm dji + r P u dji = J f(r + F»» ^ ^ / ( T ^ - h ) ^ . . . . ( 3 . 3 . D where j> i s mass d e n s i t y , U i s the i n t e r n a l energy per u n i t mass, F w i s the body for c e per u n i t mass, T w i s the surface f r a c t i o n per u n i t area S , r i s the heat supply f u n c t i o n per u n i t mass and Vi i s the f l u x of heat outwards across the area § . Conservation of mass has been assumed i n equation ( 3 . 3 . 1 ) . Then by p o s t u l a t i n g the conservation of momentum S{(F*- '^ASL + J s r m . . . . (3.3.2) and d e f i n i n g the s t r e s s i n the normal manner as t l k (Tyct = T k S . . . . ( 3 . 3 . 3 ) where i s the u n i t outward normal and a l s o d e f i n i n g the heat f l u x v e c t o r as h « f)k ^ K 0 0 S" . . . . ( 3 . 3 . ^ ) where h i s the heat f l u x across the boundary, the f i r s t law can be shown f o r an a r b i t r a r y volume to become ^ - j? V- ^ k f k -f tf^ . . . . ( 3 . 3 . 5 ) In the remainder of the development f i s assumed to be zero. A modified form of Gibb's equation (see Naghdi f o r example) i s po s t u l a t e d as • • f o N I -II T q = Q =.~c* -+Yfce.\i ( 3 . 3 - 6 ) 1 7 Here 1^ i s the t o t a l entropy r a t e . The f r a c t i o n has been introduced to define the p r o p o r t i o n of p l a s t i c work which i s converted to heat. That i s t<Hj ejj- ==> he*t t ( 3 . 3 - 7 ) and the remainder — n o n - r e c o v e r a b l e i n t e r n a l energy . . . . . ( 3 . 3 . 8 ) The f a c t o r , Y , was introduced by Lee [14] and i s g e n e r a l l y i n the range 0-9<^^l-C f o r metals. K e s t i n [ i l J considers the same phenomenon and suggests that the non-recoverable i n t e r n a l energy i s stored as the energy of l o c a l d i s l o c a t i o n s on the microscopic s c a l e . He considers the deformations which are sought as the s o l u t i o n of the problem to be on the macroscopic s c a l e . Therefore introduce U M as the i n t e r n a l energy r a t e on the macroscopic s c a l e , t h a t i s U M = D - - k l - 0 OTf eij- .... ( 3 . 3 . 9 ) f By s u b s t i t u t i n g the energy equation ( 3 . 3 - 5 ) i n t o Gibb's equation ( 3 - 3 - 6 ) i t may be shown tha t T i i = V -iflTj e,'j-J(i-y)<n,-eiy .... ( 3 . 3 . 1 0 ) and then by using ( 3 - 3 - 9 ) t h i s becomes Trj, = 6 M - j j f l l j -eij . . . . ( 3 . 3 . 1 1 ) F i n a l l y , i t i s pos t u l a t e d t h a t the entropy, Y]^ , and i n t e r n a l energy, U M , are f u n c t i o n s of s t a t e w i t h the forms •X * n (7> , & \ ) - • • • • ( 3 - 3 - 1 2 ) 18 and U M= CU ( 7 7 Cij . ( 3 . 3 . 1 3 ) Consequently >^T ?j eij beh J ( 3 . 3 . 1 5 ) S u b s t i t u t i n g (3.3.1*0 i n t o (3 .3 .11) gives I f entropy i s to be a s t a t e v a r i a b l e the rat e ^ must s a t i s f y the f o l l o w i n g requirements. (These are the same as f o r exact d i f f e r e n t i a l s of a f u n c t i o n except that they are w r i t t e n f o r r a t e s . ) 2 ( L ^ ) - l a l u ' M - J_<^ 1 . . . . ( 3 . 3 . 1 6 ) . . . . (3 .3 . 17) . . . . ( 3 . 3 . 1 8 ) These equations may be s i m p l i f i e d so that equation (3 .3«l6) becomes = £•.— T 1?V •••• ( 3 . 3 . 1 9 ) 0e'« f f 19 equation ( 3 . 3 . 1 7 ) becomes 9 U W pr^K - . . . . ( 3 . 3 . 2 0 ) and equation (3.3.18) becomes ,, = O ( 3 . 3 . 2 1 ) Equation (3«3.20) s t a t e s that the previous c o n s t i t u t i v e assmuptions are not compatible with having &fj as a v a r i a b l e f o r U M . Therefore only e ' j and T are used i n f u t u r e . In subsequent expressions f o r <fij i t w i l l be seen that equation ( 3 . 3 . 2 1 ) i s s a t i s f i e d . S u b s t i t u t i n g the above r e s u l t s i n t o ( 3 - 3 . 1 5 ) gives 'I T Qj-J J « . . . ( 3 . 3 . 2 2 ) Now d e f i n i n g Ce as the s p e c i f i c heat f o r constant e l a s t i c s t r a i n as i - ^ 3 ( 3 . 3 . 2 3 ) C E = ft-and using the e l a s t i c c o n s t i t u t i v e law f o r 67j ( 3 * 2 . 3 ) , equation ( 3 . 3 . 2 2 ) becomes £ = C e T f CijU<*ict e>ij (3-3.24) By s u b s t i t u t i n g f o r c S i j i n terms of O^i and / an a l t e r n a t i v e expression may be w r i t t e n as 1~ C^ p + C^-^cKfc^. j jT-f .... ( 3 . 3 . 2 5 ) 20 which may be used as the d e f i n i t i o n of c(p-, the c o e f f i c i e n t of s p e c i f i c heat at constant s t r e s s f o r a l i n e a r e l a s t i c m a t e r i a l , That i s , / and then v| becomes I t i s now p o s s i b l e to w r i t e an e x p l i c i t expression f o r the i n t e r n a l energy r a t e U on the macroscopic s c a l e . From ( 3 . 3 . 1 1 ) •/ p M = [T»J + <T-j ei j < < # < (3.3.28) and using the e l a s t i c c o n s t i t u t i v e law, which holds f o r t o t a l s t r e s s and e l a s t i c s t r a i n as w e l l as f o r r a t e s , t h a t i s (T^ j = GijkjL €.'kt - c ' i j k t i t ' i u (T-lt>) . . . . ( 3 . 3 . 2 9 ) S u b s t i t u t i n g f o r ^  i n ( 3 . 3.28), gives ( U „ « C e T -+ Cijki -± Cijkirt frfo €ij . . . . . ( 3 . 3 . 3 0 ) This expression may be recognised as being the same as f o r the thermo-elastic case discussed by B i o t [ 1 ] . The form of the Helmholz f r e e energy f u n c t i o n A i s now discussed. The f r e e energy on the macroscopic s c a l e i s defined as A = U M - T*i ( 3 . 3 . 3 D H E N C E A - U M - T ^ - T ^ •••• ( 3 . 3 . 3 2 ) 21 Using ( 3 . 3 . H ) A may be w r i t t e n as A ^ I f t j ^ j j ~K | T . . . . ( 3 . 3 . 3 3 ) Green and Naghdi [ 9 ] suggest that f o r i n f i n i t e s i m a l deformations A may be w r i t t e n i n a quadratic form as 4 1 J ^ " , ' O , 0 ( 3 . 3 . 3 * 0 / ) where ^  i s the hardening f u n c t i o n and , CK-U e t c . , are constant c o e f f i c i e n t s . I f A i s a f u n c t i o n of s t a t e and A , given by equation ( 3 . 3 - 3 3 ) 1 i s an exact d i f f e r e n t i a l then A i s r e s t r i c t e d to being a f u n c t i o n of C'KI and Q . Thus o, _ V . . . . ( 3 . 3 . 3 5 ) and 1*. = o . . . . ( 3 . 3 . 3 6 ) 1 / By using the i d e n t i t i e s , given by Green and Naghdi, i q , ' - - and hlm . . . . ( 3 . 3 . 3 7 ) i t may be shown th a t A has the form [A« £ C ^ M n « t t - c W « ^ i f l f i w 4 ^ Q • > > • ( 3 . 3 . 3 8 ) The i m p l i c a t i o n s of the second law of thermodynamics are now presented. Green and Naghdi [ 8 ] p o s t u l a t e an entropy 22 production i n e q u a l i t y i n the form f p/j dSl- f pr<*J2 -+f !? > 0 . . . . . ( 3 . 3 . 3 9 ) By using the energy balance equation ( 3 . 3 . 5 ) and the f r e e energy f u n c t i o n A , the above i n e q u a l i t y ( 3 * 3 . 3 9 ) may "be transformed to - P ( A - r[ T ) -+ (T^ £ u - ^ i c J U ^ . . . . ( 3 . 3 . ^ 0 ) 1 1 Then by assuming c e r t a i n f u n c t i o n a l forms f o r A , Y\ and <\ K and c o n s i d e r i n g an unloading process and then a lo a d i n g process the i n e q u a l i t y ( 3 . 3 - ^ 0 ) can be shown to reduce to In t h i s study A i s independent of » hence 4*4 & let - ^tk_ J ' k ^ O ( 3 . 3 . ^ 2 ) T S a t i s f a c t i o n of i n e q u a l i t y ( 3 . 3 . ^ 2 ) i s ensured by p l a c i n g separate r e s t r i c t i o n s on each of i t s components. F i r s t l y , r e q u i r i n g K t ^ J a ^ ° •••• ( 3 . 3 . 4 3 ) i s e q uivalent to r e q u i r i n g <Ee >Mr . r - f " " - f l f ) > ^ •••• ( 3-3-W > However, ?i and f ^ J " C . -h^J f ~) are p o s i t i v e by d e f i n i t i o n . 23 Consequently the requirement becomes QKI ^ O ( 3 . 3 . 4 5 ) This w i l l t h e r e f o r e r e s t r i c t the choice of the f u n c t i o n ^>KI . Secondly, the term - % T > * * ° . . . . (3.3-46) w i l l impose r e s t r i c t i o n s on the beat conduction equation. The l i n e a r c o n s t i t u t i v e equation f o r heat conduction ( F o u r i e r ' s law) i s adopted. That i s n*' - k M 7 a . •••• ( 3 . 3 . 4 7 ) S u b s t i t u t i o n i n t o equation ( 3 . 3 - 46 ) gives kkt T,J,( ^ , . . . . ( 3 . 3 - 48 ) Thus the necessary c o n d i t i o n i s tha t kk<, i s a p o s i t i v e quadratic form. (Not n e c e s s a r i l y p o s i t i v e d e f i n i t e . ) Before proceeding i t i s convenient to summarise the thermal and e l a s t i c - p l a s t i c c o n s t i t u t i v e p o s t u l a t e s introduced so f a r . - / / -» - , • e*j = c - i j ' ^ Kx -t ^ - i j 1 — ( 3 . 2 . 3 ) l\ f r / " 4 ^  •••• ( 3 . 2 . 1 0 ) ( 1 = C\ vet e i j •+• Cp T . . . . (3 .3 .24) 1 T £ f T .... ( 3 . 3 . 2 7 ) T 24 The e n t r o p y f l o w v e c t o r i s now i n t r o d u c e d as t h e v a r i a b l e d e s c r i b i n g h e a t f l o w . By u s i n g ^ as a v a r i a b l e , i n s t e a d o f say c^, , B i o t [ 1 Jwas a b l e t o d e v e l o p a v a r i a t i o n a l p r i n c i p l e f o r c o u p l e d t h e r m o - e l a s t i c i t y . The same advantage i s sought i n t h i s s t u d y . C o n s e q u e n t l y d e f i n e # ~ 1 ? . . . . ( 3 . 3 . 4 9 ) T Now, r e c a l l i n g GibbS; e q u a t i o n , |»Ti2 - - ^ k i ( c -+ I f?j £ij . . . . ( 3 . 3 . 6 ) and s u b s t i t u t i n g ^ i n terms o f c^. u s i n g the q u o t i e n t r u l e o f d i f f e r e n t i a t i o n g i v e s j"l = • " ^ K . K -T4^* + J ^ ^ ' •••• ( 3 . 3 . 5 0 ) Here <^fc may be i n t e r p r e t e d as t h e e n t r o p y l o s s due t o f l o w , - " T ^ C ^ / T m a y "be "the i n t e r n a l e n t r o p y p r o d u c t i o n due t o the i r r e v e r s i b l e p r o c e s s o f heat f l o w and Jf^- j &<j/T may be t h e i n t e r n a l e n t r o p y p r o d u c t i o n due t o the i r r e v e r s i b l e p r o c e s s o f p l a s t i c f l o w . T h i s i n t e r p r e t a t i o n o f t h e terms ^ K , ^ and Tiifii/'T i s t h e same as t h a t d i s c u s s e d by Fung [ 7 J i n h i s r e v i e w o f B i o t ' s work. The term 6<Rj £ij / T does n o t f i t t h e p a t t e r n o f l i n e a r p h e n o m e n o l o g i c a l laws o b e y i n g Onsager's p r i n c i p l e . T h i s p o i n t i s d i s c u s s e d by Naghdi [ l 8 j i n h i s d i s c u s s i o n o f t h e r m o - p l a s t i c i t y who s u g g e s t s t h a t , even i f Onsager's p r i n c i p l e s h o u l d a p p l y t o t h e n o n - l i n e a r phenomenon o f p l a s t i c i t y , some new v a r i a b l e s may be r e q u i r e d t o d e s c r i b e 25 the process. However, i n t h i s study the a n a l y s i s i s continued using the e x i s t i n g v a r i a b l e s . The f i n a l equations given i n Chapter 4 w i l l be shown to be unsymmetric and t h i s f e a t u r e i s a t t r i b u t e d to the ne g l e c t of Onsager's p r i n c i p l e . S u b s t i t u t i n g the c o n s t i t u t i v e law f o r heat flow (3.3.47) i n t o ( 3 . 3 . 5 0 ) gives ~ ^ k . i c + *1j ft + ^ rku ^'Ut . . . . ( 3 . 3 . 5 D For a l i n e a r theory of i n f i n i t e s i m a l deformations the non-l i n e a r term may be dropped. Hence ^ \ = - I^r.k. -fl^i-e'yi . . . . ( 3 . 3 . 5 2 ) Now, using the previous d e f i n i t i o n of Y^ as a f u n c t i o n of €-,y and T (3.3.24) and the e l a s t i c and p l a s t i c c o n s t i t u t i v e laws, ( 3 . 2 . 3 ) and ( 3 . 2 . 1 0 ) , the f o l l o w i n g r e l a t i o n may be obtained. ~ = Er f +*kl <r^ • ••• ( 3 . 3 . 5 3 ) where and <*kt^ <<\,JL- . . . . ( 3 . 3 . 5 4 ) c r = c_r _ 7> A if . . . . ( 3 . 3 . 5 5 ) T T ' 2>T -In equations ( 3 . 3 - 5 4 ) and ( 3 . 3 . 5 5 ) * I at, A„. . . . . ( 3 . 3 . 5 6 ) f ~ ^ ^ fa This r e l a t i o n may be r e w r i t t e n as a f u n c t i o n of the t o t a l s t r a i n s r a t e s © w using the e l a s t i c - p l a s t i c c o n s t i t u t i v e law 26 -/ i f Ciju has an inve r s e , £ij = CijM e.u - CijHottcT . . . . ( 3 . 3 . 5 7 ) then -ftklk~ C^w. •+ ^  T •••• ( 3 . 3 . 5 8 ) where Ce. = C<r - ^ <^ n^ ( 3 . 3 . 5 9 ) Equations ( 3 . 