## Discontinuities, Balance Laws, and Material Momentum

##### Abstract

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.

##### General Audience Abstract

One dimensional flexible bodies such as strings and rods can exhibit fascinating and counterintuitive behavior when they interact with rigid obstacles. For instance, a chain falling on a rigid surface falls faster than it would have if it were falling freely. When one end of a long chain piled up in a container placed at an elevation is pulled across the rim and let go, the chain flows out of the container like a water fountain. Discontinuities in the cross-sectional properties of an elastic rod contained in a curved frictionless channel can result in the generation of forces that propel the rod along the channel. Such counterintuitive phenomena are a consequence of the physics taking place at the point of partial contact where the flexible body comes in contact with a rigid surface. The purpose of this dissertation is to study the mechanics of such points of discontinuity. Several such phenomena where effectively one dimensional bodies interact with rigid surfaces are all around us. A familiar example is the peeling of an adhesive tape, where the peeling front qualifies as a point of discontinuity propagating through the tape as the peeling progresses. A good understanding of the mechanics of the peeling front is crucial in estimating the strength of the adhesive. Another such example of practical importance is a mooring line being placed on the seabed. In such situations, the existence of a reaction force acting at the touchdown point depends on whether or not the cable develops a kink at that point. Similar questions of importance can be asked in the context of deployment and unspooling of space tethers. In this dissertation, an analytical study of the general physics of the phenomena described above is presented. Standard theoretical tools of classical physics are employed to understand the mechanics of points of partial contact between flexible and rigid bodies. The conditions under which a flexible body could experience sharp changes in its geometry (e.g. a kink) at such points are investigated. In addition to that, we explore the implications of a nonclassical law of physics called the balance of “material momentum” in the context of such problems.

##### Collections

- Doctoral Dissertations [14334]

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