This paper introduces an inverse design framework for the precise tailoring of desired nonlinear
mechanical responses in periodic microstructures, with particular focus on prescribed nonlinear stress strain relationships. The topology optimization hinges on minimizing the error between the target and
realized properties of the microstructures. A deformation-driven homogenization framework is setup.
The periodic constraints needed for the microscale equilibrium equation are imposed through strongly
enforced periodic boundary conditions and the removal of the translational nullspace, avoiding the need
for Lagrange multipliers, greatly simplifying the implementation. Automatic differentiation is leveraged
to efficiently calculate the necessary sensitivities for the gradient-based optimization. To further aid the
design of discrete designs a intermediate density penalty constraint is proposed. Numerical examples
underscore the efficacy of our methodology, showcasing microstructures that demonstrate targeted
softening and stiffening as well as distinctive directional behaviour.