Experimental and theoretical study of longitudinal undular bores generated by fracture

Undular bores, or dispersive shock waves, are non-stationary waves propagating as oscillatory transitions between two basic states, in which the oscillatory structure gradually expands and grows in amplitude with distance travelled. They have been widely studied both experimentally and theoretically, most commonly in the context of fluids. They occur in nature and have been photographed and reported, for example, in rivers and in the atmosphere. Similar wave structures have been observed in solids during various experiments, but have not been linked to undular bores. Therefore, they have not been studied theoretically using the methods that have successfully been used in other areas. No dedicated study of the waves has been reported.

In this thesis, an important new mechanism for the generation of undular bores is reported. Using single-point and multi-point high-speed photoelasticity, the generation of undular bores in homogeneous, solid polymethylmethacrylate pre-strained bars of constant rectangular cross section by natural and induced tensile fracture is demonstrated.

An extended Boussinesq type equation is obtained from within the framework of three-dimensional dynamic nonlinear elasticity to describe the propagation of a longitudinal bulk strain wave in a nonlinear elastic bar. Such equation is necessary to capture higher order nonlinear effects that are observed at the strains encountered in the tensile fracture experiments. A Gardner equation is derived as a uni-directional model by looking for a solution in the form of an asymptotic multiple scales expansion.

The leading order viscoelastic term is introduced from a suitable spring and dashpot model, and higher order nonlinear and dispersive corrections are introduced based on the extended Korteweg - de Vries equation for a hyperelastic rod, which results in the viscoelastic extended Korteweg – de Vries (veKdV) equation.

The veKdV equation is solved using a pseudospectral method, with an initial profile that is numerically fitted to experimental measurements that are taken close to the fracture site. For the distances relevant to the experiments, the veKdV equation is shown to provide very good agreement with the key observed experimental features for suitable choice of material parameters. A robust methodology for fitting of the unknown parameters is presented, which is based on the construction of the appropriate analytical solutions for some limiting regimes.

Some local features at the front of the bore are also captured reasonably well by the linearisation near the nonzero pre-strain level. For this equation, an analytical solution is constructed in terms of a certain integral of the Airy function. From this solution, simple formulae are derived which describe the key features of the bore front in terms of material parameters and characteristics at fracture. Predictions are made with the formulae based on the speed and slope of the wave close to the fracture site obtained from experimental measurements, and later verified by comparing to the relevant experimental profile.

The experimental and theoretical approaches presented open new avenues and analytical tools for the study and applications of dispersive shock waves in solids. Such waves could be present in the signals generated by earthquakes, fracking and other similar events involving transverse fracture of an appropriately pre-strained waveguide.