Generalized Maxwell and Kelvin Models

As indicated earlier, single Maxwell or Kelvin elements are of limited utility in representing the actual stress-strain response of polymers. A more realistic mathematical model can be developed, however, by considering a series of Maxwell elements in parallel. Consider, first just two Maxwell 

Since the Maxwell elements are connected in parallel, if strain e(t) is given, one can either solve the pair of linear first order E s. (5.15) or the single second order equation (5.16b) to find the solution for a(t). As an example, consider the case of stress relaxation in which a constant strain history is applied, e(t) = oH(t). Due to the kinematic constraint, each Maxwell element sees the same global strain history and the solution for ai(t) and 02(1) from Eqs. 5.15 are as given earlier in Eq. 3.17. 

From the equilibrium constraint, the solution for the overall stress in the system is a simple superposition of the stresses in each element 

The second order differential equation (5,16b) can also be solved to obtain the same solution Eq. 5.17b. 

Three Maxwell elements in parallel would give a differential relation between stress and strain that contains first, second and third derivatives (see homework problem 5.7) as given below, 

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Give sketches for generalized Maxwell and Kelvin models. Label all elements. 

Generalized Maxwell and Kelvin models are combinations of several Maxwell elements in parallel or Kelvin elements in series respectively. They were introduced to describe discrete relaxation times. The generalized Maxwell model is written as... 

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