Characterization of dislocations and their influence on plastic deformation in single crystals

Int. J. Engng Sci. Vol. 32, No. 7, pp.

Pergamon

Copyright 0 Printed in Great

1157-1182, IYW 1994 Elsevier Science Ltd

Britain. All rights reserved 0020-7225/94 $7.00+ 0.00

CHARACTERIZATION OF DISLOCATIONS AND THEIR INFLUENCE ON PLASTIC DEFORMATION IN SINGLE CRYSTALS P. M. NAGHDI Department

and A. R. SRINIVASA

of Mechanical Engineering, 6123 Etcheverny Hall, University of California at Berkeley, Berkeley, CA 94720, U.S.A.

Abstract-After providing a rapid summary of some of the main results from a recently constructed theory of structured solids in [l], attention is focussed on the characterization of dislocations and the influence of dislocation density on plastic deformation G by crystallographic slip in single crystals. The dislocation density is identified in terms of Curl 6 p and some (but not all) of the results established for the kinematics of crystallographic slip have their counterparts in the traditional developments of crystal plasticity. A two-dimensional crystal undergoing planar motion is used to illustrate predictive capabilities of the theory presented. It may be emphasized that the theory utilized here is applicable to materials of widely varying strengths and hardening characteristics, including ordered intermetallic alloys whose mechanical behavior is primarily governed by the dislocations (through Curl GP) and not by plastic deformation itself.

1. INTRODUCTION

The purpose of this paper is to discuss a characterization of dislocations as continuous distributions and to examine their influence on crystallographic slip in single crystals. The characterization of dislocations here is based on the accepted notion of existence of a crystal lattice at the atomic scale and utilizes the basic developments of a (macroscopic) dynamical theory of structured solids constructed in two recent papers by Naghdi and Srinivasa [l, 21. The basic theory in [l] is capable of capturing features of flow and deformation up to a submicroscopic scale, i.e. a magnification scale of about X 50,000 as discussed in [l, Section 21. As is well-known, the study of the plastic deformation of single crystals, initiated by the early investigations of Ewing and Rosenhain [3] was significantly advanced by the experimental work of Taylor and Elam [4-61 and Taylor [7], who studied the behavior of aluminum and iron crystals and interpreted the kinematics of deformation with respect to the crystallography of crystals. Taylor and Elam [5] established for these crystals what is now referred to as the “Schmid law” for plastic yielding; this law, as described in [5], is valid only when the elastic deformations are neglected. However, a slighly modified version of the Schmid law is incorporated into the classical developments of crystal plasticity (see, for example, p. 40, last paragraph of Asaro [S]) by redefining the “resolved shear stress” on a slip system through the rate of working due to slip. Since this definition of the resolved shear stress on the slip system is different from the actual shear stress on the slip system, strictly speaking the classical developments of crystal plasticity recover the Schmid law only in the limiting case of rigid plasticity. Taylor and Elam [5] also studied the hardening behavior of single crystals and noted that individual slip systems harden with strain and that slip associated with one slip system may also cause hardening on other slip systems. By determining the onset of simultaneous slip on two slip systems (the so-called “double slip”), they developed a model to predict the rates of ‘self-hardening’ and cross-hardening and went on to propose the so-called “Taylor’s rule” for the onset of “double slip.” Since the advent of the electron microscope in the 195Os, the existence of crystal 1157

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P. M. NAGHDI

and A. R. SRINIVASA

defects-called dislocations-and their role in influencing crystal behavior have been documented. The continuum theories of slip in metal crystals have relied heavily on the study of dislocations (as a sort of “background” or “microscopic” model) for the purpose of formulating constitutive equations; see, for example, the review articles by Asaro [8] and Naghdi [9, Section 81, as well as a recent book by Havner [lo] which contains an extensive list of references on the subject. However, these developments do not explicitly include the effect of dislocations in that there is no measure of dislocation density or “number of dislocations per unit length” in their model; and consequently in this approach, they adopt ad hoc-albeit well-motivatedconstitutive equations for certain dependent variables such as those for the hardening parameters. In the context of the present paper, it should be noted that for some quasi-brittle crystals (such as ordered intermetallic alloys), while the plastic deformation itself defined by G, in equation (2.4) may be small (and hence negligible), the dislocation density characterized in terms of Curl G, [see equation (2.9)] may not be small. For example, a generation of a few hundred dislocation loops-while altering the physical characteristics of the crystal locallyresults only in a plastic deformation of the order of lo-‘“. Nevertheless, these dislocations play a vital role in predicting certain aspects of the mechanical behavior of ordered intermetallic alloys including their strength, hardening and fatigue characteristics; for a detailed metallurgical account of ordered intermetallic alloys and the role of dislocations in their mechanical behavior, see Liu et al. [ll]. It should therefore be clear that a theory of inelastic behavior of materials that explicitly incorporates the effect of dislocations is also relevant to the prediction of the behavior of ordered intermetallic alloys. As is well-known, different slip systems can exist in a single crystal but only a few of them are active at any instant of time. Since both active and inactive slip systems exist simultaneously at each instant, we need to utilize a suitable constrained theory that is capable of predicting which slip systems are active for a given process, or more specifically for a given boundary and initial data. Further, within the scope of such a constrained theory, the constitutive equations utilized, apart from explicit dependence of dislocation density, may be classified as Schmid-like in the sense that the condition for the initiation of slip on a given system is independent of the state of stress of all other slip systems including the inactive ones. After providing a rapid summary in Section 2 of some of the basic results derived from the theory of structured solids [l, 21, we elaborate in Section 3 on the microscopic description of single crystals, the lattice deformation ,F and plastic deformation G, on the macroscopic scale [see equations (2.3), and (2.4)], characterization of the dislocation density, the nature of the additional momentum-like balance laws which accompany the rate of plastic deformation G,, the nature of “effective mass” of dislocations and features of a special class of constitutive equations. Results for crystallographic slip with the use of constitutive equations which explicitly include dependence on dislocation density are presented in Section 4, followed by construction of a suitable constrained theory in Section 5 that is capable of predicting which slip systems are active for a given process. Finally, with the use of the constitutive results of Sections 4-5 for constrained crystallographic slip, an example of a two-dimensional crystal undergoing a motion which is designed to simulate a simple tension test is discussed in Section 6. In particular, the numerical solution obtained illustrates the evolution of the dislocation density from an assumed initial sinusoidal distribution shown in Fig. 6 and striking features of plastic strain shown in Fig. 5. This numerical solution utilizes the hardening characteristics developed in Section 5 and because of the presence of Curl G, requires a numerical analysis that is more intricate than that of a corresponding numerical solution in the usual plasticity theory in which the gradient of plastic strain is absent. The results in Section 6 are presented in nondimensional form and hence are applicable to materials of widely varying hardening characteristics and strength.

Characterization 2. SUMMARY

OF THE

BASIC

1159

of dislocations

EQUATIONS

OF STRUCTURED

SOLIDS

We provide here a summary of the basic results in the theory of structured solids developed by Naghdi and Srinivasa [l]. Thus-in the context of Cosserat (or directed) continua-consider a body 93 bounded by a boundary surface a93, embedded in a three-dimensional Euclidean space g3 and comprised of material points X endowed with a triad of directors. Let the place occupied by X and the triad of directors in a fixed reference configuration K~ of W be specified, respectively, by the position vector X and a triad of director fields DA = D,_,(X), (A = 1, 2, 3), representing the lattice vectors at X. The corresponding position vector x and the lattice triad dA at x in the current configuration K of 53 at time t are defined in terms of functions X(X, t) and gA(D,, t) (A, B = 1,2,3), by x = x(X, 07

d,s,= i%,,(Ds, t; X) = 9/,(X, t),

(A B = I,27 3),

(2.1)

where the functional form of gA in the first of (2.1)* is intended to emphasize its explicit dependence on X while in writing the second of (2.1), we have substituted DB = Ds(X) in the argument of aA. The four vector functions {X, 5BA}will be referred to as a process. The function x defines a motion of each material point, while BA represent the independent motion of the lattice triad at the same material point. We assume that (2.1),, but not (2.1),, is invertible for a fixed value of t, and for definiteness stipulate that the Jacobian of (2.1) is positive. We also assume that the triad of directors is noncoplanar. The deformation gradient F relative to X and the lattice deformation tensor [F are defined by F=z,

J=detF>O

(2.2)

and ,F=dA@DA,

,F-‘=DA@dA,

J, = det ,F > 0.

