In this article, we present a hyperelastic model for light and compliant open cell foams with an explicit correlation between microstructure and macroscopic behavior. The model describes a large number of three dimensional structures with regular and irregular cells. The theory is based on the formulation of strain-energy function accounting for stretching which is the main deformation mechanism in this type of materials. Within the same framework, however, bending, shear and twisting energies can also be incorporated. The formulation incorporates nonlinear kinematics which traces the evolution of the structure during loading process and its effects on the constitutive behavior, including the cases where configurational transformations are present leading to non-convex strain-energy functions. Also nonlinear material effects at local or beam level are introduced to accommodate a wide range of different material behaviors. Since the micromechanical formulation presented here has explicit correlation with the foam structure, it preserves in the constitutive relation the symmetries or directional properties of the corresponding structures, including the cases of re-entrant foams which exhibit negative Poisson’s ratio effects. The model captures the central features exhibit by these materials. Predictions of the model for macroscopic uniaxial strain are presented in this article.