Finite elastic–plastic deformation behaviour of crystalline solids based on a non-associated macroscopic flow rule1

Pergamon

International Journal of Plasticity, Vol. 14, No. 12, pp. 1189±1208, 1998 # 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0749-6419/98/$Ðsee front matter

PII: S0749-6419(98)00047-3

FINITE ELASTIC±PLASTIC DEFORMATION BEHAVIOUR OF CRYSTALLINE SOLIDS BASED ON A NON-ASSOCIATED MACROSCOPIC FLOW RULE1 Michael BruÈnig* and Hans Obrecht Lehrstuhl fuÈr Baumechanik-Statik, UniversitaÈt Dortmund, August-Schmidt Str 8, D-44221 Dortmund, Germany (Received in ®nal revised form 6 June 1998) AbstractÐThe present paper deals with a nonlinear ®nite element analysis of the macroscopic elastic±plastic deformation and localization behaviour of crystalline solids. The description includes elastic strains arising from the distortion of the lattice as well as crystallographic deformations due to irreversible microscopic slip along preferred lattice planes and in corresponding lattice directions. In addition, the e€ect of plastic volume changes on the microstructure is taken into account. Macro- and microscopic stress measures are related to Green's macroscopic strains via a hyperelastic constitutive law based on a free energy potential function, and the onset of plastic yielding on the microscale is described by a modi®ed yield condition which includes appropriate microscopic stress components. To be able to compute inelastic deformations from plastic potentials, the latter are expressed in terms of work-conjugate microscopic stress and strain measures thus leading to a non-associated ¯ow rule for the macroscopic plastic strain rate. The numerical integration of the rate formulation of the constitutive equations is performed using a plastic predictor-elastic corrector technique. Its implementation into a nonlinear ®nite element program is discussed, and numerical solutions of ®nite strain elastic-plastic boundary value problems involving highly localized deformations are presented. They demonstrate both the eciency of the algorithm and the in¯uence of various model parameters on the deformation and load-carrying behaviour of crystalline solids. # 1998 Elsevier Science Ltd. All rights reserved I. INTRODUCTION

An important problem in computational plasticity is the modelling and prediction of the macroscopic elastic±plastic deformation and localization behaviour of crystalline metals whose behaviour on the microscale is governed by rate-independent slip over crystallographic planes on the one hand and reversible elastic lattice distortions on the other. As discussed for example by Lin (1971), Asaro (1983), Aifantis (1987) and Havner (1992), microscopically based continuum approaches usually start from heuristic experimental observations and elementary mechanical assumptions concerning the microscopic constitutive behaviour, and the overall macroscopic response of the solid is then obtained from suitable averaging processes. Most mathematical and physical formulations of this 1

Paper presented at the Sixth International Symposium on Plasticity and Its Current Applications, Juneau, Alaska (U.S.A.), July 14±18, 1997. *Corresponding author. Fax: +49-231-755-2532; e-mail: [email protected]

1189

1190

M. BruÈnig and H. Obrecht

kind build on the original ideas of Taylor (1938), who noted that the motion of crystal dislocations represents the fundamental atomistic mechanism for the micro- and macroscopic processes associated with slip and strain hardening. The time-independent framework for modelling the ®nite deformation behaviour presented by Hill (1966, 1967), Rice (1971), Hill and Rice (1972, 1973), and Hill and Havner (1982) is based on fundamental concepts of single crystal plasticity. In particular, it involves the kinematic assumption that the plastic deformation of crystalline solids is primarily due to crystallographic slip and that the structural rearrangement of the crystal may be described by homogeneous simple shears occurring on certain slip systems. Using averaging techniques, this leads to macroscopic plastic deformations which are isochoric. The extensive tests of Spitzig et al. (1975) on the behaviour of high-strength metals undergoing uniaxial tension and compression, on the other hand, indicate that plastic deformation may also be accompanied by a permanent volume expansion and that the latter is largely insensitive to the sign of the mean stress. Moreover, their experimentsÐ which were conducted over a suciently large pressure rangeÐcast some doubt on the general validity of the macroscopic normality rule usually employed in continuum plasticity. They showed, for example, that when predictions were based on the normality rule in combination with an experimentally determined pressure-dependent macroscopic yield condition, noticeably higher values of the plastic volume change were obtained than in the experiments. This suggests that the normality concept may not always be appropriate to describe the large deformation behaviour of high-strength metals. Typically, the microscopic formulations mentioned above make use of Schmid's law, which assumes that on the microscale plastic yielding will begin on that slip system for which the resolved shear stress reaches a critical value. The latter is independent of the overall orientation of the tensile axis and, thus, of the other resolved components of the stress tensor. Generally, experimental results obtained from uniaxial tension and compression tests tend to con®rm Schmid's rule, but counter-examples suggest that nonSchmid e€ects also exist. Moreover, it is generally recognized that the ¯ow stresses of some metals are larger in uniaxial compression than in uniaxial tension. This phenomenonÐknown as the strength di€erential (SD-)-e€ectÐhas been studied extensively in recent years, and it is now well established that the magnitude of the SD-e€ect increases with the amount of carbon in the solution of the ferrite matrix of high-strength steels, and that it decreases with increasing tempering temperature. In addition, its magnitude remains relatively constant when the plastic strains increase. Spitzig et al. (1975) examined the e€ect of the hydrostatic pressure on the yield and ¯ow stresses in uniaxial tension and compression, and they showed experimentally that tests performed under high hydrostatic pressure lead to rising stress±strain curves whereas the work-hardening characteristics are not a€ected to any signi®cant extent. Surprisingly, the magnitude of the SD-e€ect, too, is not noticeably in¯uenced by the magnitude of the hydrostatic pressure. In addition, in uniaxial tension tests hexagonally close-packed (hcp) metals generally show better agreement with Schmid's law than do bcc and fcc metals. To con®rm this, Barendreght and Sharpe (1973) performed experiments on hcp single crystals of zinc which were loaded in biaxial tension and which were found to slip only in the basal plane. The resolved shear stresses thus reached critical values on the basal slip system while the normal stresses acting on the slip plane varied widely. Moreover, the experimental results showed that the critical resolved shear stress remained constant until the resolved normal stress on the active slip system became about twice as large as the resolved shear stress,

