European Congress on Computational Methods in Applied Sciences and Engineering

ECCOMAS 2000

Barcelona, 11-14 September 2000c©ECCOMAS

A THERMODYNAMICALLY CONSISTENT NON LOCALFORMULATION FOR ELASTO-DAMAGING MATERIALS:

THEORY AND COMPUTATIONS

Elena Benvenuti?, Guido Borino†, and Antonio Tralli?

? Department of Engineering, University of Ferraravia G. Saragat 1, 44100 Ferrara, Italy

E-mail: ebenvenuti@ing.unife.it, atralli@ing.unife.it

†Department of Structural & Geotechnical Engineering, University of PalermoViale delle Scienze, 90128 Palermo, Italy

E-mail:borino@unipa.it

Key words: Nonlocal models, Damage Mechanics, Thermodynamics, F.E. formulations

Abstract. A thermodynamically consistent nonlocal continuum damage model of integraltype is presented. It can be considered an extension to Continuum Damage Mechanicsof a similar approach presented by Polizzotto, Borino et al. for softening plasticity [1,2, 3]. The correspondent finite element implementation and related numerical issues arediscussed. A two-dimensional example is analyzed by finite element technique.

1

Elena Benvenuti, Guido Borino, and Antonio Tralli

1 INTRODUCTION

In a so-called elastic-damaging material, after an elastic threshold is reached, an ir-reversible process is activated which induces an irreversible degradation of the elasticproperties of the material [4]. Among several damage-induced effects, one of the mostrelevant is related to the fact that the strength of the material eventually decreases withincreasing strain. This phenomenon, known as strain softening, influences deeply theanalysis procedure.

It is well known that, either in case of plasticity generated softening or damage inducedsoftening, some specific difficulties arise due to constitutive instability [5, 6]. Local theorieslead to solutions where the strain field localizes in a band with zero width measure and,consequently, the collapse occurs with zero global dissipation of energy. This physicallymeaningless condition has revealed an intrinsic limit of traditional formulations appliedto softening materials. Such a difficulty can be alleviated by invoking regularizationtechniques based on the introduction of an internal length. The internal length can beconsidered a constitutive property of the material.

Micropolar theories, visco-plastic and visco-damaging theories, as well as gradient the-ories have been successively applied to regularize problems with softening. Alternatively,adoption of the so called fully nonlocal or integral theories offer both robust regularizationaction and physically sound interpretations [7]. In case of softening material it has beenobserved that, to overcome the previously quoted difficulties, it is possible to regularizeonly a suitable scalar measure associated to the softening process, preserving the localityof the rest of the variables involved in the damaging mechanism [5, 10]. Therefore, ameaningful simplification arises with respect to fully nonlocal theories [8, 9], where non-locality affects all the variables, because elastic constitutive equations, equilibrium andcompatibility equations are still of local type.

The first part of the present contribution is devoted to present a nonlocal formulationof integral type for elastic-damaging materials. As a consequence of the nonlocality, theclassic second principle of thermodynamics can be satisfied only in a weak form, i.e. ona finite volume, whereas its pointwise form contains an additional term, the nonlocalityresidual, which is related to the energy exchange of the mutually interacting particles.The thermodynamic forces conjugated to the internal variables which govern the damageprocess are then obtained in a thermodynamically consistent way following [1, 2, 3]. Then,a maximum dissipation principle for damaging materials (section 3) and the damage loadconditions formulated as an integral complementarity problem (section 4) will be brieflydiscussed [12].

Finally, the numerical formulation of the presented theory is considered. The incremen-tal problem is solved as usual by means of a classic implicit Euler backward technique [13].Fulfilment of non linear constitutive equations is performed through a predictor-correctoriteration scheme. Typical of the present nonlocal formulation is the iterative structure ofthe corrector phase [14]. In fact, because of the nonlocal nature of the resulting damage

2

Elena Benvenuti, Guido Borino, and Antonio Tralli

loading function, the damage consistency conditions can not be enforced localwise, sothat an additional inner iterative cycle has to be solved inside the equilibrium loop. Thepaper is then completed by finite element results relative to a two dimensional example.