3 - 5 3 ) and ( 3 * 3 . 5 8 ) may be i n v e r t e d to give T = ~} •••• ( 3 . 3 . 6 0 ) and, i f Clju e x i s t s , T = ~ £ ( ^ . k - 1 - ^ ^ ^ ) . .... ( 3 . 3 . 6 1 ) S u b s t i t u t i o n of the above equations f o r T i n t o the s t r e s s -strain-temperature r e l a t i o n s to e l i m i n a t e T and use jfi^x i n s t e a d , and the consequent questions of existence and uniqueness are discussed i n the next s e c t i o n . 27 3 . 4 Comments on the Existence of C o n s t i t u t i v e R e l a t i o n s In the previous s e c t i o n s the r e l a t i o n s • • • • (3 • 2 • 22) and T = ~ c V ^ ^ ^ ^ / S " ^ .... ( 3 . 3 . 5 3 ) were derived f o r work-hardening p l a s t i c i t y . Here, provided t h a t Cjjyu , ?i , , ^ and are s p e c i f i e d , expressions f o r <5|ft and T w i l l e x i s t . In t h i s s e c t i o n the p o s s i b i l i t y of transforming these r e l a t i o n s to give 07y and T as f u n c t i o n s of •CKI, and j^y^ i s examined. F i r s t l y , to ob t a i n an expression f o r ^ two d i f f e r e n t s u b s t i t u t i o n s are made g i v i n g the same f i n a l equation. S u b s t i t u t i n g ( 3 . 3 - 5 3 ) i n t o ( 3 . 2 . 2 2 ) y i e l d s J . . . . ( 3 - 4 . 1 ) = £trt«.j (Ti-j- - J , . 0 1 " ^ . . . . ( 3 . 4 . 2 ) Then, i f £IJ'K* has an inverse Cijki , an expression f o r oVy may be w r i t t e n as (ft,- = £ ^ l r < ^ - f X * lcA . . . . ( 3 . 4 . 3 ) A l t e r n a t i v e l y , by i n v e r t i n g equation ( 3 . 2 . 2 2 ) f i r s t , t h a t i s 28 w r i t i n g (Tjj = CijKl C u - CijuoiKiT ( 3 . 4 . 4 ) and then s u b s t i t u t i n g f o r T , gives t -+ ~ C-ij ice Urn (fi^tn • « • • ( 3 . 4 . 5 ) Ojltt eict "+ J ^ J K ^ I U ^ H . A O ( 3 . 4 . 6 ) In equations ( 3 . 4 . 3 ) and ( 3 . 4 . 6 ) two d e f i n i t i o n s f o r Cijht have "been derived. These are CijKi = ('C-'iju''^ A"/ - I ^ ij^w) •••• ( 3 . 4 . 7 ) and C i j H = d j i c A -+ ^ C ^ c ^ ^ Ctctf^^pi- (3.4.8) These two expressions are derived from the same bas i c equations by purely a l g e b r a i c manipulation and must t h e r e f o r e be equal. The question of the existence of Cijkl may now be examined by usi n g e i t h e r ( 3 . 4 . 7 ) or ( 3 . 4 . 8 ) , whichever i s more convenient. Using equation ( 3 . 4 . 7 ) a necessary c o n d i t i o n may be seen to be that the e n t i r e f u n c t i o n C<j« = C\jJ+ ^ - £ r * * j . . . . ( 3 . 4 . 9 ) must have an in v e r s e . Advantage i s taken of t h i s form i n the d i s c u s s i o n of symmetry i n Chapter 5« A l t e r n a t i v e l y , using equation ( 3 . 4 . 8 ) a s u f f i c i e n t c o n d i t i o n i s that 29 C<jU-(Ci^ ^^-/^) t (3^.10) e x i s t s . This form a l s o allows the r o l e of (2>-ij to be recognized but does not al l o w as much f l e x i b i l i t y as the previous case. In order to w r i t e an expression f o r 7 as a f u n c t i o n of C K L and <^»Wim i t i s necessary to s u b s t i t u t e f o r <Hj i n the expression ( 3 . 3 . 5 3 ) Consequently, the c o n d i t i o n s f o r the existence of a l s o apply to T . I t i s now shown tha t Drucker"s[ 5 ] no r m a l i t y c o n d i t i o n i s a s u f f i c i e n t c o n d i t i o n f o r the existence of as a f u n c t i o n of and ^ . The c o n d i t i o n i s tha t . . . . (3 .4 .11) then CijKi becomes Cijk* = + ^ .... ( 3 . 4 . 1 2 ) Now forming the quadratic form - l (C*i*<> + a ? ( r 5 j ? r K J ^ i ^ " C , i t t ^ r " .... ( 3 . 4 . 1 3 ) i -I Here Ci'jm i s p o s i t i v e d e f i n i t e by d e f i n i t i o n from e l a s t i c i t y and the term ^Wj ^ r K l r i ("« •••• fa-*- 1 * ' 30 i s p o s i t i v e . Hence the term ^-ijU i s p o s i t i v e d e f i n i t e and ther e f o r e has an i n v e r s e . The case of p e r f e c t p l a s t i c i t y g i ves the r e l a t i o n 4 B Cijw'ffij 4 o t ' w f f A . . . . ( 3 . 4 . 1 5 ) where A i s undetermined. In the case of isothermal p l a s t i -c i t y K o i t e r [ l 3 ] shows that uniquely defines <K~j even though does not uniquely determine fi-i&t . Hence a d i s -placement f o r m u l a t i o n of the problem of e q u i l i b r i u m i s not p o s s i b l e . In the case of non-isothermal p e r f e c t - p l a s t i c i t y i t may be p o s s i b l e to uniquely define Oif i n terms of and T . However, <2-*< cannot be uniquely determined and consequently a displacement f o r m u l a t i o n i s not p o s s i b l e . For t h i s reason p e r f e c t - p l a s t i c i t y i s not considered f u r t h e r i n t h i s study. 31 CHAPTER 4  FIELD EQUATIONS 4.1 I n t r o d u c t i o n In t h i s chapter the f i e l d equations of continuum mechanics are reviewed and the requirements f o r t h e i r s a t i s f a c t i o n are i n v e s t i g a t e d . Some of these equations are s a t i s f i e d i d e n t i c a l l y owing to c o n s t i t u t i v e assumptions. Others remain to he s a t i s f i e d during the s o l u t i o n of a p a r t i c u l a r problem. This outstanding set of simultaneous equations i s i d e n t i f i e d . The heat flow equation i s shown to r e q u i r e s o l u t i o n . For convenience of s o l u t i o n t h i s equation i s reduced from p a r a b o l i c to e l l i p t i c form by assuming a l i n e a r v a r i a t i o n of temperature with time and s o l v i n g i t i n c r e m e n t a l l y . In the f i n a l s e c t i o n the f i e l d equations are w r i t t e n i n terms of m a t e r i a l displacement and entropy flows r a t e s . 4.2 Review of F i e l d Equations T r u e s d e l l [ 2 2 ] gives a summary of equations and i n e q u a l i t y of thermodynamics as f o l l o w s : Conservation of mass Balance of l i n e a r momentum Balance of moment of momentum Balance of energy Entropy production i n e q u a l i t y In a d d i t i o n to the above i t i s req u i r e d i n t h i s study that 32 the f o l l o w i n g a l s o holds: F o u r i e r ' s law of heat conduction. I t i s a l s o necessary f o r the s o l u t i o n of a problem that there i s Kinematic c o m p a t i b i l i t y of m a t e r i a l and entropy displacements. The s a t i s f a c t i o n of each of the above i s now considered. Conservation of mass: In the small displacement theory considered the d e n s i t y i s considered to be constant and the i n t e g r a t e d volume of the body a l s o to be constant. Conse-quently mass i s conserved. Balance of l i n e a r momentum: This r e q u i r e s the f o l l o w i n g : ( N - . j + ^F, = m >Jl (4.2.1) and ''J VV ~ . . . . (4.2.2) These equations are r e q u i r e d to be s a t i s f i e d . A c c e l e r a t i o n s are considered to be n e g l i g i b l e and the equations w i l l be s a t i s f i e d i n a r a t e (or incremental) form f o r each time step db , t h a t i s ^ • • j m J l . . . . ( * .2 . 3 , and » * flj (ftj = 7; o« £r •••• ^•2-z+) 33 Balance of moment of momentum: This i s s a t i s f i e d by d e f i n i n g Q^ j to be symmetric through the c o n s t i t u t i v e p o s t u l a t e s and excluding body couples. Balance of energy: This r e q u i r e s the f o l l o w i n g equations to hold. p V « OTj - f <=\^\ - (r m Jl ....(4.2.5) and Il = ^ ^ ° n S ^ (4.2.6) where h i s the heat f l u x across the boundary. The f i r s t equation i s s a t i s f i e d i d e n t i c a l l y due to the method of d e r i v a t i o n of the c o n s t i t u t i v e r e l a t i o n s i n s e c t i o n (3.3). The l a t t e r equation r e q u i r e s s a t i s f a c t i o n during s o l u t i o n to the problem. Entropy production i n e q u a l i t y : I t i s shown i n s e c t i o n (3-3) that s u f f i c i e n t c o n d i t i o n s are the s a t i s f a c t i o n of the i n e q u a l i t y fiu Fit ^ ° w Jl (3.3.45) and that Knj i s a p o s i t i v e quadratic form. F o u r i e r ' s law of heat conduction: This r e q u i r e s ^ ^ k i j 9',j (r> Si (4.2.7) An incremental s o l u t i o n to t h i s equation i s sought by assuming 34 that Qj v a r i e s l i n e a r l y with time. That i s (4.2.8) where i s the time measured i n s i d e an increment of time dt . By forming the i n t e g r a l fc + St L-tdt dfr = J <*t = J k«j &,j dt •••• (4.2.9) ^ f e.«W H- ^ ^ ' j ) ^ •••• ^ • 2 - 1 0 ) , T 1 'J di = k j i ( fyjte) -+^ , J' ) d t (4.2.11) here 0,j (o) i s the gradient of 0 at the s t a r t of the time step • In f u t u r e t h i s i s denoted as . Equation (4.2.11) may be w r i t t e n i n r a t e form as <j>{ = (B'j-f i'J^O .... (4.2.12) I t w i l l be shown l a t e r that f o r uniqueness of the heat flow problem k>j' should be p o s i t i v e d e f i n i t e . In that case the inverse k-jj ' e x i s t s . The equation may then be w r i t t e n i n the form - f l . j - * 2kf'l^ = I ft} . . . . ( 4 . 2 . 1 3 ) S a t i s f a c t i o n of t h i s equation w i l l ensure the s a t i s f a c t i o n of the heat flow equation i n the i n t e g r a l over the time step subject to the assumption t h a t v a r i e s l i n e a r l y w i t h time. 35 Kinematic c o m p a t i b i l i t y ! In t h i s study the s o l u t i o n i s sought using assumed m a t e r i a l and entropy displacement f i e l d s . These w i l l be chosen to s a t i s f y c o m p a t i b i l i t y requirements i n t e r n a l l y and p r e s c r i b e d c o n d i t i o n s on the boundary. 4 . 3 Rate form of F i e l d Equations The equations which are to be s a t i s f i e d are fa-"\) ' J ' f -1 or. ^ o . . . . ( 4 . 2 . 3 ) ( 4 . 2 . 1 3 ) ( 3 . 3 . 4 5 ) with boundary c o n d i t i o n s ( 4 . 3 . 1 ) ( 4 . 3 . 2 ) ( 4 . 3 . 3 ) ( 4 . 3 . 4 ) Here a. bar, e.gjW-i , denotes a p r e s c r i b e d value of the v a r i a b l e . By using the c o n s t i t u t i v e r e l a t i o n s of s e c t i o n ( 3 . 2 ) and ( 3 . 3 ) these may be w r i t t e n f o r the general t h e r m o - e l a s t i c -p l a s t i c case as IKI _ _ l ( 4 . 3 . 5 ) at 36 wi t h n a t u r a l boundary c o n d i t i o n s ixir,/. + X C i j - W <*\CJI J£>V»,M~) =• T{ or) ST . . . . (4.3.6) Ce ~ c e t^™"^ ULwi'l° + T - 0 - on •••• (4.3.7) and forced boundary c o n d i t i o n s * • — -= L U s u (4.3.8) ~ ? 0 " S / • .... (4.3.9) The y i e l d c o n d i t i o n s assumed i n s e c t i o n (3.2) apply to the e l a s t i c - p l a s t i c c o e f f i c i e n t s . The i n e q u a l i t y ^ ° .... (3.3.45) a l s o remains to be s a t i s f i e d by the choice of ^KJ. . 37 CHAPTER 5  THE VARIATIONAL METHOD 5.1 I n t r o d u c t i o n In t h i s chapter the formal v a r i a t i o n a l method presented by M i k h l i n [16], Melosh [15] and Key [12] i s reviewed. The d e f i n i t i o n s of the operators are given and the theorems r e l a t i n g to the minimal f u n c t i o n a l method are quoted. The method i s then a p p l i e d to the case of thermo-e l a s t i c i t y . I t i s shown th a t t h i s case s a t i s f i e s the requirements of the method. The f u n c t i o n a l i s derived and the boundary value problem i s discussed. Having shown the a p p l i c a b i l i t y of the method to the thermo-elastic method the extension to the t h e r m o - e l a s t i c -p l a s t i c case i s then s t u d i e d . However, i t i s found t h a t the operator does not s a t i s f y the necessary c o n d i t i o n of formal symmetry and th e r e f o r e the formal method of minimal f u n c t i o n -a l s cannot be a p p l i e d i n the t h e r m o - e l a s t i c - p l a s t i c case. The p o s s i b i l i t i e s of choosing a flow r u l e to s a t i s f y t h i s c o n d i t i o n i s i n v e s t i g a t e d , but i t i s shown th a t t h i s i s not p o s s i b l e . An i t e r a t i v e f u n c t i o n a l method i s then proposed f o r the unsymmetric operator case. Uniqueness and convergence are discussed and the method i s a p p l i e d to the t h e r m o - e l a s t i c -p l a s t i c equations. 