# How does linear elasticity deal with Heaviside forces?

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?