Necking instabilities in elasto-viscoplastic materials

Avraham Moriel and Eran BouchbinderChemical and Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001, Israel

Necking instabilities, in which tensile (extensional) deformation localizes into a small spatial re-gion, are generic failure modes in elasto-viscoplastic materials. Materials in this very broad class— including amorphous, crystalline, polycrystalline and other materials — feature a predominantlyelastic response at small stresses, plasticity onset at a rather well-defined yield stress and rate-dependence. Necking instabilities involve a unique coupling between the system’s geometry andits constitutive behavior. We consider generic elasto-viscoplastic constitutive relations involving aninternal-state field, which represents the structural evolution of the material during plastic deforma-tion, and study necking in the long-wavelength approximation (sometimes termed the lubricationor slender-bar approximation). We derive a general expression for the largest time-dependent eigen-value in an approximate WKB-like linear stability analysis, highlighting various basic physical effectsinvolved in necking. This expression is then used to propose criteria for the onset of necking andmore importantly for the emergence of strong localization. Applications to strain-softening amor-phous plasticity, in the framework of the Shear-Transformation-Zone model, and to strain-hardeningcrystalline/polycrystalline plasticity, in the framework of the Kocks-Mecking model, are presented.These quantitative analyses of widely different material models support the theoretical predictions,most notably for the strong localization during necking.

I. INTRODUCTION

The ability of materials and structures to withstandtensile forces without failure is a fundamental physi-cal property with far-reaching practical implications. Inelasto-viscoplastic materials — a very broad class of ma-terials which feature a predominantly elastic responseat small stresses, irreversible plastic deformation uponsurpassing a rather well-defined yield stress and rate-dependence — the major process that limits this abil-ity is the development of necking instabilities, observedin a broad range of materials such as glassy alloys (bulkmetallic glasses) [1–3], many crystalline/polycrystallinematerials [4, 5], emulsions and suspensions [6–10], andamorphous bubble rafts [11–13]. This generic instabilitymanifests itself in the form of strongly localized defor-mation that results in a significant local reduction in thematerial’s cross-sectional area, which typically leads tofailure, cf. Fig. 1. Consequently, understanding and pre-dicting necking instabilities are of great importance asis also reflected by the quite extensive literature devotedto the topic, dating back at least to Considere’s 1885work [14].

In addition to the practical importance of necking in-stabilities in elasto-viscoplastic materials, this problemalso poses basic scientific challenges. One essential ele-ment of the problem is the viscoplastic constitutive rela-tion of the material, which generically involves stronglynonlinear and far-from-equilibrium dynamics, coupled tothe irreversible structural evolution of the material. Suchconstitutive relations are not yet fully developed and con-sequently their implications for necking instabilities arenot yet fully understood.

Another essential element of the problem is the macro-scopic material geometry (e.g. a long bar which is initiallycylindrical) and its dynamic evolution due to extensionaldriving forces (e.g. stretching along the cylinder’s ma-

jor axis). In particular, tensile/extensional deformation— whether spatially-homogeneous or inhomogeneous —generically leads to a reduction in the material cross-sectional area, which in turn leads to enhanced stressesthat may drive additional tensile/extensional deforma-tion. This intrinsic coupling between geometry and theconstitutive response of the material gives rise to richphysical behaviors that call for theoretical understand-ing.

The macroscopic geometry of the material and its dy-namic evolution have other important implications whichmake the problem interesting and challenging. Mostnotably, they imply that spatially-homogeneous defor-mation must be intrinsically time-dependent, since evensteady extensional deformation is accompanied by a con-tinuous (time-dependent) reduction in the cross-sectionalarea. This, in turn, implies that a conventional linearstability analysis — the standard tool for studying insta-bilities in a broad range of physical problems — does notstrictly apply to this problem; if some approximate formof it does apply and some relevant eigenvalue problemfor the growth of shape perturbations can be derived,then the emerging eigenvalues and eigenvectors are time-dependent.

Necking instabilities have been quite extensively stud-ied in various scientific contexts and communities. Whilewe cannot systematically and exhaustively review the rel-evant literature here, we would like to mention a fewworks that provide some background to our work. Wemainly focus on analytical works, which typically invokethe long-wavelength approximation (sometimes termedthe lubrication or slender-bar approximation) in whichshape perturbations along the extensional deformationaxis are characterized by a wavelength that is much largerthan the lateral dimensions of the system. In the con-text of rate-independent plasticity models, the first crite-rion for the onset of necking has been proposed by Con-

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sidere [14]. Various analyses followed, see for examplethe early review by Orowan [15] and many other laterworks [16–20]. Later, Hart extended Considere’s crite-rion to include material rate-sensitivity [21], and manyanalyses followed, e.g. [22–24]. Essentially all of theseworks focussed on the onset of necking using linearizedanalyses. A few other works [25–28], however, considereda particular class of phenomenological plasticity mod-els that enabled the development of a nonlinear stabilityanalysis.

Necking instabilities in extensional flows of non-Newtonian fluids, such as polymer melts, have also beenwidely studied, see the review papers of [29, 30]. Re-cently, necking instabilities in various models of complexfluids and soft solids have been considered [31–34]. Thevast majority of the literature on necking instabilitiesdoes not take into account the structural evolution of thematerial during plastic deformation; a contrary exampleis the very recent works of [35, 36] that considered phe-nomenological models of crystalline plasticity in whichthe dislocation density evolves with plastic deformationand affects it.

In this work, we study necking instabilities in elasto-viscoplastic materials in the long-wavelength approxima-tion, within a generic constitutive framework that ac-counts for the intrinsic rate-dependence of plastic de-formation and for the structural evolution of the mate-rial during plastic deformation through an internal-statefield. We derive a general expression for the largest time-dependent eigenvalue in an approximated WKB-like lin-ear stability analysis, allowing us to identify the variousstabilizing and destabilizing physical processes involved.The resulting expression can be used to predict neckinginstabilities in a broad range of materials and constitu-tive relations.

Unlike most of the previous works, we do not onlyconsider the onset of instability, but also extensivelydiscuss the emergence of strong localization. A cri-terion for the latter is obtained by comparing thetime-evolution of shape perturbations and the time-evolution of the spatially-homogeneous (unperturbed)state. The criterion is applied to two very differ-ent rate-dependent constitutive relations that featurean internal-state field: the Shear-Transformation-Zone(STZ) model [37–41] of strain-softening amorphous plas-ticity and the Kocks-Mecking model [42–45] of strain-hardening crystalline/polycrystalline plasticity. Com-parison of the strong localization criterion to nonlinearnumerical solutions of the two different models demon-strates favorable agreement, lending support to the pro-posed criterion.

The structure of this paper is as follows; the com-plete mathematical formulation of the necking prob-lem and its dimensionally-reduced form are presentedin Sect. II. The time-dependent linear stability analysisis presented in Sect. III, and the derivation of a gen-eral analytical expression for the largest time-dependenteigenvalue and the associated onset criterion for neck-

ing follow in Sect. IV. The condition for the emer-gence of strong localization is derived in Sect. V. Thetheoretical predictions are tested against full numeri-cal solutions for amorphous/glassy materials (using theShear-Transformation-Zone model) in Sect. VI and forcrystalline/polycrystalline materials (using the Kocks-Mecking model) in Sect. VII. Finally, some concludingremarks and future research directions are offered inSect. VIII.

FIG. 1. A schematic illustration of the development of a neck-ing instability of an initially spatially-homogeneous cylindri-cal bar of length L0 and cross-sectional area a0 (panel a). Ex-tensional deformation leads to the onset of a neck, i.e. regionin which the cross-sectional area is slightly reduced comparedto the rest of the bar (panel b). As the extensional deforma-tion proceeds, the necking instability further develops, andthe bar experiences strong localization (panel c) that is likelyto lead to catastrophic failure.