Finite deformation behavior of crystalline solids

1191

and for further increases in the resolved normal stress, the critical resolved shear stress decreased to about 70% of the value measured in uniaxial tension. Although deviations from Schmid's concept have repeatedly been observed in experiments on crystalline solids, they are generally ignored in computational plasticityÐpossibly due to the diculties involved in obtaining the necessary microscopic material properties and to the added complexities in the mechanical model and its numerical implementation. The availability of powerful computers and ecient numerical methods make it possible, however, to carry out extensive numerical simulations and thus render the second objection less and less critical. They also open up many new opportunities for the development of physically sound plasticity models for mono- and polycrystalline aggregates and are of great value in gaining a deeper understanding of the microstructural behaviour and the localization and failure behaviour of solids. Moreover, they contribute signi®cantly to a systematic development of advanced materials. An early ®nite element analysis of the nonuniform localized deformation of rate-independent elastic±plastic single crystals subjected to uniaxial tension has been presented by Peirce, Asaro and Needleman (1982). Their material model is based on a rate formulation using a particular set of stress and strain measures together with a hypoelastic constitutive law and an associated ¯ow rule. The formulation includes Schmid's concept of the critical resolved shear stress and also takes into account self-hardening and latent hardening of the slip systems. The ®nite element analysis presented by BruÈnig (1996) for the ®nite elastic±plastic deformation behaviour of ductile single crystals is based on a hyperelastic constitutive relationship and on plastic potentialsÐexpressed in terms of the generalized Schmid stressÐwhich imply a normality rule for the macroscopic plastic strain rate. Estimates of the microscopic stress and strain histories were obtained using Nemat-Nasser's (1991) plastic predictor method which permits remarkably large load increments with almost no loss in accuracy and requires only very few iterations to achieve an equilibrated solution at the global level. Nemat-Nasser (1983) and Iwakuma and Nemat-Nasser (1984) proposed an extended kinematic approach taking into account microscopic plastic volumetric deformations as well as a generalized microscopic yield condition. Their e€ective microscopic stress measure includes microscopic shear and normal stresses acting on the respective active slip system as well as the e€ect of hydrostatic stresses. BruÈnig (1997) included the in¯uence of various non-Schmid e€ects on plastic yielding and presented analyses which showed that the resulting macroscopic stress distributions are in good agreement with experimental results obtained from multiaxially loaded specimens. Moreover, ¯ow localization was found to occur at quite realistic levels of strain. This is of signi®cance for ductile crystalline solids where localization is the dominant post-failure mechanism. In addition, it also acts as a precursor of fracture (see e.g. Chang and Asaro, 1981), and the accuracy of the respective numerical predictions strongly depends on the details of the constitutive description. Analyses performed by Asaro and Rice (1977) for the case of single slip and Asaro (1979) for the case of plane strain symmetric double slip showed, for example, that the onset of shear banding is closely associated with a critical (low) value of the slip system's strain hardening rate. Furthermore, Li and Richmond (1997) discussed that in the hardening regime the nonnormality approach provides a strong destabilizing e€ect in metal polycrystals. The present paper is an extension of previous work and deals with a nonlinear ®nite element procedure for the numerical analysis of ®nite deformation elastic-plastic problems

1192

M. BruÈnig and H. Obrecht

using a generalized crystal plasticity theory which includes the in¯uence of plastic volume expansion and the SD-e€ect. The macroscopic formulation starts from a multiplicative decomposition of the material deformation gradient into elastic and plastic parts and describes inelastic strains via the local motion of the material in an intermediate con®guration. This is equivalent to an additive decomposition of the Green-type strain tensorÐ referred to an unstressed intermediate con®gurationÐinto elastic and plastic parts. Conjugate macroscopic and microscopic stress measures are related to the respective strains via a hyperelastic constitutive law based on a free energy potential function, and, in addition, deviations from Schmid's rule of the critical resolved shear stress are included by incorporating suitable microscopic stress components into the microscopic yield condition. Plastic potentials expressed in terms of the microscopic stress measures then lead to a non-associated ¯ow rule for the macroscopic plastic strain rateÐand thus to a deviation from the well-known normality rule of standard continuum plasticity theory. The development of a consistent tensor of elastic-plastic moduli as well as its implementation into a nonlinear ®nite element program is also discussed, and estimates of the local stress and strain histories are obtained via a highly stable and very accurate semi-implicit scalar integration procedure. The appearance of macroscopic shear bands is determined from a local bifurcation analysis, and continuing localized deformations as well as post-failure predictions are obtained from a global postbifurcation analysis. The numerical results for a number of plane strain problems involving large elastic and plastic strains as well as severe localization demonstrate the in¯uence of the various model parameters on the deformation and load-carrying behaviour of crystalline solids. II. FUNDAMENTAL EQUATIONS OF CRYSTAL MECHANICS

The kinematic theory of the mechanics of elastic-plastic deformations of crystals is generally based on the assumption that crystallographic slip is the sole inelastic deformation mechanism. It was ®rst presented in the pioneering work of Taylor (1938), and later extensions of this physical concept were given for example by Rice (1971), Hill and Rice (1972, 1973), and Hill and Havnver (1982). A similar kinematic approach is employed by BruÈnig (1996, 1997, 1998), where, following Lee (1969), the macroscopic formulation of the ®nite deformation behaviour of elastic-plastic materials is based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts 

~ F ˆ F F:

