macroscopic effects of micro-shear banding in plasticity of metals

Acta Mechanica 131,203-224 (1998) ACTA MECHANICA �9 Springer-Verlag 1998

Macroscopic effects of micro-shear banding in plasticity of metals

R. B. P~cherski, Warsaw, Poland

Dedicated to E. Stein on the occasion of his 65th birthday

(Received March 5, 1997; revised September 15, 1997)

Summary. Mathematical idealization of a micro-shear bands system by means of the theory of singular surfaces of order one, related to a physical model of shear strain-rate produced by active micro-shear bands and a certain averaging procedure over the representative volume element, is studied. Theoretical description of small elastic and large plastic deformations within the framework of a two-surface plasticity model, with the internal yield surface connected to kinematic hardening anisotropy and the external surface related to micro-shear banding, is proposed. The idea of the multiple potential surfaces forming a vertex on the smooth external surface is applied to display the connection with the geometric pattern of micro-shear bands. A new physical insight is given into the linear and nonlinear flow laws, in rates of deformation and stress, known in the theory of plasticity.

1 Introduction

Theoretical modelling of large plastic deformations of metals accounting for micro-shear

bands requires careful analysis of averaging procedures and proper setting of the resulting

description of the effects of micro-shear banding within continuum theory of materials. For-

mulation of a complete theory based on the precise micro-to-macro transition remains an

open question. The aim of the paper is to approach this problem from the point of view of the

contribution of micro-shear bands to kinematics o f finite elastic-plastic deformation and for-

mulation of a tentative pheomenological model of plastic flow accounting for the characteri-

stic geometric pattern of micro-shear bands. First, the macroscopic measure of the velocity

gradient, produced in the course of plastic flow with micro-shear banding, was derived. The

derivation is based on the physical model relating macroscopic shear strain rate with micro-

structural features of micro-shear bands, mathematical idealization of a system of active

micro-shear bands as a singular surface of tangential velocity discontinuities, and the known

averaging procedure applied to the new representative volume element (RVE), which is trav-

ersed by the discontinuity surface. An attempt to tackle the averaging procedure over the

RVE with micro-shear bands was presented earlier in [1]. This makes possible to derive in a more rigorous manner the macroscopic measures of the rate of plastic deformation and plastic

spin, necessary to formulate the constitutive description, which were obtained previously under certain simplifying assumptions, [2]-[4]. The double surface plasticity model, proposed in [4],

has been extended and corroborated in this study. The internal yield surface is connected with the nonlinear kinematic hardening model of Armstrong-Frederick type while the external

204 R.B. Pgcherski

surface corresponds with the saturation of the backstress effect. It appears that this phenome- non can be correlated with massive formation of micro-shear bands. Basing on the idea of the "extremal surface", presented by Hill [5], the concept of the generic micro-shear banding surface was introduced. The m.s.b, surface can be approximated by means of the class of the external limit surfaces. The model proposed shows that the contribution of active micro-shear bands with their characteristic geometric pattern transmitted to the macroscopic level produce the non-coaxiality between principal directions of stress and rate of plastic deformations. The relation for plastic spin appears in a natural way as an effect of this non-coaxiality. It transpi- res that depending on the contribution of the mechanisms involved in plastic flow, a fully active range, separated from the elastic range by a truly nonlinear zone called the partially active range, may exist. A new physical insight is given into the linear and nonlinear flow laws, in rates of deformation and stress, known in the theory of plasticity. The idea of multiple potential surfaces forming a vertex on the smooth external surface is applied here to connect the fully active range and the partially active range with the definite geometric pattern of micro-shear bands. This leads to the new form of the hypoelastic version of J2 deformation theory accounting for the effects of micro-shear banding. Consideration of the partially active range enables proper description of the unloading process. The possibilities of certain simplifi- cations and perspectives of the application of the theoretical model vis-fi-vis the results presented recently in the literature are discussed. Standard symbolic notation is used throughout. Tensors are denoted by boldface characters and the following symbolic operations are used:

ab ~ aijbj~, L : b - - L i y b k l , a : b - a i j b i j ,

with summation convention over repeated indices.

2 Physical motivation

The available results of metallographic observations reveal that in heavily deformed metals, or even at small strains if they are preceded by a change of deformation path, a multiscale hierarchy of shear localization modes progressively replaces the crystallographic multiple slip or twinning (cf. e.g. [6]). Different terminology is used depending on the level of observation. In our study, the term micro-shear band is understood as a long and very thin (of order 0.1 btm) sheet-like region of concentrated plastic shear crossing grain boundaries without deviation and forming a definite pattern in relation to the principal directions of strain. It bears very large shear strains and lies in a "non-crystallographic" position. The term "non-crystallographic" means that micro-shear bands are usually not parallel to a particular densely-packed crystallographic plane, of a conventionally possible active slip system, in the crystallites they intersect. This change of deformation mode contributes to the development of strain induced anisotropy and modifies remarkably the material properties. The experimental information about mechanical behaviour and related structural features is reviewed e.g. in [2]-[3] and [6]-[8], where comprehensive lists of references are given. The experimental observations reveal the time and spatial organization of dislocations and the hierarchy of plastic slip processes: from coplanar dislocation groups moving collectively along active slip systems, through slip lamellae and slip bands to coarse slip bands, which may further transform into transgranular micro-shear bands and form clusters (packets) of micro-shear bands of the thickness of order (10 + 100) btm. This shows that a crystalline solid subjected to plastic defor-

Micro-shear banding in plasticity of metals 205

mation is a complex, multiscale, hierarchically organized system. The clusters of micro-shear bands, produced for instance in rolling, form the planar structures, which are usually inclined by about :k 35 ~ to the rolling plane and are orthogonal to the specimen lateral face. There can be, however, considerable deviations from this value within the 15 ~ to 50 ~ range. It is worthy to stress that the problem of specifying the angle is complicated by the difficulty of distinguishing the most recently formed micro-shear bands from those that were formed earlier and subsequently rotated with material towards the rolling plane, cf. [3] and [6]-[8]. This is related with the important observation, discussed in [8] and stressed also in [3], that a particular micro-shear band operates only once and develops rapidly to its full extent, The micro-shear bands, once formed, do not contribute further to the increase in plastic shear strain. Thus, it appears that the successive generations of active micro-shear bands competing with the mechanism of multiple crystallographic slip are responsible for the process of ad- vanced plastic flow in metals. The discussion of experimental observations concerning micro- shear bands geometry leads to the hypothesis which says that it is typical of the dusters of active micro-shear bands that their planes are rotated relative to the respective planes of maximum shear stress by a certain angle B, which is usually of the order (5 + 15) ~ This deviation angle plays an essential role in the phenomenological theory of plastic deformations accounting for the effects of micro-shear banding and will be considered as a statistically averaged micro-shear bands orientation parameter transmitting to the macroscopic level the geometry of their spatial pattern. The experimental observations reveal that the spatial pattern of micro-shear bands does not change for loading conditions that deviate within limits from the proportional loading path, i.e. the load increments are confined to a certain cone, the angle of which can be determined experimentally. For instance, according to Dybiec [9], in polycrystalline Cu the critical angle @ of this cone is of the order 22 ~ A more drastic change of the loading scheme produces, however, the change of the spatial orientation of micro-shear bands. This is supported by the results presented in [10], where after cross rolling two families of micro-shear bands inclined by about �9 35 ~ to the most recent rolling direction were observed. The existence of the deviation angle/3 is characteristic for the micro-shear bands produced in the deformation processes carried out under nearly isothermal conditions. Thermal shear bands, i.e. the mode of plastic flow localization governed by a coupled thermoplastic mechanism, have also been studied by many authors (cf. [11]-[15]). In particular, the so-called ~ shear bands" are often reported to coincide with the trajectories of maximum shear stress, which result in B = 0, [14]. In our view, such a qualitative difference can be attri- buted to the influence of internal micro-stresses, which control the formation of micro-shear bands. The micro-stresses perturb locally the applied macroscopic state of stress deviating the principal axes of the stress tensor. According to the hypothesis on a micro-shear band formation presented in [6] and [7], within a suitably oriented grain the critical coarse slip band is activated, which can further transform, under appropriate dynamical conditions, in a transgranular "non-crystallographic" micro-shear band propagating in the planes that are usually deviated from the planes of applied maximum shear stress. On the other hand, the effect of micro-stresses decreases while thermoplastic coupling becomes operative and "adiabatic shear bands" develop. The recent experimental investigations, presented in [15], of the thermome- chanical coupling during a simple shear test with use of the thermovision system, which show the temperature distribution along the shearing paths and reveal a misorientation of the shear banding zone with respect to the plane of maximum shear stress for non-adiabatic conditions confirm, at least qualitatively, the aforementioned interpretation of the deviation angle ,3.

