Criteria for mesh refinement in nonlocal damage finite element analyses

itivity inlem.echanical

ntal erroron can beage growthalization

n is stillcorporatehould bence due

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European Journal of Mechanics A/Solids 23 (2004) 615–632

Criteria for mesh refinement in nonlocal damagefinite element analyses

Claudia Comi, Umberto Perego∗

Department of Structural Engineering, University of Technology (Politecnico) of Milan, p.zza L. da Vinci, 32, 20133 Milan, Italy

Received 24 April 2003; accepted 20 March 2004

Available online 18 May 2004

Abstract

The strain softening behavior due to the development of damage is well known to produce unrealistic mesh sensstandard finite element applications. In the present paper, a nonlocal continuum damage model is used to regularize the probThe accuracy of the nonlocal finite element results is assessed introducing a nonlocal error estimator based on the “merror” concept brought about by Ladevèze. To account for the dissipative nature of the problem, a time-step incremeestimator is also proposed and comparatively tested. It is shown how the obtained information about the error distributiused for the accuracy assessment of different meshes. It is also shown how mesh refinement in the late stage of damcould be effectively driven by the estimate of the process zone width, based on the analytical derivation of the locwidth. 2004 Elsevier SAS. All rights reserved.

Keywords:Isotropic damage; Nonlocal model; Error estimates; Finite elements

1. Introduction

The analysis of a large structure, like e.g. a large concrete dam, up to the ultimate state for fracture propagatioregarded as a challenging task from the computational standpoint as most of the available commercial codes do not inthe necessary technologies for this kind of analyses. In fact, a finite element code to be used for this purpose sequipped with specific procedures allowing to: (i) generate a large scale initial mesh; (ii) avoid the mesh dependeto loss of ellipticity of the governing boundary value problem and consequent strain localization in theearly stage of damagpropagation; (iii) adapt the mesh during the analysis in order to be able to resolve the strain localization band evolvdamage; (iv) account for material separation in the late stage of damage growth for vanishing material strength.

Nowadays, feature (i) is effectively implemented in most commercial codes and feature (ii) is relatively well undeven though not yet commonly implemented. Despite recent progresses concerning feature (iii) (Askes and Sluys, 200Rodriguez-Ferran and Huerta, 2000) and (iv) (Jirásek and Zimmermann, 2001; Comi et al., 2002; Simone et al., 200procedures to be used in practical engineering applications have yet to be developed. The present work addresses thethe automatic mesh adaptation (feature (iii)), proposing mechanically founded criteria for localmesh refinement. The transitiofrom a damaged continuum to a cracked continuum in the late stage of the localization process (feature (iv)) is not dhere.

In this study, mesh dependence is avoided through the formulation of a nonlocal model in the line of what originally pby Bazant and Pijaudier-Cabot (Pijaudier-Cabot and Bazant, 1987; Bazant and Pijaudier-Cabot, 1988) and more re

* Corresponding author.E-mail address:[email protected] (U. Perego).

0997-7538/$ – see front matter 2004 Elsevier SAS. All rights reserved.doi:10.1016/j.euromechsol.2004.03.006

616 C. Comi, U. Perego / European Journal of Mechanics A/Solids 23 (2004) 615–632

Comi (2001) (see, e.g., Jirásek for a comparative assessment of different nonlocal approaches (Jirásek, 1998)). To simplify thet proposediewed asently, byadevèze

mate herep by a

in the timed since itfor more

s (2000),stimateand

referencesupon meshof work

of differenta measure

, to

fective inlization,

oo coarseas damageomparingabot andtion of an thereforensideringe notched

ordiscussed

by

discussion, reference is made to the basic isotropic damage model described in Comi and Perego (2001a). The firscriterion for mesh quality assessment is based on the definition of an “a posteriori” error estimate. This can be van extension to the present nonlocal damage context of the concept of error in theconstitutive law introduced by Ladevèz(1985), Ladevèze and Moës (1997) for plasticity and viscoplasticity models with internal variables and, more receLadevèze et al. (1999) for coupled damage-plasticity problems. It is also worth mentioning the recent monograph (Land Pelle, 2001) where these concepts are expounded in a systematic way. The particular form of the error esticonsidered concerns a problem already discretized in time by a sequence of time-steps and integrated over each time-stebackward-difference scheme. This means that only space discretization errors are estimated while the errors involveddiscretization are not accounted for (Comi and Perego (2001b)). This is not a limitation for the model here considerecan be integrated exactly in time under the assumption of continuous loading. However, this is generally not the casecomplex models. A different mesh adaptation strategy for strain localization problems can be found in Askes and Sluywhere in the context of an ALE formulation an empirical remesh indicator is defined. An alternative approach for error ewith nonlocal damage models, based on a residual-type error estimator, has been recently proposed by Rodriguez-FerranHuerta (2000). As usual in this type of estimates, the error is computed comparing the current finite-step solution to asolution obtained solving a series of local problems defined on fine patches. Thenonlocality of the material response intervenein the computation inasmuch as it guarantees well posedness of the boundary value problem and hence, convergencerefinement of the finite element approximation. Following a different approach Lackner and Mang employed a rateincremental error estimator in finite element analyses of concrete structures (Lackner and Mang, 2001).

Ladevèze’s error estimator, adapted to the considered nonlocal framework, has been used to assess the accuracydiscretizations of Hassanzadeh’s four edge notched tension test (Hassanzadeh, 1991). Since the error is based onof the violation of the state equations at a given time, it turns out to be a non-monotonically growing quantity. A modified,stepwise incremental version of the estimator has therefore been introduced and critically compared to the original versionaccount for the dissipative nature of the problem.

