Rhj Ron Peerlings

Gradient enhanced damage for quasi-brittle materials

SUMMARY Conventional continuum damage descriptions of material degeneration suffer from loss of well-posedness beyond a certain level of accumulated damage. As a consequence, numerical solutions are obtained which are unacceptable from a physical point of view. The introduction of higher-order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models. A consistent numerical solution procedure of the governing partial differential equations is presented, which is shown to be capable of properly simulating localization phenomena.

A critical comparison of nonlocal and gradient-enhanced softening continua

Continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients. These gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators. Adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations. The effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting. A nonlocal damage model and two different gradient damage models are considered. In one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly, while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation. The latter, implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal. This is not true for the explicit formulation, in which the nonlocality is limited to an infinitesimal volume. This fundamental difference between the formulations results in quite different behaviour in wave propagation, localisation and at crack tips. This is shown for the propagation of waves in the models, their localisation properties and the behaviour at a crack tip. The responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations, while the explicit gradient formulation shows an entirely different and sometimes nonphysical response.

Gradient‐enhanced damage modelling of concrete fracture

Classical continuum damage theory for quasi-brittle fracture exhibits an extreme sensitivity to the fineness and orientation of the spatial discretization in finite element simulations. This sensitivity is caused by the fact that the mathematical description becomes ill-posed at a certain level of accumulated damage. The ill-posedness can be removed by the use of a gradient-enhanced damage model. In this model, higher-order deformation gradients give rise to a non-local effect, which regularizes the localization of deformation and thus renders numerical analyses mesh-objective. The mesh objectivity of the gradient-enhanced damage approach is demonstrated by the application to two concrete fracture experiments: a double-edge notched bar subjected to a uniaxial, tensile load and a single-edge notched beam under anti-symmetric four-point loading. Both the initiation and the propagation of damage can be simulated. Particularly the latter aspect calls for an appropriate definition of the strain measure which governs the evolution of damage.

Strain-based transient-gradient damage model for failure analyses

A transient-gradient enhanced damage model has been developed for the numerical modelling of the damage and fracture process within a continuum damage mechanics framework. Some deficiencies of existing gradient enhanced damage formulations for the simulation of macroscopic crack propagation are pointed out. The transient-gradient approach assumes a direct coupling between the material length parameter and the local strain state of the material, which leads to a transient behaviour of the nonlocal effect. Details of the method are presented and fully elaborated in an incremental-iterative solution scheme. Mesh objectivity and physical relevance of the method are analysed by one-dimensional and two-dimensional numerical examples.

On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials

Gradient-enhanced damage and plasticity approaches are reviewed with regard to their ability to model localization phenomena in quasi-brittle and frictional materials. Emphasis is put on the algorithmic aspects. For the purpose of carrying out large-scale finite element simulations efficient numerical treatments are outlined for gradient-enhanced damage and gradient-enhanced plasticity models. For the latter class of models a full dispersion analysis is presented at the end of the paper. In this analysis the fundamental role of dispersion in setting the width of localization bands is highlighted.

Localisation issues in local and nonlocal continuum approaches to fracture

Continuum approaches to fracture regard crack initiation and growth as the ultimate consequences of a gradual, local loss of material integrity. The material models which are traditionally used to describe the degradation process, however, may predict premature crack initiation and instantaneous, perfectly brittle crack growth. This nonphysical response is caused by localisation instabilities due to loss of ellipticity of the governing equations and—more importantly—singularity of the damage rate at the crack tip. It is argued that this singularity results in instantaneous failure in a vanishing volume, even if ellipticity is not first lost. Adding strong nonlocality to the modelling is shown to preclude localisation instabilities and remove damage rate singularities. As a result, premature crack initiation is avoided and crack growth rates remain finite. Weak nonlocality, as provided by explicit gradient models, does not suffice for this purpose. In implementing the enhanced modelling, the crack must be excluded from the equilibrium problem and the nonlocal interactions in order to avoid unrealistic damage growth.  2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.

Gradient-enhanced damage modelling of high-cycle fatigue

Continuum damage mechanics can be used to model the initiation and growthof fatigue cracks. However, finite element analyses using standard fatiguedamage formulations exhibit an extreme sensitivity to the spatialdiscretisation of the problem. The mesh sensitivity is caused by the factthat the underlying continuum model predicts instantaneous, perfectlybrittle crack growth as soon as a crack has been initiated. The growth ofdamage localises in a vanishing volume during this instantaneous growth.This localisation is not so much due to loss of ellipticity of theproblem, but is caused by the fact that the damage rate is singular at thecrack tip. The damage rate singularity can be removed by the introductionof higher-order deformation gradients in the constitutive modelling. As aresult, crack growth at a finite rate and with ! a positive amount of energydissipation is predicted. Finite element analyses converge to thissolution and are thus no longer pathologically dependent on the spatialdiscretisation.

Damage and crack modeling in single-edge and double- edge notched concrete beams

The numerical modeling of damage and crack propagation in concrete and concrete structures has evolved considerably in the past years. In this contribution, a higher order continuum model is used to model the failure behavior of single-edge notched (SEN) and double-edge notched (DEN) concrete beams loaded in four-point-shear. DiAerent types of boundary conditions, i.e. with freely rotating, fixed or constrained loading supports, are investigated and the experimentally observed curved crack paths are compared with the numerical simulations. The influence of the ratio of the compressive strength and the tensile strength is scrutinized and its relation with the failure mechanism is investigated. It is shown that an isotropic gradient-enhanced damage model permits to obtain a good agreement between experimental results and numerical simulations. # 2000 Elsevier Science Ltd. All rights reserved.

Validation and internal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene

A strain-based transient-gradient damage model is used to analyse and describe the experimentally observed failure process in a Compact-Tension test carried out on short glass-fibre-reinforced polypropylene. Several aspects regarding the nonlocal character of the damage process in the material are emphasized and the intrinsic length scale is determined using available strain fields from an experimental analysis. A good agreement between theory and experiments has been found on a global and on a local level.

Cohesive zone modeling for structural integrity analysis of IC interconnects

Due to the miniaturization of integrated circuits, their thermo-mechanical reliability tends to become a truly critical design criterion. Especially the introduction of copper and low-k dielectric materials cause some reliability problems. Numerical simulation tools can assist developers to meet this challenge. This paper considers the first bond integrity during wire bond qualification testing. During testing, metal peel off may occur. This mechanical failure mode is caused by delamination of several layers of the interconnect structure. An interfacial damage model is employed for simulating delamination. However, the fact that the considered interfaces are brittle triggers some reported numerical difficulties. This paper illustrates the potential of the interface damage mechanics approach for simulating metal peel off and it highlights the computational aspects to be developed to render a practically applicable approach.