206 R.B. Pgcherski

3 Continuum mechanics description of micro-shear banding

3.1 Macroscopic averaging in plasticity of metals

The physical constraint on any continuum mechanics approach to metal plasticity, i.e. the

physical dimension of the smallest representative volume element (RVE) of crystalline material, for which it is possible to define significant overall measures of stress and strain during plastic deformation and the assumptions of the averaging procedure were thoroughly discus-

sed by Hill [16] [17] and Havner [18] [20], as well as by Nemat-Nasser and Horii [21], where the comprehensive bibliography of the earlier papers is summarized, (cf. also [1] for a more

detailed discussion). The considerations have been restricted, typically, to quasi-static defor-

mation processes with body forces negligible. Let us assume that within the reference volume Vo of the macroscopic RVE (macro-element) the nominal stress field s~, representing micro-

stresses, and their rates g~ are self-equilibriated,

D i v s m = 0 , D i v g ~ = 0 in Vo,

and the following boundary conditions hold:

UoS = t~, Uog =t~ , on 0Vo,

( i )

(2)

where Uo is the external directed unit normal to the reference volume at a point on its boundary 0Vo. The averaging procedure and micro-to-macro transition, studied within the framework of finite strain theory by Hill [17] and Havner [18] [20] lead, in particular, to the following relations for the macroscopic measures of the deformation gadient F and its rate F,

which are expressed, with use of Gauss' theorem (divergence theorem), by means of surface

data:

F = {f} = 1 Grad X~ dVo = V~o x | uo dAo, (3)

Vo OVo

~" - {?} = ~ Grad 2o dVo = Voo 2o 0 ~'o dAo, (4)

% OVo

where the symbol X,~ denotes the microscopic field of motion of the material point X.r in the

reference configuration of the RVE into its current position x,~,

x., = X~ (X.~, ~), (5)

and the microscopic field of velocity v~ is determined in the current configuration,

v ~ : v~(x ,~ ,~ ) : v , ~ ( x ~ ( x ~ , e ) , ~) - x ~ ( X ~ , ~ ) . (6)

The Gauss' theorem applied above was specified for any suitable vector field w = w(X~) defined on the closure 12o = Vo U 0Vo and being of class C~ N CI(Vo), so that w is continuous on the closure Vo and continuously differentiable on Vo (cf. e.g. Smith [22]). Similarly, the following relations for the macroscopic measures of the suitable smooth tensor field of

nominal stress S:

1/ , / S ~ {s~} = ~o s.~ dVo = Vo X | t . dAo, (7)

Vo OVo


its rate

g = Voo dVo = Voo x o dAo, (S) Vo OVo

and the Kirchhoff stress r

T -- {~-~} = {fs,,,} = Voo x | t , dAo, (9)

can be obtained with application of Gauss ' theorem and (1), cf. [17]. The presented averaging

procedure is valid under the general assumption that the dominant mechanism of plastic deformat ion corresponds to multiple crystallographic slip. In such a case, the theory describing

kinematics and constitutive structure of finite elastic-plastic deformation of crystalline solids is

well established and the transition between the microscopic and macroscopic levels is well understood (cf. e.g. Hill [5], [17], [23], [24], Havner [18]-[20] and Mandel [25]-[26] as welt as

Hill and Rice [27], Nemat-Nasser [28], Petryk [29] and Stolz [30]). In particular, relations

between macro-measures of stress, strain and plastic work are related with the volume averages of their micro-measures. It has also been shown that certain structural features of the constitu-

tive relations, as the normali ty rule or certain constitutive inequalities are transmitted upwards through a hierarchy of observational levels unchanged (cf. Hill [5]). As it was stressed in [3], the

situation changes when an additional mechanism of micro-shear banding is taken into consideration. The next chapters are devoted to the problem of proper setting of the effects of micro-shear bands within the cont inuum theory of finite plastic deformation of metallic solids.

3.2 Physical model of shear strain rate produced by active micro-shear bands

Consider a certain RVE containing the region of progressive shear banding, depicted schematically in Fig. 1 a, where the traces of successive clusters of micro-shear bands are shown. The arrow points to the direction of the expansion of the region. According to the results of ex-

perimental observations presented in [1] and [7], at this level of observation the clusters of

a) b)

f

/

l --mS

Fig. 1. Schematic illustration of the multiscale system of shear banding: a The section of the unit cube of the RVE, traversed by the region of shear banding progressing in the direction pointed by the arrow, b The cluster of active micro-shear bands with the active zone of the thickness H,~s and the width L~,~. Beneath, the fundamental mechanism of plastic shear strain generated by the active micro-shear bands (m.s.b.) is depicted

208 R.B. Pgcherski

active micro-shear bands can be considered as elementary carriers of plastic strain. In Fig. 1 b, the "magnification" of the shear banding area is "zoomed in" and the related fundamental

mechanism of plastic shear is illustrated. The cluster of micro-shear bands has the active zone of the thickness Hr~s and the width L,~s, in which the passage of active micro-shear bands results in the local perturbation, A,~, of the microscopic displacement field, u~ = x,~ - X,,,, moving with the speed V~ as a distortion wave. Consider a set of N,~ active micro-shear

bands of similar orientation and produced within the time period At, which can be considered as an infinitesimal increment of "time-like parameter" t in the macroscopic description. As

it is depicted schematically in Fig. lb, such a system (cluster) of micro-shear bands produces

the microscopic shear strain 7,~, which is given by the following relation:

A~.s B ~ N , ~ s _ 1 ~ m i , (10) 7ras - H m s ' / k m s - L m s Xms , turns - - iVms i

where B,~ is the total displacement produced by a single micro-shear band and x,~, denotes

the average distance that N,~, micro-shear bands have moved in the active zone. The width of the active zone L,~ can be determined by the length of the path that micro-shear bands passed

with an average speed v,~ during the time period At ,

L m s = v~A~-. (11)

