13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- ratios λ=0.5, 1.0, to determine numerical solutions of constraint parameter A through a least squares fitting method suggested by Nikishkov et al. [12]. Using the obtained numerical solutions of constraint parameter, A, constraint effect near crack-tip of specimens is analyzed. Numerical solutions for constraint parameter A under biaxial loading conditions obtained in this work can be used to estimate other two commonly-used constraint parameters A2 and Q conveniently by the relationships between A and A2, Q presented in authors’ previous paper [13]. 2. Theoretical background 2.1. J-A approach As mentioned in introduction section, J-A two-parameter approach of elastic-plastic fracture mechanics, which is suggested by Nikishkov et al. [11, 12], is an alternate format of J-A2 approach developed by Yang et al. [9, 10]. In these two formats of the approach, both the constraint parameter A and A2 are the magnitude of the second term in a three-term series expansion of crack-tip stress fields. In this paper, the J-A format is used. Figure 1. Cracked specimens under biaxial loading: (a) SECP, (b) CCP, (c) DECP Considering a two-dimensional elastic-plastic specimen containing a mode I crack under plane strain condition, if the elastic-plastic behavior of the specimen material described by deformation theory of plasticity follows the Ramberg-Osgood relationship, the uniaxial stress-strain curve can be described as: n = + 0 0 0 σ σ α σ σ ε ε (1) where α is a material coefficient, n is the hardening exponent (n > 1), ε0=σ0/E, E is Yong’s modulus, σ is the stress applied on the remote end of specimen, and σ0 is the material yield stress. Table 1. Comparison of constraint parameter Q for CCP under a/W=0.5, n=10, λ=0.5 ( )) /( log 0σ J a -2.900 -2.491 -2.256 -2.074 -1.855 -1.593 -1.299 -0.991 QSSY [Ref.] -0.2107 -0.3553 -0.4822 -0.5711 -0.6497 -0.7690 -0.8223 -0.9315 QSSY -0.2196 -0.3714 -0.4849 -0.5763 -0.6758 -0.7790 -0.8739 -0.9815 Diff. (%) 4.232 4.526 0.558 0.923 4.003 1.299 6.274 5.376 In the J-A two-parameter approach suggested by Nikishkov et al. [11, 12], for the hardening exponent n ≥ 3 and under plane-strain conditions, the three-term asymptotic solution expression for
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