Issue 33

J. Fan et alii, Frattura ed Integrità Strutturale, 33 (2015) 463-470; DOI: 10.3221/IGF-ESIS.33.51 467 For the same situation, the exact u ( x , a )-solution was presented as [3]:   2 2 0 , 4 ln cos cos cos ln cos 2 2 2 2 Hu x a x x a a b b b b b                                                     (19a) Fig.1(a) presents the variations of the dimensionless crack face displacements Hu ( x , a )/( σ 0 b) for collinear cracks in an infinite plate, respectively, determined by Eq. (17a), (18a) and (19a). It is clearly shown that if the values of a / b ≤0.5, values of Hu ( x , a )/( σ 0 b) calculated through Eq. (17a) and (18a) are in good agreement with the exact u ( x , a )-solutions given by eqn. (19a). For this situation, the maximum difference between the results by Eq. (18a) and (19a) is about 0.359% occurred at a / b =0.5 and x / a =0.95; but the maximum difference between Eq. (17a) and (19a) is only 0.0071% when a / b =0.5 and x / a =0. Once the values of a / b ≥0.7, the differences between the crack face displacements by Eq. (18a) and (19a) increase sharply from 1.9% ( a / b =0.7 and x / a =0.95) to 17.06% ( a / b =0.95 and x / a =0.95). But, the differences between Eq. (17a) and (19a) are still in the small range of 0.092% to 5.1%. Therefore, it is concluded that the present expression of the crack face displacement for collinear cracks can give much better u ( x , a )-solutions than that calculated by eqn. (18a) since the higher order term √[1-(x/a) 2 ] is taken into account. Fig.1(b) gives the relationship between Hu ( x , a )/( σ 0 b) and a / b -values for collinear cracks according to eqn.(17a). It is then seen that the crack face displacements increase with the increasing values of a / b . Figure1 : Comparisons of the dimensionless crack face displacements for collinear cracks (a) Hu ( x , a )/( σ 0 b) versus x / a ; (b) Hu ( x , a )/( σ 0 b) versus a / b . According to Eq. (17a), (18a) and (19a), the dimensionless partial derivatives of u(x, a) for an array of collinear cracks are, respectively, derived as (17b), (18b) and (19b) below: 2 2 2 2 2 2 0 2 2 2 2 2 tan ( , ) 8 2 2 2 1 ln tan 1 2 1 8 tan 1 1 tan 2 2 2 1 1 tan 2 2 b a H u x a x x bx a a b a a a a a b x a a a x a b b a b x x a a a a b b                                                                                                                  3/2 2 2 3/2 2 2 2 32 tan 3 2 tan 1 tan 4 2 2 2 2 1 ln tan 1 2 3 3 tan 4 2 2 b a a b a a a x b a b b b a b a a a b b                                                                                                               (17b)