The proposed error estimator provides useful information in the pre-localization stage but tends to become less efthe late stages of damage propagation. As it is well known, to obtain meaningful results after the inception of strain locaeven with a nonlocal model the mesh has to be fine enough to resolve the localization band. When the mesh is tcompared to the bandwidth, one can either refine the mesh or introduce a discrete crack. Since the bandwidth reducesgrows, a suitable indicator for the necessity of mesh refinement can be obtained estimating the current bandwidth and cit to the size of the elements across the band. Following the bifurcation analysis procedure described in Pijaudier-CBenallal (1993), the model localization width can be derived analytically as a function of damage under the assumphomogeneous plane strain state of deformation. A critical damage, beyond which the mesh becomes inadequate, cabe computed for each element, depending on its size. A similar result was obtained in Comi and Perego (2001b) coa one-dimensional dynamic perturbation analysis. Again, the proposed indicator is tested on Hassanzadeh’s four edgtension test and critically discussed.

2. Isotropic nonlocal damage model

2.1. Material model definition

Reference is made to a local isotropic elastic-damage model based on the definition of a free energy densityΨ dependingon a scalar damage variabled and a scalar kinematic internal variableξ , as proposed in Ladevèze et al. (1999)

Ψ (ε, d, ξ) = 1

2(1− d)ε : E : ε + Ψin(ξ). (1)

Hereε is the small strain tensor,E is the undamaged elastic tensor andΨin(ξ) is the convex inelastic potential accounting fthe evolution of the domain of linear elastic behaviour as a consequence of the activation of damage mechanisms, ase.g. in Hansen (1994), Ladevèze et al. (1999), Borino et al. (1996). The static quantities, stressσ , strain energy release rateYand static internal variableχ , conjugated toε, d andξ , respectively, are obtained through the state equations

σ = ∂Ψ

∂ε; Y = −∂Ψ

∂d; χ = ∂Ψ

∂ξ. (2)

The activation and evolution of damage is governed by an activation functionf , depending on the static variables, andloading–unloading complementarity conditions

f (σ , Y,χ) 0; γ 0; f γ = 0, (3)

C. Comi, U. Perego / European Journal of Mechanics A/Solids 23 (2004) 615–632 617

whereγ is a non-negative damage multiplier.

model isi (2001).l measure

ion

n

tyisms

s

internal

ociative

e

The evolution of the kinematic quantities is governed by associative evolution equations

d = ∂f

∂Yγ ; ξ = − ∂f

∂χγ . (4)

To avoid spurious mesh dependence upon strain localization in finite element analyses, a nonlocal version of theobtained following the approach originally proposed by Pijaudier-Cabot and Bazant (1987) and more recently by ComAccording to this approach, the damage activation is assumed to depend, through the activation function, on a nonlocaY of the strain energy release rateY

Y(x) =∫Ω

W(x, s)Y (s)ds, (5)

Ω being the domain of the considered body andW a weight function which is taken here to coincide with the Gauss funct

W(x, s) = 1W(x)

exp

(−‖x − s‖2

2l2c

), (6)

lc being a material parameter, usually referred to as “characteristic length” andW(x) a normalization factor, allowing to obtaia uniform nonlocal strain energy release rateY field in the presence of a uniform strain field

W(x) =∫Ω

exp

(−‖x − s‖2

2l2c

)ds. (7)

The nonlocal variableY has to be regarded as a macroscopic homogenized quantity which accounts for material heterogeneiat the microscale. The spatial range of the microscopic interaction between different material points in the damage mechanis related to the material characteristic lengthlc.

The nonlocal variableY plays a role only in the activation conditions and evolution equations which modify as follow

f (σ , Y ,χ) 0; γ 0; f γ = 0; d = ∂f

∂Y γ ; ξ = − ∂f

∂χγ .

2.2. Specific damage model considered

In this work, the inelastic potential is assumed to be defined as a monotonically increasing function of the kinematicvariableξ as follows

Ψin = k

[n−1∑i=0

n!i! lni c + (1− ξ) lnn c − (1− ξ)

n∑i=0

n!i! lni

(c

1− ξ

)], c 1, (8)

wherek, c andn are material parameters and, by definition, 0! = 1. The stressesσ , the elastic energy release rateY and theinternal variableχ are defined through the state equations

σ = ∂Ψ

∂ε= (1− d)E : ε, Y = −∂Ψ

∂d= 1

2ε : E : ε, χ = ∂Ψ

∂ξ= k lnn

(c

1− ξ

)− k lnn c. (9)

The following, particularly simple form is assumed for the activation function and, as a consequence, for the assevolution equations

f (Y ,χ) = Y − Y0 − χ =(

1

2ε : E : ε

)− k lnn

(c

1− ξ

) 0, γ 0, f γ = 0,

d = ∂f

∂Y γ = γ , ξ = − ∂f

∂χγ = γ (10)

having setY0 = k lnn c.From the evolution equations (10) it follows that, for this particular model, the internal variableξ coincides with the damag

variabled.In one-dimension (Fig. 1), the simplicity of the model allows to carry out an analytical integration. Forε > ε0, ε0 being the

strain at the linear elastic limit (from Eq. (10),ε0 = √(2k lnn c)/E), one obtains

σ = c exp

[−

(Eε2

2k

)1/n]Eε


ends to

yenergy

ofore

followeded where,d

tial points

Fig. 1. One-dimensional stress–strain curves for isotropic damage model, with horizontal initial slope, for varying exponentn.

which shows that, forn > 0, the stress tends asymptotically to vanish with a bounded fracture energy as the strain tinfinity. The fracture energy density is given by

gf = 1

2Eε2

0 +∞∫

ε0

σ dε = 1

2Eε2

0 +∞∫

ε0

c exp

[−

(Eε2

2k

)1/n]Eε dε.

