σ σ θ φ σ β 1 0 1 2 0 / / / ( ) = ( )( ) LSY LSY g J r (5) Substituting Eq.(5) to Eq.(3), the following relation can be derived [7], J J c c m SSY LSY = φ β 1/ (6) This formulation indicates that the TSM is the one particular case of the Weibull stress criterion dependent on the combinations of n and m. The values of φ are numerically given as a function of n by Anderson and Dodds [3] and the fitting equation of φ for a bend specimen with notch depth, a/W=0.5, as follows: J J J b LSY SSY LSY = = + 1 1 1 2 0 0 φ σ γ / Φ (7) Φ=08425 2 262 . . n γ = + − × − 1 126 0 01925 8 333 10 5 2 . . . n n Assuming that the SSY stress field can be described with the HRR solution, the exponent β is replaced with 1/ (n+1). From Eq.(6) and Eq.(7), the ratio of JLSY to JSSY can be evaluated as: J J J b n m n m LSY SSY LSY = = + + + 1 1 1 2 2 2 0 0 2 2 φ σ γ / Φ (8) Figure 1 shows the results of Eq.(8), denoted as the WSSM, in comparison with the TSM. The specimen size requirement parameter, M (=b0 σ0/JLSY) is solved for JLSY/JSSY = 1.2 as a function of n. The value of M indicates the limiting value to obtain the toughness within a deviation less than 20% from the SSY condition, and corresponds to the constraint factor, M in Eq.(1). Those results suggest that the SSY toughness of the materials with large value of m and smaller value of n can be obtained in smaller specimens. 2-D AND 3-D FEM ANALYSIS In order to confirm above analytical prediction and to clarify the three dimensional size effect, 2- and 3dimensional FE analysis were performed using the WARP 3D [8]. Finite element models for toughness specimens are shown in Fig.2. 2D-SSY analysis was performed using the semi-circular model with the boundary of the SSY singularity [4]. The Ramberg-Osgood type constitutive equation with yield strength, σ0 = E/500, was adopted. The value of J-integral was calculated from both the path integral (Jd.i.) and the load-displacement Fig.1 Comparison of WSSM with TSM on specimen size requirements. 25 20 15 10 5 0 50 100 150 200 250 300 Hardening exponent, n M ( =bo σo / J ) m=10 15 20 25 30 TSM ASTM E1820-96 size requirement JLSY JSSY = 1.2 : m=2n+2 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 Jd.i./ σo , mm SSY 2D-LSY σw(B=0.25mm)/ σo σw,cr JLSY/ σo J SSY/ σo 2D (plane strain) n=5, m=10 Fig.3 Relation between the Weibull stress, σw and J integral.
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