Data-driven stress problem under purely normal homogeneous Neumann boundary conditions

TL;DR

This paper establishes a rigorous mathematical framework for a data-driven stress problem in continuum mechanics, incorporating experimental data directly and ensuring existence and uniqueness of solutions under specific boundary conditions.

Contribution

It provides the first functional-analytic foundation for the data-driven stress problem with homogeneous Neumann boundary conditions, connecting experimental data with stress field solutions.

Findings

Proves the existence of an equilibrated stress response.

Shows the finiteness of data guarantees solution proximinality.

Establishes a unique solution framework for the data-driven stress problem.

Abstract

Data-Driven Continuum Mechanics -- the continuous counterpart of Data-Driven Computational Mechanics -- is a modern paradigm that enhances classical continuum mechanics by incorporating finite sets of experimental material data directly, avoiding any form of constitutive modeling. Despite recent progress, its analytical foundations remain at an early stage. In this work, we establish a rigorous functional-analytic framework for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions. The problem is formulated as finding a stress field (satisfying the balance of linear and angular momenta and the boundary conditions) that is closest, in an $L^p$-sense, to an auxiliary stress field that is simultaneously sought and locally resembles a finite discrete set of experimental stress states. Our analysis relies on two key ingredients. First, the divergence operator induces a topological isomorphism between the space of symmetric stress fields modulo its kernel and the space of loads balanced by rigid-body motions, ensuring the existence of an equilibrated response. Second, the finiteness of the material data set guarantees proximinality in the stress space, which in turn yields a complete existence and uniqueness theory for solution equivalence classes. Together, these two properties provide a rigorous mathematical foundation for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions.