A Notion of Strain and Stress Tensor for Finite Deformation of Elastic-Plastic Continua

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 increment, 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 plasticity, 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 during 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. Currently, as far as the isotropic materials are concerned, the simplest and reasonable way εij(XK,t)=RiA(XK,t)RjB(XK,t)εAB(XK,t) (9)

81 Sigeru OGAWA, Shuichi HAMAUZU and Toshio KIKUMA

and cause the direction change of the vector embedded in the material is of our concern in this discussion. Us- εAB(XK,t)=∫t0RmA(η)RnB(η)Dmn(η)dη, (10) ing the relationship R-1=Rt where Rt denotes the transposed matrix of R, Eqs. (11) can be written as where RiA is the rotation tensor obtained by the polar decomposition of the deformation gradient, and dx1(t)=R(t)Rt(η)dx1 (η), (12) the second argument t or ƒÅ is the time variable. ƒÅ=0 dx1(η)=R(η)Rt(t)dx1(t). is associated with the reference configuration before deformation, and current configuration is the one at Since the rotation tensor R represents the local rigid η=t. It should be noted that strain tensors defined by body rotation, not only the vector dx1 but also all vec- Eqs. (9) and (10) describe strain accumulated at a typ- tors embedded in the material are subject to the rigid ical particle located by the material coordinates XK, body rotation R before or after the pure deformation. and the rate of deformation tensor Dmn is integrated Hence, if we observe the material deformation with as a function of position in the reference configura- reference to the base vectors rotating in accordance tion and time. The rotation tensors RmA, RnB are with Eqs. (12), pure deformation can be extracted and also functions of position in the reference configura- observed. It follows that, to evaluate the finite strain tion and time. In this paper, the strain tensor defined as a consequenceof the history of the rate of deforma- by Eqs. (10) is called the referential strain tensor to tion tensor, the followingentities should be integrated: distinguish from the current strain tensor or, simply, the strain tensor defined by Eqs. (9). Aim(t,η)Ajn(t,η)Dmn(XK,η), (13) This definition of strain can be seen in the work of Storen and Rice [5], in which it is introduced as not where a strain-measure but a deformation-measure because Aim(t,η)=RiA(t)RmA(η). (14) it is path-dependent. In the present work, however, it will be shown that the entities defined by Eqs. (9) and (10) can be treated as strain-measures under the Thus the definition of the finite strain tensor, Eqs. (9) certain circumstances. and (10), has been motivated.

3.1. Physical Interpretation of the Strain Ten- sor 3.2. Evaluation of the Strain Tensor for Uniax- Since the rate of deformation tensor Dmn is re- ial Isochoric Motion garded as strain rate tensor, one can expect that strain In Chapter 2, the existing notions of finite strain tensor can be obtained by integrating the rate of de- are tested by one of the simplest isochoric motions formation tensor in some way. In the context of finite described by Eqs. (4). Hence the finite strain tensors strain continuum mechanics, however, integration of defined by Eqs. (9) and (10) are tested below in the each component of Dmn with respect to time is not same manner. always physically meaningful because there may be In case of this uniaxial motion, the deformation gra- significant local rigid body rotation due to large de- dient is expressed by diagonal matrix, and the rota- formation. In Eqs. (9) and (10) the rotation tensor is tion tensor becomes the unit matrix. As a result the adopted as the adequate material spin for integrating strain tensor defined by Eqs. (9) and (10) reduces to the history of deformation at a typical particle of the time integration of each component of the rate of de- body. This definition of strain can be interpreted in formation tensor, and the followingcomponents of the the following way. strain tensor are obtained: The strain component ƒÃ11, for example, has to be the sum of deformation which has been received in ε11(t)=ln|1+t|, the pertinent period of time by the infinitesimal line ε22(t)=ε33(t)=-1/2ln|1+t|, (15) segment dx1(t) currently lying in the x1 direction. ε12(t)=ε23(t)=ε31(t)=0. Suppose dx1 is embedded in the deforming material, then the direction of the vector dx1(ƒÅ) has not always From the results in Eqs. (15), the strain tensors de- been in the x1 direction. Hence the rate of defor- fined by Eqs. (9) and (10) can be interpreted as ex- mation tensor has to be integrated while tracing the tended notion of logarithmic strain which is commonly direction change of the vector dx1(ƒÅ) to obtain the used for uniaxial finite deformation. Equations (15) correct notion of strain tensor. also indicate that, for the isochoric motion given by Generally speaking, the rotation of the vector em- Eqs. (4), ƒÃii=0 holds for the strain tensor. Starting bedded in the deforming material can be expressed by from the definition (9) and (10), this characteristic is the rotation tensor R, and then the following relations generalized for arbitrary isochoric motions as follows: are obtained:

