Modeling and Optimization for Oriented Growing Solids

Abstract

In the present paper the model for growing oriented solid is developed. Such solids have no global stress-free shape in Euclidean physical space due to the distributed defects, "recorded" into the solid during growing process. Nonetheless, in the framework of geometric approach in continuum mechanics the desired stress-free reference shape can be found in non-Euclidean space with specific (material) connection. For oriented solids, whose particles have to be identified with positions and orientations, such space can be represented by a submanifold in total space of principal bundle, defined over basic (conventional) material manifold. In such a case the deformation of growing solid can be generalized as smooth embedding from such space to total space of physical principal bundle. Invariants of connection on material principal bundle characterize the incompatibility of local deformations and can serve as cost functions in optimization problems.