Linear Viscoelastic Models#

Fig. 5 shows schematic representations of the viscoelastic models. The linear Maxwell model can be represented by a spring in series with a linear dashpot.

Schematic representations of viscoelastic rheologies — Fig. 5 Schematic representations of viscoelastic bulk rheologies available in PyLith. Note that the models with linear (Newtonian) and power-law viscous behavior are treated as separate rheologies.#

For a one-dimensional model, the response is given by

(50)#\[\frac{\mathit{d\epsilon}_{Total}}{\mathit{dt}} = \frac{\mathit{d\epsilon}_{D}}{\mathit{dt}} + \frac{\mathit{d\epsilon}_{S}}{\mathit{dt}} = \frac{\mathit{\sigma}}{\mathit{\eta}} + \frac{1}{\mathit{E}}\frac{\mathit{d\sigma}}{\mathit{dt}},\]

where \(\mathit{\epsilon}_{Total}\) is the total strain, \(\mathit{\epsilon}_{D}\) is the strain in the dashpot, \(\mathit{\epsilon}_{S}\) is the strain in the spring, \(\mathit{\sigma}\) is the stress, \(\mathit{\eta}\) is the viscosity of the dashpot, and \(\mathit{E}\) is the spring constant. When a Maxwell material is subjected to constant strain, the stresses relax exponentially with time. When a Maxwell material is subjected to a constant stress, there is an immediate elastic strain, corresponding to the response of the spring, and a viscous strain that increases linearly with time. Because the strain response is unbounded, the Maxwell model actually represents a fluid.

Another simple model is the Kelvin-Voigt model, which consists of a spring in parallel with a dashpot. In this case, the one-dimensional response is given by

(51)#\[\mathit{\sigma}(\mathit{t}) = \mathit{E\epsilon}(\mathit{t}) + \mathit{\eta}\frac{\mathit{d\epsilon}(\mathit{t})}{\mathit{dt}}.\]

As opposed to the Maxwell model, which represents a fluid, the Kelvin-Voigt model represents a solid undergoing reversible, viscoelastic strain. If the material is subjected to a constant stress, it deforms at a decreasing rate, gradually approaching the strain that would occur for a purely elastic material. When the stress is released, the material gradually relaxes back to its undeformed state.