dx1(t)=R(t)dX1, εii(t)=RiA(t)RiB(t)∫t0RmA(η)RnB(η)Dmn(η)dη dx1(η)=R(η)dX1, (11) where dX1 is the reference configuration of the vector =∫t0Dmm(η)dη . (16) dx1, and the pure deformation has been neglected be-

82 Strain and Stress Tensor for Finite Deformation

Since Dmm=0 always holds for arbitrary isochoric be recognized if one observes the deformation only motions, ƒÃii=0 also holds for any of isochoric mo- through the rate of deformation tensor, which can- tions. not distinguish the simple shear from the pure shear. It is thus concluded that the rotation tensor plays a 3.3. Evaluation of the Strain Tensor for the Mo- substantial role in calculating physically meaningful tion of Simple Shear strain tensor in the context of the finite deformation. Now consider another typical motion, simple shear, Since the referential strain tensor defined by expressed by the following equation: Eqs. (10) can be interpreted as the accumulated strain observed in the reference coordinate system, it should x1=X1+tX2, x2=X2, x3=X3, (17) accord with the Green's deformation tensor C, or more directly, with the right stretch tensor U. Re- where t is the time variable. calling that the strain tensor defined by Eqs. (10) co- For this motion, the deformation gradient F and incides with the logarithmic strain in case of uniaxial the rate of deformation tensor D are given as follows: deformation, it may be instructive to observe the referential strain tensor ƒÃ with reference to the principal axes of the right stretch tensor U, and to compare it with the logarithmic strain obtained from the principal stretches which are the eigenvalues of U. In order to do this, firstly find the eigenvalues and the eigenvec- The right stretch tensor U is obtained by extracting tors of U, and secondly diagonalize U obtaining the the square root of the Green's deformation tensor C= diagonalized right stretch tensor U. Between U and FtF. And then finding the rotation tensor R through U, the following relationship holds with the relevant the relation R=FU-1, polar decomposition of the orthogonal matrix Q: deformation gradient F completes. Thus R and U are given by U=QUQt. (22)

According to the above-mentioned procedure, U and Q are obtained as follows:

Using Eqs. (10) and Eqs. (18), (19), the referential strain tensor ƒÃ is obtained as

where

α=1/√t2+4. (25)

Hence the components of the strain tensor defined by Eqs. (9) are given as follows:

ε11=(t2-4)/(t2+4)ln(1+t2/4) +4t/(t2+4)[2tan-1(t/2)-t/2],

ε12=4t/(t2+4)ln(1+t2/4) -(t2-4)/(t2+4)[2tan-1(t/2)-t/2] (21) ,

ε22=-ε11, ε21=ε12,

ε33=ε13=ε23=ε31=ε32=0.

Equation (20) suggests that the motion of simple shear described by Eqs. (17) consists of pure shear within the X1-X2 coordinate plane, compression in the X1 direction, and extension in the X2 direction. This is readily understood by the fact that, in the X1-X2 coordinate plane, an initially regular square becomes a parallelogram after the motion of simple shear. In spite of this rather apparent observa- Fig. 1. Comparison of strain components for the tion, above-mentioned feature of simple shear cannot motion of simple shear.

