Question: In what space do convolution integrals live?

Questions from a student in the class:

Q1:

Do convolution integrals live in the C^k space or just H^k?

Q2:

Is this the reason those material models are always written in integral form?

I’m unsure how to answer this question as I’m unsure of what the student is actually trying to ask, especially with the first question. Convolution integrals are operators not members of C^k or H^k at least in the way we are thinking of in this class. There is a way to understand that question mathematically, but it is well beyond what we are talking here.

I originally asked. I have noticed most time dependent material models are written in rate form unless they have a convolution integral, then they are written in integral form. Is this done because the relaxation function only exists as an integral?

I’m not a material expert, but I would venture to say maybe. If we are in rate form, we might not have a nice continuous function and so it is more natural to consider it in integral form. However, I could also see that sometimes the update scheme might more naturally be in integral form for other reasons. Like I said, I’m not a material expert. Sorry, I couldn’t give a better answer.

Reply from an internal discussion:

Hysteretic effects can’t be captured by a simple differential operator. They rely on the configurations that the body has previously experienced. The convolution kernel provides a mechanism for describing the “memory” of the material.