Some general optimal design results using anisotropic, power law nonlinear elasticity

Recent results on optimal design with anisotropic materials and optimal design of the materials themselves are in most cases based on the assumption of linear clasticity. We shall extend these results to the nonlinear model classified as powerlaw clasticity. These models return proportionality between elastic strain energy density and elastic stress energy density. This is shown to imply localized sensitivity analysis for the total elastic energy, and for a number of optimal design problems this immediately gives practical, general results.For two- and three-dimensional problems the effective strain and the effective stress are defined from an energy consistent point of view, and it is shown that a definition generalizing the von Mises stress must be used. The optimization criterion of uniform energy density also holds for nonlinear materials, and several general conclusions can be based on this fact. Applications to size design illustrate this.For stiffness optimization the ultimate optimal material design problem is addressed. The validity of recent results are extended to nonlinear materials, and a simple proof based on constraint on the Frobenius norm is given. We note that the optimal material is orthotropic, that principal directions of material, strain and stress are aligned, and that there is no shear stiffness. In reality, the constitutive matrix only has one nonzero eigenvalue and the material therefore has stiffness only in relation to the specified strain condition. Results related to orientational design with orthotropic materials are also focused on.With respect to strength optimization, i.e. the more difficult problem with local constraints, we shall comment on the influence of the different strength criteria.