Recognizing in the above integral the incomplete function, the fracture energy can be expressed in closed form as

gf = 1

2Eε2

0 + cnk exp

[−

(Eε2

02k

)1/n](n − 1)!

n−1∑i=0

1

i!(

Eε20

2k

)i/n

.

As shown in Fig. 1, the parametern in the expression (8) of the inelastic potentialΨin is directly related to the fracture energdensity and, while keeping fixed the initial slope of the hardening/softening branch, can be identified by fitting fractureand characteristic length measurements from experimental tests. The value ofn does not affect the initial linear elastic partthe stress–strain curve. For fixedn, the parameterc 1 is related to the initial slope of the hardening/softening branch. Mprecisely,c < en/2 leads to initial hardening behavior, whilec > en/2 implies initial softening;c = en/2 corresponds to aninitial horizontal slope. The third parameter in the inelastic potential (8),k, can be exploited to adjust the strainε0 at the linearelastic limit.

2.3. Finite element nonlocal stepwise incremental implementation

Due to the nonlinear dissipative nature of the material behavior, the evolution of the structural response has to bealong the whole history of loading. According to a standard procedure, a step-by-step incremental approach is pursustarting from a known current state at timetm, the structural response at timetm+1 = tm + t is sought under the assigneload increment. To this purpose, the constitutive model can be reformulated in a stepwise incremental fashion.

Let the symbol denote the increment of a quantity over them-th time-stept = tm+1 − tm. Using Eqs. (9) one obtains

σ = (1− dm − d

)E : (εm + ε

) − σm,

Y = 1

2

(εm + ε

) : E : (εm + ε) − Ym, (11)

χ = k lnn

(c

1− ξm − ξ

)− χm.

Taking into account that, for this specific model,d ≡ ξ and solving the latter equation forξ one obtains

d = 1− c

exp[(χm + χ)/k]1/n− dm. (12)

The incrementχ of the static internal variable can be obtained from the activation condition (10)1 which has to be satisfied aall time instants. Even if a monotonic increase of a loading parameter is assigned, local monotonic loading at all mater


is not guaranteed throughout the considered time-step. In the spirit of the widely used backward-difference integration (even

onding

allading at aaccuracy

roleStartingd87;

assignedn of an

benchmarkassigned

(CEB,r elementsibution ofsh 2 has a

though no numerical integration is here required), the consistency condition is enforced at the end of the step, i.e.

f = (Ym + Y ) − (χm + χ

) − Y0 = 0, (13)

where the incrementY of the nonlocal strain energy release rate is obtained, according to Eq. (5), from the corresplocal valueY . From the consistency condition (13), one has

χ = Ym + Y − χm − Y0 (14)

and the increment of damaged is obtained substitutingχ from (14) into Eq. (12). This produces an exact result atpoints where continuous damage loading along the step occurs. Whenever global loading leads to local elastic unlotime instant within the time-step, the result has to be regarded only as an approximation of the real response, with andepending on the time-step size.

Since in the present nonlocal approach, the incrementY of the nonlocal strain energy release rate plays theof the driving quantity for constitutive computations, these can be carried out locally at each Gauss point.from the current estimateε of the strain increment, the nonlocal variableY is computed via the explicit weighteintegration (5) which does not requireinformation based on the constitutive response (Pijaudier-Cabot and Bazant, 19Comi and Perego, 2001b).

The incremental state equations (12) can also be conceived as descending from a free energy density potentialΨ m, pertinentto them-th time-step, i.e.

σ = ∂Ψ m

∂ε; Y = −∂Ψ m

∂d; χ = ∂Ψ m

∂ξ, (15)

where

Ψ m(ε,d,ξ) = 1

2

(1− dm − d

)(εm + ε

) : E : (εm + ε) − σm : ε

+ Ymd + Ψin(ξm + ξ

) − χmξ. (16)

Eqs. (16) and (15) are formally analogous to Eqs. (1) and (2), governing the current state of the system under thetotal loading, and the finite-step problem here defined. This formal analogy will be used in Section 4.2 for the definitioincremental time-step error.

3. Reference numerical example

The direct tension test on a four-edge notched specimen carried out by Hassanzadeh (1991) has been selected asexample for testing the methodologies proposed in this paper. The specimen geometry is shown in Fig. 2. Theexperimental compressive strength isfcc = 50 MPa, corresponding to a concrete grade C40 in the CEB-FIP model code1990). A plane strain, finite element discretization has been adopted using the meshes of constant strain triangulashown in Fig. 3. Meshes 0–3 have a comparable number of degrees of freedom: mesh 0 has an almost uniform distrelements; mesh 1 has a refinement in the notch region to account for the stress singularity in the elastic solution; me

Fig. 2. Hassanzadeh’s direct tension test: geometry, loading condition and material parameters. Material parameters:E = 36000 MPa,ν = 0.15,k = 5.8× 10−8 Pa,c = 405,n = 12, lc = 1.1 mm.


ligamentfine meshligament

l reactiont meshes

problem,es of twomp in theplacement

Fig. 3. Hassanzadeh’s direct tension test: used meshes of constant strain triangular elements.

