PSI - Issue 75

J. Filho et al. / Procedia Structural Integrity 75 (2025) 353–362 J.Filho, L. Wittevrongel, F. Pieron, P. Lava / Structural Integrity Procedia (2025) where is the series order, and are the rigid-body translations in x - and y -directions, R is the rigid-body rotation, ( ) and ( ) are polar functions defined by ( ) ( , ) = ⁄2 2 ( cos 2 − 2 cos( 2 −2) +{ 2 +(−1) }cos 2 ) , (3) ( ) ( , ) = ⁄2 2 ( sin 2 − 2 sin( 2 −2) +{ 2 −(−1) }sin 2 ) , (4) ( ) ( , ) = ⁄2 2 ( sin 2 + 2 sin( 2 −2) −{ 2 +(−1) }sin 2 ) and (5) ( ) ( , ) = ⁄2 2 (− cos 2 − 2 cos( 2 −2) +{ 2 −(−1) }cos 2 ) , (6) that are written in terms of the polar coordinates and as function of the crack-tip ( , ) , according to = √( − ) 2 − ( − ) 2 and (7) = tan −1 ( − ) ( − ) , (8) and can be written as function of the elastic modulus E and Poisson’s coefficient ν under plane stress conditions, which is appropriate for surface measurements, i.e. DIC, according to the respective formulations = 2(1 + ) and = 31 −+ . (9) An optimizer can be used to estimate the unknown Willliams’ coefficients ( i.e. SIFs, crack-tip position and rigid body translations and rotation), by minimizing the mean square difference between the measured and reconstructed displacement fields. The Mode I and Mode II SIFs can be extracted using the respective relations proposed by Irwin (1957), which are given by = ( 1) √2 and =− ( 1 ) √2 . (10) The equivalent J -Integral, which can be used to measure the energy release rate due to crack propagation (see Sec. 5), can also be estimated from SIFs considering linear elastic materials using the following equation = 2 ⁄ , (11) where 2 is the equivalent SIF, given by =√ 2 + 2 . (12) The Williams’ series expansion has been implemented in the MatchID post-processing module, allowing the identification of SIFs and crack-tip position. 5. J -Integral evaluation The strain energy release rate due to crack growth can be estimated by a two-dimensional path-independent contour integral criterion first introduced by Rice (1968). Later, Li et al. (1985) proposed an equivalent two-dimensional domain-independent integral as an alternative to complex paths. The path- and domain-independent J -Integrals can be calculated by means of the following equations 4

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