PHYSICAL REVIEW E 84, 021502 (2011)

Model of plasticity of amorphous materials

V. I. Marchenko and Chaouqi Misbah*

P.L. Kapitza Institute for Physical Problems, RAS, 119334, Kosugina 2, Moscow, RussiaMoscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow, Russia

Universite Grenoble 1/CNRS, Laboratoire Interdisciplinaire de Physique/UMR5588, Grenoble F-38041, France(Received 15 November 2010; revised manuscript received 21 July 2011; published 25 August 2011)

Starting from a classical Kroener-Rieder kinematic picture for plasticity, we derive a set of dynamical equationsdescribing plastic flow in a Lagrangian formulation. Our derivation is a natural and straightforward extensionof simple fluids, elastic, and viscous solids theories. These equations contain the Maxwell model as a speciallimit. This paper is inspired by the particularly important work of Langer and coworkers. We shall show thatour equations bear some resemblance with the shear-transformation zones model developed by Langer andcoworkers. We shall point out some important differences. We discuss some results of plasticity, which can bedescribed by the present model. We exploit the model equations for the simple examples: straining of a slaband a rod. We find that necking manifests always itself (not as a result of instability), except if the very specialconstant-velocity stretching process is imposed.

DOI: 10.1103/PhysRevE.84.021502 PACS number(s): 46.35.+z, 62.20.F−, 46.05.+b

I. INTRODUCTION

Plastic materials exhibit several features that are not presentin the usual liquids or solids. Their dynamics consist ina nontrivial mixture of liquid-like and solid-like behaviors.Understanding plasticity in metal industry and, in general,in technology, is of a paramount importance. Nonetheless,to date, no universal dynamical equations describing plasticmaterials, such as Navier-Stokes equations for fluids andLame equations for elastic solids, are available. Under strain,plastic material may exhibit elastic-like behaviors, yieldstress, flowing behaviors, nonlinear engineering strain-stressrelation, and so on [1]. A major goal in material science isthe description of these phenomena in terms of dynamicalevolution equations for relevant variables, namely the velocity,stress, and the analogues of strain.

There has been important contributions to the theoryof plasticity, especially for crystalline materials in termsof dislocations [2–4]. However, there is no need to evokedislocations (if ever this notion has a meaning) for amorphousmaterials, and thus the question arises of how a correspondingtheory can be built at the continuum level. This question hasknown recently an upsurge of interest [5]. An essential issuewhen addressing the question of plasticity is the distinctionbetween crystalline solids and amorphous materials. Elasti-cally deformed monocrystals are in a metastable state. Theirplastic flow takes place only upon creation of dislocationsand is thus a nonlinear process. Conversely, plastic flow ofamorphous materials (including polycrystals) should occur,in principle, linearly with respect to the applied stress. Inmonocrystals, an additional field, namely dislocation density,is introduced which couples to the elastic as well as to plasticdistortions [3,4]. It is proposed here that one can derive a plasticcontinuum theory for amorphous materials, without evokingdislocation density or an internal variable that is distinct fromthe variables describing usual kinematics of plasticity. The

*cmisbah@ujf-grenoble.fr

evolution equations can be written in a closed form in termsof the elastic and plastic distortions only.

This work is inspired by the series of important publicationsdue to Langer and coworkers [5–10]. It will be interestingto see that the set of equations derived here may, at firstglance, bear resemblance with the shear-transformation zones(STZ) model developed by Langer and coworkers [5]. A closeinspection shows that there are also some basic differences. Inthe present work, the plastic information will be automaticallyencoded in the plastic deformation tensor, as that defined inRef. [2]. Actually, the concept of distortion [2] used heredates back to Kroener and Rieder in 1956 [11] and willconstitute our basic definition of the plastic flow variable. Thepresent contribution aims to introduce the simplest dynamicalapproach by following Kroener and Rieder [11]. Our theorycan be viewed as a complementary view to the derivation ofthe STZ model.