II. PROBLEM FORMULATION:CONSERVATION LAWS, CONSTITUTIVE

FRAMEWORK AND THE LONG-WAVELENGTHAPPROXIMATION

We begin by mathematically formulating the problem,i.e. writing down the relevant conservation laws, defininga generic constitutive framework for elasto-viscoelasticmaterials with an internal-state field and stating the in-voked approximations. Mass conservation is describedby the continuity equation

∂tρ+∇·(vρ) = 0 , (1)

where ρ(r, t) is the mass density, i.e. the mass per unitvolume at point r in space at time t, and v(r, t) is thevelocity field (here and below vectors and tensors appearin bold face). Linear momentum conservation takes theform

∇·σ = ρ v , (2)

where σ(r, t) is the Cauchy (real) stress tensor and bodyforces were neglected. Note that • ≡ (∂t + v ·∇)•, is the

3

full (material) time derivative, and that v≡ ∂tu, whereu(r, t) is the displacement field. Finally, angular mo-mentum conservation is ensured by the symmetry of theCauchy stress tensor σ=σT [46].

In order to make reference to a specific class of ma-terials and to render the problem mathematically well-defined, we need to specify a relation between the stress σand the velocity v, i.e. we need a constitutive law. Moreprecisely, we need a relation between the total strain-rate (the symmetric part of the velocity gradient tensor)ε≡ 1

2 [∇v+(∇v)T ] and σ that is representative of elasto-viscoplastic materials. As viscoplasticity is an intrinsi-cally dynamic, rate-dependent class of physical phenom-ena, it must be expressed in terms of the strain-rate ε.

We proceed in two steps; first, we decompose the to-tal strain-rate into an elastic (reversible) contribution εel

and a viscoplastic (irreversible) contribution εpl,

ε = εel + εpl . (3)

Second, we discuss the structure of the elastic and vis-coplastic contributions. For the elastic part, we useHooke’s law of isotropic linear elasticity ε(σ) [46], andtake a proper time derivative to cast it in rate form [47].Consequently, εel∝ σ in the tensorial sense. For the vis-coplastic part, we leave the functional form of εpl(· · · )unspecified, and focus only on its arguments. It obvi-ously depends on the stress σ and it must depend onthe structural state of the material that is captured bya small set of properly-defined, coarse-grained internal-state fields Iα. In addition, as viscoplastic deformationmight involve thermal activation, it might depend on thetemperature T .εpl (σ, T, Iα) should be supplemented by evolution

equations for the internal-state fields Iα, which aretaken here to be scalar for simplicity (though non-scalarinternal-state fields are sometimes essential, e.g. in thecontext of the Bauschinger effect [48]). As no structuralevolution takes place in the absence of viscoplastic defor-mation, we must have

Iα = εpl:g(σ, Iα) , (4)

where g(· · · ) is a tensorial function that renders the right-hand-side of Eq. (4) a proper scalar. It is natural andphysically intuitive to expect g∝σ such that Iα evolvesdue to the viscoplastic dissipation rate εpl:σ. However,as not all internal-state viscoplastic models available inthe literature share this feature, we do not restrict thediscussion to this case. Finally, we note that Eq. (3)ε(v)= εel(σ) + εpl(σ, T, Iα) is in fact an equation for σ.

Once εpl(σ, T, Iα) and g(σ, Iα) are specified for agiven class of materials, Eqs. (1)-(4) fully describe theevolution of ρ, v, σ and Iα for any initial-boundary valueproblem with any initial geometry (if the temperature Tevolves in space and time, then the heat equation shouldbe added). Applying this general framework to the neck-ing problem, i.e. to the large elasto-viscoplastic exten-sional deformation of a cylindrical bar, is an extremely

complicated mathematical problem, even if only numer-ical solutions are considered.

In order to gain some analytical and physical insightinto this complicated problem we invoke several approx-imations. First, we note that viscoplastic deformation isquite generically a slow process compared to elastic pro-cesses (i.e. to the relevant travel time of elastic waves).Consequently, as long as we do not consider high appliedstrain-rates, we can approximate Eq. (2) by its quasi-static counterpart∇·σ=0. By so doing, we exclude fromthe discussion a class of interesting necking problems inwhich inertial effects play an important role [18, 49–56].We further assume that the material of interest is incom-pressible, i.e. its mass density is constant (and hence isomitted hereafter), which is a reasonably good approxi-mation for most elasto-viscoplastic materials.

Next, we consider a long cylinder with an initial lengthL0 and cross-sectional area a0, cf. Fig. 1a, such that itsinitial radius satisfies R0∼

√a0L0. Under these con-

ditions — which are termed the long-wavelength, the lu-brication, the slender-bar, or the shallow-water approxi-mation in various scientific communities — one needs totake in account only the spatial variation of fields alongthe main axis of the cylinder, say x, neglecting the vari-ations in the transverse directions. Consequently, in thislong-wavelength approximation incompressibility can beexpressed as a=−a ε in terms of the cross-sectional areaa(x, t) and the extensional strain-rate ε(x, t) = ∂xv(x, t),and quasi-static linear momentum balance as ∂x (a σ)=0,where σ(x, t) is the axial stress. Note that this approxi-mation may break down dynamically when strong local-ization develops. Linear elasticity, in this approximation,takes the form εel = σ/G, where G is a relevant linearelastic modulus (here it is simply the Young’s modulus).Finally, we neglect the spatiotemporal variation of thetemperature T and focus on a single internal-state fieldwhose evolution equation is I = εplg (σ, I).

Taken together, our dimensionally-reduced set of equa-tions takes the form

∂ta+ v∂xa = −a ∂xv , (5a)

∂x(a σ) = 0 , (5b)

∂tσ + v∂xσ = G[∂xv − εpl(σ, T, I)] , (5c)

∂tI + v∂xI = εplg(σ, I) . (5d)

Here, Eq. (5a) is the long-wavelength incompressiblecounterpart of Eq. (1), Eq. (5b) is the long-wavelengthquasi-static limit of Eq. (2), Eq. (5c) is the scalar coun-terpart of Eq. (3), and Eq. (5d) corresponds to the scalarcounterpart of Eq. (4) for a single internal-state field.In the remainder of the paper, we will study this set ofequations for a long cylindrical bar of length L(t) underextensional deformation generated by an imposed edgevelocity v(x=L(t)/2, t)=−v(x=−L(t)/2, t), where x=0is the middle of the bar.

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III. APPROXIMATE TIME-DEPENDENTLINEAR STABILITY ANALYSIS

The set of Eqs. (5) admits spatially-homogeneousdeformation solutions, which can be obtained onceεpl(σ, T, I) and g(σ, I) are specified. The first ques-tion we aim at addressing is the linear stability ofthese solutions. In many problems in physics, thespatially-homogeneous solutions are also temporally-homogeneous, i.e. they are time-independent. This isnot the case in the problem at hand, as is immediately in-ferred from Eq. (5a); the right-hand-side is finite (the ini-tially homogeneous cross-sectional area a is finite and ∂xvis finite because the system experiences extensional de-formation). The time-dependent nature of the spatially-homogeneous solutions has serious implications for thelinear stability problem.

