Dissipative driving force in ductile crystals and the strain localization phenomenon

Pergamon

International Journal of Plasticity, Vol. 14, Nos 10Ð11, pp. 1109±1131, 1998 # 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0749-6419/98/$Ðsee front matter

PII: S0749-6419(98)00049-7

DISSIPATIVE DRIVING FORCE IN DUCTILE CRYSTALS AND THE STRAIN LOCALIZATION PHENOMENON K. C. Le*, H. SchuÈtte and H. Stumpf Lehrstuhl fuÈr Allgemeine Mechanik, Ruhr-UniversitaÈt Bochum, 44780 Bochum, Germany (Received in ®nal revised form 26 May 1998) AbstractÐThis paper is concerned with crystals undergoing large plastic deformations. The free energy per unit volume of the reference crystal is supposed to depend on the elastic distortion as well as on constant tensors characterizing the crystal symmetry. The dissipative driving force is shown to be equal to the Eshelby stress tensor relative to the reference crystal. For single crystals obeying Taylor's equation the driving force reduces to the Eshelby resolved shear stress, which is power-conjungate to the slip rate. The Schmid law formulated with respect to the latter is used to determine the critical hardening rate at the onset of the shear band formation. # 1998 Elsevier Science Ltd. All rights reserved I. INTRODUCTION

When stresses in ductile crystals exceed some critical threshold, irreversible slip will proceed on certain active slip systems leading to plastic deformations. The fact that each slip rearrangement is governed by its associated driving force was established for small strains in the early works of Taylor and Elam (1923, 1925), Schmid and Boas (1935), and Taylor (1938a,b). Constitutive equations for ductile crystals at small strains were formulated by Mandel (1965) and Hill (1966), and extended to ®nite deformations by Rice (1971), Hill and Rice (1972), Asaro and Rice (1977), and Asaro (1983a,b). For a recent review see Havner (1992). From the thermodynamic point of view, the dissipative driving force can be derived, once the functional form of the free energy density and the entropy production inequality are speci®ed (Coleman and Noll, 1963). The present paper aims to show this derivation for crystal plasticity. We start from the assumption that the free energy per unit volume of the reference crystal for isothermal processes depends only on the lattice (elastic) distortion (Le and Stumpf, 1993, 1994). This is in contrast to the traditional internal-variable theory (Rice, 1971), which includes also slip amounts as internal variables in the list of arguments of the free energy density. The free energy density per unit volume of the initial con®guration will depend then on the Cauchy-Green deformation tensor and on the plastic distortion. Substituting the rate of the free energy density into the entropy inequality and requiring the latter to be ful®lled for arbitrary processes, one can obtain an *Corresponding author. Fax: +49-234-7094154; e-mail: [email protected]

1109

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K. C. Le et al.

`elastic part' of constitutive equations describing the lattice response, and reduce the entropy inequality to the dissipation inequality. It is shown that the dissipation is equal to the power of the Eshelby stress tensor relative to the reference crystal, regarded as thermodynamic force, on the rate of plastic deformation, regarded as thermodynamic ¯ux (Le and Stumpf, 1993). The Eshelby stress tensor relative to the reference crystal is symmetric with respect to the elastic deformation tensor de®ned on the intermediate con®guration (Epstein and Maugin, 1990). Taking this into account and choosing the elastic deformation tensor as metric of the intermediate con®guration (Maugin, 1994), it can be shown that only the symmetric part of the plastic deformation rate contributes to the dissipation. From this it follows that the symmetric contravariant driving stress tensor should enter the yield condition and the associate ¯ow rule. The Eshelby stress tensor plays a prominent role also in other continuum models with changeable microstructure such as phase transition (Truskinovsky, 1987; Knowles, 1991), brittle fracture mechanics (Le, 1989a,b; Stumpf and Le, 1990; Maugin, 1993), and dislocation theory (Le and Stumpf, 1994, 1996a,b,c). For single crystals obeying Taylor's equation the driving force reduces to the Eshelby resolved shear stress, which is power-conjungate to the slip rate. Based on this result we formulate a generalized Schmid law as follows: a slip system in a single crystal becomes active when the corresponding shear component of the Eshelby stress tensor reaches its critical value, which can depend on the amount of slips (hardening e€ect). For active slip systems we derive the constitutive equations in rate form. These equations are then applied to the strain localization phenomenon in the form of coarse slip bands or macroscopic shear bands, which is one of the most remarkable features of ductile crystals (Price and Kelly, 1964; Chang and Asaro, 1981; Spitzig, 1981; Peirce et al., 1982; Dao and Asaro, 1996). This type of instability is caused by two competing e€ects: the geometrical softening due to the elastic rotation of the conjugate slip planes, and the material hardening on the corresponding slip systems. In order to describe this phenomenon qualitatively, we need simultaneously the appropriate Hooke's and Schmid's laws. Asaro and Rice (1977) showed that Schmid's law (Schmid and Boas, 1935) in its classical form is not appropriate to explain the shear band localization in ductile single crystals undergoing large plastic deformations. However, the relationship between the constitutive equations proposed therein, which include so-called non-Schmid terms, and the entropy inequality is not clear. Duszek-Perzyna and Perzyna (1991, 1996) accepted the classical Schmid law and considered in their model the thermomechanical coupling. Unfortunately, the free energy density in their papers is not given precisely, so the reader cannot check directly, whether or not the proposed constitutive equations satisfy the entropy inequality, and it is not possible to prove the corresponding formula for the tensor of elastic moduli. In the second part of the paper we use our constitutive equations in rate form to substitute into Hill's conditions for stationary acceleration waves, regarded as the conditions of strain localization. We apply the procedure of Rudniki and Rice (1975) and Asaro and Rice (1977) to determine the critical hardening rate for single as well as double slip systems at the onset of strain localization. A comparison with the experimental data of Chang and Asaro (1981) and Spitzig (1981) shows that the theory developed in this paper can predict the onset of the shear band localization in ductile single crystals quite well. In this paper standard notations are used throughout. Bold-face letters denote vectors and second rank tensors, and letters like C,L denote fourth rank tensors. The inner product of two vectors u, v is denoted by u.v, that of covector  and vector u by (, u), and

