Localization in strain-rate-dependent solids

Computer Methods in Applied Mechanics and Engineering 90 (1991) 969-986 North-Holland

Localization in strain-rate-dependent solids Yves M. Leroy and Olivier Chapuis Koninklijke/ Shell Exploratie en Produktie Laboratorium, Postbus 60, 2280 AB, Rijswijk, ZH, The Netherlands Received 28 September 1990 Revised manuscript received 20 February 1991

The localization of the deformation in strain-rate-dependent solids evolves through three distinct phases. Its onset is signalled by the first deviation from a homogeneous mode of deformation, which we propose to pinpoint by means of a linear perturbation analysis. The second stage, marked by a sharp increase of the effective strain rate in 'shear bands' of decreasing thickness, is analysed with a special finite element scheme that uses embedded shear-band modes of deformation. The third phase is entered with the long-term evolution of the localization and the question that arises here is whether or not stationary shear bands of finite thickness are to be expected as theoretically admissible solutions.

1. Introduction

The constitutive response of most materials is known to be strain-rate-dependent for some range of temperature and rate of loading. Examples reported in the literature include soft and saturated shales [1], polymers [2], and structural steels subjected to high rates of strain [3]. We are particularly interested here in the class of rate-dependent solids whose instantaneous response is purely elastic, as is the case for crystalline solids that deform permanently by dislocation motion [4]. Mandel [5] has ~hown that, for this class of materials, the uniqueness condition for the solution of the incremental governing equations of a boundary value problem depends solely on the elastic response of the solid. As long as the characteristic stress remains small compared with the elastic moduli, uniqueness is guaranteed and the boundary value problem remains well posed [6]. In this stress range, the initiation of changes in the mode of deformation, which sometimes trigger localization [3, 7] are not associated with a bifurcation. The question therefore arises of how to pinpoint such initiation, in the absence of any loss of uniqueness. Clifton [8] and Bai [9] predicted the initiation of shear bands in a layer sustaining a simple shear mode of deformation using linear perturbation analyses. Leroy and Ortiz [10] obtained a local condition for the orientation of the bands with a similar analysis. Here, we use an extension of this type of perturbation analysis, presented by Leroy [11], to determine the modes of instability at the global or structural level. This method of assessing stability is first introduced for the illustrative example of a Shanley column, with supports having an elastic viscoplastic response, and is then applied to continuous media under multi-axial loading conditions. A finite element scheme is proposed for solving the stability problem and is applied to study the stability of the fundamental solution of a plane-strain tensile test. Elsevier Science Publishers B.V.

970

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

The second aim of this paper is to verify the validity of the predictions of the linear perturbation analysis and to follow the development of the instability, both for the column problem and the plane-strain tensile test. Of particular interest is the influence on this development of the imperfection size and the rate sensitivity of the material. In the case of the plane-strain tensile test the growth of a wavy mode of instability triggers the localization of the deformation in shear bands. The development of the localization process is analysed with a special finite element scheme with embedded shear band modes of deformation [12]. This method was shown to reduce successfully the sensitivity of the simulated bands to the mesh orientation in both two-dimensional and three-dimensional problems [10, 13]. A generalization of this method to the range of finite deformation was proposed by Nacar et al. [14]. The final part of this paper is concerned with the long-term evolution of the localization process and in particular questions the existence of shear bands of a finite thickness that could be obtained once transient effects have vanished. This question is at the core of the debate on the 'viscosity regularization'. It has been argued [6, 15, 16] that the strain-rate dependency introduces a length-scale that permits to determine the shear-band thickness. This thickness is, in general, very small compared to the length of the shear band and one-dimensional models of a layer of an infinite extent and sustaining a single shear mode of deformation has been considered repeatedly in the past [6, 8, 9, 17]. For the same model problem, we study the existence and the stability of steady states that could be seen as theoretical limit states for the shear bands.

2. Stability of strain-rate-dependent systems In this section we first summarize the results presented by Leroy [11] concerning the stability of a Shanley column [18] with supports having a rate-dependent response and the stability of strain-rate-dependent solids. For the continuous media, a finite element scheme is proposed to determine the points of neutral stability. As an application, the stability of the homogeneous solution of a plane-strain tensile test is studied. The results of these two problems are then compared with the classical results on stability and loss of uniqueness found in the literature for rate-independent solids and structures [19-22].

