PSI - Issue 42

Yuebao Lei et al. / Procedia Structural Integrity 42 (2022) 80–87 Author name / Structural Integrity Procedia 00 (2019) 000–000

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�� = 2 3 �� ( ��� ) ���

(12) In Eqn. (11), the magnitude of the square root term is less than 1 for 0 < �� < 1 . To reduce the effect of the second term, − �� , a constant factor with 0 ≤ ≤ 1 is applied to this term. 5.3. Validation of the reference stress solution Limit load/reference stress solutions may be validated by comparing the J values predicted using those limit load solutions via the reference stress estimation method with elastic-plastic FE J results. The FE J results obtained in Sections 3-4 can be used to validate the reference stress solution including the effect of the bending stress parallel to the crack plane given in Section 5.2. The reference stress J estimation scheme by Ainsworth (1984) predicts the total J from the elastic J , J e , using the reference stress, σ ref , and corresponding reference strain, ε ref , via the following relationship based on the R6 Option 2 FAC (R6 (2019)): � = ��� ��� + �� 2 ��� ��� (13) where L r is defined by � = � � , � � = ��� � (14) where P in Eqn. (14) represents primary load and P L is the limit load corresponding to the load type P . The reference stress and strain relationship in Eqn. (13) follows the uniaxial true stress-true strain curve of the material (Eqn. (6)). The normalised FE J results shown in Figs. 2(a)-4(a) are now re-plotted in Figs. 2(b)-4(b), respectively, against ��� � ⁄ , where ��� is evaluated based on Eqn. (11) with μ =0.5 to include the effect of σ 2b . The Option 2 FAC based reference stress prediction, Eqn. (13), is also plotted in each figure and denoted by “Prediction”, for comparison. From Figs. 2(b)-4(b), the newly developed reference stress solution (Eqn. (11)) with μ =0.5 could well correlate the FE J data for various values of bending stress parallel to the crack plane and various combinations of biaxial membrane stress The newly-developed reference stress solution for plates with surface cracks under combined biaxial membrane and biaxial bending stresses (Eqn. 11) is used to estimate the reference stress of the local limit load model (Lei (2018)). The model (Fig. 6) is a plate of width 2 D , which contains a rectangular surface crack of depth a and length 2 c circumscribing the real surface defect, and has the same thickness, t , as the component at the crack location. The half width of the plate, D , which should be less than the half width of the component, W , is defined as = � � + � < (15) where k 0 and k are constants which depends on geometry ( k 0 =2 and k =1.5 for plates, k 0 =2 and k =1 for cylinders, see Lei (2019b)). The model is remotely loaded by the primary stresses of the component at the crack location obtained from elastic uncracked-body stress analysis. The stresses obtained from the elastic stress analysis should be expressed as the membrane stress, σ m , and through-thickness bending stress, σ b , normal to the crack plane, the membrane stress, σ 2m , and the bending stress, σ 2b , parallel to the crack plane and the average shear stress in the crack plane, τ m . The through thickness bending stresses, σ b and σ 2b , are positive if the bending stresses tend to stretch the front surface containing the crack and may be ignored when it is judged to be a non-primary stress. The reference stress from the model is then expressed as ��� = ��3 �� + ( ��� ) � � � �� � (16) and membrane stresses parallel to the crack. 6. Updating the local limit load model