# Linear main effects and nonlinear interaction

My question is conceptual rather than one based on data/findings. I am interested to know whether a situation where the main effects of two variables are linear whereas their interaction is nonlinear makes sense from a statistical standpoint and how it might be analyzed.

Three variables: x - a trait m - a situational variable y - behavior

I expect that the relationship between x and y is linear (positive relationship) and the relationship between m and y is linear (positive relationship). X and M are not causally related.

I further expect that an interaction between x and m on y is nonlinear. Specifically, y will be highest when x is high and m is at a moderate level. Y will be moderate when x is high and m is either high or low. And last, y will be smallest when x is low and m is either high or low.

Can someone help me sort through or suggest some resources, specifically how it might be tested? It seems that most quadratic approaches assume nonlinear main effects in order to test nonlinear interactions. I am sure this is a basic question for many of you but it's giving me fits