Fig. 4. Hassanzadeh’s direct tension test: fine mesh.

refinement in the ligament region, to account for the damage development; mesh 3 has refinements in the notch andregions but, on the other hand, is coarse in the upper and lower regions, near the constrained boundary. Finally, aof 1802 elements (hereafter mesh 4) is shown in Fig. 4. As mesh 3, it has an enhanced refinement in the notch andregions but, in addition, it exhibits also a fine discretization of the rest of the structure.

The load has been imposed under displacement control. The global response of the specimen in terms of verticaversus imposed vertical displacement, obtained using the previously described nonlocal approach with the 5 differenis compared to the experimental data (dashed curve) in Fig. 5. To account for the three-dimensionalityof the real problem,the total vertical reaction has been obtained by weighting the nodal reactions with their area of influence in the realas detailed in Comi (2001). The experimental curve (dashed in Fig. 5) is obtained as the averaging of the measurclip gauges located at the notch mouths on opposite sides of the specimen. The experimental curve exhibits a busoftening branch. This is explained by the author as a consequence of the rotation of the specimen and the uneven dis


hows

ation

Fig. 5. Hassanzadeh’s direct tension test: vertical reaction versus vertical displacement for different meshes (solid curves). Dashed curve sexperimental results.

Fig. 6. Hassanzadeh’s direct tension test: contour plots of damage formeshes 0, 1 and 4. First column: after inception of damage propag(u = 0.004 mm). Second column: peak vertical reaction (u = 0.005 mm). Third column: softening regime (u = 0.0125 mm).


distribution within the fracture zone. As expected, the numerical results (solid curves in Fig. 5) do not exhibit the bump, sincel model,

en

olutionand then

ue to theocalization

d analysesy means of

error in

pplication

roblemplies thatunted for.onditions

available

y

d

devèze–Fenchel

no rotation is allowed in the numerical model. Excellent regularization properties are exhibited by the adopted nonlocawith a response converging to a single and well defined curve as the mesh is refined.

The contour plots of damage for three different levels of imposed displacementu, corresponding to a state of the specimjust after the inception of damage propagation (u = 0.004 mm), at the peak of the vertical reaction (u = 0.005 mm) and inthe softening regime (u = 0.0125 mm), respectively, are shown in Fig. 6 for the meshes 0, 1 and 4. The damage evfor meshes 2 and 3 is almost identical to that of mesh 4 and is not shown. Damage starts to develop at the notchesexpands in the whole section, simulating the fracture of the specimen along a horizontal band. It is worth noting that, dnonlocal character of the model, the evolution of the zone of severe damage does not depend on the mesh since the lbandwidth is determined by the material internal length and by the current damage value only.

4. Mechanically based error estimation

4.1. Ladevèze’s global error estimate

When large scale structural analyses have to be dealt with, accuracy cannot be conveniently checked by repeatewith different meshes. The quality of the results and the possible need for adaptive remeshing have to be assessed bdirect information as the one which can be obtained by an error estimate.

The discretization error for the nonlocal finite element model is estimated in this paper employing the concept ofthe constitutive law introduced by Ladevèze (1985), Ladevèze and Moës (1997) for plasticity and viscoplasticity models withinternal variables and, more recently, extended to coupled damage-plasticity problems by Ladevèze et al. (1999). An aof these concepts to nonlocal damage mechanics has been anticipated in Comi and Perego (2001b).

A particular form of Ladevèze’s error indicator will be introduced here with reference to the nonlocal finite-step pdefined in Section 2.3 and integrated over each time-step adopting a backward-difference-like assumption. This imonly space discretization errors are estimated while the possible errors involved in the time discretization are not accoFor the simple model here considered, no time discretization errors are involved whenever the incremental loading cimply continuous damage loading within thetime-increment at all active Gauss points since, in this case, the constitutive law isintegrated exactly in time. It should be noted, however, that for more complex models no analytical integration may beand time-integration errors should be assessed separately.

Ladevèze’s error estimate for damage models is based on the definition of a potentialΦ(σ , d, ξ), related to the free energdensityΨ (ε, d, ξ), and of its counterpartΦ∗(ε, Y,χ), conjugate through the Legendre–Fenchel transformation

Φ(σ , d, ξ) = 1

2

σ : E : σ1− d

+ Ψin(ξ) + Id1(d),

Φ∗(ε, Y,χ) = Y + ICd(ε, Y ) + Ψ ∗

in(χ),

whereId1(d) is the indicator function associated to the convex domaind 1, ICd(ε, Y ) is the indicator function associate

to the convex setCd defined as

Cd =(ε, Y )

∣∣∣ 1

2ε : E : ε − Y 0

andΨ ∗

in(χ) is the Legendre transform ofΨin(ξ), which for the simple model here considered, reads

Ψ ∗in(χ) = χ − k

n−1∑i=0

n!i! lni c + kc exp

[−

(χ + Y0

k

)1/n]n−1∑i=0

n!i!

(χ + Y0

k

)i/n

.

