Multi-field approach in mechanics of structural solids

We overview the basic concepts, models, and methods related to the multi-field continuum theory of solids with complex structures. The multi-field theory is formulated for structural solids by introducing a macrocell consisting of several primitive cells and, accordingly, by increasing the number of vector fields describing the response of the body to external factors. Using this approach, we obtain several continuum models and explore their essential properties by comparison with the original structural models. Static and dynamical problems as well as the stability problems for structural solids are considered. We demonstrate that the multi-field approach gives a way to obtain families of models that generalize classical ones and are valid not only for long-, but also for short-wavelength deformations of the structural solid. Some examples of application of the multi-field theory and directions for its further development are also discussed.Comment: 25 pages, 18 figure

Effective elastic properties of planar SOFCs: A non-local dynamic homogenization approach

The focus of the article is on the analysis of effective elastic properties of planar Solid Oxide Fuell Cell (SOFC) devices. An ideal periodic multi-layered composite (SOFC-like) reproducing the overall properties of multi-layer SOFC devices is defined. Adopting a non-local dynamic homogenization method, explicit expressions for overall elastic moduli and inertial terms of this material are derived in terms of micro-fluctuation functions. These micro-fluctuation function are then obtained solving the cell problems by means of finite element techniques. The effects of the temperature variation on overall elastic and inertial properties of the fuel cells are studied. Dispersion relations for acoustic waves in SOFC-like multilayered materials are derived as functions of the overall constants, and the results obtained by the proposed computational homogenization approach are compared with those provided by rigorous Floquet-Boch theory. Finally, the influence of the temperature and of the elastic properties variation on the Bloch spectrum is investigated

Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories

Long-wave models obtained in the process of asymptotic homogenisation of structures with a characteristic length scale are known to be non-unique. The term non-uniqueness is used here in the sense that various homogenisation strategies may lead to distinct governing equations that usually, for a given order of the governing equation, approximate the original problem with the same asymptotic accuracy. A constructive procedure presented in this paper generates a class of asymptotically equivalent long-wave models from an original homogenised theory. The described non-uniqueness manifests itself in the occurrence of additional parameters characterising the model. A simple problem of long-wave propagation in a regular one-dimensional lattice structure is used to illustrate important criteria for selecting these parameters. The procedure is then applied to derive a class of continuum theories for a two-dimensional square array of particles. Applications to asymptotic structural theories are also discussed. In particular, we demonstrate how to improve the governing equation for the Rayleigh-Love rod and explain the reasons for the well-known numerical accuracy of the Mindlin plate theory

Exact Model Reduction for Damped-Forced Nonlinear Beams: An Infinite-Dimensional Analysis

We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of the linearized beam equation. Reduction of the governing PDE to SSMs provides an explicit low-dimensional model which captures the correct asymptotics of the full, infinite-dimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The model-reduction procedure we employ also gives guidelines for a mathematically self-consistent modeling of damping in PDEs describing structural vibrations

High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice near the edges of the Brillouin zone. The leading order solution is expressed as a product between the standing wave solution and long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure

Explicit parametric solutions of lattice structures with proper generalized decomposition (PGD): applications to the design of 3D-printed architectured materials

The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-017-1534-9Architectured materials (or metamaterials) are constituted by a unit-cell with a complex structural design repeated periodically forming a bulk material with emergent mechanical properties. One may obtain specific macro-scale (or bulk) properties in the resulting architectured material by properly designing the unit-cell. Typically, this is stated as an optimal design problem in which the parameters describing the shape and mechanical properties of the unit-cell are selected in order to produce the desired bulk characteristics. This is especially pertinent due to the ease manufacturing of these complex structures with 3D printers. The proper generalized decomposition provides explicit parametic solutions of parametric PDEs. Here, the same ideas are used to obtain parametric solutions of the algebraic equations arising from lattice structural models. Once the explicit parametric solution is available, the optimal design problem is a simple post-process. The same strategy is applied in the numerical illustrations, first to a unit-cell (and then homogenized with periodicity conditions), and in a second phase to the complete structure of a lattice material specimen.Peer ReviewedPostprint (author's final draft

Multi-field continuum theory for medium with microscopic rotations

We derive the multi-field, micropolar-type continuum theory for the two-dimensional model of crystal having finite-size particles. Continuum theories are usually valid for waves with wavelength much larger than the size of primitive cell of crystal. By comparison of the dispersion relations, it is demonstrated that in contrast to the single-field continuum theory constructed in our previous paper the multi-field generalization is valid not only for long but also for short waves. We show that the multi-field model can be used to describe spatially localized short- and long wavelength distortions. Short-wave external fields of forces and torques can be also naturally taken into account by the multi-field continuum theory.Comment: 14 pages, 4 figures, submitted to International Journal of Solids and Structure