Using Comparison Theorem to Compare Corotational Stress Rates in the Model of Perfect Elastoplasticity

Using comparison theorem to compare corotational stress

rates in the model of perfect elastoplasticity

Chein-Shan Liu

, Hong-Ki Hong

a_{Department of Mechanical and Marine Engineering, National Taiwan Ocean University, Keelung 202-24, Taiwan, ROC} b_{Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan, ROC}

Received 13June 1999; in revised form 9 May 2000

Abstract

For the simple shear problem of a perfectly elastoplastic body, we convert the non-linear governing equations into a third order linear di€erential system, then into a second order linear di€erential system, and further into a Sturm± Liouville equation. Thus Sturm's comparison theorem can be employed and extended to compare the simple shear responses based on di€erent objective corotational stress rates. It is proved that the rates of Jaumann, Green±Naghdi, Sowerby±Chu, Xiao±Bruhns±Meyers, and Lee±Mallett±Wertheimer render non-oscillatory stress responses, with the Jaumann equation as a Sturm majorant for the other four equations. For an objective corotational stress rate with the general plane spin a sucient non-oscillation criterion is found to be that the plane spin must not exceed the shear strain rate. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Objective corotational stress rates; Perfect elastoplasticity; Lie algebra isomorphism; Sturm comparison theorem; Sturm majorant

1. Introduction

The study of objective stress rates is one of the major topics in constitutive modeling. Dienes (1979) was the ®rst to discover that the model of hypoelasticity based on the Jaumann stress rate may result in os-cillatory responses to simple shearing. After that, Nagteggal and de Jong (1982) applied the Jaumann rate to the kinematic hardening model and found similar oscillatory behavior. Due to this drawback of the Jaumann rate, considerable e€orts have since been made to deal with the problem of choosing an ap-propriate objective stress rate in rate type constitutive models (see, e.g., Johnson and Bammann, 1984; Atluri, 1984; Metzger and Dubey, 1987; Szabo and Balla, 1989).

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*_{Corresponding author. Address: Jung He P.O. Box 3-297, Taipei 235, Taiwan, ROC. Tel.: +886-2-2366-1931; fax:}

+886-2-2362-2975.

E-mail address: [email protected] (C.-S. Liu).

0020-7683/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S0020-7683(00)00214-6

Several objective stress rates have been suggested and shown to be plausible by subjecting the constit-utive models based on those rates to simple shear deformation. However, a convincing methodology is still lacking for analytically con®rming whether a stress rate will render the responses oscillatory or not, and criteria are also lacking for picking out appropriate stress rates. These do pose serious diculties. Usually the equations encountered are quite complicated, especially for large deformation plasticity models, which are governed by highly non-linear equations; as a consequence, numerical methods were employed to in-tegrate the rate type constitutive equations for di€erent objective stress rates, and the numerical results were then compared between the investigated stress rates (see, e.g., Nagteggal and de Jong, 1982; Moss, 1984; Atluri, 1984; Szabo and Balla, 1989). However, those comparisons gave no answers to questions such as: Why does the Jaumann stress rate used in the model of hypoelasticity render the responses oscillatory, but in the model of perfect elastoplasticity render the responses non-oscillatory? What objective stress rates will render the responses oscillatory, and what else non-oscillatory?

Liu and Hong (2000) recently proposed to use and extend Sturm's comparison theorem to examine the qualitative behavior of hypoelasticity equations. In that paper oscillation and non-oscillation criteria were set up along with the assertions that, among the corotational stress rates examined, the logarithmic stress rate advocated by Xiao et al. (1997a,b) renders its homogeneous equation to be a non-oscillatory majorant, the rate of Jaumann (1911) renders its equation to be an oscillatory minorant, and the rate of Green and Naghdi (1965) renders its homogeneous equation to be a disconjugate majorant. Along this line of thought, in this paper we intend to use Sturm's comparison theorem to study the general in¯uence of the objective corotational stress rates on the simple shear problem of perfect elastoplasticity. Before one can exploit the theorem, the equations of the problem must be subjected to three steps of conversions. They are converted ®rst to a third order linear system in Sections 3and 6, then to a second order linear system in Sections 9±11, and ®nally to a Sturm±Liouville equation in Section 11. After the conversions, the main themes of the paper are considered in Sections 7 and 11±13with Theorems 1 and 2 summarizing the main results.

2. Constitutive model

The constitutive law proposed by Prandtl and Reuss can be enlarged to take account of large defor-mation as in the following (Hong and Liu, 1999):

D ˆ De_{‡ D}p_{; …1†} sˆ 2GDe_{; …2†} s_cp _{ˆ 2s} yDp; …3† ksk 6p2sy; …4† _cp_{P 0; …5†} ksk_cp _ˆp_2s y_cp …6†

in which the two material constants, namely the shear modulus G and the shear yield stress (or strength) sy,

are determined experimentally and both are assumed to be positive. The boldfaced symbols D, De_{, D}p_{and s}

stand for ``the deviatoric1_{parts of '' the deformation rate, elastic deformation rate, plastic deformation rate,}

and Cauchy's stress, respectively, all being symmetric and traceless tensors of the second rank, whereas cp_is

a scalar, called the equivalent shear plastic engineering strain. As usual the Euclidean norm of a tensor, say s, is represented by ksk :ˆps
s, where a dot is placed between two tensors of the same ranks to denote their Euclidean inner product.

