1. Introduction: For mathematics students the prerequisite is Advanced Calculus I. ENV students should have taken Mathematics for Geophysical Science II or the ENV unit Mathematics II which ensures that they have some prior knowledge of linear elasticity. Viscoelasticity has a wide scope: both lead on roofs and glass on windows flow as very viscous liquids over large times scales but behave as solids over short time scales. The theory can be applied to various problems of current interest.

2. Timetable Hours, Credits, Assessments: 33 lectures; 20 UCU; assessment is 20% coursework and 80% examination. Example sheets will be handed out containing many examples to support and illustrate the lecture material. Some will be set as coursework and detailed solutions to exercises will be handed out.

3. Overview: The general development and broad application of the theory of viscoelasticity is of relatively recent occurrence. In fact, the activity in this field has been primarily due to the large scale development and utilisation of polymeric materials. Many of these newly developed materials exhibit mechanical response characteristics which are outside the scope of such theories of mechanical behaviour as elasticity and viscosity; thus, the need for a more general theory is quite apparent.

To be more specific, the theory of elasticity may account for materials which have a capacity to store mechanical energy with no dissipation of the energy. On the other hand, a Newtonian viscous fluid in a nonhydrostatic stress state implies a capacity for dissipating energy, but none for storing it. But, then, materials which must be outside the scope of these two theories are those for which some, but not all, of the work done to deform them can be recovered. Such materials possess a capacity both to store and to dissipate mechanical energy.

A different way of characterising these materials is through the nature of their response to a suddenly applied uniform distribution of surface tractions on a specimen. An elastic material, when subjected to such a suddenly applied loading state and held constant thereafter, responds instantaneously with a state of deformation which remains constant. A Newtonian viscous fluid responds to a suddenly applied state of uniform shear stress by a steady flow process. There are, however, materials for which a suddenly applied maintained state of uniform stress induces an instantaneous deformation followed by a flow process which may or may not be limited in magnitude as time grows. A material which responds in this manner is said to exhibit both an instantaneous elasticity effect and creep characteristics. This behaviour is clearly not described by either an elasticity or a viscosity theory but combines features of each. This more general type of material possesses a characteristic which can be descriptively referred to as a memory effect. That is, the material response is not only determined by the current state of stress but is also determined by all past states of stress and, in a general sense, the material has a memory for all past states of stress.

Even though most of the developments in viscoelasticity theory are recent the basic linear and isothermal field theory formulation has been available for a much longer time. While there were several early contributors, such as Maxwell, Kelvin, and Voigt, Boltzmann in 1874 apparently supplied the first formulation of a three-dimensional theory of isotropic viscoelasticity, whilst Volterra obtained comparable forms for anisotropic solids in 1909.

4. Recommended Reading: 

G A Holzapfel "Nonlinear Solid Mechanics" (Wiley, 2000)

A S Wineman & K R Rajagopal "Mechanical Response of Polymers" (CUP, 2000)

I M Ward & D W Hadley "Introduction to Mechanical Properties of Solid Polymers" (Wiley)

R M Christensen "Theory of Viscoelasticity. An Introduction" (2nd ed., Academic Press)

A Drozdov "Mechanics of Viscoelastic Solids" (Wiley)

A Drozdov "Finite Elasticity and Viscoelasticity, a Course on the Nonlinear Mechanics of Solids" (World Scientific)

Out of print:

D R Bland "The Theory of Linear Viscoelasticity" (Pergamon)

A C Pipkin "Lectures on Viscoelasticity Theory" (George Allen & Unwin, Springer Verlag)

Cartesian tensors and continuum mechanics. (8 lectures)

Introduction. Spring-dashpot models. (3 lectures)

Viscoelastic response in shear: hereditary laws. Relaxation, creep, energy storage and loss. (8 lectures)

Viscoelastic response in shear: differential operator laws. Links with spring-dashpot models. (3 lectures)

Tensorial stress-strain relations. Generalized Hooke's law in hereditary form. Dilatation, general response. Quasi-elastic approximation. (4 lectures)

One-dimensional dynamic response: torsional oscillations (fluids and solids), plane shear waves (fluids and solids). (3 lectures)

Nonlinear viscoelasticity. Materials of differential type, Rivlin-Ericksen tensors, convected time derivatives. Materials of the rate type. (4 lectures)