We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the existence of stress potentials. The scope of this work is for manifolds with boundary of arbitrary dimension, having constant sectional curvature. The central analytical machinery is the regular ellipticity of a boundary-value problem for a bilaplacian operator, and its consequences, which were developed in [R. Kupferman and R. Leder, arXiv:2103.16823, 2021]. One of the novelties of this work is that stress potentials can be used in non-Euclidean geometries, and that the gauge freedom can be exploited to obtain a generalization for the biharmonic equation for the stress potential in dimensions greater than two.
Bibliographical noteFunding Information:
\ast Received by the editors December 28, 2021; accepted for publication (in revised form) May 26, 2022; published electronically August 9, 2022. https://doi.org/10.1137/21M1466736 Funding: The work of the authors was partially supported by the Israel Science Foundation, grant 1035/17. \dagger Institute of Mathematics, Hebrew University, Jerusalem 9190401, Israel (firstname.lastname@example.org, email@example.com).
© 2022 Society for Industrial and Applied Mathematics.
- Saint-Venant compatibility
- differential geometry
- double forms
- partial differential equations
- stress potentials