A superimposed dot denotes (material) di€erentiation with respect to time t, that is d=dt, and a sur-mounted circle ``'' on s represents a Lie derivative of s with respect to x,

s:ˆ _s xs ‡ sx: …7†

Here x is a spin tensor to be elaborated later (see Eq. (21)). In the above rate type model (¯ow model) of perfect elastoplasticity, one crucial point is the choice of the objective stress rate s. Although the use of di€erent stress rates makes no fundamental di€erences in the mathematical structures of the models, fundamental di€erences in the material behavior modeled may occur.

3. A linear representation of the model

A linear representation of the ¯ow model formulated in Section 2 has been obtained by Hong and Liu (1999) as follows: _X ˆ AX; …8† where X ˆ X_Xs₀   ˆ X1 X2 X3 X4 X5 X0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 :ˆX0 sy a1s11‡ a2s22 a3s11‡ a4s22 s23 s13 s12 sy 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 …9†

is the augmented stress, and a1:ˆ sin u



‡p₃; a2:ˆ sin u; a3:ˆ cos u



‡p₃; a4:ˆ cos u …10†

with u being any real number. The control matrix A is subjected to the following switch criteria for plastic and elastic phases:

A ˆ As s As0 A0 s 0   if Xt_{gX ˆ 0 and} d dt …Xs†tgssXs   > 0; As s As0 015 0   if Xt_{gX < 0 or} d dt …Xs†tgssXs   6 0; 8 > > < > > : …11† in which

As s:ˆ 0 0 2a2x23 2a1x13 2…a1 a2†x12 0 2a4x23 2a3x13 2…a3 a4†x12 0 x12 x13 skew-sym: 0 x23 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ; …12† As 0ˆ A1 0 A2 0 A3 0 A4 0 A5 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 :ˆ 2 cy a1D11‡ a2D22 a3D11‡ a4D22 D23 D13 D12 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ; …13† A0 sˆ …As0†t; …14†

where c_y:ˆ sy=G is the shear yield engineering strain, xij, i; j ˆ 1; 2; 3, are the components of x,

g ˆ gss gs0 g0s g00   ˆ I5 051 015 1   …15† is the Minkowski metric, In is the identity matrix of order n, 0mn is the m  n zero matrix, and the

su-perscript t denotes the transpose.

Note that Eq. (8) is a linear …5 ‡ 1†-dimensional representation of the ¯ow model (1)±(7). The control matrix A organizes the input information of ``the deviatoric2 _{part of '' the deformation rate tensor D}

(normalized with respect to the shear yield strain c_y=2) and the spin tensor x. Because of the large number 2=c_y in As

0, the values of the components of Ass are much less than those of As0, and so the major

contri-bution to X and hence to s is made by D, not by x. 4. Objective corotational stress rates

Table 1 shows some objective corotational stress rates,s, together with the corresponding spins, x, to be investigated in this paper. First the spins are put in Eq. (7) to obtain the rates and then the rates are put in the constitutive equations (1)±(6) for further study.

The notation used here for the spins W, X, XE, and Xlogis explained as follows: W is the skew-symmetric

part of the velocity gradient tensor L :ˆ _FF 1_{, where F is the two-point tensor of deformation gradient.}

X :ˆ _RRt_{is the rate of rotation, where R is the rotation, the orthogonal tensor in the polar decomposition}

Table 1

Objective corotational stress rates and spins

Objective corotational stress rates s Spin x

Jaumann (J) W

Green±Naghdi (GN) X

Sowerby±Chu (SC) XE

Xiao±Bruhns±Meyers (XBM) Xlog

Lee±Mallett±Wertheimer (LMW) ±

F ˆ RU ˆ VR of F. XE:ˆ _RERtE is known as the Eulerian spin tensor, where RE is the diagonal

trans-formation of V, that is

V ˆ REkRtE; …16†

with k ˆ diag‰k1; k2; k3Š the diagonal tensor containing the eigenvalues, k1; k2; k3, of V. The logarithmic spin

Xlog _{was introduced recently in order that the deformation rate equals …ln V†
X}log_{ln V ‡ …ln V†X}log_,

where …ln V†
denotes the material time derivative of the Eulerian logarithmic strain tensor ln V (see Leh-mann et al., 1991; Reinhardt and Dubey, 1996; Xiao et al., 1997a,b). With the logarithmic spin Xlog_{, the}

logarithmic rate of any Eulerian symmetric tensor, say S, is de®ned by _Slog

:ˆ _S Xlog_{S ‡ SX}log

which indeed generalizes the concept of the objective stress rates in Eq. (7), but specializes x to be Xlog_{. In}

particular, if setting S to be Cauchy's stress3_{r, then one has the logarithmic rate of Cauchy's stress, r}log_,

which will be referred to later as the Xiao±Bruhns±Meyers rate. Note that r ˆ rlog _{if x ˆ X}log_{. It is}

remarkable that the deformation rate can thus be seen to be equal to …lnV†log. 5. Simple shear deformation