It should be noted that, whileΨ (ε, d, ξ) is not convex due to the presence of the damage variable, whenΨin is convex, as theone defined in (8), the new potentialΦ(σ , d, ξ), which depends on stresses rather than on strains, has been shown in (Laet al., 1999) to be also convex and therefore to enjoy the properties of convex potentials. In particular, its LegendretransformΦ∗(ε, Y,χ) is also convex and, if a functionalη(σ , ε,χ, ξ, d,Y ) is defined as follows

η(σ , ε,χ, ξ, d,Y ) = Φ(σ ,ξ, d) + Φ∗(ε,χ,Y ) − σ : ε − χξ − Yd

one has that

η(σ , ε,χ, ξ, d,Y ) 0 ∀(σ , ξ, d), (ε,χ,Y ),

η(σ , ε,χ, ξ, d,Y ) = 0 if and only if (σ , ξ, d) and(ε,χ,Y ) satisfy the state equations,(17)


where the state equations are given by

tly96)mconstantrocedure.umulated

lent

tele withouth are, in

ion 3lots havet of

nt error innt corners.at the costThe twoare betters to be finedary. The

is the onephs in

hown. At

eingsedthangion. The

e previoust al.

an initial

ε = ∂Φ

∂σ= 1

1− dE−1 : σ ; χ = ∂Ψin

∂ξ; Y ∈ ∂dΦ

or, in dual form,

σ ∈ ∂εΦ∗; ξ = ∂Ψ ∗in

∂χ; d ∈ ∂Y Φ∗.

The symbol∂ denotes the subdifferential of the argument with respect to the indicated variable.We now denote by a superposed ^ the nonlocal finite element solution, and byσ ∗ the stress field which satisfies exac

the equilibrium equations and which isconstructed according to the methodology proposed by Ladevèze and Maunder (19starting from the finite element fieldσ . The solution(σ ∗, ε, χ , ξ , d, Y ) satisfies exactly the compatibility and equilibriuequations and the time-step integrated form of the activation conditions and evolution equations. Notice that, sincestrain triangles are used for the space discretization, no additional errors are introduced by the volume integration pThe only equations which are not exactly satisfied are therefore the state equations, and, in view of Eqs. (17), the cspace discretization error at the considered time instantt can be measured by (Ladevèze et al. (1999))

e2(t) =∫Ω η(σ∗, ε, χ, ξ , d, Y )dΩ∫

Ω [Φ(σ , ξ , d) + Φ∗(ε, χ , Y )]dΩ. (18)

It should be noted that only local variables enter into the definition of the error. The nonlocal nature of the behavior plays a roin the constitutive equations accounting for damage dissipation (activation conditions and evolution equations). For the presemodel, these equations are satisfied exactly, to within the quadrature error in the evaluation ofY , and therefore do not contributo the error. Since quadrature error is independent of the mesh size and can be reduced refining the quadrature rumodifying the mesh, the only source of error related to the finite element discretization is in the state equations whicthis formulation, local.

The error distributions obtained applying the error measuree2 (18) to the Hassanzadeh’s tension test described in Sectare shown in Fig. 7 for the three different loading stages already considered in Fig. 6. The element error contour pbeen obtained plotting the integrandη(σ∗, ε, χ, ξ , d, Y ) in the expression of the error (18), computed at the centerpoineach triangle and multiplied by the area of the element. As expected, the homogeneous mesh 0 exhibits a significathe notch regions, already in the elastic range due to the elastic stress concentration in correspondence of reentraThe error grows also in the ligament as damage progresses. Mesh 1, which has been refined in the notch regionsof a rougher meshing of the ligament, exhibits good accuracy around the notches, but high errors in the ligament.main sources of discretization errors (singularity in the elastic solution and extensive damage in the softening regime)dealt with by mesh 2, which has almost the same number of degrees of freedom of meshes 0 and 1. Mesh 2 appearenough in the notch and ligament regions, but too coarse in the upper and lower regions near the constrained bounsame qualitative response is even more evident for mesh 3, where very limited error is shownin the notch region throughoutthe analysis, but significant errors appear since the beginning in the upper and lower regions. Nevertheless, mesh 3exhibiting the most uniform spatial error distribution along the whole history of loading. This is confirmed by the graFig. 8 where the element error is plotted in ascending order for a vertical displacement (a)u = 0.0025 mm and (b)u = 0.01 mm.Mesh 3 clearly shows the most uniform distribution of error, both in space and in time.

The results obtained with mesh 4, having a much larger number of spatially ad-hoc distributed elements, are not sthe same scale as in Fig. 7, no error could be observed and white maps would be obtained.

The quantitye(t) (18) represents a measure of the error cumulated up to the considered time instantt and depends onlyon the current values of the state variables and not on the previous history. This means that the errore(t) can either increasor decrease witht , as it is shown in Fig. 9a where the functione2(t) is plotted for the different meshes and for increasvertical displacement. Being based on a quadratic potential, inthe elastic range the error grows quadratically with the impodisplacement. When damage begins to develop, depending onthe mesh, the error either decreases or increases lessquadratically. Then, after reaching a peak value, it decreases as the stresses tend to vanish in the critical notch refact that the error associated to a given mesh depends on the current value of the state variables and not on thhistory seems to be somehow inconsistent with the non-reversiblecharacter of the material behavior. Correctly, Ladevèze e(Ladevèze et al., 1999) prevent the error from decreasing, defining the errore2

t as

e2t = sup

0τt

e2(τ). (19)

The resulting plot of the global error versus the imposed displacement is shown in Fig. 9(b). It can be seen that afterquadratic growth the error reaches a maximum value and then remains almost constant.


ion

uracy ofligamente proposedssessment

Fig. 7. Hassanzadeh’s tension test: total error maps. First column: elastic range (u = 0.0025 mm). Second column: peak vertical react(u = 0.005 mm). Third column: softening regime (u = 0.01 mm).

