Franco Pastrone

WAVE PROPAGATION IN MICROSTRUCTURED SOLIDS

A general model of bodies with vectorial microstructures is introduced, which includes many particular models such as scalar and affine microstructures, Cosserat continua, and liquid crystals. Assuming the existence of a strain energy function, the field equations are obtained via a variational principle as Euler-Lagrange equations of an energetic functional. Dissipative forces are also introduced. Equilibrium and stability problems are briefly discussed. Within the framework of singular surfaces theory, nonlinear acceleration and shock waves are studied and proper propagation conditions for nondissipative materials are given. The influence of dissipation is briefly discussed.

Axisymmetric equilibrium states of non-linear elastic cylindrical shells

Abstract The problem of nonuniqueness of static axisymmetric solutions for a constrained cylindrical shell under a compressive thrust is studied. Both in the elastic and hyperelastic case, we prove the existence of buckled states. For a simple choice of the elastic potential, a Hamiltonian formulation is provided.

Shear waves in micro-faulted materials

We discuss two alternative models of wave propagation in materials with a continuous distribution of microscopic faults. Dissipation is explicitly taken into account, but special exchange relations between the macroscopic deformation and the microstructure may lead to the amplification of wave amplitudes.

Waves in approximately constrained materials and applications to fiber-reinforced composites

Wave propagation in approximately constrained elastic homogeneous materials is investigated by a suitable ray method; propagation speeds and evolution equations are determined and a comparison is provided by the results of other authors. The theory is applied to isotropic and anisotropic materials and to a model for unidirectionally fiber-reinforced composites.

Nonuniqueness of equilibrium states for axisymmetric elastic shells in tension

The problem of nonuniqueness of static axisymmetric solutions for a non-linearly elastic cylindrical shell in which the ends are pulled apart with a constant traction while retaining the radii of its ends fixed is studied. In the elastic case, we prove the existence of buckled states and the possibility of necking. In the hyperelastic case a global existence and nonuniqueness theorem is proved, via the energy criterion.

Nonlinear waves and solitons in complex solids

The problem of the propagation of nonlinear waves in complex solids, namely bodies with different internal microstructures, is analyzed. In the first part, we make use of a general model of microstructured solids as introduced by Engelbrechet and Pastrone (Acc Sc Torino Mem Sc Fis 2011; 35: 23–36) and study two particular relevant models: one-dimensional solid with hierarchical microstructure and with concurrent microstructures. As expected, the hierarchical microstructure leads, with a particular but meaningful choice of the strain energy function, to a sixth-order partial differential equation (PDE) with a characteristic hierarchical structure. Hence, the case of two concurrent microstructures, as introduced by Berezovski, Engelbrecht and Berezovski (Acta Mech 2011; 220(1–4): 349–363), is studied and again for suitable explicit forms of the energy function we can obtain a fourth-order PDE and actually prove the possibility of propagation of solitary and cnoidal waves.

Travelling Waves in Non Linear Elastic Solids with Multiple Microstructures

In non classical mechanics naturally arises the problem of the propagation of nonlinear waves in solids with different internal structural scales. Here we make use of a suitable model of one dimensional microstructured solids to describe the behavior of internal structures with two different scales. Hence we have an elastic material composed by a macrostructure, a first microstructure (say a mesostructure) and a second microstructure at some smaller scale. The choice of suitable microstrains functions φ and ψ at the two levels respectively, of the microdisplacement u, of their time derivatives as strain velocities, allows us to obtain the field equations via a variational principle. In a particular case a sixth order PDE is obtained, with characteristic hierarchical structure, where the three levels hierarchy and the various coefficients may reflect the dominance of one structural level over the other ones in wave propagation. This equation is integrated in terms of elliptic functions. Using the same basic model, the case of two concurrent microstructures is studied.

Vectorial Microstructures: Applications to Fabrics and Granular Media

Textile fabrics and 2D granular media can be modelled by means of non-classical nonlinear elasticity, in particular using the theory of vectorial microstructures. The field equations are obtained via a variational principle and suitable constitutive relations are given in both cases. The microstructure takes into account the micro-ondulation of the fibres in textiles and the grains in granular media.

Axisymmetric Instabilities for Elastic Conical Shells under Compressive End Loadings

We deal with the problem of equilibrium and buckling of nonlinear elastic axisymmetric shells, whose referential shape is a truncated circular cone, subject to compressive end loadings. It is a non-trivial generalization of the cylindrical case, already studied by Cohen and Pastrone; in this particular case, suitable constitutive restrictions, which are not needed for cylinders, must be satisfied to allow the conical shell to sustain end loadings, even small, without bending. In particular, we consider thin Kirchhoff shells and prove, by means of the bifurcation theory of Poincaré, the non-uniqueness of solutions of the boundary value problem associated with the equilibrium equations, the assigned end loadings and the geometrical constraints: i.e., the axisymmetry and the inextensibility along meridians. The critical loads are determined as well as the bifurcation points. If the material is hyperelastic the equations of equilibrium are derived from a variational principle and, for some special form of the strain energy function, a Hamiltonian formulation can be provided. The possibility of a non-convex strain energy function is briefly discussed.

THERMOELASTIC STRESS ANALYSIS FOR LINEAR ELASTIC BODIES

The stress analysis for elastic cylinders made of polymethylmethacrylate (PMMA) under cyclic loading is provided by using FEM. Under the assumption that the elastic moduli depend linearly on the temperature, new fomulae, generalizing the well known Kelvins formula, are given such that we can estimate the temperature variation on the external surface of a homogeneous ideal cylinder. Using suitable numerical methods we obtain theoretical results which fit the experimental values for the temperature variation