II. KINEMATICS OF PLASTICITY

If uk denotes the kth component of the displacement field,then the total distortion tensor reads ∂iuk . If plastic flow isinvolved, the total distortion tensor ∂iuk is taken as a sum ofthe plastic flow contribution, wpl

ik, and the elastic part, wik (seeRef. [2]):

∂iuk = wplik + wik. (1)

The symmetrical part of the elastic distortion defines the straintensor:

uik = 12 (wik + wki). (2)

Note that Eq. (2) is different from (∂iuk + ∂kui)/2, which isvalid only in absence of plasticity. The constraint w

plii = 0 is

usually supposed. This is valid [2] as long as plastic flowoccurs without destroying continuity of the material. Then,from Eqs. (1) and (2) it follows that uii = ∂iui .

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V. I. MARCHENKO AND CHAOUQI MISBAH PHYSICAL REVIEW E 84, 021502 (2011)

III. DYNAMICS OF PLASTICITY

Usual elasticity can be presented in a Lagrangian formula-tion by introducing an energy and a dissipative function. Theelastic energy of a solid is given by

λ

2u2

ll + μu2ik, (3)

where λ,μ are Lame coefficients, and the kinetic energy isρv2/2, where ρ is the density of the material. The dissipationfunction reads for a viscous solid

ηs

(uik − 1

3δikull

)2

+ ζ

2u2

ll . (4)

In the presence of a plastic flow, one must introduce a newadditional dissipative part related to plasticity. The systemis described by three independent tensors, namely uik andw

pl±ik , which are the symmetric (+) and antisymmetric (−)

parts of the plastic distortion tensor, respectively. From thebasic kinematic relation Eq. (1) and the strain tensor definitionEq. (2) it follows that

wpl+ik = 1

2 (∂iuk + ∂kui) − uik. (5)

We expect the dissipation to consist of a quadratic form ofthese quantities, which we write as

2αuikwpl+ik + η

(w

pl+ik

)2 + γ(w

pl−ik

)2. (6)

This is the dissipation corresponding to plastic flow. It shouldbe mentioned here that we consider linear approximationfor plastic flow dynamics as the first step. In order to takeinto account nonlinearity, one should right the so-calledobjective time derivative (we have used ordinary derivatives)for tensors. One alterative in order to confer an objective formto expressions (4),(6) is to replace the time derivative of tensorsby the so-called corotational derivative [12], as is done inRef. [5].

The total dissipation must be positive in order to guaranteestability. This is fulfilled if

ηs > 0, ζ > 0, ηηs > α2, γ > 0. (7)

Note that there is only one dilatational viscosity constant ζ. Allother constants are related with shear motions. As we will seebelow in the liquid limit, the constant η is the usual viscosity.

The strategy now consists in performing variations of thetotal Lagrangian with respect to the independent variables.Variation with respect to δu, with δw

plik = 0, yields, upon using

2δuik = ∂iδuk + ∂kδui, the momentum conservation law,

ρui = ∂kσik, (8)

with the stress tensor σik consisting of the sum of the usualelastic part as well as the dissipative part with the usual solidviscosity terms and an additional plastic term,

σik = λullδik + 2μuik

+ 2ηsuik +(

ζ − 2

3ηs

)δikull + 2αw

pl+ik . (9)

Variation with respect to δwpl+ik , with δu = 0, and δw

pl−ik = 0

(in that case δuik = −δwpl+ik ) provides us with

σik = α(2uik − 2

3δikull

) + 2ηwpl+ik , (10)

here σik is the traceless part of σik .Finally, variation with respect to δw

pl−ik , with δu = 0, and

δwpl+ik = 0 (in that case δuik = 0) leads to w

pl−ik = 0 (a direct

consequence of the absence of dissipation for rigid rotation;note also that energy does not depend on that mode).

Note finally that in contrast to crystal plasticity where wpl+ik

is nonzero only at the dislocation line [2], this notion loses itsmeaning for amorphous materials.