Ductile crystals

1111

that of two second rank dual tensors A, B by hA, Bi=tr(ABT). Inverse and transpose are denoted by superscripts ÿ1 and T. The metric of the Euclidean space is denoted by g. In some cases we represent tensors also in component form, using for simplicity rectangular Cartesian co-ordinates. Unless otherwise speci®ed, upper, lower, and Greek indices take the values 1,2,3; the Einstein summation convention over repeated indices is used. The tensor product is denoted by . We use also the so-called push-forward and pull-back operations as well as Lie derivatives. For those readers, who are not familiar with these notions, we refer to Marsden and Hughes (1983) and Stumpf and Hoppe (1997). II. FREE ENERGY AND DISSIPATION

The local deformation of each in®nitesimal element of a crystal undergoing large plastic deformation is realized by the following sequence of operations (see Fig. 1): (i) the element is deformed by the plastic distortion under conditions for which lattice orientations are held ®xed; (ii) the lattice and the element deform as one with the elastic distortion. Thus, the multiplicative resolution of the deformation gradient F ˆ Fe Fp

…1†

holds true. Note that both the elastic and plastic distortions are in general incompatible in the sense that they are not gradients of any displacement ®elds. Denoting the determinants of Fe and Fp by Je and Jp , respectively, we stipulate that Je > 0; Jp > 0:

…2†

The free energy per unit volume of the reference crystal is assumed to be a function of the elastic distortion and of a set of constant tensors characterising the crystal symmetry (Smith and Rivlin 1957, Pipkin and Rivlin, 1959; DoÈring, 1958)* ˆ  …Fe ; a †;

…3†

where a denotes symbolically the set of anisotropic tensors. Using the principle of frame indi€erence one can show that (see Le and Stumpf, 1993, 1994) ˆ ^ …ce ; a †;

…4†

c e ˆ FeT gFe

…5†

where

is the elastic deformation tensor, and g the metric tensor of the Euclidean space. Thus, in contrast to the traditional internal-variable theory (Rice, 1971) there is no explicit dependence of the free energy (4) on internal variables. However, if one refers to the *We consider only isothermal processes here.

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Fig. 1. Deformations of an in®nitesimal element and of lattice orientations.

undeformed con®guration, then the free energy depends on Fp implicitly as shown by the following arguments. Since c e ˆ FpÿT CFpÿ1 , with C ˆ FT gF the Cauchy-Green deformation tensor, the free energy per unit volume of the undeformed con®guration reads ^ p ; C; a †: ˆ Jp ^ …ce …Fp ; C†; a † ˆ …F

…6†

Assuming the conservation of mass, the balance of momentum, moment of momentum, and energy, one can lay down the entropy inequality in the following form _  0; _ ÿ 1 hS; Ci 2

…7†

where S denotes the second Piola-Kirchho€ stress tensor. Since a contains time-independent tensors, the time derivative of is equal to + * + * ^ ^ @ @ p _ : _ ˆ ;C …8† ;F ‡ @Fp @C Substitution of (8) into (7) gives *

! + * + ^ S ^ @ @ p _ _ ÿ ; F  0: ;C ‡ @C 2 @Fp

…9†

We assume that (9) holds for arbitrary processes. Considering processes with F_ p ˆ 0 and with arbitrary F, from (9) one gets

Ductile crystals

1113

^ @ S ˆ 2 : @C Fp ;a

…10†

The inequality (9) is then reduced to the dissipation inequality in the following form * + ^ @ ; F_ p  0: …11† @Fp Introducing the tensor ^ j ˆ ÿ @ FpT @Fp C;a we transform the dissipation inequality (11) to hj; lp i  0;

…12†

…13†

where lp ˆ F_ p Fpÿ1 corresponds to the plastic deformation rate. By di€erentiation one can show (Le and Stumpf, 1993, Le and Stumpf, 1994) that (12) leads to pT j ˆ FpÿT …CS ÿ 1†F ^ :

…14†

The tensor (14) denotes the `push-forward' by Fp to the reference crystal of the well^ (see Eshelby, 1951) and is equal it in the special case known Eshelby tensor CS ÿ 1 p  F ˆ 1. We call j the dissipative driving force. The inequality (13) contains two mixed-variant tensors, for which symmetry is not de®ned. To discuss the symmetry of the tensors in the inequality (13) we transform it by using a metric tensor g of the intermediate con®guration (see Le and Stumpf, 1993) yielding …15† hgÿ1j; g lp i  0; where g lp is covariant and ^ g ÿ1j ˆ g ÿ1 c e  ÿ g ÿ1

