Hi, do you have the reference of the Al-RUB model?

I don't know what is it the Al-RUB model.

yes, i have the AL-rub model, used for the analysis of nano indentation data,

for your convenience, i am attaching the original paper. thanks

Sorry for the late answer...

I never used this model but I understand as follow.

The fit parameters are determined in two different ways for fig. 3 and fig 4.

In fig. 3 H0 was taken as the limit of hardness at high indentation depths. Hy is obtained by using Tabor's law, Hy = k*sy where sy was obtained by uniaxial traction experiments and k by taking a literature value for this kind of materials. \beta is chosen such as (H-Hy/H0-Hy)^\beta give the best alignments of points on a straight line. This means, in my opinion, that it has been changed "manually" until you obtain the best alignment of points. h* was obtained as the slope of the straight line by effectuating a linear regression (the slope is h*^(\beta/2)).

In fig. 4, in my opinion, the author simply effectuated a non linear least square optimization of experimental points on equation (19) to obtain \beta, Hy, H0 and h* as the parameters of the fit (non linear fitting using any software like Origin, Matlab or Python)

http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit

THANKSA. Benedetto,

i was waiting for your reply very anaxiously, Thanks for you kind information. 

Please go through below links. These articles might be helpful. 

http://onlinelibrary.wiley.com/doi/10.1002/app.43532/full

http://www.sciencedirect.com/science/article/pii/S0032386116301227