38 5.2 Basic D e f i n i t i o n s The method which i s reviewed i n t h i s s e c t i o n i s tha t given by M i k h l i n [ l 6 j . In order to use the method i t i s necessary to b r i e f l y review h i s basic d e f i n i t i o n s . The r e g i o n i s considered to be connected and to c o n s i s t of i n t e r n a l p o i n t s (sometimes c a l l e d an open domain) and i s denoted by S l . The boundary of the domain Jl i s denoted by S . The closed r e g i o n J t = J l - f S i s the combination of the p o i n t s on the boundary and i n the open r e g i o n . Only f i n i t e regions are considered. The s c a l a r product of two f u n c t i o n s r e l a t i v e to a given r e g i o n i s the i n t e g r a l of the product of those f u n c t i o n s over the r e g i o n . For example the s c a l a r product of two fu n c t i o n s u(p)and v ( p ) o v e r J 2 i s denoted as (a,'V) and i s (u,i / 0 = f a ( p ) v ( p ) ^ J l . . . . ( 5 . 2 . 1 ) An operator i s defined f o r some set of fu n c t i o n s i f a law i s given according to which a new set of f u n c t i o n s i s defined such t h a t to each f u n c t i o n of the given set ( f i e l d of d e f i n i t i o n , (5 A ) there e x i s t s a unique f u n c t i o n of the new set ( f i e l d of val u e s , f?^ ). The operator i s denoted by A . v An operator i s c a l l e d a l i n e a r operator i f i t s f i e l d of d e f i n i t i o n i s a l i n e a r s et and i f A ( u , •+ c<i ixt +• •• • + a 1 ) 1 U r t ) = a ( ^ i x / - + ^ z / l u i-f-- - f ^ ^ A U n • ••• (3•2 # 2) i r r e s p e c t i v e of the i n t e g e r n , <a, , et ? being constants and U, , Ui ,•• tl^ , being from the f i e l d of d e f i n i t i o n of A . 39 An operator i s c a l l e d symmetric i f f o r any two fu n c t i o n s LL and W from i t s f i e l d of d e f i n i t i o n the f o l l o w i n g i d e n t i t y i s v a l i d . ( A U , V ) = ( l i . A v ) . . . . ( 5 . 2 . 3 ) T o n t i [ 2 1 J defines a f o r m a l l y symmetric operator i f the f o l l o w i n g c o n d i t i o n holds fAtijA/") = (u,AV)-*-•[« boundary \Y)Yegm\} . . . . ( 5 . 2 . 4 ) A symmetric operator i s s a i d to "be p o s i t i v e d e f i n i t e i f f o r any f u n c t i o n u from i t s f i e l d of d e f i n i t i o n there occurs the i n e q u a l i t y ( A u , u ) . . . . ( 5 . 2 . 5 ) where the e q u a l i t y (A IL, u) = o i s only p o s s i b l e i f U=0. A symmetric operator i s s a i d to be positive-bounded  below i f f o r any f u n c t i o n a from f i e l d of d e f i n i t i o n of A the i n e q u a l i t y (AU.LL) > K |( a l l . . . . ( 5 . 2 . 6 ) .2 i s v a l i d where if i s a p o s i t i v e constant and l| u|| i s the norm of U. defined as = 'I .... ( 5 . 2 . 7 ) The energy norm f o r an operator A i s defined as ("^  u,, u) = [w,u.J = |u| ( 5 . 2 . 8 ) 40 An operator which generates a f u n c t i o n i d e n t i c a l l y equal to a constant i s c a l l e d a f u n c t i o n a l . In t h i s study quadratic f u n c t i o n a l s are of i n t e r e s t which take the form ) - f ^ U + C . . . . (5.2.9) where i s a l i n e a r operator, £ i s a l i n e a r f u n c t i o n a l and C i s a constant. 5.3 The Minimal F u n c t i o n a l Method The problem has been formulated i n the form A u = f (p) . . . . (5 .3 .D where -j~0) i s a known f u n c t i o n (with f i n i t e norm). The s o l u t i o n r e q u i r e s f i n d i n g a f u n c t i o n Lt(p) € DA which makes equation (5.3.1) an i d e n t i t y . The f a c t t h a t u(V)6 i m p l i e s t h a t the f u n c t i o n U s a t i s f i e s the boundary c o n d i t i o n s of the problem. I f A i s a l i n e a r , symmetric and p o s i t i v e d e f i n i t e operator i t i s p o s s i b l e to pose the problem as the problem of f i n d i n g the minimum of a f u n c t i o n a l . This method i s s t a t e d more p r e c i s e l y by means of the f o l l o w i n g two theorems, the proofs of which are given by M i k h l i n f l 6 ] . Theorem 1. Uniqueness. I f the operator A i s p o s i t i v e d e f i n i t e then the equation A u - f cannot have more than one s o l u t i o n . 41 Theorem 2. Minimal F u n c t i o n a l . I f A i s a p o s i t i v e d e f i n i t e operator and i f equation ( 5 . 3 . 1 ) has a s o l u t i o n , then of a l l the values which are given to the quadratic f u n c t i o n a l -ft ( 5 . 3 . 2 ) by a l l p o s s i b l e f u n c t i o n s from PA , the l e a s t i s the value given to t h i s f u n c t i o n a l by the s o l u t i o n of equation ( 5 - 3 . 1 ) . Conversely, i f there e x i s t s i n DA a f u n c t i o n which gives a minimal value to f u n c t i o n a l ( 5 . 3 . 2 ) , then t h i s f u n c t i o n i s the s o l u t i o n to equation ( 5 . 3 . 1 ) . In the above theorems M i k h l i n only considers symmetric operators. However, T o n t i [ 2 l ] shows the f o l l o w i n g necessary c o n d i t i o n . Theorem.3. Necessity of Formal Symmetry. To deduce a system of l i n e a r equations Au-f from the s t a t i o n a r i t y of a f u n c t i o n a l i t i s necessary that A be f o r m a l l y symmetric. The advantages of the minimum f o r m u l a t i o n of the problem i s that a s o l u t i o n can be found by c o n s t r u c t i n g a minimising sequence. The most important method of c o n s t r u c t -ing a minimising sequence i s the R i t z method. In t h i s study s o l u t i o n s w i l l be sought using the f i n i t e element method which has been shown by Melosh [ 1 5 ] "to be equivalent to the 42 R i t z method. By s o l v i n g the problem using minimising sequences i t i s p o s s i b l e to show convergence of approximate s o l u t i o n s . The property of convergence i s given by the f o l l o w i n g theorem of M i k h l i n ' s . Theorem 4. Minimi s i n g Sequence. I f equation ( 5 - 3 . 1 ) has a s o l u t i o n then any sequence which i s minimising f o r the f u n c t i o n a l ( 5 . 3 . 2 ) converges i n energy to t h i s s o l u t i o n . Proof of theorem 4 r e q u i r e s the operator A to be p o s i t i v e d e f i n i t e . Convergence i n energy means that the energy norm of the d i f f e r e n c e between the approximate s o l u t i o n , and the exact s o l u t i o n U Q , tends to zero. That i s I U n — L U I * ° ( 5 . 3 - 3 ) I f A i s positive-bounded-below the stronger convergence i n the mean occurs. That i s (I un- M l <=> •••• ( 5 . 3 . 4 ) M i k h l i n [ l 6 J a l s o gives the f o l l o w i n g existence theorem. Theorem 5 . Existence. The problem of the minimum of the f u n c t i o n a l has a s o l u t i o n i n the c l a s s of f u n c t i o n s with f i n i t e energy only i f the operator i s positive-bounded-below. The question of the c o n s t r u c t i o n of the minimal 43 f u n c t i o n a l needs to be discussed f u r t h e r . In order to prove p o s i t i v e - d e f i n i t e n e s s and symmetry of an operator i t i s u s u a l l y necessary to assume t h a t the boundary c o n d i t i o n s are homogen-eous. However, "by using a "change of v a r i a b l e s and making some a d d i t i o n a l assumptions, i t i s p o s s i b l e to extend the a p p l i c a b i l i t y of the f u n c t i o n a l to the non-homogeneous boundary c o n d i t i o n case. This extension i s o u t l i n e d below. In the homogeneous boundary c o n d i t i o n case the f u n c t i o n a l i s F ( u ) = (Au,u)- 2 ( u . n .... (5.3.2) In the non-homogeneous case c o n s i d e r i n g the equation PtU = f CP) .... (5-3.1) i t i s assumed t h a t A i s a l i n e a r , d i f f e r e n t i a l operator of order k . The f u n c t i o n s u are assumed to be continuous and s u f f i c i e n t l y d i f f e r e n t i a b l e i n Jl . Equation (5.3.1) i s to be solved subject to the boundary c o n d i t i o n s , G , 0 ) | = 9, > \ s = 3«J ••' •••• (5-3.5) s where G, , Gz ... are l i n e a r operators and g, » 9? ••• a r e given f u n c t i o n s . The number of f u n c t i o n s g t - depends on the order k of f\ and on W . The problem i s solved by making the f o l l o w i n g assumption. There e x i s t s a f u n c t i o n ^ ( P " ) which i s s u f f i c i e n t -l y d i f f e r e n t i a b l e and continuous, and which s a t i s f i e s the boundary c o n d i t i o n s of the problem. That i s 44 > G « C f ) | s - g t ? ••• . • ••• ( 5 . 3 . 6 ) This i s used to define a new unknown f u n c t i o n V (?) where V = a - + . . . . ( 5 . 3 . 7 ) The f u n c t i o n "V s a t i s f i e s the equation =.^(P-)^\)KI fM-JM-^ •••• ( 5 - 3 . 8 ) and the homogeneous boundary c o n d i t i o n s G , C ^ ) | 3 = o , G , C ^ ) | 5 = o , - - - . . . . . ( 5 . 3 . 9 ) I f A i s p o s i t i v e d e f i n i t e f o r the set of f u n c t i o n s - V then the s o l u t i o n of equation ( 5 . 3 . 8 ) under the homogeneous boundary c o n d i t i o n s i s equivalent to f i n d i n g the f u n c t i o n which minimises the f u n c t i o n a l = ( A v , v ) ~ 2 ( v , f J . . . . (5.3.10) S u b s t i t u t i n g f o r Y and £| gives F ( v ) « - (AIL-A + , u . - ^ ) - 2 f u - t , f - A ^ f ) .... ( 5 . 3.U) = (Aa,u> f (u,0 + (u, A ^ - ( A u , ^ + ?(^)_f A l f | V,) ^  _ ( 5 # 3 < 1 2 ) In the problems considered i n t h i s study i t i s p o s s i b l e , using the divergence theorem, to transform ("u, A M O - ( A u , ^ ) in ho f f U u , ^ ) ^ . . . . ( 5 . 3 . 1 3 ) Using the boundary c o n d i t i o n s can be w r i t t e n i n the form 45 ft ( U, = N (u) •+ M . . . . (5.3.14) where N/(u) depends only on LL and the f u n c t i o n s ^ ( , ... and M does not depend on U. but can depend on ^  . In such a case F(v] i s reduced to F ( Y ) = (Au.u) - 2 (u.-f) + £ M ( u ) d S + { e ( ^ M ) _ ( A + ^ ) - f j M ^ s } • • • • ( 5 . 3 . 1 5 ) The expression i n thejsquare brackets^ i s a constant q u a n t i t y , although perhaps unknown. Thus the minimising of the f u n c t i o n a l P(v) i s equivalent to the minimising of another f u n c t i o n a l <§(u.), where ? ( u ) - (Au,u)~2(u,$)-tfsNCu)dS . . . . ( 5 . 3 . 1 6 ) Since A i s p o s i t i v e d e f i n i t e w i t h respect to "V , then the s o l u t i o n to V w i l l be unique. Then f o r a chosen f u n c t i o n ^ t h i s w i l l ensure that vu i s a l s o unique. That i s , the non-homogeneous boundary c o n d i t i o n s o l u t i o n i s unique i f the homogeneous one i s . 5 . 4 A p p l i c a t i o n to Thermo-Elastic Equations In t h i s case the equations are 1 J e J , J m j l . . . . ( 5 . 4 . 1 ) 46 which may be represented as A u - f •••• ( 5 ' 3 - 1 ) where U> i s the v e c t o r j , <^ >k ^ and J i s the v e c t o r •£ , ?J?>H ^ Motivated by the argument of the previous s e c t i o n , the case of homogeneous boundary c o n d i t i o n s i s considered f i r s t where []• * ^ = o o r , S .... ( 5 . 4 . 2 ) Here, the operator A i s c l e a r l y l i n e a r s i n c e ^['jut • <A'ki , T , C c , k;j and are a l l constants and the d i f f e r e n t i a l operators are l i n e a r . The operator i s now t e s t e d f o r symmetry by forming the d i f f e r e n c e ( 5 . 4 . 3 ) which must v a n i s h i f A i s symmetric. In t h i s case t h i s becomes 47 ~ r r / / T r' ' r' I \ ' ^  ' .ft) 0) + 4 -t, C . , k < * w f V r v , J . ix; - ICi^u vu ?„,J . *i Cc I J J> ;> ft) r T ^' / <13 -i •/ .0) L c'c T'.< J , „ r * - [ -q r - . ' J , „ r ( 5 . 4 . 4 ) which by the quotient r u l e of d i f f e r e n t i a t i o n and the divergence theorem becomes f f f r ' , T r ' il r' 1 ^r'(•,) T / ; f £0) '[t) ALn) J ( 0 * 6 ) ^ ( ? > -U-) .(2  ..