…16†

 ˆ Fp SFpT

…17†

is contravariant. Here

corresponds to the symmetric second Piola-Kirchho€ stress tensor relative to the reference crystal. Choosing now in (17) g ˆ c e (Maugin, 1994) we obtain the symmetric tensor ^  ˆ c eÿ1j ˆ  ÿ c eÿ1 :

…18†

With (18) the dissipation inequality reduces to  d p i  0; h;

…19†

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K. C. Le et al.

where

1 d p ˆ …celp ‡ lpT c e †: 2

…20†

Thus, there is no contribution of the unsymmetric part of c elp in the dissipation inequality (19). It is easy to see that when the plastic distortion is isochoric, which means Jp ˆ 1, p  ^ in (18) does not contribute to the dissipation. The remaining trl ˆ 0, the term ÿceÿ1 1 part of the driving force reduces to  ˆ 

…21†

The inequality (19) can be used to justify the following ¯ow rule for crystals  ˆ @d p Dp ;

…22†

Dp ˆ Dp …d p ; a †  0;

…23†

with

The dissipation function Dp is supposed to be positive de®nite, convex and homogeneous of ®rst degree with respect to its argument d p . The symbol @ is used to denote the subdi€erential of convex functions (Moreau, 1970). III. CONSTITUTIVE EQUATIONS FOR SINGLE CRYSTALS

For single crystals the plastic distortion is realized by slips on the active slip systems. …k† Each slip system is characterized by a pair of covector m0 , being the unit normal to the …k† …k† …k† slip planes, and unit vector s0 denoting the slip direction, for which hm0 ; s0 i ˆ 0. For p single crystals there is only a ®nite number of slip systems. The rate of F satis®es Taylor's fundamental equation F_ p ˆ

N X p … _ …k† s0…k† m…k† 0 †F ;

…24†

kˆ1

with …k† the amount of slip on the kth slip system, where the dot denotes the material time derivative. One of the obvious consequences of (24) is that the plastic distortion is always isochoric d J_p ˆ …det Fp † ˆ 0: dt …k†

…k†

While the lattice vectors s0 and covectors m0 remain unchanged under slip, they transform by elastic distortion according to …k† ˆ FeÿT m…k† s…k† ˆ Fe s…k† 0 ;m 0 :

…25†

Ductile crystals

1115

Substituting Taylor's equation (24) into the dissipation inequality (13) we obtain N X

j…k† _ …k†  0;

…26†

kˆ1

where …k† …k† pÿT CSFpT ; s…k† j…k† ˆ hj; s…k† 0 m0 i ˆ hF 0 m0 i:

…27†

The dissipation inequality (26) will be satis®ed, if we formulate the generalized Schmid law in the following form j…k† ˆ

@D… …k† ; _ …k† † ; @ _ …k†

…28†

where D… …k† ; _ …k† † is the so-called dissipation function, which is homogeneous of ®rst degree with respect to _ …k† . For instance, we can assume that Dˆ

N X kˆ1

…1† …N† …k† j…k† j _ j; cr … ; . . . ;

…29†

…k†

where jcr … † are positive functions. Using (28), (29), one can show that …k† … † ) _ …k† ˆ 0; jj…k† j < jcr …k† …k† j ˆ jcr … † ) _ …k†  0; …k† …k†  0: j ˆ ÿj…k† cr … † ) _

…30†

This constitutive law di€ers from the well-known Schmid law (see Schmid and Boas, 1935) in having the Eshelby stress tensor (14) instead of the Cauchy stress tensor. Slip systems whose shear component of the Eshelby stress tensor reaches its critical value leading to _ …k† > 0 will be called active. Let us assume that there are M active slip systems, which can be reordered so that k ˆ 1; 2; . . . ; M. For these active slip systems with _ …k† > 0 the equations j…k† ˆ j…k† cr … †; k ˆ 1; . . . ; M

…31†

hold. Di€erentiation of eqn (31) with respect to t gives M M …k† X d …k† X @jcr …l† _ j ˆ ˆ hkl _ …l† ;

…l† @ dt lˆ1 lˆ1

where hkl ˆ is the matrix of strain hardening rates.

@j…k† cr @ …l†

…32†

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K. C. Le et al.

We are going to derive the constitutive equation in rate form for metals and alloys, whose elastic strain 1 …33† e e ˆ …ce ÿ g† 2 remains typically small. Here c e is de®ned as the push-forwards with Fp of C to the reference crystal. The free energy per unit volume of the reference crystal will be given by 1  : e e ; ˆ ee : C 2

…34†

 is the fourth rank tensor of elastic moduli, and the double dot denotes the conwhere C  satisfy the following symmetry traction of two indices. The components of the tensor C properties    ˆ C    ˆ C   :    ˆ C C According to (34) we derive Hooke's law in the form  : e e ;  ˆ C

…35†

where  is the push-forwards with Fp of S to the reference crystal. Applying the intermediate Lie-derivative L to (35) (see Le and Stumpf, 1994), we obtain  : e e :  : …L  ee † ‡ …L C† L  ˆ C

…36†

Using (24) and (33) we can show that  ee ˆ d ÿ L

M X

_ …k† p …k† ;