Assuming that the distance 2.,.~ and the number of active micro-shear bands N,~s can change

during the propagation of the active zone we have from (1 O)

B~ ( x ~ , + x~N~), (12)

where the dot denotes differentiation with respect to the "time-like parameter" t. Let us observe that under the simplifying assumption that the speed of micro-shear bands in the

active zone of the cluster is approximately the same, the rate ),~s can be identified with the speed v,~, of the head of a single micro-shear band, ) ,~ - v .... (cf. [1], where the model of a

propagating single micro-shear band was discussed). Then, the speed V~ is given by

B~ ( N ~ . ~ + ~.~sN~) (13) V~ = L ~

and the shear strain rate reads

V s (14)

In such a case, an additional evolution equation for N,~ is necessary. It is an open question,

which requires further studies on the development of micro-shear bands within the propagating active zone of a single cluster. I f the number of active micro-shear bands operating in the active zone of a single cluster can be assumed constant, the relation for the speed V.~ is given by

Bins (15)

and (14) takes the form, which is formally similar to the Orowan relation, known in the theory

of dislocations,

X~s (16) 4/,_~ = B , ~ Q , ~ v , ~ , 9 . ,~ - - L , ~ H ~ '


where c9~0,~ corresponds to the density of micro-shear bands operating within the active zone of the cluster. This is the number of active micro-shear bands that cut through a unit cross- sectional area. For N ~ assumed to be of order 100 and H ~ , L,~ being of order 100 gin, the density o ~ is of order 101~ (m-2). The experimental observations lead to the hypothesis that micro-shear bands propagate with the velocity v~. of constant magnitude, which is bounded by the elastic shear wave speed e~ (velocity of sound) in the considered metal or alloy (cf. [1]).

3.3 System of active micro-shear bands as surji2ce of strong discontinuity

The foregoing discussion of the physical nature of micro-shear banding process as well as the recent results of the microscopic observations in situ, presented by Yang and Rey [31], sup- port the following hypothesis.

Hypothesis 1. The passage of micro-shear bands within the active zone of the cluster restdts in the perturbation of the microscopic displacement.field A~.~ travelling with the speed Vs, which produces a discontinuity of the microscopic velocity field in the R VE it traverses. The progression of clusters of micro-shear bands can be idealized mathematically by means of a singular surface of oJ'cter one propagating through the macro-element (R VE) of the continuum.

The necessary mathematical formalism of the theory of propagating singular surfaces is given e.g. by Truesdell and Toupin [32], Eringen and Suhubi [33] and Kosiflski [34]. The theory allows to identify the postulated discontinuity surface of the microscopic velocity field v,~ in RVE as a singular surface moving in the region Vo of the reference configuration of the body given by the equation 2(t) : G(X,~, t) = 0, where for each instant of "time-like parameter" t c f c / ~ the surface L'(t) C Vo has the dual counterpart S(t) c V, in the spatial configuration of RVE, S( t ) :g(x ,~: t )= 0, i.e. the material points Xm E 2'(t) occupy the places x~ E S(t) at a given constant t. There exists the jump discontinuity of derivatives of the function of motion X,~, i.e. of the microscopic velocity field [2~1 r 0 and the deformation gradient Ifl r 0, which are assumed smooth in each point of Vo x I excluding the discontinuity surface

I2~1 = 22 - 2~ # 0, If] = f+ - f - # 0. (17)

This determines the singular surface of order one, called also the surface of strong discontinuity. According to [34], the considered surface of strong discontinuity of microscopic velocity field fulfills formally the properties of a non-material vortex sheet with the jump

a) b)

~-(t) s(t)

Fig. 2. The dual representation of a strong discontinuity surface of tangential velocity jump traversing RVE: a In the reference configuration, b in the current configuration

210 R.B. Pr

discontinuity of the first derivatives of )C~ given by

v~ I ;~ ]=V~s , [~f]=-~_Ns@N' for U N r (18)

where N is the unit normal vector to the discontinuity surface 2(f) and UN is the normal component of the surface velocity in the reference configuration (cf. Fig. 2). Similarly, for the spatial counterpart of a singular surface S(f) the compatibility relations take the form (el. [33], p. 96):

v~ [v,~l=V~s , I f l = - ~ - s | for U r (19)

where, as it is depicted in Fig. 2, s denotes the unit tangent vector and n is the unit normal vector to the discontinuity surface S(t), while U = u~ - v~ �9 n corresponds to the local speed of propagation of S(t).

3.4 Problem of macroscopic averaging and continuum mechanics description of micro-shear

banding

Application of the generalized form of Gauss' theorem for the gradient of the microscopic velocity field ~ , which is sufficiently smooth in each point of Vo x jr except the singular surface, where the discussed jump discontinuity, [ 2 ~ , appears, leads to

f f f / Grad 2m dVo / )~.~ @ Uo d A o - / I2~] | N dAo . J J J

Vo OVo z(t) ~(t)

(20)

Due to (20), the averaging procedure (4) of the microscopic velocity field )~,~ over the macro-element Vo can be generalized for the macroscopic RVE traversed by a singular surface of order one. The macroscopic measure of the deformation gradient ~ ' a n d its rate ~ are expressed by means of surface data in the following way:

1/ ~ Vo Xm | Uo dAo, (21)

o�89

where according to (3) ~Tr = In' and

1 / 1 / 1 / - Vo ~ @ Uo dAo = Voo Grad )~.~ dVo + Voo I)(~l | NdAo. (22)

0vo 2@ vo ~(~)

Similarly, the application of the generalized form of Gauss' theorem for the stress field s,~ over the macro-element Vo with the singular surface gives the formula for the average nominal stress

S - Voo X ~ @ t , dAo = Voo s~ dVo + ~ | [tar] dAo, (23) ovo-~(~) vo 2(~)

where

tN = Nsm(~) : ~ C ~(t). (24)


The dynamical compatibility condition for the jump of the tractions ItNl across the singular surface in the reference configuration E@) takes the form (cf. e.g. [33]. p. 34)

(25)

which can be specified for a propagating discontinuity surface of tangential velocity jump in the spatial configuration as follows (cf. [33], p. 103):

n~r~ls = - 0 U ~ . (26)

Let us consider the processes in which the jump of inertial forces across the singular surface is negligible. This corresponds to the situation in which the movement of the singular surface, being the mathematical idealization of the progressing shear banding zone, is approximated as a quasi-static process. In such a case U = UN = 0 and the jump in the tractions, ItNI, must vanish to ensure the equilibrium condition, which results in vanishing of the integral on E(t) in (23) and restores the classical averaging formula (7) (cf. Nemat-Nasser and Hori [20], p. 37).

Assuming the singular surface of order one with the velocity jump of the magnitude and accounting in (22) for (19.1),

= V~o Grad 2~, dVo + Voo ~ s | N dAo. (27) ~, z(t)

If we choose the current configuration of RVE, at time t, as the reference one, the rate of deformation gradient ,9s becomes then the rate of the relative deformation gradient, ,~'(t)(t), at time t (cf. [35], p. 54) and the averaging formula (27) will take the following spatial form:

V | = grad dV + Y e ,, dA, (2S)

ov-s(t) v s(~)

where 2~" denotes the macroscopic measure of velocity gradient ~ - ,~(~)(t) = ~ (~ ) ~ 1 (t) ,

averaged over the macro-element V traversed by the discontinuity surface S( t ) .

