A gradient theory of small-deformation isotropic

plasticity that accounts for the Burgers vector

Morton E. Gurtin∗

Department of Mathematical SciencesCarnegie Mellon UniversityPittsburgh, PA 15213, USA

Abstract

This study develops a gradient theory of small-deformation viscoplasticity based on: a systemof microforces consistent with its peculiar balance; a mechanical version of the second law thatincludes, via the microforces, work performed during viscoplastic flow; a constitutive theorythat accounts for the Burgers vector through a free energy dependent on curlHp, with Hp theplastic part of the elastic-plastic decomposition of the displacement gradient. The microforcebalance and the constitutive equations, restricted by the second law, are shown to be togetherequivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differentialequations. The first of these is an equation for the plastic strain-rate Ep in which the stressT plays a basic role; the second, which is independent of T, is an equation for the plastic spinWp. A consequence of this second equation is that the plastic spin vanishes identically whenthe free energy is independent of curlHp, but not generally otherwise. A formal discussionbased on experience with other gradient theories suggests that sufficiently far from boundariessolutions should not differ appreciably from classical solutions, but close to microscopically hardboundaries, boundary layers characterized by a large Burgers vector and large plastic spin shouldform.

Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditionsneed be supplemented by nonstandard boundary conditions associated with viscoplastic flow.As an aid to solution, a variational formulation of the flow rule is derived.

Finally, we sketch a generalization of the theory that allows for isotropic hardening resultingfrom dissipative constitutive dependences on ∇Ep.

Keywords: Strain-gradient plasticity

∗Tel.: +1-412-681-5348; E-mail address: mg0c@andrew.cmu.edu

1

April 13, 2004 2

1 Introduction

The classical theory of isotropic plastic solids undergoing small deformations is basedon the decomposition1

∇u = He + Hp, trHp ≡ 0 (1.1)

of the displacement gradient into elastic and plastic parts, where He represents rotationand stretching of the material structure, while Hp, the plastic distortion, characterizesthe evolution of dislocations and other defects through this structure. In this classicaltheory the plastic rotation Wp — the skew tensor in the decomposition

Hp = Ep + Wp

of Hp into symmetric and skew parts — is essentially irrelevant, as it may be absorbedby its elastic counterpart without affecting the resulting field equations.

Recent interest in the behavior of materials at micron length scales has led to agrowing literature concerned with strain-gradient plasticity (Fleck and Hutchinson, 1993,2001; Gurtin, 2003; Gudmundson, 2004).2 A tacit but central assumption of thesegradient theories — motivated, perhaps, by experience with classical plasticity — isthat the constitutive theory not involve the plastic rotation-field Wp; consequently,

• such theories cannot account for the Burgers vector,3

because macroscopically the Burgers vector is characterized by the tensor field (Burgers,1939; Kroner, 1960)

G = curlHp(Gij = εirqH

pjq,r

), (1.2)

a field that necessarily involves Wp, since

curlHp = curlEp + curlWp.

(We prefer to phrase the theory in terms of a Burgers vector, which is a precisely definednotion in a continuum theory, rather than in terms of geometrically necessary disloca-tions, which is not.)

To understand the issue of whether or not an isotropic theory of plasticity shouldinvolve the rotation field, it is useful to bear in mind that, typically, the Cauchy stressT expends power during plastic flow in consort with the plastic strain-rate Ep; the spinWp, whatever it may be, involves no expenditure of power and, consequently, generatesno dissipation. But the development of a higher-order (strain gradient) theory necessarilyinvolves higher-order stresses and this renders uncertain what form the underlying powerexpenditures should take; hence it is not clear whether or not intuition based on theclassical theory can be helpful. For that reason, we base our theory on the principle

1Notation. We use standard indicial notation with summation convention, when convenient. Theinner product of tensors is denoted by A :E=AijBij , while a⊗ b=aibj denotes the dyadic product ofvectors. We write A0 or devA for the deviatoric part of a tensor A,

A0 = devA = A− 13(trA)1,

((A0)ij = Aij − 1

3Akkδij

).

Given any vector u, (u×) is the skew tensor (u×)ij = εirjur. For A a tensor field,

(divA)i = Aij,j (curlA)ij = εipqAjq,p. (∗)

(In the literature one also finds curlA defined as the transpose of (∗).)2In listing these references attention is restricted to isotropic small-deformation flow theories involving

higher-order boundary conditions.3Unless one assumes that Wp ≡ 0.

April 13, 2004 3

of virtual power. Because a goal of ours is a characterization of the Burgers vector, weintroduce a microscopic stress S that performs work locally in conjunction with temporalchanges in the Burgers vector as characterized by G = curl Hp. Experience with theoriesdeveloped using the principle of virtual power then lead us to account for the powerexpended by the field Hp, which we accomplish with the aid of a second microscopicstress Tp. We complete this accounting with the assumption that power expended instretching and rotating the material structure has the form T : He. We are thereforeled to assume that power expended within any part P (subregion of the body), has theform4 ∫

P

(T : Ee + Tp : Hp + S : curl Hp

)dV. (1.3)

Consequences of the virtual-power principle (and a form of frame-indifference appro-priate to small-deformations) is that the classical macroscopic balances divT + b = 0and T = T need be supplemented by a microforce balance

T0 = Tp+ devcurl S((T0)ij = T pji + εipqSqj,p − 1

3δijεrpqSqr,p

). (1.4)

We restrict attention to a purely mechanical theory with underlying “second law” therequirement that the free energy increase at a rate not greater than the rate at whichwork is performed. Letting ψ denote the free energy per unit volume, this leads to alocal free-energy inequality

ψ −T : Ee −Tp : Hp − S : curl Hp ≤ 0 (1.5)

which is basic to our development of constitutive equations.It is important to note that the inequality (1.5) is consistent with classical plasticity

theory. Indeed, in the classical theory S vanishes, so that, by (1.4), T0 = Tp and Tp issymmetric. Consequently, the plastic spin does not enter the local free-energy inequality(4.2), which, by (1.1), reduces to the classical form ψ −T :∇u ≤ 0.

