We test and model the mechanical response of a multilayer composite structure composed of alternating layers of aligned carbon nanotubes and copper foils under compression. We employ a bistable mass-spring model to capture the three-phase hysteretic response of the loading curve with excellent agreement with the experimental observations. An in situ identification procedure is proposed to quantify the material parameters corresponding to the mesoscopic scale of the structure. We refine the results using a genetic algorithm and compare the response of two different models based on three and four springs in series. The localization of deformation can be accurately captured by these simplified models, which hold promise for the design of novel materials with tailored deformation responses.