maxwell / voight models - can someone check this for me please...?

Tutorial question was given.....

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The Creep response of a polymer corresponding to a load application for 100s can be described by a Voight Model where the Elastic element = 2GPa, and the Viscous element = 100 x10^09 Ns/m^2. If the Maxwell model has viscous element = 200 x10^09 Ns/m^2

Find the value of the Maxwell elastic element.

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I have the solution below.....but I'm not sure that I'm correct. I have, for Maxwell, Epsilon = (sigma / E) + (sigma / Viscosity).

I think it should be Epsilon = (sigma / E) + (sigma^dot / Viscosity). where sigma^dot is 'stress rate' rather than stress..?

can anyone clarify for me?? If it is 'stress rate' rather than stress' how do I calculate this - is it as a function of the time period??)

my working is this:




Strain Rate for the Voight model is found from:

Epsilon = (Sigma / E) (1- exp (-E * t)/Viscosity))

Epsilon = (Sigma / 2 x10^9) (0.864)

Epsilon = Sigma * 0.432 x 10^-9 [eqn 1]

For the Maxwell model

Epsilon = (sigma / E) + (sigma / Viscosity)

Substituting eqn 1 gives

Sigma * 0.432 x 10^-9 = (sigma / E) + (sigma / Viscosity)

Div thru by Sigma to give

0.432 x 10^-9 = (1 / E) + (1 / Viscosity)

0.432 x 10^-9 = (1 / E) + (1 / 200 x10^9)

0.432 x 10^-9 = (1 / E) + (5 x10^-12)

1 / (0.432 x 10^-9 - 5 x10^-12) = E

E = 2.34 x10^9 Pa

E = 2.34 GPa

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The dotted sigma is in the first term on rhs, i.e., sigma^dot/E.


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John Spevacek

Ok so whats the difference between sigma dot and sigma...


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I hope this goes through. I've been struggling with the Goolge server to get this post out. Oh for a good newsserver.

Sigma dot is the derivative wrt time.

The response of a Maxwell fluid in creep is easier to describe than to use the mathematical formula. Upon application of the stress, the spring immediately deforms the amount determined by the spring equation. After that, viscous motion takes over so that the deformation increases at a constant rate as determined by the fluid equation. After the stress is removed, the spring immediately snaps back by the same amount that it originally moved and the systems stays at that degree of deformation until the end of time. So the plot of deformation vs time is: straight up, than a constant positive slope, then straight down, then straight across.

You know the deformation after 100 seconds that you are trying to match. Imagine that as a point on the strain-time plot. The viscous component is a line passing through the origin, but at 100 seconds, it will be below the point that you are trying to match. Slide the line up until you hit the point and note the new intercept that you have for the line. That is how much deformation you need in your Maxwell spring. Knowing the spring's deformation and the applied stress, you can find the spring constant.


Reply to
John Spevacek

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