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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6

σ 1m , σ 1b , σ 2m and σ 2b are represented by σ 1mL , σ 1bL , σ 2mL and σ 2bL , respectively. The lower bound limit stresses may be described by the following relationship when a simple case with only σ 1m and σ 2b is considered, � ��� � � � + � 2 3 ��� � � � − � ��� � � � 2 3 ��� � � = 1 (7) Equation (7) can be equivalently rewritten as ��� � = �1 − 1 �� + ��� (8) where �� = 2 3 ��� ��� = 2 3 �� �� (9) Equation (8) may be equivalently expressed as reference stress as ��� = �� �1 − �� + ��� (10)

σ m

σ b

2c crack τ m

σ 2m

2

2 L

σ 2b

1 σ 2m , σ 2b Fig. 5 A plate under biaxial membrane and through-thickness bending stresses σ 1m , σ 1b

a

2 D

t

Fig. 6 Geometry and loads of modified local limit load model

5.2. Reference stress solution for surface-cracked plates For a plate with a surface crack under � + � + �� , the reference stress may be treated as an equivalent membrane stress applied to an uncracked body in the section parallel to the crack plane. Therefore, the effect of bending stress parallel to the crack plane on the cracked-body reference stress may be approximately estimated using Eqn. (10). For a surface crack located in the middle of the plate perpendicular to axis 1 under the crack-opening stresses σ 1m and σ 1b and the membrane stress, σ 2m , and bending stress, σ 2b , parallel to the crack plane, replacing the membrane stress σ 1m in Eqn. (10) by the reference stress of the cracked plate without the parallel bending stress, ( σ ��� ) npb , which can be evaluated from Eqn. (5), the total reference stress may now be expressed as ��� = ( ��� ) ��� �1 − �� + ��� (11) where �� now re-defined as