On the other hand, because (1.5) characterizes power expended over Hp and henceover Ep and Wp, it would here seem capricous to assume a priori that Tp is symmetric.

In appealing to the free-energy inequality (1.5), we seek a theory that accounts forthe Burgers vector and allows for energy dissipated in plastic spin, but is otherwise closein spirit to its classical counterpart. Specifically, we account for:

(i) a nonvanishing Burgers vector by allowing for a free energy Ψ(Ee,G);

(ii) the dissipative effects of plastic spin with the aid of an effective distortion-rate

dp =√∣∣Ep∣∣2 + χ

∣∣Wp∣∣2, χ ≥ 0 constant. (1.6)

In this manner, we are led to a constitutive equation for the microstress S of the form

S =∂Ψ(Ee,G)

∂G

(Sij =

∂Ψ(Ee,G)∂Gij

)in conjuntion with a constitutive equation for Tp. When substituted into the microforcebalance, these constitutive assumptions result in the flow rule

T0 − dev curl(S)︸ ︷︷ ︸

energetic backstress

= Y (dp)(Ep − χWp)︸ ︷︷ ︸dissipative hardening

, (1.7)

4Cf. Gurtin and Needleman (2004), who utilize a virtual power formulation in which the gradientterm in the internal power has the form

∫P S : curl Hp dV .

April 13, 2004 4

with Y (dp) > 0.If we neglect constitutive dependences on G, let χ → 0, and assume that Y (dp) =

Y0/|dp|, then the flow rule (1.7) reduces to the Mises condition

T0 = Y0Ep

|Ep|, (1.8)

with Y0 the yield stress.Since T0 is symmetric, taking the skew part of the flow rule (1.7) leads to a central

result of our study: for χ = 0,

• the plastic spin vanishes identically when the free energy is independent of G, butnot generally otherwise.

This result shows consistency with classical plasticity theory.Because the backstress in (1.7) depends on first and second gradients of Hp, the flow

rule has the form of a second-order partial differential equation for Hp. Consistent withthis, the virtual power formulation suggests possible microscopic boundary conditionsassociated with plastic flow. In this regard, the surface traction associated with themicrostress S has the form[

dev(S(n×)

)]iq

= Sjiεjpqnp − 13δijSjqεjpqnp,

with n the unit normal to the surface in question, and the natural boundary condition,which might be termed the microfree condition, is given by

dev(S(n×)

)= 0. (1.9)

A second boundary condition, meant to model a microscopically hard boundary, has theform

Hp(n×) = 0(Hijεjpqnp = 0

).

This condition requires that there be no Burgers-vector flow across the boundary.A formal discussion based on experience with other gradient theories suggests that

sufficiently far from boundaries solutions should not differ appreciably from classicalsolutions, but close to microscopically hard boundaries, boundary layers characterizedby a large Burgers vector and large plastic spin should form.

Because the underlying mechanics is based on the principle of virtual power, the non-local yield condition and microtraction boundary condition have a variational formulationthat should provide a useful basis for computations.

To develop a model theory amenable to analysis and computation, we consider an-tiplane shear; in this case the flow rule reduces to a pair of degenerate parabolic vector-equations.

We also sketch a generalization of the theory in which kinematic hardening induced bythe backstress is augmented by isotropic hardening ensuing from dissipative constitutivedependences on ∇Ep.

The use of plastic strain-gradients to characterize behavior at small length scales isdue to Aifantes (1984, 1987); cf. Aifantes (1992, 1999), Muhlhaus and Aifantes (1991ab),Zbib and Aifantes (1992). In discussing gradient theories one should also mention Menzeland Steinman (2000), who characterize the Burgers vector, but satisfy the dissipationinequality only globally.

Finally, in lieu of the results of this paper, the notion of relaxational isotropy, intro-duced by Gurtin (2003) to rule out dependences on Wp, seems misguided.

April 13, 2004 5

2 Characterization of the Burgers vector

We work within the kinematical framework prescribed by (1.1).

2.1 Burgers tensor G

By Stokes’ theorem, for Γ the boundary curve of a smooth oriented surface S with unitnormal e, ∫

Γ

Hp dx =∫S

(curlHp)e dA. (2.1)

This integral is generally nonzero, as the plastic distortion Hp is not the gradient of avector field, and we associate the vector measure (curlHp)e dA with the Burgers vectorcorresponding to the boundary curve of a surface-element e dA. Thus, for

G = curlHp, (2.2)

Ge provides a measure of the (local) Burgers vector for the plane Π with unit normal e,and may be viewed as the local Burgers vector, per unit area, for those dislocation linesthat pierce Π. We refer to G as the Burgers tensor and to Ge as the Burgers vectorassociated with small loops orthogonal to e, it being understood that this Burgers vectoris measured per unit area. Note that, since curl∇u = 0,

G = −curlHe.

2.2 Burgers-vector balance. Burgers-vector flow

Choose a fixed unit vector e. Then the Burgers vector Ge, associated with small loopsorthogonal to e, evolves according to a balance. To derive this balance note that

eiεijkHprk,j =

∂

∂xj

(Hprkεkijei

)

an identity whose left side represents (curl Hp)e = ˙Ge. We therefore have the Burgers-vector balance

˙Ge = −div(− Hp(e×)

)(2.3)

in which −Hp(e×) represents a tensorial Burgers-vector flux (cf. footnote 1). An in-spection of this balance shows that, given a plane Π(n) with normal n, −

(Hp(e×)

)n

represents the Burgers-vector flow across Π(n) in the direction n, a flow associated withsmall loops orthogonal to e. Thus, since

−(Hp(e×)

)n = −Hp(e × n) = Hp(n × e) =

(Hp(n×)

)e,

flow in the direction n associated with small loops orthogonal to e is equal to the negativeof the flow in the direction e associated with small loops orthogonal to n. A consequenceof this negative reciprocity is the following important result:

Hp(n×) = 0 (2.4)

at a particular point x if and only if there is no Burgers-vector flow across the plane atx with normal n; that is, if and only if the Burgers-vector flow across Π(n) vanishes forall small loops at x (Gurtin and Needleman, 2004).