Observe that for Vs = 0 the known relation is retrieved,

4 ~ (29) V

where 1 ~ f(t)(t) = {'(t) f - l ( t ) is tile microscopic velocity gradient, 1 = grad v ..... defined at the point x~ 6 V.

The averaging formula (28) enables us to account for the contribution of micro-shear banding in the macroscopic measure of velocity gradient produced at finite elastic-plastic strain. According to (28) and (29) the velocity gradient _~'. is decomposed as follows:

,1 dr" = L + L M S , L :vss = ~ V~s ~ n dA . (30)

s@)

Assuming that the singular surface S( t ) forms a plane traversing volume V, with the unit vectors s and n held constant, (30) results in

L~s ~MS S | n, (31)

212 R.B. P~cherski

where the macroscopic shear strain rate ~/MS is determined according to (14-16) by the microscopic variables as an average over the RVE,

1/ 1/ @Ms = ~ H .~x /~ dA = ~ H~B.~O.~v , .~ dA.

s(t) s(t)

(32)

Assuming for simplicity that the structural pa ramete r /3~ and the speed v,~s are constant over the surface S(t), we have

1/ 7Ms = B,~V,~s&WS, gMS = ~ Hmsg~ dA. (33)

s(O

The symbol gMs denotes the macroscopic volume density of micro-shear bands that operate within the sequence of clusters sweeping the RVE. The density gMS may change with "time- like parameter" t, for the magnitudes of H,~ and g,~ are, in general, various for different clusters. If we assume the average rate of change ~MS as the amount of the density OMs of micro- shear bands operating in the time period AT, which can be considered as an infinitesimal increment of "time-like parameter" in the macroscopic description, the following equivalent form of (33.1) can be proposed:

"J' M S = B,~s L,~s ~ M S . (34)

The order of magnitude of gMs can be estimated, for the RVE assumed as the unit cube of the length size Lo ~ 1ram, H ~ being of order 100 gm, and for the given density &~ ~ 101~ (m-2), as L)Ms ~ 109 (m 2),

The derived relations are valid for a single system of micro-shear bands. This can be gener-

alized for the case of a double shearing system,

2

v- , �9 (i) ~(i) n(i), (35) ~ = L + L % u s o | i=1

where ~/(~x is the macroscopic shear strain rate and s (~), n (~) are the respective unit vectors of the "i"th shearing system. It is worthy to note that (35) is valid under the simplifying assumption that the active micro-shear bands in both systems operate in the time period, which corresponds to a sufficiently small increment of "time-like parameter" in the macroscopic description. Otherwise, the sequence of events should be taken into considerations. The above relations provide the following macroscopic measures of the rate of plastic deformations and plastic spin produced by active micro-shear bands,

1 L D[4s = ~ ( Ms + LTs) 1

W ~ s ~ (L~/is - LT~s). (36)

The discussed averaging procedure over the RVE with the singular surface allows to account for the characteristic geometric pattern of micro-shear bands which is transmitted upwards through a multiscale hierarchy of observational levels.

Micro-shear ba~dhlg in plasticity of metals 213

4 Constitutive description

4.1 Basic concepts and relations of elasto-plasticity of structured solids

Consider a polycrystalline aggregate as a cont inuum body. An infinitesimal neighbourhood of a material point X of this body corresponds to the discussed RVE, which is sufficient for a valid cont inuum mechanics description of gross elastic-plastic behaviour. The dominant

orientation of the crystalline lattice in the RVE is represented by director vectors. We can choose an arbi trary triad of or thogonal unit vectors that will serve as a reference frame for the description of anisotropic properties and elastic behaviour of the material element. The

director vectors define the structure of the continuous body. The different visualizations of

such a triad are discussed in [3], [25] and [36]-[41], where more detailed discussion and further references can be found. In plasticity of single crystals it is usually assumed that dislocations

traversing a volume element produce a change of its shape but they do not change its lattice

orientation. The macroscopic counterpar t of such a situation in finite deformation plasticity

of polycrystals is the Mandel 's concept of the intermediate, relaxed, configuration, called isoclinic one, in which the chosen director triad keeps always the same orientation with respect to the fixed axes of the laboratory reference frame. In such a case, we can understand that the director vectors are introduced as a tool to moni tor at any instant the state of the

strain induced anisotropy, cf. [41]. Thus, the assumption that the cont inuum is endowed with structure in the form of the director vectors leads unequivocally to the concepts of the local,

relaxed, intermediate isoclinic configurations, plastic spin and structure corotat ional rate. Due to this, the decomposit ion of the deformat ion gradient ~7-becomes unique ~ = E P ,

where E denotes the elastic t ransformation f rom the intermediate isoclinic configuration to the current one and P is the plastic t ransformation from the reference configuration to the

isoclinic one. The similar decomposi t ion of ?7"was proposed earlier by Lee [44] within the context of finite elastic-plastic deformations of a continuous body without explicit definition

of a structure or director vectors and isoclinic configuration. The following basic kinematical relations hold:

~ P = I ~ E - I + E 1 5 P - 1 E -1 , 6J)=-~(_4~'~ + ~ ' T ) , ~ / / '= (_~Z'-~g?7),

co0 = D e + f / J ; D e = {l~E-i},~, CoOP -- {E:PP 1E 1}s ,

5f/.fl __ W e ~- @['.flP, W e = { ] ~ E - l } a , ~/7 p = {E15P 1E ~}~,

(37)

(38)

(39)

where ~ and </Pcorrespond to the rate of deformation and material spin, respectively, and the superscripts e and p refer to elastic and plastic, whereas the symbols {t}.~ and {t}~ denote

the symmetric and skew-symmetric parts of the second-order tensor t. Due to (30) and (36)-(39) the following kinematical relations hold:

= D r + coO; = D e + D p + D_~,zs, g/~' = W e + g//'; = W e + W ; + W~,~s , (40.1, 2)

where D p and W p correspond, respectively, to the rate of plastic deformation and plastic spin produced by crystallographic multipie slip. Observe that according to (30), if the contribution of micro-shear bands is negligible, we have D ~ s = W;MS = 0, what implies 6 ~ P = D p.�9 r = W p and ~ = D, q/2 = W .

214 R.B. Pgcherski

For the sake of brevity and simplicity, isothermal processes will be considered only. The development of the thermodynamic theory of inelastic materials at finite strain can be referred

to [36], [37], [40] and [41]. Assume the mechanical state variables (Tr, A) corresponding to the

isoclinic configuration, where a- is the second Piola-Kirchhoff stress related to the Cauchy stress a:

7r = (det E)E-I~rE T, (41)

and A represents the set of internal variables. The elastic Green strain A e = 1 /2 (ETE - 1)

can be calculated f rom the free enthalpy function H per unit mass, which may be assumed in the form H = H(rr, A) = H1 (Tr) + H2(A) ,

OH A e = --Qh &r ' (42)

where H = q5 - A ~ : (rr /&) provides the relation with the flee energy function per unit mass,

~b, depending in the case of isothermal processes in the following particular form on the state variables (A ~, A), ~ ( A ~, ,/t) = ~51 (A~) + ~ I (A) . The rate of elastic strain reads

j l e OH = - & 0 ~ : 4r = M : 7i-, (43)

where M is the elastic compliance tensor and & is the density related with the isoclinic configuration. The t ransformation of (43) to the current configuration yields

D e = ~ / [ f : 6-, (44)

where 3 / l i y = (det E) E7IEglE~*EE1MABcD and zxi ~, i ~ 'k Zgl

= e - - o U g + tr(gE- ), (45)

or for the Kirchhoff stress r = (det E) a

r = r W % + r W e D % - r D e, (46)

where the spin W e arising f rom elastic deformation and geometric constraints imposed by boundary and compatibil i ty conditions is determined according to (39.1) as W e = qg'J - gps.

