Hyperbolic Conservation Laws in Viscoelasticity.
Abstract
The equations of unsteady motion for nonlinear elastic materials are quasilinear systems of hyperbolic equations in which characteristic speeds are not constant. Thus weak initial waves are amplified and smooth solutions generally develop singularities in finite time. Particularly interesting situations arise when this destabilizing mechanism coexists and competes with dissipation; a simple example is provided by the quasilinear wave equation with linear first-order damping. A more subtle dissipation occurs in certain viscoelastic materials such as polymers, suspensions and emulsions which have memory; i.e. the stress at each material point depends not only on the present value of the deformation gradient (and/or velocity gradient), but on the entire temporal history of motion. These materials exhibit behavior intermediate between that of an elastic solid and a viscous fluid. Generally, the memory fades with time: disturbances which occurred in the distant past have less influence on the present stress than those which occurred in the recent past. The authors' is to discuss qualitative properties of the equations which model unsteady motions of such materials. In order to present some of the central ideas while avoiding serious technical difficulties, they restrict the discussion to a particular model problem for the motion of a one-dimensional viscoelastic material with fading memory.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1987
- Accession Number
- ADA194296
Entities
People
- A. E. Tzavaras
- J. A. Nohel
- R. C. Rogers
Organizations
- University of Wisconsin–Madison