2 THERMODYNAMICAL CONSIDERATIONS AND BASIC ASSUMP-

TIONS

Let us consider an elastic damaging material for which the Helmholtz free energy canbe written as

ψ(ε, d, ξ) =1

2ε : C(d) : ε+

1

2hξ2 , (1)

where ε is the (infinitesimal) strain tensor, d is the damage parameter, ξ is an internalvariable which describes the damage evolution. In this formulation the case of isotropicdamage model is considered, so that d is a scalar variable ranging from 0, for soundmaterial, to 1, for a fully damaged material. The first term represents the elastic strainenergy of the damaged material, whereas the second one is an internal energy, i.e. theenergy stored inside the microstructure because of the modifications of its natural state.The hypothesis of linear elasticity leads to write the elastic energy as a quadratic formof the strains and the constitutive tensor as a function of the damage d. More precisely,C = f(d)C0, C0 being the constitutive tensor of the sound material, where f(d) = 1− din case of linear damage, and f(d) = (1− d)2 in case of quadratic damage.

In analogy with the elastic strain energy, the internal energy (see the second term in(1)) has been assumed as a quadratic form of the internal variable, and consequently alinear hardening law follows, where h > 0 is the damaging hardening modulus.

As above indicated, the nonlocality should affect only the internal variables [2, 10]. In(1) two internal variables appear: d and ξ. Generally, three alternatives are possible: wecan regularize d, ξ or both. As already remarked in [10] one nonlocal variable is sufficientto regularize the problem. Here, ξ is treated as nonlocal. The mechanical meaning of thischoice is that the increment of strength to the further development of the damage at apoint depends not only on the damage at that point but also on the damage level reachedinside the rest of the body.

Therefore, ξ is written as a nonlocal internal variable, which is obtained as the resultof a regularization carried out on a correspondent internal local variable κ. The nonlocaloperator R transforms scalar field κ(x) in V into scalar ξ(x) in V by means of the integralrelation

ξ(x) = R(κ)|x ≡1

Vr(x)

∫

V

α(‖x− y‖)κ(y)dV (y) , (2)

where α(r) is a weight function depending only on the distance r = ‖x−y‖, and it is notincreasing (α(r1) ≥ α(r2), r1 < r2). Moreover, Vr is the reference volume

Vr(x) =

∫

V

α(‖x− y‖)dV (y) . (3)

3

Elena Benvenuti, Guido Borino, and Antonio Tralli

The weight function α(r) is generally chosen as the gaussian error function, α(r) =e−(kr/l)

2

, but can be also taken as a constant function defined over a ball with prescribedradius [12]. In any case, the definition of α(r) introduces an internal length l.

Let us impose that this constitutive model satisfies the second principle of thermody-namics. In a nonlocal framework, the Clausius Duhem inequality has to be written in aglobal form over the entire domain V

∫

V

(σ : ε− ψ) dV ≥ 0. (4)

Expression (4) can be written in a pointwise form after introduction of the nonlocalityresidual P (x), which takes into account the energy exchange between adjacent particles,

D = σ : ε− ψ + P ≥ 0 in V, (5)

where D indicates the intrinsic local dissipation. Assuming that the body is a thermody-namically isolated system, the insulation condition

∫

V

P dV = 0 (6)

holds. Substitution of (1) into (5) leads to

D = σ : ε−∂ψ

∂ε: ε−

∂ψ

∂dd−

∂ψ

∂ξξ + P ≥ 0 in V (7)

which has to hold for each possible deformation mechanism of irreversible or reversibletype. Moreover, let us introduce the state laws

σ =∂ψ

∂ε= f(d)C0 : ε (8a)

Y := −∂ψ

∂d= −

1

2

∂f

∂dε : C0 : ε (8b)

χ :=∂ψ

∂ξ= hξ = hR(κ) , (8c)

so that the explicit form of the dissipation reads

D = Y d− χR(κ) + P ≥ 0 , (8)

where Y is the thermodynamic force conjugated to the damage d and represents theenergy released for an unitary increment of damage, and χ represents the thermodynamicforce associated to the nonlocal variable ξ.