This shows that by the formulat ion of rate-type constitutive equations, describing plastic defor-

mat ion of structured solids, the specification of the relation for plastic spin is required. According to the discussion in [43] and [44], it is typical of most of the deformed metallic

solids that their distortional elastic strains remain small under arbitrary loading conditions,

whereas they can undergo large elastic dilatational changes in shape under very high pressure. Applying the earlier concept of Rice [45], who noted that the use of logarithmic measure of elastic strain enables to separate the dilatational elastic changes in shape f rom the distortionat one, Raniecki and Nguyen [43] have shown in the study of thermomechanics of isotropic elastoplastic solids at finite strain and arbitrary pressure, that the tensor of elastic moduli in Eulerian description can be expressed in terms of derivatives of the free energy as simply as in the case of infinitesimal strains, provided the logarithmic elastic strain e = in V ~ is adopted as a state variable and that the values of the ratios of principal elastic stretches U ~, f rom the polar decomposit ion E = V~R ~ = R~U ~, belong to the interval [5/6, 7/6]. In this way, the authors extended the earlier results of Willis [46], obtained under the stronger assumption that the yield stress in simple shear is much less than the elastic shear modulus. The thermo- mechanic theory of isotropic elastic-plastic solids for small distortional elastic strains but arbi-


trary elastic dilatational changes was developed in [43] and extended to account for strain induced anisotropy in [38] and [44]. We confine the further analysis to small distortional and

dilatational elastic strains. Then, the following approximate relations can be obtained, within the accuracy of O(lel2), where e is the deviatoric part of e, (cf. [43], [44] for more detailed dis-

cussion):

~ p = R e { p p - l } ~ R J ,

D e = ~ = k + E W e - w e e ,

with the elasticity equation

02~ "~ = C : D e , C & Oe Oe '

where C is the fourth-order tensor of elastic moduli.

r = R e { P P - 1 } ~ R er , (47 )

W e = R e R ~ , (4S)

= -/- _ W e t + r W e (49)

4.2 The meaning of"yield" within the context of micro-shear banding

It was recognized in [2]-[4] that in the plasticity model accounting for the effects of micro-

shear bands the meaning of "yield" is not a trivial one and requires more detailed analysis. The precise connection of the nominal yield points with intrinsic material properties was dis-

cussed earlier by Hill in [5] and [47], where the idea of an "extremal surface" was proposed.

Let us imagine that after a given prestrain of the RVE of a polycrystalline aggregate further glide hardening on the active slip systems of its constituent grains is suspended. In general,

due to constraint hardening, the incremental plastic flow under constant overall load is still

precluded. However, as it was observed in [5], special configurations of internal micro-stresses and yield vertices are possible that together admit one or more fields of strain rate which are

compatible with zero stress rate. According to Hill [48], such fields correspond to intrinsic eigenstates and are associated with incipient branching of constitutive relation between increments of objective stress and strain. This approach can be applied to the microscopic fields of

stress and strain within the macro-element (RVE). If the micro-shear bands are understood as an effect of special configuration of internal micro-stresses that accumulate at grain bounda-

ries till the glide hardening on the active slip systems is suspended and then abruptly release producing, under constant overall load, the field of plastic deformation rate D p Ms as a self- induced deformation mode, the similarity with the intrinsic eigenstates discussed in [5] and

[48] can be observed. As is emphasized in [5], the "extremal surface" is not a single surface but

is rather an assemblage of yield points for physically distinct states of the RVE, none of which can be reached from any other via purely elastic paths in the stress space. The following observation correlates the properties of the "extremal surface" with the mechanism of micro-shear banding:

The properties of the "extremal surface" conform, at least qualitatively, with the mechanism of

micro-shear banding. The yield state approaching a certain state on the "extremal swfaee" can

be related with the formation of a particular spatial pattern of micro-shear bands. Another state occupying the "extremal surface" pertains to another spatial pattern of micro-shear bands. The transition from one state to the other is not possible via a purely elastic path, for an accumulated

plastic strain is necessary to produce the new set of micro-shear bands characterized, in general, by another geometric pattern. Such an "extremal sur/ace'" forms the generic micro-shear banding surface.

216 R.B. Pr

Based on the aforementioned discussion a simplified model with two limit surfaces can be

introduced. The preliminary study of such a model of plastic flow with an external surface,

taking into account the onset o f shear-banding and the internal yield surface which is related with the backstress anisotropy, was presented in [4].

4.3 Model of plastic JTow with nonlinear evolution of kinematic hardening

It is sufficient for further study to assume the model of small elastic and finite plastic deformations with the following set of internal variables:

~. =: {&, g}, A =: {c~, s}, a = Re&R st, s = g, (50)

related with the isoclinic and current configurations, respectively. The tensor variable a, often

called the deviatoric backstress, describes kinematic hardening effect (the translation of the yield surface) and the scalar variables s =: {x,/(7, ~(i) ~MsJ, i = 1, 2, represent the isotropic hardening parameters ~ and/C, as well as the macroscopic volume density of micro-shear bands t)~s

that operate within the sequence of clusters of the 'T ' t h shearing system sweeping the RVE. The Huber-Mises criterion accounting for kinematic hardening is assumed to approximate the

internal yield surface f , and the constitutive equations at the yield point take the form:

1 f (r ' , ~, ~) = ~ (r' - c~): (r ' - c~) - ~2 = 0, (51)

D = D e+D ( c - 1 + uf us): , # j = (52)

a = ( x / 2 c / , / - mc~) "~, ce(0) = c~o, ~ = ( 2,f2DT: DP), (53)

= - 4 , = (54)

where r ' is the deviator of the Kirchhoff stress r , and z~ is a material constant representing a saturation value for z, whereas b corresponds to a constant controlling the pace of saturation.

The internal variable ~ determines the "size" of the yield surface, (i.e. ~ = l i v e R , where R is

the radius of the Huber-Mises cylinder). The combined plastic hardening modulus h reads:

1

The form of the nonlinear kinematic hardening rule (53) was proposed originally for small strains by Armstrong and Frederick [49] and applied further in the studies on cyclic plasticity (cf. e.g. Chaboche [50]). The material constant m is related with saturation of the backstress effect while the accumulated plastic strain increases. The scalar multiplier j fulfills the conditions j = 1 if # / : "~ > 0 and j = 0 if # / : ~ < 0. The objective rate of the kinematic harde-

ning parameter a reads

d = d - W ~ a + ~ W 2 W e = W - W p. (56)

The additional constitutive equation for the plastic spin W p is necessary to determine the spin W p of the rotat ion of the structure. In the study by Paulun and Pr [51] the relation for plastic spin was derived, which can be applied for the case of nonlinear kinematic hardening

rule (53.1) in the following form:

WV _ 3 (c~D p - DPa), c~ = ~1w1 c~: a . (57) + V A


More detailed discussion on the proper formulation of the relation for plastic spin for the case of kinematic hardening rule (53.1) and of its special form for m = 0, known as Prager-Ziegler !aw, is given in [38] and [51], where also pertinent references can be found.

