Paolo Maria Mariano

Multifield theories in mechanics of solids

Publisher Summary Multifield theories indicate the wide range of models in which some graphic fields must be introduced to describe the influence of material substructures on the gross mechanical behavior of solids. Solidification of metal alloys and their possible shape memory, damage states and evolution, and the influence of long chains of macromolecules on the behavior of polymers can be successfully analyzed with the help of multifield theories. Models with internal variables can be derived from multifield theories by appropriate internal constraints. The chapter focuses on the multifield description of continua that is a flexible framework to study physical situations in which the analysis of substructures is important for both practical and theoretical reasons. The chapter discusses the configurations and deduction of balance equations for bodies lacking in discontinuity surfaces, the behavior of the substructure, the influence of the substructure on the axial decay of energy in linear elastic cylinders, the derivation of thermomechanical balance at discontinuity surfaces, constitutive restrictions arising from a mechanical version, the second law of thermodynamics and the measures of substructural interactions, the analysis of the influence of substructures on configurational forces that drive the evolution of interfaces, the evaluation of the influence of material substructures on macrocrack propagation, the case where the substructure becomes latent in presence of appropriate internal constraints, and the application of the general theory to special cases.

Symmetries and Hamiltonian Formalism for Complex Materials

Preliminary results toward the analysis of the Hamiltonian structure of multifield theories describing complex materials are reported: we invoke the invariance under the action of a general Lie group of the balance of substructural interactions. Poisson brackets are also introduced in the material representation to account for general material substructures. A Hamilton-Jacobi equation suitable for multifield models is presented. Finally, a spatial version of all these topics is discussed without making use of the notion of paragon setting.

Computational aspects of the mechanics of complex materials

SummaryBodies with exotic properties display material substructural complexity from nano to meso-level. Various models have been built up in condensed matter physics to represent the behavior of special classes of complex bodies. In general, they fall within the setting of an abstract model building framework which is not only a unifying structure of existing models but—above all—atool to construct special models of new exotic materials. We describe here basic elements of this framework, the one ofmultifield theories, trying to furnish a clear idea of the subtle theoretical and computational problems arising within it. We present the matter in a form that allows one to construct appropriate algorithms in special cases of physical interest. We discuss also issues related to the construction of compatible and mixed finite elements in linearized setting, the extension of extended finite element methods to analyze the influnce of material substructures on crack growth, the evolution of sharp discontinuity surfaces in complex bodies. Concrete examples of complex bodies are also presented with a number of details.

Configurational forces in continua with microstructure

1. General remarks The action exerted on a defect or inclusion in an elastic solid by the surrounding medium can be expressed by means of the space-like part of the Noether energy momentum tensor, the so-called Eshelby tensor 1 . The theory by Eshelby is variational in essence and has been formulated for materials of grade n. It is based on Noether theorem that establishes the existence of a second order tensor quantity which is invariant along the trajectories, under conditions of homogeneity, if the equations of motion are satised. As well known, this theorem (a central result in Hamiltonian mechanics) states that every group of transformations under which the Lagrange density function is invariant generates a conserved quantity 2 . Gurtin has proved the validity of Eshelby relation in non conservative setting 3 . His argument is based on

Cracks in Complex Bodies: Covariance of Tip Balances

Abstract In complex bodies, actions due to substructural changes alter (in some cases drastically) the force driving the tip of macroscopic cracks in quasi-static and dynamic growth, and must be represented directly. Here it is proven that tip balances of standard and substructural interactions are covariant. In fact, the former balance follows from the Lagrangian density’s requirement of invariance with respect to the action of the group of diffeomorphisms of the ambient space to itself, the latter balance accrues from an analogous invariance with respect to the action of a Lie group over the manifold of substructural shapes. The evolution equation of the crack tip can be obtained by exploiting invariance with respect to relabeling the material elements in the reference place. The analysis is developed by first focusing on general complex bodies that admit metastable states with substructural dissipation of viscous-like type inside each material element. Then we account for gradient dissipative effects that induce nonconservative stresses; the covariance of tip balances in simple bodies follows as a corollary. When body actions and boundary data of Dirichlet type are absent, the standard variational description of quasi-static crack growth is simply extended to the case of complex materials.

Steady-state propagation of dislocations in quasi-crystals

We analyse the steady propagation at constant speed, lower than the shear wave-speed, of a straight dislocation in an unbounded elastic quasi-crystal with five-fold symmetry. We discuss only the ideal elastic behaviour, neglecting the dissipation associated with the atomic rearrangements. Under these conditions, we provide the expressions of phonon and phason fields in closed form. Both phonon and phason stresses appear to be singular near to the dislocation core. We also find the explicit expression of the energy per unit length around a moving dislocation.

Constitutive Relations for Elastic Microcracked Bodies: From a Lattice Model to a Multifield Continuum Description

A continuum model suitable for the description of microcracked bodies is shown. The influence of microcracks on the mechanical behavior of the body is estimated through a microstructural field added to the displacement one. This field represents the perturbation to the regular displacement field due to the presence of microcracks. It is an observable quantity; its rate must satisfy appropriate balance equations. The problem of deriving constitutive relations for such a model at least in the linear elastic case is dealt with. Constitutive equations are deduced from a lattice model using an integral identification procedure based on the equivalence in terms of virtual work, without resorting to limit processes. The discrete model considered is made of two superposed lattices; the first one is constituted of material points connected by elastic links; the second one is made of empty closed shells interacting between themselves and with the first lattice. As sample test, a one-dimensional problem is shown.

Chapter One - Mechanics of Material Mutations

Abstract Mutations in solids are defined here as dissipative reorganizations of the material texture at different spatial scales. We discuss possible views on the description of material mutations with special attention to the interpretations of the idea of multiple reference shapes for mutant bodies. In particular, we analyze the notion of relative power —it allows us to derive standard, microstructural, and configurational actions from a unique source—and the description of crack nucleation in simple and complex materials in terms of a variational selection in a family of bodies differing from one another by the defect pattern, a family parameterized by vector-valued measures. We also show that the balance equations can be derived by imposing structure invariance on the mechanical dissipation inequality.

Some remarks on the variational description of microcracked bodies

The natural minimum energy problem of the statics of a body with free discontinuities, given by a microcrack system, is replaced by a more rough two-field minimum problem on a domain without discontinuities. Equations of motion, in agreement with the classical multifield theory, are deduced for a degree one theory both in static and dynamic cases. Moreover, entropy requirements useful to describe the dissipative behavior are discussed. Brittle materials are considered.

Multifield Description of Microcracked Continua: A Local Model:

In the present paper, a continuum model used to describe brittle microcracked bodies is proposed. The configuration of the body is represented by two fields: the displacement field and an appropriate approximation of the microcrack density. The evolution of the latter field is regulated by the balance of the actions of the microcracks on each other. Moreover, the introduction of a damage entropy flux in the Clausius-Duhem inequality allows one to obtain appropriate criteria of damage growth directly from balance equations. The mathematical structure of the model thus obtained is different from those based on internal variable schemes and allows one to overcome some of their theoretical shortcomings. A simple example, although limited to an elastostatic case, allows one to recognize the basic difference between the model presented here and the models with internal variables, and to underline the potentialities of the proposed approach.