SUMMARYThe superconvergent patch recovery (SPR) technique is widely used in the evaluation of a recovered stress field r * from the finite element solution r fe . Several modifications of the original SPR technique have been proposed. A new improvement of the SPR technique, called SPR-C technique (Constrained SPR), is presented in this paper. This new technique proposes the use of the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch r * p that locally satisfy the equations that should be satisfied by the exact solution. As a result the evaluated expressions for r * p will satisfy the internal equilibrium and compatibility equations in the whole patch and the boundary equilibrium equation at least in vertex boundary nodes and, under certain circumstances, along the whole boundary of the patch coinciding with the boundary of the domain. The results show that the use of this technique considerably improves the accuracy of the recovered stress field r * and therefore the local effectivity of the ZZ error estimator.

SUMMARYThe application of the extended finite element method (XFEM) to fracture mechanics problems enables one to obtain accurate solutions more efficiently than with the standard finite element method. A component can be modelled without the need to build a mesh that matches the crack geometry, and thus remeshing as the crack grows is unnecessary. In the XFEM approach, the interpolation on certain elements is enriched with functions that make it feasible to represent the crack tip asymptotic displacement fields by using a local partition of unity method. However, the enrichment is only partial in the blending elements connecting the enriched zone with the rest of the mesh, and consequently pathological terms appear in the interpolation, which lead to increased error. In this study we propose enhancing the blending elements by adding hierarchical shape functions where appropriate; this permits compensating for the unwanted terms in the interpolation. This technique is an extension of the study of Chessa et al. (Int. J. Numer. Meth. Engng. 2003; 57:1015-1038) to fracture mechanics problems. The numerical results show that the proposed enhancement always results in greater accuracy. Moreover, enhancing the blending elements makes it possible to recover the convergence rate that is decreased when the degrees of freedom gathering technique is used to improve the condition number of the stiffness matrix.

In this work, we present two strategies for the numerical modeling of microcracks and damage within an osteon. A numerical model of a single osteon under compressive diametral load is developed, including lamellae organized concentrically around the haversian canal and the presence of lacunae. Elastic properties have been estimated from micromechanical models that consider the mineralized collagen fibrils reinforced with hydroxyapatite crystals and the dominating orientation of the fibrils in each lamella. Microcracks are simulated through the node release technique, enabling propagation along the lamellae interfaces by application of failure criteria initially conceived for composite materials, in particular the Brewer and Lagacé criterion for delamination. A second approach is also presented, which is based on the progressive degradation of the stiffness at the element level as the damage increases. Both strategies are discussed, showing a good agreement with experimental evidence reported by other authors. It is concluded that interlaminar shear stresses are the main cause of failure of an osteon under compressive diametral load.

Fuenmayor Fernández, FJ. (2014). Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models. Biomechanics and Modeling in Mechanobiology. 13(2):1-21. doi:10.1007/s10237-013-0507-y.

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