April 13, 2004 6

3 Principle of virtual power. Macroscopic and micro-scopic force balances

The theory presented here is based on the belief that the power expended by each inde-pendent “rate-like” kinematical descriptor be expressible in terms of an associated forcesystem consistent with its own balance. But the basic “rate-like” descriptors, namely u,He, and Hp are not independent, as they are constrained by (1.1), and it is not appar-ent what forms the associated force balances should take. For that reason and becauseour goal is a gradient theory, we determine these balances using the principal of virtualpower.

3.1 External and internal expenditures of power

Throughout we denote by P an arbitrary part (subregion) of the body B with n theoutward unit normal on ∂P.

In discussing the manner in which power is expended internally, bear in mind thatour goal is a theory that accounts for the Burgers vector as described by the Burgerstensor G = curlHp; for that reason we consider power expenditures associated withthe kinematic variables Hp and curl Hp. We therefore assume that power is expendedinternally by an elastic stress T work conjugate to He, a microstress Tp work conjugateto Hp, and a defect microstess S work-conjugate to curl Hp, and we write the internalpower in the form

Wint(P) =∫P

(T : He + Tp : Hp + S : curl Hp

)dV. (3.1)

Here T, Tp, and S are defined over the body for all time. The fields Tp and S rep-resent internal microforces associated with the generation, growth, and annihilation ofdislocations within the body.

The power expended on P by material or bodies exterior to P results from: (i)a macroscopic surface traction t(n) and a macroscopic body force, b, each of whoseworking accompanies the macroscopic motion of the body; and (ii) a microtraction S(n)whose working accompanies the flow of dislocations across surfaces, a flow characterizedby Hp. (Pragmatically, experience with gradient theories has shown that higher-orderkinematical gradients in the internal power are necessarily associated with concomitanttractions in the external power.) We therefore write the external power in the form

Wext(P) =∫∂P

(t(n) · u + S(n) : Hp

)dA+

∫P

b · u dV (3.2)

with t(n) and S(n) (for each unit vector n) and b defined over the body for all time.The body force b is assumed to include inertial forces.

Since the stresses Tp and S(n) enter the internal and external power expendituresthrough the terms Tp: Hp and S(n) : Hp, and since trHp = 0, we assume that trTp = 0and trS(n) = 0. Finally, we assume that T, Tp, S, t(n), S(n), and b are invariant undersuperposed (infinitesimal) rigid rotations.

3.2 Principle of virtual power

Assume that, at some arbitrarily chosen but fixed time, the fields u and He (and henceH and Hp) are known, and consider the fields u, He, and Hp as virtual velocities to be

April 13, 2004 7

specified independently in a manner consistent with (1.1); that is, denoting the virtualfields by u, He, and Hp to differentiate them from fields associated with the actualevolution of the body, we require that

∇u = He + Hp, trHp = 0. (3.3)

More specifically, we define a generalized virtual velocity to be a list

V = (u, He, Hp)

consistent with (3.3). We assume that under superposed rigid rotations the fields com-prising a generalized virtual velocity transform as their nonvirtual counterparts; i.e.,e.g.,

He → He + W, Hp and curl Hp are invariant. (3.4)

Writing

Wext(P,V) =∫∂P

(t(n) · u + S(n) : Hp

)dA+

∫P

b · u dV,

Wint(P,V) =∫P

(T : He + Tp : Hp + S : curl Hp

)dV,

(3.5)

for the external and internal expenditures of virtual power, the principle of virtual powerconsists of two basic requirement

(V1) (Power Balance) Given any part P,

Wext(P,V) =Wint(P,V) for all generalized virtual velocities V. (3.6)

(V2) (Frame-Indifference) Given any part P and any generalized virtual velocity V,

Wint(P,V) is invariant under superposed (infinitesimal) rigid rotations. (3.7)

3.3 Classical force and moment balances

To deduce the consequences of the principle of virtual power, assume that (V1) and (V2)are satisfied. In applying the virtual balance (3.6) we are at liberty to choose any Vconsistent with the constraint (3.3).

Under a superposed rigid rotation, the only field associated with the the internalpower that is not invariant is He, which transforms to He + Ω; thus a necessary andsufficient condition that this power be invariant is that∫

P

T :Ω dv = 0

for all skew tensors Ω and all all parts P, which leads to the classical result

T = T. (3.8)

Consider next a generalized virtual velocity with Hp ≡ 0, so that He = ∇u. For thischoice of V, (V1) yields∫

∂P

t(n) · u dA+∫P

b · u dV =∫P

T : He dV =∫P

T :∇u dV,

April 13, 2004 8

and we may conclude, using the divergence theorem, that∫∂P

(t(n)−Tn

)· u dA+

∫P

(div T + b) · u dV = 0.

Since this relation must hold for all P and all u, standard arguments yield the tractioncondition

t(n) = Tn (3.9)

and the local force balancedivT + b = 0. (3.10)

Thus T plays the role of the macroscopic stress, and (3.10) and (3.8) represent the localmacroscopic force and moment balances.

3.4 Microforce balance

With a view toward deriving the microscopic counterpart of the classical balances, wefirst establish the following identity: Given a part P and tensor fields A and B,

−∫∂P

(n×A) :BdA =∫P

A : curlB−B: curl (A)

dV. (3.11)

The verification of this identity is as follows:

−∫∂P

εirpnrApjBji dA = −∫P

∂

∂xr

εirpApjBji

dV

= −∫P

εirpApjBji,r + εirpApj,rBji

dV

=∫P

ApjεpriBji,r −BjiεirpApj,r

dV.