…1†

An experimentally motivated kinematic assumption in the plasticity theory of crystalline solids is that macroscopic irreversible deformations are caused by the motion of dislocations on a ®nite number of slip systems. They may also involve plastic volumetric deformations, which were observed, for example, by Spitzig et al. (1975). They reported that increasing plastic deformations may lead to a permanent volumetric expansion which is approximately proportional to the macroscopic normal plastic strain, and the underlying physical process may be an increase in dislocation density. In the mechanical model, the trace of a typical slip system  in an intermediate con®guration is characterized by the orthonormal vectors n and m , which denote the normal to the slip plane and the shear direction of the given system, respectively. The structural

Finite deformation behavior of crystalline solids

1193

rearrangement of the crystal can thus be described by homogeneous simple shears of magnitude  occurring on the various slip systems, while the relative rearrangement of the crystallographic directions is assumed to remain unchanged. The kinematic representation of an incremental motion is then taken to be a linear combination of the additional lattice strains resulting from the slip increment. This involves a volumetric expansion on the active crystallographic systems. Following the ideas of Nemat-Nasser (1983), and Iwakuma and Nemat-Nasser (1984), the crystallographic shear rate _  relative to the intermediate con®guration can be written in the form o Xn       _ ‰m n ‡ a1 G ÿ1 …n n † ‡ a2 G …m m † ‡ a3 g i g i Š _  F~ F~ ÿ1 ˆ L~ ij g i g j ˆ 

…2† 

with summations being taken over all active slip systems. In eqn (2), G represents the  metric tensor of the base vectors g i of the intermediate con®guration, and the additional parameters a1 , a2 , and a3 characterize the microscopic plastic volumetric deformation per unit rate of slip. These irreversible volume changes are usually neglected. In eqn (2) they are taken into account because recent experimental observations on super-alloys and other high-strength metals have shown that a small but ®nite plastic volume expansion may occurÐfor example due to the formation of microscopic voids on grain boundaries or at the interface of microscopic slip bands and non-metallic inclusions. In BruÈnig (1996), Green's strain measure 1 E ˆ …FT F ÿ 1† 2

…3†

is employed to describe the macroscopic material and lattice strains, and it is shown that the multiplicative decomposition of F into elastic and plastic parts (eqn (1)) is equivalent  to the additive decomposition of the strain measure E ˆ F~ ÿT EF~ ÿ1 (referred to an unstressed intermediate con®guration) into an elastic and a plastic part 





E ˆ E el ‡ E pl :

…4†

When all plastic internal variables are assumed to remain ®xed during elastic deformations, eqn (4) leads to the following expression for the objective Oldroyd rate (also referred to the intermediate con®guration) of this Green-type strain tensor r

r

r pl

E ˆ E el ‡ E

n o r _ ˆ E el ‡ symm C F~ F~ ÿ1 ;







…5†



where C denotes the elastic Cauchy-Green tensor C ˆ F T F . Using eqn (2), the inelastic part of the strain rate may also be written in the form r

E pl ˆ

X 

…N _  †;

…6†

1194

M. BruÈnig and H. Obrecht

where N is given by n     o N ˆ symm C …m n † ‡ a1 C G ÿ1 …n n † ‡ a2 C G …m m † ‡ a3 C :

…7†

III. CONSTITUTIVE EQUATIONS

As discussed by BruÈnig (1996), the elastic part of the deformation of rate-independent elastic-plastic crystalline solids may be assumed to be governed by a scalar-valued Helmholtz free energy function of the form 

el ˆ el …I^1 ; J †;

…8†

where I^1  denotes the ®rst invariant of the isochoric right-hand stretch tensor, and  J ˆ det F is the elastic part of the scalar-valued Jacobian determinant. This leads to the hyperelastic constitutive relationship 



S ˆ 

@el 

@E el

;

…9†





where  denotes the material density, S represents the macroscopic stress tensor, and both are referred to the stress-free intermediate con®guration. If the in¯uence of plastic straining on the elastic material behaviour is neglected, the associated symmetric tensor of elastic moduli may be determined from 

Cˆ 

@2 el





@E el @E

el

:

…10†

In the present case, the isotropic elastic potential function (8) is taken as  1 1   el ˆ …I^1 ÿ 3† ‡ K …ln J †2 ; 2 2

…11†

where  and K represent the shear and bulk moduli of the material. The elastic constitutive relationship 





ÿ2

 

S ˆ  J 3 Dev…G ÿ1 † ‡ K ln J C ÿ1

…12†



then follows from eqns (9) and (11), and G ÿ1 represents the inverse metric tensor of the base vectors of the intermediate con®guration, while its respective deviator is given by    1   Dev…G ÿ1 † ˆ G ÿ1 ÿ …C
G ÿ1 †C ÿ1 : 3

…13†

Finite deformation behavior of crystalline solids

1195

Using eqn (10), the isotropic elasticity tensor C then takes the form n o      2 1  4  ÿ2 C ˆ I^1 … C ÿ1 C ÿ1 ‡ G† ÿ  J 3 symm G ÿ1 C ÿ1 ‡ KC ÿ1 C ÿ1 ÿ 2K ln J G; 3 3 3 …14† where G denotes the fourth order tensor 1     G ˆ …Gik Gjl ‡ Gil Gjk † g i g j g k g l 2

…15†

and Gij represents the coecients of the metric of the contravariant base vectors of the current con®guration. In addition, the microscopic scalar stress measure  Ðwhich is assumed to be workconjugate to the microscopic scalar strain  Ðis introduced, and their product is taken to be equal to that of the macroscopic modi®ed 2nd Piola-Kirchho€ stress tensor (9) and the macroscopic plastic strain rate tensor (6). This gives 



r

 w_ pl ˆ S
E pl ˆ

X … _  †

…16†



and 



   ‡ a1 nn ‡ a2 mm ‡ a3 J 3p:  ˆ S
N ˆ nm

…17†

  denotes the microscopic shear stress, p the hydrostatic stress, and nn and In eqn (17), nm  mm represent the microscopic normal stresses acting at a material point of an active slip system . Note that eqn (17) contains the parameters a1 , a2 , and a3 , which account for the experimentally observed kinematic behaviour discussed above. As in BruÈnig (1996, 1998), the scalar stress measure (17) is used to formulate the plastic rate potential  X… _ ˆ …18†

