In this paper we consider the longitudinal vibrations of bimodular rods. After proving the equivalence between the entropy condition and Lax condition (whenever this latter is applicable), we go on to consider the Riemann problem and prove that a unique solution always exists except for the special case of no-tension materials. This procedure allows for easy determination of the explicit solution.
In this paper, we consider the problem of the free longitudinal vibrations of a beam made of a bimodular material, i.e. an elastic material whose in-tension Young's modulus is a fraction of that under compression. After recalling the exact solutions for an infinite beam and for a beam with fixed ends calculated via the characteristics method, we apply high-resolution methods based on the finite-element approach to solve the nonlinear equation of the motion. In particular, we compare the exact solutions with the numerical solutions calculated using the collocation and least-squares method developed in the present study, the space-time element method, as well as total variation diminishing (TVD) and Newmark methods.
The DJOSER analytical thermal solver for multilayer mounting structures has been tested as a useful and friendly tool for the thermal analysis of power electronic devices and their packages, able to replace the onerous programs based on the finite element method (FEM) calculations. The other problem connected with the packaging evaluation is the calculation of the thermally induced stresses and strains in the various layers composing the assembling structures. This paper deals with the first step of the implementation of a thermomechanical solver to be connected with the DJOSER program, which is able to calculate the stresses at the layer interfaces, using the same strategy, i.e., a semianalytical mathematical approach, as well as the same structural models (stepped pyramidal structures and homogeneous layers). The basic theory is briefly exposed and the method is applied to some two-layer virtual structures. The obtained results are compared with those obtained using standard FEM analyses.
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