(2.3)

In (2.2)-(2.3), the symbol @ denotes tensor product and (DA, d”) are respectively the reciprocals (or duals) of (DA, d,) defined through the relations dB * dA = DB * DA = S:, 8,” being the components of the unit tensor. A reversible process of the crystal is defined as any process in which the lattice directors behave as material line elements so that the tensor [F becomes the same as the deformation gradient F. Clearly in this case the macroscopic particles of 9, as well as the lattice directors, may be returned to their reference values in ~0 simply by reversing the motion x. It then follows that the magnitudes and orientations of the lattice directors at the same material point will be different for reversible and irreversible processes. Hence, a suitable composition of F and ,F may be regarded as measure of plastic (or permanent) deformation G, in a crystal defined as [l, Section 31; G, = ,F-‘F.

(2.4)

The tensor G, may be used to define a unique local intermediate configuration, as will be elaborated upon in Section 3 [following equation (3.2)]. The particle velocity v at x and the lattice director velocities w,_,at x are defined by v = iu,

w, = ;1/,,

(2.5)

where a superposed dot denotes material time differentiation with respect to t holding X fixed. Some additional kinematical results that will be needed later are the rate of deformation gradient P and the rate of plastic deformation GP which can be expressed in terms of the velocity gradient L and the rate of the lattice deformation tensor fi, namely fi=LF,

6,, = ,F-‘(L,F - ,fi)G,,

L=E.

(2.6)

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P. M. NAGHDI and A. R. SRINIVASA

For future reference, we also record the relative (Lagrangian) expressions for ,E and E,, namely E = ; (FTF - I),

,E = f (,FT,F - I),

strain E and the corresponding

E, = ; (G;G, - I),

(2.7)

where FT denotes the transpose of the second order tensor F and I is the second order identity tensor. From a combination of (2.4) and (2.7) follows the result (see equation (3.27) of 111): ,E = GiT(E - E,)G; ‘.

(2.8)

The skew part of the referential gradient of plastic deformation (2.4) is identified in [l, Section 31 as a measure of the dislocation density (Yper unit mass in the reference configuration (with components aAB) and can be represented as LY= -(Curl GrJT = a,,E,

@JEs,

ffAB

=

&KBMG;K,M,

(2.9)

where LYAB,GRK, sKB&,are respectively the components of (Y,G, and the permutation symbol E and EA are orthonormal basis vectors in K”. Consider now a rigid body motion of 93 from the current configuration K at time t to a configuration K+ at time t+ = t + a, where a is constant. Under such a superposed rigid body motion (henceforth abbreviated as SRBM), the material point x and the triad of directors d, at x in (2.1) move to the place xf and the triad of directors d< at x+ in K+ and we have

d;=QdA,

X +=a+Qx,

(2.10)

where a is a vector function of time and Q is a proper orthogonal tensor function of time. For quantities associated with the configuration K+, we use the same symbols as those for K but with an attached plus “+” sign. Thus, corresponding to SRBM resulting in (2.10), F’ = QF, ,F+ = Q,F and the kinematical results E, G,, a transform according to E+=E,

G;=Gp,

a+ =a.

(2.11)

We now recall the local (Lagrangian) versions of the balance laws, namely the conservation of mass and the ordinary balance of momentum in the classical theory given by Po=PJ,

poi = p,b + Div P,

(2.12)

as well as the additional balance law for the directed medium under discussion [l, Section 41, namely PO $

P@;,l) = POE

- A

(2.13)

where p& = p&f’+ Div .Y

(2.14)

In (2.12)-(2.14), p,, and p are, respectively, the mass densities in the reference and current configurations, b is the body force per unit mass, P the first (or the nonsymmetric) Piola-Kirchhoff stress tensor, 9is the body force tensor associated with the director triad, RK is the intrinsic director force tensor per unit volume, RJ6Jis the (Lagrangian) third order couple stress tensor and 9 is a fourth order tensor which arises from the director inertia and represents the inertia associated with the plastic deformation G,. Also, the notation “Div” stands for the

1161

Characterization of dislocations

divergence operator with respect to X and the relations between P and .A with their respective stress vector Rt and the second order couple stress RM are given by J = PN, RM = ,&N, where N is the outward unit normal to any surface in the reference configuration ~0, and the second (symmetric) Piola-Kirchhoff stress tensor S is defined by P = I%. For later reference, we note that under SRBM the various stresses in the equations of motion (2.12)-(2.14) transform as: P+ = QP,

,&+ = ,&,

S’ = s,

(2.15)

RK+ = RK.

The intrinsic director force tensor RK plays a significant role in the determination of the inelastic response of the crystal. Indeed, a part of RK designated below by 8, in (2.20) may be identified in special cases as the so-called “configurational force” which acts on lattice defects. This aspect of RK will be elaborated upon later in Section 3. In the remainder of this section, we recall a few results pertaining to constitutive developments for materials possessing an elastic range, i.e. constitutive results for which Gr, = 0 during purely elastic processes. Thus, in the context of the constrained theory discussed in [l, Section 51, we recall that the constraint response RKcind)[i.e. the constraint part of RK in (2.13)] is workless and satisfies a boundedness condition of the form

*(RKcind)v Q) 5 0,

(2.16)

Ou= (E, G,, Grad Gp).

(2.17)

where % stands for the set of variables

With reference to (2.16), the equation @(RKcind),%) = 0 represents a closed orientable hypersurface a3Z of dimension eight--called the yield surface--enclosing an open region X in the nine-dimensional K-space. With the use of (2.13) the yield condition can be reduced to the form @((RK)(ind), %) = @(POE- &det)(a),

Q) = @(E - K,(a),

= g(L, %?L) = 0,

Q) (2.18)

where (RK)Cdetjin (2.18) represents the determinate part of RK in (2.13). We also recall from (1, Section 61 the constitutive results A

*

=

s=p&

ww7

l?-Ic=PO

a4 d Grad G,

(2.19)

and K = ii@, y, p) = K,(Q) + iz,(% P) + K@% ‘YJP),

(2.20)

,. ii, = (Q, 0, p) = 0,

Qw=Po~,

(2.21)

P

where the second order tensor K is given by

K = L&,et) + ; Po(@$;,l) ’ tip.

(2.22)

The scalar y and the unit tensor p which occur in the arguments of the response functions in (2.20)-(2.21) are defined by Y=

with the notation

Il~pll9 p=&-

]I-)I designating the norm.

II%ll ’

(2.23)

1162

P. M. NAGHDI 3. SOME RESULTS

BACKGROUND IN

SECTION

and

A. R. SRINIVASA

INFORMATION: 2 AND

SPECIAL

FEATURES

OF

CONSTITUTIVE

THE

MAIN

EQUATIONS

A remarkably clear and concise background information pertaining to crystalline materials can be found in the books by Guinier [12] and by Guinier and Jullien [13]; and, an overview of features of single crystals at progressively finer scales, in the context of the present paper, is included in [l, Section 21. We first provide here some background information on the microscopic (or submicroscopic) scale which justify the choice of the variables dA in (2.1)2 and then proceed to discuss several important features of the basic dynamical theory summarized in Section 2, along with a summary of discussion of special constitutive results contained in [l, 21. Thus, with reference to the microscopic description of the body, let a fixed reference configurationt K: of the body 93 occupy a region %!& and, similarly, the configuration K* at time t occupy a region 9 *. Any arbitrary material volume Y* of 98 in the two configurations K: and K* occupy, respectively, the regions 9’: (s L%$)bounded by the closed surface &Y,Tand B* (SC%*) bounded by a closed surface aP. Any microscopic material point (or particle) X* within P in the configurations K: and K* is identified by the position vectors X* and x*, respectively. Further, let the location of the center of mass of Y* in the reference and current configurations be designated by X and x, respectively. The adoption of these designations here are in anticipation of identification of center of mass of Y* in ~;f and K* with the position vectors X and x of a material point in Section 2. [This, of course, means that the entire part Y* (on the microscopic scale) is associated with a single material point (or particle) X on the macroscopic scale.] We represent the periodic arrangement of the crystal lattice at each particle X* of Y* by means of vectors DX and dT, (A = 1,2, 3) in the configurations ~g* and K*, respectively. We refer to these vectors as lattice vectors and stipulate further that the lattice vectors are noncoplanar at all times. Next, we introduce the vectors D, and d, through the volume averaged of lattice structure of the crystal, namely DA=+

0 I%

DT\ dV,

d, =$

d: dv, I !?P

(A = 1,2,3),

(3.1)

where V* and Vg are the volumes of Y* in the reference and current configurations, respectively. The vectors DA and dA in (3.1) are in general different from the lattice vectors and, as noted in Section 2, are referred to as the lattice directors. Further, the two sets of vectors {Dr, DZ, DJ} and {d,, d2, d,} are each assumed to be noncoplanar. This implies that the scalar triple product for each set is nonzero, i.e. [D,D2D3] # 0 with a similar condition holding for the vectors d,_, (A = 1,2,3). These conditions allow the introduction of the reciprocal vectors (or dual vectors) DA and dA, introduced in Section 2 following (2.3). In the rest of this section we discuss several important features of the basic dynamical theory summarized in Section 2. (cx) The lattice deformation tensor and its role in defining a local intermediate configuration

While the deformation gradient F transforms a line element dX in ~~ to its image dx in lattice deformation tensor ,F transforms the lattice directors DA in &I to their image dA in dx=FdX,

dA=,FDA,

(A = 1,2,3).

tThe use of the asterisk attached to the various symbols here is to distinguish without the asterisk used for different designations in Section 2.

them from

K, K,

the i.e.