2. I. The Shanley column The geometry of the Shanley column is presented in Fig. 1 and the structure can be described as an inverted, rigid T beam of height L with two flanges separated by a distance 21. ~P

~P

,hI :e/

,., h 1

l

t Fig. I. Geometry and free-bodydiagram of the Shanleycolumn.

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

971

The column, loaded with a force that remains vertical at all times, is free to displace vertically and to rotate around its centre, the lateral displacement of which is constrained, Fig. 1. 2.1.1. Constitutive model The two flanges of the column have an elastic viscoplastic response. More precisely, we consider, for the sake of illustration, a simple linear-overstress viscosity model with a linear hardening law. The evolution of the permanent displacement u p of each flange of the column supporting a force F is thus described by the relation

li

p=~ ¼ [ F - ( F y + HU p)], t 0,

ifF>-Fy+Hu p otherwise, '

with u p = u - F/ E ,

(2.1)

in which, E, Fv and H are the elastic stiffness of a support, its initial elastic strength and a hardening modulus, respectively. The model obtained, as the inviscid plastic flow limit is approached (v/--->0-~), is an elasto-plastic model with yield criterion: ~b(F, uP)--F - (Fy + Hu p) = 0. For such a rate-independent model the evolution of the permanent displacement is governed by l i P = { 0H - l p

for $ = 0 a n d P > 0 , otherwise,

(2.2)

and the relation between increment of force and displacement at each support is P - rvti , with r r = E H / ( E + H ) ,

(2.3)

under loading conditions. In this equation we have introduced the Engesser or tangent modulus ex. When the elasto-plastic response of the column supports is given by (2.2), this modulus defines the critical load PT = eT12[L, called thereafter the tangent modulus load, at which the first loss of uniqueness occurs for the solution of the incremental governing equations. We will see next that this modulus also plays an important role in the stability analysis of the rate-dependent column with supports having a response described by (2.1). 2.1.2. Linear perturbation analysis Under quasi-static conditions, the solution of the equilibrium equations for the forces and moments (see Fig. l(b)) and the constitutive model (2.1) can be obtained for a given boundary condition at the top of the column and a set of initial conditions at any time t of the loading. The solution for an initially straight column that remains untilted as loading proceeds is called the principal or fundamental solution and is denoted by (u °, u p°, F°). We now investigate the evolution of neighbourinjz equilibrium solutions:

= / ," / +

,o

, (2.4)

LF°J that differ from the fundamental solution by the infinitesimal perturbations 5u p, 5u and 5F at time r = t + At. These perturbed solutions satisfy the equilibrium equations, the boundary conditions, as well as the constitutive equations (2.1). Here we focus on the last to obtain a

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

972

relation between the perturbation in force and displacement. Inserting (2.4) into (2.1), we obtain the following first-order system:

718~tP + HSuP=SF,

8u-~uP=(1/E)SF,

(2.5)

in which the loading condition that prevails along the fundamental path has been accounted for.

We now propose the following separation of variables: exp(A At),

(2.6)

L r(.,-) J for an evolution with time of the perturbation of an exponential type. The initial amplitudes of the perturbation (STY,8t~p, 8F) rem .,! undetermined in this linear analysis while the sign of the scalar it determines whether a growth or a decay of these perturbations occurs with time. If all solutions of (2.5) and the equilibrium equations, with structure (2.6), are found to have a negative rate it, then stability is ensured. Conversely, the existence of a solution with a positive rate signals the loss of stability of the principal solution. The transition from a stable to an unstable equilibrium is called a point of neutral stability or a stability transition point. Considering now solutions of (2.5) of the type (2.6), we obtain the following relation between perturbation in force and displacement, by eliminating 8~P: 8P=e,(it)St~,

withe(it)--E/(l+E/(itn+H)).

(2.7)

Furthermore, from the equilibrium equations [11], one can readily prove that the critical load at which a perturbation of initial rate it is admissible is nothing but P(it)= 2e(A)12/L. Note that the modulus e(it) tends to the elastic modulus E as it tends to infinity, This result indicates that as the applied load approaches the elastic buckling load or Euler load Pt.: -- 2EI"/L, a perturbation in the fundamental solution will have an initial rate of growth tending to infinity. Of more interest is the point of neutral stability corresponding to a vanishing rate it. In that limit, it is clearly seen from a comparison of the moduli e(it) and e.r, introduced in (2.7) and (2.3), respectively, that the critical modulus e ( i t ~ 0 ÷) is nothing but the tangent modulus. In other words, the first instability along the fundamental path is encountered as the applied load reaches the tangent modulus load.

