PSI - Issue 52

Vitalijs Pavelko et al. / Procedia Structural Integrity 52 (2024) 382–390 Vitalijs Pavelko/ Structural Integrity Procedia 00 (2019) 000 – 000

388

7

stresses (Fig.4, orange curve). They are strongly decreasing in cross-sections close to a plane of destruction and cause cracks stopping in this zone (if such ones were in first stage of loading). 3) Further growth of other cracks in the zone of PET where the effect of first destruction is small. In Fig.6 there is shown that in cross-sections >5 mm the difference between stresses before first destruction and after it is small and smaller as x larger. 4) Next destruction in cross-section close to one where is maximum stress ( =12.5 mm, orange curve in Fig.4) and the transition to configuration c) (yellow curve, Fig.4). 5) If at further loading the process of crushing continues (transition to configuration e), then next two cracks should be located near cross-sections with maximum stress of configuration c). Because fatigue strength is random characteristics, then at the same duration of cyclical load the number of cracks is random also. As it showed above (table 1) in the fatigue test of a panel equipped by ten piezoelectric transducers, after 60 thousand cycles of load, from 1 to 8 fatigue cracks were detected in eight transducers, and two transducers did not have any visible damage. 4.3. Distribution function of transducer fatigue strength. Estimation of parameters. The mean tensile stress of transducer which acts in the cross-section of crack initiation and corresponds to maximum of the cyclic load at intact state is accepted as a measure of fatigue strength of constrained transducer (FSCT) at the base test of 60 kilocycles. Here it is accepted that FSCT is a normal random variable with two parameters of distribution: μ mean and θ standard deviation. For estimation of these parameters the descriptive statistics is obtained from sample data in the following way. Each detected crack is matched with the value of the mentioned mean tensile stress in the corresponding cross-section. The largest number of cracks found in a single transducer after testing is 8. In other transducers, the number of cracks is less, or they are not finding out. Thus, the potential number of cracks in this experiment is at least 80, and 29 cracks were found during defect detection. It means that, so called, the uncensored statistical sample is obtained. The maximum likelihood method is used to obtain statistical estimates of the parameters of the distribution of fatigue strength of the PET built into the structure. Under considered conditions, the likelihood function ( , ) should be presented in the following form: ( , ) = [∏ 1 √2 =1 − 1 2 ( − ) 2 ] [1 − 1 √2 ∫ −12( − ) 2 ∗ −∞ ] (1) where is the value of fatigue strength confirming by the fact of occurrence of crack , =29 is the total amount of cracks in all PET, =51 is amount of potentially critical cross-sections in those destruction is not observed and ∗ =67 MPa is the value of stress in critical cross-sections. After simple mathematical transformations equation (1) can be presented as follows: ( , ) = {∑ [log 1 √2 + log − 1 2 ( − ) 2 ] =1 + log [1 − 1 √2 ∫ −12( − ) 2 ∗ −∞ ]} (2) Necessary conditions of extreme of the likelihood function ( , ) are: ( , ) =0, ( , ) =0 (3) Those conditions give a system of two transcendent equations which solution requires numerical procedure. Therefore, the simple method of direct calculation of the likelihood function ( , ) and defining the optimal solution was accepted.