The dissipative irreversible mechanism associated to the damage is governed by thecouple of variables d, κ. The dissipation D should thus be written as a bilinear form ofthe type

D = Y d−Xκ ≥ 0 in V, (8)

4

Elena Benvenuti, Guido Borino, and Antonio Tralli

where Y is defined as above and X represents the nonlocal force thermodynamicallyassociated to the increment of the local variable κ. X evaluated at the point x ∈ V repre-sents the energy that the particles of the domain V transmit to x during the irreversiblemechanism. By comparing equations (2) and (2), we can specify P as

P = χξ −Xκ = χR(κ)−Xκ . (8)

The imposition of the insulation condition (6) leads to

∫

V

(χR(κ)−Xκ) dV = 0 (8)

for each dissipative mechanism and thus for each κ. It is easy to show that the followingidentity holds (Green identity for the operator R)

∫

V

χR(κ) dV =

∫

V

R∗(χ)κ dV, (8)

where R∗ is the adjoint operator of R defined by

R∗(χ)∣

∣

∣

x

=

∫

V

1

Vr(y)α(‖x− y‖)χ(y) dV (y) . (8)

Let us note that only when the zone of the damage process Vd is sufficiently far from theboundary of the body, also considering that the weight function α(r) tends to zero forincreasing r, the characteristic volume can be considered as constant, Vr(x) ' V0 ∀x ∈ Vd,and, therefore, it follows that, in this case, the integral operator is selfadjoint, R∗ ≡ R.

By considering identity (2), equation (2) turns out to be

∫

V

[R∗(χ)−X]κ dV = 0 . (8)

Because (2) holds for any possible damage mechanism and thus for any arbitrary choiceof κ, it follows that

X = R∗(χ) in V . (8)

After substitution of (2) in (2), the dissipation D can be written in the explicit form

D = Y d−R∗(χ)κ ≥ 0 . (8)

3 MAXIMUM DISSIPATION PRINCIPLE

Let us assume the existence of a damage activation function, φd(Y,X) ≤ 0, whichcoherently with the expression of the dissipation (2), is expressed as a function of the

5

Elena Benvenuti, Guido Borino, and Antonio Tralli

thermodynamic forces X and Y dual to d and κ. Under the hypothesis of associativebehavior, so that φd identifies with damage potential, we can write

d =∂φd∂Y

λd, κ = −∂φd∂X

λd in V

φd(Y,X) ≤ 0, λd ≥ 0, φdλd = 0 in V .

These are the usual laws for generalized standard materials [4]. Let us remark that at aprescribed point x the variables d and κ can be or both zero or both different from zero.Relations (9) are completely analogous to the correspondent relations characterizing localformulations. Unlike the local case, here, X = R∗(χ) and therefore (9) form a system ofspatially coupled relations. For the sake of simplicity, a damage function of the form

φd(Y,X) ≡ Y −R∗(χ)− Y0 ≤ 0 (9)

is assumed.An extension to Damage Continuum Mechanics of the plasticity principle of maximum

dissipation has been presented in [12]. The state of the material correspondent to aprescribed damaging mechanism maximizes the function of intrinsic dissipation (or theentropy production). If the mechanism is characterized by the fields d ≥ 0 and κ ≥ 0 onV , the principle can be written as

maxY,X

∫

V

(Y d−Xκ) dV subject to φd(Y,X) ≤ 0 in V .

which expresses the analogous of the Hill Theorem of the local plasticity in the frameworkof the Damage Mechanics with nonlocal dissipation mechanisms. It can be showed thatthe stationarity conditions of the problem (3) coincide with relations (9).

Alternatively, the damaging mechanism can be described through the fields d an ξ andconsequently the principle reads

maxY,χ

∫

V

(Y d− χξ) dV subject to φd(Y,R∗(χ)) ≤ 0 in V .