2.2. Linear perturbation analysis for strain-rate-dependent solids We now discuss briefly the extension of the stability analysis presented for the Shanley column to the case of rate-dependent solids under multi-axial loading conditions. Inertia effects are now included, but the method for assessing stability remains identical with the one presented in the previous section. Consider a strain-rate-dependent solid occupying a volume O with boundary F. The dynamic principle of virtual work for a kinematically admissible virtual displacement 11 reads

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solMs

973

where o', p, b, t and/i are the Cauchy stress tensor, the mass density of the material, the body forces vector, the traction on the boundary F and the acceleration vector, respectively. The operator V ~ is the symmetric gradient of the field in argument and we assume here a linear relation between strain and displacement. Note that inertia terms were only introduced for sake of completeness and are not essential to the stability analysis presented next. Indeed, there are two time scales in this problem, the first one associated with the loading and the second with the time dependence of the following constitutive equations: o =

r = ~ . (o') ,

(v'(a)-

"/,= (l /r/)g~(tr, y)

if6>0,

"~=0

otherwise.

(2.9)

The direction of plastic flow r is derived from the potential ~,, while the effective plastic strain rate ~, is determined from the viscosity law ~b. In the limit of rate-independent plastic deformation (aq---~0 ÷), the rate law (2.9c) reduces to the yield criterion ~b(o-, 3' ) = 0. Note that the instantaneous response of the material is always purely elastic. Considering now a perturbation in the variables (o', 3', u), one can follow the procedure presented in the previous section and determine the modulus c~(a) that relates the perturbations in stress and symmetric displacement gradient [11]: = 9(A):

with~(A)-

{ D~(D~: r")(O~b/O~rJ""D~) D ~ A~l-i~tb/OTl"+i~6/i~o'J":D":r"

if ~,>0 (2.10) otherwise.

The notation adopted in this equation is identical with that used in (2.4) and (2.6). Of interest is the expression taken by this modulus 9~/l(h) in the two limits of an infinitc and a zero rate of growth. For an infinite initial rate of growth, ~(,~--, :~) is the elastic modulus, associated with the instantaneous response of the solid. Of more interest is the second limit of a vanishing rate of growth defining the point of neutral stability. In that limit, it is found that the modulus ~(h----~ 0 +) coincides with the modulus obtained from Hill's linear comparison solid [20] for a rate-independent model obtained in the limit of inviscid plasticity. These two results are similar to the ones obtained from the stability analysis of the Shanley column. Introducing (2.10) in the linearized form of the principle of virtual work (2.8), we obtain the following equation with the perturbation in displacement field B~i as unknown:

f,,

V~(~I):~(A):W(~ti)dg2+A"

j

,~l'Bfidg2=O'

(2.11)

If a solution is found with a positive rate A, then the solution of the problem is said to be unstable. Note the stabilizing influence of inertia on the modes having a non-zero initial rate of growth. Analytical solutions of (2.11) are restricted to a certain class of constitutive models and simple domain geometries. At the same time, approximate solutions are easily obtainable. For example, the finite element method with the same interpolation for the displacement and the

974

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

virtual feld yields the set of equations ~};

~I; pN N dO h ~tl h = 0,

(2.12)

where B denotes the classical strain operator based on the shape functions present in N. The sum extends over all elements involved in the discretization of the volume O. The stability analysis has been rephrased in a simple eigenvalue problem for a discrete system. A zero eigenvalue of the stiffness-like matrix in (2.12) reveals the existence of a perturbation with initial rate of growth A. The associated eigenvector ~ti h is an approximation of the mode of instability for the continuum problem.

2.3. Application to the plane-strain tensile test We now assess the stability of a rectangular specimen under plane-strain tensile loading, applying the method just presented. The specimen's aspect ratio, defined as width over length (l/L), is 1/3. Attention is focussed on perturbations of zero rate of growth, corresponding to points of neutral stability. The non-linear analysis of Section 3.2 will permit to decide whether or not these points of neutral stability correspond to stability transitions. Note also that inertia terms in (2.12) have no contribution at points of neutral stability. The fundamental path is defined by the homogeneous solution of this problem. The elastic response of the material is linear isotropic with a Young's modulus of 10 3 MPa and a Poisson's ratio of 0.3. The viscosity law is of a power type and reads 5' = ~d(o'flo',,(y))'"- 1] foro-~ ~> tr,,(3,),

zero otherwise ,

(2.13)

in which tr,. is the Von Mises effective stress, tr,,(y) a hardening function of the effective plastic strain ~,, and ~,,, a reference strain rate, set to 2 x 10 -~ during this analysis. The limit of inviscid plastic flow is obtained for an infinite value of the exponent m. In this instance, the model is equivalent to a rate-independent plasticity model with a Von Mises yield surface. The hardening function tro(y ) is obtained from the product of a power law and an exponential function: o-,,(y) = (or:, - o'~)(1 + ?/'y,)" exp(-3,/"/2) + o-~,