4.4 Approximation of the generic micro-shear banding swface

Let us observe that the nonlinear hardening rule (53.1) implicitly introduces the second limit surface, for 3~ = 0 leads to (cf. [50])

f ( r ' , I c ) = 2- : - = 0, Jc = +

where ~ is the "size" of the limit surface (i.e. K; - (1/, /2) 7-4, where 7r is the radius of the

external surface). According to the recent studies of Oliferuk et al. [52] the saturation of internal micro-stresses can be correlated with massive formation of micro-shear bands. This is rela-

ted with a certain amount of plastic strain 7 = ~vzs, accumulated along a given deformation path. Assuming that the internal micro-stresses can be represented on the macroscopic level by the backstress c~, we can determine the material constant m, which relates the saturation of

the backstress effect with micro-shear banding. In this way, the resulting limit surface 5 of

radius 7-r comes into contact, at the loading point, with the generic miro-shear banding sur-

face. The discussed results of deformation tests complemented with metallographic observations reveal that the onset of micro-shear bands is strongly dependent upon the change of

loading scheme and the resulting deformation path (cf. e.g. [6]). Therefore, the class of exter-

nal limit surfaces, determined for different loading histories, should be considered. This is depicted in Fig. 4, where the external limit surfaces 5CA and ~ s arrive at the micro-shear

banding surface. The points of contact are pertinent to the respective stress states rA' and rB', which have been reached for different loading paths/2A and/2B. According to (58) the gene-

ral functional relations ~ s = G(s and m = M (s should be used to determine the family of external surfaces, which approximate locally the generic micro-shear banding surface. Systematic experimental investigations are necessary to assess the change of ~vJs and m for different loading paths. The following evolution law for the "size" of the limit surface/C is postulated:

e = s ( I c s - 9 , i c ( 0 ) = + - , ( 59 )

ft~

where/r is a material constant representing a saturation value for/r and 13 is a material constant controlling the pace of saturation. In general, they are dependent upon the change of

loading path. The simplifying assumption should be verified experimentally, which says that the material constants, determined for the given loading path/20, can be taken the same for a class of loading paths containing/20. Observe, furthermore, that the applied stress r ~ reaching the external limit surface becomes coaxial with c~ which results, according to (57), in W p = O.

4.5 Formulation of plastic flow law accounting for micro-shear banding

According to Hill [5], the macroscopic constitutive equations describing elastic-plastic deformations of polycrystalline aggregates are either thoroughly or partially incrementally nonlinear. Depending on the contribution of the mechanisms involved in plastic flow, a region of fully active loading, called also a fully active range, separated from the total unloading (elas-

218 R.B. Pgcherski

tic) range by a truly nonlinear zone corresonding to the partially active range, may exist.

According to the works [5], [29] and [53] the following hypothesis is formulated [3].

Hypothesis 2. For continued plastic flow with the deviations from proportional loading contained within a certain cone of stress rates that corresponds to the fully active range, the incremental plastic response can be assumed as linearly dependent on the stress increment. Inside the fully

active range the spatial pattern of micro-shear bands is" fixed, whereas in the partially active

range the spatial pattern of active micro-shear bands is continuously changing following the orientation of the maximum shear stress plane. This is associated with the thoroughly nonlinear

relation between the rates of plastic deformations and stress.

The connection of the fully active range and partially active range with the geometric pattern

of micro-shear bands is necessary to specify the relation for the rate of plastic deformations

for different loading paths. Due to the fact that the multiple sources of plasticity are dealt

with, the theory of multimechanisms with multiple plastic potentials can be considered. The

concept of multiple potential surfaces forming a vertex on the smooth limit surface was stu-

died earlier by Mrdz [54] within the framework of non-associated flow laws. In our case, the

R (

f

Fig. 3. The two limit surfaces on the 7r-plane (deviatoric plane): f - internal yield surface related with the nonlinear kinematic hardening model of Armstrong-Frederick type and 5 external yield surface corresponding to the saturation of the backstress

D p

D p Fig. 4. The generic micro-shear banding (m.s.b.) surface approximated by the class of the external limit surfaces f obtained for different loading paths /2, e.g..;CA for /2X and 5B for s which relate the saturation of the backstress effect with micro-shear banding


existence of the following plastic potentials related to the mechanisms responsible for plastic

flow can be postulated (cf. [3] and [4]):

(i) the plastic potential 90 that reproduces at the macroscopic level the crystallographic multiple slips and is associated with the external surface approximated by means of the Huber-Mises locus .F = 90; ( ii) the non-associated plastic potentials 91 and 92 that approximate at the macroscopic level the multiplicity of plastic potential funetions related with the clusters of active micro-shear bands.

The plastic potential functions 9I and 92 display the geometry of the considered micro-shear

bands systems and result in two separate planes that form in the space of principal stresses n ,

i = 1, 2, 3, a vertex at the loading point on the smooth Huber-Mises sylinder 5< The planes

are defined by normals Ni, which can be expressed in terms of the unit vectors

s (i) , n (i), i = 1, 2, defining the "i"th system (cluster) of micro-shear bands

N~ 5-~(s <~/on<':)+,,(~)| (60)

The projection of the Huber-Mtses cylinder and the potential planes onto the so-called v-plane

(deviatoric plane) is displayed in Fig. 5. The normals N , ; , i - 1, 2, can be expressed in terms

of the unit normal #S and the unit tangent T to the limit surface Y(r ' , K;) at the loading point

N~ = cos 2,~?#F + sin 2!3T, N2 = cos 2fl#~ - sin 2fiT, (61)

where # s - 1/(~f2/C) r ' . The tensor T is coaxial with the tangent to the limit surface .~(r', K)

in the deviatoric plane at the loading point, (cf. Fig. 5)

O

T = ~ ' f = N [ ~ ' - (ff : #v) #F], T = ~ T ' : r ' . (62)

Due to the pressure-insensitive Huber-Mises locus Y(T', E), "?' : #v = ? : #F holds, whereas H is a normalization factor,

N I1r (# ' = - : u e ) # F I1 -~, (63)

62

Fig. 5. Geometrical representation of the projec- tions of the Huber-Mises locus .5- = 9o and the potential planes 91 and 9_9 onto the deviatoric plane and the plastic cone determined by the normals Nz and N2, within which all the possible rates of plastic deformation cjJ; are confined, where ~3 is the deviation angle of the micro-shear banding system relative to the respective plane of maximum shear stress

220 R.B. P~cherski

which according to

I1~' (~' ~' ~' (~':

is given by A/" = (]] ~ ][ sin6)-*. Due to (35)-(36), (40.1) and (60), the relation for the rate of plastic deformation takes the form

~ 2 ~ = n~ + - 2 ~-yMsi,i/i/,T. (6s)

i=1

Assuming that D p can be expressed by means of the classical J2 theory:

Dv = - @#F (66) 2

and accounting for (61) leads to

= V @*#F + ~ - tMsT, (67)

where

This is depicted in Fig. 5 showing the plastic cone, within which all possible rates of plastic deformation !;0 p are confined. Observe that only the normal component of the rate of plastic deformations contributes to the change of the radius of the limit surface f ( r ' , / C ) , and the consistency condition, J- = 0, yields

x/2 # : r e H =/3(K:, - K:). (69) ~ / * - 2 H '

