Issue 33

J. Fan et alii, Frattura ed Integrità Strutturale, 33 (2015) 463-470; DOI: 10.3221/IGF-ESIS.33.51 466 To make u ( x , a ) demonstrate the consistent behaviour for the small crack, u 1 ( a ) should have the characteristic properties: u 1 ( a )=0(1/ a ) if the half crack length a tends to be zero. Based on the above criterions, an approximate and simple expression of the crack face displacement is derived as:           3/2 2 2 * * 0 0 0 1 1 2 , 1 1 f a a g a x x x x u x a u a u u a u a a H a a a                                       (12) where g ( a ) is the only unknown function of the half crack length a . Substituting eqn. (12) into (5), and assuming the stress function σ ( x ) is equal to a constant value σ 0 , it leads to: 3/2 2 2 2 2 2 0 0 0 0 0 0 2 ( ) ( ) ( ) 1 1 a a a f a a g a x x f a ada H dx dx H a a a                                      (13) and solving Eq. (13), the unknown function g ( a ) is determined as:   2 2 0 0 0 ( ) 16 ( ) 8 ( ) 3 a g a f a ada f a a H      (14) Finally, substituting eqn. (14) into (12), the fully expression of the crack face displacements is derived as:   3/2 2 2 2 2 0 0 0 0 16 ( ) 8 ( ) 2 ( ) , 1 1 3 a f a ada f a a f a a x x u x a H a Ha a                             (15) R ESULTS AND DISCUSSIONS Calculations of u(x,a) and   , / u x a a   for collinear cracks o check the accuracy of the expression for u ( x , a ) of the central through crack, an array of collinear cracks in an infinite plate, subjected to a uniformly tensile stress field σ 0 , is taken into account. So, the correction function f ( a ) is: 2 ( ) tan 2 b a f a a b          (16) where 2 a is the full crack length; and 2 b is set as the distance between the two adjacent crack center lines. Substituting eqn. (16) into (15) and simplifying the expression, the dimensionless displacement is determined as:       2 0 3/2 2 0 , 2 2 tan 1 2 32 8 2 tan tan 1 3 2 3 2 a Hu x a a a x a b b b a a a x da a a b b b                                              (17a) Also, a generalized formula for the crack face displacements has been given by Wu and Carlsson [3]:   2 2 0 , 4 2 ln cos 1 2 Hu x a b a x b a b a                                     (18a) T