Materials Science Research International, Vol.6, No.2 pp. 80-87 (2000) General paper A Notion of Strain and Stress Tensor for Finite Deformation of Elastic-Plastic Continua Shigeru OGAWA, Shuichi HAMAUZU and Toshio KIKUMA Technical Development Bureau, Nippon Steel Corporation, 20-1 Shintomi, Futtsu, Chiba 293-8511, Japan Abstract: As a notion of strain compatible with the theory of plasticity having the plastic potential of von Mises type with the normality principle as flow rule, the strain tensor is defined by integrating the rate of deformation tensor considering the material spin with the rotation tensor. For a group of elastic-plastic continua exhibiting infinitesimal elastic and finite plastic deformation, elastic-plastic decomposition of the strain tensor is established, and it is proved that the elastic strain tensor has the exact physical meaning and that the stress tensor can be calculated through the generalized Hooke's law with the elastic strain tensor without any ambiguity. Key words: Elastic-plastic continua, Finite strain tensor, Rate of deformation tensor, Infinitesimal elastic deformation, Finite plastic deformation, Generalized Hooke's law 1. INTRODUCTION strain relationship. Present work is in the line of Watanabe's discussion. The motivation, however, lies Advantage of finite element analysis is widely rec- in the strain space plasticity proposed by Naghdi et ognized not only in the field of machine and structure al. [1-4],and the goal is to propose more rigorous and designing but also in the field of the research and engi- practical numerical scheme for an implicit finite ele- neering of mechanical working processes accompanied ment formulation than conventional ones. with large deformation. To save the computational As a preparatory step for establishing an implicit cost for such problems with large deformation, an im- finite element formulation with an explicit definition plicit finite element method is preferable because in of finite strain, this paper presents a notion of strain each time increment it seeks the solution of displace- and stress tensor which is capable of describing finite ment field which satisfies the equilibrium equations at deformation of elastic-plastic continua. the end of the time increment and hence fairly large time increment can be adopted. In order to consider 2. EXISTING NOTIONS OF FINITE the equilibrium equations at the end of the time incre- ment, rigorous notion of finite strain is needed. Theo- STRAIN retically speaking, to define the notion of finite strain, In this section, the existing notions of finite strain the relationship between displacement field and finite are investigated in view of usefulness for numerical strain should be established. However, implicit finite schemes such as finite element analysis. Classically, element codes, which are currently available commer- there are two notions of finite strain currently available cially, carefully avoid explicit definition of finite strain with clear definition associated with the displacement by directly obtaining stress tensor through integrat- field. One is the Lagrangian strain and the other is the ing constitutive equations for the rate of stress tensor Eulerian strain. Among them, the Lagrangian strain over the time increment considering the adequate ma- is of particular importance, for it is adopted as the no- terial spin. With this type of formulation, technical tion of finite strain in the theory of plasticity proposed problems arise, for example, in handling residual er- by Naghdi et al. [1-4],who have restructured the classi- ror in equilibrium equations at the end of the time cal theory of plasticity using the notion of strain space increment. Solutions to these technical problems of- spanned by the finite strain tensor. The simplest form ten go into empirical methods, because definition and of constitutive equations for plastic strain rate in the calculation of finite strain associated with displace- Naghdi's theory of plasticity can be expressed in the ment field has been bypassed. Typical one of such following form: problems is unreasonable stress behavior observed for simple shear deformation when Jaumann rate of stress tensor is utilized. Although there have been a lot of epAB=λ τSAAB, (1) discussionsmade about this problem[8], most of them seem to have been devoted for stress behavior without where epAB are plastic components of the Lagrangian considering stress-strain relationship in the context of strain rate, τSAB are deviatoric components of the sym- finite deformation. Recently Shizawa[9] made ther- metric Piola-Kirchhoff stress tensor, and λ is a scalar momechanical approach to this problem, and Watan- parameter. Throughout the paper, lower case Latin abe [10,11] made discussion in the context of stress- indices are associated with the spatial coordinates xi Received September 25, 1998 Accepted March 6, 2000 80 Strain and Stress Tensor for Finite Deformation and assume the values 