In conclusion, for the considered numerical test, the proposed error indicator provides useful information on the accthe solution obtained with different meshes. However, due to the energetic nature of the error function, the error in theregion tend to become less significant as damage grows, due to the vanishing state of stress. For this reason, therror measure seems more suitable for the definition of the global accuracy of a given mesh and for a comparative a


n for the

he

d

(a) (b)

Fig. 8. Hassanzadeh’s tension test: element error distribution for different meshes at (a)u = 0.0025 mm (elastic range) andu = 0.01 mm(softening branch).

(a) (b)

Fig. 9. Hassanzadeh’s tension test. Global error versus imposed displacement for meshes 0–4: (a) state equations errore2(t); (b) Ladeveze’serror e2

t .

of performance throughout the analysis of different meshes with a similar number of degrees of freedom, rather thadefinition of a remeshing strategy in the late stage of damage growth.

4.2. Incremental time-step error

A definition of the error reflecting the non-reversible nature of the dissipation phenomena can be obtained starting from tfree energy density potentialΨ m(ε,d,ξ) defined in Eq. (16) for each time-step. As the free energy densityΨ , also thefinite-step potentialΨ m is not convex. Following Ladevèze et al. (1999), a convex finite-step potentialΦm can be constructeas in Section 4.1 in terms of stress instead of strain increments

Φm(σ ,d,ξ) = 1

2

(σm + σ ) : E−1 : (σm + σ )

1− dm − d− εm : σ − Ymd + Ψin

(ξm + ξ

)− χmξ + Id1

(dm + d

).


For dm + d < 1, the second variation ofΦm is given by

).

Inlem,tep can be

m of the

g steps,

ation and

δ2Φm = δσ : E−1 : δσ1− dm − d

+ 2δσ : E−1 : (σm + σ )

(1− dm − d)2δd + (σm + σ ) : E−1 : (σm + σ )

(1− dm − d)3δd2 + δ2Ψin

(ξm + ξ

)= [(σm + σ )δd + (1− dm − d)δσ ] : E−1 : [(σm + σ )δd + (1− dm − d)δσ ]

(1− dm − d)3+ δ2Ψin

(ξm + ξ

)and is always non-negative for any value ofσ , d, ξ , Ψin(ξm + ξ) being convex by definition.

It is easy to verify that

ε = ∂Φm

∂σ; Y ∈ ∂dΦm; χ = ∂Φm

∂ξ. (20)

The Legendre–Fenchel transform ofΦm is given by

Φm∗(ε,Y,χ) = (1− dm

)Y + ICm

d(ε,Y) + Ψ ∗

in(χm + χ

) − σm : ε − 1

2σm : εm − (

χm + χ)ξm,

whereICmd

is the indicator function associated to the convex domainCmd defined as

Cmd =

(ε,Y)

∣∣∣ 1

2ε : E : ε + εm : E : ε − Y 0

.

The state equation forΦm∗ are given by

σ ∈ ∂εΦm∗; d ∈ ∂Y Φm∗; ξ = ∂Φm∗∂χ

. (21)

In view of the convexity of the potentialsΦm, Φm∗ the functional

ηm(σ ,d,ξ,ε,Y,χ) = Φm(σ ,d,ξ) + Φm∗(ε,Y,χ) − σ : ε − Yd − χξ

has to satisfy the following conditions

ηm(σ ,d,ξ,ε,Y,χ) 0 ∀(σ ,d,ξ), (ε,Y,χ)

ηm(σ ,d,ξ,ε,Y,χ) = 0 iff (σ ,d,ξ) and(ε,Y,χ) satisfy the state equations (20) and (21

Following the same path of reasoning as in Section 4.1, we define a fictitious stress increment fieldσ∗ which satisfies exactlyequilibrium and we denote by a superposed ^ the incremental fields obtained solving the nonlocal finite element step problem.this way, the incremental solutionσ ∗, d , ξ , ε, Y , χ satisfies exactly all equations governing the finite-step probwith the exception of the state equations. Therefore, a measure of the discretization error associated to the current sdefined as

e =∫Ω

√ηm(σ ∗,d,ξ,ε,Y ,χ)dΩ

Ω

√σ2

0/(2E)

,

whereσ0 is the stress at the elastic limit in a uniaxial test. The error cumulated at a given instant is defined as the sutime step errors up to that instant

em =m∑

i=1

ei . (22)

The proposed incremental error measure has the following properties:

• it is always non-negative, so that the cumulated discretization errorem cannot decrease;• in the linear elastic range it depends linearly on the applied load;• for a given final load, solving a problem in the elastic range applying the load just in one step or in several loadin

leads to the same final global error, both in the case of proportional and non-proportional loading histories.• being the normalization factor at the denominator a constant, the incremental error depends only on the discretiz

on the step amplitude so that incremental errors at different steps can be compared.


ported for

oseds linearlyore thanesh 4. It

gingof mesh 1

alues on(values onesh

Fig. 10. Incremental time-step error. Cumulated error versus imposed displacement for different meshes.

Fig. 11. Incremental time-step error. Contribution of each time-step to the global error for meshes 0 and 4. Cumulated error is also rereference (values on right vertical axis).