In what follows we are going to study a solid body, which is made of material described by the model of perfect elastoplasticity formulated in Section 2, under simple shear deformation, of which the deformation gradient is F ˆ 1 c 00 1 0 0 0 1 2 4 3 5; c 2 ‰0; 1†; …17†

where c is the shear engineering strain. Let

h :ˆ arctan…c=2†; _h ˆ 2_c_{c2‡ 4}; h 2 ‰0; p=2†: …18† The related kinematic quantities are listed below:

D ˆ₂_c 0 1 0 1 0 0 0 0 0 2 6 4 3 7 5; W ˆ₂_c 0 1 0 1 0 0 0 0 0 2 6 4 3 7 5; X ˆ 0 _h 0 _h 0 0 0 0 0 2 6 4 3 7 5; XE ˆ 0 ₂_h 0 _h 2 0 0 0 0 0 2 6 4 3 7 5; Xlog_ˆ _c 2 0 f …h† 0 f …h† 0 0 0 0 0 2 6 4 3 7 5; …19† where

f …h† ˆ_{2 ln……1 ‡ sin h†= cos h†}1 sin h 

‡ cos2_{h ln}1 ‡ sin h

cos h 

…20† as has been derived by Liu and Hong (2000).

The forms of W, X, XE, and Xlog lead us to consider the following general plane spin: x ˆ₂_c 0f 0 0f 0 0 0 0 2 4 3 5 _21†

which includes at least the following as special cases: 1. The Jaumann rate (1911): f ˆ 1 for W.

2. The Green±Naghdi rate (1965): f …c† ˆ 4=…c2_{‡ 4† for X.}

3. The Sowerby±Chu rate (Sowerby and Chu, 1984): f …c† ˆ 2=…c2_{‡ 4† for X} E.

4. The Xiao±Bruhns±Meyers rate (1997a,b): f …h† as in Eq. (20) for Xlog_.

5. The Lee±Mallett±Wertheimer rate (Lee et al., 1983): f …c† ˆ 2=…c2_{‡ 1†. (see also Yang et al., 1992).}

From Eq. (21), the plane spin x12ˆ f _c=2. Since c is the shear engineering strain, c=2 is the shear strain and

_c=2 is the shear strain rate, so that f is the ratio of the plane spin x12to the shear strain rate _c=2. For the

Jaumann stress rate the plane spin equals the shear strain rate, but for the other four stress rates they are not equal. Formula (20) is equivalent to4 g…x† 2 ˆ 1 2…1 ‡ x2_†‡ x 2p1 ‡ x2_sh 1_x

Eq. (157) of the article of Bruhns et al. (1999), upon making the change of variables x ˆ c=2 ˆ tan h and sh 1_{x ˆ ln…x ‡}p_{1 ‡ x2† ˆ ln……1 ‡ sin h†= cos h†.}

6. Simple shear equations Under simple shear we have

s ˆ ss1112 s12s11 00 0 0 0 2 4 3 5: …22†

Thus Eq. (8) reduces to d dt X1 X5 X0 2 4 3 5 ˆ 0 2x12 0 2x12 0 c2yD12 0 2 cyD12 0 2 6 4 3 7 5 X 1 X5 X0 2 4 3 5; …23†

where we have chosen u ˆ 11p=6 and hence a1 a2ˆ 1 in Eq. (10). Substituting Eq. (19) for D and Eq.

(21) for x in Eq. (23), we have d dc X1 X5 X0 2 4 3 5 ˆ 0f 0f 01 cy 0 1 cy 0 2 6 4 3 7 5 X 1 X5 X0 2 4 3 5: …24†

4_{The authors would like to thank one reviewer who drew the authors' attention to this equivalence. For notation, the x is neither x}

Hence the axial and shear stress components, s11_ˆsyX1

X0 ; s12ˆ

syX5

X0 ; …25†

are governed by the following di€erential equations: d dcs11ˆ f …c†s12 1 c_ysys 11_s12_{; …26†} d dcs12ˆ f …c†s11 1 cysy…s 12_†2_‡sy cy: …27†

Let the right-hand sides of Eqs. (26) and (27) equal zeros; we obtain the limiting values of the responses lim c!1s 11_{ˆ c} ysyf …1†; _c!1lims12ˆ  s2 y c2ys2yf2…1† q : …28†

The asymptotic behavior of the stresses is seen to be controlled by f …1†. The ®ve f's listed in the previous section are compared in Fig. 1, which shows that most of f's (except that of Jaumann) tend to zero as c ! 1.

