# Characteristic Times in Accumulation and Decay

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Many traditional models of stress relaxation and creep have been based on a sum of two or three exponential terms, that is, on a discrete spectrum of relaxation or retardation times. It has been demonstrated that in some cases the phenomena may also be described by nonexponential empirical models having fewer parameters. This Demonstration generates data with the conventional exponential models and fits them with a nonexponential empirical model having a single characteristic time constant, . It provides visual comparison of the two models and assists in deciding when the simplified model can be used as a substitute. This also applies to other phenomena such as charge dissipation and accumulation.

Contributed by: Mark D. Normand and Micha Peleg (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: stress relaxation with two characteristic relaxation times fitted with a nonexponential model

Snapshot 2: creep with three characteristic retardation times fitted with a nonexponential model

Snapshot 3: creep with two characteristic retardation times fitted with a nonexponential model

Snapshot 4: stress relaxation with three characteristic relaxation times badly fitted with a nonexponential model

Snapshot 5: creep with three characteristic retardation times badly fitted with a nonexponential model

Decay or dissipation phenomena have been frequently described by a series of exponential terms such as , known as the discrete generalized Maxwell model, where the , the initial value of , and the are characteristic relaxation times. Similarly, accumulation or growth phenomena have been described by the generalized Kelvin model, , where and the are characteristic retardation times.

This Demonstration generates data with or without added noise using these two models, where , and fits them with the empirical model for the decay case and for the accumulation case, where is the single characteristic time constant. It displays the generated data on which the fitted nonexponential curve is superimposed on one plot and the original (smooth) exponential curve on another for comparison. By doing so, the Demonstration assists the user in determining whether data characterized by models having two to five adjustable parameters may also be described by a model having only one parameter in the pertinent range.

The user selects the type of curve ("accumulation" or "decay") and enters the parameters , , , , and using sliders. Note that by definition and is automatically calculated. To remove a term from the exponential model either set or to zero or to values where . To remove two terms, set both and to zero. In that case and only can be varied.

For data generation using the exponential models, the user can choose whether or not random noise will be added, in which case its amplitude (in percent) can be chosen with a slider. Repeatable random noise will be produced by checking the "seed repeatable random numbers" checkbox, in which case the seed's value may be entered with a slider. The number of points to be generated and the upper limit of the time axis are also entered with sliders.

References:

[1] M. Peleg, "Linearization of Relaxation and Creep Curves of Solid Biological Materials," *Journal of Rheology*, 24(4) 1980 pp. 451–463.

[2] J. Malave–Lopez and M. Peleg, "Linearization of the Electrostatic Charging and Charge Decay Curves of Powders", *Powder Technology*, 42(3) 1985 pp. 217–223.

[3] A. Nussinovitch, M. Peleg, and M. D. Normand, "A Modified Maxwell and a Non-Exponential Model for Characterization of the Stress Relaxation of Agar and Alginate Gels", *Journal of Food Science*, 54, 1989 pp. 1013–1016.

## Permanent Citation

"Characteristic Times in Accumulation and Decay"

http://demonstrations.wolfram.com/CharacteristicTimesInAccumulationAndDecay/

Wolfram Demonstrations Project

Published: March 7 2011