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For a particular elastic material with prismatic geometry, I observed a linear relationship between stress and strain but the forces applied to the material are given by a step function where the force increments are of constant size. So I have:

$ \forall n \in \mathbb{N}, H(n) = \begin{cases} 0, n<0 \\ F_0n, 0 \leq n \\ \end{cases} $

But, given that I have $\sigma(\epsilon)=E*\epsilon$ and the derivative of the Heaviside function is the $\delta(x)$, the Dirac delta function which means that:

$\begin{cases} \frac{d\sigma}{dt}=F_0 \delta(t), \forall t \in [0,\epsilon) \\ \frac{d\sigma}{dt}=E*\frac{d\epsilon}{dt} \\ \end{cases}$

And as a result,$\forall t \in [0,\epsilon) \lim_{t \to 0} \frac{d\epsilon}{dt} \rightarrow \infty$

Mathematically, this makes sense but I'm not sure how to explain this physically. Does the linear theory of elasticity have an explanation of what might be going on within the material?

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Heaviside functions mean - physically speaking - that the response of your system is faster than your experimental time resolution, i.e., you do not see the response process because it is on a time scale so small that you are not interested in it / you are not able to measure it with the "slow" measurement apparatus you are using. This does not mean that anything is really "instanteneous" in reality - of course it is not. The change is just very fast, too fast for you to care respectively too fast for you to measure and thus a Heaviside step is a more than reasonable approximation. But it is of course not physical reality.

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