Dispersion analysis and element-free Galerkin solutions of second- and fourth-order gradient-enhanced damage models

Askes, Harm; Pamin, Jerzy; Borst, René de

doi:10.1002/1097-0207(20001030)49:6<811::aid-nme985>3.0.co;2-9

Cited by 100 publications

(79 citation statements)

References 14 publications

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“…Similar to gradient damage models, gradient effects are included in the definition of the damage parameter, [6,7,14,15,24,26]. However, in contrast to commonly used (gradient) damage models, strain gradients appear here also in the linear part of the stress-strain relation.…”

Section: Constitutive Equationmentioning

confidence: 99%

“…As is commonly done in dispersion analysis, a linearisation around a uniform strain level e 0 is carried out, [6,25,28]. It is assumed that…”

Section: Constitutive Equationmentioning

confidence: 99%

“…The above observations are in correspondence with the literature, where it has been found that the second-order term of " e e E has destabilising effects, [5,8,9], which in turn motivates the choice for this second-order model with an opposite sign, [5]. Also, in literature on softening phenomena, it has been found that second-order terms in gradient-damage formulations are needed to stabilise the solution; zeroth-order terms alone are not sufficient, [24,25].…”

Section: Remarkmentioning

confidence: 99%

“…[6,7,24,25] a distinction is made between so-called explicit-gradient series and implicit-gradient series: the former correspond to the damage-gradient series in the present paper. It should be emphasized that all individual gradient terms of the implicitgradient series are stabilising.…”

Section: Remark 5 Equationsmentioning

confidence: 99%

“…In this context, it is noted that in earlier works by the authors, the effects of gradients in the elastic response, [9], and the effects of gradients in the damage evolution, [7], were studied without mutual interaction between the two types of gradients. Although combinations of secondgradient terms in a constitutive model have been studied by the authors for static analyses in [8,29], also fourth-order terms in a damage context and dynamic problems are studied here. Series truncation is treated in Sec.…”

Section: Introductionmentioning

confidence: 99%

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A classification of higher-order strain-gradient models in damage mechanics

Askes

Sluys

2003

Archive of Applied Mechanics (Ingenieur Archiv)

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In a previous contribution, higher-order strain-gradient models for linear elasticity have been studied in statics and dynamics [9]. In this paper, the extension towards damage mechanics is made. A damage model is derived from a discrete microstructure. In the homogenisation process, higher-order strain gradients appear both in the linear and in the nonlinear parts of the constitutive equation. Similar to the elastic models, stabilising and destabilising gradients can be distinguished. The stabilising or destabilising effect of each gradient term is determined. Opposite (competing) effects on the stability are found for the gradients of the elastic and the gradients in the damage response. Various truncations of the two strain-gradient series are studied, with the aim to arrive at a continuum model that fulfills the following requirements (i) it is derivable from a discrete microstructure, (ii) it is able to describe wave dispersion in elastic and damaging media properly, and (iii) it can be used to model strain-softening phenomena, i.e. it is a regularised model. The response of the various models is studied analytically and numerically. For the analytical investigation, dispersive waves are studied and critical wave lengths are derived. Numerical simulations are carried out with the element-free Galerkin method. This combined analytical/numerical approach allows to establish the role of the critical wave length both for mechanically stable and mechanically unstable models. For stabilised models, the critical wave length sets the width of the damaging zone. On the other hand, for destabilised models, the critical wave length sets a periodicity in the response that leads to divergence of the numerical scheme. The influence of the individual gradient terms on the stability and the structural ductility is verified in static and dynamic analyses. IntroductionThe use of higher-order gradient models has gained increasing popularity during the last decades. In such models, the constitutive relations contain not only stresses and strains, but also higher-order derivatives of these fields. In cases where classical continua, which only contain stresses and strains in the constitutive equations, cannot adequately capture local deformation patterns, the proper use of higher-order continua leads to simulations that are physically realistic and mathematically sound.Below, the restriction is made to higher-order spatial derivatives, while time derivatives are left out of consideration. Examples in which the addition of higher-order strain gradients has led to an improved performance range from elasticity to damage and to plasticity models. For instance, the singularity in the strain field near a crack tip or dislocation has been removed successfully using a theory of gradient elasticity, [3,5,16,17]. Also, localisation of deformation can be restricted to localisation bands of finite width with gradient hyperelasticity, [32], We thank Akke Suiker and Andrei Metrikine of Delft University of Technology for stimulating discussions...

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Section: Constitutive Equationmentioning

confidence: 99%

“…As is commonly done in dispersion analysis, a linearisation around a uniform strain level e 0 is carried out, [6,25,28]. It is assumed that…”

Section: Constitutive Equationmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remark 5 Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A classification of higher-order strain-gradient models in damage mechanics

Askes

Sluys

2003

Archive of Applied Mechanics (Ingenieur Archiv)

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Damage, Material Instabilities, and Failure

Borst

2004

Encyclopedia of Computational Mechanics

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This chapter begins with an overview of damage mechanics. Isotropic and anisotropic models are described, as well as the coupling to plasticity, including the algorithmic treatment. Next, the implications of strain softening, which starts at a certain level of damage accumulation, are discussed as regards important notions such as material stability and ellipticity. The consequences of loss of ellipticity in terms of grid sensitivity of computations are illustrated for static and for dynamic instabilities. Next, the important role of cohesive‐zone models is highlighted, both in its original (discrete) format, a derived smeared format and as an embedded discontinuum description. To avoid loss of ellipticity, the introduction of higher‐order continua is necessary. Various higher‐order continua incorporating damage or coupled damage‐plasticity models are discussed, including aspects of numerical implementation. Finally, the numerical implementation of cohesive‐zone models in a discrete format is considered, where attention is given to conventional interface elements as well as to approaches that exploit the partition‐of‐unity property of finite element shape functions.

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