Martensitic phase transitions in crystalline solids have been studied and utilized for many technological applications, including biomedical devices. These transitions typically proceed by the nucleation and propagation of interfaces, or phase boundaries. Over the last few decades, a continuum theory of phase transitions has emerged under the framework of thermoelasticity to study the propagation of these phase boundaries. It is now well-established that classical mechanical and thermodynamic principles are not sufficient to describe their motion within a continuum theory, and a kinetic relation must be supplied to complete the constitutive description. A few theoretical techniques that have been used to infer kinetic relations are phase-field models and viscosity-capillarity based methods, within both of which a phase boundary is sharp, but smeared over a short length. Here we use a different technique to infer a kinetic relation. We discrete a one-dimensional continuum into a chain of masses and springs with multi-well energy landscapes and numerically solve impact and Riemann problems in such systems. In our simulations we see propagating phase boundaries that satisfy all the jump conditions of continuum theories. By changing the boundary and initial conditions on the chains we can explore all possible phase boundary velocities and infer kinetic relations that when fed to the continuum theory give excellent agreement with our discrete mass-spring simulations. A physical system that shares many features with the mass-spring systems analyzed in this thesis is DNA in single molecule extension-rotation experiments. DNA is typically modeled as a one-dimensional continuum immersed in a heat bath. It is also known from fluorescence experiments that some of these transitions proceed by the motion of phase boundaries, just as in crystalline solids. Hence, we use a continuum theory to study these phase boundaries in DNA across which both the stretch and twist can jump. We show that experimental observations from many different labs on various DNA structural transitions can be quantitatively explained within our model.