Discontinuities, Balance Laws, and Material Momentum

This dissertation presents an analytical study of a class of problems involving discontinuities, also referred to as shocks, propagating through one dimensional flexible objects such as strings and rods. The study entails interrogation of the classical balance laws of momentum, angular momentum, and energy across propagating discontinuities. A major part of this dissertation also concerns itself with a non-classical entity called the ``material momentum''. The balance of material momentum is studied in a variational context, where both the local and singular forms of it are derived from an action principle.

A distinguishing aspect of discontinuities propagating in continua is that, unlike in the bulk, the balance of momentum and angular momentum are not sufficient to describe their mechanics, even when the discontinuities are energy conserving. In this work, it is shown that the additional information required to close the system of equations at propagating discontinuities can be obtained from the singular form of energy balance across them. This entails splitting of the energy balance by its invariance properties, and identifying the non-invariant and invariant part of the source term with the power input and energy dissipation respectively at the shock. This approach is in contrast with other treatments of such problems in the literature, where additional non-classical concepts such as material momentum'' and configurational force'' have been invoked.

To further our understanding of the connections between the classical and non-classical approaches to problems involving discontinuities, a detailed exposition of the concept of material momentum is presented. The balance and conservation laws associated with material momentum are derived from an action principle. It is shown that the conservation of material momentum is associated with the material symmetry of the continuum, and that the conditions for the conservation of physical and material momentum are independent of each other. A new classification of the deformed configurations of the planar Euler elastica based on conserved quantities associated with the spatial and material symmetry of the rod is proposed. The manifestation of the balance of material momentum in seemingly unrelated fields of research, such as fracture mechanics, ideal fluids, and the mechanics of rods with discontinuities, is also discussed.