IV. SIMILARITY AND DIFFERENCES WITHTHE STZ THEORY

Langer and coworkers have proposed a theory for plasticity[5]. We would like to briefly recall the main steps of thattheory and will highlight the similarities and differences withour theory. The starting point for the definition of the plasticcurrent in Ref. [5] is similar to ours [Eq. (1)]. Actually, inRef. [5] the velocity field was used instead of the displacementfield (as done here). Their Eq. (3) can be formally obtainedfrom our Eq. (1) [upon differentiation of Eq. (1) with respectto time] so that we obtain

∂kvi + ∂ivk = 2uik − 2jplik , (11)

where jplik ≡ −w

pl+ik is the plastic current. This equation (see

also Ref. [2]), apart from the (conventional) minus sign in frontof j

plik , is similar in form to Eq. (3) of Ref. [5], written as

∂kvi + ∂ivk = 2DDt

[σij

2μ− p

2Kδij

]+ 2D

plasij , (12)

where K and μ are the compressibility and the shear modulus(note that a 2D geometry is assumed in Ref. [5], with planestrain condition), and σij (denoted by sij in Ref. [5]) is thedeviatoric (traceless) part of the stress tensor. The term D/Dt

denotes the Jaumann derivative, which we take, for simplicity,to be just the material derivative (as explained above). Theirplastic current D

plasij is the analog of our −j

plij , while the first

term in Eq. (12) is to be compared to our uik . The two modelswould be formally equivalent at this stage if and only if weimpose the relation (corresponding to 2D with plane straincondition)

uik = σik

2μ− p

2K(13)

and use it in our Eq. (11), where p = −σll/2 is the pressure.The above relation supposed in Ref. [5] corresponds toa Hooke’s relation. Contrariwise, we do not adopt thishypothesis, rather we define the dissipation in the Lagrangianfrom which we extract systematically the dynamical equations(precisely as one would do for the derivation of the NavierStokes equations for a simple fluid). In our model, both elasticand plastic contributions enter the total distortion tensor ∂iuk

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MODEL OF PLASTICITY OF AMORPHOUS MATERIALS PHYSICAL REVIEW E 84, 021502 (2011)

[Eq. (9)]. That equation can also be rewritten as

uik = σik

2μ− p

3K

− ηs

μuik −

(ζ − 2

3ηs)

2μδikull − α

μw

pl+ik . (14)

One sees in this equation that, on one hand, we have the STZterm in Eq. (13), which corresponds to the first line. Note thatwe have 3K in the second term due to the fact that we considerthe three-dimensional elasticity, the 2D case is a special limitin our theory, and where the 3K has to be substituted by 2K .On the other hand, we have additional contributions containingviscosity of solid and plastic deformation. In other words, ourstress tensor shows that it is a measure of both elastic andplastic deformations, as one expects. Thus, while the basicformulation bears some resemblance with the STZ theory, thismarks a first difference.

The second step of the STZ theory consists in specifyingthe plastic current D

plasij (which corresponds in our case to

specifying jplik = −w

pl+ik , as discussed above). In our case, the

plastic current is given as a function of the the other variablesdefined in the Lagrangian formulation and can be rewrittenfrom Eq. (10) as

jplik = − 1

2η(σik − 2α ˙uik), (15)

and this closes our set of equations (tilde refers to the tracelesspart).

In the STZ theory, the plastic current reads

Dplasik = ε0(σik − �ik) ≡ ε0qik, (16)

with ε0 a material-specific constant, and �ij is an internalstate variable that specifies the degree of alignment of theSTZ. By a formal comparison of our Eq. (15) and the STZEq. (16), one sees that we have the following correspondence�ik ↔ 2α ˙uik . The analog of an internal variable in our case isjust the flow rate of the traceless part of uik . This quantity hasthe following properties: it is zero for elastic solids and is equalto the deformation rate for a viscous solid, as it should clearlybe. In the general case (elasto-visco-plastic flow), uik carriesinformation both on the viscous and plastic deformations.An important point is that our internal-like variable obeysequations that naturally follow from the same Lagrangian[Eqs. (8), (9), and (10)], constituting a closed set of equations.

