What is the term for the rate of change of stress with respect to strain?

The correct term for the rate of change of stress with respect to strain is the tangent modulus. The tangent modulus represents the slope of the stress-strain curve at any given point, reflecting how the material responds to changes in stress within its elastic limit. This aspect is significant as it describes material behavior beyond the linear elastic region, where the relationship between stress and strain is not constant.

The elastic modulus, while relevant to the general relationship between stress and strain in the linear elastic region, typically refers to the initial or tangent linear response, rather than the variable rate of change seen in the tangent modulus.

Shear modulus is specifically concerned with the ratio of shear stress to shear strain, which pertains to deformation under shear loads rather than general stress-strain relationships.

Bulk modulus measures a material's response to uniform pressure applied from all directions, focusing on volumetric strain as opposed to the specific rate of change of stress concerning any axial strain.

Recognizing the distinct definitions and applications of these moduli is vital in materials science and civil engineering, as it leads to better understanding of material behavior under various loading conditions.