Lorentz Group SOo(5,1) for Perfect Elastoplasticity with Large Deformation and a Consistency Numerical Scheme

Please cite this paper as follows:

Hong-Ki Hong and Chein-Shan Liu, Lorentz Group SO

o

(5,1) for Perfect Elastoplasticity with Large

Deformation and a Consistency Numerical Scheme, International Journal of Non-Linear Mechanics,

Vol.34, pp.1113-1130, 1999.

 

Lorentz group SOM(5, 1) for perfect elastoplasticity with large

_{deformation and a consistency numerical scheme}

Hong-Ki Hong*, Chein-Shan Liu

Department of Civil Engineering, National Taiwan University, Taipei 106-107, Taiwan

Received 27 August 1998

Abstract

How to e!ectively deal with non-linearity and accurately ful"ll the consistency condition is essential for modeling and computing in plasticity. Utilizing the concepts of two phases and homogeneous coordinates, we obtain a linear representation of a constitutive model of perfect elastoplasticity with large deformation, and, furthermore, a linear irreducible representation, which contains a "ve-order spin tensor. The underlying vector space is found to be the projective realization of a composite space resulting from a surgery on Minkowski spacetime,>. The irreducible representation in the vector space admits of the projective proper orthochronous Lorentz group PSOM(5, 1) in the on (or elastoplastic) phase and the special Euclidean group SE(5) in the o! (or elastic) phase. The input path dictates symmetry switching between the two groups. Based on such symmetry a numerical scheme is devised which preserves the consistency condition for every time step. The consistency scheme is shown to be stabler, more accurate, and more e$cient than the current numerical schemes developed directly based upon the model itself, because the new scheme preserves the internal symmetry SOM(5, 1) of the model in the on phase so as to locate the stress point automatically on the yield surface at each time step without iterations at all.  1999 Elsevier Science Ltd. All rights reserved.

Keywords: Elastoplasticity; Large deformation; Irreducible representation; Lorentz group; Consistency scheme

1. Introduction

A symmetry of a constitutive model is a statement that when one makes changes in the states of the model, a particular expression for certain constitutive phenomena he formulates does not change. The changes or transformations he makes to the constitutive model which leave the form of the expression unchanged are naturally linked with the invariance of a conserved quantity. A numerical scheme which preserves symmetry and utilizes the invariance property from one time stop to the next one or few stops will be more capable of capturing key features during elastoplastic deformation and have long-term stability and much improved

*Corresponding author. Tel.: 00 886 2 2366 1931; fax: 00 886 2 2362 2975; e-mail: [email protected]

Contributed by W.F. Ames.

0020-7462/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 8 1 - X

e$ciency and accuracy. Therefore the issue of internal symmetries in constitutive laws of plasticity is not only important in its own right, but will also "nd applications to computational plasticity.

Hong and Liu recently found symmetry groups in a constitutive model of perfect elastoplasticity [1, 2] and in a constitutive model of bilinear elastoplasticity [1, 3]. The models considered were all restricted to small deformation. It is interesting to observe that the symmetry groups the models admit possess richer structures than what the models imply. For example, the symmetry group for the perfect elastoplasticity in the on (i.e. elastoplastic) phase is SOM(k, 1) in an augmented stress space. It is known that an element A of the real Lie algebra so(k, 1) of the Lorentz group SOM(k, 1) has the general form:

A"



AQQ AQ AQ 0



with the properties

(AQQ)"!AQQ, AQ"(AQ),

where the superscript t indicates the transpose. But the form of A previously found for the small deformation model of perfect elastoplasticity has zero AQQ [2] and is, therefore, less general than it may. One may generalize the model by including a non-vanishing, skew-symmetric tensor AQQ if he no longer assumes negligibly small spinning but instead considers large deformation and rotation.

As we know the numerical schemes developed up to now for the integration of the constitutive equations of elastoplasticity are executed in the stress space, for example, the tangent sti!ness-radial return method [4, 5], the radial return method [4], the elastic predictor-radial corrector method [5], the generalized midpoint rule [6], the closest-point-projection algorithm [7] and also the recently developed plastic predictor-elastic corrector method [8]. In order to enforce the consistency condition at every time step the abovementioned algorithms require some iterative calculations to force the stress point at the end of each time step to converge to the yield surface [7], which is known as a main source of numerical errors and of consumption of computational time.

In this paper we consider a large deformation constitutive model of perfect elastoplasticity (in Sections 2 and 3) and manage to put it in a more appropriate setting (in Sections 4 and 5) such that the internal spacetime structure (see Sections 6, 7 and 10) and the internal symmetries (see Sections 8 and 9) of the model are brought out. Finally, using internal symmetry inherent in the constitutive model we develop a consistency scheme (in Section 11). One direct bene"t of the scheme is that the stress point is automatically updated on the yield surface without iterative calculations for every time step. This is what the conventional constitutive numerical schemes desired and failed to achieve directly in the stress space. Another, not less important, bene"t is that the formulation of the model becomes exactly linear in the augmented stress space.