(2.14)

in which O'y, or,, n, y~, Y2 are, respectively, the initial yield strength, the residual strength of the solid once hardening has ceased, the hardening exponent and two reference strains. These five quantities are set to 1, 0,1, 3/4, 10 -3 and 5 × 10 -3, respectively, in this analysis. Note that the exponential function introduces a softening in the response of the specimen which is responsible for the initiation of instability once maximum load has been reached. The load-displacement curve for the fundamental solution of this problem is presented in Fig. 2 for a rate exponent m of 500. The domain of study, deliberately chosen to be the upper part of the specimen, is discretized with 10 x 15 four:node quadrilateral elements. Despite the homogeneous character of the fundamental solution, the modes of instability have a spatial gradient and a treatment of

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

975

~.S-

m

1.0-

0.5-

0.0

I

I

I

2

4

6

u/L

(.,.10-~)

Fig. 2. Load-displacement curve for the fundamental solution of the plane-strain tensile test. The dots and bars mark the equilibrium points at which the first eight modes of instability have been detected.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 3. The first eight modes of instability, in terms of the displacement field, for the plane-strain tensile test. Note that these modes are associated with four different wavelengths.

976

Y.M. Lerov and O. Chapuis, Localization ht strain-rate-dependent solids

the incompressibility constraints induced by the plastic deformation is required. The/~ method 123] with a stabilization procedure [24] is used. The results of the analysis are presented in Fig. 3, in which the first eight modes of instability in terms of the displacement field are presented in the chronological order of their detection. The equilibrium positions at which they were found are indicated on the two curves of Fig. 2. Note that the first instability is found at the maximum load. The eight modes are of a wavy type and are associated by pair with four different wavelengths. For each wavelength, there is a mode that is symmetric with respect to a set of axes parallel to the sides of the specimen and having for origin its centre. Interestingly enough, the first mode of a given wavelength to be detected is not always the symmetric mode, but can be the asymmetric one (Fig. 3(c, e)). Finally, the absence of shear-band modes of instability at the structural level should be noted. A local analysis [10] reveals that, at maximum load, shear-band modes are admissible. Nevertheless, in the absence of a preferential site for nucleation, no such bands are detected at the global level. A similar conclusion was reached by De Borst [25] from a bifurcation analysis for a rate-independent model and a specimen of similar geometry.

2.4. Comparison with stability criterion for rate-independent systems The results we have obtained for the stability of strain-rate-dependent systems differ from the results in the literature for rate-independent solids and structures. It is known that the first bifurcation of the rate-independent Shanley column occurs at the tangent modulus load [18, 21], and that the violation of Hill's sufficient condition for uniqueness signals the first bifurcation for solids with an associated plasticity model. The first bifurcation of the Shanley column occurs under increasing load, as is often the case with continuum problems. As a consequence the equilibrium paths beyond the first bifurcation point are stable [19, 20, 22]. For the rate-dependent column, the first loss of uniqueness is delayed up to the elastic buckling load owing to the elastic instantaneous response of the structure, but the point of neutral stability is found at the tangent modulus load [11]. This latter finding was shown there to extend to continuous media. Neutral stability in continuous media was found to be attained at the first failure of Hill's sufficient condition for uniqueness for a rate-independent comparison solid. The discrepancy between these two results on stability for rate-independent and ratedependent solids and structures arises from different definitions of stability and is not due to the presence or not of rate effects. In the spirit of the stability analysis of conservative systems, an equilibrium state for an elasto-plastic system is usually defined to be stable as long as a perturbation from that equilibrium has a bounded growth with time. A sufficient condition for stability is then furnished by Lyapunov's direct method, the Lyapunov functional corresponding to an estimate of the work supplied to the system by the perturbation [20, 22]. Nguyen and Radenkovic [26], in applying this approach to the stability analysis of ratedependent solids, obtained results identical with those that would have been obtained in the absence of rate effects. In this paper the approach to stability differs from the traditional one. We consider that loss of stability occurs when two solutions initially in an infinitesimal neighbourhood start to deviate from each other with time. This criterion was chosen so as to pinpoint the first possible change in the mode of deformation. For the Shanley column, it is thus not surprising to obtain

Y. M, Leroy and O. Chapuis, Localization in strain-rate-dependent solids

977

the tangent modulus load as the critical load, since it is the first load at which the structure can start to deviate from a straight position [18].

