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In the present work, the nonlinear dynamics of an axially functionally graded beam on a viscoelastic linear foundation is studied. The beam, modeled using Euler–Bernoulli beam theory, is axially loaded by a harmonic excitation. The motion equation are derived under two other main hypotheses, namely the beam is inextensional and it undergoes moderate large deflections.

Buckling deformations of hollow cylinders whose buckled configurations consist of inextensional deformations (folding) of their middle surface can be described as minimal states in a free gradient discontinuity problem. In the contribution this general idea is applied to a far simpler context namely the case in which the shape of the folding pattern is somehow known, that is depends on a small number of discrete or continuous parameters. The example we consider is the axial compression of a tube of rectangular hollow section. If the tube is short and thin, in a sense that can be made precise in terms of the ratio between the yield stress and the Young modulus of the material, it exhibits local buckling in the form of a rather typical pattern controlled by at most three parameters. Such parameters are identified by minimizing the total potential energy and the results are compared with test results for aluminum alloy specimens.

The paper is focused on the comparison of some numerical results with the predictions in terms of energy levels, stress intensity factor and crack orientation, of Linear Elastic Fracture Mechanics and the criterion of maximum Energy Release Rate for several Mode I, Mode II load combinations. In order to approximate solutions, the total potential energy is locally minimized via a descent method.

The paper is focused on the the safety evaluation of domes applying a modern numerical method based on the model of elastic no-tension material. The proposed method looks for a compressive thrust surface S, in equilibrium with the applied loads, contained within the extrados and the intrados surfaces of the structure. The membrane S is partitioned in two regions: one in biaxial regime, another in uniaxial regime.The boundary between the regions is unknown and depending on the external loads. The method is applied to the case study of the St. Gennaro dome. The dome has a double cap with non-parallel generatrix curves and is characterized by a fissure outline quite widespread, in both the caps, with the presence of various types of cracks.

The work is concerned with the numerical implementation of a variational model for quasi-static brittle fracture. Essentially the analysis is based on the formulation of Francfort and Marigo the main difference being the fact that we rely on local rather that on global minimization. Propagation of fracture is obtained by minimizing in a step by step process a form of energy that is the sum of bulk and interface terms. Recent attempts of producing numerical codes for variational fracture are based on the approximation of the energy, in the sense of Gamma-convergence, by means of elliptic functionals . Here instead we adopt discontinuous finite elements and search for the minima of the energy through descent methods. In constructing such a numerical model there are two main questions that must be answered: (1) The variational model for fracture requires the ability to accurately approximate the location of cracks, as well as their length. Cracks may not be restricted to propagate along the skeleton of a fixed finite element mesh. To overcome this difficulty our mesh is made variable in the sense that node positions in the reference configuration of the body are considered as further variables. (2) On adopting a Griffith type interface energy the initiation of fracture in a previously virgin body is always brutal, that is the system cannot proceed along neutral equilibrium paths or, in other words, descent directions toward local minima do not exist. To reach local minima the system must have the ability to surmount small energy barriers and then proceed through descent paths. Here we use a sort of mesh dependent relaxation of the interface energy to get out of small energy wells. The relaxation consists in the adoption of a carefully tailored cohesive type interface energy, tending to the Griffith limit as the mesh size tends to zero.