# Viscoelasticity

A viscoelastic material is one in which:

• hysteresis is seen in the stress-strain curve.
• stress relaxation occurs: step constant strain causes decreasing stress
• creep occurs: step constant stress causes increasing strain

Viscoelastic material models are frequently used to describe the behaviour of (soft) human tissue, plastics, soil, etc. Commonly used viscoelastic models are the Kelvin material and Maxwell material. Each model can be represented by springs and dash-pots set in combinations of series and parallel elements.

Linear viscoelasticity is when the function is separable in both creep response and load. All linear viscoelastic models can be represented by a Volterra equation connecting stress and strain. Like:

[itex]\epsilon(t)= \frac { \sigma(t) }{ E_{inst,creep} }+ \int_0^t K(t-t^\prime) \sigma(t^\prime) d t^\prime[itex]

or

[itex]\sigma(t)= E_{inst,relax}\epsilon(t)+ \int_0^t F(t-t^\prime) \epsilon(t^\prime) d t^\prime[itex]

where t - time, [itex]\sigma (t)[itex] - stress, [itex] \epsilon (t) [itex] -strain, [itex]E_{inst,creep}[itex] and [itex]E_{inst,relax}[itex] - instantaneous elastic moduluses for creep and relaxation, K(t) - creep function, F(t)- relaxation function.

Linear viscoelasticity usually applicable only for small deformations

Nonlinear viscoelasticity is when the function is not separable. It is usually happens when the deformations are large or if the material changes its properties under deformations

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