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A gradient viscoplasticity model has been used to analyse two- and three-dimensional shear banding. The numerical simulation of a finite width and unique oriention of the shear band is essential. It is demonstrated that the use of viscous regularisation is effective for quasi-static and dynamic problems. The role of the two length scale parameters that are introduced in the model has been investigated. The shear band width is influenced by the two interacting length scales of which the largest dominates the band width. A 3D analysis of shear banding is shown to give significantly different results than the 2D plane strain analysis under similar conditions.

In an experimental programme, impact tensile tests were performed on notched and unnotched specimens in a Split Hopkinson Bar apparatus to determine the dynamic fracture energy of concrete. The results have been compared to static experiments. In order to distinguish structural and material response the experiments were numerically simulated. That resulted in the observation that the dynamic fracture energy and the static fracture energy are approximately equal.

Summary 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.

A new model is presented for the simulation of delamination in laminated composite materials. A key feature is that the material structure and the finite element mesh are uncoupled. The displacement discontinuities that arise during the delamination process are described mathematically using discontinuous functions. This leads naturally to a set of coupled equations for the continuous and the discontinuous parts of the response. Discontinuities can pass through solid finite elements arbitrarily, with the displacement jump continuous across element boundaries. The performance of the model is demonstrated for several problems of delamination and geometric instability. Copyright © 2002 John Wiley & Sons, Ltd.

The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourth-gradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the second-gradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.

A numerical model is developed which allows the inclusion of displacement discontinuities in a strain-softening medium, independent of the finite element mesh structure. Inelastic deformations develop in the continuum and, when a critical threshold of inelastic deformation is reached, a displacement discontinuity is inserted. Discontinuities are introduced using the partition of unity concept which allows discontinuous functions to be added to the standard finite element basis. It is shown that the introduction of displacement discontinuities at the later stages of the failure process can lead to a failure mode that is fundamentally different than that using a continuum model only. This combined continuum-discontinuous model is better able to describe the entire failure process than a continuum or a discrete model alone and treats mode-I and mode-II failure in a unified fashion. Copyright © 2001 John Wiley & Sons, Ltd.

Softening solids are analysed under impact loading using a new numerical method which allows displacement discontinuities to propagate arbitrarily through a finite element mesh. The Dirac‐delta distributions that arise in the strain field of classical continuum theory in the presence of strain softening are interpreted as discontinuities in the displacement field. A new finite element procedure with Heaviside jumps added to the underlying displacement interpolation basis is able to capture displacement jumps independent of the spatial discretisation. The amplitudes of displacement jumps are represented by extra degrees of freedom at existing nodes. Numerical results for mode‐I and mode‐II failure due to impact loading are presented. The numerical results highlight the objectivity of the approach with respect to spatial discretisation under dynamic loading conditions. Copyright © 2001 John Wiley & Sons, Ltd.

A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so-called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi-brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.

The computational modelling of localization of deformation in cohesive-frictional materials must be carried out in a proper, accurate and efficient manner. A proper solution can be obtained by using an enriched material description such as a non-local damage model or Perzyna's viscoplasticity model. However, still a large number of finite elements is needed for an accurate description of the localization zone. To improve efficiency, mesh adaption is applied here by means of the Arbitrary Lagrangian–Eulerian (ALE) technique. The ALE technique must typically be applied in combination with an enriched material model. Otherwise, the use of a standard model gives a zero-width solution of the localization zone and therefore remeshing results in a continuing decrease of finite element size. Implementation of this method requires the addition of convective forces in the equations of motion, transport of the state variables and the formulation of a remeshing strategy. The remeshing strategy heavily determines the success of the ALE method. Three examples of wave propagation have been treated and it is shown that a suitable remeshing strategy in combination with an enriched material model leads to proper, accurate and efficient analyses of the localization process. © 1998 John Wiley & Sons. Ltd.

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.

Three viscoplastic approaches are examined in this paper. First, the overstress viscoplastic models (i.e. the Perzyna model and the Duvaut–Lions model) are outlined. Next, a consistency viscoplastic approach is presented. In the consistency model a rate-dependent yield surface is employed while the standard Kuhn–Tucker conditions for loading and unloading remain valid. For this reason, the yield surface can expand and shrink not only by softening or hardening effects, but also by softening/hardening rate effects. A full algorithmic treatment is presented for each of the three models including the derivation of a consistent tangential stiffness matrix. Based on a limited numerical experience it seems that the consistency model shows a faster global convergence than the overstress approaches. For softening problems all three approaches have a regularising effect in the sense that the initial-value problem remains well-posed. The width of the shear band is determined by the material parameters and, if present, by the size of an imperfection. A relation between the length scales of the three models is given. Furthermore, it is shown that the consistency model can properly simulate the so-called S-type instabilities, which are associated with the occurrence of travelling Portevin-Le Chatelier bands. © 1997 John Wiley & Sons, Ltd.

In this work a glass/epoxy composite commonly used in wind turbine blades is exposed to a humid environment at an elevated temperature. To research the anisotropic diffusion behaviour observed in unidirectional composite specimens, experimental results of slices cut along the three directional planes of the laminate immersed in demineralised water at 50deg C are coupled with numerical modelling. The weight of the slices was measured at regular intervals, from which the uptake behaviour could be deduced. The process was modelled using a 3-dimensional RVE of the material, where diffusion is modelled as steady-state and the diffusivity in each direction was measured by applying concentration gradients to the model. The experimental data shows similar water uptake behaviour for samples in both transverse directions, while the water uptake in the fibre direction was significantly faster. A proper fit according to Fick's law was obtained for the transverse direction, while this was not possible for the samples in fibre direction, suggesting a strong dependency of the diffusion behaviour on the fibre orientation. Results from the proposed numerical models show that the geometric effect of fibres acting as barriers for the water movement is indeed responsible for part of the observed anisotropy.