2. The constitutive model considered

The constitutive law of elastoplasticity of solid materials proposed by Prandtl [9] and Reuss [10] can be re-postulated (cf. [11]) and enlarged to take account of large deformation as in the following system of axioms:

D"D#D, (1)

s "2GD, (2)

 The volumetric part of the Prandtl}Reuss law is linearly elastic and was thus excluded from the present study in order to focus on the more interesting elastic}plastic behavior of the deviatoric part.

scNQ"2qWD, (3)

#s#)(2qW, (4)

cNQ*0, (5)

#s#cNQ"(2qWcNQ, (6)

in which the two material constants, namely the shear modulus G and the shear yield stress (or strength)qW, are determined experimentally and both are assumed to be positive. The bold-faced symbols D, D, D and s stand for the deviatoric parts of the deformation rate, elastic deformation rate, plastic deformation rate, and Cauchy's stress, respectively, all being symmetric and traceless tensors, whereas cN is a scalar, called the equivalent shear plastic (engineering) strain. All the D, D, D, s and cN are functions of one and the same independent variable, which in most cases is taken either as the ordinary time or as the arc length of the controlled strain path; however, for convenience, the independent variable no matter what it is will be simply called (the external) time and given the symbol t.

A superimposed dot denotes (material) di!erentiation with respect to time, that is d/dt, and a surmounted circle &&

3'' ons represents a Lie derivative of s with respect to W, that is the Jaumann rate

s :"s!Ws#sW. (7)

Here W is the spin tensor, de"ned as the skew-symmetric part of the velocity gradient. A dot is placed between two tensors to denote their Euclidean inner product, and as usual the Euclidean norm of a tensor, s say, is represented by#s# :"(s' s.

3. A non-linear representation

Let us "rst analyze the constitutive model (1)}(6). Substituting Eqs. (2), (3) and (7) into Eq. (1), we obtain 1

2G[s !Ws#sW]# 1

2qWcNQs"D. (8)

De"ne (the internal time) X :"expcN

cW, (9)

where

cW:"qWG (10)

is the shear yield (engineering) strain. Then Eq. (8) becomes 1

2G



d

dt(Xs)!XWs#XsW



"XD. (11)

The inner product of s with Eq. (8) is 1

2Gs' s # 1

Hence #s#"(2qWNqWcNQ"s'D. (13) RecallingqW'0, we have #s#"(2qWN+s'D'0 8 cNQ'0,, (14) asserting +#s#"(2qW and s'D'0, N cNQ'0. (15)

Conversely, ifcNQ'0, Eq. (6) assures #s#"(2qW, which together with Eq. (14) asserts that

cNQ'0 N +#s#"(2qW and s'D'0,. (16)

Statements (15) and (16) tell us that the yield condition#s#"(2qW and the straining condition s'D'0 are su$cient and necessary for plastic irreversibilitycNQ'0. In view of Eqs. (4), (5) and (13), the two statements are logically equivalent to the following criteria:

cNQ"1

qWs' D'0 if#s#"(2qW and s'D'0, (17a)

cNQ"0 if #s#((2qW or s'D)0. (17b)

From Eqs. (8), (17a), (17b) and (10) follows a two-phase non-linear system of di!erential equations: s !Ws#sW"!s' D

qWcW s#2GD if #s#"(2qW and s'D'0, (18a) s !Ws#sW"2GD if #s#((2qW or s'D)0. (18b) According to criteria (17a) and (17b) and the complementary trios (4)}(6) and further to the two-phase system (18a), (18b), the model of elastoplasticity has precisely two phases: the on phase in whichcNQ'0 and #s#"(2qW and the o! phase in which cNQ"0 and #s#)(2qW. In the on phase the plasticity mechanism is on so that the model exhibits elastoplastic behavior, which is irreversible, while in the o! phase the plasticity mechanism is o! so that the model responds elastically and reversibly. Thus, Eqs. (17a) and (17b) are called the on}o! switching criteria for the mechanism of plasticity, and Eqs. (18a) and (18b) is the non-linear di!erential representation of the constitutive model.

