Oscillatory tests with a Maxwell model

If we oscillate an ideal spring, at all frequencies G = G the spring modulus, and G is zero. If, on the other hand, we oscillate a perfect dashpot, the loss modulus is given by G = tio), and the storage modulus G is always zero. However, the response of the simplest combination of these two elements in series — 

At very low frequencies, G is much larger than G, and hence liquid-Kke behaviour predominates. However as the testing frequency is increased, G takes over and solid-like behaviour prevails. The determinant of which kind of behaviour is most significant is the value of the test frequency co relative to the relaxation time x. This is a simple way to define a Deborah number, De—the ratio of the relaxation time to the test time—and a measure of De in this case is cox. Hence, low Deborah numbers always indicate liquid-like behaviour, whereas high Deborah numbers means solid-like response. At the midpoint, where G goes through a maximum G = G , and this takes place at a critical crossover frequency of co = 1/x. 

As shown above, we can define the dynamic viscosity t] = G /(o, and then for the Maxwell model we see that 

This shows a linear relationship between r and G over a range of frequencies, which is a quick way to check if data really corresponds to the Maxwell model  

If we combine two Maxwell models in parallel, we begin fo see features that will later become quite familiar as we examine practical examples of oscillatory curves. The overall behaviour is seen in figure 11, and is the simple sum of the two Maxwell imits, so 

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