1, 2, 3. Similarly, upper case of formulating the constitutive equations for plastic Latin indices are associated with the material coordi- deformation is considered to adopt the plastic poten- nates XA and take the values 1, 2, 3. We also adopt tial of von Mises type with the normality principle as the usual convention of summation over repeated in- flow rule. With this type of formulation, like Eqs. (1), dices. Without any further statements, the material plastic strain rate tensor becomes proportional to the coordinates XA are assumed to indicate the position deviatoric stress tensor, and this inevitably leads trace of a generic particle of a body in the reference config- of the plastic strain tensor to zero; Equations (1) are uration, which is also assumed to be identical to the one of the simplest examples. Hence the fact that initial configuration. Eqs. (1) are not compatible with the conservation of Recalling the definition of deviatoric stress tensor, volume means that the Lagrangian strain is not suit- we can readily derive the following equation from able for describing plastic deformation accompanied Eqs. (1): by finite strain. Now, it is the Eulerian strain to be tested. For the epAA=0. (2) deformation described by Eqs. (4) when t=1, the trace of the Eulerian strain tensor is given by In the context of the infinitesimal theory of plastic- ity, Eq. (2) ensures conservation of volume throughout eE11+eE22+eE33=3/8-1/2-1/2=-5/8. (6) the plastic deformation. In the context of finite strain plasticity, however, there have been no literatures in This means that the Eulerian strain is not adequate for which the conservation of volume is confirmed start- describing plastic deformation accompanied by finite ing from Eq. (2). The conservation of volume due to strain either. plastic deformation is one of the most important stip- Recently Heidushke[12] introduced the logarith- ulations even in the context of finite strain plasticity, mic strain space description into Naghdi's theory of and hence we shall check whether Eq. (2) is compat- plasticity. The logarithmic strain tensor does not have ible with the conservation of volume. Since Eq. (2) above mentioned difficulty concerning volume preserv- always holds according to Eqs. (1), the followingequa- ing plastic deformation, and its definition has clear tion holds after any arbitrary deformation: physical background. In this paper the logarithmic strain is deemed exact strain definition, and more epAA=0. (3) practical strain definition is introduced for numerical The question now becomes whether Eq. (3) is com- purpose. patible with the conservation of volume. As one of 3. DEFINITION OF A FINITE STRAIN the simplest deformation, consider the deformation of TENSOR uniaxial elongation. Suppose we have a cubic with the side length L in the reference configuration, and the The rate of deformation tensor Dij is defined by cubic is subjected to the motion described as follows: Dij=1/2(vi,j+vj,i), (7) x1=(1+t)X1, x2=X2/√1 +t, x3=X3/√1+t, (4) where vi,j means the partial derivative of velocity vec- where t is the time variable. This motion is uniform tor vi of a typical particle in a body with respect to the elongation in the x1-direction. Though elastic strain spatial coordinates xj. The rate of deformation ten- may be induced during loading, the elastic strain com- sor is frequently used as a notion of strain rate tensor pletely vanishes after unloading due to the uniformity in the context of finite deformation. It is well known of the deformation. Hence the deformation described that zero trace of the rate of deformation tensor repre- in Eqs. (4) can be regarded as purely plastic deforma- sents isochoric motion. This fact is readily understood tion. From Eqs. (4), Jacobian is calculated as J=1 from the following relationship between the material everywhere, which assures J=0 i.e. the motion is derivative of Jacobian J and the rate of deformation isochoric. Having these observations, the motion is tensor: considered as an kinematically admissible plastic de- J=Jvi ,i=JDii. (8) formation. Considering the deformation when t=1, trace of the the Lagrangian strain is given by Therefore, in view of the conservation of volume dur- ing plastic deformation, it is confirmed that the rate of deformation tensor is a promising candidate for plas- e11+e22+e33=3/2-1/4-1/4=1. (5) tic strain rate tensor. Thus, in this paper, the rate of deformation tensor is adopted for describing plastic From this result, it is concluded that the Eqs. (1) strain rate tensor. Question now becomes how physi- are not compatible with the conservation of volume cally valid notion of finite strain is obtained from the through plastic deformation. In other words, the con- rate of deformation tensor. stitutive equations (1) for plastic strain rate are not Define the following notion of strain through inte- adequate for describing the volume preserving plastic grating the rate of deformation tensor: deformation.