3. Non-linear stability analysis To assess the validity of the linear perturbation analysis, we now wish outcome with the actual non-linear response of the systems analysed both column and the plane-strain tensile test. In particular, we shall examine the amplitude of the perturbation- introduced as initial imperfection- and that strain-rate sensitivity on the development of the instability. 3.1.

to compare its for the Shanley influence of the of the material

The Shanley c o l u m n

In this non-linear buckling analysis, it is assumed that the top of the column is displaced vertically at a constant rate V. This boundary condition introduces a characteristic loading time t L = L / V into the problem. Putting t L in relation with the relaxation time of the structure t R = ~ / E yields the dimensionless number (3.1)

T= - tR/t L - rlV[LE,

In a first series of tests, we analysed the column's response for different imperfection sizes keeping the number T constant. As may be seen from Fig. 4, the solution initially remains

e t = 10 "1°

",

/

~,,

e l = 10. 5

0 i = 10. 3

" ~ 0.50 i = 10 .2

..f..-'"

PT/ PI+ 0.0

I

0.00

I"

i

0.05

0.10

O-e i

Fig. 4. Evolution of the load-bearing capacity of the Shanley column for different values of the imperfection amplitude O~.The two dashed curves represent two equilibrium paths for the rate-independent column and they intersect the fundamental solution at the tangent modulus load and the Euler load.

978

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

close to the fundamental solution, before the bearing capacity of the column starts to decrease. For a very small initial imperfection (O~= 10-"~), the maximum load is as high as 3 / 4 of the Euler load. As the size of the initial imperfection increases, it is found that this maximum load decreases. Note also that the final load-bearing capacity of the structure is independent of the amplitude of the initial imperfection. This is at a variance with the results of an imperfection sensitivity analysis presented by Hutchinson [27], who introduced an extra non-linear spring to the system, resulting in a column whose final bearing capacity depends on the amplitude of the imperfection. In Figs. 4 and 5, the family of solutions presented are compared with two equilibrium paths obtained for the rate-independent column. These two equilibrium paths intersect the fundamental one at the Euler load and the tangent modulus load, respectively. In a second series of tests, we considered a structure with an initial imperfection 0~= 1 0 - 7 and varied the dimensionless number T by choosing a range of values for the parameter r/. The response of the column for four different values of T is presented in Fig. 5. We observe that as the viscosity 7/is reduced, the response of the column becomes indistinguishable from the equilibrium path of the rate-independent column that intersects the fundamental path at the tangent modulus load. This result indicates that, at sufficiently small values of T, the load at which buckling initiates is the tangent modulus load, i.e., the critical load detected by the linear stability analysis. It should nevertheless be noted that this limiting process pertains to small imperfection amplitude only. 1.0-

~

~- 0,5

T = 10.3

\ " ' - . " "T" "=~10 ""~ "

,/ 0.0

0,00

. .

,/ I

I

0,05

0.10

e

Fig. 5. Evolution of the load-bearing capacity of the Shanley column for different values of the dimensionless number T and an imperfection amplitude of I0 -7. As the value of 7" is reduced, the column response tends towards the equilibrium path of the rate-independent perfect column intersecting the fundamental solution at the tangent modulus load Pr and represented here by a dashed curve. The second dashedcurve leaving the principal solution at the Euler load is also an equilibrium path for the rate-independent perfect column.

