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
τ3 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.
[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.
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