In a marked difference with the present theory, the STZtheory requires us to specify an evolution equation for �ij . Tothis end, the STZ model writes

�ik = qik − 12 |qlmslm|�ik. (17)

The above equation was motivated [5] from the rate equationgoverning the population of the STZ [13]. It is not an easytask to connect this equation to the evolution equation ofour internal-like variable ˙uik . Our internal-like variable ˙uik

is linked to the other kinematic variable wpl+ik , and they are

both related to the stress tensor.In conclusion of this section, we may say that the STZ

theory shares some similarities with our theory but at thesame time there are some important differences. The major

difference lies in the fact that the STZ introduces an internalvariable motivated by some physical model from the rate equa-tion governing the population of the STZ [13]. Furthermore,the STZ theory postulates an evolution equation for the internalvariable [Eq. (17)]. Our theory is based on first defining thekinematics of plasticity (in a very similar way as that used inclassical plasticity of crystals) and then write the dissipationfunction to be a definite positive quantity, from which allequations follow at once.

V. ALTERNATIVE REWRITING OF OURSET OF EQUATIONS

It is possible to express the plastic distortion tensor in termsof other quantities. From Eqs. (9) and (10) we may express σik

in terms uik and its time derivative. It is convenient to splitthe stress tensor σik into a traceless and pressure-like term(actually, the trace of σik):

σik = δik

3σll + σik. (18)

The trace has a usual elastic (including the dissipative part)form:

σll = (3λ + 2μ)ull + 3ζ ull . (19)

The traceless parts of the stress tensor are connected with eachother and with spatial gradients of the velocity by the followingtwo relations

(η − α)σik = 2ημuik + 2(ηηs − α2) ˙uik, (20)

σik + 2(η − α) ˙uik = η

(∂ivk + ∂kvi − 2δik

3∂lvl

). (21)

The first relation is obtained by expressing ωplik from Eq. (10)

and inserting the resulting relation into Eq. (9). The secondone follows from Eq. (10) by using Eq. (5). The set ofEqs. (8), (18)–(21) defines space-time evolution of displace-ment vector u, strain uik , and the stress tensor σik . Thisconstitutes a complete set of equations for the three (vectorialand tensorial) quantities u,uik , and σik (we could, of course,alternatively use other quantities like w

pl+ik ). Note that Eq. (21)

has some similarity with the Maxwell model, used to describeplasticity with a yield stress in some models [14]. There isan important difference, however. Instead of the σik on thel.h.s., we have uik. Again, this consistently follows from theLagrangian formulation.

The Maxwell model of liquids with high viscosity canbe obtained from our equations only if one consider theincompressible limit and set α = ηs = ζ = 0. This leads,from Eq. (20), to σik = 2μuik. Equation (21) reduces to thewell-known Maxwell form

σik + μ

ησik = μ(∂iuk + ∂kui). (22)

Now we would like to exemplify our model by consideringseveral examples, such as homogeneous evolution, wavesspectra, etc.

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VI. HOMOGENEOUS PLASTIC RELAXATION

Consider a homogeneous solution of the system ofEqs. (8), (18)–(21). By setting spatial derivatives to zero, oneobtains w

plik = −uik, and

˙uik = − uik

τ, (23)

where τ is the plastic relaxation time

τ = η + ηs − 2α

μ.

Note that τ > 0 according to stability conditions Eq. (7).Equation (23) directly shows the presence of additional hy-

drodynamics modes (namely five components of the tracelesstensor uik) in the Kroener-Rieder kinematics of plasticity.

VII. SPECTRUM OF WAVES

In addition to pure plastic tensorial relaxing modes[Eq. (23)] there exist usual displacement modesu ∝ exp(−iωt + iq·r) in isotropic material. We consider inthis section an unbounded three-dimensional material. In thelow frequency limit, we obtain a liquid-type spectrum, namelylongitudinal sound waves with spectrum

ω =√

K

ρq,

where K = λ + 2μ/3 is the compressibility, and shear viscouswaves,

ω = −iη

ρq2.

Solid-type behavior is captured at high frequencies ωτ � 1,provided that η � ηs,α,ζ . There exists a range of frequenciessuch that

μ

η� ω � μ

ηs,μ

α,μ

ζ,

where we find both solid-type spectrum of longitudinal waves,

ω =√

λ + 2μ

ρq,

and transversal waves,

ω =√

μ

ρq.

If all viscosities are of the same order of magnitude,η ∼ ηs,α,ζ , then there exists no weakly decaying waves atfinite frequencies, ω > 1/τ .

VIII. PLASTIC FLOWS

Let us present a few examples where we could obtain anexact solution of the plastic dynamics.