The stationarity conditions of the above problem can be calculated by applying themethod of the Lagrange multipliers. Therefore, we boil down to calculate the saddlepoints of the functional

L(Y, χ, λd) = −

∫

V

(Y d− χξ) dV +

∫

V

λdφd(Y,R∗(χ)) dV, λd ≥ 0 in V . (9)

The first variation of (3) leads to

δL = −

∫

V

δY (d−∂φd∂Y

λd) dV +

∫

V

(δχξ +∂φd∂R∗

R∗(δχ)λd) dV +

∫

V

δλdφd dV, (10)

6

Elena Benvenuti, Guido Borino, and Antonio Tralli

which, after the replacement

∫

V

∂φd∂R∗

R∗(δχ)λd =

∫

V

R(λd∂φd∂R∗

)δχ dV , (10)

gets

δL = −

∫

V

δY (d−∂φd∂Y

λd) dV +

∫

V

δχ(ξ +R(λd∂φd∂R∗

)) dV +

∫

V

δλdφd dV. (10)

From (3), and by considering that κ = λd∂φd/∂R∗(χ), relationships (9) follow.

4 DAMAGE LOAD CONDITIONS

The response to a given increment ε at the points x ∈ Vλ ⊆ Vd where the damageprocess is active (λd > 0) is obtained by solving the equation φd = 0.

A variational formulation in λd has been provided in [12] where, under the hypothesisof associativity, the equations of the integral complementarity problem

φd(Y,X) = Y − hR∗(R(λd)) ≤ 0,

λd ≥ 0 , φdλd = 0 in Vd .

follow from the stationarity (Eulero-Lagrange) conditions of a suitable functional.For instance, in case of linear isotropic damage C = (1 − d)C0 and after substitution

of (2) and (2) into (11a), the following integral equation is obtained

φd = ε : C0 : ε− h

∫

V

1

Vr(y)2α(‖x− y‖)

∫

V

α‖y − z‖λd(z) dV (z)dV (y) = 0 in Vλ ,

(11)which is an integral equation of Fredholm of first kind.

Analogously, in case of quadratic isotropic damage, so that C(d) = (1− d)2C0, it canbe showed [12] that

φd = 2(1− d)ε : C0 : ε− ε : C0 : ελd−

− h

∫

V

1

Vr(y)2α(‖x− y‖)

∫

V

α‖y − z‖λd(z) dV (z)dV (y) = 0 in Vλ ,

which is an integral equation of Fredholm of second kind and usually can be solved moreeasily.

5 FINITE ELEMENT SOLUTION

The incremental solution procedure can be briefly summarized [13] as the ensemble of

• an incremental loading procedure,

7

Elena Benvenuti, Guido Borino, and Antonio Tralli

• an equilibrium iteration procedure, at each prescribed load increment.

Inside the equilibrium loop, the integration of the constitutive equations between theprevious equilibrium state and the new estimated one has to be carried out. Let usindicate with ∆ the increment of a given quantity between these two states. The discreteform of the consistency conditions can be written as (implicit backward Euler scheme)

φd(d+∆d) ≤ 0, ∆d φd(d+∆d) = 0, ∆d ≥ 0 (if φd(d) = 0) (11)

and must hold at the points x ∈ Vλ ⊆ Vd where the damaging process is active. Here, thecoincidence of d and λ has been exploited, thanks to the particular choice of the damagefunction (3).

The iterative procedure ends when the difference between external and internal forcesis less than a tolerance and the constitutive equations are satisfied. Typical of nonlocalmodels is the possible presence of an inner loop inside the equilibrium iterations, sincedamage consistency conditions can not be enforced pointwise. This fact will be discussedin some details in the following.

5.1 The consistency conditions

Let us suppose that the displacement field increment ∆u has been calculated andconsequently the strain increment ∆ε. It is worth noting that the procedure which leadsto satisfaction of equations (5) can be based on

• a predictor phase: at each integration point the trial value of the damage functionis evaluated

φtrd = Y tr − Y0 −X(d), (11)

where Y tr = 1/2(ε+∆ε) : C0 : (ε+∆ε) in case of linear damage orY tr = (1− d)(ε+∆ε) : C0 : (ε+∆ε) for quadratic damage, and X(d) is the valueof the thermodynamic force driving the damaging process calculated at the previousequilibrium iteration. If at some points φtrd > 0 then a corrector phase is necessary,otherwise a new equilibrium loop begins;

• a corrector phase, which, in the nonlocal formulation at hand, is constituted by afurther iterative cycle, due to the fact that the part Vλ where the damage is active isnot known in advance. The present procedure follows the scheme originally proposedby Stromberg and M. Ristinmaa [14] for nonlocal plasticity problems.

