ac finite element in gradient plasticity for localized failure ... - CiteSeerX

COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. Oñate and E. Dvorkin (Eds.) ©CIMNE, Barcelona, Spain 1998

A C0 FINITE ELEMENT IN GRADIENT PLASTICITY FOR LOCALIZED FAILURE MODES ANALYSIS Fékri Meftah† , Gilles Pijaudier-Cabot‡ , and Jean M. Reynouard¢ †

‡

¢

L.M.T-Cachan, E.N.S de Cachan / C.N.R.S / Université P. et M. Curie 61, Av. du Président Wilson, 94235 Cachan Cedex, France e-mail: [email protected] L.M.T-Cachan, E.N.S de Cachan / C.N.R.S / Université P. et M. Curie 61, Av. du Président Wilson, 94235 Cachan Cedex, France also at Institut Universitaire de France e-mail: [email protected]

U.R.G.C.-Structures / Institut National des Sciences Appliquées de Lyon 20, Av. A. Einstein, 69621 Villeurbanne Cedex, France e-mail: [email protected]

Key words: Strain softening, Localization, Regularization, Gradient plasticity, Finite element, Failures analysis. Abstract. A new variational approach in gradient plasticity is derived for the finite element analysis of quasi-brittle failure modes. A consistent integration algorithm for updating the stress and the internal state variables at integration points is formulated in an original nonlocal sense. The model includes implicitly second gradient of the effective plastic strain in the yield condition, which reduces the continuity requirements on field interpolations to the use of C0 shape functions. A one dimensional example is presented to illustrate the method.

1

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

1 INTRODUCTION Localization of deformation into narrow bands of intense straining caused by strain softening is a characteristic feature of plastic deformation. It has been experimentally observed in many engineering materials, such as concrete, rocks and soils. Localization phenomena are often associated with a significant reduction of the load-carrying capacity of the structures, hence, the onset of localization is naturally considered as the inception of failure of engineering structures. Considerable efforts have been devoted over the last decade to obtain a comprehensive understanding of the problem and to describe this behavior quantitatively. However, numerous attempts to simulate the behavior with softening plasticity or damage theories failed in the sense that the solution appeared to be fully determined by fineness and the direction of the finite element discretization. The underlying reason of this pathologically mesh dependency is the a local change of character of the governing equations, which results in a loss of well-posedness of the boundary value problem. More details and references on these aspects can be found in the papers by Pijaudier-Cabot et al.14 and Lasry and Belytschko7. To remedy the situation, generalized continuum theories should be adopted. These models incorporate an internal length parameter which plays the role of localization limiter, that is, a parameter that allows to control the localization zone size by preventing loss of ellipticity of governing equations. This internal length parameter can be introduced through the incorporation of higher order derivatives leading to the so-called gradient continuum theories. The capabilities of gradient dependence have been investigated for both softening plasticity and damage models4,10,12. Recently, formulations and algorithms for gradient dependent models have been presented in a finite element context3,5,6,9,11,12,15. In a gradient plasticity model the yield strength depends not only on the effective plastic strain but also on its Laplacian. Therefore, even if gradient dependent models bear the significant advantage of being local in a mathematical sense, the increment of the plastic strain can not be obtained at a local level since the consistency condition which governs the plastic flow becomes a second order partial differential equation. One possibility is to use a finite difference method. The algorithm is then a sequence of separate approximate solutions of the equilibrium problem using finite elements and the plastic yielding problem using finite differences1. A more general approach is to use only finite elements and to solve the two coupled problems simultaneously4. For this purpose, it was required to satisfy weakly the yield condition and to discretize the plastic strain field in addition to the standard discretization of the displacement field. Therefore, C1 continuous shape functions, for the plastic strain field interpolation, are needed in order to compute properly second gradients of the equivalent plastic strain. As a consequence, this mixed formulation leads to an oversized discretized problems making the calculations unreasonable. With the use of the gradient plastic model including implicitly second gradient of the effective plastic strain in the yield condition, a finite element method for quasi-brittle fracture will be derived in our work. The consistent integration algorithm to update the stress and the internal state variable at integration points for the gradient plasticity are formulated in an

