[NR 1] Nonlinear Kelvin Body (NKB)

Consider a Kelvin body, i.e. a spring in parallel with a dashpot:


The body is externally described by its length x and force F.

Assume that the rest length (at zero force) is zero. The dashpot has the property:



A linear spring would obey:



The two forces add up to the total force of the Kelvin body:



We can solve for the temporal change of x to obtain a diff.eq. describing the evolution of x(t) from its starting value for any given external force F(t):



A Fourier transformation yields



which allows to define a linear response function S(w) of the system:



Let the spring now have some nonlinear force-length-relation N(x):



The evolution equation then reads:



This nonlinear diff.eq. can also be solved (at least numerically) for any applied F(t). However, it is in general not possible to define a response function. This will have important consequences when we consider more complex systems build from many Kelvin bodies.