…37†

kˆ1

_ to the reference crystal and where d is the push-forwards with Fp of 1=2C 1 …k† …k† …k† p …k† ˆ ‰…m…k† 0 s0 †g ‡ g…s0 m0 †Š: 2

…38†

Note that the tensors p…k† are symmetric. Introducing the intermediate Lie derivative of  in the form the tensor of elastic moduli C X  ˆÿ …39†

_ …k† W…k† ; L C kˆ178M

we can present eqn (36) as follows  :dÿ L  ˆ C

M X kˆ1

 …k† ;

_ …k† R

…40†

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1117

 : p …k† ‡ W…k† : e e :  …k† ˆ C R

…41†

with

 …k† for elastic isotropy will be considered in the next section and The determination of W  …k† are symmetric. the result is presented in eqn (70). Note that the second rank tensors R …k† We still need to express _ in (40) through the stress and strain rates. Di€erentiating (27) and substituting the result into (32) we have  pÿT  X M d …k† dF …k† pT pÿT _ pT pÿT pT …k† _ _ j ˆ CSF ‡ F …CS ‡ CS†F ‡ F CSF ; s0 m0 ˆ hkl _ …l† : …42† dt dt lˆ1 Using (24) we calculate dFpÿT =dt and F_ pT F_ pT ˆ

M X lˆ1

…l† FpT … _ …l† m…l† 0 s0 †;

M X dFpÿT …l† pÿT ˆ ÿFpÿT F_ pT FpÿT ˆ ÿ … _ …l† m…l† : 0 s0 †F dt lˆ1

Substituting this into (42) we obtain _ ‡ CS†F _ pT ; s…k† m…k† i ˆ hFpÿT …CS 0 0

M X

h^kl _ …l† ;

…43†

lˆ1

where …l† …l† e …k† …l†  …l†  s…k† c ; h^kl ˆ hkl ÿ h0kl ; h0kl ˆ hce …m 0 so † ÿ …m0 s0 † 0 m0 i:

…44†

Note that h0kl ˆ 0 for a single slip system. Equation (43) can also be written in the form  s0…k† m0…k† i ˆ h2d  ‡ c e L ;

M X

h^kl _ …l† ;

…45†

lˆ1

By pushing (45) forward with Fe and using (25) one obtains the Eulerian form of (45) h2d ‡ gL; s…k† m…k† i ˆ

M X

h^kl _ …l† ;

…46†

lˆ1

with d the strain rate,  ˆ J the Kirchho€'s stress tensor, and L the spatial Lie-derivative. Substituting (40) into (45), one can express _ …k† as follows

_ …k† ˆ

M X lˆ1

 : d†;   ‡ c e …C  s…l† m…l† i ˆ hT …k† ; di;  h~ÿ1 kl h2d 0 0

…47†

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where T …k† ˆ

M X lˆ1

 …l† h~ÿ1 kl V ;

…48†

  …k† ˆ 2…s…k† m…k† † ‡ …ce …s…k† m…k† †† : C; V 0 0 0 0

…49†

 eR  …l† ; s…k† m…k† i: h~kl ˆ h^kl ‡ hC 0 0

…50†

Combining (40) and (47), we obtain the constitutive equation in rate form for the stress tensor   L  ˆ L : d; where

M X

 ÿ L ˆ C

 …k† T …k† : R

…51†

…52†

kˆ1

Pushing (51) forward with F e gives the Eulerian form of the rate constitutive equation L ˆ L : d; with LˆCÿ

M X

R…k† T…k† ;

…53†

…54†

kˆ1

and R…k† ˆ C : p…k† ‡ W…k† : Ee ;

T…k† ˆ

M X lˆ1

…l† h~ÿ1 kl V ;

V…k† ˆ 2…s…k† m…k† † ‡ …g…s…k† m…k† †† : C:

…55†

…56†

…57†

IV. CONDITION FOR LOCALIZATION

_ ÿ1 inside and We assume the following jump condition for the velocity gradient l ˆ FF outside a planar band ‰‰lŠŠ ˆ k n; …58† where ‰‰:ŠŠ ˆ …:†‡ ÿ …:†ÿ denotes the jump of the corresponding quantity in the squared brackets, n is the normal to the `stationary' surface of weak discontinuity and k the vector

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1119

describing the amplitude of the jump. The velocity as well as the stress are assumed to be continuous. Hill (1962) has derived the following jump condition for the stress rate at the singular surface _ ‰‰ŠŠn ˆ 0:

…59†

The time rate of the Cauchy stress tensor can be presented in the form 1 _ ˆ …L ‡ l ‡ lT ÿ trl†:  J

…60†

Substituting eqns (60) and (53) into eqn (59) and using the symmetry of , we arrive at the following equation ‰n
…C ÿ

N X kˆ1

[ R…k† T…k† sym †
n ‡ AŠk ˆ 0;

…61†

where 1 …k† ˆ …T…k†T †; A ˆ hn; nigÿ1 ; k[ ˆ gk: Tsym 2

…62†

Since k[ should not vanish, the localization condition for multiple active slip systems reads det‰n…C ÿ

M X kˆ1

R…k† T…k† sym †
n ‡ AŠ ˆ 0:

…63†

Relative to the `intermediate' description this condition takes the form  ÿ det‰n
…C

M X kˆ1

 ˆ 0:  …k† T …k† †
n ‡ AŠ R sym

…64†

In the case of isotropy one can establish the following useful relationship  …k† ˆ V  …k† : R…k† ˆ V…k† sym ; or R sym

…65†

 is speci®ed by  …k† are given by (41), while the tensor C Let us prove (65)2. The tensors R    ˆ lg ‡ …g g  ‡ g g  †; C

…66†

with l,  Lame's constants for isotropic materials. Note that M X  ˆ FpÿT d …FpT gFp †Fpÿ1 ˆ 2

_ …k† p …k† ; Lg dt kˆ1

with p …k† from (38). Therefore

…67†

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K. C. Le et al.