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Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

original nonlocal sense. The new algorithm could require either a weak or point-wise satisfaction of the yield function together with the consistency condition, while only a C0 continuity interpolation functions of the plastic problem related field is needed. The methodology to derive the finite element formulations of this model is presented. Finally, the pertinence of the finite element and the elaborated algorithm will be illustrated on a one dimensional numerical example. 2 CONSTITUTIVE RELATIONS The main feature of gradient plasticity is that the yield function F depends, beside the stress tensor σ, on both the equivalent plastic strain κ and its Laplacean ∇ 2 κ j+1 . It can take the following form4,8,11

(

)

(

)

F σ ,κ , ∇2κ = f (σ ) - τ κ , ∇2 κ = 0

(1)

where f gives the stress related part and τ is the gradient dependent yield strength which reads

(

)

τ κ , ∇2κ = τ(κ ) - g(κ ) ⋅ ∇2κ .

(2)

In relation (2) τ is the classical, local, yield strength and g is a weight function which gives the effective nonlocal contribution.

t

n ∂Ω

λ



Σ = σij

Elastic domain ( Ωe )

Plastic domain ( Ωλ) 

∂Ω λ

Σ: stress tensor Figure 1: Evolution of the plastic domain

The variational principle proposed previously by de Borst and Muhlhaus4 for gradient plasticity allows to this theory to be used as a localization limiter in a large domain of finite

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Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

element calculations8,11,15. This variational approach consists in a weak satisfaction of the yield function

∫ δλ ⋅ F(σ k +1 ,λ k +1, ∇ λ k +1) dΩ = 0 2

(3)

Ω κk +1

besides the classical weak satisfaction of the equilibrium equation. In relation (3), δλ is a virtual variation of the plastic multiplier which is assumed to be related to κ by a linear relation. The above relation means that the yield function should only be satisfied at the end of the current loading step ( k + 1) and in a distributed manner. It should also be noticed that integration in Eq.3 is restricted to the plastic domain Ω κk +1 since F < 0 in the elastic part of the body. Of course, due to Khun-Tucker conditions λ ≥ 0 , F ≤ 0 , λ ⋅ F = 0 ,

(4)

one can expect to extend the integration in Eq.3 to the whole domain, however the test function δλ should remain a general function satisfying only smoothness requirement and the boundary conditions of the approximated field. Thereafter, since the plastic domain is a priori unknown, a predictor-corrector schema is proposed4,8,11 for the solution of the algebraic equations resulting from the finite element discretization of the functional. Relation (3) becomes

j

∫ δλ ⋅ F(

j+ 1

)

σ k +1 , j + 1 λ k +1 , j + 1 ∇2 λ k +1 dΩ = 0

Ω κk +1

(5)

in which (j) is equilibrium iterations indicator in a Newton -Raphson algorithm. This means that we determine plastic flow at iteration (j+1) by integrating over the plastic domain given by all the integration points which yield (for which the stress state has violated the yield function) at the previous iteration. By this way of proceeding, one neglects the term giving the evolution process of plastic domain leading to a description of the motion of the elasto-plastic boundary ∂Ω λ which is not entirely correct. The consequence of the delay in the evolution of this boundary, is an overconstrain on the plastic flow which leads to important values of the Laplacean of the equivalent plastic strain and then causes numerical troubles during equilibrium iterations. Moreover, this variational principle gives the conditions on λ at the boundary of the plastically deforming part of the body ∂Ω λ , which must either be

( )

T δλ = 0 or ∇λ nλ = 0

(6)

nλ is the unit outward normal at the elasto-plastic boundary (Fig.1). Therefore, these conditions should be imposed at a moving boundary, a fortiori unknown, which renders the procedure impossible in a classical finite element code. Currently, the conditions are imposed

4

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

on only the external part of the elasto-plastic boundary. Finally, in order to avoid oscillations of the yield function values, the Laplacean of the plastic multiplier ∇2 λ should be properly computed by adopting C1 continuous interpolation polynomials. This continuity requirement increases significantly the number of degrees of freedom and, thus, the calculation cost. In this contribution, we propose a new variational approach that allows to prevent the main drawbacks of the previous algorithm. For this purpose we start from an earlier definition13 of a nonlocal quantity. We consider the nonlocal yield strength value τ in a material point x as weighted average of the local yield strength τ over the surrounding volume Ω s τ=