To appreciate these implications, consider a generalproblem consisting of N fields f (i), i = 1, 2, . . . N ,where the spatially-homogeneous solutions are time-dependent. We then add to these spatially-homogeneous

solutions f(i)h (t) — the subscript ‘h’ stands for

spatially-homogeneous — spatiotemporal perturbations

of wavenumber k, leading to f (i)(x, t) = f(i)h (t) +

δ(i)(t) eikx. Inserting these f (i)(x, t)’s into the completenonlinear set of equations, in our case Eqs. (5), and lin-earizing with respect to the perturbations, we obtain

∂tδ(t) = M(t, k) δ(t) , (6)

where δ(t) is a vector whose components are the time-dependent parts of the perturbation, δ(i)(t). The crucialpoint is that due to the time-dependence of the spatially-

homogeneous fields f(i)h (t), the matrix M is also time-

dependent.The time-dependence of M has two major conse-

quences that are missing in conventional linear sta-bility analyses, where it is a time-independent con-stant. To identify these, we follow the standard proce-dure and diagonalize M(t), where the dependence onk is omitted here for simplicity; here the diagonaliz-ing matrix P (t) (which consists of the diagonalizing ba-sis vectors) is also time-dependent, P−1(t)M(t)P (t) =diag(λ1(t), ..., λN (t)), as well as the eigenvalues λi(t).Defining the vector of transformed field perturbations∆(t)≡P−1(t)δ(t), Eq. (6) transforms into

∂t∆(t)=[diag (λ1(t), ..., λN (t))+

(∂tP

−1(t))P (t)

]∆(t) .

(7)The two differences compared to the conventional time-independent linear stability analysis are apparent; first,the eigenvalues λi do not exclusively determine the lin-earized evolution of the system as the time-dependenceof P (t) and its inverse play a role as well, as indicatedby the second term on the right-hand-side. Second, theeigenvalues themselves are time-dependent, λi(t).

To the best of our knowledge, Eq. (7) does not admit ageneral exact solution. To proceed, we adopt a WKB-like

approximation [57–59] in which the base vectors in P (t)are assumed to vary slowly compared to other timescalesin the problem. With this assumption, Eq. (7) can be ap-proximated as ∂t∆(t)'diag(λ1(t), ..., λN (t))∆(t). Solv-ing this equation and transforming back to the originalfields, we obtain

δ (∆t, t0)'N∑i

∆i(t0)Ei(t0) exp

[∫ t0+∆t

t0

λi(t′) dt′

],

(8)where Ei(t0) are the eigenvectors of M(t) at time t= t0,∆i(t0) are constants obtained from δ(t0) (the perturba-tion at time t0) according to ∆(t0)≡P−1(t0)δ(t0) and∆t is the time measured from t0. While Eq. (8) is anexact solution to the approximated equation, the qualityof the approximation to the exact Eq. (7) and its de-pendence on t0 and the time interval ∆t are not a prioriknown. Finally, we simplify Eq. (8) by further assumingthat the largest eigenvalue, λ+(t), dominates the sumand that we can set t0 =0 and ∆t= t, resulting in

δ(t) ' ∆+(0)E+(0) exp

[∫ t

0

λ+(t′) dt′]. (9)

Here E+(0) is the eigenvector corresponding to thelargest eigenvalue λ+ at t = 0 and ∆+(0) is the corre-sponding constant.

We now aim at applying the approximated expressionfor the time-dependent linear stability analysis in Eq. (9)to the necking problem, as formulated in Eqs. (5). First,we note that the convective derivative terms in Eqs. (5),v∂x•, vanish identically for the spatially-homogeneousdynamics or are small within the linear perturbationregime around the spatially-homogeneous state, hencethey are omitted from the analysis (though they are in-cluded in any direct numerical solutions of Eqs. (5) be-low). In their absence, Eqs. (5) become independent ofthe wavenumber of perturbations k. Note in this contextthat the velocity gradient ∂xv is finite in the absenceof perturbations and its value in this case is determinedby the applied velocity. In particular, in the spatially-homogeneous state, ∂xv is identical to its spatial average

〈∂xv〉≡ [L(t)]−1∫ 1

2L(t)

− 12L(t)

∂xv dx= L(t)/L(t) = εh(t), which

is the imposed Hencky strain-rate. In the presence of spa-tial inhomogeneity, εh(t) is in fact the average imposedHencky strain-rate, ˙ε(t)= εh(t).

Throughout this paper the applied boundary condi-tions correspond to a constant, time-independent Henckystrain-rate ˙ε, though the salient features of the resultsdo not change for other boundary conditions, e.g. con-stant velocity. For the spatially-homogeneous dynamicsεh = ˙ε is just a parameter and the time-dependent solu-tions for ah(t), σh(t) and Ih(t) can be obtained. We thenintroduce perturbations to all 4 fields, δε(t), δa(t), δσ(t)and δI(t) and linearize Eqs. (5) around the spatially-homogeneous state (recall that the convective derivativeterms are omitted and ∂xv = ε in Eqs. (5)). To pro-ceed, we eliminate δε between Eqs. (5a) and (5c), and

5

express δσ in terms of δa using Eq. (5b) (which impliesσhδa=−ahδσ to linear order).

We are then left with two fields, i.e. δ=(δa, δI)T

, andthe matrix M in Eq. (6) takes the approximate form

M(t) '

(σ∂σ ε

pl − εpl − a∂I εpl

− σa∂σI ∂I I

), (10)

where the subscript ‘h’ is omitted hereafter. In Eq. (10)we used the fact that the stress is typically muchsmaller than the elastic modulus, σ G. The time-dependent eigenvalues can be readily obtained as λ(t)=12 trM

(1±√

1− 4 detM/(trM)2)

. For constitutive re-

lations for which detM(trM)2, the largest eigenvalue— the most important physical quantity of interest inthe linearized analysis — attains a simple analytical ex-pression

λ+(t) ' trM = σ∂σ εpl − εpl + ∂I I , (11)

in case it is positive. In case the latter expression is neg-ative, the largest eigenvalue is λ+(t)' 0. Consequently,while we hereafter refer to Eq. (11) as the largest eigen-value, it should be understood that it is valid only if it ispositive; otherwise, λ+(t)=0 is used.

IV. THE LARGEST EIGENVALUE AND THEONSET OF NECKING

The previous section culminated with an analyticalapproximation, cf. Eq. (11), for the largest eigenvalueλ+(t) in the time-dependent linear stability analysis forthe necking problem. Let us briefly discuss the physi-cal meaning of the different contributions to λ+(t). Theappearance of the stress σ in the first term on the right-hand-side is a direct consequence of the coupling betweenthe system’s geometry and its mechanical response, re-sulting in stress amplification due to shape perturbations.The stress σ multiplies the variation of the plastic strain-rate with the stress, ∂σ ε

pl, at constant internal structureI and temperature T . As the stress is the driving forcefor plastic deformation, we expect εpl to generically in-crease with increasing σ, at constant internal structure Iand temperature T . Consequently, the first term on theright-hand-side of Eq. (11) is positive, σ∂σ ε

pl> 0, i.e. itgenerically promotes a necking instability.

The second term on the right-hand-side of Eq. (11)is the (extensional, hence positive) plastic strain-rate εpl

with a minus sign, i.e. this contribution is intrinsicallystabilizing. That is, homogeneous plastic deformation,by itself, tends to limit the growth of shape perturba-tions. We note that the ratio of the second to first termson the right-hand-side of Eq. (11) is the dimensionlessstrain-rate sensitivity m−1≡∂ log(σ)/∂ log(εpl), which issmall for many materials [27, 43]. Hence we expect thefirst term to dominate the second one in many cases.