_  d ;



and the inelastic part of the deformation is described by the ¯ow law r

E pl ˆ

_ @



@S

ˆ

X @ 

  ; @S 

…19†

which, making use of eqn (17), leads to the expression r

E pl ˆ

X …N _  †:

…20†



Equation (20) depends on the entire current state of stress and deformation, and it is completely equivalent to the plastic strain rate expression (6), which was derived from

1196

M. BruÈnig and H. Obrecht

kinematic considerations and also accounts for both frictional e€ects and plastic volume changes. In computational plasticity Schmid's law is usually used to determine the onset of plastic yielding (see, for example, BiuÈmig, (1996)). It depends exclusively on the resolved shear stress whereas the remaining microscopic stress components are neglected. On the other hand, as already mentioned, experiments on multiaxially loaded specimens have shown that these additional stresses may signi®cantly in¯uence the yielding process. The experiments performed by Barendreght and Sharpe (1973) on biaxially loaded tension specimens of crystalline zinc, for instance, indicate that Schmid's rule is valid only as long as the microscopic normal stress components remain suciently small, whereas increasing values of these stresses may lead to a decrease in the critical resolved shear stress of up to 30%. Following the ideas of Nemat-Nasser (1983), Iwakuma and Nemat-Nasser (1984), and BruÈnig (1997), deviations from Schmid's rule are modelled via the e€ective microscopic stress measure   on the slip system . In particular,   is expressed in the form 

  ‡ b1 nn ‡ b2 J 3p;   ˆ nm

…21†

which may also be written as 

where

  ˆ S
M ;

…22†

n   o M ˆ symm C …m n † ‡ b1 C G ÿ1 …n n † ‡ b2 C :

…23†

In eqns (21) and (23), the constitutive parameter b1 describes the magnitude of the microscopic sliding frictionÐthat is, the amount of slip resistance of the material caused by the acting normal stressesÐwhereas b2 accounts for the in¯uence of the hydrostatic pressure on the motion of the crystallographic dislocations. Note that in the special case of pressure-insensitive crystalline solids the coecients b1 and b2 vanish, and eqn (21) then reduces to the well-known Schmid law. Expression (21) therefore accounts for a variety of Schmid and non-Schmid e€ects. To arrive at a hyperelastic constitutive expression which relates the e€ective micro scopic stress   to the macroscopic elastic strain tensor E el , the di€erential of eqn (22) 



d  ˆ S
dM ‡ M
dS

…24†

is employed, where the di€erential of M given by n  o  dM ˆ symm 2dE el …m n † ‡ b1 G ÿ1 …n n † ‡ b2 1Š

…25†

and all internal variables are considered to be held ®xed at their respective current values. Expression (24) may also be written in the form 



d  ˆ A
dE el ‡ M
dS ;

…26†

Finite deformation behavior of crystalline solids

1197

n  o A ˆ 2symm …m n † ‡ b1 G ÿ1 …n n † ‡ b2 1ŠS :

…27†

where A is given by

Introducing the elastic constitutive relationship (12) into eqn (26), one obtains the following scalar microscopic elastic constitutive law 



d  ˆ …M C ‡ A †
dE el zel ˆ L
dE el :

…28†

For rate-independent crystalline materials it is also assumed that slip occurs on a glide system  when the elective microscopic stress equals the critical value F , that is,   ˆ F ;

…29†

where   is initially taken to be equal for all slip systems. Furthermore, slip is assumed to continue on that system if the microscopic yield stress remains at the current critical value F , and its rate is then given by X …h _ †; …30† _F ˆ

where h denotes the respective slip hardening rates. Alternative models which include self and latent hardening are described, for example, by Zhou et al. (1993) or Asaro (1983), who also discusses their ability to reproduce experimental observations. For active sliding on systems  and , the increment of the microscopic resolved stress is given by the equality _ ˆ _F ; _  > 0;

…31†

whereas for an inactive slip system  the inequality _ < _F ; _  ˆ 0

…32†

holds, and for noncritical systems one has   < F :

…33†

Note that the microscopic yield condition (29) is formulated in terms of the e€ective microscopic stress measure   (22), which accounts for the complete microscopic stress state acting on the respective slip system , and it thus incorporates all e€ects in¯uencing the onset of plastic yielding in multiaxially loaded solids. The macroscopic ¯ow rule (20), on the other hand, is based on the respective plastic potential function, which in turn depends on the work-conjugate microscopic stress measure  (17) and also accounts for inelastic volume expansion e€ects. Moreover, due to the inclusion of frictional and hydrostatic e€ects in the micromechanical model, the ¯ow rule (20) is not associated to the current yield surface (29) and thus deviates from the classical normality rule.