1 φ(ξ ) ⋅ τ(x + ξ ) dΩ = 0 , with Ω s = φ(ξ ) dΩ Ωs

∫

∫

Ω

(7)

Ω

−ξ2  is a gaussian weight function and ξ denotes the relative position in which φ(ξ ) = exp  2 ⋅ l2    vector pointing to the infinitesimal volume dΩ . The parameter l is the internal length governing the contribution on the yield strength (additional carrying capacity) of the surrounding volume at the given material point x. A gradient formulation can be derived from this non local theory10. The local yield strength is then expanded into Taylor series according to τ( x + ξ ) ≈ τ + ∇τ • ξ +

1 2 1 1 2 3 ∇ τ • ξ ( ) + ∇ 3 τ • ξ ( ) + ∇ 4 τ • ξ 4 + ⋅⋅⋅ 2! 3! 4!

(8)

where, ∇ n is the nth order gradient operator, the dot • denotes inner product between nth order tensors and ξ ( n) denotes the n factor dyadic product ξ ⊗ n - times⊗ξ . By substituting Eq.8 into Eq.7 and considering that the domain Ω s remains small comparing to Ω, an exact integration of relation (7) is feasible leading to the vanishing of the odd terms such that τ ≈ τ + c 1 ( l ) ⋅ ∇ 2 τ + c2 ( l ) ⋅ ∇ 4 τ + ⋅⋅⋅ .

(9)

The gradient parameters ci are of the dimension of a length to an even power, so that the internal length scale is present in this gradient formulation. Thus, neglecting higher order terms in expression (9), we simply assume that the yield condition reads τ + l 2 ⋅ ∇2τ − τ = 0 .

(10)

In the next section, we will show that this definition is suitable for numerical analysis since it enables a straightforward C0-continuous finite element interpolation. Furthermore, a very important feature of this relation is that it holds on the whole domain. Indeed, in this differential equation τ acts as a source term that drives the variation of the local quantity τ . Thereafter, a change in the values of τ , notably at the elasto-plastic boundary, will provide

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Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

non zero values of τ in former elastic integration points close to the plastic domain. Thus, the localization zone can extend and the regularization effect is ensured. In this contribution, a weak form of this equation will then be considered instead of Eq.1 (which is valid in the plastic domain only) to derive the set of equations governing both of equilibrium and plastic flow. It is important to note that the model does not differ fundamentally from the usual gradient plasticity one derived by de Borst and Muhlhaus4. Nonlocality is expressed in a slightly different way only. 3 FINITE ELEMENT IMPLEMENTATION The weak form of the partial differential equations which govern deformation processes are derived according to the weighted residuals approach16. The equilibrium equation at the current load step ( k + 1) L T σ k +1 = 0

(11)

in which L is a differential matrix operator8 and where body forces have not been included for simplicity, is multiplied by the vectorial weight function δu T (virtual displacement field) and subsequently integrated on the domain Ω T T ∫ δu (L σ k +1) dΩ = 0.

(12)

Ω

The weak form of the averaging equation (10) is obtained in a similar way. Multiplication with the weight function δτ and integration yields

∫ δτ (τ k +1 + l

2

)

∇2 τ k +1 − τ k +1 dΩ = 0 .

Ω

(13)

With the aid of the divergence theorem, the resulting equation can be transformed to

∫ [(Lδu)

T

Ω

]

dσ dΩ =

∫ [δu

T

]

t k +1 dΣ −

∂Ω

∫ [(Lδu)

T

Ω

]

σ k dΩ .

(14)

and

∫ (δτ dτ − l

Ω

2

(∇δτ)T ∇dτ

) dΩ − ∫ δτ dτ dΩ = ∫ δτ τ Ω

k

dΩ −

Ω

∫ (δτ τ k − l

Ω

2

(∇δτ)T ∇τ k ) dΩ (15)

in which d denotes increment of the variable ( k +1 = λ k + σd , τ k +1 = τ k + dτ , τ k +1 = τ k + dτ ), and t is the external stress vector acting on the configuration boundary ∂Ω , on which the non standard conditions

δτ = 0 or (σdτ ) n ∂Ω = 0 T

6

(16)

