Mathematical modelling and analysis of sheared energetic materials

Abstract

The large stores of chemical energy within energetic materials mean that their improper handling poses a serious safety concern, and so appropriate safety protocols need to be put in place. To do so requires understanding of the materials, but their mechanical properties cannot be easily determined through rheometric testing precisely because of the safety risks involved. To proceed, mathematical models can be created that simulate the mechanical behaviour. The core objective of the present work has been to develop reduced and thus mathematically tractable models that, by focusing on general physical principles, capture the essential aspects of the behaviour of sheared energetic materials. In particular, a dynamic model has been developed to examine the phenomenon of shear banding, motivated by its ability to cause extreme shear rates within shrinking regions and the resultant likelihood of hotspot generation. In the model, dynamic shear banding is treated by considering simple shear of a slab of shear-softening, bi-viscous fluid, producing a parabolic free-boundary problem for the diffusion of shear stress through the shear band and surrounding unyielded material. The behaviour of the system is analysed using precise asymptotic methods, and expansions involving material parameters are obtained for the growth of the shear band over time, and the spacetime variation of shear stress within the band and the surrounding region. In turn, the latter allows for determination of the spacetime variation of the rate of local heating due to mechanical dissipation within the band and the hard region, giving order-of-magnitude predictions for temperature increases likely to occur within the material. Since the expansions are in terms of material parameters, an opportunity for comparison against experiment arises. We expect good qualitative agreement of the model with empirical results after assignment of suitable parameter values for a given material sample.