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

979

3.2. The plane-strain tensile test

The linear stability analysis of the plane-strain tensile test (Section 2.3) has revealed that there is a point of neutral stability as the maximum load is reached. Beyond that critical load, the deformation could thus cease to be homogeneous, owing to the growth of wavy modes of instability. These modes may be viewed as being already present on the virgin specimen as initial geometrical or material imperfections and the linear perturbation analysis reveals ~vhich of these are critical to the stability of the specimen. It remains to determine how the evolution in time of these modes differs from the prediction of the linear perturbation analysis and what the influence of the material rate sensitivity and the amplitude of the imperfections is on that evolution. A quasi-static non-linear analysis is needed to answer these two questions. In the absence of inertia effects, modes of long wavelength are known to have the fastest growth, as, for example, during the tensile test of cylindrical bars [28] or the stretching of thin plates [29]. Accordingly, we choose a combination of the two modes presented in Fig. 3(a, b) as the initial imperfection. The amplitude of the imperfection is defined as the maximum deviation from the perfect geometry measured on the surface of the specimen. Note at this point that shorter-wavelength modes have an influence on the position of the shear bands that will be formed on the specimen [11, 30]. We shall disregard this influence in this analysis. The growth of the wavy imperfection is expected to trigger the initiation of a localization of the deformation in shear bands. Isoparametric elements are known for inhibiting the growth of such shear bands, owing to the continuous strain interpolation over each element [30, 31]. To improve the performance of the four-node quadilateral, we choose here a finite element method introduced by Ortiz et al. [12]. The aim of this method is to complement the conventional strain interpolation of the element by specialized modes of deformation such that shear banding is fully accommodated at the local level. The resulting enhanced element was observed to reduce successfully the sensitivity of the development of shear bands to the mesh orientation, in both two- and three-dimensional problems [10, 13]. A generalization to the range of finite deformation was proposed by Nacar et al. [14]. A general formulation of this finite element method for transient problems is found in the work of Leroy and Ortiz [10] and is adopted here. To complete the description of the numerical scheme, note that the displacement of the top of the specimen is prescribed as a function of time. This function is initially linear, delining a constant nominal strain rate, set to 2 x 10 -~ in this analysis. During the localization process, the strain rate becomes much larger than this nominal value in the bands of intense shearing. To control this evolution, we choose to modify the loading function such that the maximum strain rate remains at all times equal to the initial nominal one. This indirect displacement control was found to be necessary to follow the equilibrium path. The resulting mixed finite element method has been used in various contexts by [30, 32, 33] and is close to the one proposed recently by Chen and Schreyer [34]. In a first series of tests we vary the amplitude of the imperfection from 2 x 10 --~ to 10-3, for a constant exponent m of 250. The evolution of the load-bearing capacity of the specimen is recorded in Fig. 6. During an initial period, the response of the specimen is indistinguishable from the homogeneous solution. As the maximum load is approached, the long-wavelength imperfection grows and triggers the initiation of two shear bands which cross the specimen. The critical nominal strain at initiation of localization corresponds to the beginning of the drastic reduction in load-bearing capacity of the specimen (Fig. 6). Note that this critical strain

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

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1.0-

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immnmmmmmnmN~1) immmmmmmmmmmm~ ,mmmmmmm|nmmmu Immmmmmnmmmmm| Immnmmmmmmmmmm mmmmmmmmmmnmm mmmmnmmmmmmmmm immmmmmmmmmnmn mmmummnmmmmmnm |mmmmmmmmmmmmm mmmmmmnmmnmmmm

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set to 500.

depends on the amplitude of the imperfection. The larger the imperfection, the smaller is the nominal strain at the initiation of the localization process. This result is of course in agreement with the analysis of Hutchinson et al. [29] on necking during sheet-metal forming. During the development of the localization, a snap-back is reported on the load-displacement curve plotted in Fig. 6. It is necessary to reduce the displacement at the boundary to control the evolution of the strain rate in the shear bands and to remain on the equilibrium path. Concerning the position of the shear bands on the specimen, it is found that the two bands intersect at an angle of 90° and at a quarter of the width of the specimen. This result, found to be independent of the amplitude of the imperfection, can be inferred from Fig. 7(a). In Fig. 8(a) we present, in each element where plastic flow is still taking place at the end of the simulation, the normals to the two directions of shear banding. This information, obtained from a local stability analysis, is used to improve the strain interpolation of each element [10]. In a second series of tests we have varied the rate-sensitivity exponent m and kept constant the amplitude of the imperfection, A = 10 -3. The results are presented in Fig. 9 for three ~,tiucs of :i~e exponent m. As the rate sensitivity of the material is increased, the initiation of the localization process is delayed. This is in good agreement with the results of Hutchinson and Neale [35, 36] on the influence of the strain-rate sensitivity on the development of necking on cylindrical bars and during sheet metal forming. The orientation and the position of the shear bands are found in this series of tests to coincide with those obtained for the first series of tests and presented in Figs. 7(a) and 8(a). Note that for a very small rate-sensitivity (m = 500), the critical strain is just at maximum load, showing the relevance of the predictions of the linear stability analysis.