A. A semi-infinite plate with an oscillating displacementfield at the free surface

We consider the case of a semi-infinite material in the y

direction with a planar-free surface. In the x and z directions,

the material is supposed to be infinite (actually, by infinitewe mean that we are interested in modes with wavelengthsmuch smaller than the system size). We consider a planestrain condition. The free surface is located at y = 0, andwe assume that that plane is subject to an oscillation of thedisplacement field along the x direction. More precisely, theonly nonzero component of the imposed displacement is the x

component and it is given by ux(t,y = 0) = u cos ωt. In thiscase, it is easy to check that the nonzero components of thefields in the material are ux,uxy, wxy, wyx, w

pl+xy , and σxy.

We will only determine the expressions of uxy and ux . Indeed,σxy is related to uxy by Eq. (20), and wxy, wyx can then beextracted from Eq. (1), while w

pl+xy follows from Eq. (10). Their

precise expression is not of special interest, so that we onlyprovide the expression of ux, uxy (the subsequent calculationis straightforward). In order to determine ux, uxy, we useEqs. (8), (20), (21) and easily obtain

ρux = ∂yσxy = ∂y

2ημuxy + 2(ηηs − α2)uxy

η − α, (24)

μuxy + (η − 2α + ηs)uxy = η − α

2∂yvx. (25)

It is then found that each field is a linear combination of thecomplex modes ∝ exp(−iωt + iκy), where κ is defined by

κ2 = (κ ′ + iκ ′′)2 = iωρ

η

[1 − iω(η − α)2

ημ − iω(ηηs − α2)

]. (26)

For example, the displacement field isux = u cos(ωt − κ ′y) exp(−κ ′′y). The low-frequencylimit recovers a known Stokes result for a shear viscous modein liquids (see Sec. 24 in Ref. [15]). Elastic solid behavior(an emission of shear sound) corresponds to the limit of highplastic viscosity, η → ∞.

IX. NECKING INSTABILITY IN THELONG-TIME REGIME

We would like to point out some results that can be capturedanalytically in some special limit. The long-time behavior of aslab under tension is expected to be dominated by plastic flow.Ultimately, the plastic flow should look like a hydrodynamicalflow. Let us concentrate on this limit.

Consider first a plate of a plastic material with a free surface(Fig. 1). The plate has a finite extent L(t) along the x direction,has a thickness in the vertical y direction, and is assumed tobe infinite along the z direction, so that the two-dimensionalassumption makes sense. This is a similar geometry to thattreated in Ref. [5]. The 2D assumption here corresponds to

FIG. 1. (Color online) A sketch of the geometry under consider-ation.

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MODEL OF PLASTICITY OF AMORPHOUS MATERIALS PHYSICAL REVIEW E 84, 021502 (2011)

the plane strain condition, where uxz = uyz = uzz = 0. Theplate is stretched along the x direction. For a flat geometry,we have obtained an exact solution with the plate thicknessh(t) that depends only on the t variable. This type of solutionexists only in the case where the stretching occurs at a givenconstant velocity.

Let us first motivate the solution on the basis of symmetries.Because of the axial symmetry with respect to the y axis atx = 0, vx must be zero on that line. For constant h, there isa simple solution that fulfills that symmetry, vx = cx, wherec(t) is for the moment an arbitrary function of time. Fromincompressibility condition we have

vy = −cy + g(x,t),

where g is a priori an arbitrary function of x and t. Symmetrywith respect the middle line, y = 0, enforces g = 0.

We straightforwardly obtain from the Navier-Stokes equa-tion the pressure field

p = −ρ(c + c2)x2

2+ ρ(c − c2)

y2

2+ f (t),

where f (t) is a function of time to be determined below. At thefree surface, the normal component of the stress (the tangentialvanishes automatically) must vanish. This is easily computedfrom the above result by using the definition

σyy = −p + 2η∂yvy = −p − 2ηc.

