The Weibull function is a stretched exponential function that is often used for reliability and failure analysis using mechanical properties data. The equation is difficult to include in this reply, but readily available through Wikipedia. The Weibull function/distribution is available in Excel with some guides on its use and comparisons in Excel help and through other sources such as:

http://www.qualitydigest.com/magazine/1999/jan/article/using-microsoft-excel-weibull-analysis.html

and through: weibull.com/basics

Another function that has the same equation is called: Kohlrausch (stretched exponential) or Kohlrausch-Williams-Watts (KWW) function, originally derived for alternating voltage dielectric data.

I have not specifically answered your question about statistical validation, however this is available in the links provided. Weibull is used for the task you have rather than bimodal distribution.

bi is 2, as in bipedal, bisexual, bicycle, binoculars, bifocals, etc... tri would be three, as in trimodal, tripod, tricycle...

thank you pr Lawrence for your answer, now i request another answer for quatromodale

I write to add to or clarity my answer about Weibull or stretched exponential functions. The exponent (often given the symbol k or beta in the Weibull equation) is called the shape parameter, or skewness. The skewness or stretching can be due to two (or more) overlapped mechanisms, maybe defects; one with a faster time, lower stress or strain, while the other(s) has a longer time. The two (or more) events are both occurring during the same test. Some deformation modes only become available when others are underway. There are several distributions that are used for this analysis, however Weibull is common for mechanical tests.

If the defects cause events that are independent of each other then the skewness can give two peaks in the response, a bimodal distribution of data (break stress as a mechanical property). If so a superposition of two distributions could be used to separate their contributions. However in a mechanical test two defects or structures with several failure modes are more likely to give a skewed distribution, since the modes are not independent. As one part of the structure yields or fails, stress is transferred to remaining parts until the specimen breaks. If defects can be classified as bi-, tri-, tetra- … poly- then you may still observe a Weibull distribution of failure data from a mechanical test.

The number of parameters increase with distribution type from normal (Gaussian, statistically random so data are symmetrically distributed about a mean), Weibull (add skewness), Poisson (skewness and Kurtosis or sharpness), superimposed distributions will have a separate set of parameters for each distribution). What a challenging question you have posed, with answers of increasing speculation. 

Thank you dear colleague

The problem arises on the level of the tests of validation

Generally tests of Khé2, least square, Smirnov Kolmogorov etc

These tests validate fracture tensile for bimodal distributions with 2 and three parameters. Thus which are the criteria of precision of the results?

Quattro or Cuatro is spanish for 4, so although the word Quatromodal does not exist, if it did it would suggest a distribution with 4 modes.

dear Lawrence Margulies

please, i request if there are software with I can confirm my results

Dear Mohamed

Thank you

I research a software more power than easyfit, for exemple like Weibull 7