7. Closed-form solutions of the Jaumann equation

Di€erentiating the second equation of Eq. (24) and using the other two equations thereof, we obtain d2_X5 dc2 ˆ f0 f dX5 dc f0 c_yf X0‡ 1 c2 y f 2 ! X5 _29†

in which the primes stand for di€erentiation with respect to c. For the Jaumann stress rate, f ˆ 1 and f0_{ˆ 0, so Eq. (29) reducing to}

d2_X5 dc2 ‡ 1 1 c2 y ! X5_{ˆ 0: …30†}

This equation indicates that, unless cy > 1, X5and hence s12will not oscillate. However, for most materials,

the theoretical upper limit of cy is 1=15 (see e.g., Hirth and Lothe, 1982); therefore, cy < 1. From Eqs. (30),

(24) and (25) the responses can be derived as follows: s11 sy ˆ mc_yc4‡ cyc1exp‰m…c con†Š cyc2exp‰ m…c con†Š mcyc3‡ c1exp‰m…c con†Š c2exp‰ m…c con†Š ; …31† s12 sy ˆ mcyc1exp‰m…c con†Š ‡ mcyc2exp‰ m…c con†Š mc_yc3‡ c1exp‰m…c con†Š c2exp‰ m…c con†Š; …32† where m ˆ  1 c2 y 1 s ; c1ˆ_2mc1 y‡ s12_c on† 2sy s11_c on† 2msy ; c2ˆ 1 2mc_y‡ s12_c on† 2sy ‡ s11_c on† 2msy ; c3ˆ 1 _mc1 y 1 mcy " s11_c on† msy # ; c4ˆc_c3 y

in which c ˆ c…t† and c_onˆ c…ton†, where t and ton are the current time and the switch-on time of plastic

deformation. For c_on6 c < 1 the body is in the plastic phase, while for 0 6 c < c_onit is in the elastic phase. Note that c_on6ˆ c_y unless the initial stresses are zeros and Eq. (2) is replaced by _s ˆ 2GDe_{. Formulae (31)}

and (32) (and also Fig. 2) con®rm that the axial and shear responses are non-oscillatory.5

Note that Eq. (24) is for perfect elastoplasticity; on the other hand, for hypoelasticity, it becomes d dc X1 X5 X0 2 4 3 5 ˆ 0f 0f 01 cy 0 0 0 2 4 3 5 X 1 X5 X0 2 4 3 5 _33†

which, for the Jaumann stress rate, leads to X0_{ˆ 1 and}

d2_X5

dc2 ‡ X5ˆ 0: …34†

Obviously, this equation indicates that X5 _{and hence s}12_{for hypoelasticity are oscillatory.} 6_{The contrast}

between Eqs. (34) and (30) (and between Eqs. (33) and (24)) reveals that the fundamental di€erence of the mathematical structures between hypoelasticity and elastoplasticity and the physical limit 0 < cy< 1 of the

plastic yield are the major factors which render the responses qualitatively di€erent for the Jaumann rate. 8. Consistency scheme

From Eq. (25) it follows that …s11_†2_{‡ …s}12_†2_{ˆ s}2

y () …X1†2‡ …X5†2 …X0†2ˆ 0 …35†

5_{The precise meaning (de®nition) of non-oscillatory will be given in Section 11.} 6_{The precise meaning (de®nition) of oscillatory will be given in Section 11.}

and so the yield condition in the two-dimensional stress space …s11_{; s}12_{† is converted to a null cone condition}

in the three-dimensional Minkowski space …X1_{; X}5_{; X}0_{† 2 M}3_{, which endows the inde®nite metric}

g ˆ I2 021

012 1

 

: …36†

The control matrix A ˆ _c 0 f 0 f 0 1 cy 0 1 cy 0 2 6 4 3 7 5; …37† in Eq. (24) satis®es At_{g ‡ gA ˆ 0; …38†}

and hence is an element of the Lie algebra so…2; 1†. Correspondingly, the fundamental solution G of Eq. (24) satis®es (see e.g., Hong and Liu, 1999, 2000)

Fig. 2. Axial and shear stress responses for the simple shear problem based on the Jaumann rate. Those calculated by the group preserving scheme agree very well with the closed-form solutions.

Gt_{gG ˆ g; …39†}

det G ˆ 1; …40†

G0

0P 1 …41†

and hence is an element of the proper orthochronous Lorentz group SOo…2; 1†. This leads us to use the

following group preserving scheme for Eq. (24):

Xn‡1ˆ …I3 sA† 1…I3‡ sA†Xnˆ ‰I3‡ 2s…I3 sA† 1AŠXn; …42†

where Xndenotes X…cn† and s :ˆ Dc=2 ˆ …cn‡1 cn†=2. It is easy to check that this transform preserves the

properties (39)±(41). This 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 step in the plastic phase without any iterative calculations; that is, the consistency condition is ful®lled automatically and exactly.

In Fig. 2 the closed-form solutions of the Jaumann equation are plotted to check the numerical solutions using the above scheme. It can be seen that this scheme is very accurate. Fig. 3shows the ®ve s11_{'s and s}12_'s

for the equations of Jaumann, Green±Naghdi, Sowerby±Chu, Xiao±Bruhns±Meyers and Lee±Mallett± Wertheimer. Before further comparing the ®ve corotational stress rates, let us manage to reduce the order of system (24) in the following two sections.

Fig. 3. Axial and shear stress responses for the simple shear problem based on the rates of Jaumann, Green±Naghdi, Sowerby±Chu, Xiao±Bruhns±Meyers, and Lee±Mallett±Wertheimer.

9. Spinor map from SL…2; R† onto SOo…2; 1†

Any 2  2 real symmetric matrix H can be written as H ˆ x0_x‡ x₂ 1 _x0x2_x1

 

; …43†

where x0_{, x}1_{and x}2 _{are real numbers.}7_{The determinant of H is …x}0_†2 _x1_†2 _x2_†2_.

