Geometry of interactions in complex bodies

Abstract

We analyze geometrical structures necessary to represent bulk and surface interactions of standard and substructural nature in complex bodies. Our attention is mainly focused on the influence of diffuse interfaces on sharp discontinuity surfaces. In analyzing this phenomenon, we prove the covariance of surface balances of standard and substructural interactions.

Introduction

Bodies are called complex when their material substructure (i.e. the texture from nano-level to meso-level) has a prominent influence on their gross behavior and there is a not negligible occurrence of interactions due to substructural changes. Examples are liquid crystals, elastomers, ferroelectric and microcracked bodies, spin glasses. Above all, soft condensed matter displays complex behavior. Applications in nanotechnology, smart structures and various fields of technology open basic theoretical and experimental problems that challenge, in a certain sense, even some aspects of the foundational concepts of standard continuum mechanics.

Basically, the standard paradigm of Cauchy’s format of continuum mechanics, prescribing that the material element is a sort of indistinct sphere that we collapse in a point in space seems to be not sufficient to account for the articulated substructural nature of a complex body.

In fact, for complex bodies the material element is rather a ‘system’ and one needs the introduction of an appropriate morphological descriptor $\nu$ of such a system (order parameter), at least at a coarse grained level, that describes the essential geometrical features of substructural shapes.

Physical circumstances of disparate nature suggest many possible choices of $\nu$, each one characterizing special models. Moreover, the selection of morphological descriptors is strongly related with the representation of substructural interactions arising within each material element and between neighboring material elements as a consequence of substructural changes. Interactions are represented in fact by objects conjugated in the sense of power with the rates of the quantities describing the geometry of the body and its changes. In this sense, since placement and order parameter fields are involved, the description of complex bodies adopted here is called multifield. It has basic differences with standard internal variable models. In a multifield approach, morphological descriptors enter directly the geometrical representation of the body and its kinematics; true interactions are associated with their rates and balanced. Information about the material substructure are then introduced already at the level of geometrical description of the body. On the contrary, in standard internal variable models, the geometrical description of the body is of Cauchy’s type: the material element is morphologically equivalent to an indistinct sphere described just by its place in space. There, internal variables come into play to describe just the removal from thermodynamical equilibrium (a detailed treatment of these classes of models can be found in [34]).

Here, by following the general unified framework of multifield theories proposed by Capriz in 1989 [4] (see also [3]) and then developed further on in its abstract structure [32], [23], [8], [9], we do not specify the nature of $\nu$. We require only that $\nu$ be an element of a finite-dimensional differentiable paracompact manifold $M$ without boundary to cover a class as large as possible of special theories. Our attention is then focused on the general setting which contains as special cases prominent theories interpreting problems typical of condensed matter physics. In a certain sense our work deals with a model of models. However, in our general point of view, we face the basic difficulty that $M$ does not coincide with a linear space in general. Moreover, we cannot consider a priori $M$ embedded in a linear space. In fact, since the possible embedding of a finite dimensional manifold in a linear space is not unique, its potential choice is of constitutive nature.

We focus our attention on conservative processes. For them, the relevant appropriate Hamiltonian formalism has been developed in [9] as a natural evolution of the Hamiltonian formalism in classical non-linear elasticity (the one discussed in [27]). We start from the results in [9] and analyze some of the rather subtle geometrical questions induced by the abstract nature of $M$.

Our essential point of view is as in what follows: “Geometry and mechanics associated with maps between manifolds are a general framework for condensed matter physics and are also a tool to construct new models of unusual and perhaps unexpected phenomena” [25].

Here, we focus the attention on the interaction between diffuse interfaces and additional sharp discontinuity surfaces. From one hand, in fact, in complex bodies there is a frequent occurrence of branching of substructures between domain walls and/or homophase gradient effects. The presence of the gradient of the morphological descriptor $\nu$ in the list of entries of the Lagrangian density allows us to account for these effects ‘smearing’ them as due to diffused interfaces. However, from the other hand, additional macroscopic surfaces of discontinuity may also occur and evolve. They are due to defects such as crack, shock or acceleration waves; their evolution is influenced by the presence of diffused interfaces due to substructural arrangements. As an example one may consider a polarized ferroelectric material in which external loads induce a shock wave: the shock front encounters walls of polarized domains and interact with them. Also, such domain walls influence the propagation of surface defects like cracks, as experiments point out.

In general, when sharp discontinuity surfaces are endowed with own energy, they are referred to as structured; on the contrary they are unstructured. We analyze here both cases paying attention to the nature of interface balances of standard and substructural actions that involve the jump of bulk stresses and, in the structured case, surface stresses. Really, interface balances involving standard surface stresses have been obtained in [17] while surface substructural measures of interactions have been introduced in [22], [23] and the relevant balance equations derived there (see also [8]). However, there is no proof of their covariance. Such a proof is provided here in Theorem 2 and is the main result of this paper. It implies a non-standard notion of observer, which is, for us, not only the representation of the ambient space and the time scale, but also the representation of the manifold of substructural shapes (see also [25]).

The technique of the proof is based on the validity of an integral balance of energetic nature. In the case of unstructured discontinuity surfaces, such a balance is just the integral version of Noether theorem in the bulk and arises naturally from invariance properties of the Lagrangian density. In the structured case, such a balance is augmented by the energetic contribution of the discontinuity surface given in terms of a superficial flux of energy the physical nature of which is discussed in Section 4.

In Section 2 we discuss the natural way to represent the morphology of complex bodies (a way necessary when the prominence of substructural interactions renders not efficient standard homogenization techniques). We describe the use of morphological descriptors to represent the geometry of substructural shapes and discuss up to a certain extent the nature of the space of maps assigning to each material element its morphological descriptor. In Section 3 we start to construct mechanics by assigning a Lagrangian density in which substructural gradient effects are taken into account and recall from [9] the version of Noether theorem appropriate to multifield descriptions of complex bodies. Finally, Section 4 contains the main result, i.e. the proof of the covariance of interfacial balances of standard and substructural interactions.