(3.2)

the corresponding

symbols

Characterization

of dislocations

1163

The role of ,F between the lattice directors DA and dA is seen to be similar to that of F between the line elements dX and dx. Whereas (3.2), is integrable, (3.2), is not and of course implies that ,F is not integrable. Given the definition G, through the relation (2.4), a unique intermediate configuration K exists at every material point and can be realized by mapping the body from K to K by means of a local process (see Fig. 1) such that: (1) the lattice directors behave as material line elements and (2) the lattice directors $ in K coincide with the reference lattice directors D,. The intermediate configuration E can be achieved by means of the following linear transformations dX, = ,F-’ dx,

;i, = ,F- ‘dA,

(3.3)

between a line element dx and the infinitesimal vector d% and between the lattice directors dA and the vector a,. Substitution of (3.2),,2 into (3.3)1,2 results in dx = G, dX,

&=D,&

(3.4)

which show that during the deformation from ~~ to K, G, acts on the material line element dX but not on the reference directors DA representing the lattice structure. It should be noted that K is a collection of tangent spaces, i.e. a collection of local configurations, which do not continuously fit together to form a global configuration. This is because G, need not in general satisfy any integrability condition and hence no position vector can be associated with the intermediate local configuration K. When a crystal is subjected to a process during the time interval [t], t], (tl < t), a relatiue plastic deformation tensor H, can be defined from a comparison of the local intermediate configurations at times t and t,. Such a relative measure of plastic deformation, namely Hp(f) = (G,(t) - Gp(fJ)G&) = G&)G;‘(t,)

- I,

(3.5)

will be utilized later in this paper. It can be easily verified that the tensor (H, + I) represents a mapping of the line element in the local intermediate configuration at time r, to the line element in the local intermediate configuration at time t. Indeed, using (3.4) we may write d%(r) = G&) dX,

d,(r,) = G,(t,) dX.

(3.6)

Then, after solving for dX from (3.6)2 and substituting the result into (3.6),, with the help of (3.5)2 we arrive at d%(r) = G,(t)G,‘(t,)

d%(t,) = (H, + I) d%(t,).

(3.7)

(p) Characterization of dislocation density and the Burgers vector

As we noted in Section 1, the presence of imperfections in a crystal lattice plays a significant role in the inelastic behavior of the crystal. Indeed, the motion of certain imperfections called “dislocations” (which may be seen in the form of criss-crossing lines in single crystals under high magnification) gives rise to the plastic deformation of the crystals, while the interaction between arrays of such dislocations has been identified (beginning with the work of Taylor [7]) as the primary mechanism of strain hardening in a single crystal. Thus, in order to construct a physically-based dynamical theory where the effect of lattice imperfections are taken into account, we need to characterize these imperfections in the context of the continuous lattice structure introduced earlier. Although the vector field dA or equivalently ,F describes the lattice structure of the entire ES 32:7-I

1164

P. M. NAGHDI

and A. R. SRINIVASA

crystal, it is not in itself an explicit measure of the field “defectiveness” of the lattice. However, focussing attention on the primary type of crystal imperfection, namely the dislocations, it is clear an explicit characterization of dislocations is essential in order to identify their effect on the inelastic behavior of the crystal. We now provide some background and discuss the main features of the development in [l] which led to a measure of dislocation density (2.9) representing the number of dislocation lines crossing a given area element in the current configuration of a deformed crystal. Following the procedure adopted by Bilby [14], we define a Burgers circuit as a closed sequence of lattice steps (or segments) in a deformed crystal lattice. The corresponding steps (or segments) in the reference lattice, called the associated path, can be easily realized with the help of (3.2) relating the triad of the reference lattice directors to the triad of the current lattice directors. In general, the associated path is not closed, but begins at a lattice starting point S and ends at a lattice final point F (see, for example, Hirth and Lothe [15, Figs 1.20 and 1.21, pp. 22-231 and Guinier [12, Fig. 2.81). We define the positive normal to the areaenclosed by the Burgers circuit by the right-hand screw convention. In this way, the vector FS in the reference lattice is called the Burgers vector and is a measure of the number of dislocation lines treading the circuit. The description in the above paragraph has its roots in regarding a single crystal as a discrete set of lattice points and a finite number of discrete dislocation lines. We now proceed to define the Burgers circuit and its associated path for the continuous lattice structure. It should be emphasized that while the lattice steps in a discrete lattice are small but finite, the corresponding “steps” in a continuous lattice are infinitesimal, i.e. tangent vectors at a material point. With this background, consider a simple closed curve Ce, referred to as a circuit and parametrized by A E [0, 11, in the current configuration K at some time t. The place of any material point on % is then specified by x = f(A), and f(0) can be taken as the material point corresponding to the starting point S (referred to in the preceding paragraph). Further, let the closed curve Ce,,be the inverse image of Y in the reference configuration K,) such that x = X(h) = x_‘@(h), t).

(3.8)

By virtue of the invertibility of (2.1) X = x-‘(x, t) and any entity defined as a function of X can be expressed as a different function of (x, t). Applying this to the reference director triad D, in K(, and to d, = iBA(X, t) in K, we have D, = DA(X) = D&‘(x,

t)] = &(x, t),

d, = aA(X, t) = 9,_,[x-‘(x, t), t] = ;i,(x, t).

(3.9)

of D, and dA, The right-hand sides of (3.9)1,2 are the spatial (Eulerian) representation respectively. When a crystal is undergoing a process, the current lattice directors dA change with time, but the reference lattice directors remain unchanged in the sense that the material derivative of DA is zero. Returning to the discussion of Burgers circuit, we first observe that the components dWAof the increment d% with respect to the lattice directors & in (3.9)* are dp(A, t) = d% . dA(x(A, t), t).

(3.10)

These components d? represent the incremental distance travelled along each of the lattice vectors and hence are the infintesimal “lattice steps” corresponding to the lattice steps in the discrete description of the lattice points. We obtain the “associated path” in the continuum description by performing the same sequence of infinitesimal lattice steps with respect to the reference lattice directors at the material points, i.e. via the increments d%= dFASA(x(A, r), t).

(3.11)

Since ;i, and DA are different in general, it should be clear that while the sum of the increments df (=tiA&,) form a closed path, the same is not true of the sum of the increments df

Characterization

of dislocations

1165

=d?fiA). The path formed by the increments d% is called the associated path in the continuous description of the lattice. It then follows that (3.12) The first of (3.12) follows from the fact that V is a closed path, the tangent vector to which is dx while the left-hand side of the second of (3.12) defines the Burgers vector B and follows dh’ from the fact that 6, # ;i, in general. With the use of transformations (3.2),,2, the line integral on the left-hand side of (3.12)2 can be rewritten as B(%, t) = f j-’ ‘G = f 4,

dx = $ ,F-‘FdX ‘G

,F-‘F dX =

(3.13)

G, dX, f%

where ,fi is the spatial form of ,F(X, t) and the closed circuit %$,is the inverse image of %’in the reference configuration K,,. Finally, with the use of Stokes’ theorem, the line integral in (3.13) can be converted into a surface integral and the result can be expressed in terms of a referential surface integral over the inverse image A of the current area associated with (3.13) in the reference configuration. The result can be stated as (3.14) where N is the outward unit normal to the reference surface A and (Yrepresents the dislocation density defined by (2.9). (y) The additional momentum-like

balance law

The additional momentum-like balance law, which results in the additional equations of motion (2.13) associated with the rate of plastic deformation GP, plays a significant role in the construction of the basic dynamical theory in [l, Section 41. To elaborate, we recall that the kinetic energy per unit mass in K is assumed to have the form