The group SL…2; R† consists of all real 2  2 matrices U with unit determinant U ˆ U11 U12

U2 1 U22

 

; det U ˆ 1;

and is a subgroup of SL…2; C†, elements of which are often called spin transformations, so that U is a spin transformation (matrix). Since H is symmetrical, it is obvious that UHUt _{is also a 2  2 real symmetrical}

matrix. This leads us to write ^ H ˆ UHUt_{; …44†} namely, ^x0_{‡ ^x}1 _^x2 ^x2 _^x0 _^x1 " # ˆ U11 U12 U2 1 U22   x0_{‡ x}1 _x2 x2 _x0 _x1   U1 1 U21 U1 2 U22   : …45†

Taking the determinants of both sides and using det U ˆ det Ut_{ˆ 1, one readily obtains ^x… †}1 2_‡

^x2

… †2 _†^x0 2_{ˆ x… †}1 2_{‡ x… †}2 2 _†x0 2_{, ensuring that the length (more precisely, the Minkowski separation) of}

the …2 ‡ 1†-vector x ˆ …x1_{; x}2_{; x}0_†t _{is preserved by the spin transformation U : H 7! ^H. Indeed, the spin}

transformation U : H 7! ^H induces a proper orthochronous Lorentz transformation G : x 7! ^x, which is an element of SOo…2; 1†. Eq. (43) may be rearranged as

H11 H12 H22 2 4 3 5 ˆ C x 1 x2 x0 2 4 3 5; …46† where C :ˆ 10 0 11 0 1 0 1 2 4 3 5: …47†

Then Eq. (45) is equivalent to ^ H11 ^ H12 ^ H22 2 4 3 5 ˆ R HH1112 H22 2 4 3 5; …48† where R :ˆ …U 1 1†2 2U11U12 …U12†2 U1 1U21 U12U21‡ U11U22 U12U22 …U2 1†2 2U21U22 …U22†2 2 4 3 5: …49†

Applying C 1_ˆ1 2 1 0 1 0 2 0 1 0 1 2 4 3 5 _50†

to both sides of Eq. (48) and noting Eq. (46) we have ^x1 ^x2 ^x0 2 4 3 5 ˆ C 1_RC x 1 x2 x0 2 4 3 5: …51†

Comparing this with the proper orthochronous Lorentz transformation ^x1 ^x2 ^x0 2 4 3 5 ˆ G x 1 x2 x0 2 4 3 5 _52† we have G ˆ C 1_{RC: …53†}

Substituting Eq. (50) for C 1_{, Eq. (49) for R, and Eq. (47) for C in the above equation, we obtain the spinor}

map G1

1ˆ …U

1

1†2 …U21†2 …U12†2‡ …U22†2

2 ; …54† G1 2ˆ U11U12 U21U22; …55† G1 0ˆ …U 1

1†2 …U21†2‡ …U12†2 …U22†2

2 ; …56† G2 1ˆ U11U21 U12U22; …57† G2 2ˆ U11U22‡ U12U21; …58† G2 0ˆ U11U21‡ U12U22; …59† G0 1ˆ …U 1

1†2‡ …U21†2 …U12†2 …U22†2

2 ; …60† G0 2ˆ U11U12‡ U21U22; …61† G0 0ˆ …U1

1†2‡ …U21†2‡ …U12†2‡ …U22†2

2 : …62†

This map is from the group SL…2; R† onto the group SOo…2; 1† and is a two-valued representation in the

sense that the two spin transformations U map to the same proper orthochronous Lorentz transformation G. Fig. 4 summarizes the procedure for obtaining the spinor map from SL…2; R† onto SOo…2; 1†.

With formulae (54)±(62) the 2  2 matrix U suces to determine the 3  3matrix G. Therefore, instead of solving the third order fundamental solution equation

_G ˆ AG; G…0† ˆ I3; …63†

we may solve the second order fundamental solution equation

But we still lack a conversion formula from B 2 sl…2; R† to A 2 so…2; 1†. This will be formulated in the Section 10.

10. Lie algebra isomorphism of sl…2; R† onto so…2; 1†

This section relates A of the third order system (63) to B of the second order system (64), and vice versa. The relation is indeed a Lie algebra isomorphism of sl…2; R† onto so(2,1). Let us recall that G 2 SOo…2; 1†

and U 2 SL…2; R† satisfy Eqs. (63) and (64) with A 2 so…2; 1† and B 2 sl…2; R†. For the spin transformation U 2 SL…2; R†, parametrizing Eq. (44) as

H…t† ˆ U…t†H…0†Ut_{t†; …65†}

di€erentiating Eq. (65) with respect to the parameter, and using Eqs. (64) and (65) again, we obtain _H11 _H12 _H12 _H22   ˆ B11 B12 B21 B22   H11 H12 H12 H22   ‡ H11 H12 H12 H22   B11 B21 B12 B22   : …66†

Also parametrizing Eq. (48) as H11…t† H12…t† H22…t† 2 4 3 5 ˆ R…t† HH1112…0†…0† H22…0† 2 4 3 5 _67†

taking time derivative of Eq. (67), and then using Eq. (67) again, we have _H11 _H12 _H22 2 4 3 5 ˆ _RR 1 H_H11 12 H22 2 4 3 5:

Comparing the above equation with Eq. (66) yields _RR 1_ˆ 2B_B11 2B12 0 21 B11‡ B22 B12 0 2B12 2B22 2 4 3 5: …68†

For the proper orthochronous Lorentz transformation G 2 SOo…2; 1†, taking the time derivative of Eq.

