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A note on tensile instabilities and loss of ellipticity for a fiber-reinforced nonlinearly elastic solid

Merodio J., Neff P.

Archives of Mechanics

2006

Vol. 58, nr 3

293-303

In this paper we examine the loss of ellipticity and the associated failure of fiber-reinforced compressible nonlinearly elastic solids under deformations leading to fiber extension. In particular, the analysis concerns a material model that consists of an isotropic base material augmented by a reinforcement depending on the fiber direction and referred to as a reinforcing model. We examine a reinforcement that introduces additional stiffness under simple shear deformations in the fiber direction. In previous contributions it was shown for this material that loss of ellipticity under uniaxial tensile loading in the fiber direction requires a non-convex reinforcing model. Here we generalize this result and show that loss of ellipticity under plane deformations not associated with uniaxial loading in the fiber direction but also creating fiber extension may occur for convex reinforcing models.

Numerical simulation of atomic positions in quantum dot by means of molecular statics

Dłużewski P., Traczykowski P.

Archives of Mechanics

2003

Vol. 55, nr 5-6

393-406

Deformation of a crystal structure is considered here in terms of constitutive modelling based upon both the atomistic and continuum approaches. Atomistic calculations are made by using the Stillinger-Weber potential for the GaAs and CdTe structures. The stress-strain behaviour of the best-known anisotropic hyperelastic models are compared with the behaviour of the atomistic one in the uniaxial deformation test.

Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 2, The strongly curved shells

Hamdouni A., Millet O.

Archives of Mechanics

2003

Vol. 55, nr 2

177-219

In the first part of this paper we have deduced a classification of asymptotic shallow shell models with respect to the level of applied forces, from the non-linear three-dimensional elasticity. We have used a constructive approach based on a dimensional analysis of the non-linear three-dimensional equilibrium equations, which naturally makes appear dimensionless numbers characterizing the applied forces (... and ...) and the geometry of the shell (... and C). To limit our study to one-scale problems, these dimensionless numbers are expressed in terms of the relative thickness ... of the shell, considered as the perturbation parameter. In the first part, we have studied the case of shallow shells corresponding to C=.... In the second part of this paper, we will study the case of strongly curved shells for which C=.... The classification that we obtain is then more complex. It depends not only on the force levels, but also on the existence of inextensional displacements which keep invariant the metric of the middle surface of the shell.

Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Pt. 1, The shallow shells

Hamdouni A., Millet O.

Archives of Mechanics

2003

Vol. 55, nr 2

135-175

The purpose of this paper is to construct a classification of asymptotic shell models (inferred from the non linear three-dimensional elasticity) with respect to the applied forces and to the geometrical data. To do this, we use a constructive approach based on a dimensional analysis of the nonlinear three-dimensional equilibrium equations, which naturally gives rise to the appearance of dimensionless numbers characterizing the applied forces and the geometry of the shell. In order to limit our study to one-scale problems, these dimensionless numbers are expressed in terms of to the relative thickness ... of the shell, which is considered as the perturbation parameter. This leads, on the one hand, to distinguish shallow shells from strongly curved shells which have a different asymptotic behaviour, and on the other hand, to fix the applied force level. For each of these two classes of shells, using the usual asymptotic method, we propose a complete classification of two-dimensional shell models based on decreasing force levels, from severe to low. In the first part of this paper, we present the classification for shallow shells. We obtain successively the nonlinear membrane model, another membrane model, Koiter's non linear shallow shell model, and the linear Novozhilov-Donnell one, respectively for severe, high, moderate and low forces.