How does linear elasticity deal with Heaviside forces?

Question

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?

Answer 1

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.