The interaction between two or more independent causes of materials degradation results in a duplex mode of materials degradation. The materials degradation rate whether expressed in loss of material or loss of mechanical strength is often faster than the sum of each form of materials degradation acting separately. For this reason, duplex modes of materials degradation are of great practical importance. Synergisms between unrelated forms of materials degradation are not immediately obvious and have in some instances been only recently recognized. In this paper a chemically aggressive environment over a geo-synthetic polymeric membrane is investigated. The membrane free energy depends upon the mono-axial strain and the degree of reaction with an external agent. We suppose chemical potential assigned and the minimization problem is formulated. Practically the minimum problem is non-convex, and coexistence of equilibrium phases is possible. The proposed general model is able to describe the complex phenomenon when polymeric geo-synthetic membrane are undergoing to heterogeneous and complex action of the package system built in dump of urban waste. Finally a FEM simulation is compared with minimization problem theoretical results.