Let us calculate indeed ∆d at the integration points where φd = 0 by solving φd(d+∆d) =0 with respect to the unknown ∆d. To this purpose, because the value at a point dependson that (unknown) of the surrounding points, a trial distribution of ∆d has to be assumed.The result of the calculation is that a new distribution of ∆d is found. If we evaluate againφd(d +∆d) at each integration point, a new volume Vλ emerges: some additional points

8

Elena Benvenuti, Guido Borino, and Antonio Tralli

can be now involved into the dissipative process, and some others can cease to be active(φd(d+∆d) < 0). A new loop is necessary in order to restore fulfillment of equations (5)everywhere. But this means that we need a further iterative cycle inside the equilibriumiterative cycle. The cycle stops when relationships (5) hold at each integration point.

Now, let us focus our attention on the numerical strategy adopted to solve the con-sistency conditions (corrector phase). Let us denote with δ the increment of a quantitybetween two iterations of the loop for the integration of the consistency equations.

According to expression (2), X is evaluated at each integration point x through thenonlocal relation

X(d+∆d)(x) =

∫

V

α(‖x− y‖)

V 2r (y)

∫

V

α(‖y − z‖) (d(z) + ∆d(z)) dV (z) dV (y) , (11)

so that the equation

φtrd (d+∆d) = Y tr(d+∆d)− Y0 −X(d+∆d) = 0, (11)

is a Fredholm integral equation (of the first kind in case of linear damage, and of the secondkind, in case of quadratic damage law). To avoid to solve them, which is quite awkward,a modified Newton Raphson technique is applied. From linearization of equation (5.1),

φtr(d+∆D + δd) = φtr(d+∆D) +∂φtr

∂∆d

∣

∣

∣

d+∆Dδd = 0 , (11)

so that

∆d = ∆d+ δd = ∆d− φtr(

∂φtr

∂∆d

)−1

, (11)

follows. Instead of performing the inversion of the matrix ∂φtr/∂∆d, the following ap-proximation is introduced

∂φtr

∂∆d= h I, (11)

in case of linear damage, and

∂φtr

∂∆d= [h+ (ε+∆ε) : C0 : (ε+∆ε)]I, (11)

in case of quadratic damage, where I is the identity matrix. Therefore, at each iterationthe vector of increments of d is obtained, and substituted into the nonlocal expression(5.1), by solving an (approximated) linearized version of the consistency conditions (5).This is equivalent to consider at each iteration the body as local and turns out to representa small approximation to the true solution of the integral equation.RemarksThe iterative loop inside the corrector phase could not be required. For instance, in a very

9

Elena Benvenuti, Guido Borino, and Antonio Tralli

recent paper [15], the damage process is governed by the vector of the total strain andtherefore the active damage process zone Vλ is known in advance once the displacementvector has been calculated, with the appreciable advantage of saving of memory andcomputational time. This corresponds to an alternative version of the formulation athand, where instead of making recourse to an inner loop, only one iteration is performedand nonlocality is enforced at the level of the equilibrium iterations.

6 NUMERICAL EXAMPLES

In a first step, the (local) constitutive laws correspondent to assumption of linear andquadratic damage have been compared, in view of a good agreement between the stress-strain response of the model and the presumed behavior of the class of materials we aredealing with. As Figure 1 reveals, a linear damage relation could be employed in orderto describe quasi-brittle materials as certain rocks, whereas a quadratic damage basedformulation seems to apply better to materials whose residual strength decays slowly tozero for increasing strain as, for instance, some kind of concrete, see the local stress-strainrelation in Figure 1.1. Analogously, the damage-strain relation (Figure 1.2) is very sharpin the former case in comparison with the correspondent relation associated to the latterone.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ε

σ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

0

0.2

0.4

0.6

0.8

1

Comparison beween local damage evolution for different models

Figure 1: Comparison between linear, continuous line, and quadratic damage, dotted line: local consti-tutive relations and damage evolution.