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

have been considered in equation (15). The last condition can be interpreted as zero history parameter flux through the boundary of the body. Therefore, if these extra-boundary conditions are considered, Eq.15 shows no need for C1 interpolation for the local yield strength. It remains however to treat the source term dτ such that its explicit dependency on the second order gradient can be avoided. During plastic flow, the stress tensor increment dσ can be related to both the strain and the local yield strength increments dε and dτ , respectively. For the sake of simplicity, an associated flow law is considered dσ = De [dε − dλ n]

(17)

in which De is the elastic stiffness matrix and n = ∂F ∂σ is the gradient vector to the yield function. Further, a linear relation between and dλ = η dκ

(18)

and the maximum plastic work assumption in a local manner dκ =

dτ(κ )

(19)

h

are adopted in which h is the softening modulus. This last equation is valid during the softening flow only. We do not consider here the case where the material strength has been completely exhausted. By considering Eqs.17-19 together with Eq.14, one obtains the following equation

∫ [(Lδu)

T

Ω

]

η   T De dε dΩ − (Lδu) De n dτ  dΩ = h   Ω

∫

∫ [δu

T

]

t k +1 dΣ −

∂Ω

∫ [(Lδu)

T

Ω

]

σ k dΩ (20)

in which the strain and the local yield strength appear as the unknown fields. In the same way, the source term dτ in equation (15) should be expressed as a function of these two quantities. For this purpose, we make call to the non local yield function that reads F (σ , τ ) = f (σ ) − τ = 0

(21)

and in which the gradient dependency is made implicit through τ . Therefore, the consistency condition gives T

∂F  ∂F  dF (σ , τ ) =   dσ + dτ = n T dσ − dτ  ∂σ  ∂τ

(22)

and substitution of Eqs.17-19 into Eq.22 provides the relation giving the increment of the nonlocal yield strength dτ = − dF + nTDedε −

7

η T e n D n dτ h

(23)

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

in which dF represents the difference of values of the yield function at the current (j+1) and previous (j) iteration within the load step (k+1)

(

) (

)

dF = F σ j +1 , τ j +1 − F σ j , τ j .

(24)

The implementation of expression (23) in the second variational equation (15) governing the problem leads to η     T − δτ n T D e dε dΩ + δτ  1 + n T D e n dτ − l 2 (∇δτ ) ∇dτ  dΩ  h    Ω Ω

∫

∫

∫

= δτ τ j dΩ − Ω

∫ (δτ τ j − l

2

(∇δτ)

T

Ω

)

∫

.

(25)

∇τ j dΩ − δτ dF dΩ Ω

where it can be clearly noticed that all the implied fields only require a C0 continuity. Departing from the weak forms given by Eqs. (20) and (25), the finite element discretization is rather straightforward. Following the Galerkin approach, the displacement and strain fields are discretized according to the normal finite element procedure u=Na , ε =Ba.

(26)

where N is the interpolation matrix containing the shape functions, B is the matrix of the shape function derivatives ( B = L N ) and a is the nodal displacement vector. Further, a separate interpolation of the local yield strength is introduced τ = H Τ , ∇τ = Q Τ

(27)

where H denotes the interpolation functions, T is the nodal values vector of the local yield strength and Q = [∇Hi ] containing the gradient of interpolation functions. It is emphasized that the interpolation polynomials of u and t do not need to be of the same order. Both discretization only need to satisfy C0-continuity requirements. To avoid stress oscillations, the use of an interpolation for the displacements which is one order higher than that of the local strength seems advisable (the Babuska-Brezzi conditions for mixed finite elements in compressible solids). Substituting the above identities in Eqs. (20) and (25) and requiring that these equations hold for any admissible variations δ a and δ Τ lead to the following set of algebraic equations which describe the incremental process in the discretized gradient enhanced elastoplastic continuum K aa K  τa

K aτ  K ττ 

 da  f ek +1 − f ik   dΤ  =  k ~k     f τ − f τ 

(28)

where the elastic stiffness matrix K aa , the external force vector fe and the internal force vector fi are defined conventionally as

8

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

∫

K aa = B T D e B dΩ , f ek +1 = Ω

∫N

T

∫

(29)

n De B dΩ ,

(30)

t k +1 dΣ , f ik = BT σ k dΩ

∂Ω

Ω

the off-diagonal matrices K aτ and K τa are K aτ = −

η

∫ h B D n H dΩ , K τa = − ∫ H T

e

Ω

T T

Ω

and the gradient-dependent matrix K ττ is K ττ =



η

∫ 1 + h n

Ω

T

  De n H TH − l 2Q TQ  dΩ .  

