Inelastic Models and Linear Viscoelasticity

Abstract

We have seen some of the classical constitutive equations introduced in the last three centuries, and explored the general formulation of constitutive equations in the last chapter. There, we mention that the general constitutive principles should be taken as guidelines only; they should emerge from the physics of the fluids’ microstructures.

Problems

Problem 5.1

Show that, with the relaxation modulus function (5.13), the relation (5.11) is equivalent to

$$\begin{aligned} \mathbf {S}&=\sum \limits _{j=1}^{N}{\mathbf {S}^{(j)}}, \\ \mathbf {S}^{(j)}+\lambda _{j}\dot{\mathbf {{S}}}^{(j)}&=2\eta _{j}\mathbf {D} ,\;\;\;\eta _{j}=G_{j}\lambda _{j}. \nonumber \end{aligned}$$

(5.46)

This relation is called the linear Maxwell equation. Equation (5.46) is equivalent to (5.9).

Problem 5.2

Show that the shear stress for (5.11) in an oscillatory flow, where the shear rate is $\dot{\gamma }=\dot{\gamma }_{0}\cos \left( \omega t\right) ,$ can be expressed as

$$\begin{aligned} S_{12}=\left| {G^{*}}\right| \sin \left( {\omega t+\phi }\right) ,\;\;\;\tan \phi =\frac{{G^{\prime \prime }}}{{G^{\prime }}}, \end{aligned}$$

(5.47)

where $\tan \phi$ is called the loss tangent.

Problem 5.3

Verify that for the spectrum

$$ H\left( \lambda \right) =\cos ^{2}\left( {n\lambda }\right) , $$

$$ \eta ^{\prime }\left( \omega \right) =\int _{0}^{\infty }{\frac{{H\left( \omega \right) }}{{1+\lambda ^{2}\omega ^{2}}}d\omega }=\frac{\pi }{{4\omega }} \left( {1-e^{-2n/\omega }}\right) . $$

At large n,  the data $\eta ^{\prime }$ is smooth, but the spectrum is highly oscillatory. Conclude that the inverse problem of finding $H\left( \lambda \right) ,$ given the data $\eta ^{\prime }$ in the chosen form is ill-conditioned – that is, a small variation in the data (in the exponentially small term) may lead to a large variation in the solution.

Problem 5.4

For the Maxwell discrete relaxation spectrum (5.13), show that

$$\begin{aligned} G\left( t\right) =\sum \limits _{j=1}^{N}{G_{j}e^{-t/\lambda _{j}}} ,\;G^{\prime }\left( \omega \right) =\sum \limits _{j=1}^{N}{\frac{{ G_{j}\omega ^{2}\lambda _{j}^{2}}}{{1+\omega ^{2}\lambda _{j}^{2}}}},\;\eta ^{\prime }\left( \omega \right) =\sum \limits _{j=1}^{N}{\frac{{G_{j}\lambda _{j}}}{{1+\omega ^{2}\lambda _{j}^{2}}}}. \end{aligned}$$

(5.48)

In particular, with one relaxation mode $\lambda =\lambda _{1},$

$$\begin{aligned} \tan \phi =\frac{1}{{\omega \lambda }} \end{aligned}$$

(5.49)

deduce that as $\omega =0\rightarrow \infty ,$ the response goes from fluid ($\phi =\pi /2$) to solid behaviour ($\phi =0$).

Problem 5.5

Suppose we have a Maxwell material with one relaxation time,

$$ G\left( t\right) =\frac{{\eta _{0}}}{\lambda }e^{-t/\lambda }. $$

and $\varOmega _{i}=\text {constant.}$ Show that the solution to the circular Couette flow problem considered in Sect. 5.3.2 is

$$\begin{aligned} \frac{{M\left( t\right) }}{{M_{N}}}=1-e^{-t/\lambda }. \end{aligned}$$

(5.50)

Problem 5.6

Working in Laplace transform domain, show that the mechanical analog of Fig. 5.9c leads to (5.44).