In this paper, a quadratic damage evolution law has been considered, characterized byYoung modulus E=20000 N/mm2, a Poisson ratio ν = 0.3, a hardening modulus h =0.0008 N/mm2, and a damage threshold Y0 = 0.0001 N/mm2.

The example of a square plate in plane stress condition with dimension L = 200 mmis studied, where the nodes at the bottom are fixed, and a traction load is uniformly dis-

10

Elena Benvenuti, Guido Borino, and Antonio Tralli

tributed on the nodes at the top. The plate is discretized by means of 8-noded serendipityelements with 4 integration points. The strain localization is triggered by lowering by 5%the damage energy release rate Y0 in a square zone with surface equal to 1/4 of the totalsurface of the plate, placed at the right bottom corner. The loading procedure is controlledby an arc-length technique [17].

Figure 2: From left to right: Mesh a), b), c)

In order to establish the effective mesh independence of the proposed model, the load-displacements relations obtained by considering the meshes a), b), c) indicated in Figure2 have been compared both for the cases of local and nonlocal constitutive equation(quadratic damage). The displacement which has been considered is that of the nodeplaced on the symmetry axis at the loaded side, where the load is the resultant of thedistributed load. A Gaussian function of the form e−4r

2/lc2 with lc = 30 mm assigns theweights to points placed in the same influence volume. It may be of some interest toobserve that, in this case, the presence of four Gauss points for each element leads tohighlight the nonlocal effect even with a coarse mesh. The dominating behavior of thenonlocal model at hand, which emerges from Figure 3, is of snap back type for each mesh,whereas the local model would exhibit a global softening behavior for the coarse meshand snap back after refining the discretization.

To explain such a result, the particular constitutive law should be considered with at-tention. As it is well known, when the constitutive law can be approximated as bilinear,the probability that a global snap back response is measured increases for increasing val-ues of the ratio between the strain correspondent to the peak stress εp and the ultimatestrain correspondent to completely damaged material εu (d=1). Even if here a quadraticdamage law is used, and thus the (local) strength tends asymptotically to zero for in-creasing value of the strain, the influence of the last part of the constitutive law, doesnot influence the global load displacement response. The sharply descending first partof the post-peak branch dominates (brittleness prevails on ductility) and the incrementalprocedure interprets the behavior as brittle with ultimate strain εu ∼= 3εp. This effect hasbeen already observed in literature [16] for a gradient type regularization technique andshould be interpreted as a consequence of the choice of treating as nonlocal the force ther-modynamically conjugated to the internal variable responsible for the damaging process

11

Elena Benvenuti, Guido Borino, and Antonio Tralli

0 0.02 0.04 0.060

50

100

150

200

250

300

350

u [mm]

P [N

]

l,a)

l, b)

nl, a), b)

nl, c)

Figure 3: Load displacement comparison for different meshes (a,b,c): local (l) and nonlocal (nl) response.

κ and of the particular form of the damage function which has been assumed, regardlessof the relation f(d) (e.g. linear or quadratic) which relates C to C0.

More extensive numerical experimentations and a comparative analysis with a gradientformulations are still in progress.

7 CONCLUSIONS

In this paper some theoretical and numerical aspects are presented of a thermody-namically consistent nonlocal formulation of integral type for elastic-damaging materials.Only the internal variable, which describe the isotropic damage hardening, is regularized,and the second principle of Thermodymamics is imposed in a nonlocal form. In particu-lar, this formulation represents an extension to nonlocal elastic-damaging materials of atheory recently developed by Polizzotto, Borino et al. [1, 2, 3] in the framework of theThermodymamics with internal variables applied to nonlocal plastic materials with soft-ening. Moreover, the theorem of maximum dissipation is extended to the case of nonlocaldamaging material and the incremental response to a prescribed field of increments ofthe total strain, both in cases of linear and quadratic damage, is studied. Finally, thenumerical formulation of this thermodynamically consistent nonlocal model is presented.In particular, due to the choice of the damage loading function, the process zone is notknown in advance so that an additional inner loop is required inside the equilibrium itera-tions cycle. The load displacement relations in the cases of local and nonlocal constitutiverelationships are then compared in a two-dimensional case.

