Uniaxial tensile tests were carried on a ceramic. The following strength values

zachutnat4o 2021-11-20 Answered
Uniaxial tensile tests were carried on a ceramic. The following strength values were obtained:
310, 380, 400, 415, 448, 455, 475, 489, 500, and 502.
a. Calculate the probability of survival or probability of failure for these data. What is the Weibull modulus? Also calculate the characteristic strength.
b. What is the tensile strength at \(\displaystyle{50}\%\) survival probability? If the volume was increased three fold, what would be the new strength value at \(\displaystyle{50}\%\) survival probability?

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Expert Answer

Xyle1991
Answered 2021-11-21 Author has 1726 answers

If N samples are tested, we rank their strength in ascending order to obtain the failure probability.
\(\displaystyle{F}{\left({V}\right)}={\frac{{{i}}}{{{N}+{i}}}}\)
\(\begin{array}{|c|c|}\hline Test&F\left(V\right)&Strenght\left(\sigma\right) \\ \hline1&0.05&310\\ \hline2&0.1&380\\ \hline3&0.14&400\\ \hline4&0.19&415\\ \hline5&0.24&448\\ \hline6&0.29&455\\ \hline7&0.33&475\\ \hline8&0.38&489\\ \hline9&0.43&500\\ \hline10&0.48&502\\ \hline \end{array}\)

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Steven Arredondo
Answered 2021-11-22 Author has 1214 answers
Now plowing \(\displaystyle{\ln{{\ln{{\left[{\frac{{{1}}}{{{1}-{F}{\left({V}\right)}}}}\right]}}}}}\)
\(\displaystyle={m}{\left({l}{g}{\left\lbrace\sigma\right\rbrace}-{l}{g}{\left\lbrace\sigma_{{{0}}}\right\rbrace}\right)}\)
\(\displaystyle{m}={5.9}\) and \(\displaystyle\sigma_{{{0}}}={476}\)
\(\displaystyle{P}{\left({V}\right)}={\exp{{\left[-{\frac{{{V}}}{{{V}_{{{0}}}}}}{\left({\frac{{\sigma}}{{\sigma_{{{0}}}}}}\right)}^{{{m}}}\right]}}}\)
\(\displaystyle{0.5}={\exp{{\left[-{\frac{{{60}}}{{{20}}}}{\left({\frac{{\sigma}}{{{476}}}}\right)}^{{{6}}}\right]}}}\)
\(\displaystyle\sigma={373}{M}{P}{a}\)
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