SUMMARYNew methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described. These methods, called multiscale aggregating discontinuities, are based on the concept of 'perforated' unit cells, which exclude subdomains that are unstable, i.e. exhibit loss of material stability. Using this concept, it is possible to compute an equivalent discontinuity at the coarser scale, including both the direction of the discontinuity and the magnitude of the jump. These variables are then passed to the coarse-scale model along with the stress in the unit cell. The discontinuity is injected at the coarser scale by the extended finite element method. Analysis of the procedure shows that the method is consistent in power and yields a bulk stress-strain response that is stable. Applications of this procedure to crack growth in heterogeneous materials are given.

SUMMARYWe present a new multiscale method for crack simulations. This approach is based on a two-scale decomposition of the displacements and a projection to the coarse scale by using coarse scale test functions. The extended finite element method (XFEM) is used to take into account macrocracks as well as microcracks accurately. The transition of the field variables between the different scales and the role of the microfield in the coarse scale formulation are emphasized. The method is designed so that the fine scale computation can be done independently of the coarse scale computation, which is very efficient and ideal for parallelization. Several examples involving microcracks and macrocracks are given. It is shown that the effect of crack shielding and amplification for crack growth analyses can be captured efficiently.

SUMMARYIn this paper, the modified or corrected extended finite element method originally presented in Fries (Int. J. Numer. Meth. Engng. 2008; 75:503-532) for the 2D case is extended to 3D including different remedies for the problem that the crack front enrichment functions are linearly dependent in the blending elements. In the context of this extension, we address a number of computational issues of the 3D XFEM, in particular possible quadrature rules for elements with discontinuities. Also, the influence of finite deformation theory for crack simulations in comparison to linear elastic fracture mechanics is investigated. A number of numerical examples demonstrate the behavior of the presented possibilities.

A comparison between the recently developed Cosserat brick element (see [9]) and other standard elements known from the literature is presented in this paper. The Cosserat brick element uses a director vector formulation based on the theory of a Cosserat point. The strain energy for a hyperelastic element is split additively into parts for homogeneous and inhomogeneous deformations respectively. The kinetic response due to inhomogeneous deformations uses constitutive constants that are determined by analytical solutions of a rectangular parallelepiped to the deformation modes of bending, torsion and hourglassing. Standard tests are performed which typically exhibit hourglassing or locking for many other finite elements. These tests include problems for beam and plate bending, shell structures and nearly incompressible materials, as well as for buckling under high pressure loads. For all these critical tests the Cosserat brick element exhibits robustness and reliability. Moreover, it does not depend on user-tuned stabilization parameters. Thus, it shows promise of being a truly user-friendly element for problems in nonlinear elasticity.

SUMMARY The extended finite element method (XFEM) is by now well‐established for crack calculations in linear elastic fracture mechanics. An advantage of this method is its discretization independence for crack simulations. Nevertheless, discretization errors occur when using the XFEM. In this paper, a simple recovery based error estimator for the discretization error in XFEM‐calculations for cracks is presented. The method is based on the Zienkiewicz and Zhu error estimator. Enhanced smoothed stresses incorporating the discontinuities and singularities because of the cracks are recovered to enable the error estimation for arbitrary distributed cracks. This approach also allows the consideration of materials with generally inelastic behaviour. The enhanced stresses are computed by means of a least square fit problem. To assess the quality of the error estimator, global norms and the effectivity index for the global energy norm for examples with known analytical solutions are presented. Copyright © 2012 John Wiley & Sons, Ltd.

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We investigate the effect of crack shielding and amplification of various arrangements of microcracks on the stress intensity factors of a macrocrack, including large numbers of arbitrarily aligned microcracks. The extended finite element method is used for these studies. In some cases the numerical XFEM simulation provides results that are more accurate than currently available analytical approximations because the assumptions are less restrictive than those made in obtaining analytical approximations. Stress intensity factors for the tip of a macrocrack under the influence of nearby microcracks are calculated under far field mode 1 boundary conditions. For a microcrack aligned with the macrocrack the numerical results agree quite well with the analytically exact stress intensity factors. The influence of the distance to the macrocrack tip and the rotation angle is investigated for unaligned microcracks, and it is shown in several examples with many randomly distributed microcracks that the influence of those microcracks which are not in close proximity to the macrocrack tip is on the order of 5%.