This research aims at understanding the conditions that lead to reaction initiation of polymer-bonded explosives (PBXs) as they undergo mechanical and thermal processes subsequent to impact. To analyze this issue, a cohesive finite element method (CFEM) based finite deformation framework is developed and used to quantify the thermomechanical response of PBXs at the microstructure level. This framework incorporates the effects of large deformation, thermomechanical coupling, failure in the forms of micro-cracks in both bulk constituents and along grain/matrix interfaces, and frictional heating. A novel criterion for the ignition of heterogeneous energetic materials under impact loading is developed, which is used to quantify the critical impact velocity, critical time to ignition, and critical input work at ignition for non-shock conditions as functions of microstructure of granular HMX and PBX. A threshold relation between impact velocity and critical input energy at ignition for non-shock loading is developed, involving an energy cutoff and permitting the effects of microstructure and loading to be accounted for. Finally, a novel approach for computationally predicting and quantifying the stochasticity of the ignition process in energetic materials is developed, allowing prediction of the critical time to ignition and the critical impact velocity below which no ignition occurs based on basic material properties and microstructure attributes. Results are cast in the form of the Weibull distribution and used to establish microstructure-ignition behavior relations.