Carmine Trimarco

Pseudomomentum and material forces in nonlinear elasticity : variational formulations and application to brittle fracture

SummaryThis work examines critically the various formulations of the balance of linear momentum innonlinear inhomogeneous elasticity. The corresponding variational formulations are presented. From the point of view of the theory of elastic inhomogeneities, the most interesting formulations are those which, being either completely material or mixed-Eulerian, exhibit explicitly the inhomogeneities in the form ofmaterial forces. They correspond to the balance ofpseudomomentum, a material covector which is seldom used but which we show to play a fundamental role in the Hamiltonian canonical formulation of nonlinear elasticity. The flux associated with pseudomomentum is none other than theEshelby material tensor. Applying this formulation to the case of an elastic body containing a crack of finite extent, the notion of suction force acting at the tip of the crack follows while afracture criterion à la Griffith can be deduced from a variational inequality. Possible extensions to higher-grade elastic materials and inelastic materials are indicated as well as the role played by pseudomomentum in the quantization of elastic vibrations.

Elements of Field Theory in Inhomogeneous and Defective Materials

So-called configurational forces, also called material forces in modern continuum mechanics, and more generally energetic driving forces, are those « forces »which are associated by duality to the displacement or motion of whatever may be considered a defect in a continuum field theory. Conceptually simple examples of such « defects » are dislocations in ordered crystals, disclinations in liquid crystals, vortices in fluid mechanics, cracks and cavities in materials science, propagating fronts in phase-transition problems, shock waves in continuum mechanics, domain walls in solid-state science, and more generally all manifestations, smooth or abrupt, of changes in material properties. In such a framework, the material symmetry of the physical system is broken by the presence of a field singularity of a given dimensionality (point, line, surface, volume). Until very recently all these domains were studied separately but a general framework emerged essentially through the works of the authors and co-workers, basing initially on inclusive ideas of J.D.Eshelby (deceased 1985) — hence the coinage of Eshelbian mechanics by the authors for the mechanics of such forces. In this framework which is developed in a somewhat synthetic form, all configurational forces appear as forces of a non-Newtonian nature, acting on the material manifold (the set of points building up the material whether discrete or continuous) and not in physical space which remains the realm of Newtonian forces and their more modern realizations which usually act per quantity of matter (mass or electric charge). That is, configurational forces act on spatial gradients of properties, on field singularities, etc. They acquire a true physical meaning only in so far as the associated expanded power is none other than a dissipation; accordingly, configurational forces are essentially used to formulate criteria of progress of defects in accordance with the second law of thermodynamics. Within such a general vision, in fact, many irreversible properties of matter (e.g., damage, plasticity, magnetic hysteresis, phase transition, growth) are seen as irreversible local rearrangements of matter (material particles in an ordered crystal, spin layout in a ferromagnetic sample, director network in a liquid crystal) that are represented by pure material mappings. This is where some elements of modern differential geometry enter the picture following earlier works by Kroner, Noll, and others.

On Material and Physical Forces in Liquid Crystals

Abstract Various variational formulations, applications of Noethers theorem, and direct manipulations of field equations are used to unambiguously exhibit and contrast the various forms of physical forces (those acting on mass elements) and so-called material (configurational) forces (which are the relevant agents of the motion of defects according to Eshelby) in the case of nematic liquid crystals. Although a material formulation and due attention paid to a clear distinction between fields and parameters in variational formulations do help to clarify the matter, some ambiguity remains due to the inherent mixed, fluid-ordered medium , nature of liquid crystals and this, to some extent, justifies the mixed feelings expressed by J. D. Eshelby and E. Kroner in previous attempts at establishing a distinction between the two classes of forces—an ambiguity which does not survive in solid crystals.

Pseudomomentum and material forces in inhomogeneous materials

Abstract The balance of pseudomomentum (covariant material , canonical momentum) is established in different ways for both pure finite-strain anisotropic elasticity and electromagnetoelasticity in the Galilean approximation for materially inhomogeneous solids. This balance law relates pseudomomentum. Eshelbys energy-momentum tensor, and the material inhomogeneity force. The relationship with the Hamiltonian canonical formulation of finite-strain elasticity is outlined and consequences for the evaluation of energy release rate via path-independent integrals in electro- or magnetoelasticity are drawn. This work, of a fairly general nature, builds on the pioneering results of the Stanford group around G. Herrmann.

Material Mechanics of Electromagnetic Solids

Eshelby [1–5] introduced the notion (and the naming) of Maxwell stress tensor of Elasticity having in mind the Maxwell energy-stress of electromagnetism.

Driving Force on Phase Transition Fronts in Thermoelectroelastic Crystals

Starting from known physical balance laws and constitutive equations, this work develops in a rational number the canonical balance laws of momentum and energy at both regular and singular material points. This is achieved in the framework of the quasi-electrostatics of thermoelectroelastic crystals, with a view to providing a sound basis for the study of electroelastic fracture, and of the propagation of phase transition fronts. The latter receive special attention with the formulation of a true thermomechanics in which it is shown that a hot surface heat source develops at thermodynamically irreversible points of progress of such fronts. The expression of the driving force acting on such fronts in these conditions is obtained in terms of the relevant Hugoniot-Gibbs functional. This provides the basis for the formulation of simple criteria of progress of the front.

Configurational Forces and Coherent Phase-Transition Fronts in Thermoelastic Solids

The progress of a thermoelastic phase into another one of different symmetry may be viewed as the progress of a defect or a material inhomogeneity (one phase) into a thermoelastic solid (the other phase) [l], [2]. Furthermore, the condition of coherence between phases at the front (continuity of the lattice sites in microscopic terms) is best expressed on the material manifold and, dynamically, using the so-called inverse-motion description of which G. Piola was a pioneer; while the notion of transformation strain associated with the phase transition fits perfectly in the formalism introduced by Epstein and Mangin [3]. Within the framework of the theory of material inhomogeneities [4] and for arbitrary nonlinear thermoelastic conductors of heat, the power expended by the inhomogeneity force at the interface is computed. This is equivalent to a heat source at the interface. An application of irreversible thermodynamics allows then a closure of the system of field-jump equations.

Theory of elastic inhomogeneities in electromagnetic materials

Abstract A field theoretic formulation of nonlinear anistropic inhomogeneous elasticity and electromagnetoelasticity is presented in the Galilean approximation with a view toward capturing the essential material properties which play an active role in the fracture mechanics of electromagnetic materials. The notions of pseudomomentum, Eshelby stress, and inhomogeneity force are the essential ingredients in this formulation.

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.

Stresses and momenta in electromagnetic materials

Abstract The notion of energy–momentum, or energy–stress, pertains typically to electromagnetism. Eshelby transferred such a notion into elasticity in 1951 and, afterward, into continuum mechanics, in order to account for the force acting on a material defect. Similarities and differences between the Maxwell tensor of electromagnetism and the Eshelby tensor are shown and commented hereby. Basing on a Lagrangian approach to electromagnetic materials, canonical momenta are shown to emerge in a natural way. On the basis of these canonical quantities, one can introduce the material momentum (or pseudomomentum) along with the classical momentum and the material stress (Eshelby stress) along with the Maxwell stress, which is a Cauchy-like stress.