13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( ) − − = − − α 0 min min 1 exp K K K K F K Jc Jc (6) where the Weibull modulus, α, takes the value of 4. Now, following a standard maximum likelihood estimate, the scale parameter, 0K , corresponding to the 63.2% cumulative failure probability, is given by ( ) ( ) 20 MPa m 0.3068 20 1 4 1 4 ( ) 0 + − − = ∑ = N i Jc i r K K (7) where N denotes the total number of specimens tested and r represents the number of valid tests (uncensored data). The median toughness at the tested temperature follows simply as ( ) 0.9124 20 20 MPa m 0 ) ( − + = K KJc med (8) The master curve of median toughness, ) ( Jc med K , for 1-T specimens over the transition range for the material has the final form ( ) [ ] C, MPa m 20 70exp 0.019 0 ) ( o T T KJc med − = + (9) where T is the test temperature and 0T is the reference (indexing) temperature The above expression for the median toughness applies throughout the lower part of the DBT range prior to the occurrence of significant ductile tearing. ASTM E1921 [10] test standard outlines procedures to construct various tolerance bounds based on the above representation for the toughness distribution. 4. Computational Procedures and Finite Element Models Calibration of the Weibull modulus for the tested pressure vessel steel is conducted by performing detailed finite element analyses on 3-D models for the SE(B) and precracked Charpy specimens. described in Section 5.1. The analysis matrix includes conventional, plane-sided SE(B) specimens with = aW 0.2 and 0.5, and precracked Charpy (PCVN) specimens with = aW 0.5. Figure 1 shows a typical finite element model constructed for the 3-D analyses of the SE(B) specimen with = aW 0.5. A conventional mesh configuration having a focused ring of elements surrounding the crack front is used with a small key-hole at the crack tip where the radius of the key-hole, 0ρ , is 0.0025 mm. Symmetry conditions permit modeling of only one-quarter of the specimen with appropriate constraints imposed on the remaining ligament and symmetry planes. A typical quarter-symmetric, 3-D model has 22 variable thickness layers with ~27000 8-node, 3-D elements (~31000 nodes) defined over the half-thickness 2 B ; the thickest layer is defined at 0 = Z with thinner layers defined near the free surface ( 2 Z B= ) to accommodate strong Z variations in the stress distribution. The finite element models are loaded by displacement increments imposed on the loading points to enhance numerical convergence with increased levels of deformation. The next section addresses cleavage fracture predictions using the Weibull stress model based on toughness data measured at different lower-shelf temperatures. The numerical computations for the cracked configurations at the test temperature reported here are generated using the research code WARP3D [11]. The analyses utilize an elastic-plastic constitutive model with 2J flow theory and conventional Mises plasticity in large geometry change (LGC) setting. The plastic stress-strain
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