Imposing σyy = 0 on the free surfaces at y = ±h, at any x,

we obtain c + c2 = 0. This provides us with c = (t − t0)−1,

where t0 is a constant of integration. It is convenient to measurethe time from the moment t0, so we will set t0 = 0. For thelength of the strip L, one has L = cL, so that L = st, withconstant velocity s. Due to mass conservation, one obtains thath = γ /t, where the total volume is defined as γ s. The solutioncorresponds to stretching plastic flow for t > 0 and s,γ > 0,

while for t < 0 and s,γ < 0 it represents a contracting flow.Reporting this solution into the above-mentioned boundary

conditions fixes the function:

f (t) = ργ 2

t4− 2η

t.

The xx component of the stress tensor is

σxx = −p + 2η∂xvx = −p + 2ηc.

The total force of stretching is defined as

F2d ≡∫ h

−h

dyσxx = −4ργ 2h

3t4+ 8ηh

t. (27)

The first part is inertial, while the second one is viscous. Theviscous part dominates at long time such that t � γ 2/3ν−1/3,

where ν = η/ρ is the kinematic viscosity.It is a simple matter to extend the calculation to a cylindrical

3D geometry (stretching of a rod) following precisely the sameline. We only give the results: the velocity components are

vz = z

t, vr = − r

2t.

The pressure reads

p = 3ρ

8t2

(γ 2

t− r2

)− η

t.

We obtain for the length and the radius L = st,a = γ /√

t .

The volume is given by πa2L = πsγ 2. The total axial forcein the rod stretching problem is

F3d ≡ 2π

∫ a

0rdrσzz = − 3

16

ργ 2

t3S + 3

η

tS, (28)

where S = πa2 is a rod cross-section area.The above solution exists only for a constant velocity

stretching. The question thus naturally arises of what happensif an other process is imposed. This is what we wouldlike to investigate now. Following Ref. [5], if the lateralboundary of the plate moves at a predetermined strain rateL/L = � = const., then our result shows that a homogeneousthinning of the strip is not possible. Thus, a modulated stripprevails. This is a precursor of the necking problem. Thus,necking appears here as a natural phenomenon due to materialflowing [16] whenever the stretching is not performed at aconstant speed. The necking is not related with an instability[5] but rather the fact that a homogeneous thinning does notexist [except if L(t) ∝ t] in plastic dynamics.

Let us investigate the stability of the homogeneous so-lutions. We consider deviations with large wavelengths ascompared with the layer thickness. By analogy, with the theoryof shallow water (see Ref. [15], Sec. 108), we derive effectivehydrodynamic equations in terms of the thickness h, andvelocity v along x. Mass conservation yields

h + ∂x(hv) = 0. (29)

Note that when the viscous term dominates, one can rewritethe total force Eq. (27) as follows: F2d = −8η∂th. In a generalcase of inhomogeneous shape, one should have

F2d = −8η(h + v∂xh).

Consequently, the momentum conservation law in this purelyviscous limit has the form

h(v + v∂xv) = −8ν∂x(h + v∂xh). (30)

From Eqs. (29) and (30), one obtains upon linearizationabout the 2D homogeneous solution

tδh + x∂xδh + γ ∂xδv + δh = 0,

γ (tδv + x∂xδv + δv) = −8νt∂x(tδh + x∂xδh).

Introducing the new coordinate x = x/t one arrives at:

t2δh + γ δv′ + tδh = 0,

γ (tδv + δv) = −8νtδh′.

Taking the Fourier transform ∝ exp(iqx), we find for a givenq mode of one of the fields, say δh :

t2δh + 4(t + τq)δh + 2δh = 0,

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V. I. MARCHENKO AND CHAOUQI MISBAH PHYSICAL REVIEW E 84, 021502 (2011)

here, τq = 2νq2. This equation has a first integral

t2δh + 2tδh + 4τqδh = C,

where C is a constant. One obtains finally

δh = C

t2exp

(4τq

t

) ∫ t

t∗exp

(−4τq

t

)dt,

where t∗ is a second constant of integration. At large time,t � τq, the deviation amplitude is

δh = C

t

(1 − 4

τq

tln

t

et∗

).

It decays mostly as t−1 like the strip thickness h. Consequently,one may say that the solution is marginally stable for largewavelength fluctuations.

The 3D problem of the rod stretching with small inho-mogeneity can be formulated in the same manner [Eqs. (29)and (30)]. The results are identical to the 2D ones, if one makesthe substitutions h → S and 8ν → 3ν.

