Abstract A model for a nonlinear elastic body with a double porosity structure is proposed. We generalize previous work which has concentrated mainly on linear double porosity elastic materials or has had a particular structure in the nonlinearities. Here we allow the strain energy function and the effective permeabilities associated to the macro and micro porosity structures to be general nonlinear functions of the deformation gradient, macro and micro pressure gradients and of the macro and micro pressures themselves. A fully nonlinear acceleration wave analysis is developed. The general three-dimensional propagation condition is derived and it is shown how the permeabilities and pressure gradients alter the acoustic tensor with inevitable consequences for wavespeeds and for propagation conditions. The amplitude equation for an acceleration wave is derived and is completely analysed in the one-dimensional situation for a wave moving into a state with constant strain and constant macro and micro pressures.