K =

;

{V ’

V +

~p(dsl[~p])},

where the inertia tensor coefficient Q referred to an appropriate reference configuration K” is defined by

(3.15) fixed orthonormal

basis in the

It can be readily verified that the coefficients %‘ABcDhave the physical dimensions of [L-‘1 with the symbol [L] denoting the dimension of length. Then, the ordinary momentum per unit mass and the momentum associated with GP per unit mass are defined by aK -=v

av

’

(3.17)

1166

P. M. NAGHDI

and A. R. SRINIVASA

The first of (3.17) arises from the first term on the right-hand side of (3.15) which is an expression for the ordinary kinetic energy per unit mass, while the second of (3.17) may be interpreted as the magnitude of the additional inertia arising from the kinetic energy of the microscopic particles surrounding the center of mass which contribute to the rate of permanent deformation &. The second term inside the bracket on the right-hand side of (3.15) can be derived from the inner product of the difference vector [a, - (&),I with itself at a given instant of time, where the notation (d,& designates the special value of & for reversible processes only (see [l] for details of the derivation). Clearly, the difference vector [& - (&Jr] represents the rate of change of the lattice directors in excess of the rate of change of material line elements which coincide with them instantaneously such that the excess kinetic energy vanishes for purely elastic processes. It should be evident from the foregoing remarks that the additional momentum-like balance equations (2.13) have their origin in a momentum-like law for the lattice director velocities (2.5)* in excess of the velocities of material line elements with which instantaneously coincide. (6) The interia tensor and the inertia coeficient

The lattice directors, as well as the lattice vectors, have the physical dimensions of length so that the dual vectors dB and DB have the physical dimensions of l/length. It should be evident from (2.22) that the term &@cr,]) - & corresponds (in the submicroscopic description) to the effect of the increase in inertia of the nonmaterial region influenced by the moving dislocation. Since plastic deformation of a crystal takes place by the motion of individual dislocations close to the atomic scale, the inertia associated with plastic deformation may be viewed as that due to the movement of dislocations. We recall that the notion of the kinetic energy and hence a and “equivalent mass” associated with discrete definition of the “crystal momentum” dislocations has been discussed in the past by considering the linear elastic behavior of single dislocations: see, in this connection, p. 484 and p. 493, paragraph 2, lines 4, 5 of Nabarro [16] where it is shown that a screw dislocation (idealized as a line defect) moving in uniform rectilinear motion has a kinetic energy per unit length of the form (the notation here differs slightly from that in equations (7.3) and (7.6) of [15]): (3.18)

where V is the uni-directional dislocation velocity, R,, is the dislocation core radius, R is the radius of the region influenced by the by the moving dislocation, p is the shear modulus, a, is the shear wave velocity, and b is the magnitude of its Burgers vector for the dislocation. The crystal momentum of a single dislocation of “effective mass” m corresponding to (3.18) has the form (3.19)

The expressions (3.18) and (3.19) suggest that its equivalent mass is not a constant but a function of the kinematical variables, although for a slow-moving dislocation (3.18) can be approximated to m =$

In f 0

(per unit length).

(3.20)

This result is in support of the physical dimensions of the inertia coefficients 3ABcD in (3.16)

Characterization

of dislocations

1167

and confirms that the inertia coefficient 41rin (2.13) plays a significant role in the determination of response functions in the constitutive equations such as that for K in (2.22). (e) Features of constitutive results

The basic constitutive developments summarized in Section 2 provide important clarification pertaining to the structure of the constitutive response functions for both viscoplasticity and the more usual (rate-independent) plasticity. The forms of the constitutive response functions indicated in (2.19)-(2.21) may be motivated by assuming that the external rate of work on the crystal is equal to the rate at which energy is stored plus the rate of energy dissipation in the crystal. More precisely, the results (2.19)-(2.21) involving the stress potential $ can be derived in the context of a more general thermomechanical procedure after specialization to the isothermal case. For details of this thermomechanical procedure, which also makes use of an entropy balance, see Green and Naghdi [17]. It should also be noted that in the discussion of constitutive equations, it becomes convenient to define a new dependent variable K [see equation (2.22)] as the sum of RK and a part of the inertia term that represents the “effective mass” of dislocations. The response function 4 in the constitutive assumption (2.19), depends on the set of variables Q which includes G, and its gradient Grad G, and hence also dislocation density. Once the response function 3 is known, S, ,&+!,K, are determined by (2.19)2,3 and (2.21), and we need to specify constitutive equations for K:,, K3 and the inertia tensor 8 in (2.22) which also depends on the basic kinematical variables. Continuity assumptions of the response functions for K and ,& at the initiation of yield permits a geometrical interpretation of the function KZ (see subsection 5.2 of [l]) and leads to the condition that @&(oU, p), “u) = 0 must be identically satisfied for all values of p. This result then implies that the function fi;, always lies on the y:?ld surface @ in K-space. It is convenient to also recall for later reference some special constitutive equations for the rate-independent terms S, .A and K, in (2.19)-(2.21). Thus, under the assumption that the Cauchy stress tensor T depends only on the lattice deformation tensor ,F, i.e. T = ?(,F), a relatively simple expression can be obtained for I,$in the form [2, subsection 2.11: $ =

J&(P) + b(G,, Grad Gp),

(3.21)

where Jp = det G, # 0. Setting & = 0, it can be shown that

,_1a+1

FT

J, a,F ’

’

/pu=o

(3.22)

JT,

(3.23)

and ,F-r&F

= p&I-

where J, = det(,F) # 0, J = det FZ 0. It should be noted that the right-hand side of (2.25) corresponds to Eshelby’s [18] “energy-momentum” tensor, i.e. the elastic force acting on a defect.

4.

CRYSTALLOGRAPHIC

SLIP

It is well-known that when a single crystal is subjected to finite deformation it remains quite uniform on the macroscopic scale. Evidently this is true for an extension of the order of 70% (see [5, p. 47, 1st paragraph]). However, under a higher magnification, one can observe a large

1168

P. M. NAGHDI

and A. R. SRINIVASA

number of microscopic steps on the surface of the crystal, clearly revealing the nonuniformity of the motion on the microscopic scale (see Fig. 4.12 of [13, p. 1931 and Fig. 2.18 of [19, p. 701). It is an accepted fact that the formation of these steps is due to the relative sharing motion of the microscopic particles with respect to each other on specific crystal planes (called slip planes) along certain directions (called slip directions) as reported by Taylor and Elam [4, p. 1661 and subsequently summarized by Taylor [7, p. 362, 2nd paragraph] for homogeneous deformations. This hearing motion leaves the lattice structure of the crystal unaltered. Each slip plane with any of its allowable slip directions forms the so-called slip system. From crystallographic studies of any given crystal in a given configuration (usually considered to be stress-free), the set of all possible slip systems may be found and are tabulated in standard texts on the subject (see, for example, Barrett [20]). Such slip systems, in the context of modern understanding of the structure of matter, may be regarded as arising from the atomic lattice. The equivalent lattice structure representing the volume averaged properties of the atomic lattice in the sense of (3.1) was introduced in the opening paragraphs of Section 3. In this section we provide a rapid summary of the standard notations, together with some derived results, relevant to the kinematics of crystallographic slip in single crystals. Readers who are familiar with the subject of multiple crystallographic slip in crystals are aware of the fact that the notation in this area is naturally cumbersome. This is because a large number (up to 12 for FCC crystal) of different slip systems can exist in a single crystal of which only a few are active at any given time. Thus, in addition to notations for identifying the possible slip systems in a crystal, we also need to introduce separate notations for those that are active at a given instant. With this in mind, at each macroscopic particle (or material point) of the crystal we admit the existence of N (N 2 1). slip systems. Each slip system consists of vectors sck) and n’k’-not necessarily unit vectors-which represent the slip direction and the normal to the slip plane in the current configuration. Further, let ($‘, n{f)) be the values of (#‘), &)) in K,,. The reference vectors (s&“‘,n&“‘)are assumed to be orthonormal for each k, i.e. (k)

(k) =

%I * %I

,#’

.

&d = 1 ,

(k) so .n{f)=()

(k=l,...,N).

(4.1)

Here the index k signifies a member of the N possible slip systems which are assumed to be numbered in some sequential manner. Recalling that the slip systems in the microscopic description of the crystal arise from the atomic lattice, we suppose that the macroscopic slip systems (sck), rick)))are defined with respect to the lattice directors alone. Since the lattice directors in K() are acted upon by only the lattice deformation tensor ,F as in (3.2),, it can be shown that s(k) = ,F@‘,

rick)= (det ,F)(,F-‘)I@’

(k = 1, 2,

. , N).