X. NUMERICAL RESULTS

We have solved Eqs.(29) and (30) numerically in the fullynonlinear regime. We first make the following change ofvariable: X = x/L(t) (in order to map the time-dependentinterval L into [0,1], where L(t) is the actual length of thebar). We denote in the new variable the thickness by f (X,t).It follows that Eqs. (29) and (30) take the form

Lf − XL∂Xf + ∂X(f v) = 0 (31)

and

f L2∂tv − LLXf ∂Xv + Lf v∂Xv

− ∂Xf ∂Xv − f ∂XXv = 0. (32)

The boundary conditions are

v(X = 1,t) = L, (33)

expressing the fact that the right end of the bar is pulled atvelocity L=, while the left end is fixed (zero velocity). Thetwo other boundary conditions concern the spatial derivativesof the thickness f at the two ends. We assume that at thetwo ends the bar is horizontal (zero slope, in order avoid anyspurious cusp at the two end):

∂Xf (X = ±1,t) = 0. (34)

Note that the thickness itself at the two ends is free to evolve.We then have to specify the imposed stretching law L(t).

Several cases have been considered.(i) Example 1: Linear stretching lawWe have considered the case where L(t) increases linearly

with time. For definiteness, we have set L(t) = 0.1 + 0.1t .We have considered several initial conditions. First, we havetaken f (X,t = 0) = 0.1 + 0.01 sin(πX/2) and v(X,t = 0) =L(t = 0)X. The set of the two partial differential equationswith the above boundary and initial conditions has beensolved by using the software Maple 10, code “pdsolve” (option“numeric”). The initial sinusoidal height decayed in the courseof time until f (X,t) becomes a constant. We have repeated

the simulation with several initial amplitudes in front ofthe “sin” function. We have also considered initial heightsexhibiting several oscillations; for example, f (X,t = 0) =0.1 + 0.01 sin(5πX). We have reached the same final state:a straight bar (constant thickness).

(i) Example 2: Nonlinear stretching lawWe have examined the behavior of the numerical solution

with a nonlinear dependence in the stretching law. Thefirst considered law is an exponential one, L(t) = e0.1t .We have considered the same initial conditions as in theabove example. After some transients, the height showeda necking instability, as shown in Fig. 1 (which is not aschematic view but a solution of the problem). We havealso considered a constant initial height f = 1, with thesame final solution, a fact that confirms the linear natureof the instability. The necking instability persists also forany nonlinear stretching law that we have examined, forexample, an algebraic law; a typical studied example isL(t) = 1 + t3.

As a summary of the various numerical study, we can statethe following: (i) If one imposes a constant stretching, wefind that the ultimate stage is a homogeneous thickness thatdecreases in time as 1/t, in agreement with our analyticalresults. (ii) Starting from a small perturbation (of sine type),we observe stability of the straight bar. (iii) Most importantly,we have found that if the stretching velocity is not constant,the ultimate stage is a modulated thickness of necking type.In fact, Fig. 1 represents the result of our numerical solution,which exhibits necking in the case of initially flat plate andL/L = const. (same stretching law as in Ref. [5]). This behav-ior is found for various initial conditions, and (nonconstant)stretching laws. Thus, necking seems to be a robust featurethat takes place whenever the stretching is nonconstant.

XI. CONCLUSION

We have presented a simple picture of plastic theory for anamorphous material based on a Lagrangian formulation. Ourwork has been inspired by that of Langer and coworkers. Ourtheory shares some features with the STZ theory, but we havepointed out some differences. We have exemplified the modelby taking simple specific examples. The next step consists ina full numerical solution of the set of equations in variousgeometries and configurations. The model should be extendedto take into account the notion of material indifference, whichconsists in replacing the time derivative of tensors by anobjective derivative of tensors. A natural choice would be tochoose the corotational derivative [5], also known under thename of the Jaumann derivative [1]. It would also be importantto make a systematic comparison with the STZ theory, whichhas been successful in describing several important features[5–10]. We hope to investigate these questions in a futurepaper.

ACKNOWLEDGMENT

We acknowledge CNRS, Univ. J. Fourier, and CNES forfinancial support.

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