Marco Rovati

Extrema of Young’s modulus for cubic and transversely isotropic solids☆

For a homogeneous anisotropic and linearly elastic solid, the general expression of Youngs modulus EðnÞ, embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which EðnÞ attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Youngs modulus on direction n are given as well.

On the negative Poisson’s ratio of an orthorhombic alloy

Abstract The existence of the counterintuitive property for which a material laterally expands when stretched, is described with reference to the orthorhombic CuAlNi alloy. In particular, it is shown that there is a set of planes for which Poisson’s ratio is always negative.

Stationarity of the strain energy density for some classes of anisotropic solids

Homogeneous, anisotropic and linearly elastic solids, subjected to a given state of strain (or stress), are considered. The problem dealt with consists in finding the mutual orientations of the principal directions of strain to the material symmetry axes in order to make the strain energy density stationary. Such relative orientations are described through three Eulers angles. When the stationarity problem is formulated for the generally anisotropic solid, it is shown that the necessary condition for stationarity demands for coaxiality of the stress and the strain tensors. From this feature, a procedure which leads to closed form solutions is proposed. To this end, tetragonal and cubic symmetry classes, together with transverse isotropy, are carefully dealt with, and for each case all the sets of Eulers angles corresponding to critical points of the energy density are found and discussed. For these symmetries, three material parameters are then defined, which play a crucial role in ordering the energy values corresponding to each solution. 2003 Elsevier Ltd. All rights reserved.

Optimal orientation of the symmetry axes of orthotropic 3-D materials

Assuming the elastic energy as a meaningful measure of the global stiffness (or flexibility) of an elastic body, in this paper the interest is paid to the determination of those local orientations of the material symmetry axes in an orthotropic solid which correspond to extreme values of the energy density.

Optimal Orientation of Orthotropic Properties for Continuum Bodies and Structural Elements

In this paper some notes on recent results obtained in the field of optimal design of orthotropic bodies and components are given. After a brief recall of the mechanical behaviour of orthotropic bodies and structural elements, a general formulation of the optimization problem is given: assuming the elastic energy as a meaningful measure of the global stiffness (or flexibility) of a three dimensional elastic body, the interest is focused on the determination of those local orientations of material symmetry axes in an orthotropic solid, which correspond to extreme values of the energy density. The stationarity conditions are computed, and a mechanical interpretation is given as well. Then, the two dimensional case is dealt with in details, both for the elastic and plastic cases, defining also a unified approach for approximate optimal solutions. Some simple examples of application are given as well.

Effect of Internal Length Scale on Optimal Topologies for Cosserat Continua

Micropolar field theory represents an extension of the classical Cauchy continuum theory. In this paper, a topology optimization procedure for maximum stiffness is applied to two dimensional structural elements described in terms of micropolar (Cosserat) solids. The role of the characteristic length of bending on the optimal configurations is highlighted in the applications.

Closed-form Solutions in Optimal Design of Structures with Nonlinear Behavior

ABSTRACT Optimal design problems for flexural systems with a nonlinear constitutive law are considered, in the presence of constraints on displacements. A general nonlinear holonomic moment-curvature relationship is assumed and a direct variational method is applied in order to obtain optimality criteria. Accordingly, a general method of solution is proposed and some examples are solved.