£,) s 4 ^ f l ^ r . f f l f r r e f C ^ ^ K . . . . ( 5 . 4 . 5 ) 4 8 r1 U'} CO ^Cz) ^  Here L i j k t a n d nfj are symmetric and U n- =• a .,• cp; ~ y>^ = o> on S , hence ( M « U W ) - ( U W . * 0 = O . . . . ( 5 . 4 . 6 , Thus the thermo-elastic operator i s symmetric. I t f o l l o w s that the weaker c o n d i t i o n of formal symmetry i s a l s o s a t i s f i e d . The p o s i t i v e - d e f i n i t e n e s s of the operator i s now in v e s t i g a t e d by forming the energy product and checking the i n e q u a l i t y ( A * , u ) ^ c O . . . . ( 5 . 2 . 5 ) This i n t e g r a l becomes 4 [ I, C\]kio<^#^ . u . + [ 1 , cL<< ^ / L , ^ 4 ( 5 . 4 . 7 ) which, using the qu o t i e n t r u l e of d i f f e r e n t i a t i o n , the divergency theorem, il± ~ cf>\ = o onS and s u b s t i t u t i n g f o r 0 , becomes jfCyu « M « ^ ^ 6 ^ 2 - k $ f r f i ) d J L • • • • ( 5 . 4 . 8 ) Here, CijM i s a p o s i t i v e d e f i n i t e form by d e f i n i t i o n and Ce and T are p o s i t i v e . Also c{tr i s r e s t r i c t e d to be p o s i t i v e . , -I I t i s necessary to req u i r e nfj to be p o s i t i v e d e f i n i t e , t h i s i s 49 a more r e s t r i c t i v e c o n d i t i o n than t h a t r e q u i r e d by the entropy production i n e q u a l i t y which r e q u i r e d i t to he p o s i t i v e . Consequently > o ( 5 . 4 . 9 ) I t remains to he proved that the e q u a l i t y holds only when u. ^  - = ° • Consider the term ii[KX t*M . .... ( 5 . 4 . 1 0 ) When w.t' = o everywhere i n JZ ; then U j t j - 0 inJZ a l s o . Consequently when u{ = o , U-j.j LLK, fe = ° • (5 .4 .11) Conversely, t h i s q uadratic form can only equal zero when Ui,j=0 . In t h i s case since U^fywhere kv are constants. But Ut=o on the boundary hence 0 i n JZ . Therefore, w i t h t h i s boundary c o n d i t i o n Cijta >o when LL^O 0 — o when uv s o I (5 .4 . 12) The term i > 2 i s p o s i t i v e , but not p o s i t i v e d e f i n i t e . The term ( 5 . 4 . 1 3 ) (5.4.14) 50 w i l l be p o s i t i v e d e f i n i t e i f kij i s req u i r e d to be p o s i t i v e d e f i n i t e . I t has th e r e f o r e been proved that the operator A f o r t h e r m o - e l a s t i c i t y i s p o s i t i v e d e f i n i t e . The operator has been shown to s a t i s f y the requirements f o r s o l u t i o n of the equations by the minimal f u n c t i o n a l method. In the case of homogeneous boundary c o n d i t i o n s the f u n c t i o n a l F((X)= ( A n , u ) - 2 ( u j ) . . . . ( 5 . 3 . 2 ) becomes where and ( 5 . 4 . 1 5 ) ( 5 . 4 . 2 ) ( 5 . 4 . 1 6 ) t By using the quotient r u l e of d i f f e r e n t i a t i o n and the divergence theorem i t i s p o s s i b l e to transform the B,{ term so that only f i r s t d e r i v a t i v e s of the f i e l d v a r i a b l e s j £ and U{ are req u i r e d to form the f u n c t i o n a l (here 0 w i l l be the * sum of the previous increments of & plus the i n i t i a l v a l u e ) . Thus the a l t e r n a t i v e f u n c t i o n a l i s obtained 51 VI 1 d i (5.4.17) The f u n c t i o n a l i s now extended to the case of non-homogeneous boundary c o n d i t i o n s . The problem can be s t a t e d as M i - £ n or> Su-fix = fx °^ £ ? V (5.4.18) As discussed i n the previous s e c t i o n i t i s necessary to assume ,—^  — the existence of the f u n c t i o n s tX^  and cj>-% i n J l such that . , A  = ?> . . . . (5.4.19) Ti ( i ^ ) = T o^, s T . Then d e f i n i n g the new v a r i a b l e s and where Y a - = U,' - ^ = • • • • (5.4.20) and which s a t i s f y the f o l l o w i n g boundary c o n d i t i o n s 1 "V"i •= o or) STu. . ° ° " ^ V .... (5.4.21) The operator A i s p o s i t i v e d e f i n i t e f o r the v a r i a b l e s 1A- and 52 s i n c e the "boundary c o n d i t i o n s are homogeneous. The r e q u i r e d f u n c t i o n a l i s $ C O = ( A u , u ) ~ 2(u ,0 -t ( u, A ^ ) - ( A M ) By.breaking the boundary i n t o i t s component parts £=Su-t£r and S~ S^ -+S"e and s u b s t i t u t i n g the boundary values where appropriate the above i n t e g r a l becomes 53 Jl - Z\ nj (T-- ii dS - Z [*< * fr dS •+ J(<T?y, fr, U , (5.4.24) Neglecting the constant terms i n J the f u n c t i o n a l i s - 2 ( 5 . 4 . 2 5 o Again, the term 6 7 . may be transformed to give the a l t e r n a t i v e f u n c t i o n a l ' f i ( 5 . 4 . 2 6 ) The min i m i s a t i o n of t h i s f u n c t i o n a l leads to a unique s o l u t i o n to the f i e l d equations which s a t i s f i e s the non-homogeneous boundary c o n d i t i o n s . I f approximate s o l u t i o n s are sought using a 54 sequence which i s minimising f o r the f u n c t i o n a l then, si n c e the therm o - e l a s t i c operator i s p o s i t i v e d e f i n i t e , these s o l u t i o n s w i l l converge i n energy. That i s | U n - U0 I ^ ° (5.4.27) Denoting the d i f f e r e n c e (Un-U 0) as and ^ then energy convergence means t h a t JL ' t rft J J _ .... (5.4.28) where /\ ^ Q J J  1 ' .... ( 5 - 4 . 2 9 ) The stronger convergence i n the mean cannot be expected s i n c e i t has not been proved t h a t the thermo-elastic operator i s positive-bounded-below. The above convergence c o n d i t i o n r e f e r s to the s o l u t i o n of equations ( 5 . 4 . 1 ) . I t should be r e c a l l e d that these equations c o n t a i n approximation t h a t the temperature v a r i e s l i n e a r l y with time during each increment. The convergence considered above i s th e r e f o r e w i t h respect to the space v a r i a b l e s ><^  and not time t . A more complete convergence d i s c u s s i o n would r e q u i r e i n c l u s i o n of t h i s approximation but t h i s i s not attempted i n t h i s study. 55 5.5 A p p l i c a t i o n to T h e r m o - E l a s t i c - P l a s t i c Equations In t h i s case the equations are ( 5 . 5 . D where the y i e l d c o n d i t i o n s are assumed to be s a t i s f i e d i n the e n t i r e r e g i o n . The above equations are represented as <4u - f ( 5 . 3 . D where & i s the v e c t o r { Uu^, <Pw j and ^ i s the v e c t o r [ ? <?&;°K } T -The character of the operator i s again i n v e s t i g a t e d w i t h respect to the homogeneous boundary c o n d i t i o n case Ui^ — <fi^ = o c n S . . . . ( 5 . 4 . 2 ) The operator i s c l e a r l y l i n e a r s i n c e the parameters CiJR^ i t^ut e t c . are constant f o r the incremental time step. This i s a basic p o s t u l a t e of incremental p l a s t i c i t y . The operator f\ i s now t e s t e d f o r formal symmetry by forming the s c a l a r ( A l / ? W ^ ) - ( J fa b < W * r y m f e g i a i j ^ . . . . ( 5 . 2 . 4 ) which should v a n i s h i f A i s f o r m a l l y symmetric. Forming t h i s i n t e g r a l i n the same manner as i n the thermo-elastic case 56 and using the quotient r u l e of d i f f e r e n t i a t i o n and the divergence theorem, leads to the i n t e g r a l 4 ; i },s (5.5.2) Thus f o r formal symmetry i t i s necessary that n i'j — ^Cj i , . . . ( 5 . 5 « 3) - Oirt-i j .... (5.5.4) and " ^ .... (5.5.5) Condition (5.5*3) i s s a t i s f i e d s i n c e kfj i s symmetric by d e f i n i t i o n . Conditions (5.5*4) and (5«5«5) are not obviously s a t i s f i e d and the p o s s i b i l i t y of choosing a flow r u l e , t h a t i s choosing j£w , so that they are both s a t i s f i e d i s now discussed, Rewriting c o n d i t i o n (5.5-5) using (3.2.24) and (3.3.54) f o r o i ^ and o(HI gives the requirement t h a t *'ki + = <i - > p % r •••• (5-5-6) 57 Using ( 3 . 3 . 5 6 ) f o r ^  , t h i s becomes Pkl ~ (T ( J " . . . . ( 5 . 5 . 7 ) Here K and T are both p o s i t i v e by d e f i n i t i o n and ( flnm ( T ^ .) i s r e q u i r e d to be p o s i t i v e from equation ( 3 « 3 . 4 5 ) due to the second law of thermodynamics. From p r a c t i c a l c o n s i d e r a t i o n s i t appears reasonable that i f AUI = ( a sc*Ur) • ( 5 - 5 . 8 ) then the s c a l a r m u l t i p l i e r should be p o s i t i v e . This i s necessary i f i n the isothermal case i t i s des i r e d to make Gijut p o s i t i v e d e f i n i t e . A l s o , i f the s c a l a r was negative t h i s would c o n t r a d i c t Drucker's hypothesis which appears to give good r e s u l t s i n some instances. Consequently, i t i s necessary that If < O n.-r . . . . ( 5 . 5 . 9 ) This c o n t r a d i c t s t e s t r e s u l t s f o r some common metals, see Timoshenko [20 J f o r instance, and i s considered to be an unacceptable c o n d i t i o n . Thus i t i s not p o s s i b l e to s a t i s f y c o n d i t i o n ( 5 . 5 « 5 ) « The question of the s a t i s f a c t i o n of ( 5 . 5 . 4 ) i s no longer r e l e v a n t since i t i s necessary that a l l of the co n d i t i o n s ( 5 . 5 - 3 ) . ( 5 . 5 . 4 ) and ( 5 . 5 - 5 ) are s a t i s f i e d . I t i s th e r e f o r e concluded that i t i s not p o s s i b l e to make the t h e r m o - e l a s t i c - p l a s t i c operator f o r m a l l y symmetric i n the general case. Consequently the minimal f u n c t i o n a l method discussed in. s e c t i o n ( 5 - 3 ) cannot be a p p l i e d . However, i n s e c t i o n ( 5 . 6 ) an i t e r a t i v e method i s proposed f o r f o r m a l l y 58 unsymmetric operators. Before l e a v i n g t h i s d i s c u s s i o n on symmetry some s p e c i a l cases may he noted. F i r s t l y , Drucker's hypothesis t h a t ^ f fa ^ 1)^1 . . . . ( 3 . 4 . 1 1 ) c o n t r a d i c t s equation ( 5 . 5 . 9 ) . Consequently the use of t h i s c o n d i t i o n w i l l lead to a f o r m a l l y unsymmetric operator. Secondly, i f -|\ i s zero then from equation ( 5 « 5 . 6 ) i t i s necessary that p i s a l s o zero. I f p i s zero then e i t h e r ^ or ( ) must be zero. The c o n d i t i o n t h a t e i t h e r X or ( [3mr ) i s zero i s u n r e a l i s t i c . Therefore i t i s not p o s s i b l e to make the operator symmetric when 5.6 An I t e r a t i v e Method f o r Unsymmetric Operators The equation to be solved i s A U, =- f ( 5 . 6 . 1 ) where /\ may be an unsymmetric operator. The proposed i t e r a t i o n scheme r e q u i r e s that the operator A i s s p l i t i n t o i t s symmetric and anti-symmetric p a r t s , (\ and A r e s p e c t i v e l y . These are defined as A S - KA-H A*") . . . . ( 5 . 6 . 2 , A * = K A - A O . . . . ( 5 . 6 . 3 , where A i s the a d j o i n t of the operator A . Then equation ( 5 . 6 . 1 ) i s w r i t t e n as 59 . . . . (5.6.4) In) (n-i) I f an i t e r a t i o n scheme i s introduced where u, and 11 are the s o l u t i o n s of the f\ and (fl-i) i t e r a t i o n s , the process may be w r i t t e n as AV n ) = j - A V * - 0 (5.6.5) The advantage of w r i t i n g the equation i n t h i s form i s that the , In) s o l u t i o n of U i s the s o l u t i o n of an equation w i t h a fo r m a l l y symmetric operator and may the r e f o r e be found from the s t a t i o n a r i t y of a f u n c t i o n a l (Tonti's theorem). Further-more i f (\% i s p o s i t i v e d e f i n i t e the s o l u t i o n may be found by minimising a f u n c t i o n a l of the form F ( U W ) = ( 4 Y n > u ( " > ) - 2 ( u t " ! f - / u " , - ' ) ) . . . . (5.6.6) However, i t i s not p o s s i b l e to s t a t e a convergence c r i t e r i o n f o r a s e r i e s of approximate s o l u t i o n s s i m i l a r to that given by theorem 4 i n s e c t i o n (5.2) f o r the minimal f u n c t i o n a l method. A c r i t e r i o n f o r the convergence of the i t e r a t i o n scheme may be developed by c o n s i d e r i n g the d i f f e r e n c e of successive s o l u t i o n s . T-P AS (V)) r (i-/-) I F A U = J - A U . . . . (5.6.7) . . . . (5.6.8) then A t U- - U J = - A (. U - U / . . . . (5.6.9) M ^ . (n-') Wr i t i n g V ^ U, — U. t t h i s becomes and A5 U ^ 0 = f - A \ 6o /\s y ( n ) = - 1? V * " * ^ ( 5 . 