 ÿ1 ˆ ÿgÿ1 …Lg†g  ÿ1 ˆ ÿ2 Lg

M X

_ …k† q …k†;

…68†

kˆ1

where

1 …k† …k† …k† ÿ1 q …k† ˆ gÿ1 p …k† gÿ1 ˆ ‰gÿ1 …m…k† 0 s0 † ‡ …s0 m0 †g Š: 2

…69†

According to (39) and with (68)-(69) we obtain  …k†  ˆ 2l…q …k† g  ‡ g q …k†  † ‡ 2…q …k† g  ‡ g q …k†  ‡ q …k† g ‡ g q …k† †: …70† W Substituting now eqns (66) and (70) into eqns (41), (35) and (66) into eqn (49), and using  and q , one can prove the validity of (65)2. Equation (65), the symmetry properties of C can be obtained by push-forward with Fe . Thus, for isotropic materials the localization condition (64) reduces to  ÿ det‰n:…C

M X M X kˆ1 lˆ1

 ˆ 0:  …k† R  …l† †
n ‡ AŠ h~ ÿ1 kl R

…71†

If only one slip system is active, one can further transform the localization condition (63) or (64) to a more convenient form. Denoting B ˆ n
C
n and multiplying (61) with Bÿ1 to get ‰1 ÿ Bÿ1 …n
…R Tsym †
n† ‡ Bÿ1 AŠk[ ˆ 0:

…72†

We drop here the index of this active slip system for short. Denoting H ˆ 1 ‡ Bÿ1 A and multiplying the last equation with Hÿ1 , we obtain ‰1 ÿ Hÿ1 Bÿ1 …n
…R Tsym †
n†Šk[ ˆ 0;

…73†

or, using the symmetry of R, ‰1 ÿ …Hÿ1 Bÿ1 Rn† …Tsym n†Šk[ ˆ 0:

…74†

Upon multiplying (74) with h:; Tsym ni, we obtain …1 ÿ hHÿ1 Bÿ1 Rn; Tsym ni†hk[ ; Tsym ni ˆ 0:

…75†

Since the second factor hk[ ; Tsym ni in (75) cannot vanish, the only condition allowing a non-zero k[ is that the ®rst factor in (75) vanishes, which gives the critical h at localization as  s0 m0 i ‡ hHÿ1 Bÿ1 Rn; Vsym ni: h ˆ ÿhce R;

…76†

Pulling back all tensors in the second term of the right-hand side of (76) with Fe and using the fact, that the operation of taking inverse can be interchanged with that of pull-back, we can present (76) in the form

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 s 0 m 0 i ‡ hH  n; V  sym n i;  ÿ1 B ÿ1 R h ˆ ÿhce R;

…77†

 given by (49) and n ˆ FeT n. with V V. ASYMPTOTIC ANALYSIS FOR SINGLE SLIP

Since the elastic strains e e are assumed to be small, let us keep in (77) the principal terms of orders O…1† and O…ee † and neglect all other terms of order O…ee2 † and higher.  can be written as According to (41) the tensor R  1;  ˆR 0 ‡R R

…78†

 : p ; R  : e e : 1 ˆ W 0 ˆ C R

…79†

where

 from (49) in the form Using the de®nition (30) we can represent V  1;  ˆV 0 ‡V V

…80†

 ˆR  : e e † ‡ 2…ee …s0 m0 †† : C:   0; V  1 ˆ 2…s0 m0 †…C  ˆ …g…s0 m0 †† : C V

…81†

with

 is of the order O…E e †, we can approximate H  ÿ1 as follows Since A   ÿ1 ˆ 1 ÿ B ÿ1 A: H

…82†

Taking (78)±(82) into account, the asymptotic expansion for h reads h ˆ h0 …n† ‡ h1 …n†;

…83†

 0sym n i;  0 n ; V  0 ; s0 m0 i ‡ hB ÿ1 R h0 …n† ˆ ÿhgR

…84†

 1 ‡ 2ee R  1sym n i ‡ hB ÿ1 R  0sym n i  0 ; s0 m0 i ‡ hB ÿ1 R  0 n ; V  1 n ; V h1 …n† ˆ ÿhgR ÿ1 ÿ1  B R  0sym n i:  0 n ; V ÿ hB A

…85†

where

and

In order to calculate the critical orientation n , as ®rst approximation we replace (83) by h ˆ h0 …n†:  0 into (84), we obtain  0; V Substituting R

…86†

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K. C. Le et al.

