Second-order elasticity of polymeric hydrogels with chemically coupled effects

•A fully coupled chemo-mechanical formulation is obtained for elastic constants of hydrogel.•The influence of chemical potential is analytically studied on the elastic constants.•An isotropic swelling of hydrogel is analyzed, and the relationship between chemical potential and stretch is revealed.•The classical relation is obtained among the Lamé constants, Young's modulus and Poisson's ratio, where all the parameters are dependent on the chemical potential.•A stiffening effect is observed subject to tension for the effect of second-order elastic parameters.