The third contribution to λ+(t) in Eq. (11), ∂I I, con-cerns the evolution of the material’s structure — de-scribed by the coarse-grained internal-state field I —which inevitably accompanies plastic deformation. Todiscuss the physical meaning of this contribution oneneeds to be a bit more explicit about the physical na-ture of the scalar internal-state field I. To that aim,it would be natural to assume that I corresponds tothe density of plasticity carriers in the material, forexample dislocations in crystalline materials [42–45] orShear-Transformation-Zones (STZs) in amorphous ma-terials [37, 60, 61]. With this physical picture in mind,

∂I I represents the variation of the rate-of-change of thedensity of plasticity carriers with their density. For ma-terials in which locked-in plasticity carriers hamper sub-sequent plastic deformation, we expect ∂I I < 0. Thisis the case for strain-hardening materials such as met-als, for which this contribution is stabilizing. On theother hand, for strain-softening materials such as bulkmetallic glasses [62–70], the activation of plasticity car-riers facilitates subsequent plastic deformation and wehave ∂I I > 0. Consequently, and not quite surprisingly,Eq. (11) predicts that strain-softening materials are moreprone to necking instabilities, while in strain-hardeningmaterials necking instabilities may be at least delayed tolarger strains.

The largest eigenvalue in Eq. (11) can be used to de-rive the onset conditions for necking, i.e. λ+(t)>0. In thenext section we will argue that more significant, physi-cally meaningful and predictive information about neck-ing can be extracted from the time-dependent eigenvalueλ+(t). In the meantime, however, we do consider the on-set condition λ+(t) > 0 and compare it to conventionalonset conditions available in the literature. Apparentlythe first, and quite extensively used, criterion for the on-set of necking is Considere’s criterion dσ/dεpl<σ, whichassumes that the stress σ can be expressed solely as afunction of the plastic strain εpl. As is evident fromthe discussion in Sect. II, this is not a physically real-istic formulation — the plastic strain itself is not a le-gitimate independent physical field, but rather can becalculated from a time-integral over the plastic strain-rate εpl. As such, we cannot directly compare our onsetcriterion λ+(t) = σ∂σ ε

pl − εpl + ∂I I > 0 to Considere’scriterion.

An extensively used generalization of Considere’s cri-terion, taking into account the rate-dependence of plas-tic deformation — i.e. considering a relation of the formσ(εpl, εpl) — has been proposed by Hart [21]. The Hartcriterion reads

1

m+

∂σ

σ∂εpl< 1 , (12)

where the strain-rate sensitivity m−1 has been definedabove. By dividing by σ∂σ ε

pl> 0 and rearranging, ouronset criterion λ+(t) =σ∂σ ε

pl− εpl+∂I I > 0 can be cast

6

in a form somewhat reminiscent of Eq. (12)

1

m− ∂I Iσ∂σ εpl

< 1 . (13)

Comparing Eq. (13) to Eq. (12) we observe that

−σ−1∂I I/∂σ εpl appears to play the role of the di-mensionless strain-hardening/softening coefficientσ−1∂σ/∂εpl (which is positive for strain-hardeningmaterials and negative for strain-softening materi-als). Indeed, as discussed above, for strain-hardening

materials we expect ∂I I < 0, i.e. −σ−1∂I I/∂σ εpl > 0

and for strain-softening materials we expect ∂I I > 0,i.e. −σ−1∂I I/∂σ εpl < 0. While this analogy mightappear suggestive, there is no fundamental reason toexpect −σ−1∂I I/∂σ εpl to correspond to the measuredstrain-hardening/softening coefficients extracted fromstress-strain curves. Instead, we believe that Eq. (13)offers a more physically sound criterion for the onset ofnecking, going beyond the criteria that are available inthe literature.

V. STRONG LOCALIZATION CRITERION

The onset criterion in Eq. (13) predicts the time, orequivalently the strain, at which shape perturbationsstart to grow, λ+(t) > 0. As such, it does not provideinformation about the evolution of the neck and in par-ticular about the emergence of macroscopic shape local-ization. The approximate time-dependent linear stabilityanalysis developed in Sect. III is, in fact, capable of pro-viding more extensive and quantitative predictions aboutnecking dynamics, as we show next. Combining the re-sults in Eqs. (9) and (11), we obtain the time-evolutionof the cross-sectional area (shape) perturbation as

δa(t)'∆+(0)E+(a)(0) exp

[∫ t

0

(σ∂σ ε

pl−εpl+∂I I)dt′],

(14)where E+(a) is the component of the vector E+ that cor-responds to δa and all of the functions in the integrandare understood to depend on the time t′.

We now aim at using δa(t) of Eq. (14), obtained fromthe approximated linear analysis, to predict the onset ofstrong, nonlinear shape localization. Such a strong lo-calization will be macroscopically manifested and mighteventually lead to catastrophic failure (e.g. by shearbanding, cavitation, crack propagation or the like), whichgoes beyond the scope of this paper. Strong localiza-tion is expected to initiate once the perturbation δa(t)becomes non-negligible compared to the homogeneouscross-sectional area ah(t) = a0 exp(− ˙ε t), which is thespatially-homogeneous solution of Eq. (5a) for a constant˙ε. That is, we are looking for the time in the necking dy-namics at which δa(t)/ah(t) = κ ∼ O(10−1). While theexact value of κ, i.e. the ratio of the perturbation to thehomogeneous solution at which non-negligible nonlinear-ities set in, is not a priori known, we show below that

the results are only weakly dependent on κ in this range.Note also that both the growing perturbation after theonset of necking and the exponential time decay of thehomogeneous solution tend to increase the relative mag-nitude of the perturbation in the ratio δa(t)/ah(t).

Applying this criterion, the onset of strong necking-mediated localization is predicted to occur at t= t`, whichis a solution of the following integral equation∫ t`

0

(σ∂σ ε

pl + ∂I I)dt′ ' log

(κ

ζ

), (15)

where the negligibly small average elastic strain-rate˙ε − εpl has been omitted from the integrand and we de-fined ζ ≡ ∆+(0)E+(a)(0)/a0. To make use of Eq. (15)we need to evaluate the right-hand-side, which involvesκ ∼ O(10−1) and the relative magnitude of the initial(t = 0) perturbation. The exact value of κ ∼ O(10−1)does not make a significant quantitative difference dueto the logarithmic dependence. ζ is determined eitherby the prescribed perturbation in a deterministic calcu-lation or by the typical noise level in a realistic physi-cal situation; as such, ζ in Eq. (15) is interpreted as acharacteristic (relative) amplitude of perturbations, be-ing more general than its formal definition given abovein the framework of the linear stability analysis.

We believe that the prediction in Eq. (15) for the emer-gence of strong localization is more physically significantand relevant compared to the onset of necking predictedin Eq. (13). While both are based on a linearized analy-sis, the former in fact provides an estimate for the onset ofnonlinearities. Indeed, as has been mentioned in Sect. I,several works highlighted the limitations of linearized on-set criteria, trying to circumvent them by studying con-stitutive relations that allow some nonlinear progress andinvoking several simplifying geometrical assumptions. Inparticular, Hutchinson and Neale [27] performed sucha nonlinear analysis for rate-dependent viscoplastic ma-terials and Fressengeas and Molinari [28] extended theanalysis to include also thermal effects on the localiza-tion process, though some other simplifying assumptionshave been made (e.g. a constant applied force has beenassumed). While these works have provided some an-alytical insight into the nonlinear evolution of necking,they seem to lack the degree of generality of Eq. (15),both in terms of the adopted constitutive framework andin terms of the additional simplifying assumptions in-voked. Note also that [27, 28], like the present work,invoke the long-wavelength (lubrication/slender bar) ap-proximation, which is anyway likely to break down whenstrong nonlinearities set in.

In the reminder of this paper we aim at quantita-tively testing Eq. (15) for two widely different elasto-viscoplastic constitutive relations, describing differentclasses of materials, against fully nonlinear numerical so-lutions of Eqs. (5). Before discussing these quantitativeanalyses in detail in the next sections, we would like tobriefly demonstrate how such analyses are performed. Toassess the quality of the proposed criterion, we need to

7

obtain fully nonlinear solutions of Eqs. (5). Such nu-merical solutions, once εpl(σ, T, I) and g(σ, I) are speci-fied, are obtained by first transforming the equations tothe Lagrangian frame of reference, in which the integra-tion domain is time-independent. Then the equationsare solved by a conventional technique, see Appendix fordetails.

