13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- fibers, since the crack is traction free; and 2) it contains the surface energy term of the crack faces: 0 tr cr WZ PZ G V . (5) Finally, the total Gibbs potential can be expressed as follows: 0 0 ( , ; , , ) ( ; , ) ( , ) AZ tr cr AZ tr cr cr tot dr AZ dr WZ G G L W V V L . (6) We compute the volume VAZ of AZ material in the reference state by integrating the AZ width, which is expressed in terms of total COD in the specimen with PZ cutoff (Eq. (2)): ( ) ; , , ; , 1 cr cr L L eff AZ cr AZ eff o tot dr B V B x dx x W dx . (7) The total COD tot in Eq. (7) is obtained as the superposition of COD ( , )L due to remote load and crack “closure” ( ; , ) cr AZ dr dr dr by the traction dr acting along AZ boundary as shown in Figure 8. The thermodynamic forces XAZ and XCR responsible for AZ (ℓ ) and crack (ℓ ) growth are expressed as the partial derivative of total Gibbs potential Gtot with respect to corresponding parameters ℓ and ℓ , since they are the only independent geometrical characteristics of CL: ; CR AZ tot tot cr AZ eff eff G G X X B B . (8) The derivative with respect to crack length ℓ of the first term of Gtot decomposition Eq. (6) is zero, when CL length L is fixed. The derivative of the second and third terms of Eq. (6) is the energy release rate (ERR) 1 cr J due to crack extension into PZ, when PZ is stationary (L is constant); and the derivative of the fourth term is simply 2. Thus, the thermodynamic force reciprocal to crack length is: 1 2 CR cr X J . (9) We compute the thermodynamic force reciprocal to AZ in a similar manner considering the specimen with PZ cutoff as a linear elastic solid. The derivative of the first term of decomposition Eq. (6) gives the energy release rate in the specimen with crack of length L loaded by remote load and traction dr applied on ℓ part of : 2 0 1 ' tot AZ eff G K G E B , (10) where ′ is the plain strain Young’s modulus and Ktot is the sum of SIFs due to remote load and traction along the AZ boundary: ( , , ) ( , , , ) AZ tot dr dr K K LW K W L , (11) The derivative of the (Eq. (7)) with respect to ℓ is expressed as the following integral: 1 ; , , ; , 1 cr L AZ cr AZ tot dr AZ AZ eff V x W F dx B . (12) It is evaluated numerically and found to be approximately equal to COD at the root of PZ with a reasonable accuracy (any estimation of accuracy?):
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