Antonio Maria Cazzani

Rotations in computational solid mechanics

SummaryA survey of variational principles, which form the basis for computational methods in both continuum mechanics and multi-rigid body dynamics is presented: all of them have the distinguishing feature of making an explicit use of the finite rotation tensor.A coherent unified treatment is therefore given, ranging from finite elasticity to incremental updated Lagrangean formulations that are suitable for accomodating mechanical nonlinearities of an almost general type, to time-finite elements for dynamic analyses. Selected numerical examples are provided to show the performances of computational techniques relying on these formulations.Throughout the paper, an attempt is made to keep the mathematical abstraction to a minimum, and to retain conceptual clarity at the expense of brevity. It is hoped that the article is self-contained and easily readable by nonspecialists.While a part of the article rediscusses some previously published work, many parts of it deal with new results, documented here for the first time.

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.

Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes

A family of new 4-noded membrane elements with drilling degrees of freedom and unsymmetric assumed stresses is presented; it is derived from a mixed variational principle originally formulated for finite strain analysis and already used in the literature to develop a purely kinematic membrane model. The performance of these elements, investigated through some well established benchmark problems, is found to be fairly good and their accuracy is comparable with that given by models with a larger number of nodal parameters.

ON THE EVALUATION OF THE SHAKEDOWN BOUNDARY FOR TEMPERATURE-DEPENDENT ELASTIC PROPERTIES

ABSTRACT A computational procedure for estimating the shakedown limit in the presence of temperature-dependent elastic moduli is proposed. Strictly speaking, only upper bounds are provided; however, the correct value is obtained when the actual inadaptation modality is identified and the procedure indicates a way for detecting it.