6 . 1 0 ) I f A i s p o s i t i v e d e f i n i t e i t s inverse $ e x i s t s , hence Y = - A h X ( 5 . 6 . U ) I f the operator (A* A ) i s bounded then the norm H ^ l l , (I A 5 V v M - ' | l . . . . (5.6.12) may be w r i t t e n as | | V H „ < n -^Vfl- I W ' " ' ° H | . . . . ( 5 . 6 . 1 3 ) where the norm of the operator A convergence c r i t e r i o n may be w r i t t e n as < P W all n.idhete P<\-o 6 l 4 ) || v ( * ^ \ \ f 1 r * (5.6.W) which, using the previous i n e q u a l i t y ( 5 . 6 . 1 3 ) . becomes || A^'/C 'I < f •••• ( 5 - 6 . 1 5 ) Thus, i f the bound of the operator (ft A ) can be evaluated then the above method may be used. However, si n c e A and A are f u n c t i o n s of the s t a t e of s t r e s s and w i l l vary over the re g i o n J l , t h i s appears to be a formidable task. The above method was not used i n t h i s study. I t may be seen from the d e f i n i t i o n of the a n t i -symmetric operator A that i f Drucker's hypothesis i s adopted then the l a c k of symmetry of /4 i s due to the p l a s t i c c o u p l i n g terms A and . For common metals under normal temperatures i t i s known that c o u p l i n g e f f e c t s are s m a l l . Consequently, under these c o n d i t i o n s convergence was expected. In the 61 s o l u t i o n of some t y p i c a l problems, which w i l l be discussed f u r t h e r i n Chapter 7, the method was found to converge i n l e s s than 5 c y c l e s . In order to estimate the range of a p p l i c a b i l i t y of the method an a n a l y s i s s i m i l a r to the convergence proof o u t l i n e d above w i l l be r e q u i r e d . A uniqueness theorem was developed using the decomposition of A • Theorem 5« Uniqueness f o r Unsymmetric Operators. I f the symmetric part of the operator A i s p o s i t i v e d e f i n i t e then the s o l u t i o n of the equation A U = f (5.6.1) i s unique i f H i s r e s t r i c t e d to be r e a l . Proof i Assume the existence of two s o l u t i o n s U,(, and Lit to the equation A a * j~ ( 5 . 6 . 1 ) W r i t i n g the d i f f e r e n c e U ~ Uj- Ut and s u b t r a c t i n g the two equations gives A \ x ^ O ( 5 . 6 . 1 6 ) Forming the energy product w i t h li and decomposing A i n t o symmetric and anti-symmetric parts gives (A U, U ) = ( As K, u )-f ^ V , u ) . . . . ( 5 . 6 . 1 7 ) • ~ / A ^ ^ ^ ' \ « . . k0^ but i f u i s r e a l then ^ f\ J=0 by d e f i n i t i o n of A . Therefore (A S U , U ) — o .... (5.6.18) 62 A O i s p o s i t i v e d e f i n i t e and the r e f o r e U - O i s the only s o l u t i o n . I t i s therefore proved t h a t the s o l u t i o n i s unique. When the i t e r a t i o n scheme i s used i t may ther e f o r e be s t a t e d i f h i s p o s i t i v e d e f i n i t e that i f the scheme converges then t h i s s o l u t i o n i s unique. 5.7 A p p l i c a t i o n of the I t e r a t i v e Method to Thermo-Elastic- P l a s t i c Equations The f i e l d equations i n t h i s case are 1 Ct 1 'I J'j L Q / J'j k&rW*1 °^**ifi ^K,4 — ( 5 . 7 . 1 ) ( 5 . 7 . 2 ) which are subject to the boundary c o n d i t i o n s U{ — IXi <z>r> >^u. cf>i = <jb+ °n j-( it*, (fa') = T i The operator /\ i s s p l i t i n t o i t s symmetric and anti-symmetric parts AS and A {- U ^ ^ ^ - l l ^ A ^ ^ } (5.7.3) 63 e . . . . ( 5 . 7 . 4 ) I f A i s p o s i t i v e d e f i n i t e then the s o l u t i o n equations ( 5 . 7 . 1 ) f o r each i t e r a t i o n may be accomplished by minimising the f u n c t i o n a l Jl La 6 + i r , (n) r J s 9 dt ( 5 . 7 . 5 ) 64 ("0 ( n - O I f the i t e r a t i o n process converges then the s o l u t i o n u. = u w i l l s a t i s f y the "boundary c o n d i t i o n s i n a d d i t i o n to the f i e l d equations i n Jl . I t may be noted t h a t during the i t e r a t i o n (w) / (n-0 process when u. =f= U then the boundary c o n d i t i o n s w i l l not be s a t i s f i e d . The question of whether A* i s p o s i t i v e d e f i n i t e i s now discussed. The energy product may be w r i t t e n i n the form 'j \l (A* U, M . ) =• J ^ [ | ( C i j ^ 4 Cif/fj) + 1 C i j m n ( o { m - e i w n \ ( ^ f y ^ f ^ C ^ 4 r ( fw>n + u M ) *K * -' . . . . (5.7.6) As I f ft i s to be p o s i t i v e d e f i n i t e then Cc = ( Ce-* T CK^KI °it<^iw ") > o (5.7.7) and [ K ^ / K * ) + J H** - ^ X ^ - ^ ^ w f ^ ] (5.7.8) T -e must be p o s i t i v e d e f i n i t e . In the s p e c i a l case then h>- = ^ •••• «•*•"> the second term becomes . . . . (5.7.9) 65 These c o n d i t i o n s do not e a s i l y a l l o w any d e t a i l e d conclusions to he drawn. As may he expected from a p h y s i c a l p o i n t of view, the c o n d i t i o n s depend on the e x i s t i n g s t a t e of s t r e s s and temperature which i n turn a f f e c t the r a t e of heat generation during the increment. I t would appear t h a t i f the heat generation i s excessive then l a c k of uniqueness may occur. In the s o l u t i o n of some problems u s i n g the f i n i t e element method i t was found to be more convenient to t e s t the complete operator f o r p o s i t i v e d e f i n i t e n e s s r a t h e r than use the above c o n d i t i o n s . For the problems discussed i n Chapter 7 the operator was found to be p o s i t i v e d e f i n i t e and t h e r e f o r e the s o l u t i o n s were unique. give a mathematical c r i t e r i o n f o r the convergence of a s e r i e s of approximate s o l u t i o n s found by minimising the f u n c t i o n a l ( 5 . 7 . 5 ) . However, i n problems where the co u p l i n g e f f e c t s are s m a l l i t may be s a i d i n t u i t i v e l y that the operator i s "nearly symmetric". I t i s known p h y s i c a l l y that small c o u p l i n g has l i t t l e i n f l u e n c e on the s o l u t i o n s . I t may t h e r e f o r e be suggested t h a t a s e r i e s of s o l u t i o n s to a s l i g h t l y coupled problem, obtained from a sequence of f i e l d s which would be w i l l probably possess convergence p r o p e r t i e s s i m i l a r to the symmetric case. the convergence discussed above i s with respect to the s o l u t i o n of the r a t e form of the f i e l d equations f o r an increment As mentioned i n s e c t i o n ( 5 « 6 ) , i t i s not p o s s i b l e to minimising f o r a symmetric uncoupled case As was the case f o r the the r m o - e l a s t i c equations, 66 ( 5.7.1). These equations c o n t a i n the approximation to the heat flow equation as i n the thermo-elastic case and a l s o the c o n s t i t u t i v e p o s t u l a t e of piecewise l i n e a r p l a s t i c i t y . 67 CHAPTER 6 A FINITE ELEMENT FORMULATION 6•1 I n t r o d u c t i o n In t h i s chapter a f i n i t e element i s developed f o r the case of plane s t r a i n f o r a p a r t i c u l a r choice of c o n s t i t -u t i v e r e l a t i o n . The theory developed i n the previous chapters i s a p p l i c a b l e i n the general case. However, f o r the purpose of i l l u s t r a t i o n a r e l a t i v e l y simple case i s considered. The element i s developed using F e l l i p p a ' s [ G ] work i n which the method i s f o r m a l i s e d . The simples t element i s used, w i t h a l i n e a r i n t e r p o l a t i o n f u n c t i o n , which gives a constant s t r e s s and temperature t r i a n g l e . The y i e l d f u n c t i o n and flow r u l e w i l l t h e r e f o r e be constant over each element. A modified y i e l d c o n d i t i o n i s proposed, which i s convenient f o r the incremental method. I t was shown i n Chapter 5 that the convergence theorems f o r symmetric, p o s i t i v e d e f i n i t e operators do not apply i n the p l a s t i c case. Consequently, advantage cannot be taken of the a p p l i c a t i o n of these theorems to f i n i t e elements by Melosh [ 15] and O l i v i e r a [19 ] . Some comments are made about the computer program which was developed f o r the s o l u t i o n of the equations. 68 6 . 2 Choice of C o n s t i t u t i v e R e l a t i o n s In order to develop a f i n i t e element i t i s necessary to s p e c i f y the form of the c o n s t i t u t i v e r e l a t i o n s . In t h i s study they were chosen as i s discussed below. The e l a s t i c behavior was chosen to be i s o t r o p i c , t h e r e f o r e Cijkl = '-^ % ~ % tk* ( 6 . 2 . 1 ) and A'jj = *< £<ij ( 6 . 2 . 2 ) The c o n d u c t i v i t y c o e f f i c i e n t s f o r F o u r i e r ' s law were chosen f o r i s o t r o p i c heat flow, hence /f,y = k Snj ( 6 . 2 . 3 ) The p l a s t i c behavior was chosen to be s p e c i f i e d by an i s o t r o p i c y i e l d f u n c t i o n . The hardening f u n c t i o n was a l s o assumed to be i s o t r o p i c . Green and Naghdi [" 8 j propose a quadratic form f o r the y i e l d f u n c t i o n i n the case of i n f i n i t e s i m a l displacements. They w r i t e 2 . . . . (6.2.4) Reducing t h i s to the i s o t r o p i c case and a l s o assuming th a t ^ i s independent of , gives 4 e > , < < -4 5 * ^ ^ + ^ ^ T l „ i i ( 6 . 2 . 5 ) 69 By making the f u r t h e r assumption that £ i s independent of the h y d r o s t a t i c component of s t r e s s , a common assumption f o r metals, the f u n c t i o n becomes J~ = k rKi -t i ^ . . . . ( 6 . 2 . 6 ) Here the f a c t o r on the f i r s t term has been dropped si n c e i t would be absorbed i n ^  and ^ . Hence, using ( 6 . 2 . 6 ) , - - uxi a.r\cl J •= b I ( 6 . 2 . 7 ) When s p e c i f y i n g the flow r u l e Drucker*s[ 5 ] normality c o n d i t i o n was adopted. That i s 3kt= - ( 3 . 4 . U ) With t h i s c o n d i t i o n f o r pkt and the above form f o r £ given by ( 6 . 2 . 6 ) then the entropy production i n e q u a l i t y fuiKi^O ( 3 . 3 . 4 5 ) i s s a t i s f i e d since h r^ =4u^ ^ ~ °*J'^^° (6;2,8) S u b s t i t u t i n g the form ( 6 . 2 . 7 ) and using ( 3 . 4 . 1 1 ) i n t o the flow r u l e gives e'ij - * *j'(<rj r^ifr) . . . . ( 6 . 2 . 9 ) Now ?) can be determined from the equation * h ( h i j - ? t > ) = \ • • • • < 3 - 2 - 1 6 ) 70 when the hardening f u n c t i o n hjj has been s p e c i f i e d and using the f a c t t h a t % ry^ll. ( 6 . 2 . 1 0 ) Green and Naghdi suggest the f o l l o w i n g form f o r p 0^ f o r i n f i n i t e s i m a l deformations. Ki = Kv\ 4 ^ / ^ e l ^ -f H i « T . . . . (6 .2 .11) In the i s o t r o p i c hardening case, and assuming h w independent ^ Il . ' of g,^ , t h i s becomes W» r w + H* Kl^^Z h i 1 • • • • ( 6 . 2 . 1 2 ) Hence, using equations ( 3 . 2 . 1 6 ) , ( 6 . 2 . 1 0 ) and ( 6 . 2 . 1 2 ) , W ^wiw Hi &KI ^ ) = H <K< <Tk< ( 6 - 2 - 1 3 ) The p l a s t i c flow r u l e ( 6 . 2 . 9 ) then becomes « 1 ; ( Put <Kt + t>TT ) . . . . (6.2.14) Hence the terms }) A;; H and 3/3,7 - i n equation ( 3 . 2 . 2 2 ) I i ^ r; V become " r'i n T t r ~ -L - ( 6 . 2 . 1 5 ) "let i ) and n Tl -If ~~ J — ] . . . . ( 6 . 2 . 1 6 ) h t m fat The parameters h and may be i d e n t i f i e d by w r i t i n g the equations f o r the u n i a x i a l s t r e s s case. • H i l l [ * 1 0 j and others 71 have shown t h i s method f o r the isothermal case. Here i t i s shown t h a t , w i t h the above p a r t i c u l a r c o n s t i t u t i v e assumpt-ions , the method may be e a s i l y extended to the non-isothermal case. For u n i a x i a l s t r e s s r„ =r , (rlz = ^ - =r <r%% - rsz = a . . . . ( 6 . 2 . 1 7 ) and (6.2.18) S u b s t i t u t i n g these s t r e s s e s i n t o equation (6.2.14) gives Cn = • — (T + k l T ( 6 . 2 . 1 9 ) For the u n i a x i a l case the e l a s t i c law ( 3 . 2 . 3 ) becomes Cg = ~, F~ -h oi'l ( 6 . 2 . 