On material-uniform elastic bodies with disclinations and their homogenization
Cy Maor
TL;DR
This work analyzes material-uniform hyperelastic bodies containing discrete defects (disclinations and dislocations) and shows that, when the symmetry group $\mathcal{G}$ is discrete, disclination content cannot be made arbitrarily small, hindering homogenization of disclinations in crystalline materials. Utilizing a framework of implant maps and a locally-flat material connection, the authors connect disclination content to $\mathcal{G}$ and examine homogenization via $\Gamma$-convergence, highlighting that isotropic materials may homogenize but without recovering the defect content from the limit energy. The results contrast disclinations with dislocations, clarifying the limitations of extending the geometric defect paradigm beyond dislocations and illustrating how the limiting constitutive response can depend only on the metric $G$. These findings have implications for modeling distributed defects and interpreting homogenization limits in crystalline versus isotropic elastic media.$
Abstract
In this note, we define material-uniform hyperelastic bodies (in the sense of Noll) containing discrete disclinations and dislocations, and study their properties. We show in a rigorous way that the size of a disclination is limited by the symmetries of the constitutive relation; in particular, if the symmetry group of the body is discrete, it cannot admit arbitrarily small, yet non-zero, disclinations. We then discuss the application of these observations to the derivations of models of bodies with continuously-distributed defects.