(53), and using Eqs. (63), (68) and (53) again, we obtain A ˆ C 1 2B_B11 2B12 0 21 B11‡ B22 B12 0 2B12 2B22 2 4 3 5C: …69†

The inverse relation is

2B11 2B12 0 B21 B11‡ B22 B12 0 2B12 2B22 2 4 3 5 ˆ CAC 1_{: …70†}

Substituting Eq. (47) for C and Eq. (50) for C 1 _{in Eq. (69) yields}

A11 A12 A10 A21 A22 A20 A01 A02 A00 2 4 3 5 ˆ B21 0 B12 B12 0 B21 BB1211‡ BB2221 B11 B22 B12‡ B21 0 2 4 3 5 _71†

in which B11‡ B22ˆ 0 has been used. Similarly, from Eq. (70) it follows

B11 B12 B21 B22   ˆ 14…A11‡ A00‡ A10‡ A01† 12…A12‡ A02† 1 2…A21‡ A20† 14…A11‡ A00 A10 A01†   : …72†

These formulae relating the matrices A and B express explicitly the Lie algebra isomorphism sl…2; R†  so…2; 1† of the two Lie algebras sl…2; R† and so…2; 1†. Both the Lie algebras have three inde-pendent parameters.

11. Jaumann's U and G are disconjugate Substituting Eq. (37) for A in Eq. (72) gives

B ˆ₂_c 0 1 cy‡ f 1 cy f 0 " # : …73†

Thus we may examine the second order system

_Y ˆ BY; …74†

where Y ˆ …Y1_{; Y}2_†t_{, instead of the third order system (24). Eq. (74) can be further recast to the Sturm±}

Liouville form …p…c†u0_c††0_{‡ q…c†u…c† ˆ 0 …75†} either with u…c† ˆ Y1_{c†; p…c† ˆ} 2cy 1 ‡ cyf …c†; q…c† ˆ 1 2 f …c† 1 cy ! ; …76† or with u…c† ˆ Y2_{c†; p…c† ˆ} 2cy cyf …c† 1; q…c† ˆ 1 2 f …c†‡ 1 cy ! : …77†

Let c denote a real constant. A function u…c† de®ned on the interval I :ˆ ‰a; b†; 0 6 a < b 6 1 or a so-lution u…c† of Eq. (75) on I is oscillatory (resp. non-oscillatory) if u…c† is real-valued and nontrivial … 6 0† and u…c† c has an in®nite (resp. a ®nite) number of zeros on I. Speci®cally, a non-oscillatory solution u…c† of Eq. (75) is said to be disconjugate if u…c† c has at most one zero. Eq. (75) is said to be oscillatory on I if at least one real-valued nontrivial solution u…c† is such that u…c† c has an in®nite number of zeros on I. Conversely, when every real-valued nontrivial solution u…c† is such that u…c† c has at most a ®nite number of zeros on I, u…c† is said to be non-oscillatory on I. Furthermore, in the latter case, Eq. (75) is said to be disconjugate on I if every real-valued nontrivial solution u…c† is such that u…c† c has at most one zero on I.

We call a fundamental solution U satisfying Eq. (64) disconjugate (resp. non-oscillatory, oscillatory) on the interval I if every (resp. every, at least one) component of U is disconjugate (resp. non-oscillatory, oscillatory) on the interval I. Similarly, a fundamental solution G satisfying Eq. (63) is said to be dis-conjugate (resp. non-oscillatory, oscillatory) on the interval I if every (resp. every, at least one) component of G is disconjugate (resp. non-oscillatory, oscillatory) on the interval I. We call stress s non-oscillatory (resp. oscillatory) on the interval I if every (resp. at least one) component of s is non-oscillatory (resp. oscillatory) on the interval I, but for stress we will not speak of disconjugacy because the stress responses may depend heavily on the initial stresses; thus the condition of disconjugacy for stress would have been too restricted and tedious to make sense.

Now let us turn our attention to the Jaumann equation and seek the closed-form formulae for U and the corresponding G for the plastic phase. Let us substitute f ˆ 1 into Eq. (73) and solve Eq. (64) for

UJ…c† ˆ coshmc 2  1‡cy 1 cy q sinhmc 2  1 cy 1‡cy q sinhmc 2 coshmc2 2 4 3 5: …78†

For such UJ…c† we check that det UJ…c† ˆ 1 and that UJ…c† 2 SL…2; R†. Then substituting Eq. (78) into Eqs.

