In continua with ‘active’ microstructure stability competes with multi-field destabilizing effects

We say that bodies extended in space possess an ‘active’ microstructure when they have internal degrees of freedom with related peculiar interactions that cannot be completely described by the standard stress. These interactions may alter the material stability. Internal (observable) degrees of freedom are described by phase fields additional to the deformation. When they are linked to the macroscopic strain by an internal constraint, the emerging scheme reduces to higher-gradient extensions of standard continuum mechanics, which describe second-neighbour or higher-order interactions. These can strengthen or weaken the material stability. To explore these effects, comparing different but linked modeling approaches, we conduct a skeletal analysis referring to the Hadamard stability of an elastic one-dimensional body, described under varying assumptions about its internal structure. The analysis refers to the long wavelength with respect to the microstructural length. We show that in continua with active microstructure stability competes with multi-field destabilizing effects. Then, looking at a three-dimensional (3D) setting and general manifold-valued phase fields, we also extend to the multi-field setting a uniqueness theorem by Ericksen & Toupin on the displacement boundary value static problem of small deformations superposed on finite strains. While the mathematical treatment is intentionally rather elementary, it provides conceptual insights into the description of multi-phase materials with active microstructures.