FIG. 2. The minimal normalized cross-sectional area amin/a0,as a function of the accumulated Hencky strain ε (measuredfrom the onset of plasticity), for various applied strain-rates ˙ε(see legend). Minimal cross-sections were obtained from fullynonlinear numerical solutions of Eqs. (5) for the amorphousplasticity model discussed in Sect. VI, with an initial inho-mogeneity imposed on the internal-state field. The solutionin which spatial homogeneity is imposed is added for refer-ence (dashed gray line), as it is strain-rate independent. Thecurves corresponding to the perturbed solutions significantlydeviate from the spatially-homogeneous solution at character-istic strains marked by the arrows and denoted by ε∗` .

The system is driven by an applied Hencky strain-rate ˙ε and the spatially-homogeneous initial conditionsare supplemented with a perturbation δI(t = 0) =ζ I0 cos(2πx/L0) of the internal-state field I, where I0

is its homogeneous initial value. In addition to trackingthe time-evolution of the perturbed system, we also solveEqs. (5) by enforcing spatial homogeneity; this time-dependent spatially-homogeneous solution serves as a ref-erence for evaluating the relative importance of spatial in-homogeneity. For convenience, we use the Hencky strainε≡ ˙ε t as the independent variable instead of the time t,which also renders the homogeneous cross-sectional areaevolution, ah(ε)=a0 exp(−ε), strain-rate independent.

In Fig. 2 we present an example of such solutions usinga constitutive relation to be discussed in detail in Sect. VIand a relative perturbation amplitude ζ=10−6. To quan-tify the evolution of the system, we plot the spatial min-imum of the cross-sectional area, amin(ε)≡min [a(x, ε)],for various applied strain-rates ˙ε, together with the homo-

geneous solution ah(ε). We observe a generic behavior inwhich the perturbed solutions appear indistinguishablefrom the homogeneous solution (this point will be fur-ther discussed below) over a certain range of extensionalstrains ε, followed by a rather abrupt drop in the minimalcross-sectional area of the perturbed solutions relative tothe homogeneous one, marking the onset of strong lo-calization. The latter, denoted by ε∗` and schematicallymarked on the figure, exhibits rate-dependence.

As explained above, the ultimate goal of the developedtheory is to quantitatively predict the strong localizationstrain ε∗` using Eq. (15), which results in the predictionε` ≡ ε(t`). In Fig. 3 we plot the integrand of the in-tegral relation in Eq. (15) (normalized by the appliedstrain-rate ˙ε) as a function of the strain ε (instead ofthe time t) for homogeneous solutions (parameters asin Fig. 2). The first important observation we makeis that the integrand — which is closely related to thelargest eigenvalue of Eq. (11) — is positive and increaseswith ε in the very same strain interval in which the per-turbed systems appear to be indistinguishable from thespatially-homogeneous system in Fig. 2. This observa-tion clearly demonstrates that while the onset conditionis important, it provides only partial information aboutthe macroscopic development of instability and the onsetof strong localization. Obviously, the relation betweenthe onset of instability and the onset of strong localiza-tion depends on the typical magnitude of perturbationsquantified by ζ (e.g. set by the noise level in the system),as is clear from Eq. (15).

The onset of strong localization can be obtained byintegrating the integrand shown in Fig. 3 until a valuedetermined by κ on the right-hand-side of Eq. (15) isreached. In Fig. 3 we used vertical lines to mark thestrain values ε` for which Eq. (15) is satisfied for κ =0.15. In what follows, we compare the predictions forthe strong localization strain ε` to the strong localizationstrain ε∗` obtained directly from fully nonlinear solutionsfor two widely different elasto-viscoplastic constitutiverelations, describing different classes of materials, andfor various physical conditions and material parameters.

VI. APPLICATION I: AMORPHOUSMATERIALS AND THE STZ MODEL

The analysis up to now remained fairly general, andas such could be applied to any constitutive law thatinvolves a single internal-state field (the generalizationto more internal-state fields is straightforward, but willbe more mathematically involved). We derived an insta-bility onset criterion as the zero-crossing in time of thelargest time-dependent eigenvalue in Eq. (11), discussedits relations to existing criteria, and obtained a stronglocalization criterion in Eq. (15). In order to quantita-tively test these criteria, one should specify a constitutivelaw, i.e. εpl(σ, T, I) and g(σ, I). In this and the follow-ing sections, we consider two widely different constitutive

8

FIG. 3. (σ∂σ εpl + ∂I I)/ ˙ε (the integrand of Eq. (15) normal-

ized by the applied strain-rate ˙ε), as a function of the accumu-lated Hencky strain ε (measured from the onset of plasticity)for various strain-rates (for homogeneous solutions with theparameters of Fig. 2). The solutions of Eq. (15), ε` = ε(t`),are marked by the vertical lines.

laws — one for amorphous/glassy materials and one forcrystalline/polycrystalline materials — and use them toquantitatively test our predictions.

We start by considering amorphous/glassy materialswhich lack the long-range order of crystalline materials.The elasto-viscoplastic response of amorphous/glassymaterials has attracted a lot of interest in the last fewdecades [41, 66, 71–74]. It is now widely acceptedthat plastic deformation in these materials is mediatedby spatially-localized immobile rearrangements of pre-dominantly shear nature — Shear-Transformation-Zones(STZs) —, which are qualitatively different from disloca-tions in crystalline materials. While various fundamen-tal questions about the nature and properties of plasticdeformation in amorphous/glassy materials are still de-bated and intensively investigated, a few quite successfulcoarse-grained phenomenological models have emerged inthe literature such as the Soft Glassy Rheology (SGR)model [75–77] and the Shear-Transformation-Zone (STZ)model [37–41].

We will consider here the STZ model. The reasonfor this choice is that this model includes an internal-state field and is most suitable for studying transient,far from steady-state elasto-viscoplastic deformation ofamorphous/glassy materials. Moreover, this relativelysimple model, which is formulated within a nonequi-librium thermodynamic framework, has been shown tocapture various salient features of the elasto-viscoplasticdeformation of amorphous/glassy materials [37–41, 78–89]. In particular, it has been shown to exhibit shear-banding [90, 91], necking [92, 93], and to predict a

brittle-to-ductile-like transition in the fracture toughnessof glasses as a function of the their preparation proto-col [94, 95].

Plastic deformation in amorphous/glassy materials re-sults from the macroscopic accumulation of microscopicirreversible strain at the cores of STZs, which corre-sponds to a transition between the STZ internal statesthat are separated by a barrier. The coarse-grainedplastic strain-rate in the framework of the STZ model,adapted to our long-wavelength approximation, takes theform [41]

εpl(σ, T,Λ) = Λ(χ) τ−10 C(σ, T )

(1− σy

σ

), (16a)

Λ(χ) = exp

(− ezkBχ

). (16b)

In Eq. (16a), Λ(χ) is the density of STZs, τ−10 C(σ, T ) is

the stress σ- and temperature T -dependent barrier tran-sition rate between the STZ internal states (τ0 is a mi-croscopic vibration time-scale) and σy is the yield stress.We consider relatively low temperatures T such that nospontaneously thermally-activated creation/annihilationof STZs takes place and the yield stress σy representsa rather sharp transition. Consequently, Eq. (16a) de-scribes the post-yielding regime σ>σy, where as εpl = 0for σ ≤ σy. In addition, heat generation is assumed tobe sufficiently slow such that the temperature T remainsnearly equilibrated with the heat reservoir at any time.

