Computational aspects of the mechanics of complex materials

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

“…Examples are the polarization vector for ferroelectrics, the degrees of freedom exploited at low scales by the atomic rearrangements in quasicrystals (what is collected in the so-called phason field), the peculiar direction of stick molecules with head-to-tail symmetry in liquid crystals, etc. A unified view on the matter foresees that, besides the deformation through which we reach a placement taken as reference-say B, and other macroscopic shapes B a -we have a field, defined over B itself, which takes values on a differentiable manifold M (see [6,37,39,42]) that we call the manifold of microstructural shapes. The common assumption that M is finite-dimensional is sufficient to include in the framework, as special cases, models that we know in solid-state physics (example those for ferroelectrics, magnetoelastic materials, quasicrystals, elastomers) and also more abstract schemes such as the Cosserat one [10] (called also micropolar), used for models of beams and shells (among many, see the basic papers [21] and [62] on the matter, the first one being that opening the application of Cosserat ideas to the description of the elastic structural elements) or liquid crystals in smectic order, and the micromorphic one (either considering microstrain or deformable directors, see [22,29,45]), adopted for polymers or models of strain-gradient plasticity, which have also been the playground for several analytical and geometrical investigations (see, example, [11,30,47,51,64]).…”

“…For a lattice where we replace mass points with small rigid bodies and add rotational springs (or, alternatively, we substitute the springs with beams suffering just elongation and bending), an appropriate use of the Cauchy-Born view allows us to connect the discrete structure with the Cosserat continuum. Moreover, when we consider a discrete structure composed of two superposed and connected lattices, the first made of mass points and springs, the second one by deformable shells connected by springs, another adaptation of the Cauchy-Born rule allows us to derive the micromorphic scheme or the continuum with stretchable vectors (see [42]). …”

Ground states in complex bodies

“…The requirement of objectivity, that is the invariance with respect to the action of SO (3) (here on both the ambient spaceR 3 and M), implies that in the range of large deformations the energy density e (x, y, F, ν, N ) cannot be convex with respect to F , see pertinent comments in [52]), once one fixes the other arguments, while it may be a convex function of N (see [42]). …”

Internal structures and internal variables in solids

“…The mathematical structure of such theories [Capriz 1989] includes a coarse-grained morphological descriptor introduced to describe the morphology of the material element [Mariano and Stazi 2005], which represents certain additional independent fields [Mariano 2002]. For example, considering the material element as a cell able to deform independently of the rest of the body, Mindlin [1964] in fact introduced a second-order symmetric tensor as a morphological descriptor.…”