1198

M. BruÈnig and H. Obrecht

Using, ®nally, eqns (5) and (6) together with the rate formulation of the microscopic elastic material law (28), one arrives at the scalar constitutive equation r

_ ˆ …M C ‡ A †
E ÿ

X …K _ †;

…34†



where K is given by K ˆ M C
N ‡ A
N :

…35†

Alternatively, eqn (34) may be written in the form Xh ÿ
 i Kÿ1 _ ˆ _ ÿ _  ;

…36†



where _ ˆ

Xh ri …Kÿ1 † …M C ‡ A †
E

…37†



can be interpreted as a ®ctitious scalar measure of the total strain rate in each slip system . IV. INTEGRATION OF THE CONSTITUTIVE EQUATIONS

The scalar eqn (34) describes the entire elastic±plastic rate behaviour of the material for arbitrary loading paths, and the total stress and deformation quantities may thus be evaluated from a direct numerical integration of (34). Following Nemat-Nasser (1991) and BruÈnig et al. (1995), this is done using an integration algorithm which is based on the observation that for ductile metals undergoing large deformations, the elastic part of the strain increments tends to be comparatively small. Therefore, at the beginning of each incremental step the scalar measure of the total strain rate is assumed to be entirely due to plasticity, and the associated error is subsequently corrected by determining the corresponding elastic contribution from the above constitutive relations. When, as described in some detail in BruÈnig (1996), t is taken as a general evolution parameter and the crystallographic slip rate _  is assumed to be constant during the interval t  t^  t ‡
t, a numerical integration of eqn (34) leads to   …t ‡
t† ÿ   …t† ˆ
  ˆ

X ‰K …
 ÿ
†Š;

…38†



where the ®nite increments
 and
are given by

t‡
t …


 ˆ t

_ d

…39†

Finite deformation behavior of crystalline solids

and
ˆ

t‡
t …

_ d:

1199

…40†

t

Since in the plastic predictor step the total incremental deformation is assumed to be plastic, the predictor strain increment
p may be obtained from (37), which leads to
p

t‡
t …

ˆ t

Xh

ri …Kÿ1 † …M C ‡ A †
E d:

…41†



Similarly, the conjugate microscopic predictor stress at the end of the increment may be determined from the expression  X @  F
…42† p …t ‡
t† ˆ F … …t† ‡
p † ˆ F … …t†† ‡ p ; @ which gives the relationship between the e€ective microscopic stress and the crystallographic strain. Ignoring the elastic part of the strain increment implies that both the change in the microscopic plastic strain and the current e€ective microscopic stress are overestimated, and the respective errors at the end of the increment are given by
er   ˆ p …t ‡
t† ÿ   …t ‡
t† and 

t‡
t …


er ˆ

…43†

_ er d;

…44†

t

where _ er represents the mean value of  ˆ _ …t† ÿ _  …t† ˆ

_er

X

…Kÿ1 † _



…45†



in the interval t  t^  t ‡
t. To be able to relate
er  to
er   , it is assumed that the microscopic hardening rule
er   

X …h
er †

…46†



holds, so that substituting eqns (43)±(46) into eqn (38) gives the following estimate for the error in the crystallographic slip increment
er  ˆ

X ‰…K ‡ h †ÿ1 …p …t ‡
t† ÿ  …t††Š:

…47†

1200

M. BruÈnig and H. Obrecht

Using, now
 ˆ
p ÿ
er  ;

…48†

one ®nds the following expressions for the e€ective microscopic stress and crystallographic strain in each slip system  at the end of the increment   …t ‡
t† ˆ   …t† ‡

X …h
†

…49†



and

 …t ‡
t† ˆ  …t† ‡
 :

…50†

Corresponding estimates of the respective macroscopic tensorial quantities may be derived from the above fundamental relationships. In particular, following Weber and Anand (1990), the plastic part of the deformation gradient may be determined from ~ ~ ‡
t† ˆ exp…Lpl †F…t†; F…t

…51†

where Lpl is given by pl

t‡
t …

L ˆ t

_ F~ F~ ÿ1 d ˆ

o Xn     ‰m n ‡ a1 G ÿ1 …n n † ‡ a2 G …m m † ‡ a3 g i g i Š
 

…52† and the tensor exponent may be evaluated using a generalized midpoint rule. This gives exp Lpl  …1 ÿ Lpl †ÿ1 ‰1 ‡ …1 ÿ †Lpl Š

…53†

with 0 <   1 (see e.g. Steinmann and Stein, 1996). From the kinematic relations as well as eqn (51) one obtains the macroscopic plastic  strain tensor E pl …t ‡
t†, and the corresponding macroscopic elastic strain tensor may then be computed from 





E el …t ‡
t† ˆ E …t ‡
t† ÿ E pl …t ‡
t†:

…54†



The associated current stress tensor S …t ‡
t† follows from the hyperelastic constitutive relationship (12), while eqn (14) yields the corresponding tensor of the current elastic moduli C…t ‡
t†, and the consistent tensor of the current elastic-plastic moduli is ®nally obtained from 

C

ep

ˆ

dS 

dE

jt‡
t :

…55†

Finite deformation behavior of crystalline solids

1201

V. ANALYSIS OF PLASTIC FLOW LOCALIZATION

At a certain stage in the loading process, the initially smooth displacement distribution may develop into a pattern which involves highly localized deformations. This may lead to the formation of cracks and to rapid fracture at an overall strain which is only slightly larger than that which corresponds to the onset of localization. In general, the value of this critical strain is quite sensitive to the constitutive description of the material, and it is also well known that classical elastic-plastic theories based on smooth yield surfaces and the normality rule may not be able to predict localization at all. Material models which di€er from these classical ones, on the other hand, - such as those which develop vertices on the yield surface or include dilatational plastic e€ects as well as devations from the normality rule - may be better suited to predict localization at realistic levels of strain. Fundamentals of the analysis of localization in elastic-plastic solids have been developed by Hill (1962) and Rice (1977), and its main purpose is to determine the conditions under which bifurcation into a localized mode can occur. In addition, bifurcation analyses give a good indication of the ability of various constitutive descriptions to realistically predict the elastic-plastic deformation and failure behaviour of materials and structures. In general, localization is characterized by the fact that within a thin planar band of  orientation n (referred to the initial con®guration) the rate ®eld quantities di€er from those outside the band. In particular, at the onset of localization the rate of the deformation gradient is discontinuous across the band interfaces and, in addition, must satisfy the compatibility condition  F_ a ÿ F_ b ˆ q_ n ;