(54)±(62) we obtain GJ…c† ˆ 1 c2 ycosh…mc† 1 c2_y c_{1 c}y 2 y p sinh…mc† 1 1‡cy‰1 sinh…mc†Š cy  1 c2 y p sinh…mc† cosh…mc† _{}1 1 c2 y p sinh…mc† 1 1‡cy‰sinh…mc† 1Š 1  1 c2 y p sinh…mc† cosh…mc† c2y 1 c2_y 2 6 6 6 6 4 3 7 7 7 7 5: …79†

We check that the above GJ…c† satis®es properties (39)±(41); i.e., GJ2 SOo…2; 1†. In view of formulae (78)

and (79), and also of formulae (31) and (21), we obtain immediately the following:

Lemma 1. For the simple shear problem of perfect elastoplasticity using the Jaumann rate, the fundamental solutions UJand GJare disconjugate for 0 6 c < 1 and the stress responses s11and s12are non-oscillatory for

0 6 c < 1.

12. Non-oscillation criterion

Consider the following two di€erential equations:

…pj…c†u0…c††0‡ qj…c†u…c† ˆ 0; j ˆ 1; 2; …80a;b†

where pj…c† and qj…c† are real-valued continuous functions of c on an interval I ˆ ‰a; b†; 0 6 a < b 6 1. If

p1…c† P p2…c† > 0; q1…c† 6 q2…c†; …81†

then Eq. (80b) is called a Sturm majorant for Eq. (80a) on I, and Eq. (80a) is a Sturm minorant for Eq. (80b) on I.

The following lemma is an extension of Sturm's comparison theorem (see Liu and Hong, 2000). Lemma 2. Let Eq. (80b) be a Sturm majorant for Eq. (80a). If Eq. (80b) is disconjugate on the interval I, then Eq. (80a) is also disconjugate on the interval I.

Now let us consider an objective corotational stress rateswith the general plane spin x of Eq. (21) and give a sucient condition for the stress responses to be non-oscillatory.

Theorem 1 (Non-oscillation criterion). For the simple shear problem of a perfectly elastoplastic body, suppose that the function f …c† in Eq. (21) is continuous on the interval R‡_{:ˆ ‰0; 1†. If f …c† 6 1 on the interval R}‡_{, then}

the fundamental solution U satisfying Eq. (64) with B given by Eq. (73) is disconjugate on the interval R‡_{, and}

the fundamental solution G satisfying Eq. (63) with A given by Eq. (37) is non-oscillatory on the interval R‡_.

Hence the stress responses s11_{and s}12 _{for such f …c† are non-oscillatory on the interval R}‡_.

Proof. Recall that for the Jaumann rate, f …c† ˆ fJ…c† ˆ 1. If f …c† 6 fJ…c†, according to Eq. (76) (or Eq. (77))

we have p…c† P pJ…c† and q…c† 6 qJ…c†. Then the comparison made between Eq. (76) or (77) with f …c† ˆ fJ…c†

and those with f …c† 6 fJ…c† shows that the Jaumann equation is a Sturm majorant for the equation with

f …c† 6 fJ…c†. Because, according to Lemma 1, UJ and GJ are disconjugate, by Lemma 2 the fundamental

solution U of Eq. (74) for f …c† 6 fJ…c† is disconjugate and the G for f …c† 6 fJ…c† obtained from Eqs. (56)±

(62) via the multiplication, addition, and subtraction of the disconjugate functions is non-oscillatory, not necessarily disconjugate, however. The solution of Eq. (24) can be expressed in the following transition formula:

X…c† ˆ ‰G…c†G 1_c

on†ŠX…con†; 8c P con; …82†

where X :ˆ …X1_{; X}5_{; X}0_†t _{is the augmented stress. We solve Eq. (39) for the inverse}

G 1_{ˆ gG}t_{g …83†} and partition G as G ˆ Gss Gs0 G0 s G00   ; …84† where Gs

s, Gs0and G0s are of order 2  2, 2  1 and 1  2, respectively. Thus Eq. (82) can be partitioned as

follows: Xs_c† X0_c†   ˆ Gss…c†…Gss†t…con† Gs0…c†…Gs0†t…con† Gs0…c†G00…con† Gss…c†G0s…con† G0 s…c†…Gss†t…con† G00…c†…Gs0†t…con† G00…c†G00…con† G0s…c†G0s…con†   Xs_c on† X0_c on†   …85† which is valid for the plastic phase. Substituting the above X…c† into Eq. (25) we obtain the stresses s11_and

s12_{. Because X}0_{> 0 is a monotonic increasing function of c and the components of G are non-oscillatory on}

the interval ‰c_on; 1†, it follows that s11_{and s}12_{are non-oscillatory on the interval ‰c}

on; 1†. On the interval

‰0; c_on†, the body is in the elastic phase, and the governing equations reduce to hypoelasticity equations, _s xs ‡ sx ˆ 2GD, which render the stress responses non-oscillatory as proved in Liu and Hong (2000). Therefore, the stress responses are non-oscillatory for c 2 ‰0; 1†. 

