L.J. Sluys

Viscoplasticity for instabilities due to strain softening and strain-rate softening

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 e⁄ects, but also by softening/hardening rate e⁄ects. A full algorithmic treatment is presented for each of the three models including the derivation of a consistent tangential sti⁄ness 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 e⁄ect 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.

Localisation in a Cosserat continuum under static and dynamic loading conditions

Abstract Localisation studies have been carried out for an unconstrained, elasto-plastic, strain-softening Cosserat continuum. Because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement. A finite, constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions. Because of the existence of rotational degrees-of-freedom in a Cosserat continuum additional wave types arise and wave propagation becomes dispersive. This has been investigated analytically and numerically for an elastic Cosserat continuum and an excellent agreement has been found between both solutions.

From continuous to discontinuous failure in a gradient-enhanced continuum damage model

Abstract A computational framework for the description of the combined continuous–discontinuous failure in a regularised strain-softening continuum is proposed. The continuum is regularised through the introduction of gradient terms into the constitutive equations. At the transition to discrete failure, the problem fields are enhanced through a discontinuous interpolation based on the partition of unity paradigm of finite-element shape functions. The inclusion of internal discontinuity surfaces, where boundary conditions are applied without modifications of the original finite-element mesh, avoids the unrealistic damage growth typical of this class of regularised continuum models. Combined models allow for the analysis of the entire failure process, from diffuse microcracking to localised macrocracks. The discretisation procedure is described in detail and numerical examples illustrate the performance of the combined continuous–discontinuous approach.

Wave propagation, localization and dispersion in a gradient-dependent medium

A continuum model that incorporates a dependence upon the Laplacian of the inelastic strain is used to regularize the initial value problem that results from the introduction of strain softening or non-associated flow. It is shown that the introduction of this gradient dependence preserves well-posedness of the initial value problem and that wave propagation in the enhanced continuum is dispersive. An analysis of the dispersive wave propagation reveals the existence of an internal length scale. Numerical analyses of one-dimensional and two-dimensional problems confirm that this internal length scale sets the localization zone and show that the results are insensitive to the fineness of the discretization and to the direction of the grid lines. This holds true with respect to the strain profiles, the energy dissipation and the extent of wave reflection.

On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture

In this paper, strong discontinuities embedded in finite elements are used to model discrete cracking in quasi-brittle materials. Special attention is paid to (i) the constitutive models used to describe the localized behaviour of the discontinuities, (ii) the enforcement of the continuity of the crack path and (iii) mixed-mode crack propagation. Different constitutive relations are adopted to describe the localized behaviour of the discontinuities, namely two damage laws and one plasticity law. A numerical algorithm is introduced to enforce the continuity of the crack path. In the examples studied, an objective dissipation of energy with respect to the mesh is found. Examples of mode-I and mixed-mode crack propagation are presented, namely a double notch tensile test and a single-edge notched beam subjected to shear. In the former case different crack patterns are obtained depending on the notch offset; in the latter case special emphasis is given to the effect of shear on the global structural response. In particular, both the peak load and the softening response of the structure are related to the amount of shear tractions allowed to develop between crack faces. The results obtained are compared to experimental results. As a general conclusion, it is found that crack path continuity allows for the development of crack patterns similar to those found in experiments, even when reasonably coarse meshes are used.

Wave propagation and localization in a rate-dependent cracked medium-model formulation and one-dimensional examples

Abstract A smeared-crack model is proposed in which the stress after cracking is not only a function of the crack strain (softening function), but also of the crack strain rate. This rate dependence is shown to introduce an internal length scale into initial value problems. As a result, solutions of initial value problems that involve smeared cracking remain well posed. The width of the localization zone. where high strain rates are present, is proven to be independent of the finite element discretization. Convergence upon mesh refinement is also obtained for the consumption of energy in a cracked bar and the extent of wave reflection on a cracked zone. This is because in a rate-dependent medium, waves are dispersive and propagate with a real wave speed. Numerical results have been obtained for one-dimensional localization problems to substantiate these analytical findings.

Three-dimensional embedded discontinuity model for brittle fracture

Abstract Three-dimensional elements with discontinuous shape functions have been developed to model brittle fracture in unstructured meshes. Through the incorporation of discontinuous shape functions, highly localised deformations can be captured in a coarse finite element mesh. By carefully examining the variational formulation, traction continuity is assured in a weak sense across discontinuities and a discrete constitutive (traction–separation) model is applied. Using discrete constitutive models avoids the need for numerical parameters to achieve mesh objective results with respect to energy dissipation. Further, kinematic enhancements allow objective analysis in unstructured meshes. Examples of three-dimensional analysis are shown to illustrate the performance of the model.

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.

Modeling of crazing using a cohesive surface methodology

A novel cohesive surface model for crazing in polymers is developed. The model incorporates the initiation, growth and breakdown of crazes based on micromechanical considerations. The initiation of crazes is controlled by the stress state, in particular by the hydrostatic stress and cohesive surface normal traction. The widening of a craze is based on a rate-dependent viscoplastic formulation and failure of the craze occurs when the fibrils reach a material-dependent maximum extension. Crazing is simulated using a high density of cohesive surfaces immersed in the continuum. The finite element method is used to discretize both cohesive surfaces and continuum separately. The capabilities of the method to describe multiple crazing is demonstrated with an example.

Continuum Models for the Analysis of Progressive Failure in Composite Laminates

The performance of continuum material models for mesolevel modeling of progressive failure in composite laminates is examined. Two different continuum models are used: a continuum damage model and a softening plasticity model. It is shown how mesh-independent results can be obtained with both models by introducing a viscosity term. The capability of the models to represent complex intraply failure behavior is assessed by means of an analysis of a notched plate, where interface elements are used to model delamination. Proper results from different modeling approaches are shown, but upon calibration a limitation of the continuum approach in the representation of matrix failure is encountered. With a second example, this limitation is emphasized further and explained as a consequence of the homogenization that is inherent in continuum models, irrespective of the applied failure criteria and material degradation laws.