…56†

where the subscripts a and b denote the ®eld quantities inside and outside the localized band, respectively. Moreover, in eqn (56) the vector 

q_ ˆ q_ m

…57†

 _ and the unit vectors n and m de®ne the represents the amplitude of the jump in F,   orientation of the discontinuity. When n and m are orthogonal, the material within the   band deforms in simple shear, whereas when n and m are parallel, the band undergoes an extension normal to the planes of discontinuity. The ®rst case represents the familiar shear band localization, while the second describes splitting failure, which may be viewed as an idealized progressive failure mechanism. Mixed failure modes are associated with combi  nations of n and m which are neither orthogonal nor parallel, and the respective localized deformation mechanisms are characteristic of elastic-plastic solids which exhibit microscopic friction and volume changes. Rate equilibrium across the band interfaces is satis®ed when  n …P_ a ÿ P_ b † ˆ 0

…58†

where P_ denotes the rate of the 1st Piola-Kirchho€ stress tensor, and the associated constitutive relations may be expressed in the form _ P_ T ˆ C~ ep F;

…59†

1202

M. BruÈnig and H. Obrecht

where the components of the tensor of elastic-plastic moduli     C~ep ˆ …Fim Cep;mjnl Fkn ‡  jl Gik † g i g j g k g l

…60†

follow from (55). Using eqns (56), (58) and (59), the fundamental equation governing the localization process becomes ~ep _ q ˆ n …C~ep … n C~ep b n †_ b ÿ Ca †Fa : 





…61†

Note that eqn (61) may be used to determine the onset of localization and the initial band orientation but not the post-localization behaviour. The latter must be evaluated from the global solution of the complete nonlinear boundary value problem. When the current material properties inside and outside the band are assumed to be identical, the right hand side of eqn (61) vanishes, and one thus obtains the homogeneous linear bifurcation problem   qˆ0 … n C~ep b n †_ 

…62†



for q_ (or m ) and m . Localization therefore occurs at that point of the deformation history  at which eqn (61) has a nontrivial solution, and for the associated band direction n the determinant   …63† det… n C~ep b n† ˆ 0 

vanishes. A useful numerical procedure for computing the eigenvalues n and the eigenmodes q_ has been given, for example, by Ortiz et al. (1987). As the localized deformation progresses, the material properties inside and outside the band change, and in the initial post-localization regime the ®eld quantities must satisfy eqn (61). It represents an inhomogeneous set of equations for the components of q_ , which may be integrated numerically to give the initial evolution of the ®eld quantities inside and outside the band. Subsequently, the development of the failure mode must be determined from a complete solution of the governing global equations (for an overview see e.g. Needleman and Tvergaard, 1992). VI. VARIATIONAL FORMULATION AND FINITE ELEMENT IMPLEMENTATION

The analyses presented below were obtained from a nonlinear ®nite element program which is based on the principle of virtual work in the form …



…

Gradu
FSd v ÿ

…u† ˆ Bo

u
t0 d a ˆ 0; 

…64†

@Bo

where Bo and @Bo denote the volume and surface of the body in the initial con®guration, and the ®rst integral in eqn (64) represents the variation of the current stored energy density while the second accounts for the contribution of the prescribed surface tractions t0 .

Finite deformation behavior of crystalline solids

1203

Incorporating the above global and local displacement quantitiesÐtogether with the local stresses and strainsÐinto eqn (64), and then using a consistent linearization procedure, choosing suitable shape functions and nodal degrees of freedom for the unknown displacements, carrying out the appropriate integrations and ®nally assembling the individual element sti€ness matrices and load vectors, one arrivesÐas usualÐat a set of linearized algebraic equations for the nodal displacement increments, which may be written in the familiar abbreviated form KT
V ˆ R:

…65†

In eqn (65), KT denotes the global tangent sti€ness matrix (which depends explicitly on the current elastic-plastic moduli as well as the current state of stress and deformation), R corresponds to the residual unbalanced force vector, and
V represents the vector of incremental displacements. Equation (65) is solved recursively until a suitable norm of
V falls below a suciently small limit, thus indicating that a converged equilibrium solution has been reached. VII. NUMERICAL SIMULATIONS

The numerical analyses discussed in this section were obtained for uniaxially as well as biaxially loaded crystalline solids under plane strain conditions using crossed-triangle elements built up of four triangles with linear displacement ®elds. Inelastic macroscopic deformations are thus restricted to the loading plane, and the corresponding two-dimensional version of the above general formulation is similar to Asaro's (1983) planar double slip model. The con®gurations considered are similar to those in the experiments of Spitzig et al. (1975), and, in addition. their measured values of the elastic material properties (E ˆ 198000 N=mm2 and v ˆ 0:29) were used in the numerical calculations. To describe the rate-independent plastic behaviour it is ®rst assumed that the yield strength merely  and that the microscopic yield condition depends on the microscopic normal stresses nn (21) may therefore be expressed in the form   ‡ 0:015nn : F ˆ nm

…66†

Furthermore, the initial e€ective microscopic yield stress was chosen as F ˆ 0 ˆ 500 N=mm2 . Alternatively, a pressure sensitivity of the critical resolved shear stress was modelled by changing the microscopic yield condition (21) to 

 ‡ 0:03 J p: F ˆ nm

…67†

For a rectangular specimen subjected to both uniaxial tension and compression, the material's work-hardening behaviour is described by the exponential hardening law h



