Technische Universitat Chemnitz Sonderforschungsb ereich Numerische Simulation auf massiv parallelen Rechnern Detlef Michael Mathias Meisel Some remarks to large deformation elastoplasticity continuum formulation Preprint SFB Abstract The continuum theory of large deformation elastoplasticity is summarized as far as it is necessary for the numerical treatment with the FiniteElementMethod Using the calculus of mo dern dierential geometry and functional analysis the fundamental equations are derived and the pro of of most of them is shortly outlined It was not our aim to give a contribution to the development of the theory rather to show the theoretical background and the assumptions to b e made in state of the art elasto plasticity PreprintReihe des Chemnitzer SFB SFB Septemb er Contents Intro duction Some Dierential Geometry Kinematics of Finite Deformations The stress tensor and balance of momentum Balance of energy and principle of virtual work The second law of thermo dynamics Linearization of nonlinear elasticity Multiplicative Elastoplasticity at Finite Strains App endix Authors addresses Detlef Michael TU Chemnitz Fakultat fur Mathematik D Chemnitz Mathias Meisel Gesellschaft fur Fertigungstechnik und Entwicklung EV Institut fur Werkzeugtechnik und Qualitatsmanagement Geschatsb ereich Berechnen und Prufen Lassallestr D Chemnitz httpwwwtuchemnitzdesfb Intro duction From a theoretical p oint of view a physical body can b e treated as a set B of material points representing the atoms or molecules of the bodys material Considering this body as a continuum the theory shortly outlined in the following applies For more details see Tri First we intro duce a system U of op en subsets of B with the following prop erties a U and B U b Finite intersections of subsets from U are again subsets of U c Any unions of subsets from U are again subsets of U U is called a topology of B and A U is called neighbourhood of a p oint x B if x A x x B is called a Hausdor space if for each two p oints x B and y B there exist neighbour T hoods A and A with A A ie the topology of B is rich enough to separate the x y x y p oints of B The Hausdor space B is said to have the dimension n if for each x B there exists a n neighbourhood A that can b e mapp ed onto an op en subset of R by a bijective mapping x k called chart or local coordinate system with comp onents fx g k n 1 If this mapping can b e cho osen to b e C and orientation preserving the ndimensional 1 k Hausdor space B equipp ed with the chart fx g is called a ndimensional C manifold The mathematical assumptions made ab ove corresp ond to the common physical under standing of bodies and physical spaces So from a general p oint of view including also shells ro ds and exotic materials like liquid crystals a physical body B and the physical space S containing B can b e considered to b e sp ecial cases of manifolds To have a unied approach as well as for conceptual clarity it is usefull to think geomet rically and to represent b o dies in terms of manifolds MH Some Dierential Geometry The tangent space 1 Let M b e a ndimensional C manifold and let x M Then the tangent space T M to x n M at x is the vector space R of vectors regarded as emanating from x So vectors like the velocity and the acceleration see ch can b e understo o d as elements of T S to S at x 1 x S where S denotes the physical space describ ed as C manifold M will b e M to T M is called the cotangent space Its elements T The dual space T x x x M is dual called covectors A covector is a functional x T M R In this sence T x x to T M A covector is said to b e covariant and a vector to b e contravariant x 1 1 This is not the most general denition of a C manifold but it should b e suitable for the understandig of the present sub ject for more details see Tri 2 After the intro duction of cotangent spaces T M a vector v also can b e dened as a functional x vx T M R x Tensors p at x M is a multilinear mapping A tensor t of the typ e q p copies z t R T M T M T M T M x x x x z q copies It is said that t is contravariant of rank p and covariant of rank q i M b e the base vectors in T M and the dual base vectors Let fe g T M and fe g T x j x x in T M resp ectively Then the comp onents of t are dened by x i i p 1 i i p 1 t te e e e j j j j q 1 q 1 and i i p j p 1 p j 1 q tw w v v t w w v v q j j i i q q p 1 1 i i a b holds for all w w e T M and v v e T M j b x a x j The Riemannian metric For x M let g b e a covariant tensor of rank ie a tensor of typ e with the prop erties gu v gv u u v