13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 0 1 0 0 0 0 2 3 (1 ) σ σ σ α δ σ σ ν σ σ ν ε ε ij n e ij kk ij ij s − + = + − , (3) where ν is the Poisson’s ratio, δij is the Kronecker delta, sij is the deviatoric stress, and σe is the von Mises effective stress defined as 3 /2 ij ij e s s = σ . The asymptotic stress field can then be expressed as ( ) ( ) ( ) + + = n L r n A L r n A L r A ij s ij s ij s ij , ~ , ~ , ~ (3) 2 2 (2) 2 (1) 1 0 3 2 1 θ σ θ σ θ σ σ σ , (4) The angular stress functions and the stress power exponents are functions solely of the hardening exponent and are independent of the applied load and any material properties. The characteristic length, L, in Equation 4 is typically assigned one of the primary dimensions of the specimen under investigation; possibilities include the crack length, the specimen width, the specimen thickness, or even unity. For the current study, L = 1 mm for all cases. The parameters A1 and s1 from Equation 4 are related to the HRR solution (Eq. 1) by 1 0 0 1 s nI L J A − = σ αε , 1 1 1 + =− n s , (5) Comparing Eq. 4 and 1, it can be seen that the first term of the two-parameter equation (Eq. 4) is the same as the HRR solution (Eq. 1). Further, the additional stress power exponents can be calculated using s3 = 2s2 – s1 for n ≥ 3. Thus, the final stress power exponent, s2, can be calculated numerically. An apparent application of the J-A2 method is to use the two parameters to predict failure of flawed components. Chao et al. [9] described the development of Material Failure Curves by plotting J-integral values versus the absolute value of A2 (J vs. |A2|), as A2 is always a negative value. Using a modified version of Equation 4, ( ) ( ) ( ) + + = n L r n A L r n A L r A ij s c ij s c ij s c c , ~ , ~ , ~ (3) 2 2 (2) 2 (1) 1 0 3 2 1 θ σ θ σ θ σ σ σ , (6) the aforementioned critical stress (σc) and corresponding critical radial distance (rc) from the RKR model are substituted and J and A2 can be determined and plotted. According to the RKR model, the location where failure initiates, rc, occurs in the range of 2-4 grain diameters ahead of the crack. This equates to 0.12mm to 0.24mm for A533B, or 1 < r/(J/σ0) < 5 where r/(J/σ0) is the normalized radial distance ahead of the crack. Once σc and rc have been determined, the critical values can be used in Equation 6 to plot a Material Failure Curve as shown in the left portion of Fig. 1.
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