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Abstract: Various macroscopic models to describe instability pattern formation are discussed in this paper. They are similar to the Ginzburg–Landau envelope equation, but they can remain valid away from the bifurcation and are based on the technique of Fourier series with slowly varying coefficients. We focus on two questions: the need to take phase changes into account and the boundary conditions to be associated with macroscopic models. The analysis is carried out on the basis of numerical simulations for the problem of a compressed beam on a nonlinear foundation that is quite similar to the well known Swift–Hohenberg equation. The first macroscopic model involves a real envelope so that the phase is assumed to be constant. The second model is also macroscopic and it is a sort of Ginzburg–Landau equation with a complex envelope. The third one follows from a multi-scaled approach with a numerical bridging between the full model near the boundary and a macroscopic model in the bulk. [Copyright &y& Elsevier]

Abstract: In this work, we have assessed the results of the local and nonlocal versions of Rousselier’s damage model, which have been used here for simulation of ductile crack growth. There are several issues regarding the accuracy of the results which has been addressed in this paper, e.g., accuracy in simulation of crack path, extent and width of the damaged region, fracture resistance behaviour in situations such as symmetric vs. non-symmetric boundary-value problems, mixed-mode loading vs. mode-I loading of the crack-tip, etc. It was also observed that the shape and orientation of the elements at the crack-tip, in addition to their size, influence the results of the local damage model. In this work, it was shown that the above issues can be resolved through the use of nonlocal damage models. The predictions of the nonlocal model are also consistent with the experimental observations unlike its local counterpart. Several examples were presented, where the results as obtained by both the local and nonlocal models were compared. From this experience, it is recommended that the local damage models should not be used blindly by the analysts for all kinds of mesh design, loading, boundary conditions, etc. [Copyright &y& Elsevier]