Returning to the principle of virtual power, choose u ≡ 0. Then, by (3.3), He = −Hp,and (3.6) reduces to the microscopic virtual power relation∫

∂P

S(n) : Hp dA =∫P

(Tp −T) : Hp + S : curl Hp

dV. (3.12)

Using the identity (3.11) on the integral of S : curl Hp, we find that∫∂P

S(n) +

((n×)S

) : Hp dA =∫P

Tp −T + (curl (S))

: Hp dV. (3.13)

This relation must be satisfied for all deviatoric Hp and all P, and, since T is symmetricand Tp deviatoric, a standard argument yields the microforce balance

T0 = Tp+ devcurl (S)((T0)ij = T pji + εipqSqj,p − 1

3δijεrpqSqr,p

)(3.14)

and the microtraction condition

S(n) = dev(S(n×)

) ((S(n))iq = Sjiεjpqnp − 1

3δijSjqεjpqnp

). (3.15)

April 13, 2004 9

4 Free-energy imbalance

Let P be an arbitrary part of the body. We consider a purely mechanical theory basedon the requirement that the temporal increase in free energy of any part P be less thanor equal to the power expended on P. Letting ψ denote the free energy, measured per unitvolume, this requirement takes the form of a free-energy imbalance

˙∫P

ψ dV ≤ Wext(P) =Wint(P). (4.1)

Since ˙∫Pψ dv =

∫Pψ dV and T : He = T : Ee (as T is symmetric), we may use (3.1)

augmented by (2.2) to localize (4.1); the result is the local free-energy imbalance

ψ −T : Ee −Tp : Hp − S : G ≤ 0. (4.2)

5 Constitutive theory

We base the theory on an energetic constitutive equation

ψ = Ψ(Ee,G) (5.1)

that accounts for elastic strain energy as well as energy associated with a nonzero Burgersvector. As is clear from the local free-energy imbalance (4.2), to complete the constitutivetheory we need supplement (5.1) with constitutive equations for the stresses T, Tp, andS. With this in mind, we note that

ψ =∂Ψ(Ee,G)∂Ee

: Ee +∂Ψ(Ee,G)

∂G: G

and hence that (4.2) takes the formT− ∂Ψ(Ee,G)

∂Ee

: Ee +

S− ∂Ψ(Ee,G)

∂G

: G + Tp : Hp ≥ 0. (5.2)

Conditions sufficient that (5.2) be satisfied in all “processes” are that the free energydetermine the macroscopic stress T and the defect microstess S through the relations

T =∂Ψ(Ee,G)∂Ee

,

S =∂Ψ(Ee,G)

∂G,

(5.3)

and that the constitutive relation between the microstress Tp and the plastic distortion-rate Hp satisfy

Tp: Hp ≥ 0. (5.4)

The field Tp : Hp represents the dissipation, per unit volumeOne would expect that, plastically, the material response to spin differs from that

to straining, and that straining and spin each incur dissipation. With this in mind, weintroduce an effective distortion-rate

dp =√∣∣Ep∣∣2 + χ

∣∣Wp∣∣2, χ ≥ 0 (constant), (5.5)

April 13, 2004 10

with χ a constitutive modulus that measures the relative importance of plastic spin.Then, guided by (5.4) and standard theories of viscoplasticity, we consider a constitutiverelation for Tp of the form5

Tp = Y (dp)(Ep + χWp

), Y (dp) > 0, (5.6)

with Y (dp) a nonlinear viscosity. Note that the dissipation (5.4) has the simple formY (dp)(dp)2.

The specific choice Y (dp) = Y0/dp renders the constitutive theory rate-independent;

therefore, to include this important special case, we do not require that Y (dp) be definedwhen dp = 0, only that Y (dp)dp have a limiting value as dp → 0.

6 The flow rule and its symmetric and skew parts

The microforce balance T0 = Tp+ devcurl (S), augmented by the constitutive equa-tions for Tp and S, plays the role of a flow rule:

T0 = Y (dp)(Ep − χWp

)+ dev curl (S), S =

∂Ψ(Ee,G)∂G

. (6.1)

Here devcurl (S) represents a backstress due to energy stored in defects, while the termY (dp)

(Ep−χWp

)represents dissipative hardening due to flow (cf. (1.7)). In components,

(T0)ij = Y (dp)(Epij − χWpij) + εipqSqj,p − 1

3δijεrpqSqr,p, Sij =∂Ψ(Ee,G)∂Gij

. (6.2)

The flow rule may be split into symmetric and skew parts, with symmetric part

T0 = Y (dp)Ep + sym0 curl (S), sym0 = symdev , (6.3)

and, since T is symmetric, with skew part

χY (dp)Wp = skw curl (S). (6.4)

In components, (6.3) and (6.4) take the form

(T0)ij = Y (dp)Epij + 12 (εipqSqj,p + εjpqSqi,p)− 1

3δijεrpqSqr,p,

χY (dp)W pij = 1

2 (εipqSqj,p − εjpqSqi,p).

Note that (6.4) does not involve the macroscopic stress T. Because of this a free energyindependent of G (and hence independent of the Burgers vector) renders S ≡ 0 and, forχ = 0, leads to the result

Wp ≡ 0.

In contast, our theory shows that a free energy dependent on G is accompanied bydissipative effects induced by plastic spin; in our theory

• plastic spin is a direct response to a nonvanishing Burgers vector.5We could allow Y to depend also on G and an internal state variable ϕ meant to represent the

accumulated plastic distortion; such a generalization would not change the presentation appreciably.

