The nonlinear elasticity of hyperelastic models for stretch-dominated cellular structures

For stretch-dominated cellular structures with arbitrarily oriented cell walls made from a homogeneous isotropic hyperelastic material, recently, continuum isotropic hyperelastic models were constructed analytically, at a mesoscopic level, from the microstructural architecture and the material properties at the cell level. Here, the nonlinear elastic properties of these models for structures with neo-Hookean cell components are derived explicitly from the strain-energy function and the finite deformation of the cell walls. First, the nonlinear shear modulus is calculated under simple shear superposed on finite uniaxial stretch. Then, the nonlinear Poisson's ratio is computed under uniaxial stretch and the nonlinear stretch modulus is obtained from a universal relation involving the shear modulus as well. The role of the nonlinear shear and stretch moduli is to quantify stiffening or softening in a material under increasing loads. Volume changes are quantified by the nonlinear bulk modulus under hydrostatic pressure. Numerical examples are presented to illustrate the behaviour of the nonlinear elastic parameters under large strains.