4. A linear representation

In this section and the next one we will manage to put the constitutive model in such a form as to reveal internal symmetry. For these purposes let us introduce

X "



X Q X



:" X qW s/(2 s/(2 s/(2 s s s qW . (19)

The components of X are the homogeneous coordinates of the vector space of (s/(2, s/(2, s/(2, s, s, s)/qW. Here and henceforth the index s refers to the totality of the &&internal space'' coordinates, while the index 0 refers to the_{`internal timea coordinate; a bold-faced symbol even with the index(es) denotes a tensor} (and sometimes the matrix of the components).

Thus Eq. (11) becomes d dtX Q"AQQXQ#AQX, (20) in which A QQ:" 0 0 0 0 (2= (2= 0 0 0 (2= 0 !(2= 0 0 0 !(2= !(2= 0 0 !(2= (2= 0 ! = ! = !(2= 0 (2= = 0 ! = !(2= (2= 0 = = 0 . (21)

Mehrabadi et al. [12] had shown that the A QQ in the form of Eq. (21) is a spin tensor in a six-dimensionalspace. In view of Eq. (12) and the positivity of G,qW, cW and X, the on}o! switching criteria (17a) and (17b) become XQ "AQXQ'0 if XgX"0 and d dt[(X Q)gQQXQ]'0, (22a) XQ "0 if XgX(0 or d dt[(X Q)gQQXQ])0, (22b) where g "



g QQ gQ g Q g



"



I 0 ;_ 0;_ !1



, (23)

in which IG is the identity of order i, i being a natural number, and

A Q"(AQ), (24) in which A Q:"2 cW D/(2 D/(2 D/(2 D D D . (25)

Organizing Eqs. (20), (22a) and (22b) we have d

where A "



A QQ AQ A Q 0



if X gX"0 and d dt[(X Q)gQQXQ]'0, (27a) A "



A QQ A Q 0;_ 0



if X gX(0 or d dt[(X Q)gQQXQ])0. (27b) This is a linear, (6#1)-dimensional representation of the constitutive model (1)}(6).

5. A linear irreducible representation

Due to the vanishing traces of the deviatoric tensors s and D, i.e.,

s"!s!s, D"!D!D, (28)

the third component of the vector Eq. (26) can be obtained from the "rst two components. To delete this redundancy, let us introduce

X"



XQ X



" X X X X X X :"X qW as#as as#as s s s qW , (29) where

a:"sin

h#n3



, a:"sinh, a:"cos

h# n

3



, a :"cos h, (30)

withh being any real number. The on}o! switching criteria turn out to be XQ "AQXQ'0 if XgX"0 and d dt[(XQ)gQQXQ]'0, (31a) X_{Q "0 if XgX(0 or} d dt[(XQ)gQQXQ])0, (31b) where g"



gQQ gQ gQ g



"



I 0; 0;_ !1



, (32) AQ"(AQ), (33)

 A special case of Eq. (29) in which h"0 can be viewed as the homogeneous coordinates (non-dimensionalized with respect to the tensile yield strength(3qW) of the Il'yushin stress space (s, (s#(3s, (3s , (3s, (3s).

in which AQ" A A A A A :"2 cW aD#aD aD#aD D D D . (34)

Then Eq. (26) is reduced to

X_{Q "AX,} (35) where A"



AQQ AQ AQ 0



if XgX"0 and d dt[(XQ)gQQXQ]'0, (36a) A"



AQQ AQ 0;_ 0



if XgX(0 or d dt[(XQ)gQQXQ])0. (36b) in which AQQ:"

0 0 _{2a= 2a= 2(a!a)=}

0 0 _{2a= 2a= 2(a!a)=}

2 (3(2a!a)= 2 (3(a!2a)= 0 != != 2 (3(a!2a)= 2 (3(2a!a)= = 0 != ! 2 (3(a#a)= 2 (3(a#a) = = = 0 . (37)

However, we may go further to prove that AQQ as in Eq. (37) is skew-symmetric, i.e. a"!₍₃1(2a!a), a"!₍₃1(a!2a), a!a"!₍₃1(a#a),

a"!₍₃1(a!2a), a"!₍₃1(2a!a), a!a"!₍₃1(a#a). (38) It is obvious that the "rst two equations imply the last four equations, so only the "rst two equations need to be checked. To this end substitution of the "rst two of Eq. (30) into the "rst two of Eq. (38) gives

sinh"!1 (3



2 cos

h# n 3



!cosh



, sin

h# n 3



" !1 (3



cos

h# n 3



!2 cosh



, which obviously hold for any real number ofh, thereby completing the proofs. That is

Indeed, by extending the reasoning for the six-dimensional case by Mehrabadi et al. [12], the AQQ can be shown to be a spin tensor occurring in the "ve-dimensional internal space, representing the spin tensor W occurring in the three-dimensional external space.