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

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6

u/L(~to

Fig. 8, Normals to the shear bands detected at the local level using a linear stability analysis, in all elements in which plastic flow is still taking place at the end of the simulations. The two meshes have (a) 10 x 15 elements and (b) 14 x 21 elements. The imperfection amplitude is 10 -3.

~ )

Fig. 9. Load-displacement curves for three values of the strain-rate exponent m and comparison with the fundamental solution represented by the dashed curve. Mesh of l0 x 15 elements and imperfection amplitude of 10 -~.

I,S-

nlog~ll¢OUS

m

1+0-

tO>` a.

0.5-

0.0

I

I

I

2

4

6

u/L

txl0 .+)

Fig. 10. Load-displacement curves for two meshes of 10 x 15 and 14 × 21 elements and comparison with the fundamental solution represented by the dashed curve. Note the mesh sensitivity during the development of the localization but not at the initiation. The imperfection amplitude is 10-~.

982

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

To complete this work, we now study the influence of the mesh size on the simulation of the initiation of the instability and the development of the shear bands. The rate-sensitivity exponent m is set to 250 and the imperfection amplitude is 10 -3. A mesh of 14 x 21 elements is considered and the results of the analysis are presented in Figs. 7(b), 8(b) and 10, in which they are compared with the results obtained with the coarser mesh. Upon examining the evolution of the load-bearing capacity of the specimen in Fig. 10, we see that the initiation of the instability is well captured since the drop in load, which defines the initiation of the localization, is obtained at the same nominal strain for the two meshes. Nevertheless, a mesh dependence exists during the development of the localization process, the snap-back in the load-displacement curve being more important for the finer mesh. We now compare the shear bands obtained at the end of the two simulations shown in Figs. 7 and 8. In both cases, the thickness of the bands varies between two and three elements. These results indicate that the band thickness becomes governed by the mesh size at some point in the simulations.

4. Existence and stability of stationary shear bands In this section we wish to question the long-term evolution of the localization process and the possibility for shear bands to attain a finite thickness once transient effects have vanished. The existence of shear band with a finite thickness is essential to validate finite element simulations since it ensures that the results are indeed independent of the mesh size. The introduction of a strain-rate dependency has been proposed repeatedly as a means to define a shear band thickness [6, 15, 16] that was absent in elasto-plastic formulations. Here, noting that the shear band has, in general, a thickness which is small compared to its length, we propose to study the one-dimensional model of a layer of an infinite extent sustaining a simple shear mode of deformation (Fig. 11). We study the existence of steady states that could be seen as theoretical limit states for the shear bands. A similar setting was considered by Needleman [6] in studying the relation between the band thickness and the imperfection size, and by Leroy and Molinari [17] in studying stationary solutions for the flow of thermoviscoplastic materials in shear zones. The thickness of this layer, set to two in this dimensionless analysis, could be the size of the specimen or could be related to the size of a defect that has triggered the localization of the deformation in a sub-region of the specimen at an earlier

Y

÷v

I

0

S

-V Fig. 11. A layer of infiniteextent under a simple-shearmode of deformation. The stationary velocityprofiles are linear across the layer.

Y.M. Leroy and O. Chapuis, Localization in strain-rate-dependent solids

983

stage of the loading. The equation of motion and the constitutive response of the material in this sheared layer are written in a one-dimensional form as v,y = p O ,

3~P),

4" = p,(V,y -

~"= (3'P)",

(4.1)

in which z, v, 7 p are the shear stress, the velocity and the effective plastic strain, respectively. This constitutive model (4.1b, c) is a one-dimensional version of the constitutive equations (2.9). The dimensionless numbers/z and p are associated with the elastic shear modulus of the layer and its mass density, while m is the strain-rate exponent. Note the absence of a hardening function in the constitutive equation (4. lc). This is a characteristic of materials that have already sustained a large amount of permanent deformation and is a necessary condition for obtaining steady states. We assume that the velocity has the prescribed value of V at the boundary y -- 1 and that a symmetry condition holds at the origin of the y axis, perpendicular to the infinite layer (Fig. 11): vl. = V ,

vl,, = 0 .

