Linear Vs Non linear

All structural problems are nonlinear in nature. However most of the times a linear analysis is a good approximation. 

Linear analysis is based on the following assumptions that stress and strain follows Hooke’s Law (i.e. linear relationship between stress and strain), deformations are covered by small deflection theory (i.e. small geometric difference between the initial and deformed shape) and other material properties (e.g. CTE, etc) are constant. 

Nonlinear analysis is based on the following assumptions that stress and strain does not follow Hooke’s Law (i.e. nonlinear relationship between stress-strain due to material plasticity), deformations are covered by large deflection theory (i.e. large geometric difference between the initial and deformed shape) and material properties that are temperature dependent. 

Any reason causing a variation in stiffness of the assembly being analyzed is potentially a source of non-linearity and therefore requires a non-linear analysis to be captured.

It is widely accepted that the three main sources of non-linearity are:

- Plasticity of material (variation of the material Young's modulus will cause the stiffness of the structure to change)

- Large displacements (Stiffness varies as a result of large geometric difference between the initial and deformed shape)

- Contact: if two parts or bodies of the assembly come into contact, or lose contact, or the extent of their contact patch changes, then the stiffness of the assembly also varies.

Examples: A bar distributed with axial load having constant young's modulus in linear FEA assumptions cross-section of the bar does not change after loading and the same bar with material constants varying from initial to final E, is non linear.

Another good example where (material) nonlinear analysis is a must is when you run linear FEA on a steel part for example and you find out that stress in a location has exceeded the yield, in this case all the linear stress results that exceed yield are useless because after yield steel does not behave per Hook's law