The following active micro-shear bands fractions, r(i) g2) , of the rate of plastic shearing ~/*, a M S ~ a MS are introduced, respectively

@~) cos 2fl = JMSZ~(1) x*, _x( 2).~ex cos 2/3 = J Ms- r(2) ~'*, (70.1, 2)

where due to (70.1) and for ~* > 0 the following constraints hold:

= r(2) [0, 1] ~(1) ,e(2) [0, 1]. (71) ~(1) ~_ e(2) 1, f(M 1) + a MS e J Ivzs, a lvzs E S + J MS - J Ms

Based on the observation that the micro-shear bands can be active only in the case of continued plastic flow, i.e. when the loading condition is fulfilled, it is assumed that for ~* = 0, fM(1) ~(2) = 0. All possible special cases resulting from the conditions (71) are discussed in S = ~MS [3]. Accounting for (34) and (70) allows us to relate the fractions f (~s with the pertinent macroscopic densities of micro-shear bands ~(i) i = 1, 2. The density of micro-shear bands MS 0(0 operating in the 'T ' th shearing system (cluster) can be considered as an internal variable MS that is given by the following evolution equation:

~ = r(~)(~, ~,1c, ~, ~(~)'eMs~ ~*, 0~s(0) = 0!~), (7~)

which results in

Micro-shear banding in plasffcity of metals 221

Proper specification of the evolution functions F (i) remains an open problem and needs further studies. Certain microscopic models lead to the conjecture that shear banding may contribute to the rate of plastic deformation as a sequence of the generations of active micro-shear bands governed by the logistic equation (Verhulst equation), taken from population dynamics

(cf. [55> According to (62) and (66)-(69), the rate of plastic deformation takes the form, which is

formally similar to the hypoelastic version of J2 deformation theory, studied earlier by Budi- ansky [56] and St6ren and Rice [57] (cf. [2]):

- 2 H - ( 7 4 )

{ ~ : #r fMstan23 6 ~r 1 = ~ ] [ e l [ s i n 6 ' 6 E ( c , ~ ] - p a r t i a l l y a c t i v e r a n g e

Hz 1 H tan 6 ~ ,/~zs tan 2/3, 6 E [0, @] - fully active range.

(75)

The corresponding relation for plastic spin in the case of micro-shear banding reads ([3])

{ ctan 6, 6 E C,~- ef)~s -- fMSX(5) : (~r - - ~-~), X(6) = (76)

4 rH cos 2/3 c tan@, 6El0 ,@].

Observe that accounting for the partially active range enables proper description of the unloading process. The second term in (74) is responsible for the non-coaxiality between the principal directions of stress and rate of plastic deformations, and //1 plays the role of the non-coaxiality modulus, which in the case of tbe futty active range, i.e. for 6 ~ [0, 5c,], is formally similar to the Mandel-Spencer non-coaxiality modulus discussed earlier within the context of plastic behaviour of single crystals and geological materials (cf. [3]). The symbol f:~fs = JMs~(1) _ JMs,~(2) ~a.fs~, E [-1, 1], denotes the net fraction of the active micro-shear bands that contribute to the total rate of plastic shearing. The fraction fa(x is the controlling parameter of the non-coaxiality modulus H1. If .f~1s = 0, (74) transforms into the J2 flow law. According to (70) and (71), the magnitude of f~s can fluctuate within the limits fMS E [--1, 1]. However, the assumption that fMS is kept constant during the deformation process can be considered as a useful first approximation.

5 Concluding remarks

In [2] the possibility of an elastic-plastic material model accounting for the double shear system with related yield planes, intersecting at the Huber-Mises yield locus, was briefly men- tioned as an alternative approach. In such a case, the yield planes can be associated with plastic potentials which might result in a nonlinear associated flow law of corner theory. Notwith- standing the fact that the additional yield conditions related with shear banding have no firm physical and experimental foundations, such an approach could be considered as an attractive approximation which simplifies the theoretical and computational analysis of boundary value problems. This finds confirmation in recent studies, [58], where a similar approach, based on the additional yield condition related with a single shear system and the assumption that the contribution of the mechanism of crystallographic multislip, represented by the J2 flow law,

222 R.B. P~cherski

and micro-shear banding allows the additive composi t ion of pert inent rates of plastic deforma-

tion, was presented independently. The authors proposed % dual yield constitutive model

involving both the J2 flow and a threshold shear stress based-flow", which was incorpora ted

in a finite element p rogram capable of handl ing large strains and rotations. The results of

s imulat ion of shear band localization occurring in a uniaxially loaded plane strain specimen

show that a relatively simple phenomenological approach, captur ing the most essential

features of shear banding, can lead to sat isfactory approximat ion of mater ial behavior.

References

[1] P~cherski, R. B.: Macroscopic measure of the rate of deformation produced by micro-shear banding. Arch. Mech. 49, 385-401 (1997).

[2] P~cherski, R. B.: Physical and theoretical aspects of large plastic deformations involving shear banding. In: Finite inelastic deformations. Theory and applications, Proc. IUTAM Symposium Hanno- ver, Germany 1991, (Besdo, D., Stein, E., eds.), pp. 167-178. Berlin Heidelberg New York Tokyo: Springer 1992.

[3] P~cherski, R. B.: Modelling of large plastic deformations based on the mechanism of micro-shear banding. Physical foundations and theoretical description in plane strain. Arch. Mech. 44, 563- 584 (1992).

[4] Pgcherski, R. B.: Model of plastic flow accounting for the effects of shear banding and kinematic hardening. ZAMM 75, 203 204 (1995).

[5] Hill, R.: The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 779-795 (1967).

[6] Korbel, A.: The mechanism of strain localization in metals. Arch. Metall. 35, 177-203 (1990). [7] Korbel, A.: Mechanical instability of metal substructure catastrophic plastic flow in single and

polycrystals. In: Advances in crystal plasticity (Wilkinson, D. S., Embury, J. D., eds.), pp. 42 86. Canadian Institute of Mining and Metallurgy 1992.

[8] Hatherly, M., Malin, A. S.: Shear bands in deformed metals. Scripta Metall. 18, 449-454 (1984). [9] Dybiec, H.: Private communication.

[10] Piela, K., Korbel, A.: The effect of shear banding on spatial arrangement of the second phase partic- les in the aluminum alloy. Mat. Sci. Forum 217-222, 1037 1042 (1996).

[11] Duszek, M., Perzyna, P.: The localization of plastic deformation in thermoplastic solids. Int. J. Solids Struct. 27, 1419-1443 (1991).

[12] Bai, Y., Dodd, B.: Adiabatic shear localization. Oxford: Pergamon Press 1992. [13] Nguyen, H. V., Nowacki, W. K.: Simple shear of metal sheets at high strain rates. Arch. Mech. 49,

369-384 (1997). [14] Dodd, B., Bai, Y.: Ductile fracture and ductility with applications to metalworking. London: Acade-

mic Press 1987. [15] Gadaj, S. P., Nowacki, W. K., Pieczyska, E. A.: Changes of temperature during the simple shear test

of stainless steel. Arch. Mech. 48, 779-788 (1996). [16] Hill, R.: The mechanics of quasi-static plastic deformation in metals. In: Surveys in mechanics (Bat-

chelor, G. K., Davies, R. M., eds.), pp. 7-31. Cambridge: Cambridge University Press 1956. [17] Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lon-

don Ser. A 326, 131-147 (1972). [18] Havner, K. S.: On the mechanics of crystalline solids. J. Mech. Phys. Solids 21, 383 394 (1973). [19] Havner, K. S.: Aspects of theoretical plasticity at finite deformation and large pressure. ZAMP 25,

765 781 (1974). [20] Havner, K. S.: Finite plastic deformation of crystalline solids. Cambridge: Cambridge University

Press 1992. [21] Nemat-Nasser, S., Hori, M.: Micromechanics: overall properties of heterogeneous materials.