2 0 ) The t o t a l s t r a i n then i s e„~ (L -t-fu 1^+ I*'* ™ ) T . . . . ( 6 . 2 . 2 1 ) I E 3>h I \ h ( r J which may be w r i t t e n as C n ~ -Z °^ "+ ^ T . . . . ( 6 . 2 . 2 2 ) where £ 7 i s the isothermal tangent modulus and ^  i s the c o e f f i c i e n t of thermal expansion. I f £7 and are s p e c i f i e d f o r a m a t e r i a l then the parameters h and h may be found. However, i t should be r e a l i s e d t h a t these w i l l s p e c i f y the y i e l d f u n c t i o n . Conversely i f the y i e l d f u n c t i o n and, say ET , are s p e c i f i e d then the f u n c t i o n f o r oL w i l l have been f i x e d . A modified form of the y i e l d c o n d i t i o n given i n s e c t i o n ( 3 . 2 ) 72 was used f o r the incremental process i n order to s i m p l i f y the method. The y i e l d c o n d i t i o n of s e c t i o n (3«2) may be w r i t t e n as 7\^0 iJ f £ / cr>d$>0 . . . . ( 6 . 2 . 2 3 ) if £ ^ a ^ f ^ O . . . . ( 6 . 2 . 2 4 ) * = C P V . . . . ( 6 . 2 . 2 5 ) where J ~ ^ " • •••• (6.2.26) In equation ( 6 . 2 . 2 3 ) i t i s necessary that ^>0 . However, i n the incremental method of s o l u t i o n J i s unknown at the s t a r t of the increment. Consequently i t would be necessary to guess where § > O and then to solve the increment and check the assumption. I f the guess was wrong another guess would be r e q u i r e d . This t r i a l and e r r o r r o u t i n e would be req u i r e d to be s a t i s f i e d f o r every p o i n t i n the body. Instead of using t h i s time consuming method the f o l l o w i n g y i e l d c o n d i t i o n was used. ^ 7 ^ ° l'f f ^ / .... (6 . 2 . 2 7 ) and , r .... (6.2.28) Here and ^  are the values of the y i e l d f u n c t i o n and hardening f u n c t i o n a t the beginning of the increment. Then i f equation (6.2.27) a p p l i e d ( ?> f o ) was computed using equation (6.2.14). I f equation (6.2.28) a p p l i e d {?>=rO) then 73 e-»y was set equal to zero. I t may be seen that i t i s p o s s i b l e to have j ^/ ' " i and consequently & <j fO f o r an increment during which the m a t e r i a l i s unloading, i . e . J< O . This would v i o l a t e the y i e l d c o n d i t i o n of s e c t i o n ( 3 . 2 ) . However, the e r r o r may be minimised by making t h i s increment of s t r e s s e s and s t r a i n s s m a l l , using a small increment of loads. The hardening f u n c t i o n ^ was updated at the end of each increment according to the f o l l o w i n g r u l e where ^\ and = i n i t i a l values of |" andy^ f o r the increment and ff and / f = f i n a l values of and at the end of the increment. Case 1: E l a s t i c increment, D = O , j \ ' < T^ 1' • Set" / ; = ~f-i ( 6 . 2 . 2 9 ) Case 21 P l a s t i c increment I f ^ > ^ set ^ = ( 6 . 2 . 3 0 ) I f J f s e t / f 7 ^ ( 6 . 2 . 3 D 6 . 3 D e r i v a t i o n of the F i n i t e Element Equations In t h i s s e c t i o n a f i n i t e element i s derived using the format of F e l l i p p a [ 6 J . In order to s i m p l i f y the c a l c u l a t i o n of the y i e l d f u n c t i o n and c o n s t i t u t i v e law a l i n e a r displacement f u n c t i o n was used. This gave constant 74 s t r e s s e s and temperature over each element. Consequently the e l a s t i c - p l a s t i c moduli, C-t'jW » ij e^c- and the y i e l d f u n c t i o n ^ were constant f o r each element. I t was recognized that there would be d i s c o n t i n u i t i e s of s t r e s s and temperature between elements. However, i t was f e l t t h a t t h i s element was s u f f i c i e n t to i l l u s t r a t e the method. Accuracy could be improved by reducing the g r i d s i z e . The chosen displacement f i e l d s a t i s f i e s C- O c o m p a t i b i l i t y . Consequently the convergence theorems of Melosh [15 J and O l i v e i r a [19] may be a p p l i e d to the thermo-e l a s t i c case. However, i n the p l a s t i c case there i s no convergence c r i t e r i o n f o r the f u n c t i o n a l , as discussed i n the previous chapter, and consequently the r e s u l t s of Melosh and O l i v e i r a cannot be a p p l i e d . In the f o l l o w i n g d e r i v a t i o n t r i a n g u l a r co-ordinates are used. These are defined by F e l l i p p a £6 ] . The m a t e r i a l displacement r a t e s UL^, 1 = 1,2. , and the entropy displacement r a t e s fri , 7 = l,Z are l i n e a r f u n c t i o n s of the co-ordinates. The domain of the element i s s e l e c t e d as a t r i a n g l e and the three corners are used as nodal p o i n t s . I t then f o l l o w s t h a t the displacements may be w r i t t e n as ' 1 f1 V o t o 3 o f. 7. P. j . u ( 6 . 3 . 1 ) . . . ( 6 . 3 . 2 ) 75 S i m i l a r l y Here <j>- i s the i n t e r p o l a t i o n matrix which i s a l i n e a r f u n c t i o n of the t r i a n g u l a r co-ordinates T7- . The g e n e r a l i s e d • u co-ordinates f o r m a t e r i a l displacement r a t e s «£t* and entropy displacement rates *>\ are the values of H\ and at the nodes These are shown i n Figure 2 which i s r e f e r e d to c a r t e s i a n co-ordinates ( X,, X9_ ) . c4 Figure 2 Generalised Co-ordinates The s t r a i n r a t e s e i j may then be w r i t t e n as — 0 O 1 ST s ; 0 0 O <3J • p Q, &z a* bi b » "2 « "z "2 9 OK) n "2 a b. •2 b» 2^ . *: . . . . ( 6 . 3 . 4 ) — ( 6 . 3 . 5 ) 76 and the entropy divergence ^ w , m a s 4 • • • t = i ( 6 . 3 . 6 ) ( 6 . 3 . 7 ) In the above equations A =• J x„ = area of t r i a n g l e with nodal co-ordinates ( X * 9 X^ ) i = 1 , 3 . . . . . (6.3«8) and = ( V - tf); 4 ^ (*',-*?) J «*=(x*-x/) . . . . ( 6 . 3 . 9 ) .... (o.3 . 1 0 ) The i t e r a t i v e f u n c t i o n a l given i n Chapter 4 may now be formed f o r the element by s u b s t i t u t i n g the above i n t e r -p o l a t i o n f u n c t i o n s and i n t e g r a t i n g of the area of the element S.t and i t s boundary St . The f u n c t i o n a l i s 77 JJZ, ft) . Ln) ----- • , ^ . ^ - 2 n Js j (Tfj U n o(S - 2 r Te "fee ( 6 . 3 . 1 1 ) where Cijm and ^»y^  are obtained from equation ( 5 - 7 . 5 ) as Cijw = 4 ( C*iK« ^  C w f j + ^ 6 *^**^  i^w.^ H") Cj"nw Geep^  ) .... ( 6 . 3 . 1 2 ) a n C * <\ 'j ~ Z ( ^'/^ " ^WfJ +X ( ^ n ^ ^ - ^ H ^ ) CfjlMM Grtfy) ( 6 . 3 . 1 3 ) In the above f u n c t i o n a l kij has been replaced by (£rj k ) f o r the i s o t r o p i c case from equation ( 6 . 2 . 3 ) . The isothermal e l a s t i c - p l a s t i c c o n s t i t u t i v e matrix Cijm i n equation ( 3 . 2 . 2 2 ) must be modified f o r the case of plane s t r a i n . In the case of plane s t r a i n , (6.3.14) and consequently 23 ( 6 . 3 . 1 5 ) 78 then the r e l a t i o n -/ e,j - C i ) M <Tki i,}.k,i*l,3 ( 6 . 3 . 1 6 ) becomes hence -, . • \ J i z ( 6 ' 3 - 1 7 ) -/ • = - ^ # _ (6.3.18) '3333 which may be s u b s t i t u t e d to give * W < V « " C4£*)tu . . . . ( 6 . 3 . 1 9 ) Consequently f o r the plane s t r a i n case the matrix C-ijv*. i n equation ( 6 . 3 . 1 1 ) should be replaced by L,-jkt =• ( Ujki - ) . . . . ( 6 . 3 . 2 0 ) where the summations on are from /,2 . In the above 6>7;= f!»>K'*fy -ijfa+Ti&rjtt . . . . ( 6 . 3 . 2 1 ) -) = -risr + p i A i j t i . . . . ( 6 . 3 . 2 2 ) and j (T»» = 4 ^  V 5 1 ^ •••• < 6 ' 3 . 2 3 ) S u b s t i t u t i n g the i n t e r p o l a t i o n formulae gives the f o l l o w i n g component i n t e g r a l s of ( 6 . 3 . 1 1 ) 79 J l , ' i r r (6.3.24) , J Ze 1 ' 1 ' ' ' . . . . ( 6 . 3 . 2 5 ) Jit L« J Q 1 ( 6 . 3 . 2 6 ) I Q ( 6 . 3 . 2 7 ) 1 .... (o.3•2o) . . . . (6.3.29) where ^ • t €^ , and £^ are the t o t a l s of the previous increments of <p-^ , e^i , 2>J and r e s p e c t i v e l y . The i n t e g r a l of the body fo r c e s Ft- i s omitted f o r s i m p l i c i t y . 80 J r O it it iSl . . . . ( 6 . 3 . 3 0 ) where i s given by F e l l i p p a i n h i s t a b l e of i n t e g r a l s of i n t e r p o l a t i o n v e c t o r s as 2 I I I 2 I o o o o O O 2 o o o o o o 2 / o o o o l o o — ( 6 . 3 . 3 1 ) I 2 The surface i n t e g r a l s were evaluated f o r the case where the t r a c t i o n T and temperature 6 were constant along each side of the t r i a n g l e as shown i n Figure 3. 8 1 Figure 3 Surface T r a c t i o n s and Temperatures . ft The convention has been adopted that the i side i s • f t - r 1 ' i . opposite the i node. Also I j where f = ' , 3 and J - M Z • f t • rti r e f e r s to the t r a c t i o n r a t e on the 1 s i d e i n the J ' * i * i d i r e c t i o n . S i m i l a r l y f o r 9- , rtj and ttj . Using the t r i a n g u l a r co-ordinates the U,,- may be w r i t t e n as ui - -u , =3 O ?. y, O o o o o o o f> 5, IX, r. o o o o , z a 4 o o o *>. o y. % o o o o O ?. o ~ Cj& SJS . . . . ( 6 . 3 . 3 2 ) The surface i n t e g r a l f o r the t r a c t i o n s then becomes G^SOt ^  •••• ( 6 - 3 - 3 3 ) S - .1*0 / r\ T, U , . «S - ( j 82 where -r* -[j, i't V t't't;} •••• ( 6 . 3 . 3 4 ) hence 2 J T i a - - . . . . ( 6 . 3 . 3 5 ) ST where W)}-— < 6 , 3 ' 3 6 ) S i m i l a r l y f o r the temperature case, c a l l i n g Q - = the i n t e g r a l becomes ' \ t * c * " . . . . ( 6 . 3 . 3 7 ) where (nB), - f ^ 8 ^ ( 6 . 3 . 3 8 ) By assembling the previous i n t e g r a l s to form the i t e r a t i v e f u n c t i o n a l f o r the element and then considering a v a r i a t i o n of the generalised co-ordinates S, and z>j< the fo l lowing twelve simultaneous equations are obtained: 8 3 A (-tap* CijK^ VM)p . (»») V sy mm, i t o j ' o i , ! i 1 • ( 6 . 3 . 3 9 ) 84 For convenience of d i s c u s s i o n the previous equations may he w r i t t e n as . S (ri) r- p - (<n->) ^p dp = i p - Bp- Sp ( 6 . 3 . 4 0 ) At the s t a r t of an i t e r a t i o n scheme, 0= 1 , i f ^  i s assigned some value, probably zero f o r convenience of automatic computation, i f the i n i t i a l temperature d i s t r i b u t i o n i s known i n terms of the t o t a l v e c t o r s Sp and %^ and i f §p i s p r e s c r i b e d then c l e a r l y the problem i s reduced to the s o l u t i o n of Kdp &p =• j-p (6.3.41) This i s a symmetric s e t of equations which i s convenient f o r s o l u t i o n . The above d i s c u s s i o n has been l i m i t e d to the equations f o r the element. The complete s t r u c t u r e m a trix may be assembled by s u p e r p o s i t i o n of the element matrices i n the normal manner of the d i r e c t s t i f f n e s s method. The requ i r e d s o l u t i o n i s then e x t r a c t e d from the complete set of equations. For convenience of d i s c u s s i o n i n the next s e c t i o n the complete s t r u c t u r e equations are w r i t t e n as K^Ap = Fp - ®p - Ko/p Ap - F •••• < 6 -3.42) 85 6.4 Comments on the Computer Program f o r S o l u t i o n of the  E q u a t i o n s In t h i s s e c t i o n some comments a re made r e g a r d i n g the computer program which was w r i t t e n and used to s o l v e some p r o b l e m s . The comments a re r e s t r i c t e d to the o r g a n i s a t i o n o f the program i n c l u d i n g t e s t s o f y i e l d i n g , i n c r e m e n t a l and i t e r a t i v e l o o p s . Q u a l i t a t i v e remarks about the answers are postponed to Chapter 7 where some s p e c i f i c examples a re d i s c u s s e d . A f low c h a r t f o r the computer program i s g i v e n i n F i g u r e 4 . T h i s i s i n o u t l i n e o n l y and i l l u s t r a t e s the p l a c e o f the i t e r a t i o n scheme i n the i n c r e m e n t a l p r o c e s s . I t may be noted t h a t the i t e r a t i o n scheme does not r e q u i r e more than one i n v e r s i o n o f the mat r i x K ^ s pe r i n c r e m e n t . I t i s o n l y n e c e s s a r y to do a m a t r i x m u l t i p l i c a t i o n f o r the s o l u t i o n o f s u c c e s s i v e i t e r a t i v e s o l u t i o n s A^"\ The convergence t e s t f o r the i t e r a t i o n scheme was t h a t Ln) L*) Ln-i) 4 (A* Aui - A" A" ) , - 4 and ,(ri) ufn) A A*(n-» A f C ~ » < , ' ° J 0 • • • • { 6 A ' 2 ) where and a re the v e c t o r s o f the m a t e r i a l and en t ropy d i s p l a c e m e n t s from the A i t e r a t i o n c y c l e . 