The Richtmyer-Meshkov instability of small perturbed single-mode interfaces between an elastic-plastic solid and an inviscid liquid is investigated by theoretical analysis and numerical simulation in this work. A modified model including the Atwood number effect is proposed to describe the long-term behaviors of small perturbations at the solid-liquid interface. In contrast to an effective theoretical model at the solid-vacuum interface, this model is appropriate at different Atwood numbers. Owing to the effect of elastic-plastic characteristics and the density ratio, the evolution of the spike amplitude exhibits nonlinear mechanical behavior. As the absolute value of the Atwood number decreases, the maximum spike amplitude also decreases. To validate this model, an Eulerian finite-difference multicomponent code is adopted to study the time evolution of the spike amplitude at different Atwood numbers. The model coefficients are obtained by analyzing the relevant characteristic statistics collected from the numerical results. Under different initial conditions such as Atwood number and shock strength, the applicability of this modified model is verified by comparing the numerical results with the model profile. The consistency in results signifies that the modified model is not only suitable for specific shock intensity and Atwood number, but also adaptable within a certain range.