ˆ h ˆ h0 sech

2



h0 s ÿ 0

 …68†

1204

M. BruÈnig and H. Obrecht

(see e.g. BruÈnig, 1996), where ho ˆ 2800 N=mm2 denotes the plastic hardening parameter, s ˆ 580 N=mm2 the saturation value of the ¯ow stress, and latent hardening e€ects are neglected. In multislip situations which include latent hardening, is taken to be the sum of the slips in all currently active slip systems. In BruÈnig (1997, 1998) it was shown that when microscopic yield conditions are based on Schmid's rule of the critical resolved shear stress, the resulting macroscopic stressengineering strain curves are nearly identical for both uniaxial tension and compression. When, on the other hand, the microscopic yield conditions include frictional as well as hydrostatic e€ects, the numerical simulations lead to a strength di€erential (SD-) e€ect in the sense that the predictions of the macroscopic yield and ¯ow stresses are larger in uniaxial compression than in uniaxial tension. At the same time the resulting macroscopic plastic strains are found to be isochoric, so that microscopic models which predict the SDe€ect do not automatically predict a plastic volume expansion as well. BruÈnig (1998) therefore also studied the in¯uence of various macroscopic ¯ow rules on the deformation behaviour of crystalline solids. For a macroscopic ¯ow rule which is associated to the yield conditions (66) or (67), for example, (that is, for a1 ˆ b1 and a3 ˆ b2 ) one obtains the results given in Fig. 1. It shows that, depending on the values of a1 , and a3 , the computed values of the plastic volume increase may be up to ten times as large as the magnitudes observed experimentally. On the other hand, the corresponding true stress-engineering strain curves are practically identical to those obtained from the assumption that the plastic strains are isochoric. Thus, including plastic volume expansion e€ects into the macroscopic ¯ow rule does not signi®cantly in¯uence the predictions of the overall stress-carrying behaviour of crystalline solids. It does, however, in¯uence the plastic volume expansion. For associated plastic ¯ow laws, for instance, the latter is severely overestimated, whereas for non-associated ¯ow laws this is not the case. This can be seen from the results in Fig. 1, where the predictions obtained for a1 ˆ 0:0022 and a2 ˆ a3 ˆ 0 agree quite well with the experimental results reported by Spitzig et al. (1975). The extent to which microscopic yield conditions which include non-Schmid terms and macroscopic ¯ow rules which incorporate plastic volume expansion e€ects may in¯uence

Fig. 1. Plastic volume increase vs Green's plastic strain: numerical results and Spitzig et al.'s (1975) experimental data.

Finite deformation behavior of crystalline solids

1205

the macroscopic localization behaviour of crystalline solids was also investigated by BruÈnig (1997, 1998). His numerical simulations show that variations in the microscopic yield condition may have a noticeable e€ect on the predicted onset of macroscopic plastic yielding as well as on the magnitude of the SD-e€ect. The in¯uence on the localization behaviour, however, was found to be insigni®cant. Also, localization generally occurred beyond the maximum load, and when the plastic strains were assumed to be isochoric, the localization mode was an ideal shear band involving ®nite plastic deformations in the  direction orthogonal to the planar band with orientation n . In the cases considered, the associated shear plane normals were found to be inclined at 34 (relative to the loading axis) in uniaxial tension and 26 in uniaxial compression, and the latter values were also largely independent of the constitutive parameters bi in eqn (21). When, on the other hand, the macroscopic ¯ow rules included plastic volume expansion e€ects, the parameters ai in eqn (7) had a very signi®cant in¯uence on the ®nite deformation and localization behaviour. This can be seen from Fig. 2, which shows the true stressengineering strain curve for a prescribed uniaxial extension of a crystalline specimen. If an isochoric ¯ow rule with zero values of the constitutive parameters ai in eqn (7) is taken into account, bifurcation into a localized mode is found to occur beyond the maximum load, whereas for increasing values of the parameter a3 localization is predicted prior to reaching the load maximum. The fundamental true stress-engineering strain curve, however, is not a€ected and remains unchanged. Similarly, increasing values of a3 lead to continuously decreasing values of the bifurcation stresses and strains, and the latter, in particular, decrease rapidly for increasing values of a3 . This indicates a signi®cant reduction in the ductility of the respective material.

Fig. 2. Cauchy stress vs Green's strain: in¯uence of the kinematic parameter a3 on the onset of localization.

1206

M. BruÈnig and H. Obrecht

Fig. 3. Plastic volume increase vs kinematic parameter a3 .

The in¯uence of a3 on the plastic volume change at the onset of localization may be seen from Fig. 3. It shows that the change in plastic volume associated with the bifurcation load increases roughly linearly with increasing values of a3 until a fairly sharp maximum is reached at about a3 ˆ 0:038. Subsequently, the increase in plastic volume corresponding to the onset of localization diminishes nearly linearly and eventually even vanishes. Comparing Figs 2 and 3 it is seen that plastic volume changes tend to be larger when localization occurs beyond the maximum load, whereas when bifurcation occurs prior to the load maximum, the associated plastic volume expansion tends to be smaller. Hence, for a3  0:038 the plastic volume change attains a maximum and the respective bifurcation point roughly coincides with the maximum in Fig. 2. Figure 4, ®nally, shows the dependence of the inclination angle of the localization plane on a3 . For a3 > 0:038 the inclination angle is 45 , whereas for decreasing values of a3 it also decreases until for vanishing a3 (which corresponds to isochoric plastic deformations) it reaches a value of 34 .

Fig. 4. Normal of plane of localization vs kinematic parameter a3 .