83 Sigeru OGAWA, Shuichi HAMAUZU and Toshio KIKUMA

The referential strain tensor observed with reference the referential strain tensor does not exactly coincide to the principal axes of the right stretch tensor U is with the logarithmic strain tensor for the motion of then given by ε'=QtεQ, which renders simple shear. The above discussion suggests that the referential ε'11=-α{4tan-1(t/2)-t[1-ln(1+t2/4)]}, strain tensor holds the exact physical meaning in case

ε'12=α{t[2tan-1(t/2)-t/2]-2ln(1+t2/4)}, of infinitesimal deformation with finite rotation, because in this case the shear deformation, which has ε'22=-ε'11, ε'21=ε'12, caused the difference between the embedded coordi- ε'33=ε'13=ε'23=ε'31=ε'32=0. nate axes and the corotational coordinate system, is (26) infinitesimal. The proof of this is given below. Let the coordinate system XA be defined by the Consider arbitrary infinitesimal deformation with fi- eigenvectors of the right stretch tensor U. Then the nite rotation. Let the right stretch tensor U be ex- logarithmic strain in X2 direction is given by ln(U22), pressed by and it should be compared with ƒÃ'22. The shear com- U=I+ε, (27) ponent ƒÃ'12 should be zero to have exact accordance with U. Figure 1 shows the result of the compari- where I is the unit tensor and all components of the son. Although the referential strain tensor does not strain tensor ƒÃ is infinitesimal because the deformation exactly coincide with the logarithmic strain calculated is assumed to be infinitesimal. Substituting Eq. (27) through the right stretch tensor, it gives fairly good into the equation of the polar decomposition, the de- approximation of the logarithmic strain. It gives prac- formation gradient is given by tically the same value of strain as the logarithmic F=R(I+ε). (28) strain tensor does until the time t=1 when the deformation is quite large, ln(U22)=0.5. Using Eq. (28), the velocity gradient tensor L is given as follows: 3.4. Evaluation of the Strain Tensor for In- finitesimal Deformation with Finite Rotation L=FF-1=RRt+Rε(I+ε)-1Rt. (29) Before proceeding to the evaluation of the strain tensor for arbitrary infinitesimal deformation with fi- Since the strain tensor ƒÃ is composed of infinitesimal nite rotation, discussion would be worthwhile about components, neglecting the higher order of ƒÃ, the fol- the reason why the strain tensor defined by Eqs. (10) lowing equation can be obtained: does not exactly coincide with the logarithmic strain tensor for the motion of simple shear. Figure 2 is a (I+ε)-1=I-ε+O(ε2)=I-ε. (30) schematic diagram of the reference and the current configurations of a small generic part of a body. The Substituting Eq. (30) into Eq. (29), and neglecting the deformation is well analyzed according to the polar higher order of ƒÃ, the velocity gradient can be written decomposition of the deformation gradient F=RU. as The eigenvectors A, B expressing the directions of principal stretches of the right stretch tensor U are L=RRt+Rε(I-ε)Rt=RRt+RεRt. (31) considered to be embedded in the part of the body. They rotate for the amount given by the rotation ten- Since R is orthogonal, ƒÃ is symmetric, and RRt is sor R after receiving the stretches according to the skew-symmetric, the rate of deformation tensor D is eigenvalues of the right stretch tensor U, and then given by they become the vectors a, b in the current configu- D=(L+Lt)/2=RεRt. (32) ration. On the other hand, if the coordinate axes X1, x2 in the reference configuration are considered to be Now using the definition of the referential strain ten- embedded in the body, before rotating due to the ro- sor, Eqs. (10), and Eq. (32), the referential strain ten- tation tensor R, they receive not only the stretches sor ƒÃ is calculated as follows: but also some shear deformation because, in general, their directions in the reference configuration do not ε=∫t0Rt(RεRt)Rdη=∫tεdη=ε. (33) coincide with the ones of the eigenvectors of the right stretch tensor U. Hence the relative angles between the original coordinate axes and the eigenvectors ob- It is concluded that the referential strain tensor co- served in the current configuration differ from the ones incides with the strain tensor defined from the right observed in the reference configuration. It means that stretch tensor in case of infinitesimal deformation with the angle between the embedded coordinate axes X1, finite rotation, and hence the current strain tensor X2 changes due to the deformation. It is, therefore, defined by Eqs. (9) is just another expression of the concluded that, even if the rate of deformation tensor strain tensor with reference to the current coordinate is observed with reference to the corotational coor- system. dinate system defined by the rotation tensor, which Moreover it is found from Eq. (32) that one cannot is the idea of the definition of the referential strain capture the pure deformation if one observes the de- tensor given by Eqs. (10), the corotational coordinate formation only through the rate of deformation tensor system does not exactly coincide with the coordinate because it contains not only the correct strain but also axes embedded in the body. This is the reason why the rigid body rotation.

