Shape and Anisotropy Optimization in Nonlinear Elasticity

In this Chapter the optimization for nonlinear elastic and plastic solids is investigated from the Noether standpoint. The optimization criteria are provided by integral and local functionals. A procedure is proposed for deriving the necessary optimality conditions for the problems. The derivation makes essential use of a formula that expresses the variation of a functional in terms of the variations of the boundaries and of the state functions. The role of the necessary optimality conditions plays the trace of the generalized Eshelby’s energy–momentum tensors. The traces of the Eshelby’s energy–momentum tensor should be constant on the corresponding parts of the varying boundary. For the problems of limit plasticity, the role of the coaxiality of stress and distortion tensors in linear anisotropic elasticity plays the coaxiality of stress tensor and flow tensor.