(4.2)

The stationary solutions of this problem are readily obtained and correspond to linear velocity profiles and constant ~tress: (o = V y ; ~,P= V; z = Vm). The existence of a unique steady state is guaranteed for any prescribed velocity V. We now have to question the physical existence of these steady states and therefore study their stability. The first-order system for (4.1) and (4.2) reads Sr, y =

p SO ,

Sol, = 0 ,

S4. = ~ ( S v , s

SSP),

--

Sr =

mV'"-1

S~tp ,

Sol(, = 0 .

(4.3)

This system is autonomous and the time dependence of the perturbation is exactly represented by the exponential function exp(At), in which A is the initial rate of growth of the perturbation. The sign of the real part of A will determine the stability of the stationary solutions. We now eliminate the perturbations in effective plastic strain and in stress from (4.3) and obtain

so,:,y _ ( a + m v,_,,,) --ifpa St~ = 0 ,

sol, =0,

sol,,=o,

(4.4)

in which the same notation as in (2.6) is employed. The solution of this system for bounded perturbations in velocity is 8t3(y) = sin(Ay)

,

A 2 --

-

/ 1~A

+ -- V m

\ pA i-,,,)

= ~ " ( 1 + k ) "~

"~

k E N '

(4.5) "

We have thus found a series of perturbations with a discrete spectrum of wavelengths. Two perturbations are associated with each wavelength and their rates of growth are A = - V t-''' t~ _+ 2m

Vt-,,, ~ 2m

)2

_ IZ ~r-'(1 + k) 2 p '

(4.6)

984

Y.M, Leroy and O. Chapu&, Localization h7 strain-rate-dependent solids

The scalar ~. is a complex number when k > V t - " v ~ / ( m 2 ~ r ) - l and is real otherwise. Independently of the wavelength, the real part of h is always negative. All perturbations are thus decaying with time, with short-wavelength perturbations having an oscillatory evolution. Note that A is never a pure imaginary number, precluding any Hopf bifurcation that would have signalled the existence of periodic solutions. We can thus conclude that the steady states for this simple model problem are stable. 5. Discussion and conclusion

Linear perturbation analyses have been presented for assessing the stability of a Shanley column and of a continuous medium under multi-axial loading conditions. In both problems the constitutive model was describing a strain-rate-dependent solid or structure with an instantaneous elastic response. We have found that the point of neutral stability coincides with the first failure of Hill's sufficient condition for uniqueness for a comparison solid that is, obtained by considering the limit of inviscid plasticity. Non-linear numerical analyses were conducted for the Shanley column and a plane-strain tensile test. The structural modes of instability for the plane-strain test are all of a wavy type. For the two problems, it was observed that a decrease in the amplitude of the mode of instability, introduced as an initial imperfection, resulted in a lag between the time of loss of stability, as predicted by the linear perturbation analysis, and the time of first observation of a change in the mode of deformation. It was also found that a decrease of the rate sensitivity of the material reduced this lag. In the limit of inviscid plasticity, the prediction of the linear stability analysis does correspond with the actual initiation of the instability. For the plane-strain tensile test, the growth of the long-wavelength mode of instability, introduced as an initial imperfection, triggers the initiation of a set of two shear bands that defines the final failure mode of the ~pecimen. The orientation and the position on the specimen of these two bands are not influenced by the amplitude of the imperfection nor the strain-rate sensitivity of the material. During the development of the localization it was necessary to vary the rate at which the top of the specimen was displaced to follow an equilibrium path and to keep the strain rate in the band bounded. This result indicates that inertia effects are important during localization even in a displacement-controlled set-up. The final part of this paper has dealt with the long-term evolution of the localization process and explored the possibility of obtaining shear bands of a finite thickness. For the plane-strain tensile test of a solid whose strain-rate dependency was modelled with a power-law viscosity function, it was not possible to determine a definite shear-band thickness from the results of a mesh-sensitivity analysis. A different conclusion was reached by Leroy and Ortiz [10] from the results of an analysis with a linear-overstress viscosity model. To further investigate the possibility for the shear bands to attain a finite thickness in the long term, we explored the existence and the stability of steady states for a layer of infinite extent, subjected to a simple shear mode of deformation. The conclusion is that, in the absence of any further degradation of the mechanical properties of the material, a stable stationary shear band of a finite thickness can be attained. This thickness, which remains undetermined in this analysis, depends on the whole history of the localization process, the size of the initial defect that triggered the instability, and is of course sensitive to the structure of the viscosity function considered.

Y.M. Leroy and O. Chapuis. Localization in strain-rate-dependent solids

985

Acknowledgment This paper is published by the permission of Shell Internationale Research Maatschappij B.V.

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