(31)

The vectors of non-standard residual forces which emerge from the inexact fulfillment of the equation (10), defining the non local yield function, read f τ = H T (τ k − dF ) dΩ and f ~τ =

∫

Ω

∫ [H

T

]

τ k − l 2Q T∇τ k dΩ

Ω

(32)

Note that the tangent stiffness operator in the system (28) is non-symmetric due to the expression of the off-diagonal matrices K aτ and K τa . 4 ALGORITHM This algorithm is base on a return mapping schema using the well known elastic predictorplastic corrector procedure. We give here the different steps followed during the iterative process. 1. Solve for the nodal increments dε = B da and dτ = H dΤ according to Eq. (28) and update using a total incremental approach ∆ε j +1 = ∆ε j + dε , ∆τ j +1 = ∆τ j + dτ

(33)

2. Compute the strain and the local yield strength at integration points ε j +1 = ε 0 + ∆ε j +1 , τ j +1 = τ 0 + ∆τ j +1

(34)

3. Compute the elastic predictor and estimate the loading function

(

tr σ trj+1 = σ 0 + D e ∆ε j+1 , F j+1 = F σ trj+1 , τ j

tr 4. If Fj+1 ≥ 0 then compute

4.1 The new stress state

9

)

(35)

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

 η ∂F ∆σ j +1 = De ∆ε j +1 − ∆τ j+1 ∂σ h τ j +1 

( )

  ; σ j +1 = σ 0 + ∆σ j +1 σ = σ tr  

(36)

4.2 the approximate increment of nonlocal yield strength

(

) (

)

∆F = F σ j +1 , τ j +1 − F σ j , τ j ; ∆τ j +1 = − ∆Fj +1 + n TDe ∆ε j +1 −

η T e n D n ∆τ j +1 (37) h

In a first approximation the value of the local yield strength is used for evaluating Fj+1 . else τ j +1 = τ j + ∆τ j +1 , n = 0 , η h( τ ) = 0 , ∆Fj+1 = 0

(38)

For elastic points the source term τ varies by leading to a variation of the local yield strength in the surrounding elastic domain near the plastic zone and thus provides an extension of the localization zone. Indeed, as a consequence of Eq.38, in the set of equations (28) the submatrices K aτ K τa vanish, but the term f τ − f ~τ ≠ 0 such that non zero values of the local yield strength are obtained. 5. Update the internal and non standard residual forces. If the process has not converged, compute the new tangential stiffness operator and enter a new iteration. 5 NUMERICAL EXAMPLE y

d σ x

L Figure 2: Imperfect bar in tension

The performance of the model and the algorithm is illustrated with a simple one dimensional problem of bar in tension (Fig.2). In the calculations, the length of the bar is L = 100 mm , Young’s modulus E = 20000 N mm2 , the tensile strength f t = 2 N mm2 and the softening modulus h = 2000 N mm2 (a linear softening is considered). The internal length scale is set to l = 5 mm . At the center of the bar, an imperfection zone (d = 5 mm) with a 10% smaller value of ft is assumed. Calculations have been carried out for meshes consisting of 40, 80 and 160 one dimensional elements with a quadratic interpolation for the displacements and a linear interpolation for the local yield strength. They are stopped before the local yield strength vanishes.

10

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

[

σ N mm 2

2.0

]

[ ]

ε 10 −4 9.0

1.5 6.0

40 elements 80 elements 160 elements

1.0 0.5

[

u mm × 10 −3

0.0

]

x [mm] 0.0

0 2.2

3.0

4

[

τ N mm 2

8

12

16

20

0

]

20

[

τ N mm 2

40

60

80

100

]

2.0 1.8

1.0

1.4

x [mm]

1.0 0

20

40

60

80

x [mm]

0.0 0

100

20

40

60

80

100

Figure 3: Load elongation diagrams for 40, 80 and 160 elements (a). Evolution of the strain (b), local (c) and non local (d) yield strengths along the bar for progressive loading.

