PSI - Issue 2_B

Moslem Shahverdi et al. / Procedia Structural Integrity 2 (2016) 1886–1893 Shahverdi et al./ Structural Integrity Procedia 00 (2016) 000–000

1888

3

P  

P

C (2) where  P is the load-point displacement. If a polynomial function as Eq. (3) is used for fitting compliance-crack length curves, then the total strain energy release rate can be calculated by Eq. (4). 3 0 C C ma   (3) 2 2 3 2 G mP a B  (4) The ECM method can be used for mode partition in symmetric crack propagation where it is possible to determine the Modes I and II components of displacement along with the Modes I and II components of load. However, ECM cannot be used for the mode partitioning of mixed-mode results as is the case for the asymmetric crack propagation. 2.2. Extended global method Williams (Williams 1988) developed beam theory based equations for calculating the energy release rate from the values of bending moments and loads in a cracked laminate. In the present work, the equations have been modified in order to solve a crack propagation problem in which the crack is asymmetric and lies between two different orthotropic layers under the bending moments M 1 and M 2 , as shown in Figure 1. According to linear-elastic analysis, the total strain energy release rate is: 2 2 2 1 2 1 2 2 3 3 3 1 1 2 2 1 2 ( ) 6 ( ) M M M M G B E h E h E h h            (5)

Figure 1. Schematic illustration of an asymmetric crack in a composite joint subject to bending moments where the bending moments (assumed positive when counterclockwise) are evaluated at a section of the specimen surrounding the crack tip. According to the “global method” pure Mode I exists when symmetric moments act on the joint arms, i.e. M 1 = M I and M 2 = – M I , and pure Mode II requires equal curvature of both arms, i.e. M 1 = M II and M 2 = ψ M II . Furthermore,  is defined based on the ratio of the joint arm thicknesses h 1 and h 2 . However, because the curvature of the orthotropic-layered arms depends on the bending stiffness rather than just the thickness, in the present work this definition is replaced by the equivalent bending stiffness ratio:

( ) ( ) EI EI

eq2 eq1

(6)

 

Therefore, under mixed-Mode I/II loadings are

1 I II M M M   and

(7)

2 I II M M M   

Substitution of Eqs. (7) into Eq. (5) leads to the partition of G into G I and G II as: 2 I I eq1 1 2 ( ) G M B EI           and 2 2 II II eq1 (1 (1 ) ) 2 ( ) G M B EI       

(8)

With

( ) ( ) EI EI

eq1 eq

(9)

 