4. 4 Maximum likelihood method

4.1. 4.1 Principle

Let’s note \({p}_{f}({\sigma }_{w})\) the probability density associated with the cumulative rupture probability \({P}_{f}({\sigma }_{w})\):

\({p}_{f}({\sigma }_{w})=\frac{m}{{\sigma }_{u}}{(\frac{{s}_{w}}{{\sigma }_{u}})}^{m-1}\text{exp}\left[-{(\frac{{\sigma }_{w}}{{\sigma }_{u}})}^{m}\right]\)

The quantity \({p}_{f}({\sigma }_{w})d{\sigma }_{W}\) is equal to the probability of breaking a test piece subjected to a stress corresponding to a stress of WEIBULL included in the interval \(\left[{\sigma }_{W},{\sigma }_{W}+d{\sigma }_{W}\right]\). The probability that all the test tubes in the base have broken is therefore:

\(p(m,{\sigma }_{u})d{\sigma }_{w}=\prod _{i}{p}_{f}({\sigma }_{W}^{i})d{\sigma }_{w}\), eq 4.1-1

p being the likelihood function. The maximum likelihood method then consists in choosing the parameters of the WEIBULL model so that the likelihood function defined by [éq 4.1-1] (in practice rather its natural logarithm) is maximum.

4.2. 4.2 Resolution

An iterative method is used again. Again, at iteration \((k)\), (\({m}_{k},{\sigma }_{u(k)}\)) as well as the \({\sigma }_{W(k)}^{i}\) are known. For these fixed WEIBULL constraint values, the maximization of \(\text{Log}(p)\) leads to a new pair (\({m}_{k+1},{\sigma }_{u(k+1)}\)) given by:

\(f({m}_{k+1})=\frac{N}{{m}_{k+1}}+\sum _{i=1}^{i=N}\text{Log}({\mathrm{\sigma }}_{W(k)}^{i})-N\frac{\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W(k)}^{i}{)}^{{m}_{k+1}}\text{Log}({\mathrm{\sigma }}_{W(k)}^{i})}{\sum _{i=1}^{i=N}{({\mathrm{\sigma }}_{W(k)}^{i})}^{{m}_{k+1}}}=0\) eq 4.2-1

\({\sigma }_{(k+1)}=\sqrt[{m}_{k+1}]{\frac{1}{N}\sum _{i=1}^{i=N}({\sigma }_{W(k)}^{i}{)}^{{m}_{k+1}}}\). eq 4.2-2

At each step, the resolution of [éq 4.2-1] can be achieved using Newton’s method, the \(f(m)\) gradient being given by:

\(\frac{\text{df}}{\text{dm}}(m)\text{=-}N\left(\frac{1}{{m}^{2}}+\frac{\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}{\text{Log}}^{2}({\mathrm{\sigma }}_{W}^{i})\right)\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}\right)-{\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}\text{Log}({\mathrm{\sigma }}_{W}^{i})\right)}^{2}}{{\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}\right)}^{2}}\right)\).

Note:

If \(m\) is set, \({\sigma }_{u(k+1)}\) is given by [4.2-2]. On the other hand, if \({\sigma }_{u}\) is fixed, \({m}_{k+1}\) is no longer a solution to [4.2-1] but to:

\(f({m}_{k+1})=\frac{N}{{m}_{k+1}}+\sum _{i=1}^{i=N}\text{Log}(\frac{{\sigma }_{W(k)}^{i}}{{\sigma }_{u}})(1-(\frac{{\sigma }_{W(k)}^{i}}{{\sigma }_{u}}{)}^{{m}_{k+1}})=0\) .

This equation can be solved again using Newton’s method, the gradient now being given by:

\(\frac{\text{df}}{\text{dm}}(m)\text{=-}\frac{N}{{m}^{2}}-\sum _{i=1}^{i=N}(\frac{{\sigma }_{W}^{i}}{{\sigma }_{u}}{)}^{m}{\text{Log}}^{2}(\frac{{\sigma }_{W}^{i}}{{\sigma }_{u}})\) .