Viscoelasticity: Difference between revisions

Revision as of 17:24, 27 February 2024

Revision as of 13:58, 2 March 2024

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==Background==

In the nineteenth century, physicists such as [[James Clerk Maxwell|Maxwell]], [[Ludwig Boltzmann|Boltzmann]], and [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] researched and experimented with [[Creep (deformation)|creep]] and recovery of [[glass]]es, [[metal]]s, and [[rubber]]s. Viscoelasticity was further examined in the late twentieth century when [[synthetic polymer]]s were engineered and used in a variety of applications.<ref name=McCrum>McCrum, Buckley, and Bucknell (2003): "Principles of Polymer Engineering," 117-176.</ref> Viscoelasticity calculations depend heavily on the [[viscosity]] variable, ''η''. The inverse of ''η'' is also known as [[Viscosity#Fluidity|fluidity]], ''φ''. The value of either can be derived as a [[Temperature dependence of liquid viscosity|function of temperature]] or as a given value (i.e. for a [[dashpot]]).<ref name=Meyers/>

[[Image:Non-Newtonian fluid.svg|thumb|350px| Different types of responses {{nowrap|(<math>\sigma</math>)}} to a change in strain rate {{nowrap|(<math>d\varepsilon/dt</math>)}}]]

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* <math>\lambda_1</math> is the relaxation time;

* <math>\lambda_2</math> is the retardation time = <math> \frac{\eta_s}{\eta_0}\lambda_1 </math>;

* <math> \stackrel{\nabla}{\mathbf{T}} </math> is the [[Upperupper convected time derivative]] of stress tensor:<math display=block> \stackrel{\nabla}{\mathbf{T}} = \frac{\partial}{\partial t} \mathbf{T} + \mathbf{v} \cdot \nabla \mathbf{T} -( (\nabla \mathbf{v})^T \cdot \mathbf{T} + \mathbf{T} \cdot (\nabla \mathbf{v})); </math>

*<math>\mathbf{v}</math> is the fluid velocity;

*<math>\eta_0</math> is the total [[viscosity]] composed of solvent and polymer components, <math> \eta_0 = \eta_s + \eta_p </math>;

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Whilst the model gives good approximations of viscoelastic fluids in shear flow, it has an unphysical singularity in extensional flow, where the dumbbells are infinitely stretched. This is, however, specific to idealised flow; in the case of a cross-slot geometry the extensional flow is not ideal, so the stress, although singular, remains integrable, although the stress is infinite in a correspondingly infinitely small region.<ref name="c"/>

If the solvent viscosity is zero, the Oldroyd-B becomes the [[Upper Convected Maxwell model|upper convected Maxwell model]].

=== Wagner model ===

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times of the original data. Finally, fit the pseudo data with the Prony series.

==Effect of temperature on viscoelastic behavior==

{{main|Time–temperature superposition}}