12

Elena Benvenuti, Guido Borino, and Antonio Tralli

8 ACKNOWLEDGEMENTS

E. Benvenuti and A. Tralli gratefully acknowledge the financial support of the researchproject MURST 1998 Composite materials in civil constructions: emerging applicationsand structural modeling.

G. Borino gratefully acknowledges the financial support of the research project MURST1998 Structural integrity assessment of large dams.

13

Elena Benvenuti, Guido Borino, and Antonio Tralli

REFERENCES

[1] C. Polizzotto, G. Borino, P. Fuschi, “A thermodynamically consistent formulation ofnonlocal and gradient plasticity”, Mech. Res. Comm., 25, 75-82 (1997).

[2] G. Borino, P. Fuschi, C. Polizzotto, “A Thermodynamic Approach to nonlocal Plas-ticity and Related Variational Principles ” J. of Appl. Mech., 66, 952-963 (1999).

[3] C. Polizzotto, G. Borino, “A Thermodynamics-based formulation of gradient depen-dent plasticity”, Eur. J. of Mech. A/Solids, 17, - 4399-4419 (1998).

[4] J. Lemaitre, J.L. Chaboche, Mechanics of Solid Materials, Cambridge UniversityPress, (1990).

[5] Z.P. Bazant, G. Pijaudier Cabot, “Nonlocal continuum damage, localization insta-bilities and convergence”, J. of Appl. Mech., 55, 287-293 (1987).

[6] Z. P. Bazant, F.-B. Lin, “Non-Local Yield Limit Degradation”, I. Jour. Num. Meth.Engrg., 26, 1805-1823 (1988).

[7] Z. P. Bazant, “Nonlocal damage theory based on micromechanics of crack interac-tions”, J. of Engrg. Mech. ASCE, 120, 593-617 (1994).

[8] D.B.G. Edelen, N. Laws, “On the Thermodynamics of Systems with Nonlocality”Arch. Rational Mech. Anal., 43, 24-35 (1971).

[9] A.C. Eringen, “Theories of Nonlocal Plasticity”, Int. J. Engng.Sci., 21, 741-751(1983).

[10] G. Pijaudier-Cabot, Z. P. Bazant, “Nonlocal damage theory”, J. of Engrg. Mech.ASCE, 113, 1512-1533 (1987).

[11] E. Benvenuti, A. Tralli, “On Nonlocal F.E. Models for Strain Localization and Frac-ture Analysis”, in CD Proc. of Eur. Conf. on Computational Mechanics, ECCM ’99Munchen, Germany (1999).

[12] E. Benvenuti, G. Borino, C. Polizzotto, A. Tralli, Formulazioni nonlocali di tipointegrale per materiali elasto-danneggiativi, in CD AIMETA ’99, Como 6-9 Sept.1999.

[13] J.C. Simo and T.J.R. Hughes, Computational inelasticity, Springer Verlag New York(1998).

[14] L. Stromberg and M. Ristinmaa, “F.E. formulation of a nonlocal plasticity theory”,Comp. Methods Appl. Mech. Engrg. 136, 127-144 (1996).

14

Elena Benvenuti, Guido Borino, and Antonio Tralli

[15] C. Comi, “A nonlocal model with tension and compression damage mechanics”, sub-mitted Eur. J. Mech. A/Solids (2000).

[16] R. de Borst, O.M. Heeres, A. Benallal, “A gradient enhanced damage model: theoryand computation”, Computational Plasticity, Owen, Onate, Hinton Eds. (1997).

[17] R. de Borst, Non linear analysis of frictional materials, Ph.D thesis, Delft Universityof Technology, Delft (1986).

15