Issue 33

J. Fan et alii, Frattura ed Integrità Strutturale, 33 (2015) 463-470; DOI: 10.3221/IGF-ESIS.33.51 464 Nevertheless, the weight function is strongly dependent on the solution of the crack face displacement whose functional dependence on the crack is undetermined. Therefore, it is necessary to make appropriate assumption on the opened crack shape. The present paper is aimed at developing an approximate and simple expression of the crack face displacement for collinear cracks and center crack subjected to mode I loading conditions. W EIGHT FUNCTION METHOD he weight function method has been widely applied to determine the SIFs of cracked structures since it is able to take complex loading conditions into consideration. It has been shown that, if the SIF K ( a ) (1) and the crack face displacement u (1) ( x , a ) of any linearly elastic cracked solid are known as functions of the crack length a for a symmetrical load system (1), then for the same cracked solid subjected to any other symmetrical load system (2), the SIF K ( a ) (2) can be obtained by the simple integration of the weight function h ( x , a ) and the stress function σ (2) ( x ): (2) (2) 0 ( ) ( ) ( , ) a K a x h x a dx     (1) where the weight function, independent of σ (2) ( x ), is defined as: (1) (1) ( , ) ( , ) ( ) u x a H h x a K a a    (2) In Eq. (1) and (2), a is the half or full crack length for edge cracks and center cracks, respectively; H is a material constant, H = E for plane stress condition and H = E /(1- v 2 ) for plane strain condition with E the Young’s modulus and v the Poisson’s ratio; K ( a ) (1) and u (1) ( x , a ) are, respectively, the known reference stress intensity factor and the crack face displacement in mode I loading conditions for the known load system (1); and σ (2) ( x ) is the stress distribution function across the plane of the crack in the crack free solid subjected to the load system (2). From Eq. (1), it is known that if the weight function is set to be a known function for a particular crack geometry, the SIF for any stress distribution is able to be calculated by integrating Eq. (1). Therefore, the main problem is to exactly determine the weight function h ( x , a ) and the corresponding partial derivative (1) ( , ) / u x a a   of the crack face displacement. For a reference SIF K ( a ) (1) , it is possibly available in the SIF handbooks for a wide range of crack configurations and loading conditions. However, there are often no detailed databases about the crack face displacement u (1) ( x , a ), and it is difficult to rigorously calculate u (1) ( x , a ) since the functional dependence of u (1) ( x , a ) on both x and a is undetermined. Thus, approximate u (1) ( x , a )-solutions across the crack line of a cracked solid are of primary importance for determining the weight function h ( x , a ) and stress intensity factor K ( a ) (2) [5]. C RACK OPENING DISPLACEMENTS ACROSS THE CENTER CRACK n Eq. (1) and (2), if K ( a ) (1) = K ( a ) (2) = K (a), and σ (1) ( x )= σ (2) ( x )= σ ( x ). Then, substituting (2) into (1), it leads to: 2 0 ( , ) ( ) ( ) a u x a K a H x dx a       (3) So, the crack face displacement u ( x , a ) in eqn. (3) is the only unknown function dependent on the crack line x and half of the crack length a . As is well known that u ( x = a , a )=0 for any crack tip. As a result, Eq. (3) becomes: 2 0 ( ) ( ) ( , ) a K a H x u x a dx a       (4) Integrating Eq. (4) over the half crack length a , we have: 2 0 0 ( ) ( ) ( , ) a a K a da H x u x a dx      (5) T I