Issue 33

J. Fan et alii, Frattura ed Integrità Strutturale, 33 (2015) 463-470; DOI: 10.3221/IGF-ESIS.33.51 465 Based on the previous works [3], the crack face displacement u ( x , a ) is assumed to be dependent on x and a :   * * 0 0 1 1 , ( ) ( ) x x u x a u a u u a u a a               (6) where u 0 ( a ) and u 1 ( a ) are the unknown functions relative to the half crack length a ;   * 0 / u x a describes the deformation of the center crack;   * 1 / u x a is assumed to be the higher order of   * 0 / u x a . To approximately determine the crack face displacements u ( x , a ) across the whole crack line of a central through crack, u ( x , a ) should also follow the given criterions below [3, 5]: (I) exhibiting proper limiting behavior near the crack tip; (II) deforming as a shape of the ellipse when the cracked solid is subjected to a remotely uniform stress field; (III) demonstrating the consistent behavior for the small crack; (IV)   0 , | 0 x u x a a     . If an infinite solid with a central through crack of the length 2 a is subjected to a remotely uniform tensile stress filed σ 0 , the crack face displacements and the SIF are, respectively, presented as:       0 0 2 , 2 and y u x a a K a a H          (7) where ξ =0 is the coordinate with its origin at one of the crack tips, and ξ =2 a is the other crack tip. Based on eqn. (7), if a finite solid, with a center crack of the length 2 a , is also undergoing a remotely uniform tensile stress σ 0 , the crack face displacements and the SIF are able to be written as:            0 0 2 , and y f a u x a a x a x K a f a a H         (8) where x is the coordinate with its origin at the crack center; x =± a are set to be the two crack tips of the crack; 2 b is the width of the elastic cracked solid; f ( a ) is defined as the correction function which is dependent on the crack geometry and the size of the cracked solid. In the criterion (II), it is assumed that the crack face displacement of the center crack deforms as a shape of the ellipse when the cracked body is subjected to a uniformly distributed stress field perpendicular to the crack line. So, the shape of the opened crack can be expressed as an elliptic function:   2 * 0 1 x x u a a         (9) As a consequence, from Eq. (8) and (9) the first term in the right hand side of Eq. (6) is determined as: 2 * 0 0 0 2 ( ) ( ) ( ) 1 f a a x x u a u a H a          (10) Here, the first term u ( x , a ) satisfies the criterions (I) and (II). So, the second term u ( x , a ) should make the full expression of u ( x , a ) satisfy all the four criterions. Upon that, * 1 x u a       is taken as a higher order of * 0 x u a       :   3/2 2 * 1 1 x u x a a             (11)