(4.2)

The first of (4.2) readily follows from the fact that ,F acts on so(k), but the second of (4.2) is less obvious and can be derived from the fact that rick) represents the normal to a plane (which is defined with respect to the crystal lattice) and hence transforms like the normal to an area element under the deformation ,F. The details of the derivation of (4.2)2 are similar to that discussed in Naghdi [9, p. 3841. We now turn our attention to the set of linearly independent active slip systems and identify them in the following manner: let there be R (21) active slip systems consisting of the pairs of sectors {(@I), ti”‘), (sP’*‘,nPc2’), (sP’3’,nPc3’), . . . , (@?‘, I@“‘)} arranged in some order, where the indices p(l), . . . , p(R) are drawn from the numbering given to the N possible slip systems. Thus, the p(k)th slip system is the kth active slip system. For example, a face centered cubic crystal has 12 possible slip systems which are numbered (in some sequential way) from 1 to 12. Further, suppose that at a given instant there are only three active slip systems (R = 3), say the lst, the 7th and the 9th. Then, p(l) = 1, p(2) = 7, p(3) = 9. Having disposed of the above preliminaries, we now address the question of relating the concept of crystallographic slip on the microscopic scale to the kinematics of the crystal on the

Characterization

1169

of dislocations

macroscopic scale. We recall that the relationship between crystallographic slip and the macroscopic deformation of the crystal in the work of Taylor and Elam [4] was established by neglecting the elastic deformation and assuming that the deformation is homogeneous on the macroscopic scale. They arrived at the conclusion that the relative motion of the material points (i.e. the deformation gradient tensor F) is composed of shearing motions along each slip system. Subsequently, results for inhomogeneous elastic-plastic deformations were written down by analogy; see Havner [lo] for details of both the work of Taylor and Elam [4], as well as more recent efforts in crystal plasticity. In all these cases (owing to the long-range ordering of the atomic lattice), both the atomic and macroscopic descriptions of slip systems are regarded to be identical and the effect of dislocations (either as discrete entities at the submicroscopic level or as continuous distributions on the macroscopic level) are entirely neglected. The question naturally arises whether a parallel development can be pursued in which the microscopic (or submicroscopic) description of the crystal is characterized in terms of the independent variables d, (representing the effect of the lattice vectors) through which the presence of dislocations is explicitly accounted for. As it turns out, some of the main kinematical results derived are similar to those utilized in “classical” crystal plasticity. A rapid outline of our derivation is as follows. The lattice directors d, form a convenient set of three basis vectors located at the center of mass x of the region Y* in the microscopic description of the crystal. With this in mind, the relative position vectors U* with respect to the center of mass x of any microscopic particle in Y* may be expressed as u* = zAdA. The scalars zA are the “lattice components” of the relative position vector u *. Taking the material time derivative of u* and t-toting that a part of it, i.e. iAdA, represents the relative velocity of the microscopic particle with respect to the underlying lattice, we stipulate that the term iAdA must be composed of the sum of shearing motions on each macroscopic slip system. This is the fundamental notion of crystallographic slip, after taking an appropriate limit as the size of Y* tends to zero, and leads to the result (for details see [21])

(k)shk) C3mhk))G,, k=l

(4.3)

where jCk) represents the rate of slip on the kth slip system. It may be noted here that upon suitable reinterpretation of the symbols, the expression (4.3) is similar in form to that in Havner [lo, p. 37, equations (3.2)]. Indeed, by neglecting the effect of lattice deformation such that G, = F, we recover the result of Taylor and Elam [4]. By a standard results from linear algebra, we also have

$ (det G,,) = [(det G,)G;y

-t$,= 0,

(4.4)

where in obtaining (4.4) we have also used (4.3). By integrating (4.4) and using the result that G, = I in K”, it follows that det G, = det(I) = 1. In terms of the index notation introduced at the beginning of this section, we may rewrite (4.3) as

$, = 5 jP(k)Ap(k) (R 5 81, k=l

where for convenience

y(k) # 0

(4.5)

we have set A p(k)- b gck)@ II;‘~‘]G,.

(4.6)

The tensors ApckJmay be regarded as a covariant basis for the set of active slip systems, the truth of which is easily verified by postmultiplication of (4.6) by Gp’. Since the tensor field G, is an element of a 9-dimensional Euclidean space, we may introduce a set of 9 - R linearly

1170

P. M. NAGHDI

and

A. R. SRINIVASA

independent tensors A:, (i = 1,. . . , 9 - R), which form a basis for the orthogonal of the subspace spanned by Aptkj such that2t

A,+*Apck) = 0,

complement

(k = 1,. . . , R; i = 1,. . . ,9 - R).

(4.7)

Then, with the use of (4.7), the inner product of (4.5) and Al yields A’.&,=O,

(i = 1,. . . ,9 -R).

(4.8)

The results (4.5), (4.4) and (4.8) are the fundamental equations in the kinematics of crystallographic slip. The first of these represents the expression for the rate of the plastic deformation by the process of shearing on each slip system and is accompanied with no local plastic volume change in accordance with (4.4). The condition (4.8) imposes a constraint on the rate of plastic deformation of a crystal undergoing crystallographic slip.

5. CONSTRAINED

THEORY

FOR

CRYSTALLOGRAPHIC

SLIP

The condition (4.8) imposes a constraint on the rate of plastic deformation ei, along the active slip systems in the current configuration of the crystal. In the context of the theory summarized in Section 2, this constraint on the plastic deformation is accompanied by a constraint response. It should be emphasized that this feature is in contrast to the usual classical theories of crystal plasticity where there is no additional balance of momentum and consequently no constraint response. Preliminary to discussing the consequences of the contraint (4.8) and its effect on the constitutive results, it will be helpful to briefly recall the procedure for dealing with internal constraints in the context of a purely mechanical theory of deformable media (see Truesdell and No11 [22]). Thus, in the presence of a constraint condition of the form (4.8), the constitutive equations are determined to within an additive constraint response such that each response function such as RK can be expressed as ( )= (

)ind +

(

)det*

(5.1)

The determinate parts ( )det in (5.1) are specified by constitutive equations while the indeterminate parts ( )ind, which play the role of Lagrange multiplers, are arbitrary functions of (X, t) and are furthermore workless. With the above background in mind, we recall that when a crystal underoges a reversible process, i.e. the lattice directors behave as material line elements, the rate of plastic deformation vanishes and we have

cl;,=0

for reversible processes.

(5.2)

The condition (5.2) may be regarded as a special case of (4.8) or (4.5) with all the -j~~‘~’ set to zero. In view of this, all the constraints may be treated in a unified manner and in what follows we adopt this approach. It is convenient to define a second order tensor B by (5.3)

B = (RK)ind + Div&%d,

where (RK)ind and (R&)ind are respectively the constraint responses of the intrinsic director force tThe superscript in Al should emphasize the orthogonality

not be confused with the common of the two tensors on the left-hand

notation for edge dislocations; side of (4.7).

here,

it is used to

1171

Characterizationof dislocations and the third order director stress tensor in the balance law (2.13)-(2.14). that (R-&d = O,

ii = (RK)ind= arbitrary finite,

It can then be shown

for reversible processes

(5.4)

while B * Apw = 0,

(k = 1,. . . , R)

(5.5)

when slip takes place on the p(k) independent slip systems. The tensor g may be viewed as the effective force necessary to maintain the constraints (4.8). In the more familiar setting of constrained theories of deformable media (such as incompressibility or inextensibility), the restrictions imposed by internal constraint conditions persist throughout the entire process (or motion). By contrast, the constraint conditions (4.8)-which also include (5.2) as a special case-may hold only partially during crystallographic slip, i.e. only some of the slip systems may be active throughout relevant processes. Given this premise, we stipulate that the constraints hold only as long as the effective force maintaining the constraints, namely B, lies within a closed and bounded region in the appropriate space of constraints. For example, the constraint $ = 0, valid for reversible processes, is stipulated to hold as long as the constant force B (=(RK)ind) lies within a bounded region 3iTin the nine-dimensional K-space. The boundary X%!of this region is represented by the equation (2.18) and is called the yield surface. In a similar manner, for each set of constraints of the form (4.8), we may introduce various regions where the constraints are stipulated to hold. In view of the form (4.8) for the rate of plastic deformation in terms of the active slip systems, the constitutive equations for the determinate parts & and & in (2.20) must now be slightly modified to explicitly include their dependence in the active slip systems Aptkj, i.e. (5.6) The inclusion of Apckj in the arguments of (~5.5)~,zfor & and & suggests that the determinate responses are different when different slip systems are active. This implies that certain restrictions must be imposed on the constraint responses in order to ensure that the overall responses (which include the sums of the determinate and constraint responses) are continuous at the instant or transition from one set of constraints to another. Examples of such transitions are: (a) transition from elastic behavior-+ single slip; and (b) transition from single slip + multiple slip, To continue the discussion, consider a crystal undergoing a reversible process in the time interval [to, tl]. At time c = t, a transition occurs and R (21) slip systems become active for I > cl. Then, the continuity of the director implies that $

