Reference shape of bodies with enhanced kinematics. Part I. Geometric methods

The work develops differential-geometric methods for modeling finite incompatible deformations of hyperelastic solids. They are based on the representation of a body as a smooth manifold, on which a metric and a non-Euclidean connection are synthesized. The resulting geometric space is interpreted as global stress-free shape, and the physical response and material balance equations are formulated relative to it. Within the framework of the geometric approach, deformations are modeled as embeddings of a non-Euclidean shape in physical space. Measures of incompatibility are represented by invariants of the affine connection, namely, curvature, torsion and nonmetricity, and the connection itself is determined by the type of physical process.
This article is the first part of the study. The proposed geometric approach is applied to bodies whose response depends on the first deformation gradient. Compatibility conditions are obtained and their geometric interpretation is proposed.