Issue 31

J. Xavier et alii, Frattura ed Integrità Strutturale, 31 (2015) 13-22; DOI: 10.3221/IGF-ESIS.31.02 15 in which / C P   is the compliance. From the Timoshenko beam theory and Castigliano theorem an expression for the compliance of the DCB specimen can be obtained. This equation can be solved for the flexural modulus ( f E ) using an initial compliance 0 ( ) C and the corrected initial crack length 0 ( ) a   as [4]     1 3 0 0 0 3 LR 12 8 5 f a a E C BhG Bh              (2) where  accounts for root rotation effects and can be determined from finite element analysis [4], and G LR is the shear moduli of the material. The CBBM is based on eq a , which is considered to account for the FPZ effect at the crack tip as given by: eq FPZ a a a      [7, 10]. Finally, the application of CBBM to the DCB test yield the following expression for the strain energy release rate in mode I (resistance or R –curve) [4] 2 2 eq I 2 2 LR 2 6 1 5 f a P G B h E h G           (3) It is worth noting that this procedure is less sensitive to experimental errors. This is supported by the fact that the measurement of the crack length during the fracture test is not required. Besides, the inherent elastic properties variability among specimens is taken into account by computing a flexural modulus for each P   curve Eq. (2). In mode I loading, strain energy release rate ( I G ) and CTOD ( I w ) can be related by the following expression [9] I I I I I ( )d w o G w w    (4) The cohesive law ( I I f( ) w   ) can then be directly obtained by differentiating the above equation I I I I ( ) G w w     (5) This data reduction scheme, however, requires the accurate evaluation of the I I f( ) G w  relationship. Moreover, a suitable differentiation algorithm must be used to avoid noise amplification in the reconstruction of the constitutive cohesive law. In order to solve Eq. (5), it is proposed here to fit the I I G w  data by a continuous function described by the following expression (logistic function) 1 2 I 2 I I,0 1 ( / ) p A A G A w w     (6) where 1 A , 2 A , p and I,0 w are constants to be determined in least-square sense. In this function, the 2 A parameter must provide an estimation of the critical strain energy release rate: I 2 I Ic lim w A G G    . Figure 2 : Finite element model of the DCB test (cohesive law, mesh and boundary conditions).