Ie + (RK)ind - Div(~~~~~l=

FJy Ig + fi - DivW&J

(5.7)

where & and ~~~}~~~are given, respectively, by (2.201, (2.21) and (2.19&. Now, making use of the continuity of G,, &,, and $, at c = t, and using the constitutiv~ results (2.20), (2.21) and (2.19)3, we arrive at the result lim (RK)&j = !iLy (ii;! + B)

fth

(5.8)

where fi satisfies (5.5) and & is given by (5.6),. Observing that plastic deformation begins at t = t,, so that lim, T,, (Rfoind satisfies the yield condition (2.18), we conclude from (5.8) and (2.18) that the right-hand side of (5.8) satisfies the yield condition (2.18) identically for all values of & satisfying (5.5). It can then be shown that the condition (5.8) implies that: (a) The outward unit normal to the yield surface in K-space lies in the vector subspace spanned by the active slip systems Apckp

1172

P. M. NAGHDI

and A. R. SRINIVASA

(b) The yield function Q, is independent of the inactive slip systems and hence is of the form of a hypercylindrical section with its generators parallel to the inactive slip directions. In particular, for the case of single slip, the yield surface is in the form of a hyperplane normal to the active slip system. It is worth mentioning that in spite of the assumption of crystallographic slip on distinct slip systems, the yield surface does not exhibit corners, contrary to the usual developments in crystal plasticity. Instead, the general yield surface in K-space is in the form of flat hyperplanes connected by rounded corners. The presence of sharp corners is the result of special constitutive equations for K* and fi;,. In the remainder of this section, within the framework of the constrained theory, we focus attention on special constitutive equations that will be utilized in Section 6. A rapid summary of some of the results established previously [2, subsection 2.31 pertaining to the rate-independent constitutive equations is given at the end of Section 3 [see equations (3.19),,* and (3.20)]. We consider now special forms of the rate-dependent constitutive equations KZ and &, as well as detailed discussions of constitutive equations for the work hardening response of single crystals. It should be emphasized that the special constitutive equations developed here are Schmid law-like in the sense that the response of the material to slip on any one slip system is independent of the stress state on all other slip systems (including the inactive slip systems). We consider now special forms for the response functions in (5.6). In view of the fact that the overall response K is determined to within an additive constraint response and also that the constraint response satisfies the condition (5.5), it is sufficient to specify the constitutive equations for only the components of fi, and c(;, along the active slip systems. With this in mind, we stipulate that .p_APw)_ Kp;kdW K2 IIApck,ll Kp(!f)(~) ,.

when jgCk)> 0, when qpCk)< 0,

8, - ,‘$,(k) = qypck’

(5.9) (5.10)

where i,“(k) is the rate of slip on the kth active slip system, n is a constant “viscosity coefficient” and K;(k) are scalar functions of the variable 021. Two features of the above constitutive equations must be noted: (i) the components of the are independent of the other slip systems, and (ii) responses K? and Kj along a Slip SySteIII A,,(k) A is dependent upon while the functions K&) are independent of rates, the function K(;,. pO II&k, II the direction (sense) of the rate of slip on the p(k)th slip system but not upon its magnitude. In view of the above features, the constitutive response functions (5.9) and (5.10) may be considered “Schmid-like” in line with the remarks made in the Introduction. Further, recalling the continuity condition (5.8) and the paragraph following it, it can be demonstrated that the yield function Q in K-space is of the form of intersecting hyperplanes each at a distance K&) from the origin. The edges and corners of the yield surface represent the “degenerate” regions where multiple slip takes place. Owing to the presence of flat yield surfaces with many corners, the yield surface may be considered as Tresca-like. We now proceed to obtain a specific form for the function K&). In view of the fact that they represent the distance from the origin of each of the hyperplanes forming the yield surface, the functional form of K~(~) reflects the hardening characteristics of the crystal, which has received considerable attention in the material science literature. It is a well-established fact that the presence of dislocations and other impurities strongly affect the hardening behavior of the crystals. Indeed, as Van Bueren [23, p. 160] states: “... work hardening is principally due to the interaction between dislocations....” He also identifies three ways in which impediments to dislocation motion (and hence work hardening) can take place. These are:

Characterization

of dislocations

1173

(4 long-range tb) dislocations

interaction between dislocations; which intersect the glide plane of the moving dislocation, particularly by intersecting screw dislocations; and (cl the action of immobile dislocations that are more or less parallel to the moving dislocations. In this case the Burgers vectors are not in the glide direction. In the present development in which dislocations are modelled as a continuous distribution, the effect of elastic interactions of dislocations is already taken into account by the fact that the Cauchy stress T is assumed to be dependent only on the lattice deformation ,F, in order to account in a “cartoon-like” manner for the remaining contributions (b) and (c), we first recall that at a given material point, the total Burgers vector of all the dislocation lines crossing a unit area in the reference configuration IC(,is given by B = a[N]

(5.11)

where N is the outward unit normal to the surface area in ~0. With this in mind, we specify that the work hardening parameters K: for the kth slip system are given by /c:=

foK

+

,K(G~*~~‘~n~~‘)

f

_LK IfOf

+

jK

I(@

’ ol[I@])l

f

c,K

I(l$’

’ a[~[/)

* (I@‘@n&“))i

X qY’])l

(no sum on k).

(5.12)

6. AN EXAMPLE In order to examine the predictive capabilities of the theory of Sections 4 and 5 for crystallographic slip, we consider here a two-dimensional rectangular crystal undergoing a plane motion. In its reference configuration h, depiected in the X-Y plane of the rectangular Cartesian coordinates as indicated in Fig. 2, the crystal is of length L and width w with only two possible slip systems each oriented at an angle 0, ( LY= 1,2) with respect to the positive X-axis in? e. The two-dimensional crystal is assumed to have a perfect lattice structure with no defects in its reference configuration so that the reference directors (Q, D,) coincide with the basis vectors (E,, E2) along the (X, Y) coordinate axes in 4. While the reference configuration is assumed to be stress-free, the same is not necessarily true of the initial configuration due to

Fig. 1. A schematic diagram representing (on the macroscopic scale) the elastic-plastic deformation of a singIe crystal resulting from cr~tallographic slip on a single slip system from a reference configuration IC,)to a current configuration K. The motion is described by the deformation gradient F of the material points and the lattice deformation tensor ,F which represents the motion of the lattice. Also shown is a local intermediate configuration C obtained by the application of the tensor to the material points. Note that the lattice is undeformed in the configuration ii such that the lattice deformation tensor from K,, to 12is the identity tensor I. tFigure 3 representing the shape of the deformed crystal is placed next to Fig. 2 for easy comparison. However, we defer elaborating on the features of deformed crystal until later in this section.

1174

P. M. NAGHDI

and A. R. SRINIVASA

Fig . 2. A two-dimensional rectangular crystal of length L and width W depicted in the X-Y plane of the reference configuration. Also shown are two possible slip systems oriented at 45 and loo” relative to the positive X-axis.

Fig. 3. Reformed configuration of the crystal after it is subjected to an extension of 27.5% along the boundary Y = L, with the remaining boundary conditions as described in Section 6.

