Mechanochemical reactions are driven by the direct absorption of mechanical energy by a solid (often crystalline) material. Understanding how this energy is absorbed and ultimately causes a chemical transformation is essential for understanding the elementary stages of mechanochemical transformations. Using as a model system the energetic material LiN3 we here consider how vibrational energy flows through the crystal structure. By considering the compression response of the crystalline material we identify the partitioning of energy into an initial vibrational excitation. Subsequent energy flow is based on concepts of phonon-phonon scattering, which we calculate within a quasi-equilibrium model facilitated by phonon scattering data obtained from Density Functional Theory (DFT). Using this model we demonstrate how the moments (picoseconds) immediately following mechanical impact lead to significant thermal excitation of crystalline LiN3, sufficient to drive marked changes in its electronic structure and hence chemical reactivity. This work paves the way towards an ab initio approach to studying elementary processes in mechanochemical reactions involving crystalline solids.