Correlation between two parameters that vary with time?

Essentially, you have one input function, energy, and an output, damage index. You can try first a simple AR model. If it does not give any significative resutl, try a nonlinear model. A simple approach that can give you a fair idea is the cross-correlation. You can also compute the time-frequency representation, for example using the wavelet transform.

I think we should first isolate the minimum earthquake energy that would not cause any structural damage. The excess energy that would occur may be perhaps random or vary in pulses or spikes. Each quantum of excess energy can be related with a single time variable. Of course then the damage supposed to occur is consequential and would be cumulative. But if failure itself changes with each damage occurrence, then the prediction will be painstaking.

Mutual information is used for identification of depedencies between non parametric distributions