Poincaré path integrals for elasticity

Abstract

We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators $\mathcal{P}$ for elasticity satisfying $\mathcal{DP}+\mathcal{PD}=\mathrm{id}$ and ${\mathcal{P}}^2=0$, where the differential operators $\mathcal{D}$ correspond to the linearized strain, the linearized curvature and the divergence, respectively. In general we derive path integral formulas in the presence of defects. As a special case, this gives the classical Cesàro-Volterra path integral for strain tensors satisfying the Saint-Venant compatibility condition.

arXiv commentrevised version