The classical theories of Linear Elasticity and Newtonian Fluids, though trium phantly elegant as mathematical structures, do not adequately describe the defor mation and flow of most real materials. Attempts to characterize the behaviour of real materials under the action of external forces gave rise to the science of Rheology. Early rheological studies isolated the phenomena now labelled as viscoelastic. Weber (1835, 1841), researching the behaviour of silk threats under load, noted an instantaneous extension, followed by a further extension over a long period of time. On removal of the load, the original length was eventually recovered. He also deduced that the phenomena of stress relaxation and damping of vibrations should occur. Later investigators showed that similar effects may be observed in other materials. The German school referred to these as "Elastische Nachwirkung" or "the elastic aftereffect" while the British school, including Lord Kelvin, spoke ofthe "viscosityofsolids". The universal adoption of the term "Viscoelasticity", intended to convey behaviour combining proper ties both of a viscous liquid and an elastic solid, is of recent origin, not being used for example by Love (1934), though Alfrey (1948) uses it in the context of polymers. The earliest attempts at mathematically modelling viscoelastic behaviour were those of Maxwell (1867) (actually in the context of his work on gases; he used this model for calculating the viscosity of a gas) and Meyer (1874).

Describes general mathematical modeling of viscoelastic materials as systems with fading memory. Discusses the interrelation between topics such as existence, uniqueness, and stability of initial boundary value problems, variational and extremum principles, and wave propagation. Demonstrates the deep connection between the properties of the solution to initial boundary value problems and the requirements of the general physical principles. Discusses special techniques and new methods, including Fourier and Laplace transforms, extremum principles via weight functions, and singular surfaces and discontinuity waves.

Initial-Boundary Value Problems and the Navier-Stokes Equations

This paper presents a solution to the boundary value problem arising when a rigid punch of arbitrary asymmetric profile is pressed against the surface of a linear viscoelastic half space. The solution, which is valid for a continuously varying contact area with an arbitrary number of montonic increasing maxima and null minima, is expressed in terms of the one parameter family of solutions to the corresponding problem for an elastic half space. The shape of the contact area is always the same as that of some member of the one parameter family of contact areas associated with the corresponding elastic problem. It is shown how the solution may be used to generate specific results for the case of an axisymmetric punch of arbitrary smooth profile. Finally a generalization to viscoelasticity theory of Hertz's theory of the contact of smooth elastic bodies valid for a variable contact area with any number of maxima and minima is given.