Characterization

of dislocations

1175

the possible presence of dislocations in that configuration. From the initial configuration, the crystal is subjected to a two-dimensional motion in accordance with the following boundary data: (i) the lateral edges of the crystal (X = 0, W) are always stress-free, (ii) the bottom edge (Y = 0) is held fixed such that the relative displacements ul and u2 are zero at Y = 0 and (iii) the relative displacements u1 and u2 at the top edge (Y = L) may be arbitrary functions of time. Because of the two-dimensional nature of the problem, in general there are two nonzero components of the dislocation density OL,namely (Y,3 and (~23. However, an initial value for (Y may be specified in terms of only one nonzero component in the initial configuration. For purposes of illustration, we select the orientation of the slip systems in N,,to be 8, = 45”, B2= 100”(see also Fig. 2) and assume the following sinusoidal form of a dislocation distribution in the initial con~guration: a13

= a0

sin sin 6n;X, t

12?& L ’

ffAB

-0

if A f 1, B Z 3,

-

(6.1)

where the coe~~ient fyo in (6.1), is a constant. The above nonzero component of ff,+S respresents a distribution of positive and negative edge dislocations throughout the crystal. Recalling the statement of the boundary conditions in the opening paragraph of this section, we prescribe the displacements along the line segment Y = L to be u, = 0 and specify u2 by

9

[2f - toI,

t()C:tst,,

u2 = V 7

\,

t2 -

F

F

t - to + tl + -

ft* - to + fl],

t1

?r

sin

[ G-1

0

t ,]I

’

t, s t s t2,

t > t2,

(6.2)

-

where urnaxis the maximum velocity of the boundary. According to (6.2), the boundary Y = L is accelerated from 0 to v,, in the time interval [0, to], followed by a period of constant velocity v,, in the interval [to, tl]; and finally, the boundary is decelerated to a state of rest in the interval [t,, t2]. In order to predict the response of the crystal to the loading (6.2), which is designed to simulate a displacement controlled tension test, we need to specify a special form for the strain energy + in (2.19), as well as for the hardening constants ,,,K, (m = 0, 1, . . . ,4), in (S.ll).t Thus, recalling (3.18), ssetting $2 = 0 and invoking the incompressibility condition Jp = det G, = 1, we assume a special case of (3.18) in the form (6.3) which is quadratic in the lattice strain ,E [defined by (2.7),]. [Although the special form (6.3) can be used in the presence of finite strain, its adoption here is also motivated by the fact that we expect the lattice deformations to be small.] The components of the constant elasticity tensor C in (6.3) are taken to be in the form CAB,

=

~~A&k’D

+

d~AC&D

+

~AdBC]

(A, B, C, D = 1,2)

(6.4) where A and p are the elasticity constants and the indices A, B, C, D range over the values 1,2 only. tin view of the fact that the only nonzero components of (,K, ,K and q~ in (5.11).

a

are 1y13 and crz3, it suffices to specify only the coefficients

1176

P. M. NAGHDI

and A. R. SRINIVASA

The governing equations for the boundary-value problem under consideration are the equations of motion (2.12)-(2.14), together with (6.3) and the constitutive equations (2.19),,,, (2.20)-(2.22) with c(;*,& given by (5.9)-(5.11). The boundary conditions are; x=x

forO
Y = 0,

x = X + R.H.S. of (6.2) Rf = 0,

for OsXsw,

andX=O,

X=w

Y= L,

04YsL.

(6.5)

Clearly, owing to the complexity of the problem, analytical solutions are prohibitive and instead we seek numerical solutions. Prior to discussion of the numerical procedures, we first nondimensionalize the basic equations by introducing the following nondimensional variables (i)

(nondimensional

(ii)

fi = 6

(nondimensional

(iii)

I_ %I,, L (nondimensional

coordinates),

velocity),

time),

l/2

(nominal Mach number),

(iv) (4

ti = 441nondimensional L(

width).

(6.6)

We also normalize the kinetical variables by means of the nominal “yield stress” of the crystal in K(, such that

8 director inertia g=-= ordinary inertia per unit length ’ PoL2

(6.7)

By the first of (6.7) and (5.10) the nondimensional components of the viscoplastic response function & in the direction of the active slip systems can be expressed as

(6.8) where the nondimensional rate of slip on the p(k)th slip system yp’k’ and the coefficient p representing the nominal strain rate associated with the process are defined by p(k)L ,-p(k) = -Y

Y

u max

’

(6.9)

M2 ~ovrnaxL Finally, we introduce the ratio p = ~ which is analogous to the Reynolds number (in n ’ viscous fluid flow) associated with the viscoplastic response. In terms of these nondimensional variables and in the absence of both the ordinary and director body forces, it is desirable to record the component form of the ordinary momentum equations (2.12)2 as (6.10)

Characterization

while the director momentum

1177

of dislocations

balance (2.13) can conveniently

be written in the form

;(M”~[~(G,)])-$42~~)[~GP]=-[i(,+i
(6.11)

In order to reduce the system of equations (6.10)~(6.11) to a more manageable form, we make the following order-of-magnitude assumptions about the nondimensional variables: P&p, &, where E is assumption momentum assumptions

ii21

-

{n/l

O(l),

PI

-

O(E)

<<

(6.12)

1,

a small nondimensional parameter. It should be clear from (6.12)2 that the M - O(E) is equivalent to neglecting the inertia effects in both the ordinary and the director momentum balance laws. With this observation and the remaining in (6.12)1,2, the equations (6.10) and (6.11) can be further reduced to aPiA = 0(c2) ax‘4

and

(6.13)

i - 2, + ii;? + & = 0 + O(E)‘.

Taking the inner product of (6.13)2 with Apfkjr rearranging and (4.5) we obtain

the terms and remembering

(5.5)

(6.14) The results (6.13), and (6.14) are the basic equations of motion for the single crystal under discussion. It should be emphasized that while (6.13)i is an elliptic partial differential equation for the determination of the components x^,, (A = 1,2), of the position vector x of the crystal, the director momentum equation (6.14) is a first order hyperbolic partial differential equation for the determination of G,. The differential equation (6.14) may at first sight appear to be quite simple in form, but this is not the case: this is because g2 on the right-hand side of (6.14) depends not only on G, but also on Curl G, which, in turn, gives rise to the hardening features of the crystal as is evident also from the hardening parameters K&) in (5.9) and (5.12). Keeping these remarks in mind, we utilize the Galerkin formulation of the finite element method (FEM) in dealing with the numerical solution of (6.13), and finite difference method to solve (6.14). The set of nonlinear equations which results from the application of the FEM to (6.13)i is coupled with the set of ODE’s which result from the finite difference approximation to the calculation of the dislocation density on the right-hand side of (6.14). The resulting system of equations is coupled across elements due to the presence of the Curl G, terms on the right-hand side of (6.14). This considerably complicates the numerical scheme since the discretized equations can no longer be formulated element by element in the manner of the usual formulations of the FEM. We use nine-noded quadrilateral elements for the FEM and a third order central difference method for the finite difference approximation of the dislocation density in (6.14). The matrix method resulting from the use of the Newton-Raphson method for the solution of (6.13) was inverted by using the Cholesky factorization available in the NAG library of FORTRAN subroutines. The ODE which results from the finite difference approximation to the right-hand side of (6.14) was solved using the variable order, variable step Adams method which is also available in the NAG library. The values used in the numerical solution for the nondimensional material constants are: x = 1- S.Z30.0, OK OK

p = 0.1,

A plot of the calculated nondimensional

LO.1 OK

* = 2.0. OK

“average normal stress”

P,,/,,K

(6.15) at the top edge of

1178

P. M. NAGHDI

and A. R. SRINIVASA

2.0

/-MO1 .5 h -

% e : .c tl t

’

. -

.sl.O 5 z ,P f -

B . s = 0.5 ‘-

i-

o.olca a “B”““c 0.000

0.050

I”0.100

0.150

* IiS

I’#

0.200

Nondimensional boundary displacement u/L

Fig. 4. A plot of the average nondimensional component P2J,,u versus the nondimensional displacement u2/L. Note that in the range of deformation shown, there is a region of softening (from approx. 8.9 to 18.7% extension) followed by a region of hardening.

the crystal versus the nondimensional boundary extension uz/L is shown in Fig. 4. A noteworthy feature of the numerical solution for the normal stress in Fig. 4 is that localized plastic deformation begins at an extension of about 0.8%, even though the average stress pz2/()K continues to increase linearly beyond this extension. Moreover, as is evident from Fig. 4, the crystal hardens rapidly after 0.8% extension followed by a period of almost constant average stress. The crystal then softens during the period of extension from approx. 8.9 to 18.7% beyond which it again hardens. Figure 4 clearly shows that most of the hardening takes place within the range of 0.8-l% of the total elongation during which the plastic deformation is small and hence negligible but in this range the dislocation density is not small. This type of result is likely to be significant in the calculation of the production of dislocations during cyclic fatigue and near crack tips, as well as their role in the propagation of cracks in ordered intermetallic alloys. With reference to the shape of the deformed crystal after a boundary extension of 27.5% in Fig. 3, first it should be noted that the middle portion is considerably narrower than the ends and that the narrower region is nonuniform and unsymmetrical. (The nature of deformation of the crystal as a consequence of 27.5% extension of its top boundary is particularly striking if Fig. 3 is viewed at an oblique angle to the plane of the figure.) The contour plots of the components of the relative plastic deformation H, [see equation (3.5)] are shown in Figs 5(a)-(d), where the algebraic values of the components HpAs increase as the colors range from blue to red. Moreover, the relative plastic deformation is localized along a band which is oriented at an angle of aobut 45” with respect to the positive X-axis. It is this region which undergoes most of the plastic deformation and consequently becomes narrower than the top and bottom portions of the crystal. The triangular shaped regions near the top and bottom edges of the crystal remain elastic throughout the deformation and are reminiscent of the “dead-zones” encountered in the rigid-plastic analysis of metal forming. These regions, which correspond to HpAs= 0 are red in Figs 5(a) and (c) and deep blue in Figs 5(b) and (d). This difference in color is due to the fact that in the central band, the crystal undergoes lateral

Characterization

1179

of dislocations

-0.03575

0.398114

-0.0715

0.362568

-0.10725

0.327022

-0.143

0.291477

-0.17615

0.255931

-0.2145

0.2203Q5

-0.25025

0.194839

-0.296

0.144293

-0.32175

0.113747

-0.3575

0.076201

-0.39325

0.042655

-0.429

0.007109

0.425855 0.387669 II ,.

