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.

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