Stress-Strain/linear-non linear in Engineering

In tensile loading, the applied stress is linearly proportional to the induced strain, and it call as elastic deformation. The relationship between the applied stress s and the strain being induced e is as:

                                                                            s = E e

When the applied stress exceeds the elastic region, plastic deformation takes place, i.e. the applied stress is no longer proportional to the strain. The point where the non-linearity of the stress-strain relationship begins is known as the proportional limit. The applied stress is related to the  induced strain in the plastic deformation region by the following equation: 

                                                                            s = K e ⁿ

The ultimate tensile strength is the maximum stress level on the engineering stress-strain curve and it is the maximum stress that can be withstand by a structure in tension. All deformation before this point is uniform throughout the narrow region of the material. After which, subsequent deformation is confined to a small constriction or neck and as the area on which the load is acting on reduces, a smaller load is required to produce a greater deformation. Ultimately, fracture occurs at the neck.