Note that Eq. (35) is a linear, irreducible, (5#1)-dimensional representation of the constitutive model (1)}(6), in which X and A are the augmented stress vector and the control tensor, respectively. The control tensor A organizes the input information of the deviatoric deformation rate tensor D (normalized with respect to the shear yield straincW/2) and the spin tensor W. If Eq. (35) is viewed as a matrix representation, the (5#1);1 matrix X contains the contravariant components of the augmented stress vector X and the (5#1);(5#1) matrix A contains the mixed components of the control tensor A.

6. The cone and the discs

In this section we study the properties of the augmented stress space of X induced by the constitutive model (1)}(6). The model formulated in the deviatoric stress space of s may be converted into a model in the augmented stress space of X; the "rst "ve equations of (35) and Eqs. (6), (4) and (5) thus become successively



I 0;_ 0;_ XgX



XQ "



AQQ AQ 0" 0



X, (40) XgX)0, (41) XQ *0, (42)

in terms of the Minkowski metric g (in the space-like convention) of Eq. (32). The vector space of augmented stresses X endowed with the Minkowski metric tensor g is referred to as Minkowski spacetime and designated as,>.

Thus a deviatoric stress point s on the yield hypersphere#s#"(2qW in Euclidean space $ corresponds to an augmented stress point X on the right circular cone+X#XgX"0, emanating from X"0 of Minkowski spacetime,>, henceforth referred to as the cone, while an s within the yield hypersphere corresponds to an X in the interior+X"XgX(0, of the cone. The exterior +X "XgX'0, of the cone is uninhabitable since #s#'(2qW is forbidden according to Eq. (4). Even though admitting an in"nite number of Riemannian metrics, the yield hypersphere2\ of s does not admit a Minkowskian metric, nor does the cylinder of (s, X). It is the cone in the X-space which admits the Minkowski metric. When X is frozen in the o! phase as indicated by Eq. (31b), the augmented stress X stays in the closed 5-disc# (i.e. closed 5-ball !) on the hyperplane X" constant in the space of (X, X, X, X, X, X); the hyperplane is identi"ed to be Euclidean 5-space$, which is endowed with the Euclidean metric I. In summary, the augmented stress X either evolves on the cone when in the on phase, or stays in the discs of simultaneity, which are stacked up one by one in the interior of the cone and are glued to the cone, when in the o! phase. This inspires us to remould the spacetime by removing the interior of the cone and gluing (identifying) a continuously in"nite number of stacking Euclidean closed 5-discs to the cone. This surgery results in a composite space endowed with the Minkowski metric (32) on the cone and the Euclidean metric I on the closed discs.

Spacetime of this sort underlying the plasticity theory may be called internal spacetime, because we can think of it as having to do with the intrinsic nature of the mechanical behavior of the solid materials which

 The reason for adopting the space-like convention is that the nature of elastoplasticity rejects time-like paths, as to be elaborated in Section 7, so that the space-like convention favorably results in non-negative squared lengths exclusively, as shown in Eq. (49). Otherwise the time-like convention would have resulted in embarrassing negative squared lengths.

the constitutive model describes, rather than their position or motion in ordinary (external) space and time. Thus the &&temporal'' component X and the &&spatial'' components XQ"(X, X, X, X, X) may be thought of as the internal time and the internal space, respectively. Thus the internal time X is distinguished from time t, which is the independent variable of the constitutive model and may therefore be deemed as the external time. The external time t is dictated by the external input whereas the internal time X is coined for the age of the internal state of the model. Since Eqs. (1)}(6) is one of the so-called_{`locala constitutive laws,} the external space does not appear explicitly in the model. In such a point of view X and XQ are no more disparate and incompatible as the usual (5#1)-dimensional vector in Euclidean spacetime$>; they are now being organized to an integrated object in Minkowski spacetime,>, which is not a simple exten-sion of ordinary Euclidean 5-space to 5#1 dimenexten-sions, with X as just one more dimension. Because the corresponding entries in the metric (32) have di!erent signs, !1 versus positive de"niteness, the internal time coordinate X is not on the same footing as the"ve internal space coordinates (X, X, X, X, X), and the structure built on the spacetime consequently has internal symmetries (see Sections 8 and 9) quite unlike those on Euclidean space.

7. No time-like paths in internal spacetime

From Eqs. (3), (5) and (6) andqW'0, it is not di$cult to prove that

cNQ"(2#D#, (43)

which indicates thatcN/(2 is the arc length of a path in the plastic strain space. Criteria (17a) and (17b) ensure that

cNQs' s "0 (44)

no matter whether in the on or in the o! phase. Substituting Eqs. (2), (3) and (7) into the above equation and considering the positivity of G andqW, we obtain

D' D"0. (45)

This and Eqs. (1) and (43) give

cNQ)(2#D# (46)

no matter whether in the on or o! phase. This inequality tells us that the maximum value of the speci"c dissipation power""qWcNQ an admissible path in the state space may discharge is (2qW#D#. (On the other hand, axiom (5) tells us that the minimum value of the speci"c dissipation power an admissible path may discharge is zero.)