The evolution of the cumulative errorem for the five meshes shown in Figs. 3 and 4 is plotted in Fig. 10 versus the impdisplacement. It can be observed that the error is monotonically increasing. In the linear elastic range, the error growwith the imposed displacement. As the elastic limit is exceeded, for the coarser meshes the error initially grows mlinearly and then the rate of growth decreases progressively. Obviously, the minimum error is obtained with the finest mis interesting to note that while mesh 1 is more accurate thanmesh 0 in the elastic range and in the initial part of the damarange, thanks to the finer discretization around the notches, as damage grows, the error in the coarser ligament regionbecomes predominant and the cumulated error becomes higher than with mesh 0.

The value of the incremental error at each time step for meshes 0 and 4 is shown in Fig. 11 (bar diagrams with vthe left vertical axis). In the same graph are also plotted the corresponding curves of cumulated error as a referencethe right vertical axis). As expected, the incremental error is constant in the elastic range, then increases dramatically for m


age beginsligamentr a more

o

chedoon as theg history.damagingze et al.allows for

d to asensionallizationb).

Fig. 12. Incremental time-step error. Comparison with Ladeveze’s total error measure.

0 when damage starts to propagate and finally decreases as damage tends to 1. Conversely, with mesh 4, when damto propagate, the incremental error almost immediately begins to decrease, thanks to the finer discretization of theregion. In this way, the evolution of the incremental error provides a better understanding of error sources and allows fofocussed mesh refinement strategy.

The proposed cumulated time-step error and the Ladevèze’s error in its original definition cannot be compared for twreasons: (i) presence of the square root of the integrand function at the numerator ofem, (ii) different normalization ofeandem. To make the comparison meaningful, the modified definitionet of Ladevèze’s error

et =∫Ω

√η(σ∗, ε, χ , ξ , d, Y )dΩ

Ω

√σ2

0 /(2E)

(23)

has been used in Fig. 12 where results for meshes 0 and 4 are shown. The dashed horizontal lines mark the higher values reaby Ladevèze’s estimator in the line of what done in Eq. (19). Since at each step a nonnegative error is added, as selastic limit is exceeded, the cumulated incremental error becomes larger and keeps growing until the end of the loadin

It should be noted however, that Ladevèze’s total error measure has the advantage that, for local elastoplasticmaterial behavior, it may allow for a combined estimate of the space-time discretization error, as shown in Ladevè(1999). The same concept does not seem to be applicable in the case of the incremental error here proposed, whichan assessment of the space discretization error only.

5. Critical localization width

5.1. Localization width estimate

The nonlocal nature of the material model brings into the problem an intrinsic material length, commonly referre“internal length”. The width of the localization zone depends on this length and can be estimated for three and two-dimproblems following the bifurcation approach proposed by Pijaudier-Cabot and Benallal (1993). A derivation of the locawidth for the same nonlocal material model in a one-dimensional case has been proposed in Comi and Perego (2001

The rate form of Eq. (9)1 is given by

σ = (1− d)E : ε − dE : ε.


Assuming continuous loading (f = 0), from Eq. (10)1 one obtains( )∫

solutions

ld

n ing

ntith

an

only as

σ = (1− d)E : ε − E : ε 1− d

knln1−n c

1− dΩ

W(x, s)ε(s) : E : ε(s)ds. (24)

Assuming zero body forces and small strains, the equilibrium and compatibility equations in rate form read

div σ = 0

ε = 1

2

(gradu + gradT u

). (25)

The idea is now to look for bifurcated solutions of the rate equations (24), (25) starting from a homogeneous state. Theare sought in the form

u = v0 exp(−iqn · x), (26)

hereq is the wave number andn is the propagation direction. For an infinitebody, direct substitution of the above defined fieu into the equilibrium equations (25)1, account taken of (24) and (25)2, leads to[

n · H (q) · n] · v0 = 0, (27)

whereH (q) is the fourth order tensor of tangent moduli of the nonlocal rate problem:

H (q) = (1− d)E − W (q)1− d

knln1−n

(c

1− d

)E : ε ⊗ ε : E. (28)

This tensor depends on the wave numberq through the Fourier transformW(q) of the weight functionW(x). Nontrivialsolutions of (27) exist if and only if

det[n · H (q) · n] = 0. (29)

The normalncrit and the wave numberqcrit satisfying Eq. (29) can be obtained following a geometrical method as showPijaudier-Cabot and Benallal (1993), to which we refer for the details. Forn = ncrit, Eqs. (28) and (29) lead to the followinexpression forW(qcrit)

W(qcrit) = kn lnn−1(c/(1− d))

4µ((ε1 − ε3)/2)2 + 4µ2/(λ + µ)((ε1 + ε3)/2+ λ(ε1 + ε2 + ε3)/(2µ))2, (30)

whereµ and λ are the initial undamaged elastic Lamè constants andε1 ε2 ε3 are the principal strains at the currehomogeneous state. Under plane strain conditions(ε2 = 0), the denominator in Eq. (30) can be easily verified to coincide wthe double of the elastic energy release rate and the critical condition becomes

W(qcrit) = kn lnn−1(c/(1 − d))

2Y. (31)

This expression can be further simplified using the loading conditionf = 0 and noting that for a homogeneous stateY = Y

W(qcrit) = n

2ln−1

(c

1− d

). (32)

Choosing as the weight function the normalized Gauss function in Eq. (6), the Fourier transform ofW(x) turns out to be

W(q) = exp

(−q2l2c

2

).

