4 B.R. Davis et al. The unique contribution of the current implementation is the inclusion of the mode III energy release rate distribution along the crack front. The remaining terms in Eq. 1.8, excluding Eqs. 1.9 and 1.10, comprise a pure mode III component, 1 2 uIII T ıK ıai L uIII, and three modal-interaction coupling terms. Understanding the influence of the coupling-mode terms is critical in determining their respective contributions to the 3-D mixed-mode energy release rates. Consider the following coupling mode terms from Eq. 1.8: GI=II i D 1 `i uI T ıK ıai L uII ; (1.11) GII=III i D 1 `i uII T ıK ıai L uIII ; (1.12) GIII=I i D 1 `i uIII T ıK ıai L uI : (1.13) Using symmetric and anti-symmetric arguments, it can be shown and verified numerically that calculating the coupling terms in Eqs. 1.11 and 1.13 about a symmetric domain lead to a cancellation effect. The result is GI/IIi DGIII/Ii D0. However, Eq. 1.12 has an additive effect that results in GII/IIIi ¤0. The only contributing non-zero components to GII/IIIi are out-ofplane shear. Therefore, any addition to the total energy release rate fromGII/IIIi must be a factor of the out-of-plane mode III energy release rate: GIII i D 1 `i 1 2 uIII T ıK ıai L uIII CGII=III i : (1.14) With the individual mixed-mode terms determined and the modal-interaction coupling terms accounted for, the VCE total energy release is successfully decomposed, satisfying the following summation: Gi DGI i CGIIi CGIIIi : (1.15) 1.2.3 Numerical Examples In this section two verification analyses are presented to demonstrate the accuracy of the new, 3-D, mixed-mode energy release rate implementation using the VCE method. The mixed-mode VCE results are compared with analytical and Mintegral methods. Each model and crack front geometry is meshed using the FRANC3D software. The first problem considered a 45ı-inclined circular crack centrally embedded within a rectangular isotropic body, as shown in Fig. 1.1. The model geometry was appropriately sized to approximate crack behavior within an infinite body. Analytical expressions for the mixed-mode I/II/III stress intensity factors for an inclined penny crack under remote tension [23] are used as a reference solution. The second numerical example is a half-penny-shaped surface crack in an isotropic cylindrical specimen [24, 25]. Figure 1.2 shows the global geometry and loading conditions that induce mode II/III behavior along the crack front. The local, in-plane geometry of the crack front is depicted in Fig. 1.2. The VCE energy release rate results are compared with M-integral calculations using the FRANC3D software. Figures 1.3 and 1.4 display the mixed-mode and total energy release rate distributions calculated by the VCE implementation for each example. The results compare extremely well with the reference solutions. The average percent differences between the VCE and reference results are 0.11 % and 0.23 % for the inclined circular crack and the surfacecracked cylinder, respectively.
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