What does this important observation imply for a path in the augmented stress space of Minkowski spacetime? From Eqs. (46), (5) and (9) it follows that

2(X)D' D!cW(XQ)*0, via Eq. (34) which further reduces to

(X)AQAQ!(XQ)*0.

Substituting Eq. (35) into the above equation, we obtain

where

Xs Q :"XQQ!AQQXQ. (48)

(Compare de"nitions (7) and (48).) Thus

(dX) :"dXg dX*0. (49)

This de"nes the Minkowskian length dX of a di!erential element dX of a path. We should be cautious to distinguish dX from dX, the latter of which is the&&temporal'' component of dX. Recalling that a path such that dXg dX'0 (resp."0, (0) is called a space-like (resp. null, time-like) path in ,>, we thereby conclude that the curve+X(t) "tG(t)t, in the augmented stress space is a space-like or null path in the Minkowski spacetime,> no matter whether in the on phase or in the o! phase. Here t

Gdenotes an initial time and t the current time. Indeed, Eq. (49) conveys an important message that no time-like paths may exist in the internal spacetime of elastoplasticity.

Moreover, from Eqs. (1), (2), (43) and (45) it follows that ds :"#ds#"2G(#D#!(dcN)/2,

which, with the aid of Eqs. (11), (10), (29), (32) and (49), gives dX

X" ds

(2qW (50)

no matter whether in the on phase or in the o! phase. In a word, the Minkowskian length dX of a di!erential element of a path in the augmented stress space,> of (X, X, X, X, X, X) is X/((2qW) times the Euclidean length ds of the corresponding di!erential element of the corresponding path in the deviatoric stress space$ of s.

The vector X(t)!X(tG) and the path +X(t)"tG(t)t, are said to be future-pointing if X(t)'X(tG) strictly. Therefore, the solution to Eq. (35) with the on-phase A of Eq. (36a) can be viewed as a future-pointing space-like or null path on the cone+X"XgX"0,, while the solution to Eq. (35) with the o!-phase A of Eq. (36b) is a space-like path on a closed disc of simultaneity+X"XgX)0 and XQ"0,. It is worth stressing that the interior of the cone is sliced into stacking discs of simultaneity tagged with di!erent values of X; therefore, the admissible augmented stress space can be reached either along the paths in the discs of simultaneity when in the o! phase or along the future-pointing space-like or null paths on the cone when in the on phase.

8. PSOM(5, 1) symmetry in the on phase

The solution of Eq. (35) may be expressed in the following transition formula from the augmented stress X(t) at time t to the augmented stress X(t) at time t:

X(t)"[G(t)G\(t)]X(t), (51)

in which G (t) is the fundamental solution of Eq. (35), that is a transformation tensor (represented by a square matrix of order (5#1) containing the mixed components) satisfying

GQ (t)"A(t)G(t), (52)

From Eqs. (5), (9) and (29) it follows that

X(t)*X(t)*X(tG), ∀t*t*tG, (54)

applicable to both the on and o! phases.

In the remainder of this section we concentrate on the on phase to bring out internal symmetry inherent in the model in the on phase. Denote by I an open, maximal, continuous time interval during which the mechanism of plasticity is on exclusively. From Eqs. (36a) and (32) it is easy to verify that the control tensor A in the on phase satis"es

Ag#gA"0. (55)

Hence, the corresponding transformation G satis"es (see, e.g. [3])

GgG"g, (56)

det G"1, (57)

G*1. (58)

Thereby the on-phase control tensor A is an element of the real Lie algebra so(5, 1) and generates the on-phase transformation G, which is thus an element of the proper orthochronous Lorentz group SOM(5, 1). See, for example, Cornwell [15]. So the function G(t) of time t3I may be viewed as a connected path of the Lorentz group and the algebraic and topological properties of the proper orthochronous Lorentz group are shared by the constitutive model in the on phase.

From Eq. (31a), XQ '0 strictly when the mechanism of plasticity is on; hence,

X(t)'X(t), ∀t't, t, t3I, (59)

which means that in the sense of irreversibility there exists future-pointing time-orientation from the augmented stress X(t) to X(t). (Compare Eq. (59) with Eq. (54).) Moreover, such time-orientation is a causal one, because the augmented stress transition formula (51) and inequality (59) establish a causality relation between the two augmented stresses X(t) and X(t) in the sense that the preceding augmented stress X(t) in#uences the following augmented stress X(t) according to formula (51). Accordingly, the augmented stress X(t) chronologically and causally precedes the augmented stress X(t). This is indeed a common property for all models with inherent symmetry of the proper orthochronous Lorentz group. By this symmetry a core connection among irreversibility, the time arrow of evolution, and causality has thus been established for plasticity in the on phase.

