Nonlinear systems are those for which the principle of superposition does not hold. Nature abounds with nonlinear systems; in fact, they are the rule rather than the exception. The sources of nonlinearities can be material or constitutive, geometric, inertia, body forces, or friction. The constitutive nonlinearity occurs when the stresses are nonlinear functions of the strains. The geometric nonlinearity is associated with large deformations in solids, such as beams, plates, frames, and shells, resulting in nonlinear strain-displacement relations (e.g., mid-plane stretching, large curvatures of structural elements, large strains, and large rotations of elements). The inertia nonlinearity may because by the presence of concentrated or distributed masses; in a Lagrangian formulation, the kinetic energy is a function of the generalized coordinates as well as their rates and, in fluid flow, the acceleration includes a nonlinear convective term. Other examples include Coriolis and centripetal accelerations. The nonlinear body forces are essentially magnetic and electric forces. The friction nonlinearity occurs because the friction force is a nonlinear function of the displacement and velocity, such as dry friction and backlash.

The nonlinearities may appear in the governing partial-differential equations, or the boundary conditions, or both. To some extent, the form of the nonlinearity appearing in the equations and boundary conditions depends on the coordinate system used and the orientation of the body forces, such as gravity. Examples of nonlinear boundary conditions include free surfaces in fluids and deformation-dependent constraints.

Cited from 2004 Linear and Nonlinear Structural Mechanics

Usually when you have higher order coupling between two or more variables that depend on each other. Simplest example would be the non linear (quadratic) terms of the incompressible Navier-Stokes equations. They are commonly referred to as the convection terms. As your colleague Mr. Ma specified above, these terms disrupt the applicability of the superposition method.

Dear Prof. Ma Zhoayang, A good example of a non-linear mechanics is the (attached) book by Dr. Jeremy Dunning-Davies and myself (see especially Chapter 4 and its appendix). Our new mechanics is a variable (particle) inertial mass mechanics that is based upon the experimental discovery of variable particle inertial mass by (the late) British Prof. Eric Laithwaite who invented the trains in Germany & Japan and float on magnetic fields and so do not touch the rails ... and upon the experiments of Russian Prof. Alexander Dmitriev concerning the falling accelerations of horizontally spinning rotors whose accelerations turn out to be a function of the rotor horizontal spin ... and so then the equivalence principle of Einstein is violated by these falling & spinning rotors.

What are the fundamental reasons for linearity? I don't see them. It is simply the limiting case and the first aproximation sometimes valid.

Dear Eugene F. Kislyakov, Yes, that's right. Linearity IS just a first mathematical approximation, isn't it?

God Bless, Denny

The only our problem (and it is the mathematical problem!), Dennis, is that we can not solve non-linear differential equations in more than two dimensional configurational space. Einstein's equations are of this kind. If somebody (as says George Dishman) has solved them for two merging black holes (or neutron stars), then he must be awarded by Nobel prize, but not the diggers of LIGO's tunnel.

Hi,

I agree with Prof. Eugene F Kislyakov.

With my best regards!

Dr. Adel OUESLATI

Taken from the famous book on PDE, as written by Walter Strauss, you can see a proper mathematical explanation as to what constitutes a nonlinear term in a mathematical equation. Then depending on what physical model you choose to use, many different behaviors can then be modeled by different non-linear terms.