I don't quite fully understand the derivation of the Maxwell model for prediction of stress relaxation for viscoelstic materials. usually the governing equation can be described in terms of strain rate:

$$ \frac{d\epsilon}{dt} = \frac{d\sigma}{dt} \cdot \frac{1}{E} + \frac{\sigma}{\eta} \tag{1} $$

where the equation is then rearranged and integrated to get an equation in terms of stress and time.

This may be a silly question but why is it incorrect for the equation to be written using strain rather than stress rate like:

$$ \epsilon = \frac{\sigma}{E} + \frac{\sigma}{\eta}t\tag{2} $$

And then rearranged for stress, $\sigma$, in terms of time, t like:

$$ \sigma = \frac{\epsilon}{\frac{1}{E} + \frac{t}{\eta}}\tag{3} $$


1 Answer 1


If we integrate both sides of:

$$\frac{d\epsilon}{dt} = \frac{d\sigma}{dt} \cdot \frac{1}{E} + \frac{\sigma}{\eta} \tag{1}$$

with respect to time, we get: $$\int_0^t \frac{d\epsilon}{dt^\prime} dt^\prime= \int_0^t \frac{d\sigma}{dt^\prime} \cdot \frac{1}{E} dt^\prime+ \int_0^t \frac{\sigma}{\eta} dt^\prime\tag{2}.$$

It's fair to say that: $$\int_0^t \frac{d\epsilon}{dt^\prime} dt^\prime= \epsilon\tag{3},$$

and if $E$ is constant with respect to time, it's fair to say that:

$$\int_0^t \frac{d\sigma}{dt^\prime} \cdot \frac{1}{E} dt^\prime= \frac{\sigma}{E}\tag{4}.$$

For the last term, if $\sigma$ and $\eta$ are constant with respect to time, then it's fair to say that:

$$\int_0^t \frac{\sigma}{\eta} dt^\prime = \frac{\sigma}{\eta}t\tag{5},$$

but if $\sigma$ is constant with respect to time, then the original equation might as well be:

$$\frac{d\epsilon}{dt} = \frac{\sigma}{\eta} \tag{6}$$

since we have:

$$\frac{d\sigma}{dt} = 0.\tag{7} $$

Based on Eq. 1, it's fair to assume that $\sigma$ depends on time, so Eqs 2-4 lead us to the following formula:

$$\epsilon(t)= \frac{\sigma(t)}{E} + \frac{1}{\eta}\int_0^t \sigma(t^\prime) dt^\prime\tag{8}.$$

If you know how $\sigma(t)$ depends on time, then you may be able to put Eq. 8 in "closed form", but if that were the case, Eq. 1 would likely be presented in a more simple way. The fact that Eq. 1 is written in the way that it's written, suggests that $\sigma$ has some time-dependence which prevents us from using simpler equations like the ones that you proposed.


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