Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 375   2 1 n p k w w   (3) whereas the continuously distributed shear tractions p t are consistent with the assumption of perfect connection in the longitudinal x -direction imposing that 1 2 2 1 1 2 0 2 2 h h u u       (4) It is worthwhile noting that the interface condition expressed by Eqn. (4) makes the axial and bending problems of the two elastically connected semi-infinite beams coupled and it needs to be imposed in order to analyze the elementary end loading condition with N 0 ≠ 0 and related compensating moment, which would otherwise generate unrealistic sliding displacements at the interface in asymmetric specimens. In order to solve the problem, it is convenient to rearrange the basic equations described above. Such a rearrangement, based on differentiating Eqn. (3) four times and Eqn. (4) three times with respect to x , leads to the following two coupled linear differential equations with constant coefficients for the unknowns p n and p t : " ' ' 1 ˆ ˆ ˆ 0 ˆ ˆ 0 iv n n n t t n p p p a bp cp gp k c            (5) having defined, according to Eqn. (1), 1 2 2 1 2 1 2 ˆ w x s xz s xz k E h h a k G h h G h       , 2 3 3 1 2 3 3 4 1 2 1 24 ( 1) 12( ) ˆ w x k h h b k E h h h         (6a) 2 2 2 1 2 2 3 1 2 1 12 (1 ) ˆ 6 w x k h h c k E h h h              and 1 2 1 2 1 4( ) 4(1 ) ˆ x x h h g E h h E h      (6b) where 1 2 h h   , z x E E   and / (2 ) x z xz xz zx E E G      (6c) define relative thickness and orthotropy ratios in the layers and 2( / / ) x xz xz E G      . It is worthwhile noting that all the coefficients in Eqn. (6) are positive, except for the coefficient ˆ c which is positive only for h 2 > h 1 and vanishes in the case of symmetrical specimen with a mid-thickness interface ( h 1 = h 2 ). Furthermore, under the assumption of infinite shear stiffness of the layers ˆ 0 a  and Eqn. (5) so simplified is proved to govern the problem of the two elastically connected semi-infinite beams according to Euler-Bernoulli beam theory, for which the effects of the shear deformations are neglected. Appendix A presents this simplified formulation and a comparison with the model first proposed in [16] and recently reconsidered in [10] with reference to a DCB specimen. The solution of the coupled differential problem (5) requires further rearrangement of the equations: firstly, the second of Eqn. (5) is solved for the first derivative of the interfacial shear tractions; secondly, this result is substituted in the first of Eqn. (5) leading to the following fourth-order linear differential equation with constant coefficients for the interfacial normal tractions " ˆ ˆ 0 iv n n n p p a ep    (7) with 2 3 1 1 2 1 2 3 3 4 1 2 1 6 (1 ) ( ) ˆ ˆ ˆ ˆ 3 w x k h h e b g k c k E h h h            (8)