 : p ‡ h…p : C†  n; B ÿ1 …C : p †ni ˆ ÿp : N  : p ; h0 …n† ˆ ÿp : C

…87†

 are given by where the components of the fourth rank tensor N    n  :    ÿ C   l n  …B ÿ1 † C    ˆ C N l

…88†

 and that  satis®es the same symmetry properties as C Note that N  ˆ N
 n ˆ 0: n
N

…89†

 is the tensor of elastic moduli governing plane stress states Now we want to show that N in a plane perpendicular to n and that, therefore, the quadratic form (87) is nonpositive de®nite. Let the unit covectors u , v , n form an orthogonal triad. We de®ne f ˆ 2eenu u ‡ 2eenv v ‡ eenn n ;

…90†

e e? ˆ eeuu u u ‡ eeuv …u v ‡ v u † ‡ eevv v v :

…91†

and

According to (90) and (91) 1 e e ˆ e e? ‡ …n f ‡ f n †: 2

…92†

Substituting (92) into the free energy density (34), we have 1  : e e ‡ hf;  : e e †ni ‡ 1 hf; B fi:   …C ˆ e e? : C ? ? 2 2

…93†

 Then the following condition should We minimize this quadratic form with respect to f. be ful®lled  : e e †n ˆ 0: B f ‡ …C ?

…94†

Solving this equation and substituting f into (93), we ®nd min f

1  e 1 e   : e e †ni ˆ 1 e e : N  : e e ; ˆ e e? :C : e ? ÿ h…e? : C†n; B ÿ1 …C ? ? 2 2 2 ?

…95†

 from (88). Equation (94) means that the stress tensor  has no components assowith N ciated with the normal n . It is also evident that the quadratic form (95) is positive de®nite in e e? . On the other hand, by (89) we have  : e e ‡ …N
 n †
f ˆ N  : ee ;  : e e ˆ N N ? ? and thus the quadratic form (95) can also be written as

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1 e  e e : N : e : 2 We see, therefore, that (95) is a positive de®nite quadratic form of e e? but a nonnegative de®nite quadratic form of e e . In particular  : e e ˆ 0 if and only if e e ˆ 0: e e : N ? Using these results in (86), (87), we see that the critical value of h for localization on a plane of normal n must be either negative or zero, the latter occurring when …i† n ˆ m0 or …ii† n ˆ s[o

…96†

Thus, when we approximate (83) by (86), we ®nd that the critical hardening rate at the inception of localization is hcr ˆ 0;

…97†

and the plane of localization is either the slip plane (i.e. case (i), n ˆ m0 ) or a plane that we shall refer to as the kink plane (case (ii), n ˆ s[0 ). Next, let us analyze in (83) the in¯uence of the terms of order O…ee †. We expand (83) in a series with respect to n , ®rst about n ˆ m0 and later about n ˆ s[0 . For the perturbation about n ˆ m0 we write  n ˆ m0 ‡ ;

…98†

where  is assumed to be small and should be chosen so that n is a unit covector. We introduce also the second rank tensors  0 ; E ˆ m0
C
 ;  ;  ˆ 
 ˆ m0
C
m  H  C
 M

…99†

and we observe that by the representation (38) for p  : p ˆ s[ Ms  : p †n ˆ …M  [;  ‡ E†s  [ ; p : C …C 0 0 0

…100†

 n . Using (98), (99), one can see where s[0 ˆ gs0 . We calculate now the inverse of B ˆ n
C
that  ‡ E ‡ E T ‡ H†  ÿ1 ˆ ‰1 ‡ M  ÿ1 …E ‡ E T ‡ H†Š  ÿ1 M  ÿ1 : B ÿ1 ˆ …M

…101†

 we expand (101) to obtain Since  is small, so also are E and H,  ÿ1 ÿ M  ÿ1 …E ‡ E T †M  ÿ1 ÿ M  ÿ1 HM  ÿ1 …E ‡ E T †M  ÿ1 …E ‡ E T †Mÿ1 ‡ . . . ;  ÿ1 ‡ M B ÿ1 ˆ M …102†  After some standard matrix where the neglected terms are of third and higher orders in . calculations we ®nd that

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 ÿ1 E†s  [ i:  ÿ E T M h0 ˆ ÿhs[0 ; …H 0

…103†

Using (98) and (99), we can rewrite (103) in the form  [ †…n ÿ m0 †i; h0 ˆ ÿhn ÿ m0 ; …s[0
M
s 0

…104†

 are where the components of the fourth rank tensor M    ÿ C   l m0 …M    m0    ˆ C  ÿ1 †l C M

…105†

 in (88) when n is set equal to m0 . Since n ÿ m0 has and correspond to the components of N no component in the direction of m0 to the order considered (n and m0 are unit covectors), one can see that (104) is a negative de®nite quadratic form. We consider now the expression for h1 of (85). We collect ®rst all terms of the order O…ee †, which can be obtained by setting in (85) n ˆ m0  1 ‡ 2ee R0 ; s0 m0 i ‡ hMÿ1 R  1sym m0 i ‡ hM  ÿ1 R  0 m0 i  0 m0 ; V  1 m0 ; R h10 ˆ ÿhgR  0M  ÿ1 R  0 m0 ; R  0 m0 i;  ÿ1 A ÿ hM where

…106†

 0 ˆ hm0 ; m  0 igÿ1 : A

 1; V  1 in (106) and using the symmetry properties of  0; R Substituting the expressions for R   C and W, one can prove that h10 ˆ 0:

