PSI - Issue 32

Valery Vasiliev et al. / Procedia Structural Integrity 32 (2021) 124–130 Vasiliev and Lurie / Structural Integrity Procedia 00 (2019) 000–000

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plays a fundamental role, because the value of s obtained for the plates made of the given material is assumed to be independent of the crack length, which is confirmed experimentally. 1. As an example, first consider a bendable rectangular plate b L  ( L is the length, and b is the width) with a side crack at three-point bending. The plate is loaded by a force applied in the middle, between the supports. The stress near the end of the crack is calculated by the formula 2 3 / (2 ) PL hb   , where h is the thickness of the plate. Introducing the stress concentration factor / yy k     , the following expression for the breaking load can be obtained 2 2 / (3 ), / u u u u u P b h k L k      , where u  is the ultimate strength of the plate material. Three plates were tested with the ultimate strength of the plate length. The first plate with parameters 60.2 , b mm  2.96 h mm  and 17 l mm  failed at a force 230 P N  . In this case, we get 14.5 k   . Using the theoretical dependence ( ) k k     , we find 1470   and / 0.0116 s l mm    . For the second plate with parameters 60.2 b mm  , 2.97 h mm  , and 10 l mm  we get 860   . Further, assuming that 0.0116 s mm  and 860   using the theoretical dependence ( ) k k     , we find 11.1 k   and 301 u P N  . Similarly, for the third plate with parameters 60.4 b mm  , 2.96 h mm  and 22 l mm  at 0.0116 s mm  , we find 1900   , 16.4 k   and 203 u P N  . Comparison of the results obtained for the ultimate load with the experimental ones indicates a high prediction accuracy. 2. Let us consider shortly the application of the concept of stress concentration for the plastic materials. Here the stress factor must be calculated using the magnitude of the stress intensity An experimental study was carried out on plates of aluminum alloy, brass and steel. A crack was defined as a side cut in a strip that was loaded according to a three-point bending test. As an example, we briefly discuss the experiment and calculation for aluminum alloy plates. For a plate with a crack 5mm long, the experimentally established proportionality limit is 65   MPa. Thus, the stress concentration coefficient / 1.154 u k      and according to (6) and (8) we have 5.5   mm and then / 0.91 s l    . Further, assuming that the value of 0.91 s  mm is unchanged for the material under consideration, we give a forecast of the proportionality limit for the strip with different long cracks. For a plate with a crack of length 10 l  mm at mm, we have 11   , 1.56 k   and predict using ultimate stresses (proportionality limit) 48.1   MPa. For 15 l  mm, we get 16.5   , 1.93 k   and 38.9   MPa. Finally, for 20 l  mm, we also find 22   , 2.22 k   and 33.8   MPa. As a result, it turned out that these values of the ultimate stresses obtained theoretically coincide with experimental accuracy with high accuracy (the correlation coefficient is less than 6%). 3. To assess the strength of an orthotropic plate with a crack, we use the quadratic strength criterion     2 2 / / 1 m m x x y y s s     , where x s and у s are the ultimate tensile strength in the direction of the 0 x and 0 y axes, respectively. The stresses acting in the plate near the crack top under tension are calculated using the formulas (4), (5) and are represented as follows m x x o k    and 0 m y y k    , where x k and y k are the stress concentration factors in the vicinity of the crack. Using the written criterion, the following dependence can be obtained for the ultimate stress tensile of the plate:     2 1/ 2 [ / / ] x x y y k s k s     . With the help of the obtained regular solutions (6) and (4) for the considered plate, theoretical dependences ( ) x k  and ( ) x k  on the parameter  and, finally, the dependence ( )   on the parameter  are constructed. Further, the experimental values of the breaking stresses obtained for a specific plate with a given crack length are used. Using the theoretical dependence ( )     , we find the corresponding parameter value  and the scale parameter / s c mm   . The main idea of the further calculation is that the parameter s is considered to be independent of the crack length. Then, for a plate with a crack of a different length, we obtain the   max max i   2 2   ˆ   ˆ   0 0 ˆ r r l  , / , / l s m xx yy xx yy     r r     (8) instead of max m yy    as it was assumed for the brittle materials.