When stress reaches the proportional limit, what is the strain-energy density called?

When stress reaches the proportional limit, the strain-energy density is referred to as the Modulus of Resilience. This term specifically denotes the maximum amount of strain energy per unit volume that a material can store while still behaving elastically, meaning that it can return to its original shape once the applied stress is removed. At this point, the material's behavior is linear and reversible, as per Hooke's Law, and it is important for understanding the elasticity of materials under load.

Modulus of Resilience is defined up to the yield point where the material deforms elastically and does not undergo permanent deformation. It is a crucial parameter in engineering and materials science because it helps predict how materials will behave under various loading conditions.

The other terms in the question relate to different material properties: Ultimate Strength refers to the maximum stress a material can withstand before failure; Modulus of Toughness measures the total energy a material can absorb before fracturing; and Elastic Modulus quantifies stiffness by relating stress to strain in the elastic region but does not specifically address energy storage in the same way that Modulus of Resilience does. Understanding these distinctions is key in analyzing material performance under stress and strain conditions.