The load-elongation curves for the three discretizations are depicted in Fig.3 and show a convergence towards a solution with a mesh independent, finite energy dissipation. The same slope of the stress-displacement diagram is obtained upon mesh refinement. Furthermore, this figure shows the evolution of the strain, local and non local yield strengths in the bar for progressive loading. It can be observed that the algorithm allows to the source term τ to vary outside the imperfect zone such that the local yield strength follows. The localization zone can therefore extend to reach a finite size.

11

Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

6. CONCLUSION A new variational approach in gradient plasticity has been derived. It consists in a weak satisfaction of the partial differential equation giving the non local history parameter, which is obtained from the averaging integral procedure. An advantage of this approach is that boundary conditions with regard to the second field can be taken into account in a natural way, and at the fixed external boundary of the body. Further, this algorithm requires only a C0 continuous interpolation of the two fields. The incorporation of the gradient term in the mathematical description introduces an internal length scale in the model. The one dimensional example illustrates that this internal length scale preserves the well-posedness of the problem in the softening regime, same as in usual gradient plasticity models. Finally, some extension of the model should be considered, namely the incorporation of non linear softening flow and the treatment of the elastic zone and when the material strength has been exhausted. REFERENCES [1] T. Belytschko and D. Lasry, A study of localization limiters for strain-softening in statics and dynamics. Comp. & Struct., 33, 707-715 (1989). [2] Z.P. Bazant and G. Pijaudier-Cabot, Non-local continuum damage, localization instability and convergence. ASME J. Appl. Mech., 55, 287-293 (1988). [3] A. Benallal, R. de Borst and O.M. Heeres, A gradient enhanced damage model: theory and computation. In : D.R.J. Owen, E. Onate and E. Hinlton (Eds.) : Proc. fifth. Int. Conf. on Computational Plasticity , Theory and Applications. Barcelona : CIMNE, 373-380 (1997). [4] R de Borst and H.B. Muhlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects. Int. J. Num. Meth. Eng., 35, 521-539 (1992). [5] R. de Borst, L.J. Sluys, H.B. Muhlhaus, and J. Pamin, Fundamental issues in finite element analyses of localization of deformation. Eng. Comput., 10, 99-121 (1993). [6] C. Comi and U. Perego, A generalized variable formulation for gradient dependent plasticity softening plasticity. Int. J. Num. Meth. Eng., 39, 3731-3755 (1996). [7] D. Lasry and T. Belytschko, Localization limiters in transient problems. Int. J. Solids Struct., 24, 581-597 (1988). [8] F. Meftah, Contribution à l’etude numerique des modes localisés de rupture dans les structures en bétons de type poutre. approche multicouches par la plasticité au gradient. Ph.D. Thesis, INSA of Lyon, France, 220, (1997). [9] F. Meftah and J. M. Reynouard, A multilayered beam element in gradient plasticity for the analysis of localized failure modes. In Int. J. Mech. of Coh. and Frict. Mat., Accepted for publication September 1997. [10] H.B. Muhlhaus and E.C. Aifantis, A variational principle for gradient plasticity. Int. J. Solids Struct., 28, 845-857 (1991).

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Fékri Meftah, Gilles Pijaudier-Cabot, and Jean M. Reynouard

[11] J. Pamin, Gradient-dependent plasticity in numerical simularion of localization phenomena. Dissertation, Delft University of Technology, Delft, The Netherlands, 134, (1994). [12] R.H.J Peerlings, R. de Borst, W.A.M. Brekelmans and J.H.P. de Vree, Computational modelling of gradient enhanced damage for fracture and fatigue problems. In : D.R.J. Owen, E. Onate and E. Hinlton (Eds.) : Proc. Fourth Int. Conf. on Computational Plasticity , Theory and Applications. Swansea : Pineridge Press, 975-986 (1995). [13] G. Pijaudier-Cabot and Z.P. Bazant, Nonlocal damage theory. ASCE J. Eng. Mech., 113, 1512-1533 (1987). [14] G. Pijaudier-Cabot and Z.P. Bazant and M. Tabbara, Comparison of various models for strain softening. Eng. Comput., 5, 141-150 (1988). [15] L.J. Sluys, Wave propagation, localization and dispersion in softening solids. Dissertation, Delft University of Technology, Delft, 173 (1992). [16] O.C. Zienkiewicz O.C. and R.L. Taylor, The Finite Element Method, Fourth edition. London : McGraw-Hill, Vol. 1, (1991).

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