Figure 1: Schematic graph showing the expected behavior of the correlation length \(\xi\), normalized by its initial value \(\xi_{0}\), as strain \(\epsilon\)increases towards failure at critical strain \(\epsilon_{c}\) for the three phase transition cases described in the text. The Griffith crack (blue line) and the exponential (orange line) are both examples of first-order transitions, exhibiting abrupt failure evident as the discontinuity in \(\xi\) at \(\epsilon_{c}\). The inverse power-law (green line) shows the smooth, continuous evolution to failure characteristic of a second-order transition.