April 13, 2004 11

Let ωp denote the plastic-spin vector

Wp = ωp×, W pij = εirjωpr , (6.5)

so thatεqijW

pij = εqijεirjωpr = −2ωpq . (6.6)

Thus, by (6.4),

2χY (dp) ωpq = −εqij(εiαβSβj,α − εjαβSβi,α) = −2(Sqα,α − Sαα,q),

and the skew part of the flow rule has the simple form

χY (dp) ωp = ∇trS− divS(χY (dp) ωpq = Sαα,q − Sqα,α

). (6.7)

7 Microscopically simple boundary conditions.

7.1 The projection P(e)

Given any unit vector e, we write P(e) for the projection onto the plane perpendicularto e:

P(e) = 1− e⊗ e(P (e)ij = δij − eiej

). (7.1)

Then, for A a tensor,AP(e) = 0 ⇔ A(e×) = 0. (7.2)

To verify (7.2), note first that, since P(e)(e×) = (e×), it follows that AP(e) = 0 ⇒AP(e)(e×) = 0 ⇒ A(e×) = 0. On the other hand,

(e×)(e×) = −P(e), (7.3)

so that A(e×) = 0 ⇒ A(e×)(e×) = 0 ⇒ AP(e) = 0.

7.2 Microfree and microhard boundary conditions

We now focus on the boundary ∂B, with outward unit normal n. The external powerexpended on B is (3.2) and, by (12.5), the microscopic portion of this power is given by

Wext(B) =∫∂B

S(n)︷ ︸︸ ︷dev

(S(n×)

): Hp︸ ︷︷ ︸

M(n)

dA. (7.4)

Since (n×)P(n) = (n×) and P(n) is symmetric,

dev(S(n×)

): Hp =

(S(n×)

): Hp =

(S(n×)P(n)

): Hp =

(S(n×)

):(HpP(n)

),

and we may write M(n) alternatively as:

M(n) = dev(S(n×)

): Hp =

(S(n×)

):(HpP(n)

). (7.5)

M(n) represents the microscopic power expended, per unit area, on ∂B by the materialin contact with the body.

April 13, 2004 12

We limit our discussion to boundary conditions that result in a null expenditure ofmicroscopic power in the sense that M(n) vanishes on ∂B. Specifically, we considermisroscopic boundary conditions asserting that at each point of ∂B and each time:

either dev(S(n×)

)= 0︸ ︷︷ ︸

microscopicallyfree (microfree)

or HpP(n) = 0︸ ︷︷ ︸

microscopicallyhard (microhard)

. (7.6)

The indicial form of the microfree condition is expressed in (1.9), while the microhardcondition expressed in that form is Hij(δjk − njnk) = 0.

The microfree condition would seem consistent with the macroscopic condition

Tn = 0,

the microhard condition withu = 0.

As a consequence of (7.2), the microhard condition may be written as

Hp(n×) = 0(Hijεjpqnp = 0

), (7.7)

so that there is no Burgers-vector flow across a microhard boundary (cf. (2.4)). Further,a consequence of the microhard condition in the form Hp

P(n) = 0 is that Hp = Hp(n⊗n)and hence that, at a microhard point of the boundary, Hp has the form Hp = b⊗n withb ⊥ n (Gurtin and Needleman, 2004).

8 Variational formulation of the flow rule and micro-scopically simple boundary conditions

We now consider specific boundary conditions in which a portion Shard of ∂B is micro-scopically hard and the remainder Sfree microscopically free:

Hp(n×) = 0 on Shard, dev(S(n×)

)= 0 on Sfree (8.1)

(cf. (7.7)). We begin with the microscopic virtual power relation (3.12) applied toB. Because the boundary conditions (8.1) render the power expenditure M(n) null on∂B, we consider (3.12) with the boundary term omitted and, for convenience, with Hp

replaced by V: ∫B

(Tp −T) :V + S : curlV

dV = 0. (8.2)

We refer to V as a test field and assume that V is kinematically admissible in the sensethat V is deviatoric and

V(n×) = 0 on Shard. (8.3)

The next step mimics the steps leading from (3.12) to (3.14) and (12.5). Applying theidentity (3.11) with A = S and V = Hp, we find, upon using the condition V(n×) = 0,that∫

B

S : curlV dV =∫B

V :[curl (S)

]dV +

∫∂B

(S(n×)

):V dA

=∫B

V :[curl (S)

]dV +

∫Sfree

(S(n×)

):V dA −

∫Shard

S:

(V(n×)

)︸ ︷︷ ︸=0

dA. (8.4)

April 13, 2004 13

Thus (8.2) holds if and only if∫Sfree

(S(n×)

):V dA+

∫B

Tp −T +

[curl (S)

]:V dV = 0, (8.5)

and therefore, using standard arguments, (8.2) holds for all kinematically admissible testfields V if and only if the microforce balance (3.14) and the microfree boundary condition(8.1)2 is satisfied. Finally, since the microforce balance — when supplemented by theconstitutive equations (5.3) and (5.6) for S and Tp — is equivalent to the flow-rule (6.1),we are led to the

Variational formulation of the flow rule Suppose that the constitutive equa-tions

S =∂Ψ(Ee,G)

∂G, Tp = Y (dp)

(Ep + χWp

)are prescribed. Then the flow rule (6.1) on B and the boundary condition

dev(S(n×)

)= 0 on Sfree

are together equivalent to the requirement that (8.2) be satisfied for all kinemati-cally admissible test fields V.

9 Separable free energy. Global defect balance. Cold-working

We now consider a free energy that is separable in the sense that

Ψ(Ee,G) = Ψe(Ee) + Ψp(G), (9.1)

with elastic energy Ψe(Ee) and defect energy Ψp(G). For such an energy the macroscopicand microscopic stresses are uncoupled, with microscopic stress given by

S =∂Ψp(G)∂G

. (9.2)

Assume that the boundary is microscopically dissipationless, so that

S(n×) : Hp = 0 on ∂B.

