Optimization of the Stress Intensity in Controlled Evolution of a Continuously Inhomogeneous Solid

Abstract

Control and optimization of manufacturing processes play essential role in LbL (layer-by-layer) technologies. Optimal control allows one, in particular, to reduce the residual stresses in produced details. The present paper provides an example of such optimization. The LbL-structure under study is represented as a hyperelastic hollow cylinder with finite thickness. This item is produced by a successive joining of infinitesimally thin layers (laminae) to a cylindrical substrate. Deformations assume to be finite in the course of production. The assembled structure has residual stresses due to layerwise shrinkage. It means that the LbL-structure does not have stress-free shape in Euclidean space. In this regard, a non-Euclidean approach is used: the body is embedded into a space with non-Euclidean connection. This space is considered as a stress free shape for the body. The optimization problem is solved to reduce residual stresses. The control is the hydrostatic loading applied to the internal surface of the cylinder, while the cost function is the maximum of stress intensity.