84 Strain and Stress Tensor for Finite Deformation

(a) Reference configuration. (b) Current configuration.

Fig. 2. Schematic diagram of the reference and the current configurations of a small part of a body.

4. ELASTIC-PLASTIC DECOMPOSITION Since U is symmetric, the relationship Up=(Up)t OF THE STRAIN can be obtained from above two equations. Thus elastic-plastic decomposition of the deformation gra- The elastic-plastic decomposition of the strain ten- dient can be written as sor is discussed in this section, which is indispensable for elastic-plastic finite element analysis. The rate of F=RU=RUeUp=R(Up)tUe. (38) deformation tensor D is assumed to be decomposed into the elastic component De and the plastic compo- Generally speaking, since Up is not symmetric, it is nent Dp as follows: not consistent with the definition of the right stretch tensor. However if infinitesimal elastic strain is ne- D=De+Dp. (34) glected, Up becomes symmetric. Hence the asymmet- Lee [6] and Lubarda and Lee [7] derived Eq. (34) as- ric component of Up is negligible compared to the suming the existence of the plastically deformed con- plastic strain which is assumed to be finite. figuration for infinitesimal elastic deformation with fi- Lubarda and Lee[7] derived the following equation nite plastic deformation. Here in this paper infinitesi- of the elastic-plastic decomposition: mal elastic deformation with finite plastic deformation F=VeFp. (39) is also assumed as is the case for most elastic-plastic continua. However we interpret Eq. (34) in a slightly Equation (38) differs from Eq. (39) only in describing different way from Lee [6] and Lubarda and Lee [7]. elastic deformation with the right stretch tensor in- Deformation is decomposed into pure deformation and rigid body rotation through the polar decompo- stead of the left stretch tensor. This is readily understood if one applies the polar decomposition theorem sition theorem. The pure deformation which is de- to Fp in Eq. (39). scribed by the right stretch tensor U is assumed to be decomposed into elastic component Ue and plas- Starting from Eq. (35) and Eq. (38), the velocity tic component Up. Let the elastic strain tensor with gradient L=FF-1 is given by reference to base vectors in the reference coordinate system be denoted by ƒÃe. Then the following equa- L=RRt+Rεe(I+εe)-1Rt tion holds because ƒÃe is infinitesimal: +R(I+εe)Up(Up)-1(I+εe)-1Rt. (40)

Ue=I+εe, (35) Neglecting ƒÃe compared with I, the following equa- and then Ue turns out to be symmetric. There are tion can be obtained: following two ways of decomposing the right stretch tensor U into the elastic component Ue and the plas- L=RRt+RεeRt+RUp(Up)-1Rt. (41) tic component Up: Using Eq. (41), the rate of deformation tensor can be U=UeUp=(I+εe)Up, (36) written as

U=UpUe=Up(I+εe). (37) D=RεeRt+RεpRt=De+Dp, (42)

