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Dependence of relaxation time on the polymer length (via a string of viscoelastic springs)
Dear all,
According to polymer theory, the relaxation time depends on its length (or unit number), and there exist some relationships to describe such dependence. The smaller number of polymer units, the smaller the relaxation time of the polymer is. How to describe this via a simple model of visco-elastic springs?
Some Kelvin-Voigt visco-elastic springs are distributed in series, and have same elastic and viscous constants, as shown in the figure. The right end of the polymer is subjected to a step displacement, and the left end is fixed. Based on the mechanical equilibrium throughout the polymer all the time, I can deduce the equilibrium equation for every visco-elastic spring, and then obtain the final equation for the displacement in the red circle, such as
(disp-xn(t))*n= xn(t)+dxn(t)/dt
xn(t)=n*disp/(n+1)*(1-exp[-(n+1)t])
where disp is the step displacement, n is the unit number. It is easy to get the second equation, but it seems that the longer polymer needs a much shorter relaxation time, and doesn’t agree with the truth. I am puzzled for such a result, is the assumption of the mechanical equilibrium wrong or any other problems?
Thank you very much!
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may be useful...
Dear Kong Dong,
As so far, i can expect. Maybe it is better for you to derive the equation via the theory of minimum potential energy for the system (the model is similar with series-wound multiple linear springs).
Hope it will help
Regards
Liu Gang
Dear Gang, Many thanks.
Dear Gang, Many thanks.
Welcome, Dong
Dear Dong, Welcome
To my understanding (may be very limited), the physical variables are depend on the dimensional coordinates and time, at asy P(X,t), here X can be three dimensional.
For the static problems, we can set P just related to dimensial coordinates, it means P(X). When we take large deformation or irregular geometry boundary conditions into account, one effective way is transform through the dimensional coordinates.
For the dynamic problems P(X,t), the fourier transformation is a good option for us to transform the governing equation between time domain and physical domain.
For this problem, it is one dimensional condition and time dependent, we can set P(x,t). May be, the governing equation should be second order ODE though it will be time dependent (may be can seperate the variables)...
All the best