Relaxation due to the Sqrt(t)-Law in Comparison with the MAXWELL-Fluid
Univ.-Prof. Dr.-Ing. habil. Josef BETTEN
RWTH Aachen University
Templergraben 55
D-52056 A a c h e n
betten@mmw.rwth-aachen.de
This worksheet is concerned with the relaxation functions (11.72 a, b) taken
from BETTEN's book: Creep Mechanics, Third Edition, Springer-Verlag 2008.
In the following example, the parameters b = B = [1/2, 1] have been assumed.
The solid lines in this Figure refer to the equation S(t, B), while the dashdot lines
represent the relaxation function s(t, b). The difference between these two functions
(11.72a,b) in a range of t = [0, 2], also called the distance of the two relaxation
functions, can be expressed by the L-two-error norm defined as:
For example, for b = B = 1 on [0..1], [1..2], and [0..2] we calculate the following
values:
These distances are based upon the assumption b = B. However, one can find an optimal
parameter b[opt] as a function of B , so that the simple function s(t, b) can be considered
as the best "best approximation" to the sqrt(t)-law S(t, B). Thus, one can use the simple
evolutional equation
instead of the evolutional equation
based upon the sqrt(t)-law S(t, B).
One can find an optimal parameter b[opt] by minimizing the L-two-norm,
as shown in the following MAPLE worksheet:
Minimizing the integral, we arrive at the following results:
From these data we find a cubic parabola by using the least-squares method:
The next Figure represents the relaxation curves S(t, B) and s(t, b) for the
parameter combinations {[b = B = 2], [b = 3.04, B = 2]} and {[b = B = 1/2], [b = 0.45, B = 1/2]}
In this Figure the thick "solid lines" refer to the parameter B = [2, 1/2], the thin
"solid lines" represent b = B , and the "dashdot lines" are the approximations characterized by b[opt] = [3.04, 0.45].
The above mentioned parameter combinations yield to the following L-two-norms:
Appendix
A full-length output of the calculation of the optimal parameter b[opt] as a function
of the first parameter B is listed in the following Worksheet:
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