Then, since G = curlHp, we may use (3.11), (9.2), the microforce balance (3.14), andthe symmetry of T0 to conclude that

˙∫B

Ψp dV =∫B

S: curl Hp dV =∫B

Hp :[curl (S)

]dV

=∫B

Hp : (T0 −Tp) dV =∫B

(T0 : Ep −Tp : Hp) dV. (9.3)

Thus we have a global defect balance (Gurtin, 2000, 2003)

˙∫B

Ψp dV =∫B

T0 : Ep dV

︸ ︷︷ ︸cold working

−∫B

Tp : Hp dV

︸ ︷︷ ︸dissipation

(9.4)

April 13, 2004 14

asserting that the net defect energy changes at a rate balanced by the difference betweenthe coldworking and the dissipation. By (5.4)), the dissipation is nonnegative; thus thenet defect energy decreases at a rate not greater than the cold working. Moreover, if theenergy stored in defects is neglected, then the cold working and the dissipation coincide.

10 Quadratic free energy. Power-law viscosity

10.1 General forms

The most general quadratic, isotropic free energy Ψ(Ee,G) has the form

Ψ(Ee,G) = µ|Ee0|2 + 12κ|trE

e|2

+ 12k1|symG0|2 + 1

2k2|trG|2 + 1

2k3|skwG|2 + k4Ee0 :G0 + k5(trEe)(trG) (10.1)

and leads to stresses

T = 2µEe0 + κtrEe + k4G0 + k5(trG)1,

S = k1symG0 + k2(trG)1 + k3 skwG + k4Ee0 + k5(trEe)1.

(10.2)

For a separable free energy the constants k4 and k5 vanish, so that

Ψp(G) = 12k1|symG0|2 + 1

2k2|trG|2 + 1

2k3|skwG|2,

T = 2µEe0 + κtrEe,

S = k1symG0 + k2(trG)1 + k3 skwG,

(10.3)

generalizing the energy (10.6).A specific choice for Y , often used to model metals, has power-law form

Y (dp) = Y0|dp|m−1, (10.4)

with m typically small. For m > 0, this constitutive relation is meaningful even whendp = 0, although (for m < 1) Y (dp) would not be differentiable at dp = 0, a defect easilyremoved by considering instead the regularized power-law

Y (dp) = Y0|λ+ dp|m−1, (10.5)

with λ > 0 small.

10.2 L2 defect energy. Material length scale

With a view toward better understanding the nature of the flow rule (6.1), consider thefree energy (10.3)1 with k = k1 = k3 = 1

3k2,

Ψ(Ee,G) = µ|Ee0|2 + 12κ|trE

e|2 + 12k|G|

2︸ ︷︷ ︸L2 defectenergy

, (10.6)

for which the stresses T0 and S have the simple form

T0 = µEe0, S = kG.

By (1.1) and (2.2), the flow rule (6.2) then becomes

(T0)ij = Y (dp)(Epij − χWpij) + k

[Hpjt,ti −H

pji,pp − 1

3δijHprt,tr

]. (10.7)

April 13, 2004 15

The gradient term, which is multiplied by k, involves second spatial derivatives of Hp;comparing this term to the macroscopic term T0, bearing in mind that T0 = µEe0, leadsus express k in the form

k = µ"2, (10.8)

where " > 0 represents a material length scale associated with the Burgers tensor, andto write the symmetric and skew parts of (10.7) as follows:

(T0)ij = Y (dp)Epij + µ"2[

12 (Hp

jt,ti +Hpit,tj)− E

pij,pp − 1

3δijHprt,tr

],

χY (dp)W pij = µ"2

[12 (Hp

jt,ti −Hpit,tj)−W

pij,pp

].

(10.9)

Experience with other gradient-plasticity theories (e.g., Bittencourt, Needleman, Gurtin,and Van der Giessen, 2003) then suggests that, given a boundary-value problem associ-ated with length scales that are large compared to ":

(i) away from boundaries the underlined terms in (10.9) would be small, so that,consistent with standard plasticity theory,

T0 ≈ Y (dp)Ep, Wp ≈ 0;

(ii) sufficiently close to microhard boundaries the underlined gradient terms would notbe negligible and a boundary layer characterized by a large Burgers vector andlarge plastic spin would form.

11 Antiplane shear

11.1 Kinematics

A class of deformations that simplifies the theory, but yet affords insight, is antiplaneshear. Within this class the displacement is given by

u(x, t) = u(x1, x2)e, e ≡ e3, (11.1)

so that

∇u = e⊗ γ, γ = ∇u, γ · e = 0, γ is independent of x3.

We require that, in addition,

Hp = e⊗ γp + ν ⊗ e, He = e⊗ γe − ν ⊗ e, (11.2)

with γp, γe, and ν orthogonal to e and independent of x3. The decomposition ∇u =He + Hp then implies that γ = γp + γe, so that

∇u = γp + γe. (11.3)

Further, a consequence of (11.2)1 is that Wp = skw(e⊗(γp − ν)

); hence (6.6) yields

ωp = 12 e× (γp − ν). (11.4)

The difference γp − ν therefore determines the plastic rotation-vector.Next, by (11.2) and the properties of γp and ν,

G = curlHp = (curlγp)⊗ e− (e×)(∇ν), (11.5)

April 13, 2004 16

and since (curlγp)⊗ e is orthogonal to (e×)(∇ν), it follows that

|G|2 = |curlγp|2 + |∇ν|2. (11.6)

Further,curl curlγp = −(γp −∇divγp)

and therefore

dev(curl curl (G)

)= curl curl (G)= (curl curlγp)⊗ e− e⊗ν= −(γp −∇divγp)⊗ e− e⊗ν, (11.7)

where denotes the Laplacian(e.g., (ν)i = ∂2νi/∂xj∂xj

).