85 Sigeru OGAWA, Shuichi HAMAUZU and Toshio KIKUMA

where ƒÃp is defined by where ƒÂij is the Kronecker symbol. Using these no- tations, the generalized Hooke's law can be described

εp=[Up(Up)-1+{Up(Up)-1}t]/2. (43) by τij=2μ γeij, σ=3kεe, (52) Since it is apparent that De and Dp satisfy the invari- where ƒÊ is the shear modulus of elasticity and k is ante requirement under superposed rigid body mo- the bulk modulus. Since it is assumed that the plastic tion, the elastic-plastic decomposition of the rate of deformation is volume preserving, the following equa- deformation tensor completes. tions hold: According to the elastic-plastic decomposition of the rate of deformation tensor, the definitions of the Dpii=0, εp=εpii/3=0, (53) referential elastic strain tensor εeAB, the referential plastic strain tensor εpAB, the elastic strain tensor εeij, where ƒÃp is the plastic mean normal strain. Utilizing and the plastic strain tensor εpij are given as follows: Eqs. (53), the following equations are readily obtained:

ε=(εeii+εpii)/3=εe, (54) εeAB=∫t0RmA(η)RnB(η)Demn(η)dη, (44) γeij=εij-εpij-δijε=γij-εpij, (55)

where ε is the total mean normal strain, and γij are εpAB=∫t0RmA(η)RnB(η)Dpmn(η)dη, (45) the deviatoric components of the total strain tensor εij. With Eqs. (54) and (55), the generalized Hooke's law is rewritten as εeij=RiARjBεeAB, (46)

τij=2μ(γij-εpij), σ=3kε. (56) εpij=RiARjBεpAB. (47)

Using Eqs. (10),(34), (44) and (45),the followingre- Equations (52) or Eqs. (56) are the constitutive lationshipholds: equations observed with reference to the current coordinate system. The same constitutive equations observed with reference to the reference coordinate sys-

εAB=∫t0RmARnB(Demn+Dpmn)dη=εｅAB+εpAB. tem are derived below. The following equations define the referential stress (48) tensor ƒÐAB: As for the current strain tensor, the following relation- σAB=RiARjBσij. (57) ship is readily obtained using Eqs. (9) and (48): The Cauchy stress tensor discussed so far may be called the current stress tensor in contrast to the ref- εij=RiARjB(εeAB+εpAB)=εeij+εpij. (49) erential stress tensor. The referential mean normal stress ƒÐ and the refer- Hence it is confirmed that, with the definition of strain ential deviatoric stress tensor ƒÑAB are defined as tensor given so far in this paper, the additivity of the elastic strain tensor and the plastic strain tensor holds σ=σAA/3, τAB=σAB-δABσ. (58) for infinitesimal elastic deformation accompanied by finite plastic deformation. These definitions readily lead the following relation- ships: 5. CONSTITUTIVE EQUATIONS FOR σ=σ, τij=RiARjBτAB, (59) STRESS EVALUATION ε=εe=εe=ε, γeij=RiARjBγeAB, (60)

We confine our attention to a group of elastic- where the mean normal component ε and the devia- plastic continua which exhibit infinitesimal elastic de- toric components γAB of the referential strain tensor formation while the plastic deformation could be fairly εAB are introduced. large. For such material, as is the case for most met- Substituting Eqs. (59) and (60) into Eqs. (52), the als, stress is determined by the elastic strain through generalized Hooke's law with reference to the reference the generalized Hooke's law with sufficient accuracy. coordinate system is obtained as Hence in this paper the generalized Hooke's law is utilized for stress evaluation excluding non-linear elastic τAB=2μ γeAB, σ=3kεe=3kε. (61) behavior. The deviatoric components ƒÑij of the Cauchy stress According to the generalized Hooke's law, it is ap- tensor ƒÐij, the mean normal stress ƒÐ, the deviatoric parent that the elastic strain tensor plays a substan- components ƒÁeij of the elastic strain tensor ƒÃeij, and the tial role in calculating the stress tensor, and the to- elastic mean normal strain ƒÃe are defined as follows: tal strain and the plastic strain tensor are used only for obtaining the elastic strain tensor starting from δ=σii/3, τij=σij-δijσ, (50) the elastic-plastic decomposition of the rate of deformation tensor described by Eq. (34). The valid- εe=εeii/3, γeij=εeij-δijεe, (51) ity of these constitutive equations for stress tensor,

