Effective balance equations for elastic composites subject to inhomogeneous potentials

Abstract

We derive the new effective governing equations for linear elastic composites subject to a body force that admits a Helmholtz decomposition into inhomogeneous scalar and vector potentials. We assume that the microscale, representing the distance between the inclusions (or fibers) in the composite, and its size (the macroscale) are well separated. We decouple spatial variations and assume microscale periodicity of every field. Microscale variations of the potentials induce a locally unbounded body force. The problem is homogenizable, as the results, obtained via the asymptotic homogenization technique, read as a well-defined linear elastic model for composites subject to a regular effective body force. The latter comprises both macroscale variations of the potentials, and non-standard contributions which are to be computed solving a well-posed elastic cell problem which is solely driven by microscale variations of the potentials. We compare our approach with an existing model for locally unbounded forces and provide a simplified formulation of the model which serves as a starting point for its numerical implementation. Our formulation is relevant to the study of active composites, such as electrosensitive and magnetosensitive elastomers.

Effective balance equations for electrostrictive composites

“…, which is not fulfilling condition (29). The systems of PDEs (15) is nonetheless shown to be homogenisable [64] by embracing micro-scale periodicity, as done in the remainder of this work.…”

“…(see also [64,70]). By noticing that the macro-scale variable x spans a bounded domain of R 3 , that is the representation of a bounded domain of the three-dimensional Euclidean space S , and by virtue of Eq.…”

“…For the case studied in [64] and in this work, Eq. (30) yields a locally unbounded force and the differential problem given by (9-10) is to be considered as being driven by a locally unbounded generalised forcẽ…”

“…Here we follow a different approach via considering the electric field as given, and addressing the asymptotic homogenisation of a linear elastic composite whose deformations are driven by the divergence of a tensor, referred to as generalised Maxwell stress tensor and supplied from the outset. As such, we provide a novel formulation in the context of asymptotic homogenisation by following an approach consistent with [64], where the Authors deduce the effective balance equations for a material composite subjected to a continuous volume force given by the Helmholtz decomposition. The two formulations are both addressing homogenisation of a locally unbounded problem and, for completeness, are shown to coincide under a consistent set of simplifying assumptions.…”

“…The two formulations are both addressing homogenisation of a locally unbounded problem and, for completeness, are shown to coincide under a consistent set of simplifying assumptions. In particular, the framework in [64] in absence of a vector potential can be recovered if the electric force is identified with the gradient of the potential introduced in [64] and if the electric properties are assumed to be continuous across the interface of the composite's sub-phases. The formulation reported in [47] can be likewise considered as a particular case of our model for an appropriate specific choice of the imposed electric field and in absence of free charges.…”