We propose a theoretical model for the nonlinear mechanical response of Langmuir lipid monolayers subjected to a dilational in-plane deformation. Lateral diffusion in conjunction with free convection has been considered to drive nonlinear mass transport in Langmuir lipid monolayers. The present model combines the conservative dynamical equations for lipid transport along the monolayer plane together with a material relationship accounting for nonlinear hypoelasticity, as experimentally observed from high-strain rheological measurements [Hilles, Adv. Colloid Interface Sci. 122, 67 (2007)]. The dynamical equations have been resolved for oscillatory nonlinear motion, the theoretical spectral amplitudes being found in quantitative agreement with the experimental values obtained from surface rheology experiments performed in Langmuir monolayers of two different lipid systems, namely DPPC and native E. Coli lipids. The presence of micrometer-sized phase coexistence domains in these lipid systems has been claimed to pump diffusive transport along the monolayer plane. This dynamical scenario defines a relaxation regime compatible with the observed nonlinear mechanical behavior.