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. 

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In the following example, the parameters  b = B = [1/2, 1]  have been assumed. 

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  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: 

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For example, for  b = B = 1  on  [0..1], [1..2], and [0..2]  we calculate the following  

values:  

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  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    

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instead of  the evolutional equation 

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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: 

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Minimizing the integral, we arrive at the following results: 

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From these data we find a cubic parabola by using the least-squares method:  

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  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]} 

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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: 

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                                                                   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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