Finite deformation behavior of crystalline solids

1207

VIII. CONCLUSIONS

A nonlinear ®nite element procedure for the macroscopic rate-independent analysis of crystalline solids has been presented. The crystal's deformation was assumed to arise from elastic deformations due to the distortion of the lattice and from plastic deformations due to crystallographic slip in certain lattice planes. In addition, plastic volume changes are taken into account, a non-associated ¯ow rule for the macroscopic plastic strain rate is used, and particular attention is given to the in¯uence of deviations from Schmid's rule of the critical resolved shear stress on the stress-strain behaviour of crystalline solids in uniaxial tension and compression. The numerical results show that variations in the microscopic yield condition (21) have a signi®cant in¯uence on the true stress-engineering strain behaviour of the material as well as on the onset of macroscopic plastic yielding in tension and compression. The amount of plastic volume expansion and the localization behaviour, on the other hand, were found to be largely una€ected by the choice of the parameters bi in the microscopic yield condition (21), whereas variations of the constitutive parameters ai have a strong in¯uence on the predicted plastic volume expansion. Furthermore, the use of associated ¯ow laws leads to an overestimation of the plastic volume increase, so that non-associated ¯ow rules should be used and the plastic potential function (18) should be chosen independently of the microscopic yield condition. Finally. whereas generalized macroscopic ¯ow rules which account for irreversible volume increases may have a remarkable in¯uence on the localization behaviour of crystalline solids, they do not a€ect either the macroscopic true stress-engineering strain curves or the predicted onset of plastic yielding. REFERENCES Aifantis, E. C. (1987) The physics of plastic deformation. Int. J. Plasticity, 3, 211. Asaro, R. J. (1979) Geometrical e€ects in the inhomogeneous deformation of ductile single crystals. Acta Metall., 27, 445±453. Asaro, R. J. (1983) Micromechanics of crystals and polycrystals. Adv. Appl. Mech., 23, 2±115. Asaro, R. J. and Rice, J. R. (1977) Strain localization in ductile single crystals. J. Mech. Phys. Solids, 25, 309±338. Barendreght, J. A. and Sharpe, W. N. (1973) The e€ect of biaxial loading on the critical resolved shear stress of zinc single crystals. J. Mech. Phys. Solids, 21, 113±123. BruÈnig, M. (1996) Macroscopic theory and nonlinear ®nite element analysis of micromechanics of single crystals at ®nite strains. Comput. Mech., 18, 471±484. BruÈnig, M. (1997) Numerical modelling of ®nite elastic±plastic deformations of crystalline solids including nonSchmid e€ects. In Computational Plasticity 5ÐFundamentals and Applications, ed. D. R. J. Owen, E. OnÄate and E. Hinton, pp. 907±912. Barcelona: CIMNE. BruÈnig, M. (1998) Microscopic modelling of volume expansion and pressure dependence of plastic yielding in crystalline solids. Archive Appl. Mech., 68, 71±84. BruÈnig, M., Obrecht, H. and Speier, L. (1995) Finite deformation elastic-plastic analysis based on a plastic predictor method. In Computational Plasticity 4±Fundamentals and Applications, ed. D. R. R. Owen, E. OnÄate, pp. 141±152. Swansea: Pineridge Press. Chang, Y. W. and Asaro, R. (1981) An experimental study of shear localization in aluminum-copper single crystals. Acta Metall., 29, 241±257. Havner, K. S. (1992) Finite Plastic Deformation of Crystalline Solids. Cambridge: University Press. Hill, R. (1962) Acceleration waves in solids. J. Mech. Phys. Solids, 10, 1±16. Hill, R. (1966) Generalized constitutive relations for incremental deformation of metal crystals by multislip. J. Mech. Phys. Solids, 14, 95±102. Hill, R. (1967) The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids, 15, 79±95. Hill, R. and Havner, K. S. (1982) Perspectives in the mechanics of elastic-plastic crystals. J. Mech. Phys. Solids, 30, 5±22.

1208

M. BruÈnig and H. Obrecht

Hill, R. and Rice, J. R. (1972) Constitutive analysis of elastic-plastic crystals at arbitrary strains. J. Mech. Phys. Solids, 20, 401±413. Hill, R. and Rice, J. R. (1973) Elastic potentials and the structure of inelastic constitutive laws. J. Appl. Math., 25, 448±461. Iwakuma, T. and Nemat-Nasser, S. (1984) Finite elastic-plastic deformation of polycrystalline metals. Proc. R. Soc. London A, 394, 87±119. Lee, E. H. (1969) Elastic-plastic deformations at ®nite strains. J. Appl. Mech., 36, 1±6. Li, M. and Richmond, O. (1997) Intrinsic instability and nonuniformity of plastic deformation. Int. J. Plasticity, 13, 765±784. Lin, T. H. (1971) Physical theory of plasticity. Adv. Appl. Mech., 11, 255±311. Needleman, A and Tvergaard, V. (1992) Analyses of plastic ¯ow localization in metals. Appl. Mech. Rev., 45, 3± 18. Nemat-Nasser, S. (1983) On ®nite plastic ¯ow of crystalline solids and geomaterials. J Appl. Mech., 50, 1114± 1126. Nemat-Nasser, S. (1991) Rate-independent ®nite-deformation elastoplasticity: A new explicit constitutive algorithm. Mech. Mater., 11, 235±249. Ortiz, M., Leroy, Y. and Neddleman, A. (1987) A ®nite element method for localized failure analysis. Comp. Meth. Appl, Mech. Eng., 61, 189±214. Peirce, D., Asaro., R. J. and Needleman, A. (1982) An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall., 30, 1087±1119. Rice, J. R. (1971) Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids, 19, 433±455. Rice, J. R. (1977) The localization of plastic deformation. In Proc. 14th Int. Congr. Theoret. Appl. Mech., ed. W. T. Koiter, pp. 207±220. North-Holland, Amsterdam. Spitzig, W. A., Sober, R. J. and Richmond, O. (1975) Pressure dependence of yielding and associated volume expansion in tempered martensite. Acta Metall., 23, 885±893. Steinmann, P. and Stein, E. (1996) On the numerical treatment and analysis of ®nite deformation ductile single crystal plasticity. Comp. Meth. Appl. Mech. Eng., 129, 235±254. Taylor, G. I. (1938) Plastic strains in metals. J. Inst. Met., 62, 307. Weber, G. and Anand, L. (1990) Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comp. Meth. Appl. Mech. Eng., 79, 173±202. Zhou, Y., Neale, K. W. and Toth, L. S. (1993) A modi®ed model for simulating latent hardening during the plastic deformation of rate-dependent fcc polycrystals. Int. J. Plasticity, 9, 961.