Paolo Podio-Guidugli

Configurational forces and the basic laws for crack propagation

This paper develops a framework for dynamical fracture, concentrating on the derivation of basic field equations that describe the motion of the crack tip in two space-dimensions. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law.

A regularized equation for anisotropic motion-by-curvature

For realistic interfacial energies, the equations of anisotropic motion-by-curvature exhibit backward-parabolic behavior over portions of their domain, thereby inducing phenomena such as the formation of facets and wrinkles. In this paper, a physically consistent regularized equation that may be used to analyze such phenomena is derived.

Hypertractions and hyperstresses convey the same mechanical information

A strengthened and generalized version of the standard Virtual Work Principle is shown to imply, in addition to bulk and boundary balances, a one-to-one correspondence between surface and edge hypertractions and hyperstress fields in second-gradient continua. When edge hypertractions are constitutively taken null, the hyperstress is shown to take the form it has for a relevant example of second-gradient fluid-like material, referred to as a Navier–Stokes—α fluid.

Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving

Abstract This paper develops a framework for dynamical fracture, concentrating on the derivation of balance equations and constitutive equations that describe the motion of the crack tip in two space-dimensions. The theory is based on a configurational force balance and a mechanical version of the second law of thermodynamics. Kinking and curving of the crack are allowed under the assumption that the crack will propagate in a direction that maximizes the rate at which it dissipates energy.

Discontinuous energy minimizers in nonlinear elastostatics: an example of J. Ball revisited

Simple direct methods of the Calculus of Variations, together with milda priori restrictions of a constitutive nature, are exploited to show that a uniform radial displacement at the boundary fails to induce a homogeneous radial expansion of a compressible elastic ball when the boundary displacement reaches a critical value (of which an explicit lower bound is offered): rather, above the critical value, energy minimizers are radial deformations with a hole at the center, and are accompanied by a stress field with radial stress vanishing, and circumferential stress unbounded, at the surface of the hole.

A JUSTIFICATION OF THE REISSNER–MINDLIN PLATE THEORY THROUGH VARIATIONAL CONVERGENCE

We provide a justification of the Reissner–Mindlin plate theory, using linear threedimensional elasticity as framework and Γ-convergence as technical tool. Essential to our developments is the selection of a transversely isotropic material class whose stored energy depends on (first and) second gradients of the displacement field. Our choices of a candidate Γ-limit and a scaling law of the basic energy functional in terms of a thinness parameter are guided by mechanical and formal arguments that our variational convergence theorem is meant to validate mathematically.

On Configurational Inertial Forces at a Phase Interface

The evolution equations for a phase interface are discussed. Interfacial structure is neglected, as are thermal and compositional variations. The focus is on a new treatment of the inertial forces at the interface.

EXISTENCE AND UNIQUENESS OF A GLOBAL-IN-TIME SOLUTION TO A PHASE SEGREGATION PROBLEM OF THE ALLEN–CAHN TYPE

We study a model of phase segregation of the Allen–Cahn type, consisting in a system of two differential equations, one partial and the other ordinary, respectively interpreted as balances of microforces and microenergy; the two unknowns are the order parameter entering the standard A–C equation and the chemical potential. We introduce a notion of maximal solution to the o.d.e., parametrized on the order-parameter field; and, by substitution in the p.d.e. of the so-obtained chemical potential field, we give the latter equation the form of an Allen–Cahn equation for the order parameter, with a memory term. Finally, we prove the existence and uniqueness of global-in-time smooth solutions to this modified A–C equation, and we give a description of the relative ω-limit set.

Elasticity for geotechnicians

Description based on online resource; title from PDF title page (ebrary, viewed October 10, 2013).

Thickness waves in electroelastic plates

Abstract Within the linear theory developed in [J. Struct. Control 5 (2) (1998) 73] for coherently oriented, transversely isotropic electroelastic plates capable of thickness changes, the general boundary-value problem uncouples into a “membrane” problem and a “flexure” problem. When progressive waves for the membrane problem are investigated, the relative propagation condition reveals that three different types of solutions exist involving oscillatory thickness distension and contraction accompanied by in-plane motion. In the special case of no electroelastic coupling in the material response the propagation condition can be explicitly solved; one purely electrical and two purely mechanical waves obtain. A simple argument based on kinematical similarities indicates that the two mechanical waves can be regarded as two-dimensional counterparts of the first equivoluminal and dilatational modes of the three-dimensional Rayleigh–Lamb theory.