This work approaches the fields of homogenization and of materials design for the linear and nonlinear mechanical properties with prescribed properties-profile. The set of achievable properties is bounded by the zeroth-order bounds (which are material specific), the first-order bounds (containing volume fractions of the phases) and the second-order Hashin-Shtrikman bounds with eigenfields in terms of tensorial texture coefficients for arbitrarily anisotropic textured materials.