11.2 Macroscopic equations

We restrict attention to separable materials. By (11.2)2 and the elastic stress-strain law(10.3),

T = 2 sym(e⊗ τ ), τ = µ(γe− ν), (11.8)

and the macroscopic balance divT = 0 is equivalent to

divτ = 0. (11.9)

11.3 Flow rule

We consider the L2 defect energy (cf. (10.6), (10.8), (11.6))

Ψp(G) = 12µ"

2|G|2 = 12µ"

2(|curlγp|2 + |∇ν|2

), (11.10)

so that, by (5.3)1,

S = µ"2G = µ"2((curlγp)⊗ e− (e×)(∇ν)

). (11.11)

By (11.2)1,

2(Ep − χWp) = e⊗[(1− χ)γp + (1 + χ)ν

]+

[(1 + χ)γp + (1− χ)ν

]⊗ e; (11.12)

therefore, using the projection (7.1),

P(e)(Ep − χWp

)e = 1

2

[(1 + χ)γp + (1− χ)ν

],

P(e)(Ep − χWp

)e = 1

2

[(1− χ)γp + (1 + χ)ν

],

(11.13)

and, by (11.7),P(e)

[curl curl (G)

]e = −(γp −∇divγp),

P(e)[curl curl (G)

]e = − ν.

(11.14)

By (11.11) and (11.12)–(11.14), the flow rule (6.1) reduces to a pair of equations:

τ = 12Y (dp)

[(1 + χ)γp + (1− χ)ν

]− µ"2

(γp −∇divγp

),

τ = 12Y (dp)

[(1− χ)γp + (1 + χ)ν

]− µ"2 ν.

(11.15)

April 13, 2004 17

Equations more revealing than (11.15) may be obtained by first adding and then sub-tracting the two equations (11.15). The resulting system,

Y (dp)(γp + ν) = µ"2( (γp + ν)−∇divγp

)+ 2τ ,

χY (dp)(γp − ν) = µ"2( (γp − ν)−∇divγp

),

(11.16)

which represents the flow-rule for antiplane shear, consists of an equation involving themacroscopic shear τ supplemented by what is essentially an equation for the plastic spinωp = 1

2 e× (γp − ν). Here, by (5.5) and (11.12),

dp =√

12

[|γp + ν|2 + χ2|γp − ν|2

]rendering the regularized power-law (10.5) of the form

Y (dp) = Y0

(λ+

√12

[|γp + ν|2 + χ2|γp − ν|2

] )m−1

. (11.17)

11.4 Microscopic boundary conditions

We identify the body B with a region in the cross-sectional plane, so that the normal nto ∂B satisfies n · e = 0. We restrict attention to the microfree and microhard boundaryconditions defined in (7.6).

Microfree condition

By (11.11) and the identity (n×)(e×) = e⊗n, the microfree condition dev(S(n×)

)= 0

is equivalent to the condition

dev((n×)S

)=

mu"2dev[(n× curlγp)⊗ e− (e⊗ n)(∇ν)

]= µ"2dev

(#)︷ ︸︸ ︷[(n× curlγp)⊗ e− e⊗ (∇ν)n

]= 0.

Since e is perpendicular to both n× curlγp and (∇ν)n, the term (#) has zero trace; themicrofree condition therefore takes the form

(n× curlγp)⊗ e− e⊗ (∇ν)n = 0. (11.18)

Operating with this equation on e gives n × curlγp = 0, which leaves e ⊗ (∇ν)n = 0;hence (∇ν)n = 0. Thus the microfree boundary condition takes the form

n× curlγp = 0 and (∇ν)n = 0. (11.19)

Microhard condition

By (11.2), the microhard condition in the form (7.7) becomes

(e⊗ γp + ν ⊗ e)(n×) = −e⊗ (n × γp)− ν ⊗ (n × e) = 0.

Since e · ν = 0, if we operate on e with the transpose of the underlined terms, we arriveat n × γp = 0. Thus ν ⊗ (n × e) = 0 and operating with this equation on (n × e) wearrive at ν = 0. Thus the microhard condition takes the form

n × γp = 0 and ν = 0. (11.20)

April 13, 2004 18

11.5 Discussion

Because of the thermodynamic framework one would expect that, for τ known, (11.16),considered as a coupled system of partial differential equations for γp and ν, is parabolic.But the parabolicity would, in a sense, be degenerate, as the defect energy vanisheswhenever both curlγp and ∇ν vanish.

A numerical study of the anti-plane shear problem in an L×L unit cell with microhardconditions on top and bottom and microfree conditions on the sides, subject to simple-shear displacement boundary-conditions on top and bottom and to zero macroscopictraction conditions on the sides, should prove interesting. Instabilities of the partial dif-ferential equations (11.16) for γp and ν might not appreciably hinder a computationalprogram, since, as suggested in (i) of §10.2, it is only near a hard boundary that insta-bilities should become important. The questions that one might attempt to answer arethe following:

(i) Are the behaviors suggested in (i) and (ii) at the end of §10.2 seen?

(ii) How does the solution vary with the constant χ?

(iii) Is there a strong length-scale effect; that is, how does the solution change with celldimension, all other parameters being fixed?

12 Generalized theory with gradient-dissipation

Plastic strain-gradients enter the constitutive theory discussed in §9 through an energeticdependence on curlHp that affects the flow rule through a backstress involving secondgradients of Hp. We now sketch a generalization of this theory in which kinematichardening induced by the backstress is augmented by isotropic hardening ensuing fromdissipative constitutive dependences on ∇Ep.

12.1 Microforce balance. Free-energy imbalance

We begin by replacing the internal power expenditure (3.1) by the more general relation

Wint(P) =∫P

(T : He + Tp : Hp +K ...∇Hp

)dV, (12.1)

with K a third-order tensor, so that

K ...∇Hp = KijkHpij,k.

Since Hp is deviatoric, we may, without loss in generality, require that K be deviatoricin its first two subscripts; i.e., Kppk = 0. We do not change the external power, whichremains (3.2), but to avoid confusion we write the term S(n) as K(n).

