PSI - Issue 42

J. Lhonneur et al. / Procedia Structural Integrity 42 (2022) 513–521 J. Lhonneur et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 4. Mechanical responses.

The Weibull law is often used for representing the stochastic nature of the ultimate strength measured on fragile materials. In the Weibull statistical theory of strength, the stochastic nature of the failure behavior is explained by a weakest link effect. Let us consider that flaws are localized in the solicited sample volume, each having the same probability ( ) of failure for an imposed force less than . The sample survival probability ( ) for an imposed force is equal to the probability that each of the flaws would not fail for a force less than : ( ) = (1 − ( )) (1) As function is positive and less than 1 , it may be expressed as follows: ( ) = (− ( )) (2) where is a positive, not decreasing function which tends to zero when tends towards zero. Weibull proposed to use the following form for : ( ) = ( ) (3) where is a shape parameter and is a scale parameter. The probability ( ) of failure of a sample for a loading force less than is then expressed as: ( ) = 1 − ( ) = 1 − (− ( ) ) = 1 − (− ( ) ) (4) with an equivalent scale parameter depending on the solicited sample volume. Equation (4) may be rewritten as follow: (− (1 − ( ))) = ( ) − ( ) (5) Values of Weibull parameters and obtained for each type of samples by a linear regression are displayed in Error! Not a valid bookmark self-reference. . Table 2. Weibull parameters for the three types of samples.

Sample type Cement paste Cement-steel ( ) 20.1 2.35 × 10 −43 0.978 7.13 5.57 × 10 −16 0.957 Cement-silica 6.72 8.91 × 10 −16 0.989 2