…107†

Note that eqn (107) holds true in the most general case of anisotropy. We collect now terms of the order O…ee † in (85). To do this, we should use the asymptotic expansion (102) and  ˆA  0 ‡ 2h;  0 igÿ1 :  m A  after Keeping in (85) all terms of the order O…ee † and making use of the symmetry of C, some algebraic transformation we have  ÿ1 Es  [; V  1 m 0 ; s[ i  1sym i  1sym m  1 ;  ÿ1 R  ÿ hM  0 i ‡ hR  s[0 i ÿ hE M h11 ˆ hs[0 ; V 0 0  1 s[ ; s[ i:  0 s [ ; s [ i ÿ hA  ÿ1 A ‡ 2hE T M 0

0

0

…108†

0

For isotropic materials the identities (65) hold true, so (108) reduces to  1 s[ ; s[ i:  0 s [ ; s [ i ÿ hA  1 m0 ; s[ i ‡ 2hE T M  ÿ1 A  1 ;  ÿ1 R  s[0 i ÿ 2hE M h11 ˆ 2hR 0 0 0 0 0

…109†

Let us choose the Cartesian coordinates aligned with the directions of the slip system. The indices s, m, z denote then projections on the corresponding coordinates (no sum on  ÿ1 is given by repeated indices of form s, m, or z). In matrix form M

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01



 ÿ1 ˆ B M @0 0

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1 0 0C A:

0 1 l‡2

1 

0

…110†

 1 , are proportional to the identity tensor  0 and A The tensors A  1 ˆ 4…ee s ‡ ee z †1;  0 ˆ …lI ‡ 2ee †1; A A mm sm mz

…111†

where I ˆ eess ‡ eemm ‡ eezz : Calculation of E according to (99)2 gives 0 0 E ˆ @ ls 0

s 0 z

Finally, the tensor R1 is obtained as 0 …2l ‡ 4†eesm  1 ˆ @ lI ‡ 2…ee ‡ ee † R ss mm 2eemz

1 lI ‡ 2…eess ‡ eemm † 2eemz …2l ‡ 4†eesm 2eesz A: e 2esz 2leesm

1 0 lz A: 0

…112†

…113†

Substitution of these formulae into eqn (109) leads to h11 ˆ 4eesm s :

…114†

Combining eqns (104), (107) and (114), we obtain ®nally h ˆ ÿ…2z ‡ 42s † ‡ 4eesm s ‡ O…ee2 ; 3 ; ee 2 †;

…115†

where  ˆ …l ‡ †=…l ‡ 2†. Thus, the critical hardening rate hcr is achieved at s ˆ

1 e e ; z ˆ 0 2 sm

…116†

and is equal to 1 hcr ˆ …eesm †2 : 

…117†

Corresponding calculations can be carried out for the case (ii), in which we apply a perturbation about the kink-plane n ˆ s[0 . The critical orientation is given by  n ˆ s[0 ‡ ;

…118†

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with m ˆ

1 e e ; z ˆ 0; 2 sm

…119†

and the critical hardening rate by the same formula (117). VI. ASYMPTOTIC ANALYSIS FOR DOUBLE SLIP

Outside of certain hexagonal metals, single slip is the exception rather than the rule in single crystals. Even when a favorable single slip is activated initially, lattice rotations rotate the slip planes towards the stress axis so that at a certain moment double slip occurs in conjugate systems. In order to analyze this situation, we should consider the localization condition relative to the current con®guration, which. for isotropic materials, is given by (71). We keep in this equation terms of orders O…1† and O…ee †. Let us recall that Cabcd ˆ l…ceÿ1 †ab …ceÿ1 †cd ‡ ‰…ceÿ1 †ac …ceÿ1 †bd ‡ …ceÿ1 †ad …ceÿ1 †bc Š:

…120†

This equation is obtained from (66) by the push-forward operation with Fe . Since e e is assumed to be small, the same is true of its counterpart ee within the current description 1 …ee †ab ˆ …Feÿ1 †a …ee † …Feÿ1 †b ˆ …gab ÿ …ce †ab †  1: 2

…121†

Using the smallness of ee , one can show that ceÿ1  gÿ1 ‡ 2gÿ1 ce gÿ1 :

…122†

 1;  ˆC 0 ‡C C

…123†

 abcd ˆ lgab gcd ‡ …gac gbd ‡ gad gbc †; C 0

…124†

From (120) and (122) it follows

with

 abcd ˆ 2l‰…ee] †ab gcd ‡ gab ee] †cd Š ‡ 2‰…ee] †ac gbd ‡ gac …ee] †bd ‡ …ee] †ad gbc ‡ gad …ee] †bc Š: …125† C Here ee] ˆ gÿ1 ee gÿ1 denotes the tensor associated to ee . Let us analyze now R…k† . Due to (65) and (57) we have  R…k† ˆ …s…k† m…k† † ‡ …m…k† s…k† † ‡ …g…s…k† m…k† †† : C:

…126†

~ …k† by Let us introduce the unit vectors s~ …k† and covectors m s~ …k† ˆ

s…k† m…k† …k† ~ ; m ; ˆ js…k† j jm…k† j

…127†

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1127

where js…k† j and jm…k† j are the lengths of s…k† and m…k† , respectively. It is easy to show that …k† ~ …k† …1 ÿ ee…k† ˆm s…k† ˆ s~ …k† …1 ‡ ee…k† ss †; m mm †;