The virtual-power principle as expressed by (3.6) and (3.7) here yields the macroscopicbalances (3.9) and (3.10) and leads to the microscopic virtual power relation∫

∂P

K(n) : Hp dA =∫P

(Tp −T) : Hp +K ...∇Hp

dV (12.2)

(cf. (3.12)). Applying the divergence theorem to the integral of K · ∇Hp, we find that∫∂P

K(n)−Kn

: Hp dA =

∫P

Tp −T− divK

: Hp dV, (12.3)

April 13, 2004 19

which yields the microforce balance

T0 = Tp − divK((T0)ij = T pij

)(12.4)

and the microtraction condition

K(n) = Kn((K(n))ij = Kijknk

). (12.5)

Further, as a consequence of (12.1) the free-energy imbalance becomes

ψ −T : Ee −Tp : Hp −K ...∇Hp ≤ 0. (12.6)

12.2 Constitutive equations

We continue to assume that ψ = Ψ(Ee,G) is the constitutive equation for the free energy(cf. (5.1)), we define

S =∂Ψ(Ee,G)

∂G, (12.7)

and, as before, we let Te = ∂Ψ/∂Ee. Then, since S : G = SijεirsHjs,r, (12.6) takes theform (

Kjsr − Sijεirs)Hjs,r + T pijH

pij ≥ 0. (12.8)

Thus R defined byRjsr = Kjsr − Sijεirs − 1

3δjsSipεirp (12.9)

is deviatoric in its first two subscripts and (12.8) becomes

Tp : Hp + R ...∇Hp ≥ 0. (12.10)

Our next step is to give constitutive equations for Tp and R consistent with (12.10).The constitutive equation for Tp should not differ drastically from (5.6). In deciding ona relation for R we follow the gradient-plasticity theories of Fleck and Hutchinson (2001),Gurtin (2003), and Gudmundson (2004) and restrict attention to a dependence on thegradient of the plastic strain-rate. Consistent with this and guided by (Gurtin, 2000,§15), we replace the effective distortion-rate (5.5) by

dp =√∣∣Ep∣∣2 + χ

∣∣Wp∣∣2 + h2

∣∣∇Ep∣∣2,

where χ and h are nonnegative constants with h a length scale, with Y (dp) > 0 andconsider constitutive equations for Tp and R of the form

Tp = Y (dp)(Ep + χWp

), R = Y (dp)h2∇Ep. (12.11)

Then R is symmetric in its first two subscripts (Rijk = Rikj), so that R ...∇Hp = R ...∇Ep,and the dissipation (12.10) has the simple form Y (dp)(dp)2.

12.3 Microforce balance revisited. Flow rule

By (2.2) and (12.9),divK = divR−

[devcurl (S)

], (12.12)

and the microforce balance (12.4) takes the form

T0 = Tp +[devcurl (S)

]− divR (12.13)

April 13, 2004 20

(cf. (3.14)). Since R is symmetric in its first two subscripts, the skew part of (12.13)is (6.4), so that, as before, plastic spin is a direct response to a nonvanishing Burgersvector.

The balance (12.13) and the constitutive equations yield the flow rule

T0 − dev curl(S)︸ ︷︷ ︸

energetic backstress

= Y (dp)(Ep − χWp) + div[Y (dp)h2∇Ep

]︸ ︷︷ ︸dissipative hardening

, (12.14)

with S = ∂Ψ/∂G.

12.4 Microscopically simple boundary conditions. Variationalformulation of the flow rule

By definition, the microscopic boundary conditions are those that yield K(n) : Hp = 0on ∂B (cf. (3.2) with S(n) = K(n)). By (12.5), (12.9), and the steps leading to (7.5),

K(n) : Hp = Kn : Hp = dev(S(n×)

): Hp + Rn : Ep

=(sym0

(S(n×)

)+ Rn

): Ep + skw

(S(n×)

):Wp. (12.15)

Thus, since the underlined term can also be written as(S(n×)

):(Wp

P(n)), we consider

microscopic boundary conditions asserting that, at each point of ∂B and each time,

either Ep = 0 and WpP(n) = 0, or

sym0

(S(n×)

)+ Rn = 0 and skw

(S(n×) = 0;

(12.16)

(12.16)1,2 are microhard conditions; (12.16)3,4 are microfree conditions.Our next step is to establish a variational formulation of the flow rule based on bound-

ary conditions in which a portion Shard of ∂B is microscopically hard and the remainderSfree microscopically free. We begin with the microscopic virtual power relation (12.2)applied to B. Because the boundary conditions (12.16) render the power expenditure nullon ∂B, we consider (12.2) with the boundary term omitted and with Hp = V:∫

P

(Tp −T) :V +K :∇V

dV = 0. (12.17)

As before, we refer to V as a test field and assume that V is kinematically admissible inthe sense that V is deviatoric and

symV = 0 and (skwV)P(n) = 0 on Shard. (12.18)

Using the divergence theorem, (12.12), (12.15) and the sentence following it, and (12.18),

∫B

K :∇VdV = −∫B

V : divKdV +∫∂B

Kn :V dA = −∫B

V :

divR−[curl (S)

]dV

+∫Sfree

[sym0

(S(n×) + Rn

]: (symV) +

(skw

(S(n×)

): (skwV)

dA

+∫Shard

[sym0

(S(n×) + Rn

]: (symV) +

((S(n×)

):((skwV)P(n)

)︸ ︷︷ ︸

=0

dA.

April 13, 2004 21

Thus (12.17) holds if and only if∫Sfree

Kn :V dA+∫B

Tp −T +

[curl (S)

]− divR

:V dV = 0, (12.19)

and therefore, using standard arguments, (12.17) holds for all kinematically admissibletest fields V if and only if the microforce balance (12.4) and the microfree boundary con-ditions (12.16)3,4 are satisfied. Finally, since the microforce balance, when supplementedby the constitutive equations, is equivalent to the flow-rule (12.14), we are led to the

Variational formulation of the flow rule Assume that the constitutive re-lations (12.7), (12.9), and (12.11) are satisfied. Then the flow rule (12.14) on Band the microfree boundary conditions (12.16)3,4 are together equivalent to therequirement that (12.17) be satisfied for all kinematically admissible test fields V.

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