Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physical-viscosity solution. Numerical experiments illustrating the robust performance of these schemes are presented.

The fracture of glassy polymers

The fracture of glassy polymers starts with the separation of molecule bundles which then form a craze; this is followed by the fracture or rupture of the craze by sliding of the molecule bundles. The first process has the approximate characteristics of brittle fracture, the second those of viscous flow; at low velocities, therefore, the crack extends by the essentially viscous mechanism in the craze layer, whereas at higher velocity the stress required for this rises so high that either a quasi-brittle fracture occurs between the craze and the adjacent bulk polymer (for example, in polystyrene), or patches of craze arise in the bulk ahead of, and away from, the propagating fracture front, as in cast polymethylmethacrylate (PMMA). When the craze wedge ahead of the ‘ viscous ’ crack in the craze layer suddenly peels off the adjacent bulk polymer, either multiple crazes and cracks arise, radiating from its edge, or a new craze wedge is initiated. In either case only one craze wedge propagates, and it drops off the adjacent bulk when the rate of stretching of the craze (normal to its plane) reaches a critical magnitude. The repetition of this process results in the well known striation of the surface of fracture in polystyrene and other polymers. Since the fracture mechanism includes an essentially velocity-dependent viscous process, the Griffith theory cannot be applied to glassy polymers even as an approximation. The work of fracture oscillates by orders of magnitude within microseconds in the region of striations.