Wam Marcel Brekelmans

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.

Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling

An accurate homogenization method that accounts for large deformations and viscoelastic material behavior on microscopic and macroscopic levels is presented. This method is based on the classical homogenization theory, assuming local spatial periodicity of the microstructure. Consequently, the microstructure is identified by a representative volume element (RVE) with conformity of opposite boundaries at any stage of the deformation process. The local macroscopic stress is obtained by applying the local macroscopic deformation (represented by the deformation tensor) on a unique RVE through imposing appropriate boundary conditions and averaging the resulting RVE stress field. If the assumption of local periodicity of the morphology is valid, this homogenization procedure supplies a consistent objective relationship between the local macroscopic deformation and the microstructural deformation. The homogenization method was implemented in a multi-level finite element program with meshes on macroscopic level (mesh of entire structure) and microscopic level (meshes of RVEs). The performance was successfully verified by the comparison of the deformation of a perforated macroscopic sheet to the response of a homogenized sheet.

Multi-scale computational homogenization: Trends and challenges

In the past decades, considerable progress had been made in bridging the mechanics of materials to other disciplines, e.g. downscaling to the field of materials science or upscaling to the field of structural engineering. Within this wide context, this paper reviews the state-of-the-art of a particular, yet powerful, method, i.e. computational homogenization. The paper discusses the main trends since the early developments of this approach up to the ongoing contributions and upcoming challenges in the field.

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.

Comparison of nonlocal approaches in continuum damage mechanics

Abstract A local continuum damage theory and two distinct nonlocal variants are applied to model the failure behaviour of a construction made of macroscopically brittle material. In the nonlocal formulations a material characteristic length parameter is introduced associated with the width of the microstructural damage zone. The numerical implementation of the approaches has been performed in a finite element code. Simulation results calculated for a plane stress configuration are compared. The local approach solutions show severe lack of mesh objectivity, whereas both the nonlocal solutions converged after mesh refinement. By adequate tuning of the nonlocal descriptions mutually similar responses can be obtained, although intrinsic differences are present in the resulting damage distributions.

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

Abstract A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall–Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.

Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling

Homogenisation methods provide an efficient way to model the mechanical behaviour of heterogeneous materials. In this paper, a homogenisation procedure is adopted that allows to determine apparent properties for Perzynas elastoviscoplastic constitutive law for arbitrary microstructures. Numerical simulations on a representative volume element (RVE) are performed, from which the volume averaged state variables are acquired, necessary to establish the constitutive equations for the equivalent homogeneous medium. The applicability of mixed and periodic boundary conditions has been assessed. In addition, the difference between uniform and irregular distributions of the microstructural constituents is discussed. To substantiate our findings, a comparison is made between the global response of a heterogeneous and the corresponding homogenised structure.

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

Abstract A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress–strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading–reloading cycles are presented.

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.