…128†

where e …k†  …k† e …k†] ~ ; S i; ee…k† ~ ;m ~ …k†] i: ee…k† ss ˆ he s mm ˆ he m

substituting (128) into (126) and recalling that   ee , we have

where

R…k†  R0…k† ‡ R…k† 1 ;

…129†

~ …k† †† : C0 ; s…k† m R…k† 0 ˆ …g…~

…130†

and e…k† ~ …k†  ‡ …m ~ …k† s~ …k† † ‡ …g…~s…k† m ~ …k† †† : C0 …ee…k† ~ …k† †† : C1 : s…k† m s…k† m R…k† ss ÿ emm † ‡ …g…~ 1  …~

…131† We analyze further the matrix h~kl given by (50), which can be rewritten relative to the current con®guration as follows h~kl h^kl ‡ hgR…l† ; s…k† m…k† i:

…132†

Taking (128)±(131) into account, we can get the following approximation …l† e…k† ~ …k† i ‡ hR…l† ~ …k† i…ee…k† ~ …k† i: …133† s…k† m s…k† m s…k† m h~kl ˆ h^kl ‡ hR…l† ss ÿ emm † ‡ hR1 ; g~ 0 ; g~ 0 ; g~

Finally. the tensor A is given by eqn (62). Let us consider now for simplicity the two-dimensional model illustrated in Fig. 2. We assume that in the current con®guration there are two active slip systems symmetrically oriented about the tensile axis where  is the angle each slip plane makes with it; the crystal is subject to the stress along the x2 axis. The angle between the direction of the shear band and the x2 axis is denoted by . It is easy to see that only the in-plane components of stress   0 0 ˆ ; …134† 0  and strain e

e ˆ



?

ÿ 8 0

0



3 ? 8

…135†

are involved, with  ? ˆ = (Poisson's coecient is taken to be  ˆ 1=3). We assume furthermore that h11 ˆ h12 ˆ h21 ˆ h22 ˆ h;

…136†

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Fig. 2. Con®guration of double slip systems.

This assumption simpli®es considerably the solution of the bifurcation problem. Due to (134) and the symmetry of the double slip system in the current con®guration, one can …k† show that the matrix (44) vanishes, h0kl ˆ 0. Calculating now the matrices h~ÿ1 kl , B, R , and A and ®nally the determinant (71), one gets the following analytical solution h sin2 2…cos 2 cos 2 ÿ cos2 2 † : ˆ  2 sin2 2 cos 2

…137†

which corresponds to the result of Asaro (1979). Note that we do not use here the incompressibility condition as in Asaro's analysis. Taking for example  ˆ 30 , we plot in

Fig. 3. Hardening rate versus angle .

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1129

Fig. 4. Misorientation angle cr versus .

Fig. 5. Critical hardening rate versus .

Fig. 3 the graph of the function (137) versus . The maximum of h= in (137) is found at ˆ 37; 2288 and equal to hcr = ˆ 0:0502405. Duszek-Perzyna and Perzyna (1996) found for double slip hcr = ˆ 0:417 at ˆ 36:3 . The experimental data of Spitzig (1981) and Chang and Asaro (1981) give the critical value for h= in the interval 0.030.06 and the angle of misorientation between 5 6 , which are in excellent accord with our theoretical prediction. In Fig. 4 the graph of the misorientation angle  ˆ cr ÿ  is plotted. Finally, in Fig. 5 we show the dependence of …h=†cr on the angle . VII. CONCLUDING REMARKS

In this paper we have shown how to formulate the constitutive equations for crystals undergoing large plastic deformation. We propose also the correction of the Schmid law

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for single crystals. In those problems, where small terms play an important role, such as in the problem of strain localization, it is shown that this correction can improve essentially the prediction of the theory, maintaining at the same time the logical consistency of the constitutive equations with the entropy production inequality. AcknowledgementsÐThe authors gratefully acknowledge the constructive proposition of the ®rst reviewer.

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Rice, R.J. (1971) Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433. Rudinki, J.W. and Rice, J.R. (1975) Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371. Schmid, E. and Boas, W. (1935) KristallplastizitaÈt, Springer Verlag, Berlin. Smith, G.F. and Rivlin, R.S. (1957) The anisotropic tensors. Quarterly of Appl. Math. 15, 308. Spitzig, W.A. (1981) Deformation behavior of nitrogenated Fe±Ti±Mn and Fe±Ti single crystals. Acta Metall. 29, 1359. Stumpf, H. and Hoppe, U. (1997) The application of tensor algebra on manifolds to nonlinear continuum mechanics. ZAMM 77, 327. Stumpf, H. and Le, K.C. (1990) Variational principles of nonlinear fracture mechanics. Acta Mechanica 83, 25. Taylor, G.I. (1938) Plastic strain in metals. J. Institute of Metals 62, 307. Taylor, G.I. (1938b) Analysis of plastic strain in a cubic crystal. In Stephen Timoshenko 60th Anniversary Volume, pp. 218±224. McMillan Co., New York. Taylor, G.I. and Elam, C.F. (1923) The distortion of an aluminum crystal during a tensile test. Proc. Royal Soc. London. A102, 643. Taylor, G.I. and Elam, C.F. (1925) The plastic extension and fracture of aluminum single crystals. Proc. Royal Soc. London A108, 28. Truskinovsky, L. M. (1987) Dynamics of non-equilibrium phase boundaries in a heat conducting nonlinearly elastic medium. J. Appl. Math. Mech. (PMM) 51, 777.