The dimensionless transition rate is given by a con-ventional stress-biased thermal activation expression

C(σ, T ) = exp(− ∆kBT

)cosh

(ΩσkBT

), where ∆ is a typical

activation barrier and Ω is the product of a typical micro-scopic STZ volume and a typical local strain contributedby any STZ transition. If the stress becomes very large,e.g. near the tip of a crack or inside a narrow neck, ac-tivation barriers are expected to be completely washedout and other dissipative mechanisms kick in. In par-ticular, when the stress satisfies σ > ∆/Ω, we take thestress-biased thermal activation expression to cross overto the linear relation C(σ, T )=Ωσ/(2∆).

The density of STZs, Λ, is given in Eq. (16b) in termsof a Boltzmann-like factor in which χ is an effective tem-perature and ez is the typical STZ formation energy (kBis Boltzmann’s constant) [41]. The effective tempera-ture χ is a thermodynamic temperature that describesthe slow configurational degrees of freedom of an amor-phous/glassy material which fell out of equilibrium withthe vibrational degrees of freedom described by T [41].Within the two-temperature thermodynamic theory ofdeforming glasses, χ evolves according to the followingeffective heat equation

χ=2σεpl

c0σy(χ∞ − χ) , (17)

where c0 is the effective heat capacity, χ∞ is the steady-state effective temperature (the initial value of χ, χ0,

9

typically satisfies χ0 < χ∞) and no χ-diffusion is takeninto account.

Equation (17) suggests that χ is driven by the plasticdissipation power σεpl and that the fraction of the dis-sipation power that is stored in the material’s structure(“stored energy of cold work”) is inversely proportionalto the yield stress σy (the complementary part is irre-versibly transferred into the heat reservoir). The increaseof χ with plastic deformation, which leads to enhancedplastic deformation, represents a “flow induces flow” pro-cess, characteristic of strain-softening materials. The ini-tial value of χ, χ0, depends on the history of the glass,e.g. the cooling-rate through the glass transition or theaging time in the vicinity of the glass temperature, andhence describes the initial non-equilibrium state of theglass. As Λ is uniquely related to χ through Eq. (16b),

Eq. (17) serves as the relevant I equation and the con-stitutive law is fully specified.

We first consider the spatially-homogeneous solutionof Eqs. (16), (17) and (5c), where the latter reads σ/G=˙ε − εpl(σ, T, χ). All calculations start at the onset ofplasticity, σ(t = 0) = σy, and the strain ε is measuredrelative to the elastic strain at this point. The stress-strain curves for various applied strain-rates ˙ε, spanningnearly 3 orders in magnitude, are shown in Fig. 4a. Thecalculations were performed with the following set of pa-rameters: G = 37 GPa (shear modulus), τ0 = 10−13 s,σy = 0.85 GPa, ez ' 1.8 eV, ∆ ' 0.69 eV, T = 400 K,

Ω = 90 A3, c0 = 0.4, χ∞ = 900 K, obtained in [96] for

Vitreloy 1 (Vit1), the first commercial and widely-usedbulk metallic glass. We observe that the stress typicallyexhibits a maximum that is followed by relaxation tosteady-state, where the magnitude of the stress overshootincreases with increasing strain-rate. The stress peak isof elastic origin, i.e. upon yielding the plastic strain-rateεpl cannot keep up with the applied strain-rate ˙ε and thelatter is accommodated mainly by the elastic strain-rate,and the subsequent relaxation is of plastic origin as εpl in-creases with increasing plastic deformation until reachingsteady-state εpl= ˙ε.

A stress peak followed by stress relaxation is charac-teristic of amorphous/glassy materials which are strain-softening. As discussed in Sect. IV, for such materials weexpect ∂I I>0, which implies that the largest eigenvaluein Eq. (11) might be positive already at the early stagesof the elasto-viscoplastic deformation process. This isdirectly demonstrated in Fig. 4b, which also shows thatthe largest eigenvalue undergoes a significant evolutionwith strain, hence the onset conditions (i.e. when λ+ firstbecomes positive) cannot reasonably predict the stronglocalization process.

The results presented in Figs. 4a-b can be used to testour criterion for strong localization in Eq. (15) againstfully nonlinear numerical solutions. To that aim, wenumerically solved Eqs. (5), together with the constitu-tive law given by Eqs. (16)-(17) (see Appendix for de-tails). Such solutions have already been presented inFig. 2, where a sharp drop in the minimal cross-sectional

area amin has been observed to occur at a strain ε∗` (de-fined as the strain in which the slope is largest). Weare interested in the dependence of the strong localiza-tion strain ε∗` on the history of the glass or its initialage, as quantified by χ0, and on the imposed strain-rate ˙ε. Previous work [95] has shown that transient(i.e. far from steady-state) elasto-viscoplastic deforma-tion, which is of interest here, crucially depends on acompetition between a timescale characterizing the ex-ternal driving rate, τext≡ 1/ ˙ε, and the initial plastic re-laxation timescale, quantified by τpl≡τ0 exp(−eZ/kBχ0)(cf. Eq. (16)). Consequently, the dimensionless ratioξ ≡ τext/τpl is expected to properly characterize tran-sient elasto-viscoplastic deformation in this constitutivemodel.

In Fig. 4c we plot ε∗` as a function of ξ for variousapplied strain-rates ˙ε, where ξ is varied by up to 4 or-ders of magnitude by varying χ0 for each ˙ε, and ˙ε spansnearly 3 orders of magnitude. It is observed that as theapplied strain-rate ˙ε is increased, strong localization oc-curs at larger strains ε∗` for a fixed χ0 (i.e. fixed initialstructure and hence density of STZs), as observed experi-mentally for athermal amorphous systems such as bubblerafts [12], bulk metallic glasses [3], and in previous com-putational studies [92, 93]. In addition, as χ0 is increasedfor a fixed applied strain-rate ˙ε— i.e. the initial viscoplas-tic response is stronger — the system can sustain a largerstrain ε∗` prior to strong localization. This trend in factdepends on the strain-softening degree which is deter-mined by the proximity of χ0 to the limiting value χ∞(cf. Eq. (17)); indeed, for the material parameters usedhere, when χ0 is sufficiently close to χ∞, ε∗` starts todecrease with increasing χ0 for the lowest applied strain-rate ˙ε, see lowest curve in Fig. 4c.

The major question we are now ready to address iswhether the strong localization strain ε∗` measured infully nonlinear extensional deformation solutions over awide range of physical conditions, as presented in Fig. 4c,is quantitatively predicted by the theoretical predictionin Eq. (15). The latter is solved for t` based on thespatially-homogeneous solution used in Figs. 4a-b, lead-ing to ε`=ε(t`) once κ∼O(10−1) is specified. In Fig. 4dwe plot the numerically measured ε∗` against the pre-dicted strong localization strain ε` using κ=0.15. It is ob-served that the theory quantitatively predicts the stronglocalization strain over a broad range of parameters us-ing a single number κ, lending significant support to theproposed theoretical framework. In the next section weperform a similar analysis for a very different constitutiverelation describing crystalline/polycrystalline materials.

