Rheology - Viscoelasticity

Viscoelasticity

The classical theory of elasticity deals with the behaviour of elastic solids under small deformations, for which,(1) according to Hooke's Law, stress is directly proportional to the strain — but independent of the rate of strain, or how fast the deformation was applied, and (2) the strains are completely recoverable once the stress is removed. Materials that can be characterized by classical theory of elasticity is known as linear elastic materials, even for such materials the linear relationship between stress and strain may be valid only for a certain range of strains. A large number of solids show non-linear relationship between stress and strain even for small stresses (such as rubber), but if the strains are still recoverable they are known as non-linear elastic materials. The classical theory of fluid mechanics, governed by the Navier-Stokes equation, deals with the behaviour of viscous fluids, for which, according to Newton's Law, the stress is directly proportional to the rate of strain, but independent of the strain itself. These behaviour are, of course, generally observed for ideal materials under ideal conditions, although the behaviour of many solids approaches Hooke's law for infinitesimal strains, and that of many fluids approaches Newton's law for infinitesimal rates of strain. Two types of deviations from linearity may be considered here.

1. When finite strains (larger strains, as opposed to infinitesimal strains) are applied to solid bodies, the stress-strain relationships are often much more complicated (i.e. Non-Hookean). Similarly, in steady flow with finite strain rates, many fluids exhibit marked deviations in stress-strain rate proportionality from Newtons law.
2. Even if both strain and rate of strain are infinitesimal, a system may exhibit both liquid-like and solid-like characteristics. A good example of this is when a body which is not quite an elastic solid (i.e. an inelastic solid) does not maintain a constant deformation under constant stress, but rather continues to deform with time – or "creeps" under the same stress at constant temperature. When such a body is constrained at constant deformation, the stress required to hold it at that stretch level gradually diminishes—or "relaxes" with time.

Similarly, a fluid while flowing under constant stress may show some elastic properties as well, such as storing some of the energy input instead of dissipating it all as heat and random thermal motion of its molecular constituents or having some recovery of strains after stresses are removed, although it may never recover all of its deformation upon removal of the initial applied stress. When such bodies are subjected to a sinusoidally oscillating stress, the strain is neither exactly in phase with the stress (as it would be for a perfectly elastic solid) nor 90 degrees out of phase (as it would be for a perfectly viscous liquid) but rather exhibits a strain that lags the stress at a value between zero and 90 degrees: i.e., Some of the energy is stored and recovered in each cycle, and some is dissipated as heat. These are viscoelastic materials.

Thus, fluids are generally associated with viscous behaviour (a thick oil is a viscous liquid) and solids with elastic behaviour (an elastic string is an elastic solid). A more general point of view is to consider the material behaviour at short times (relative to the duration of the experiment/application of interest) and at long times.

• Fluid and solid character are relevant at long times:
  We consider the application of a constant stress (a so-called creep experiment):
  • if the material, after some deformation, eventually resists further deformation, it is considered a solid
  • if, by contrast, the material flows indefinitely, it is considered a fluid

• By contrast, elastic and viscous (or intermediate, viscoelastic) behaviour is relevant at short times (transient behaviour):
  We again consider the application of a constant stress:
  • if the material deformation strain increases linearly with increasing applied stress, then the material is linear elastic within the range it shows recoverable strains. Elasticity is essentially a time independent processes, as the strains appear the moment the stress is appliled, without any time delay.
  • if the material deformation rate increases linearly with increasing applied stress, then the material is viscous in the Newtonian sense. These materials are characterized due to the time delay between the applied constant stress and the maximum strain.
  • if the materials behaves as a combination of viscous and elastic components, then the material is viscoelastic. Theoretically such materials can show both instantaneous deformation as elastic material and a delayed time dependent deformation as in fluids.

• Plasticity is the behaviour observed after the material is subjected to a yield stress:
  A material that behaves as a solid under low applied stresses may start to flow above a certain level of stress, called the yield stress of the material. The term plastic solid is often used when this plasticity threshold is rather high, while yield stress fluid is used when the threshold stress is rather low. However, there is no fundamental difference between the two concepts.