The unusual properties of shape memory alloys (SMAs) are due to solid-to-solid martensitic phase transformations (MPTs) which correspond to a lattice level instability of the crystal structure. The high temperature phase is usually a high symmetry structure and is called the austenite phase whereas the low temperature phase has a low symmetry and is called the martensite phase. Currently, there exists a shortage of material models of MPTs based on the material's atomic composition and crystal structure that would lead to computational discovery of new improved SMAs. The present work develops a lattice dynamics model using a first-order self-consistent approach based on statistical perturbation theory that aims to capture the qualitative and ultimately quantitative behavior of MPTs. In particular, the atomic interactions are modeled using Morse pair potentials. The effects of atomic vibrations on the material properties are captured by renormalizing the frequencies of atomic vibration using self-consistent equations. These renormalized frequencies are dependent on both configuration and temperature. The model is applied for the case of a one dimensional bi-atomic chain. The constant Morse pair potential parameters are chosen to demonstrate the usefulness of the current model. The resulting model is evaluated by generating stress-free equilibrium paths with temperature as the loading parameter. These plots are generated using branch-following and bifurcation techniques. A second-order phase transformation (PT) is predicted which involves transformation from a high symmetry phase to a low symmetry phase as the temperature is decreased. Thus, the current model is able to capture the important aspect in MPTs, i.e., transformation from high symmetry phase to low symmetry phase as temperature is decreased. We believe that this model applied to three dimensional structures will be able to capture first-order MPTs that occur in SMAs. This qualitative prediction of a temperature-induced PT indicates the likely hood that the current model can be used for the computational discovery of new shape memory alloys. Such an undertaking would involve, first, determining the potential parameters of new alloys from first-principles calculations and, second, using these parameter values with the current self-consistent model to evaluate the shape memory behavior of the new previously unstudied materials.