The XFEM for high‐gradient solutions in convection‐dominated problems

Convection‐dominated problems typically involve solutions with high gradients near the domain boundaries (boundary layers) or inside the domain (shocks). The approximation of such solutions by means of the standard finite element method requires stabilization in order to avoid spurious oscillations. However, accurate results may still require a mesh refinement near the high gradients. Herein, we investigate the extended finite element method (XFEM) with a new enrichment scheme that enables highly accurate results without stabilization or mesh refinement. A set of regularized Heaviside functions is used for the enrichment in the vicinity of the high gradients. Different linear and non‐linear problems in one and two dimensions are considered and show the ability of the proposed enrichment to capture arbitrary high gradients in the solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust and direct evaluation of J2 in linear elastic fracture mechanics with the X‐FEM

The aim of the present paper is to study the accuracy and the robustness of the evaluation of J_k‐integrals in linear elastic fracture mechanics using the extended finite element method (X‐FEM) approach. X‐FEM is a numerical method based on the partition of unity framework that allows the representation of discontinuity surfaces such as cracks, material inclusions or holes without meshing them explicitly. The main focus in this contribution is to compare various approaches for the numerical evaluation of the J₂‐integral. These approaches have been proposed in the context of both classical and enriched finite elements. However, their convergence and the robustness have not yet been studied, which are the goals of this contribution. It is shown that the approaches that were used previously within the enriched finite element context do not converge numerically and that this convergence can be recovered with an improved strategy that is proposed in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Eshelby’s inclusion problem in the gradient theory of elasticity: Applications to composite materials

We extend Eshelby’s integral representations for elastic inclusion problems to the case of gradient theory of elasticity. Gradient elastic effects associated with the existence of an interphase layer, within a simple and robust gradient model whose properties are described by the harmonic equation, are discussed. The decomposition of the corresponding solution into “classical” and “gradient” components is established. It is shown that the aforementioned Eshelby-type integral formulas for gradient elasticity can be expressed in the same form as in the standard theory of elasticity, but only for the “classical” part of the solution. The implementation of Eshelby’s approach in determining the effective properties of composites by the three-phase method requires the derivation of a complete solution for the gradient model. An example of application of the so-obtained generalized gradient method for determining the effective properties of composites with size effects due to cohesion and surface forces is given.     

A numerical method to simulate ductile failure of tensile plates with interacting through‐wall cracks

This paper proposes a simple numerical method to simulate ductile failure behaviours of tensile plates with interacting through‐wall cracks. The method is based on the finite element damage analysis using the stress‐modified fracture strain damage model. To validate the proposed method, simulated results are compared with a total of 23 published experimental data of flat tensile plates with interacting through‐wall cracks. Despite its simplicity, the proposed method well predicted the experimental maximum loads of tensile plates with interacting cracks, including the loads for crack coalescence. Systematic analyses are also performed to investigate the effect of the element size used in the finite element damage analysis.     

A Unit‐Cell Approach of Finite Element Analysis for Transverse Impact Damage of 3‐D Biaxial Spacer Weft‐Knitted Composite

Abstract: This paper evaluates the transverse impact damage of a 3‐D biaxial spacer weft‐knitted composite using experimental results and complimentary finite element analysis. The load–displacement curves and damage morphologies during impact loading were obtained to analyse energy absorption and impact damage mechanisms of the knitted composite. A unit‐cell model based on the microstructure of the 3‐D knitted composite was established to calculate the deformation and damage evolution when the composite is impacted by a hemisphere‐ended steel rod. An elastoplastic constitutive equation is incorporated into the unit‐cell model and the critical damage area failure theory developed by Hahn and Tsai has been implemented as a user‐defined material law (VUMAT) for commercial available finite element code ABAQUS/Explicit. The load–displacement curves, impact damages and impact energy absorption obtained from ABAQUS/Explicit are compared with those FROM experiments. The good agreement of the comparisons supports the validity of the unit‐cell model and user‐defined subroutine VUMAT. The unit‐cell model can also be extended to evaluate the impact crashworthiness of engineering structures made out of the 3‐D knitted composites.