An example of when you want a nonlinear material is the copper workpiece in your example.

A material tensile test is performed on a sample of material in a specific "dog bone" shape that is clamped into the testing machine. A Stress-Strain curve is computed from the Force-Displacement data that is measured by sensors in the tensile testing machine. The Yield Strength is the Stress when the curve has deviated by 2% strain from the straight line portion of the linear elastic part of the curve.  There is often another straight line portion at a much flatter slope after yield, and that slope is the Tangent Modulus.  Sometimes the slope is practically flat and the Tangent Modulus is zero.