Amsterdam: North-Holland 1993. [22] Smith, D. R.: An introduction to continuum mechanics. Dordrecht: Kluwer 1993. [23] Hill, R.: On the micro-to-macro transition in constitutive analyses of elastoplastic response at finite

strain. Math. Proc. Camb. Phil. Soc. 95, 481-494 (1984).


[24] Hill, R.: On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Math. Proc. Camb. Phil. Soc. 98, 579-590 (1985).

[25] Mandel, J.: Plasticit~ classique et viscoplasticitd. C.I.S.M. Wien New York: Springer 1972. [26] Mandel, J.: Mdcanique des solides andlastiques. - Gdndralisation dans R9 de la r+gle du potentiel pla-

stique pour un ~ldrnent polycrystaltin. C. R. Acad. Sc. Paris 290 B, 481-484 (1980). [27] Hill, R., Rice, J. R.: Constitutive analysis of elastic-plastic crystals at arbitrary strain. J. Mech. Phys.

Solids 20, 401-413 (1972). [28] Nemat-Nasser, S.: Micromechanically based finite plasticity. In: Plasticity today, modelling, methods

and applications (Sawczuk, A., Bianchi, G., eds.), pp. 85-95. London: Elsevier 1985. [29] Petryk, H.: On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface

vertex. J. Mech. Phys. Solids 37, 265-291 (1989). [30] Stolz, C.: On relationship between micro and macro scales for particular cases of nonlinear beha-

viour of heterogeneous media. In: Proc. of IUTAM/ICM Symposium on Yielding, Damage and Failure of Anisotropic Solids (Boehler, J.-P., ed.), pp. 617-628. London: Mechanical Engineering Publications 1990.

[31] Yang, S., Rey, C.: Shear band postbifurcation in oriented copper single crystals. Acta Metall. 42, 2763 2774 (1994).

[32] Truesdell, C., Toupin, R. A.: The classical field theories. Encyclopaedia of Physics, III/1 (Flt~gge, S., ed.). Berlin Gdttingen Heidelberg: Springer 1960.

[33] Eringen, A. C., Suhubi, E. S.: Elastodynamics. Vol I: Finite motions. New York: Academic Press 1974.

[34] Kosifiski, W.: Field singularities and wave analysis in continuum mechanics. Warszawa: PWN - Polish Scientific Publishers and Chichester: Ellis Horwood 1986.

[35] TruesdelI, C., Noll, W.: The non-iinear field theories of mechanics, Encyclopaedia of Physics, III/3 (Fltigge, S., ed.). Berlin G6ttingen Heidelberg: Springer 1965.

[36] Mandel, J.: Thermodynamics and plasticity. In: Foundations of continuum thermodynamics (Del- gado Domingos, J. J. et al., eds.), pp. 283-304. London: McMillan 1974.

[37] Kleiber, M., Raniecki, B.: Elastic-plastic materials at finite strains. In: Plasticity today, modelling, methods and applications (Sawczuk, A., Bianchi, G., eds.), pp. 3-46. London: Elsevier 1985.

[38J P~cherski, R. B.: The plastic spin concept and the theory of finite plastic deformations with induced anisotropy. Arch. Mech. 40, 807-818 (1988).

[39] Raniecki, B., Mrdz, Z.: On the strain-induced anisotropy and texture in rigid-plastic solids. In: Inela- stic solids and structures. A. Sawczuk Memorial Volume (Kleiber, M., Kdnig, A., eds.), pp. 13-32. Swansea: Pineridge Press 1990.

[40] Cleja-Tigoiu, S., Sods, E.: Elastoviscoplastic models with relaxed configurations and internal state variables. Appl. Mech. Rev. 43, 131-151 (1990).

[41] Besseling, J. F., Van Der Giessen, E.: Mathematical modelling of inelastic deformation. London: Chapman & Hall 1994.

[42] Lee, E. H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1-6 (1969). [43] RanieckJ, B., Nguyen, H. V.: lsotropic elastic-plastic solids at finite strain and arbitrary pressure.

Arch. Mech. 36, 687-704 (1984). [44] Nguyen, H. V.: Constitutive equations for finite deformations of elastic-plastic metallic solids with

induced anisotropy. Arch. Mech. 44, 585 594 (1992). [45] Rice, J. R.: Continuum mechanics and thermodynamics of plasticity in relation to microscale defor-

mation mechanics. In: Constitutive equations in plasticity (Argon, A. S., ed.), pp. 23 -79. Cambridge/ Mass.: MIT Press 1975.

[461 Willis, J. R.: Some constitutive equations applicable to problems of large dynamic plastic deformation. J. Mech. Phys. Solids 17, 359-369 (1969).

[47] Hill, R.: Theoretical pIasticity of textured aggregates. Math. Proc. Camb. Phil. Soc. 85, 179-191 (I 979). [48] Hill, R.: On intrinsic eigenstates in plasticity with generalized variables. Math. Proc. Camb. Phil.

Soc. 93, 177 189 (1983). [49] Armstrong, P. J., Frederick, C. O.: A mathematical representation of the multiaxial Bauschinger

effect. G.E.G.B. Report RD/B/N 731 1966. [50] Chaboche, J. L.: Time-independent constitutive theories for cyclic plasticity. Int. J. Plasticity 2,

149-188 (1986). [51] Paulun, J. E., Pr R. B.: On the relation for plastic spin. Arch. Appl. Mech. 62, 386-393

(1992).

224 R.B. Pr Micro-shear banding in plasticity of metals

[52] Oiiferuk, W., Korbei, A., Grabski, M. W.: Mode of deformation and the rate of energy storage during uniaxial tensile deformation ofaustenitic steel. Mat. Sci. Eng. A 220, 123 128 (1966).

[53] Christoffersen, J., Hutchinson, J. W.: A class of phenomenological corner theories of plasticity. J. Mech. Phys. Solids 27, 465 487 (1979).

[54] Mrdz, Z.: Non-associated flow laws in plasticity. J. M6c. 2, 21-42 (1963). [55] P~cherski, R. B.: A model of plastic flow with an account of micro-shear banding. ZAMM 72,

T250 T254 (1992). [56] Budiansky, B.: A reassessment of deformation theories of plasticity. J. Appl. Mech. 26, 259-264

(1959). [57] St6ren, S., Rice, J. R.: Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421-441 (1975). [58] Ramakrishnan, N., Atluri, S. N.: Simulation of shear band formation in plane strain tension and

compression using FEM. Mech. Mat. 17, 30~ 317 (1994).

Author's address: R. B. P~cherski, Center of Mechanics, Institute of Fundamental Technological Re- search, Polish Academy of Sciences, Swietokrzyska 21, PL-00-049 Warsaw, Poland

macroscopic effects of micro-shear banding in plasticity of metals

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