Why is plastic deformation a non-linearity if I am using Bilinear hardening to model it in FEA?

Question

Attached you may find a picture for isotropic bilinear hardening up until the ultimate tensile strength. The first line is the elastic region while the second one is plastic. I couldn’t understand that why is plasticity even a non-linearity in FEA? I mean if I just consider the graph below, then a point in a FEA model will just move over this graph only. If it switches from first line to the other, then there is just a change in the elastic modulus. If I am conducting a geometrically linear analysis, then how could inputting a material plasticity model, like the one showed below, make the analysis still be non-linear?

r13 · Answer

In the plastic range, the stress-strain relationship is non-linear as shown in the graph below. With consideration of geometric changes after yield, the true stress-strain curve (dotted line in the graph) shall be used instead of the normal curve, and non-linearity needs to be considered in the analysis.

Note: Due to the shrinking of section area and the ignored effect of developed elongation to further elongation, true stress and strain are different from engineering stress and strain.
$delta_t = delta(1 + epsilon)$, and
$epsilon_t = ln(1 + epsilon)$

Why is plastic deformation a non-linearity if I am using Bilinear hardening to model it in FEA?

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