We solve Eq. (56) for the inverse

G\"gGg (60)

and partition G as G"



GQQ GQ

GQ G



, (61)

where GQQ, GQ and GQ are of order 5;5, 5;1 and 1;5, respectively. Thus, Eq. (51) is partitioned into the following matrix equation:



XQ(t) X(t)



"



GQQ(t)(GQQ)(t)!GQ(t)(GQ)(t) GQ(t)G(t)!GQQ(t)GQ(t) GQ(t)(GQQ)(t)!G(t)(GQ)(t) G(t)G(t)!GQ(t)GQ(t)

 

XQ(t) X(t)



, (62) which is valid for the on phase.

How can one determine the deviatoric stress tensor s(t) once he has the augmented stress vector X(t)? From Eq. (29) it follows that

s s s s s " a !a ! a a 0; 0;_ (3 2 I 2qW (3XXQ. (63)

This is indeed a projective realization of the response. By this and the on-phase transition (62) one can map s(t) to the current response s(t).

9. SE(5) symmetry in the o4 phase

Contrary to Eq. (62) of the on phase, the transition for the o! phase is very simple. To "nd it, recall that Eqs. (51)}(53) are still applicable but with

G(t)"



R (t) T(t) 0;_ 1



, (64)

where R and T are respectively of order 5;5 and 5;1 and governed by RQ "AQQR, TQ"AQQT#AQ,

in which AQQ is given in Eq. (37) and AQ in Eq. (34). Thus it is easy to show that R3SO(5), T3¹(5), and G3SE(5); therefore, the constitutive law in the o! phase has an internal symmetry characterized by the special Euclidean (or proper motion) group SE(5), which is the semi-direct product of the translation group ¹(5) with the proper rotation group SO(5). Even such an o!-phase transformation (64) exists and is invertible, it is no longer an element of the Lorentz group because such G does not satisfy Eq. (56) although det G"1 remains to hold and G"1.

The inverse of Eq. (64) is given by G\"



R

!RT 0;_ 1



. (65)

Thus according to Eq. (51) we obtain



XQ(t) X(t)



"



R (t)R(t) !R(t)R(t)T(t)#T(t) 0;_ 1



XQ(t) X(t)



, (66)

which is valid for the o! phase. In a similar way the stress response in the o! phase can be realized by invoking Eq. (63) and the o!-phase transition formula (66).

In summary the stress response is the projective realization (63) of the on-phase transition formula (62) or the o!-phase transition formula (66); switching between the two depends upon the control tensor A and obeys the on}o! switching criteria (31a) and (31b). The switching from a transformation of the Lorentz group in the on phase to a non-Lorentzian transformation in the o! phase indicates that internal symmetry switches from one kind to another, and vice versa. As a result the constitutive model in the deviatoric stress space of s has symmetry switching between the special Euclidean group SE(5) acting on the closed 5-ball of

admissible states and the projective proper orthochronous Lorentz group PSOM(5, 1) acting on the yield hypersphere.

10. Basis responses to basis controls

In this section we restrict ourselves to the on phase. First let us consider the stress response to a control tensor of single shearing:

AG" $ $ 2 . 2 1 $ $ 2 1 2 . (67)

for 1)i)5; namely AG is equal to 0_;

except that the i0th and 0ith entries are both #1. To such an elementary control the fundamental solution can be readily found to be

GG" 1 \ cosh t 2 sinh t $ $ sinh t 2 cosh t (68)

for 1)i)5, where t31; namely GG is equal to I except that the iith, 00th, i0th and 0ith entries are cosht, cosh t, sinh t and sinh t, respectively. Then the stress response is calculated via Eqs. (62) and (63), in which X(t) plays the role of the prescribed initial condition (at time t).

For example, if the material specimen is sheared homogeneously and uniformly in the xx plane of the external space (x, x, x) at the constant rate of cW/2 per unit time, then A of Eqs. (36a) and (36b) is equal to A of Eq. (67) with i"5. As a result the on-phase transformation G of Eq. (62) is equal to G of Eq. (68).

If the specimen is subjected to a rigid-body rotation at the constant rate:

AGH" $ $ 2 . 2 1 2 $ $ 2 !1 2 . 2 $ $ (69)

for 1)i(j)5; namely AGH is equal to 0_;

 except that the ijth and jith entries are #1 and !1, respectively, then the fundamental solution is

 For notation do not mix up the basis control tensor AGH with the mixed component AGH of the control tensor A, neither the basis response tensor GGH with the mixed component GGH of the transformation tensor G.

