In Linear analysis, forces and deformations are directly proportional to each other.
For example axially loaded bar has extension equal to PL/AE...more P so more extension. This is linear analysis (inclined line in graph between P and delta).
In non-linear analysis forces and deformations are not directly proportional. For example, even after removing load, strain continues to increase.
There are three types of non-linearaties;
1. Geometric; for example p-delta (second order effects)
structural stiffness can change as the structure deforms, and therefore the displacement will not be proportional to the applied load
2. Material; for example stress is not directly proportional to strain anymore after Hook's Law limit
3. Boundary conditions;
contact between components depends on the load between the components, which in turn affects the stiffness of the contacting parts, and therefore produces a displacement response that is not proportional to the applied load.
These 3 types of non-linear behavior can be studied in either
A. Static domain or
B. Dynamic domain
One need to perform non-linear analysis if either of the above problem needs to be analysed and for those structures where one of the above 3 problems post sufficient doubt in linear analysis range.