0.349634 0.311eOE 0.273733 0.235758 0.197732 0.159707 0.121881 0.083855 0.045630

I

Fig. 5. Contour plots of the components

0.007605.

HsB ofthe relative plastic deformation H, for a boundary extension of 27.5%.

1180

P. M. NAGHDI

and A. R. SRINIVASA

9.219793 e.176696 0.133399 0.099302 0.047406 9.094309 -9.m -9.991993 4.12.M 4.166677 -9.211173 I

-0.23427

(a)

I

e.939394

9.964497

0.77777Q

tM94913

0.616162

0.323424

Q.634349

0.359932

0.292929

9.196441

0.1313l3

0.016949

4.039393

-9.lS2362

-e.l91919

4.332934

-9.333533

a.491525

.WlS192

.9.661017

-4.676763

-9.839309

4.9363&(

.1

-1

Fig. 6. Contour plots of the nonzero components of the dislocation density tensor: (a) initial dislocation density al3 as spec :ified in equation (6.1); (b) dislocation density component CY,~after a boundary extension of 27.5%; (c) dislocation density component LY~?after a boundary extension of 27.5%.

Characterization

of dislocations

1181

contraction along with shear in the negative X-direction (see Fig. 3) so that H$‘, and H$, are negative in these regions. Thus, in keeping with the coloring scheme mentioned earlier, the algebraic values of H$‘, and Hg, are larger in the top and bottom triangular regions; and, hence these regions are red, while the central band is blue. In contrast, the crystal extends and shears in the positive Y-direction so that HPI* and H$, are positive along the central band. They are thus displayed as red regions in Figs 5(b) and (d). A contour plot of the initial dislocation density specified by (6.1), is shown in Fig. 6(a). The red regions represent a high density of positive edge dislocations, while the blue regions represent a high density of negative edge dislocations. The evolution of dislocation density, i.e. the nonzero components (Y,~and crz3 of OL,after a boundary extension of 27.5%, are exhibited in Figs 6(b), (c). It can be clearly seen that the dislocations have localized into two prominent bands near the top and bottom edges of the crystal, although there are also indications of initiation of additional bands between the two prominent ones. The values of LY,~and ff23 dislocation density along these bands are substantially higher than the initial dislocation density in Fig. 6(a). The region between the bands have undergone significant lattice rotations as compared to the top and bottom of the crystal. Moreover, the dislocation bands significantly harden the crystal leading to the second region of hardening shown in Fig. 4. Finally, it is desirable to make a few comments regarding comparisons of the predictions of the theory with experiments. In this connection, it should be noted that no quantitative measurements of the dislocation density tensor have been made to date, partly because of the difficulty involved in making dynamic measurements on large specimens of a size of about 1 cm. Indeed, a very recent review article by Tanner and Bowen [24] addresses this issue and discusses the promise of synchrotron X-ray topographical techniques for the measurements of a wide range of microstructural phenomena. Given the information compiled by Tanner and Bowen [24] (see especially their Figs 9, 10, 16 and 17), it is clear that quantitative measurements of dislocation density tensor may be forthcoming and could then be used to compare with the predictions of the theory. Acknowledgment-The results reported here were obtained in the course of research supported by the Solid Mechanics Program of the U.S. Office of Naval Research under contract NOOO14-90-J-1959,R&T 4324-436 with the University of California at Berkeley.

REFERENCES

[l] P. M. NAGHDI and A. R. SRINIVASA, A dynamical theory of structured solids. Part I. Basic developments. Phil. Trans. R. Sot. Lond. A. 345,425 (1993). ]2]_ P. M. NAGHDI and A. R. SRINIVASA, A dynamical theory of structured solids. Part II. Special constitutive _ equations and special cases of the theory. Phil. Trans. R. Sot. Land. A. 345,459 (1993). . 131 J. A. WEING and W. ROSENHAIN. Phil. Trans. R. Sot. Lond. 193.353 (1900). [4j G. I. TAYLOR and C. F. ELAM, Prac. R. SOC. Land. A 102,643 (1923). ' ' [S]G. I. TAYLOR and C. F. ELAM, Proc. R. Sot. Lond. A 108,28 (1925). [6] G. I. TAYLOR and C. F. ELAM, Proc. R. Sot. Lond. A l&337 (1926). [7] G. I. TAYLOR, Proc. R. Sot. Lond. A 145,332 (1934). [8] R. J. ASARO, Micromechanics of crystals and polycrystals. In Advances in Applied Mechanics, Vol. 23, pp. 1-115. Academic Press, New York (1982). [9] P. M. NAGHDI, J. Appt. Math. Phys. (ZAMP) 41, 315 (1990). [lo] K. S. HAVNER, Finite Plastic Deformation of Crystalline Solids. Cambridge Univ. Press, Cambridge (1992). [ll] C. T. LIU, R. W. CAHN and G. SAUTHOFF (Eds), Ordered Interrnetallics-Physical Metallurgy and Mechanical Behaviour. NATO AS I Series E: Applied Science Vol. 213. Kluwer (1991). (121 A. GUINIER, The Structure of Matter-From the Blue Sky to Liquid Crystals (translated from the 1980 French edition by W. J. DUFFIN. Edward Arnold, London (1984). [13] A. GUINIER and R. JULLIEN, The Solid State: From Superconductors to Superalloys (translated from the French La MatiereCre a I’Etat Solide: des Supraconducteurs aux Superalliages by W. J. DUFFIN), IUCr. Texts of Crystallography 1, Oxford Univ. Press, New York (1989). [14] B. A. BILBY, Continuous distributions of dislocations. In Progress in Solid Mechanics (Edited by I. N. SNEDDON and R. HILL), Vol. 1, pp. 331-398. North-Holland, Amsterdam (1960). (151 J. P. HIRTH and L. LOTHE, Theory of Dislocarions, 2nd Edn. Wiley, New York (1982). ES 32:7-J

1182

P. M. NAGHDI

and A. R. SRINIVASA

[16] F. R. N. NABARRO, Theory of Cry& Dislocations. Dover, New York (1987). (This is a slightly corrected version of the book first published in 1%7 by Oxford University Press.) 1171 A. E. GREEN and P. M. NAGHDI, Proc. R. Sot. Land. A 357,253 (1977). [IS] J. D. ESHELBY, The continuum theory of lattice defects. In Solid State Physics (Edited by F. SEITZ and D. TURBULL). Vol. 3. DD. 79-144. Academic Press. New York (1956). [19] R. W. HERTZBERG,‘Deforrnation and Fracture Mechanics of Eniineering Materials. Wiley, New York (1989). 1201 C. S. BARRETT, Crystal structure of metals. In Merufs Handbook, 8th Edn, pp. 233-250. ASM, Metals Park, Ohio (1973). [21] P. M. NAGHDI and A. R. SRINIVASA, Some general results in the theory of crystallographic slip. In preparation. [22] C. TRUESDELL and W. NOLL, The non-linear fields theories of mechanics. In S. Fliigge’s Handbuch der Pysik, Vol. 111/3. Springer, Berlin (1965). [23] H. G. VAN BUEREN, Imperfections in Crystals, 2nd Edn. North-Holland, Amsterdam (1961). (241 B. K. TANNER and D. K. BOWEN, Mater. Sci. Reports 8,369 (1992). (Received II May 1993; accepted 13 July 1993)