VII. APPLICATION II: CRYSTAL PLASTICITYAND THE KOCKS-MECKING MODEL

Encouraged by the success of the main theoreticalresult in Eq. (15) to quantitatively predict the stronglocalization strain in a constitutive model of amor-

10

FIG. 4. (a) The normalized stress σ/σy vs. uniaxial strain ε corresponding to the spatially-homogeneous solution of the STZmodel (see text for details) for 4 imposed strain-rates, spanning nearly 3 orders in magnitude (see legend). (b) The eigenvalueλ+ of Eq. (11) for the imposed strain-rates specified in panel (a). (c) The localization strain ε∗` as obtained from full numericalsolutions as a function of ξ, which quantifies the competition between the external loading rate and the initial plastic relaxationcontrolled by the glass structure/age (see text for details). (d) The numerically-obtained localization strain ε∗` vs. the predictedlocalization strain ε` obtained from solutions of Eq. (15) with κ=0.15.

phous/glassy materials, we set out to further test thedegree of generality of the prediction. To that aim,we repeat the analysis of the previous section for avery different constitutive model. In particular, we con-sider the Kocks-Mecking model which has been ratherwidely used to describe plastic deformation in crys-talline/polycrystalline materials [42–45]. We choose tostudy this phenomenological model both because of itsimpact on the metallurgical literature and because it isone of the few models which explicitly invoke an internal-state field, an essential element in our theoretical frame-work.

The Kocks-Mecking constitutive model takes the form

εpl(σ, ρ) = τ−10

(σ − σ0

σT (ρ)

)m, (18a)

σT (ρ) = α0M bG√ρ , (18b)

which is somewhat analogous to Eqs. (16) of the STZmodel of amorphous plasticity, yet it is very different interms of the underlying physics. As the dominant micro-scopic mechanism of plastic deformation in crystalline

materials is dislocation slip, the fundamental internal-state field in Eqs. (18) is the areal density of disloca-tions ρ (not to be confused with the mass density usedearlier). It affects the plastic strain-rate in Eq. (18a)through the Taylor-like stress σT (ρ) in Eq. (18b), whichquantifies the increase in the resistance to plastic de-formation with increasing dislocation density ρ [97, 98].The physical idea is that dislocations remain locked-inthe material and hamper the subsequent motion of otherdislocations, which is believed to be at the heart of theimportant phenomenon of strain-hardening [45]. σT (ρ)in Eq. (18b), which scales with

√ρ, is proportional to a

constant α0 that depends on dislocation-dislocation in-teractions, the Taylor factor M , the magnitude of theBurger’s vector b and the shear modulus G (these pa-rameters depend on the temperature T , but we do not in-clude this dependence explicitly). The plastic strain-rateεpl(σ, ρ) in Eq. (18a) is inversely proportional to a micro-scopic time τ0 and is driven by the applied stress σ shiftedby a reference flow-stress σ0. Finally, the rate-sensitiverelation between εpl(σ, ρ) and the dimensionless stress(σ − σ0)/σT (ρ) is phenomenologically captured by the

11

FIG. 5. (a) The normalized stress σ/σ0 vs. uniaxial strain ε corresponding to the spatially-homogeneous solution of the Kocks-Mecking model (see text for details) for 4 imposed strain-rates, spanning nearly 3 orders in magnitude (see legend). (b) Theeigenvalue λ+ of Eq. (11) for the imposed strain-rates specified in panel (a). Note that on the scale used, the 4 differentcurves appear to overlap. (c) The localization strain ε∗` as obtained from full numerical solutions as a function of ρ0/ρ∞ fora single strain-rate, ˙ε = 1 s−1 (results for the other strain-rates are not shown as they are practically indistinguishable). (d)The numerically-obtained localization strain ε∗` vs. the predicted localization strain ε` obtained from a solution of Eq. (15) withκ=0.1.

exponent m, consistently with the notation in Sect. IV.The density of dislocations ρ is a dynamical quantity

that evolves with plastic deformation [45]. Hence, tofully define the constitutive model, we need a dynam-ical ρ-evolution equation analogous to Eq. (17). In theframework of the Kocks-Mecking model, it takes the form

ρ = εpl (k1√ρ− k2 ρ) = k2

√ρ εpl(

√ρ∞ −

√ρ) , (19)

where ρ∞ ≡ (k1/k2)2. Here k1√ρ corresponds to the

storage of dislocations and k2 ρ corresponds to dynamicrecovery, assuming the initial value of ρ satisfies ρ0<ρ∞.Similarly to Eq. (17), the evolution of the material’s in-ternal structure is driven by plastic deformation, i.e. it isproportional to εpl in agreement with Eq. (4) and henceit also vanishes when εpl = 0. Note, though, that unlikeEq. (17), ρ is proportional to εpl, not to the plastic dissi-pation power σεpl, which raises some questions about theproper generalization of Eq. (19) to non-unidirectionaland to fully tensorial viscoplastic flows, as discussedin [85].

Equations (18)-(19) have been recently analyzed in

a similar context in [35, 36]. The analysis in [35, 36],however, exclusively focussed on the onset conditions fornecking. Indeed, applying our general expression for thelargest eigenvalue in Eq. (11) to Eqs. (18)-(19), we re-cover the onset criterion derived in [36] for the Kocks-Mecking model (cf. Table 2 in the main text of [36] andEq. (A11) in its Appendix/Supplementary data). Yet,our analysis goes well beyond that of [35, 36], not onlyin providing a general onset criterion that is not specificto a particular constitutive relation, i.e. Eq. (11), butmore importantly, in predicting the emergence of stronglocalization, which is not discussed at all in [35, 36].

To test our prediction for the emergence of strong lo-calization in the Kocks-Mecking model, we repeated theanalysis of the previous section with Eqs. (18)-(19) in-stead of Eqs. (16)-(17). All calculations start at theonset of plasticity (see Appendix for details), and thestrain ε is measured relative to the elastic strain at thispoint. The calculations were performed with the fol-lowing set of parameter: G = 48 GPa, τ0 = 10−6 s,m = 167, σ0 = 13.4 MPa, α0Mb = 3 × 10−10 m,

12

k1 =139× 106 m−1, k2 =5, characteristic of copper pro-cessed by equal channel angular extrusion [36, 99]. Thespatially-homogeneous stress-strain curves for various ap-plied strain-rates ˙ε, shown in Fig. 5a, exhibit a strain-hardening behavior characteristic of crystalline materi-als. That is, the stress in these curves increases mono-tonically with strain, a behavior that is qualitatively dif-ferent from the strain-softening behavior typically exhib-ited by amorphous materials, cf. Fig. 4a. Note that asthe strain-hardening “rate” dσ/dε in the plastic regimeis significantly smaller than the elastic modulus G, theelastic branch in the stress-strain curves appears to benearly vertical in Fig. 5a. Also note that the exhibitedstrain-rate sensitivity is quite small.

As discussed in Sect. IV, strain-hardening materialstypically feature ∂I I < 0 (here I is the dislocation den-sity ρ), which implies that the expression in Eq. (11)may be negative during the elasto-viscoplastic deforma-tion process, in which case the largest eigenvalue is takento be λ+ = 0. As shown in the examples in Fig. 5b, λ+

corresponding to Eq. (11) is indeed negative in the earlystages of the extensional deformation; consequently, wetake the integrand of Eq. (15) to vanish in this range ofstrains. At larger strains, the largest eigenvalue becomespositive and sizable, as shown in Fig. 5b, and a neck-ing instability develops (in this range the integrand ofEq. (15) is used as is).

The strong localization strain ε∗l is shown in Fig. 5c as afunction of the initial density of dislocations ρ0 for a sin-gle strain-rate, ˙ε=1 s−1 (results for the other strain-ratesare not shown as they are practically indistinguishable).Interestingly, and somewhat nonintuitively, ε∗l is a de-creasing function of ρ0 in this model, a result that shouldbe tested against experimental data (we are not aware ofexisting data that quantify the variation of neck evolu-tion with the initial density of dislocations). In Fig. 5d weplot the numerically measured ε∗` against the predictedstrong localization strain ε` using κ=0.1. It is observedthat yet again the theory reasonably well predicts thestrong localization strain using a single number κ. Theseresults, together with the results of the previous section,seriously support the theoretical framework developed inthis paper and open the way to better understandingnecking instabilities in elasto-viscoplastic materials.