86 START Read s t r u c t u r e m a t e r i a l and geometric data Set t o t a l s t r e s s e s , temperature, o d i s p l a c ements and ®& = O Incremental loop Read load increments and dt I Element loop Test y i e l d f u n c t i o n 1 E l a s t i c P l a s t i c B u i l d symmetric element matrix - i f p l a s t i c use current t o t a l s t r e s s e s and temperatures f o r moduli Add Kif, matrices to form symm-e t r i c s t r u c t u r e matrix K*f>  I Set Ap*0 Figure 4 Computer Program Flow Chart - Part (a) 87 ^ I t e r a t i o n loop^«-Test whether any elements are p l a s t i c B u i l d element anti-symm. k^ Solve Kp ACA = i Add k^ to form s t r u c t u r e Form Q using /\^ Test whether any elements are p l a s t i c AO0 is*"1 -Ft*) Evaluate ZAc<= J\>*A Yp u s i n g iS.jp from c y c l e Some p l a s t i c Store PC^A I—' l^ n~i y L ^ A l l e l a s t i c I f 0>l t e s t f o r convergence of ASM-i n <£/p,riot converged^ Converged J ( fl»!o. not converged EXIT Figure 4 Computer Program Flow Chart - Part (b) v B u i l d increment of load v e c t o r f o r r e s i d u a l temperature gradient using A^. 0Add to previous t o t a l to get new term. Evaluate element by-element a f t e r checking y i e l d cond-i t i o n and using current t o t a l s t r e s s e s and temps, f o r p l a s t i c moduli Calc. increment of s t r e s s e s and temp-erature using Agi. Evaluate a f t e r checking y i e l d c o n d i t i o n and using current t o t a l s t r e s s e s and temps, f o r p l a s t i c moduli. Add to current t o t a l s to give new t o t a l s t r e s s e s and temperatures. P r i n t Incremental displacements T o t a l displacements T o t a l s t r e s s e s and temps. Y i e l d f u n c t i o n s and y i e l d l i m i t s Go to s t a r t of next increment Completed increments Z T Z ' EXIT Figure k Computer Program Flow Chart - Part (c) 89 I t was found t h a t i n the problems discussed i n the next chapter that convergence occured w i t h i n 5 c y c l e s i n a l l cases. The program was run on the U.B.C. computer, an IBM 3 6 O - 67, and computation times and costs appeared r e a l i s t i c . For instance a s t r u c t u r e w i t h 144 degrees of freedom cost $10 to solve J l increments of l o a d i n g . 90 CHAPTER 7 SOLUTION OF SOME ILLUSTRATIVE PROBLEMS 7 .1 I n t r o d u c t i o n In t h i s chapter s o l u t i o n s are given to some problems to i l l u s t r a t e the v a l i d i t y of the method. The r e s u l t s show that (a) The c o n s t i t u t i v e p o s t u l a t e s , p a r t i c u l a r l y the thermo-p l a s t i c c o u p l i n g terms i n c l u d i n g the temperature dependence of the y i e l d f u n c t i o n and heat generation due to p l a s t i c s t r a i n , give r e s u l t s which are q u a l i t a t i v e l y i n agreement with p h y s i c a l expectations. (b) The minimal f u n c t i o n a l method of s o l u t i o n gives good r e s u l t s f o r both the symmetrical t r a n s i e n t heat flow problem and the unsymmetrical f u l l y coupled thermo-plastic problem. Owing to the s c a r c i t y of e x i s t i n g s o l u t i o n s to e l a s t i c - p l a s t i c s t r a i n - h a r d e n i n g problems and the complete l a c k of s o l u t i o n s to f u l l y coupled thermo-plastic problems, i t was not p o s s i b l e to d i r e c t l y compare the numerical r e s u l t s of t h i s study to closed form s o l u t i o n s . The f o l l o w i n g problems were chosen f o r a n a l y s i s due to t h e i r s i m i l a r i t y to e x i s t i n g s o l u t i o n s } Case 1 - E l a s t i c - p l a s t i c work-hardening with an i n i t i a l l y uniform temperature f i e l d . This case shows q u a l i t a t i v e 91 agreement with e x i s t i n g s o l u t i o n s and i s used f o r comparison w i t h Case 3 "to show the r o l e of thermal c o u p l i n g . Case 2 - Transient heat flow, no mechanical c o u p l i n g . This case shows good agreement between numerical r e s u l t s and closed form s o l u t i o n . Case 3 - E l a s t i c - p l a s t i c work-hardening w i t h i n a non-uniform temperature f i e l d . This case shows q u a l i t a t i v e agreement wit h an e x i s t i n g s o l u t i o n and a l s o the i n f l u e n c e of thermal co u p l i n g . Symmetrically loaded c y l i n d e r s i n plane s t r a i n were chosen f o r a n a l y s i s due to the a v a i l a b i l i t y of s o l u t i o n s by Bland [2] f o r the isothermal and e l a s t i c a l l y coupled thermo-p l a s t i c cases, by D'Isa [ 4 ] f o r the isothermal case and by Carslaw-Jaeger [3 ] f o r the uncoupled t r a n s i e n t heat flow case. S o l u t i o n s f o r the f u l l y e l a s t i c - p l a s t i c a l l y coupled case were not a v a i l a b l e . Numerical s o l u t i o n s were found f o r the f u l l y coupled case using the smae c y l i n d e r s as i n the e l a s t i c a l l y coupled case to a l l o w a q u a l i t a t i v e comparison. The comparisons are summarised i n Table I . 92 I Case 1 - E l a s t i c - P l a s t i c Work-Hardening w i t h an i n i t i a l l y Uniform Temperature F i e l d 1 ( i ) 1 ( i i ) 1 ( i i i ) 1 ( i v ) 1 (v) Charlwood, v.Mises y.c. i n c r . flow, cf = V f fo Charlwood, v.Mises y . c , i n c r . flow.^'/o^|i= jr-o Charlwood, v.Mises y . c , i n c r . f l o w , * ' , V^o Bland [ 2 ] , Tresca y.c. assoc. flow , dC-o D'Isa [ 4 ] , v.Mises y . c , deformation theory Case 2 - Transient Heat Flow, no Mechanical Coupling 2 ( i ) 2 ( i i ) Charlwood, incremental temp. load , 4 1 *• |i - X1 - o Carslaw & Jaeger [ 3jt step f u n c t i o n temperature load Case 3 - E l a s t i c - P l a s t i c Work-Hardening w i t h i n a Non-Uniform Temperature F i e l d 3 ( i ) 3 ( i i ) 3 ( i i i ) Charlwood, v.Mises y . c , i n c r . f low, «-'^ <?;^ i-V= o Charlwood, v.Mises y . c , i n c r . f low, *J , y ^ o Bland [ 2 ] , Tresca y . c , assoc. flow, <x'-£o Table I - Table of Comparative S o l u t i o n s 93 M a t e r i a l p r o p e r t i e s were chosen to approximate the behavior of m i l d s t e e l u sing the model discussed i n s e c t i o n (6.2). A u n i a x i a l y i e l d l i m i t of 40 ks'x at absolute temp T = 450°F was chosen, reducing to 25 Ms'[ at T = 950°F. D e t a i l s are given i n Table I I . The s t r e s s e s and temperatures from the f i n i t e element s o l u t i o n s were taken as the average of the values from p a i r s of elements as shown i n Figure 5 . The average values were assumed to apply at the c e n t r o i d of the p a i r s . Displace ments were taken at the nodes of the elements. Figure 5 Averaging Technique f o r Stresses and Temperatures 94 i Uncoupled Case E l a s t i c a l l y Coupled Case General P l a s t i c Case i E' k / i n 2 3 0 , 0 0 0 3 0 , 0 0 0 3 0 , 0 0 0 V 0.3 0.3 0.3 k/in* 1,000 1,000 1,000 h k / i n 2 689.65 689.65 689.65 b k 2 / i n 4 - G F 0 . 0 0 . 0 0.00107 k / i n 533.46 533.46 641 .80 i n - k / i n - °F 0.3 0.3 0 . 3 k k/°F - sec 0.00432 0.00432 0.00432 y 0 . 0 0 . 0 0 . 9 Table I I M a t e r i a l P r o p e r t i e s f o r Computer S o l u t i o n s 95 7 . 2 P r e s e n t a t i o n of S o l u t i o n s Case 1 - E l a s t i c - P l a s t i c Work-Hardening with an I n i t i a l l y  Uniform Temperature F i e l d A hollow c y l i n d e r with i n t e r n a l pressure was used as i l l u s t r a t e d i n Figure 6. The problem was analysed f i r s t l y f o r the uncoupled isoth e r m a l , work-hardening case i n order to q u a l i t a t i v e l y check the r e s u l t s and provide a b a s i s of comparison f o r the coupled s o l u t i o n s . In the computer s o l u t i o n s the c y l i n d e r was i d e a l i s e d as shown i n Figure 7 and the parameters <*' , and X were set to zero. In the isothermal case the y i e l d f u n c t i o n reduced to the von Mises c o n d i t i o n . The l o a d i n g diagram i s as shown i n Figure 8. The computer s o l u t i o n was compared wi t h the r e s u l t s of Bland [ 2 ] who used Tresca's y i e l d c o n d i t i o n and i t s a s s o c i a t e d flow r u l e , and a l s o w i t h D*Isaf 4 ] who used von Mises' y i e l d c o n d i t i o n with a deformation theory flow r u l e . Bland's r e s u l t s were evaluated f o r k»= £j and K 0= 2 where ^ - 0 ^ = h0 ~ i n i t i a l y i e l d c o n d i t i o n and ^ i s the u n i a x i a l y i e l d s t r e s s . The l a t t e r case causes y i e l d i n g to s t a r t i n the c y l i n d e r at the same i n t e r n a l pressure f o r von Mises and Tresca y i e l d c o n d i t i o n s . These were the best comparisons which could be found. The r e s u l t s f o r t h i s isothermal case are shown i n Figure 9 f o r the s t a t e when the e n t i r e s e c t i o n had j u s t become p l a s t i c , (C-b ). The comparisons of the computer s o l u t i o n 96 w i t h those of Bland and D'Isa are shown. Considering the d i f f e r e n c e s i n the flow r u l e i n the case of D'Isa and the y i e l d f u n c t i o n i n the case of Bland, i t was f e l t t hat the r e s u l t s f o r t h i s i s o t h e r m a l , uncoupled, work-hardening case compared s u f f i c i e n t l y w e l l to show tha t the s o l u t i o n technique gave acceptable answers. This problem was then analysed on the computer wit h e l a s t i c coupling,<y o , ?l •= If-0 , and then w i t h e l a s t i c - p l a s t i c c o u p l i n g , , Yo , i n order to i l l u s t r a t e the e f f e c t s of thermo-mechanical c o u p l i n g . Short time steps were used so that the heat would not have time to d i s s i p a t e . The s t r e s s e s and displacements f o r these cases were so s i m i l a r to the isothermal case t h a t they could not he e a s i l y d i s t i n g u i s h e d on a graph to the s c a l e shown i n Figure 9 . However, the temperature r i s e i s shown f o r the f u l l y p l a s t i c s t a t e i n Figure 10. In the e l a s t i c coupled case i t may be seen that the temperature rose to +0.5°F on the i n s i d e and f e l l to -0.5°F on the outside due to volumetric compression and t e n s i o n r e s p e c t i v e l y . In the p l a s t i c coupled case the temperature rose to 4.1°F on the i n s i d e and f e l l to -0.5°F on the o u tside. The more s i g n i f i c a n t increase on the i n s i d e i s due to the p l a s t i c s t r a i n generating heat. The outside temperature i s the same as i n the e l a s t i c coupled case as there had not been any p l a s t i c s t r a i n there. These r e s u l t s are thus i n agreement, q u a l i t a t i v e l y , with expected behavior. 