The smallest feasible bifurcation mode wavelengthλcrit = 2π/qcrit can be obtained substituting this expression ofW (q) intoEq. (32)

λcrit = 2π

qcrit= √

2πlc

[− ln

(n

2ln−1

(c

1− d

))]−0.5. (33)

While for a local model, sinceH does not depend on the wave numberq, the same path of reasoning would lead toundetermined critical wavelength (and thusλcrit = 0 is a feasible value), for the nonlocal continuum the wavelengthλcrit ofthe bifurcation mode has the specific value (33) which is directly related to the characteristic lengthlc = 0 of the model. Thislength, defines the width of the localization zone and is a positive, decreasing function of damage, tending to zero


shown in

one-

ogeneoush, which

the currentin a mesh-stant strainomequired.5

nds one

ovesalization

stageuate and

Fig. 13. Critical wavelength versus damage.

damage tends to one. The evolution with damage, which does not depend on the plane-strain loading path followed, isFig. 13 forn = 12, c = 405 and different values of material lengthlc. Note that with these values ofn andc, the tangent at thepeak of the stress–strain curve (ford = 0) is zero and, accordingly, the critical wavelength is infinite (non-softening behavior).

It is worth noting that the expression (33) ofλcrit is identical to the one derived in Comi and Perego (2001b) for adimensional case which therefore proves valid also in two dimensions.

5.2. Critical damage

Even though the hypotheses under which the expression (33) has been derived are restrictive (infinite body, hominitial state), the bifurcation approach described in the previous section provides a useful estimate of the bandwidtcan be used to assess whether the assigned mesh is fine enough to resolve the damage process zone in relation tovalue of damage. In fact, the nonlocal continuum approach can accurately reproduce the damage localization processindependent band only as long as the element size is small enough to resolve the current band (say at least three conelements within the band). According to Eq. (33) (see Fig. 13), as damage grows, the element size will eventually bectoo large if compared to the reducing bandwidth. At that point, remeshing or transition to a discrete model will be reImposing that the critical wavelength be bigger thenr times the element sizele (with r = 3 − 5, meaning that at least 3–elements are needed across the localization band), i.e.

λcrit rle = Le

and solving (33) ford, one can define a critical damage value

d dcrit = 1− c exp

[−n

2exp

(2π2l2c

L2e

)](34)

above which remeshing is necessary to resolve the bandwidth. Fig. 13 shows schematically the definition ofdcrit. This valuedepends on the typical size of the chosen mesh, throughLe, tends to one as the element size tends to zero and also depethe material internal lengthlc of the nonlocal model. As proposed in Comi et al. (2002),dcrit can also be used to trigger thtransition from a nonlocal continuum model to a discrete propagating crack.

For r = 5 (i.e. at least 5 elements across the localization band), Fig. 14 shows, marked in black, the elements whered > dcritfor the different meshes and for increasing values of the imposed displacement. Obviously, while the coarser mesh 0 pralmost immediately inadequate, with a growing number of black elements, the finest mesh 4 is able to resolve the locband well beyond the displacement corresponding to the peak reaction.

While the map ofdcrit shown in Fig. 14 does not provide any information on the accuracy of the results in the earlyof the analysis, it gives a clear indication of the time and spatial position at which the given mesh becomes inadeq


agein a

avior hasave been

has beenr has also

Fig. 14. Maps of finite elements whered > dcrit (marked in black) for meshes 0, 2, 3, 4 and for increasing imposed displacement.

remeshing is necessary. In view of the considerations expounded at the end of Section 4.1, the definition of the critical damprovides an information which is complementary to the error measure defined in (22) and could be conveniently integratedremeshing strategy focussed on the late stages of damage localization.

6. Conclusions

The problem of the accuracy of finite element analyses in the presence of nonlocal elasto-damaging material behbeen discussed. Making reference to a simple nonlocal isotropic damage model, two criteria for mesh refinement hanalyzed. In the first case, a “mechanically based” error estimator originally proposed by Ladevèze et al. (1999)adapted to the specific nonlocal model here considered. A modified, stepwise incremental version of the error estimatobeen proposed to account for the non-reversible nature of the dissipative phenomena occurring during the analysis.


With the aid of a numerical test, the proposed estimator has been shown to provide useful information on the accuracy of aHowever,cant in the

damagealande isOn the otherthe

mbinationhe whole

ontract

.es.

rence

e

s

ents.

0,

5,

.

ppl.

g.

.28.t.

given mesh throughout the history of loading, and to allow the comparative accuracy assessment of different meshes.being based on a measure of the violation of the state equations, the error estimate becomes less and less signifilocalization region as damage grows and, consequently, stresses tend to vanish.

The second criterion studied is based on the analytical derivation of the critical wavelength, i.e. the width of thelocalization band. Following the bifurcation analysis procedure proposed by Pijaudier-Cabot and Benallal (1987), the criticlength of the used nonlocal model has been derived analytically. Under the assumption of a homogeneous initial straindamage state, the critical length is shown to decrease progressively with damage. Hence, when a critical value of damagexceeded, the current mesh size cannot resolve anymore the damage bandwidth and remeshing becomes necessary.hand, this criterion does not provide information on the accuracyin the early stage of the analysis, when damage is belowcritical value. In this sense the second criterion can be viewed as being complementary to the first one.

Even though more extensive numerical experiments are certainly required, the results of this study suggest that a coof the two criteria may provide a useful and complete guidance for the definition of a mesh refinement strategy along thistory of loading of the structure to be analyzed.

Acknowledgements

This work has been carried out within the framework of a research project partially supported by MIUR c2002087915.

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Criteria for mesh refinement in nonlocal damage finite element analyses

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