Title: Generalized Stress Potentials in Incompatible Elasticity
 
Abstract:
The methods of stress-functions are an important analytical tool in classical elasticity. They are motivated by the observation that in two-dimensional systems, the fact that the stress tensor at equilibrium is symmetric and divergence-free leads to its representation as the “curl-curl” of a scalar function. By combining this with so-called “constitutive relations”, one obtains a biharmonic equation for this potential.
Recent work by physicists, motivated by problems in incompatible elasticity, suggested a generalization of the stress function approach to nonlinear and non-Euclidean systems with arbitrary constitutive relations. In that respect, mathematical difficulties arose in both understanding the mechanisms that allow the stress function approach to be extended thus, and furtherly in establishing the analytic foundations beyond the two-dimensional case.
In this talk, I present a rigorous framework for obtaining stress potentials in such a generalized setting, with emphasis on suitable choice of gauge for solving problems. I shall also describe the line of reasoning which allows a representation for the stress to be produced, based on newly developed geometric-analytical tools – namely, the regular ellipticity of a boundary value problem for a bilaplacian operator acting on “double forms”.


This talk will be based on elements of a joint work with Raz Kupferman:
https://arxiv.org/abs/2103.16823
https://arxiv.org/abs/2104.05794