GGH" 1 \ cos t 2 sin t $ $ !sin t 2 cos t \ 1 (70)

for 1)i(j)n, where t31; namely GGH is equal to I except that the iith, jjth, ijth and jith entries are cost, cos t, sin t and !sin t, respectively. Similarly the stress response is calculated via Eqs. (62) and (63). In fact a rigid-body rotation induces no deformation and stress; therefore, Eq. (70) just re#ects this fact and con"rms that the constitutive model is frame-indi!erent.

The proper orthochronous Lorentz group SOM(5, 1) acting on the Minkowski spacetime ,> endowed with the canonical metric (32) contains "ve subgroups of the type (68) and "fteen subgroups of the type (70). The basis elements of the real Lie algebra so(5,1) are de"ned by

AGH"d

dtGGH(t)



_R. (71)

Hence the real Lie algebra so(5, 1) possesses a convenient basis composed of "ve basis elements of the type (67) and "fteen basis elements of the type (69).

Any control tensor A3so(5, 1) is thus a linear combination of the basis elements (67) and (69): A" ) GH) L AGHAGH# ) G) L AGAG"[AGH], (72)

with the mixed components of A being AGH"!AHG for 1)i)j)5, AG"AG for 1)i)5 and A"0. This is consistent with Eq. (36a) together with Eq. (39) and (33).

The basis element AGH may generate a one-parametersubgroup GGH(t), which corresponds to a rotation in the (XG, XH) plane of the Minkowski spacetime (X, X, X, X, X, X) with metric g, while the basis element AG may generate a one-parameter subgroup GG(t), which correspondsto a hyperbolic rotation in the (XG, X) plane of the Minkowski spacetime (or a boost in the XG-direction). The problems encountered in applications are often posed as follows: Given a control tensor A, which contains the information of the normalized deviatoric deformation rate 2D/cW and the spin W, "nd the deviatoric stress response s. Thus, we may deem AGH, i, j"1, 2, 3, 4, 5, 0, as (constant-valued) basis controls so that GGH(t), i, j"1, 2, 3, 4, 5, 0, are the resulting basis responses. Since the basis controls AGH, i, j"1, 2, 3, 4, 5, 0, span the whole control space of A, the resulting basis responses GGH(t), i, j"1, 2, 3, 4, 5, 0, su$ce to constitutively characterize the mechanical properties of the material.

11. Consistency scheme

The simplest scheme for Eq. (35) is a time-centered Euler scheme:

XL>"XL#qA(XL>#XL), (73)

where XL denotes the value of X at the discrete time stop tL and q is one half of the time increment, that is q :"*t/2"(tL>!tL)/2. Using the Cayley transform we have

It is easy to check that this transform preserves the properties (56)}(58) of the Lorentz group, i.e., Cay(qA)3SOM(5, 1) if q is small enough.

A numerical algorithm is called a group preserving scheme if for every time increment the map from XL to XL> preserves the group properties (56)}(58). Now let us investigate what Cay(qA)3SOM(5, 1) implies as a numerical scheme for the constitutive law of plasticity. From Eqs. (29), (56) and (74) it follows that

XL>gXL>"XLgXL"(XL>)

#sL>#

2qW !1



"(XL)

#sL#2qW !1



"0. (75) Because XL>*XL'0, the equalities (75) says nothing but for every time increment the points sL and sL> are located on the yield hypersphere, i.e., #sL>#"#sL#"(2qW. In other words, the consistency condition is ful"lled exactly for every time stop in the on phase. This is what the conventional schemes of computational plasticity desired and failed to achieve directly in the stress space. Therefore, the new numerical scheme may be speci"cally called a consistency scheme.

Using Eq. (36a) and the following formula



b a a b



\ " b\# 1 b!ab\ab\aab\ ! 1 b!ab\ab\a ! 1 b!ab\aab\1 1 b!ab\a , (76)

whereb is a non-singular 5;5 matrix, a is a 5;1 matrix, b is a scalar such that bOab\a, we "nd the inverse of I!qA to be

(I!qA)\"



B#ccqBAQAQB cqBAQqAQB c



, (77) where

B :"(I!qAQQ)\, c:"1!qAQBAQ1 . (78) Substituting Eqs. (77) and (36a) into Eq. (74) we obtain the discretized transition matrix in the on phase,

Cay(qA)" I#2cqBAQAQBAQQ

2cqBAQAQBAQ #2qBAQQ#2cqBAQAQ #2qBAQ 2cqAQBAQQ#2cqAQ 1#2cqAQBAQ

. (79)

The mapping in the o! phase is obtained by Eqs. (74) and (36b) as in the following: XL>"



I#2qBAQQ 2qBAQ_0;

 1



XL,

(80) from which it is obvious that XL>"XL in the o! phase.