97 Figure 7 F i n i t e Element I d e a l i s a t i o n of Hollow C y l i n d e r f o r Cases 1 and 3 98 Figure 8 Load Diagram f o r Case 1 - I n t e r n a l Pressure 10, CD i_ CL 6 <y \-Legend: — Charlwood-elastic coupling Charlwood-plastic coupling 7.5 10.0 123 15.0 17.5 20.0 ~ 5 Rod i us r(ins.) Figure 10 Graph of Temperatures f o r Case 1 Figure 9 Graph of Stresses and Displacement f o r Case 1 100 Case 2 - Transient .Heat Flow, No Mechanical Coupling A s o l i d c y l i n d e r was used as i l l u s t r a t e d i n Figure 11. This problem was analysed i n order to assess the accuracy of the incremental s o l u t i o n f o r heat flow. In the computer s o l u t i o n s the c y l i n d e r was i d e a l i s e d as shown i n Figure 12 and the parameters , and X were set to zero. The l o a d i n g diagram i s shown i n Figure 13. The computer s o l u t i o n s were compared with the r e s u l t s of Carslaw and Jaeger f 3 J f o r the case where the surface temperature i s represented as a step f u n c t i o n . By applying the temperature load i n the computer s o l u t i o n over a small time step (one second was used) i t was f e l t t h a t a good comparison could be made f o r times i n excess of one second. The r e s u l t s f o r t h i s case are shown i n Figure 14:, f o r the computer s o l u t i o n and a l s o Carslaw and Jaeger's s o l u t i o n . I t may be seen t h a t the shape of the temperature curves at each time agree w e l l , i n d i c a t i n g that the s p a t i a l i d e a l i s a t i o n of the f i n i t e elements i s reasonable. There i s a s l i g h t time l a g between the computer s o l u t i o n and Carslaw and Jaeger's s o l u t i o n . However, t h i s time delay i s s m a l l and i t i s f e l t t h a t the l i n e a r i s a t i o n of the heat f l o w equation i n the time domain i s acceptable. Case 3 - E l a s t i c - P l a s t i c Work-Hardening w i t h i n a Non- Uniform Temperature F i e l d A hollow c y l i n d e r was analysed as i l l u s t r a t e d i n 101 Figure 12 F i n i t e Element I d e a l i s a t i o n of S o l i d C y l i n d e r f o r Case 2 102 <? <? o — 9 o 0 12 5 10 20 50 , 100 1000 ^ t(sec) Figure 13 Load Diagram f o r Case 2 - Transient Temperature Radius r(ins) Figure 14 Graph of Temperature f o r Case 2 103 Figure 6. The problem was analysed f i r s t l y f o r e l a s t i c c o upling and then f o r p l a s t i c c oupling as w e l l . In both cases there was a steady s t a t e temperature gradient across the c y l i n d e r . Comparison of these two cases shows the i n f l u e n c e of temperature on the y i e l d f u n c t i o n . In the computer s o l u t i o n the c y l i n d e r was i d e a l i s e d as shown i n Figure 7. For the e l a s t i c case the parameters T£L and % were set to zero. For the p l a s t i c case o(' , and if were a l l non-zero. The l o a d i n g diagram i s given i n Figure 15. In the e l a s t i c a l l y coupled case the computer s o l u t -ions were compared wi t h Bland's [ 2 ] r e s u l t s . Bland gives s o l u t i o n s f o r Tresca's y i e l d c o n d i t i o n w i t h a steady s t a t e temperature gradient and e l a s t i c c o u p l i n g . The y i e l d c o n d i t i o n i s independent of temperature although temperature s t r e s s e s are i n c l u d e d . The s t r e s s e s and displacements f o r t h i s case are given i n Figure 16 f o r the computer s o l u t i o n s to the e l a s t i c coupled and p l a s t i c coupled problems and a l s o f o r Bland's s o l u t i o n f o r the e l a s t i c coupled case with k.~ (Ty and a l s o K - 2 ^ //3 . The temperature d i s t r i b u t i o n s for' the computer s o l u t i o n s are given i n Figure 17. I t may be seen that the comparison between the computer s o l u t i o n and Bland's s o l u t i o n f o r the e l a s t i c a l l y coupled case i s q u i t e good. The d i f f e r e n c e s are s i m i l a r to the r e s u l t s of Case 1. I t may be noted t h a t the pressure to cause t o t a l y i e l d i s almost i d e n t i c a l f o r Cases 1 and 3. "the 104 temperature gradient having l i t t l e e f f e c t . However, the pressures a t the s t a r t of y i e l d i n g were found to be d i f f e r e n t . These are given i n Table I I I . Computer Bland [2 ] I n i t i a l F u l l Y i e l d Y i e l d c = a c = b I n i t i a l F u l l Y i e l d Y i e l d c = a c = b Uncoupled, isothermal: Case 1 E l a s t i c c o u p l i n g : Case 3 / P l a s t i c c o u p l i n g : Case 3 21.6 82 29.1 85 2 8 . 3 81 21.6 72 30.1 70 Table I I I - Comparison of Pressures f o r Y i e l d i n g Comparison of the curves f o r the computer s o l u t i o n s i n Figure 16 f o r the e l a s t i c and p l a s t i c coupled cases shows that the p l a s t i c coupled case y i e l d e d at a s l i g h t l y lower pressure. This i s due to the temperature dependence of the y i e l d f u n c t i o n i n the p l a s t i c coupled case. The c o u p l i n g e f f e c t s are small f o r the temperature gradient considered. This i s i n agreement with p r a c t i c a l experience. More important i s the r e s u l t t h a t the trend of the r e s u l t s i s c o n s i s t e n t with p h y s i c a l reasoning, i n d i c a t i n g that the model i s capable of rep r e s e n t i n g the c o u p l i n g e f f e c t s . Figure 15 Load Diagram f o r Case 3 I n t e r n a l Pressure and Temperature Radius r(ins) Figure 17 Graph of Temperature f o r Case 3 106 Radius r(ins) Figure 16 Graph of Stresses and Displacement f o r Case 3 10? CHAPTER 8  SUMMARY A t h e o r e t i c a l model has heen given f o r thermo-e l a s t o - p l a s t i c behavior which i s c o n s i s t e n t with the laws of thermodynamics. The theory includes a general form f o r the p l a s t i c c o n s t i t u t i v e p o s t u l a t e s , i n c l u d i n g dependence on s t r e s s , s t r a i n and temperature, F l e x i b i l i t y of choice of flow r u l e , y i e l d and hardening f u n c t i o n s i s maintained. The problem may be formulated with f o r c e , m a t e r i a l displacement , temperature and heat flow boundary c o n d i t i o n s . I t has been shown that the f i e l d equations may be posed i n v a r i a t i o n a l form using a conventional minimal f u n c t i o n a l f o r the thermo-elastic case and using a new i t e r a t i v e f u n c t i o n a l f o r the general t h e r m o - e l a s t o - p l a s t i c case. F u n c t i o n a l s have been presented f o r both cases. Approximate s o l u t i o n s have been obtained using the f i n i t e element method. Results f o r a simple c o n s t i t u t i v e law using a constant s t r e s s and temperature f i n i t e element have been presented. These have been compared w i t h e x i s t i n g s o l u t i o n s f o r some l i m i t i n g cases and are shown to give good r e s u l t s . The features of temperature dependence of the y i e l d f u n c t i o n and heat generation due to p l a s t i c s t r a i n are di s p l a y e d i n the r e s u l t s . The cou p l i n g e f f e c t s are s m a l l f o r the m a t e r i a l and temperature ranges considered. However, f o r other m a t e r i a l s they may be more s i g n i f i c a n t . In the s o l u t i o n 108 of problems w i t h thermal boundary c o n d i t i o n s i t appears to be more convenient to solve the problem i n coupled form than to use the more common uncoupled method. The f i n i t e element method was found to be s u i t a b l e f o r s o l u t i o n of the equations by the v a r i a t i o n a l methods. The i t e r a t i v e scheme converged r a p i d l y i n a l l the problems considered. The computer times were p r a c t i c a l . I t appears p o s s i b l e to use a more r e f i n e d f i n i t e element to obtain more accurate s o l u t i o n s i f de s i r e d . However, i f s t r e s s e s and temperature are v a r y i n g i n each element then the y i e l d and hardening f u n c t i o n s w i l l need to be recorded as v a r i a b l e s over the domain of each element. This may be p o s s i b l e although i t w i l l probably be i n v o l v e d . The advantages of the use of a more r e f i n e d element over sub-d i v i s i o n of simple elements appears s l i g h t . Using the method of t h i s study i t i s p o s s i b l e to study the e f f e c t s of v a r i o u s c o n s t i t u t i v e p o s t u l a t e s . Using the analogy between heat flow and f l u i d f l o w i n a porous medium i t appears p o s s i b l e to adapt the method to solve problems of f l u i d f low i n porous e l a s t o - p l a s t i c bodies. 109 BIBLIOGRAPHY B i o t , M. A., "T h e r m o - e l a s t i c i t y and I r r e v e r s i b l e Thermodynamics", J o u r n a l of Appl i e d P h y s i c s , v. 27, n. 3, PP. 240-253, March, 1956. Bland, D. R., " E l a s t o - P l a s t i c Thick-Walled Tubes of Work-Hardening M a t e r i a l Subject to I n t e r n a l and E x t e r n a l Pressures and Temperature Gradients", J o u r n a l of Mechanics and Physics of S o l i d s , v. 4, pp. 204-229, 1956. Carslaw, H. S., and J . C. Jaeger, Operational Methods  i n Applied Mathematics, Dover, 1945, p. 123. D'Isa, F. A., Mechanics of Metals. Addison Wesley, 1968, pp. 212-215. Drucker, D. C.,"Some I m p l i c a t i o n s of Work Hardening and Id e a l P l a s t i c i t y " , Quarterly J o u r n a l of Appl i e d  Mathematics, v. 7, n. 4, pp. 411-418, Jan. 1950. F e l l i p p a , C. A., Refined F i n i t e Element A n a l y s i s of Li n e a r and Non-linear Two-Dimensional S t r u c t u r e s Report No. 6 6 - 2 2 , Department of C i v i l Engineering, U n i v e r s i t y of C a l i f o r n i a , 1966. Fung, Y. C , Foundations of S o l i d Mechanics, P r e n t i c e H a l l , 1965, p. 360. Green, A. E., and P. M. Naghdi, "A General Theory of an E l a s t i c - P l a s t i c Continuum, Archives of R a t i o n a l  Mechanics and A n a l y s i s , n. 18, pp. 251-281, 1965. 110 9. Green, A. E., and P. M. Naghdi, "A Thermodynamic Development of E l a s t i c - P l a s t i c Continuum", I.U.T.A.M. Symposium Proceedings. Vienna, 1966. pp. 117-131. y ' 10. H i l l , R., The Mathematical Theory of P l a s t i c i t y . Oxford, 1950. J L 11. K e s t m , J . , "On the A p p l i c a t i o n of the P r i n c i p l e s of Thermodynamics to St r a i n e d S o l i d M a t e r i a l s " , I.U.T.A.M. Symposium Proceedings. Vienna, 1966. pp. 177-212. i 2 . Key, S.W., A Convergence I n v e s t i g a t i o n of the D i r e c t S t i f f n e s s Method. Ph. D. Thesis, U n i v e r s i t y of Washington, 1966. 13. K o i t e r , W. H., "General Theorems i n the Theory of P l a s t i c i t y " , Progress i n S o l i d Mechanics I. Ed. R. H i l l , North-Holland. I960. 14. Lee, E. H., " F i n i t e - S t r a i n E l a s t i c - P l a s t i c Theory with A p p l i c a t i o n to Plane-Frame A n a l y s i s " , J o u r n a l of Applie d Physics, v. 38, n. l , pp. 19-27, Jan. 1967. 15. Melosh, R. J . , Development of the S t i f f n e s s Method to  Define Bounds on E l a s t i c Behavior of S t r u c t u r e s . Ph. D. Thesis, U n i v e r s i t y of Washington, 1962. 16. M i k h l i n , S. G., V a r i a t i o n a l Methods i n Mathematical Physics. Pergamon Press, 1964. 17. M i k h l i n , S. G., The Problem of the Minimum of the Quadratic F u n c t i o n a l . Holrien Day r 1 0 6 5 . I l l 18. Naghdi, P. M., " S t r e s s - S t r a i n R e l a t i o n s i n P l a s t i c i t y and T h e r m o p l a s t i c i t y " , Proceedings of 2nd  Symposium on Naval S t r u c t u r a l Mechanics, Pergamon, i960, pp. 121-169. 19. O l i v e i r a , E. R. de A., " T h e o r e t i c a l Foundations of the F i n i t e Element Method", I n t e r n a t i o n a l J o u r n a l of S o l i d s and S t r u c t u r e s , v. 4, pp. 929-952, 1968. 20. Timoshenko, S. P., Strengths of M a t e r i a l s I I , van Nostrand, 3rd Ed., 1956, p. 519. 21. T o n t i , E., " V a r i a t i o n a l Formulation f o r L i n e a r Equations of Mathematical Ph y s i c s " , Some Papers  about V a r i a t i o n a l P r i n c i p l e s , P o l i t e c h n i c o d i Milano, 1968. 22. T r u e s d e l l , C , "Thermodynamics of Deformation", Non-Equilibrium Thermodynamics; V a r i a t i o n a l  Methods and S t a b i l i t y . Ed. R. J . Donnelly et a l . Proceedings of Symposium at U n i v e r s i t y of Chicago, U. of Chicago Press, 1966. 

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