Hi, I have done some stress relaxation tests to study the viscoelastic behaviour of hydrogels. I am just wondering how to use stress-time data to find prony series constants required for material characterization. Please help.
The Prony method is described in detail in the attached entry of Wikipedia (see also the Prony series under Viscoelasticity - link also attached). In essence, the sampled data (the equidistant sampled function of interest in time) form the vector F, whose components are Fn, n = 0, 1, ..., N-1. From these components one constructs a matrix and a vector that together describe a linear matrix equation, the solution of which is a vector P whose components are Pm, m = 1, 2, ..., M. These components form the coefficients of an M-th order monic polynomial whose roots are used to construct matrices in matrix equations described in part by the coefficients {Fn | n}. The solutions of these equations combined with the roots of the above-mentioned monic polynomial, provide one with the necessary constants from which to determine all the parameters in the Prony series corresponding to N equidistant samplings in time of one's property of interest.
https://en.wikipedia.org/wiki/Prony%27s_method
https://en.wikipedia.org/wiki/Viscoelasticity#Prony_series
https://www.researchgate.net/post/How_do_you_define_viscoelastic_material_in_ABAQUS
Dear Arzmandi,
There are numerous methods for determining the Prony series from relaxation and/or creep data. An early method involved constructing log-log plots of relaxation data in which straight line approximations for the data on the log-log graph yield the time constants (i.e. torr i 's) from the slopes, and the exponential coefficients (i.e. alpha i 's) are obtained from the intercepts. Other methods have can be employed. For example, Johnson and Quigley determined a relaxation time constant for a nonlinear model which is similar to a one-term Prony series model.
The hereditary integral method can be employed to obtain an analytical representation of material response when it is subjected to rate dependent loading. it can be used for schedules in which the material is not allowed to relax between subsequent loading changes. The analytical representation can be used in a nonlinear regression analysis, with measured data, to evaluate the Prony series constants. Several regression analyses can be performed using different weight functions.
Load versus time test data for a sequence of different rate loading segments is least-squares fitted to a Prony series hereditary integral model of the material tested. A nonlinear least squares regression algorithm is employed. The measured data includes ramp loading, relaxation, and unloading stress-strain data. The resulting Prony series, which captures strain rate loading and unloading effects, produces an excellent fit to the complex loading sequence.
A common form for these constitutive equations N employs a Prony series (i.e., a series of the form N∑i αi . e-t/torri )
The Prony constants of the regression(Weighted Nonlinear Regression for Relaxation Test ) can be calculated.
Please follow these references for further understanding of creep relaxation prony step series
1.F10gge, W. "Viscoelasticity", Blaisdell Publishing Co. , Massachusetts, 1975.
2. Cristensen, R. M. "Theory of Viscoelasticity" 2 na Edition, Academic Press, New York, 1982.
3. Johnson, A. R., and Quigley, C. J., "A Viscohyperelastic Maxwell Model for Rubber Viscoelasticity", Rubber Chemistry and Technology, Vol. 65, No. 1, pp 137-153 (1992).
Hope it is useful for you.
Ashish
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