Gianpietro Del Piero

Macro- and micro-cracking in one-dimensional elasticity

Abstract In classical fracture mechanics, the equilibrium configurations of an elastic body are obtained by minimizing an energy functional containing two contributions, bulk and surface. Usually, the bulk energy is convex and the surface energy is concave. While this type of minimization successfully describes macroscopic cracks, it fails to model micro-defects forming a so-called process zone. To describe this phenomenon, we consider, in this paper, a model with a non-concave, “bi-modal” surface energy, which allows the formation of both macro- and micro-cracks. Specifically, we consider the simplest one-dimensional problem for a bar in a hard device and show that if the surface energy is not subadditive, the solution exhibits a new mode of failure with a finite number of micro-cracks coexisting with one fully developed macro-crack. We present an explicit example of a “quantized” micro-cracking with a subsequent development into a single macro-crack.

On the analytic expression of the free energy in linear viscoelasticity

Two definitions of free energy for a linear viscoelastic material, due to Graffi and to Coleman and Owen, are considered, and the compatibility of these definitions with some expressions of the free energy proposed in the literature is examined. For the expressions of Staverman and Schwarzl and of Breuer and Onat, the two definitions are proved to be equivalent, and the set of all relaxation functions for which the two expressions are indeed free energies is determined. Two more expressions, proposed by Volterra and Graffi and by Morro and Vianello, are taken into consideration. For them, only the classes of relaxation functions for which they are free energies according to the first definition, is completely characterized. All results are established under regularity assumptions weaker than those usually made in the literature.

On the completeness of the crystallographic symmetries in the description of the symmetries of the elastic tensor

It is shown that all symmetries possible for the elastic tensors can be reduced to the twelve symmetries already used in the description of the crystal classes. Each symmetry can be characterized by a group of rotations generated by no more than two rotations. The use of a canonical basis related to such rotations considerably simplifies the component forms of the elasticity tensor. This result applies to non-symmetric tensors; for symmetric tensors, the number of independent symmetries reduces from twelve to ten.After the present work was submitted, the following paper came to our attention: 14. S.C. Cowin and M.M. Mehrabadi, On the identification of material symmetry for anisotropic elastic materials. Q. Jl. Mech. appl. Math.40 (1987) 451–476. This paper contains an independent analysis of the partial ordering ≺ among the crystallographic elastic symmetries. However, it does not deal with the problem of the completeness of these symmetries.

A Variational Approach to Fracture and Other Inelastic Phenomena

The aim of the present article is a description of the many phenomenological aspects of fracture and of their relations with other inelastic phenomena, such as the formation of microstructure and the changes in the material’s strength induced by plasticity and damage.

STRAIN LOCALIZATION IN OPEN-CELL POLYURETHANE FOAMS: EXPERIMENTS AND THEORETICAL MODEL

Confined compression tests performed by the authors on open-cell polyurethane foams reveal the presence of strain localization. After a brief description of the experiments, a theoretical model is proposed. In the model, the foam is represented as a chain of elastic springs with a two-phase strain energy density, and the strain localization is due to a progressive collapse of the springs. The collapse is a sort of continuum instability, which can be attributed to phase transition. An appropriate choice of the material constants leads to a close reproduction of the experimental force-elongation response curves.

Lower bounds for the critical loads of elastic bodies

SummaryWe discuss briefly a sufficient condition for hadamard stability, proposed by Holden and applied by the same Author to the evaluation of lower bounds for the critical load of elastic bodies subject to a prescribed deformation process.A refinement of the method leads to an improved condition of stability, which is also more simple in form. When applied to a body in simple compression, the new condition yields the same estimate of the critical load as Holdens condition, whereas a substantial improvement is found in the case of simple extension.RiassuntoSi discute brevemente una condizione sufficiente per la stabilità secondo Hadamard, proposta da Holden e dallo stesso applicata alla delimitazione inferiore del carico critico di un corpo elastico soggetto a un processo deformativo preassegnato.Viene successivamente dedotta una condizione sufficiente più semplice e, nello stesso tempo, più larga della precedente. Nel caso di un corpo soggetto a compressione uniforme, essa dà una stima del carico critico coincidente con quella di Holden, mentre, nel caso di trazione uniforme, dà luogo ad una stima nettamente migliore.

The Energy of a One-Dimensional Structured Deformation

Starting from an assumed expression for the energy of a simple deformation, a representation formula for the energy of a one-dimensional structured deformation is obtained. If w and 0 denote the bulk and the interfacial energy density for a simple deformation, the corresponding densities for a structured deformation are determined by the lower semicontinuous and convex envelope of w and by the subadditive envelope of 0. This result holds under a specific type of convergence assumed for approximating sequences of simple deformations; as discussed briefly in the final remarks, other physically meaningful notions of convergence may lead to different expressions for the energy.

A Class of Fit Regions and a Universe of Shapes for Continuum Mechanics

A new class of fit regions is proposed as an alternative to those available in the literature, and specifically to the class defined by Noll and Virga in their paper [12]. An advantage of the proposed class is that of being based mostly on topological concepts rather than on less familiar concepts from geometric measure theory. A distinction is introduced between fit regions and shapes of continuous bodies. The latter are defined as equivalence classes of fit regions, made of regions all with the same interior and with the same closure. In the final part of the paper the axioms for a universe of bodies, formulated by Noll and incorporated in Truesdells book [15], are re-discussed and partially re-formulated.

Interface Energies and Structured Deformations in Plasticity

I discuss the model of an elastic bar with interface energy at the jump points of the displacement function. If the interface energy is convex near the origin, the energy minima are attained by uniform limits of sequences of functions with a number of jumps growing to infinity, while their amplitudes decrease to zero. The limit elements are identified with mathematical objects called structured deformations. The model determines three regimes for the bar, which I call elastic, plastic, and fractured, and is able to describe the phenomenon of elastic unloading.

Foundations of the Theory of Structured Deformations

To introduce the theory of structured deformations, a good starting point is perhaps to illustrate a situation in which such objects arise naturally, although unexpectedly, from a problem in fracture mechanics.1