VIII. CONCLUDING REMARKS

We have presented a rather general analysis of neck-ing instabilities in rate-dependent elasto-viscoplastic ma-terials in the long-wavelength approximation, within ageneric constitutive framework that accounts for thestructural evolution of the material during plastic de-formation through an internal-state field. Developing anapproximated WKB-like time-dependent linear stabilityanalysis, we derived in Eq. (11) an analytical expressionfor the largest time-dependent eigenvalue in the stabil-ity problem, which reveals the various destabilizing and

stabilizing physical processes involved in necking insta-bilities. The onset of necking criterion, associated withthe strain/time in which the largest eigenvalue of Eq. (11)becomes positive, is discussed in relation to onset criteriaavailable in the literature.

Our theoretical framework, however, allows to go sig-nificantly beyond the onset condition in order to derivea criterion for the emergence of strong localization, whenthe cross-sectional area reduction associated with neckingbecomes macroscopically appreciable. The analytical cri-terion, presented in Eq. (15), is tested against fully non-linear numerical solutions of two widely different elasto-viscoplastic constitutive relations involving an internal-state field, one for strain-softening amorphous/glassy ma-terials (using the Shear-Transformation-Zone model) andthe other for strain-hardening crystalline/polycrystallinematerials (using the Kocks-Mecking model). Both anal-yses demonstrate that the criterion in Eq. (15) quantita-tively predicts the onset of strong localization, exhibitingweak (logarithmic) dependence on the typical (relative)magnitude of initial perturbations ζ and a dimensionlessparameter κ of O(10−1). Consequently, we believe thatour theoretical results can be used to predict necking in-stabilities in a broad range of materials and constitutiverelations.

Our results open up the way for several future researchdirections, some of which are mentioned here. First, ourgeneral results should be applied to more sophisticatedconstitutive relations, for example those that are capableof quantitatively describing the elsato-viscoplastic flowsof glasses near their glass temperature. This is highlyrelevant, for example, for many bulk metallic glass pro-cessing methods, such blow molding or stretch rolling,where necking instabilities play a critical role. Second,our predictions should be tested against experimentaldata for various materials, where more refined measure-ments of the evolution of the material microstructureduring plastic deformation should be performed in or-der to guide the choice of the relevant of internal-statefields and their dynamics. Finally, the range of valid-ity and limitations of the long-wavelength (lubrication orslender-bar) approximation should be elucidated. To thisaim, one needs to consider two-dimensional and probablyalso three-dimensional extensions of the present analysis,which most likely entail large-scale numerical calculationsinvolving advanced computational techniques. Such cal-culations should also resolve the role of boundary effects,e.g. those associated with clamped boundary conditions.Obviously, these issues are not only related to the rangeof validity of the presented theory, but also to quantita-tive comparisons to experiments.

ACKNOWLEDGMENTS

We acknowledge support from the Richard F. Good-man Yale/Weizmann Exchange Program, the WilliamZ. and Eda Bess Novick Young Scientist Fund and the

13

Harold Perlman Family. We are grateful to Y. Bar-Sinaifor his valuable help in developing the numerical tools, toE.A. Brener for fruitful discussions and a critical readingof the manuscript, and to C.H. Rycroft and J. Schroersfor their comments on the final version of the manuscript.

Appendix: Fully nonlinear numerical solutions

During extensional deformation, the material changesits shape, mathematically implying a time-dependent,evolving domain. In order to obtain fully nonlinear solu-tion of the problem at hand, we performed a coordinatetransformation from the Eulerian (spatial) frame of ref-erence x to the Lagrangian (material) frame of referenceX. In the latter frame of reference, the domain is fixed(i.e. time-independent), but the equations contain addi-tional nonlinearities. In particular, Eqs. (5) take the form

0 = J−1∂X (AΣ) ≡ J−1∂XF , (A.1a)

∂tA = −A∂tJJ

, (A.1b)

∂tF = GA

(∂tJ

J− Epl

)+ F

∂tA

A, (A.1c)

∂tI = Epl G(

Σ, I). (A.1d)

In Eqs. (A.1), J(X, t)≡∂x/∂X is the Jacobian of thetransformation (the scalar counterpart of the deforma-

tion gradient tensor), and A(X, t) ,Σ(X, t), I(X, t) andV (X, t) are the Lagrangian counterparts of the Eulerianfields a(x, t) , σ(x, t), I(x, t) and v(x, t) in Eqs. (5). Whilethe latter are defined over a time-dependent spatial do-

main x ∈[−L(t)

2 , L(t)2

], the former are defined over a

fixed (time-independent) spatial domain X ∈[−L0

2 ,L0

2

].

Epl(X, t) is the Lagrangian counterpart of the Eule-

rian plastic strain-rate εpl(x, t), and G(Σ(X, t), I(X, t))is the Lagrangian counterpart of the Eulerian functiong(σ(x, t), I(x, t)) in Eq. (5d). The convective derivative∂t + v∂x in the Eulerian formulation is simply replacedby a partial time derivative ∂t in the Lagrangian for-mulation. Finally, note that ∂XV = ∂tJ , which implies∂xv→∂tJ/J .

Equation (A.1a) is simply solved by a space-independent force, F (t)=A(X, t) Σ(X, t). The remainingequations, Eqs. (A.1b)-(A.1d), are solved numerically.

The fields A(X, t), J(X, t) and I(X, t) are discretizedin space using the Method of Lines [100]. At each pointin time, each field is represented by N discrete valuesdefined at N spatial points and F is characterized by asingle space-independent value. The total 3N + 1 de-grees of freedom is constrained by only 3N equations,where each of Eqs. (A.1b)-(A.1d) contributes N equa-tions. The additional equation emerges from the bound-ary condition. In particular, as we impose the velocityat the bar’s edges, v(x=L(t)/2, t) =−v(x=−L(t)/2, t),in the Eulerian formulation we have

〈∂xv〉 =1

L(t)

∫ L(t)2

−L(t)2

∂xv(x, t) dx =2v(x=L(t)/2, t)

L(t)= ˙ε .

(A.2)Transforming Eq. (A.2) to the Lagrangian frame of ref-erence, we obtain

1

L0

∫ L02

−L02

∂tJ(X, t) dX = ˙ε e˙εt . (A.3)

This integral relation, which adds spatial coupling to theordinary time differential Eqs. (A.1b)-(A.1d), is the ex-tra equation we needed. Equation (A.3) was discretizedusing a simple sum in the numerical implementation.

The spatially discretized system of 3N + 1 equationswas integrated in time using a standard differential-algebraic equation solver in Mathematica [101] with thefollowing initial conditions

A (X, 0) = 1 ,

F (0) = A(X, 0) Σ(X, 0) ,

I (X, 0) = I0 (1 + ζ cos (2πX/L0)) ,

(A.4)

where Σ(X, 0) should be specified (see below), ζ is the

relative amplitude of the perturbation and I0 is a con-stant initial background level of the internal-state field.The time t = 0 is set to the onset of plastic deforma-tion, which is also used to define the Lagrangian frameof reference, J (X, 0)=1 (i.e. the small initial elastic de-formation is neglected). Consequently, the initial forceF (t=0) is set by Σ(X, 0)=σy for the STZ model and by

Σ(X, 0) that satisfies ˙ε= Epl (Σ(X, 0),R0), where R0 isthe Lagrangian counterpart of initial dislocation densityρ0, for the Kocks-Mecking model. In all presented re-sults we used N=1000 (higher values of N were used toverify the appropriate numerical convergence associatedwith this choice).

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