T M x gu u u T M x With this symmetric and p ositive denite and therefore invertible metric tensor g the functional hu vi gu v is an inner product on T M Written in comp onents it reads x a b hu vi u v g ab Intro ducing such a tensor g for each x M we get a Riemannian metric on M Tensor and vector elds The asso ciated tensor and vector elds b a For e T M the associated vector eld e T M is dened by its b a x x a ab ab comp onents g with g denoting the comp onents of g The associated covector b a b b eld u u e T M to u u e T M is dened by u g u Similarly t means a b x a ab x the tensor associated to t with all indices lowered and t with all indices raised esp ecially g g and g g Scalar pro ducts Using this notation and equation we dene the dual paring b etween elements of T M and T M and the scalar product i h in T M by u v hu vi for u T M x x x x v T M and iu vh hu v i for u v T M These denitions yield to the identity x x hu v i u v iu vh for any u v T M as well as hu vi u v iu v h for any x u v T M x The dual base e e a c b In two or three dimensions the dual base is dened by the vector pro duct e e e e a c b a a Due to this denition the equation e e holds and once the base fe g is given the b b b a base vectors e are uniquely dened Since we deal here with spaces of arbitrary dimenson we have to go another way a Let the base fe g in T M b e given On page we just intro duced and used a base fe g b x a a a This M From now on we p ostulate that the base vectors fe g fulll e e in T b b x a uniquely determins the e since in case of n dimensions n conditions have to b e fullled by cho osing n comp onents of n covectors Note that this do esnt mean to have any orthonormalized system in T M or in T M x x The covariant derivative Let x M The covariant derivative of a vector v T M along a vector w T M is a x x bilinear mapping dened by g r ad v T M T M T M fullling x x x w g r ad v f g r ad v w f w g r ad f v f g r ad v rf w v w w f f a a for all scalar functions f where rf e T M and rf w w Dening a a x x x this for all x v and w have to b e regarded as vector elds and we get a connection on c the manifold M The Christoel symbols of this connection on M with the coordinate ab c c a system fx g are dened by g r ad e e Using the prop erties for v v e and a c a ab e b h i a b b a v g r ad follows Intro ducing the w w e the identity g r ad v w e rv e e b a b a e w b Christoel symbols we get c v b b c a c w e w e v v g r ad v c c jb ab w b x c v a c c v with v ab jb b x Dening for each v T M another tensor g r ad v of typ e with comp onents x a v a c a a v v g r ad v jb cb b b x the covariant derivative can b e expressed by g r ad v g r adv w w We dene the covariant derivative of a covector u T M along a vector w T M via the x x 3 T T 1 T 2 T As an example in two dimensions e e e and e 1 2 fullls this conditions f f f 4 a b ac b b ca b c rf w hrf wi rf w g g w g w g g w ab c ab c ab c b x x x 5 Here and in the following we use the symb ol originally dened for the dual paring also in the a a b sense of g r adv w g r adv w since the comp onents of the resulting vector can b e regarded as b dual parings of the columns of g r adv with w associated vector elds a b u u e w w e g r ad u g r ad u a b w w ac c Using the rules from page the identity g g the prop erties and the repre ab b f sentation some calculus shows that the comp onents r of r g r ad u r e will b e f f w cd u g f c c ad b Applying to g and taking for the g g g u r w cf cf cf d f ab ab b b x x cd d cd g g cf g g f cf cd rightmost term into account that g g we nd cf b b b b x x x x g g u u g af bf f f ad ad c b ab b again and u g u g g w r w a d d ca f f b b f b b x x x x x the symmetry of g was used So nd the analogon to for covectors u c b c b c a g r ad u w e u w e u cjb a cb w b x u a c u with u a cjb cb b x p Regarding a tensor t of typ e as a multivector of p vectors and q covectors it is natural q to dene the comp onents of the covariant derivative of a tensor by a generalization of and ab e ab v g r ad t t cdje v cd ab t ag f b g b a ab f ab e cd t t t t v e g e cd f e cd de cg ce f d x The Riemannian space If a connection is dened on a manifold M M is called an ane space Intro ducing a Riemannian metric g on an ane space M M b ecomes torsion free ie the Christoel c c symbols are symmetric A torsion free ane space is called a Riemannian ab ba space As can b e shown Tri in a Riemannian space M for each x M a lo cal Carte i and sian coordinate system with the base vectors i T M can b e found with j x j k gi i Therefore