A second-order nonlinear elastic model is developed for biogels, whereby the elastic constants are estimated from the free energy densities for polymer networks and polymer-water mixtures. For typical values of the constants, important physical insights are revealed: a biogel stiffens under tension, contracts longitudinally under torsion (inverse Poynting effect), and experiences a negative normal stress under simple shear. This approach of extracting nonlinear elastic constants works for any constitutive law, and has the advantage of exploring physical phenomena without recourse to computationally intensive simulations.