13. The Jaumann equation is a Sturm majorant

Now let us compare the ®ve stress rates listed in Section 5. From the expressions for the ®ve f 's in Section 5 (and Fig. 1), we have the following inequalities:

fGN6 fJ; 8c 2 R‡;

fSC< fJ; 8c 2 R‡;

fXBM6 fJ; 8c 2 R‡;

Then from Eq. (76) or (77) it follows that pGNP pJ; qGN6 qJ; 8c 2 R‡;

pSC > pJ; qSC < qJ; 8c 2 R‡;

pXBMP pJ; qXBM6 qJ; 8c 2 R‡;

pLMWP pJ; qLMW6 qJ; 8c P 1;

con®rming that the Jaumann equation is a Sturm majorant for the equations of Green±Naghdi, Sowerby± Chu and Xiao±Bruhns±Meyers on the interval R‡_{, and of Lee±Mallett±Wertheimer in the range c P 1.}

Then using Lemmas 1 and 2 and Theorem 1 we obtain the following

Theorem 2. For the simple shear problem of a perfectly elastoplastic body, the Jaumann rate renders the stress responses non-oscillatory for c 2 R‡_{. Moreover, the Jaumann equation is a Sturm majorant for the equations}

of Green±Naghdi, Sowerby±Chu and Xiao±Bruhns±Meyers for c 2 R‡_{, and of Lee±Mallett±Wertheimer for}

1 6 c < 1. Therefore, the rates of Green±Naghdi, Sowerby±Chu, and Xiao±Bruhns±Meyers render the stress responses non-oscillatory for c 2 R‡_{, and the Lee±Mallett±Wertheimer rate renders the stress responses}

non-oscillatory for 1 6 c < 1.

Fig. 3displays the stress response curves of the model of perfect elastoplasticity with the ®ve objective corotational stress rates, which as can be seen are located in a narrow strip between Jaumann's and Sowerby±Chu's curves. The strips have a width of about 0:07syfor the axial stress s11and about 0:002syfor

the shear stress s12_{, indicating that the di€erences of f's have merely a minor in¯uence on the responses.}

Indeed, as already mentioned, in Eq. (24) f is much less than 1=cy; therefore, the control matrix A in the

plastic phase (and also in the elastic phase) is dominated by 1=cy, not by f. Fig. 3(b) shows that the Lee±

Mallett±Wertheimer rate renders slightly oscillatory responses in the initial stage. This may be attributed to the fact that fLMW> fJfor c < 1. However, its responses from c ˆ 1 on must not oscillate as guaranteed by

Theorem 2 and do not oscillate as shown in Fig. 3(b).

Moss (1984) was the ®rst to point out that for the stress rates of Jaumann and Green±Naghdi ``insta-bility'' may occur in the shear stress component at the elastic±plastic transition point of the model of perfect elastoplasticity (see Fig. 3(b)). He also noted that the instability was due to the ``inaccuracy'' of the structure of the mathematical theory. However, as illustrated in Fig. 5, where the responses based on the Green±Naghdi rate with two sets of initial conditions are compared and, in particular, the alleged ``in-stability'' disappears in the responses of the model which has been pre-stressed to s11_{ˆ 200 MPa before}

simple shearing, it would be clear that oscillation or the so-called instability depends in fact on the initial conditions prescribed, but not at all on the ``inaccuracy'' of the mathematical theory.

14. Conclusions

For the simple shear problem of a perfectly elastoplastic body, we conclude that:

1. The rates of Jaumann, Green±Naghdi, Sowerby±Chu, and Xiao±Bruhns±Meyers render the stress re-sponses non-oscillatory for the shear engineering strain c 2 R‡_{, while the Lee±Mallett±Wertheimer rate}

renders the stress responses non-oscillatory for 1 6 c < 1.

2. The Jaumann equation is a Sturm majorant for the equations of Green±Naghdi, Sowerby±Chu, and Xiao±Bruhns±Meyers for c 2 R‡_{, and of Lee±Mallett±Wertheimer for 1 6 c < 1, and so the}

3. For an objective corotational stress rate with the general plane spin the condition 1 P f …c† 2 C…R‡_{† is}

sucient to ensure the stress responses to be non-oscillatory on the interval R‡_{. In other words, to obtain}

Fig. 5. In¯uences of initial stresses on the responses (of the model of perfect elastoplasticity using the Green±Naghdi stress rate): zero initial stresses versus pre-stress s11_{ˆ 200 MPa. Notice that the alleged ``instability'' of the shear stress response in the elastic±plastic}

transition does not exist for the pre-stress case.

non-oscillatory stress responses, the plane spin x12in the objective corotational stress rate used must not

exceed the shear strain rate _c=2 exerted to the body.

4. For the Jaumann rate the fundamental di€erence of the mathematical structures of the augmented stress equations between hypoelasticity and elastoplasticity and the physical limit of the shear yield engineering strain 0 < c_y < 1 are the major factors causing their stress responses qualitatively di€erent.

To reach such conclusions we performed a Sturm-type qualitative analysis of the related equations re-sulting from a series of exact conversions summarized in Fig. 6. We converted the constitutive equations into a third order linear system, and developed a consistency scheme to calculate the responses. Then, using the Lie algebra isomorphism of sl…2; R† onto so(2;1) we found a second order linear system representation, which in turn was converted into a Sturm±Liouville equation. This series of conversions involves no ap-proximations at all and is essential for the success of the qualitative analysis.

The main tools of this paper were the exact linearization of the ¯ow model, the reduction of the third order linear system to the second order linear system, and the extension of the Sturm comparison theorem at the ®rst time to qualitatively assess the e€ects of di€erent corotational stress rates on the responses of a perfectly elastoplastic body. This methodology may be extended to investigate the simple shear problem of other more complicated models.

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