Note that AQQ satis"es

(AQQ)#5w(AQQ)#4wAQQ"0, (81)

where

Fig. 1. In (a)}(e) the stress responses for the three straining cycles calculated by the consistency scheme are compared with the closed-form solutions. In (f ) the errors of the ful"llment of the yield condition by using the consistency scheme and the Runge}Kutta}Fehlberg method are compared for the case where the initial and subsequent stresses are all located on the yield surface.

is the Euclidean norm of the axial vector of W. By formula (81), B in terms of the powers of AQQ can be expressed as follows: B"(I!qAQQ)\"I#dAQQ#d(AQQ)#d(AQQ)#d(AQQ), (83) where d :"1#5wq#5wqq#4wq, d :" q#5wq 1#5wq#4wq, d :"1#5wq#4wqq , d :" q 1#5wq#4wq. (84)

Once XL is calculated at each time stop formula (63) gives the value of the response sL at each time stop. In order to gain the insight of the consistency scheme, and to understand what makes a di!erence in applying the new scheme from applying the Runge}Kutta}Fehlberg method [16] to solve Eq. (18a) directly,

let us consider the case where both D and W are constant matrices. For such inputs the closed-form solution of the responses can be derived, so we can compare the results calculated by the two schemes with the closed-form solution. The material constants used in the calculations were G"50 000 MPa and qW"500 MPa. We applied three cycles, each cycle consisting of constant D and W for one second and then negative constant D and W for another one second, with

D" 0.002 0.009 0.005 0.009 !0.001 0.004 0.005 0.004 !0.001 , W" 0 0.001 0.002 !0.001 0 !0.005 !0.002 0.005 0 .

Fig. 1a}Fig. 1e show that the consistency scheme gave very accurate responses in all respects, where the initial stresses were taken to be s"50 MPa, s"20 MPa, s"!100 MPa, s"!40 MPa, and s"30 MPa. Furthermore, Fig. 1f shows that the consistency scheme supplied a more acurate result of the yield condition, #s#"(2qW, than the one calculated by the Runge}Kutta}Fehlberg scheme, where the initial stresses were taken to be located on the yield surface with s"300 MPa, s"0 MPa, s"400 MPa, s"0 MPa, and s"0 MPa.

12. Conclusions

In this paper we have investigated internal symmetry inherent in a constitutive model of perfect elastoplasticity with large deformation. Even though the constitutive equations are non-linear in the deviatoric stress space of s, they can be converted to a linear system XQ "AX in the (5#1)-dimensional augmented stress space of X. In the augmented stress space not only the non-linearity of the model is unfolded, but also an internal spacetime structure of the Minkowskian type is brought out. The control tensor A for the on phase was proved to be an element of the real Lie algebra so(5, 1) of the proper orthochronous Lorentz group SOM(5, 1), and the fundamental solution G of the system XQ"AX with the on-phase A was shown to be an element of the proper orthochronous Lorentz group, so that the causality relation of the augmented stresses was veri"ed. To account for both the on and o! phases we constructed a composite space endowed with a Minkowskian metric on the cone but with a Euclidean metric on each of the discs inside the cone. As a result we found that the perfect elastoplastic model with large deformation possesses two kinds of symmetry } SE (5) in the o! phase and PSOM(5, 1) in the on phase } and has symmetry switching between the two depending on the control input. The (basis) control tensors of single shearing and of rigid-body rotation were found to form a convenient basis of the real Lie algebra so(5, 1), and the corresponding (basis) responses to be the corresponding subgroups of SOM(5, 1).

Based on the symmetry study, a numerical scheme which preserves the group properties for every time increment was developed. This group preserving scheme may be speci"cally called a consistency scheme, since it is capable, among other bene"ts derivable from the group properties, of updating the stress point to be automatically located on the yield surface at the end of each time increment in the on phase without any iterative calculations, that is, the consistency condition is ful"lled automatically and exactly. Moreover, another bene"t derived from the group properties is that the new scheme is exactly linear in the augmented stress space, because SOM(5, 1) is a subgroup of the general linear group G¸(6, 1). In this regard, the conventional numerical schemes typically do not share the group properties so that perform less accurate than the consistency scheme. Since the new scheme is easy to implement numerically and has high computational e$ciency and accuracy, it is highly recommended especially for engineering applications which demand intensive calculations.

Acknowledgements

The "nancial support provided by the National Science Council under Grant NSC 87-2211-E-002-035 is gratefully acknowledged.

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