86 Strain and Stress Tensor for Finite Deformation

therefore, solely depends on the validity of the elastic nite rotation. strain tensor. The notion of the plastic strain tensor Confining our attention to a group of elastic-plastic is needed only for obtaining the elastic component of continua which exhibit infinitesimal elastic deforma- the strain tensor, and rigorous validity of the notion tion while the plastic deformation could be fairly large, of the plastic strain tensor is irrelevant to the correct- elastic-plastic decomposition of the strain tensor is ness of the constitutive equations for stress tensor. established in accordance with the definition of the Now it is already confirmed that the definition of fi- strain tensor and the elastic-plastic decomposition of nite strain tensor proposed in this paper holds precise the rate of deformation tensor. validity when it is used to describe a motion com- Since the elastic strain tensor is to describe a mo- posed of finite rotation and infinitesimal deformation, tion composed of finite rotation and infinitesimal de- and the elastic strain dealt in this paper can be re- formation within our scope, the stress tensor can be garded as the strain corresponding to such a motion. calculated through the generalized Hooke's law with Hence it is concluded that the constitutive equation the elastic strain tensor defined here without any am- for stress tensor obtained so far is valid in the context biguity. of finite deformation as far as the elastic deformation is infinitesimal. REFERENCES Utilizing the notion of finite strain tensor and the 1. A.E. Green and P.M. Naghdi, Archive for Rational constitutive equations of stress tensor, a useful de- Mechanics and Analysis,18 (1965) 251. scription for plastic strain tensor and stress tensor can 2. P.M. Naghdi and J.A. Trapp, Int. J. Eng. Sci., 13 be obtained through analytically integrating the con- (1975) 785. stitutive equations for plastic strain rate tensor, which 3. P.M. Naghdi and J.A. Trapp, Quart. J. Mech. Appl. will be shown in the subsequent paper. Math., 28 (1975) 25. In addition it can be readily shown that the cur- 4. J. Casey and P.M. Naghdi, Trans. ASME, J. Appl. rent formulation is theoretically equivalent to the rate Mech., 48 (1981) 285. type formulation with the Green-Naghdi rate of the 5. S. Storen and J.R. Rice, J. Mech. Phys. Solids, 23 Cauchy stress tensor. (1975) 421. 6. E.H. Lee, Trans. ASME, J. Appl. Mech., 36 (1969) 1. 6. CONCLUSION 7. V.A. Lubarda and E.H. Lee, Trans. ASME, J. Appl. Mech., 48 (1981) 35. As a notion of strain compatible with the theory 8. M. Goto, J. Jpn. Soc. Tech. Plasticity, 27 (1986) 25. of plasticity having the plastic potential of von Mises 9. K. Shizawa, Trans. Jpn. Soc. Mech. Eng., Ser.A, 62 type with the normality principle as flow rule, the (1996) 121. strain tensor is defined by integrating the rate of de- 10. O. Watanabe, Trans. Jpn. Soc. Mech. Eng., Ser.A, 56 formation tensor considering the material spin with (1990) 203. the rotation tensor. 11. O. Watanabe, Trans. Jpn. Soc. Mech. Eng., Ser.A, 56 It has been proved that the strain tensor coincides (1990) 213. with the strain tensor defined from the right stretch 12. K. Heiduschke, Int. J. Solids Structures, 32 (1995) tensor in case of the infinitesimal deformation with fi- 1047.