PSI - Issue 42

Yuebao Lei et al. / Procedia Structural Integrity 42 (2022) 80–87 Author name / Structural Integrity Procedia 00 (2019) 000–000 � ( ⁄ , ⁄ , , � , � | � ⁄ |) for � ≠ 0 2 3 � � ( ⁄ , ⁄ , � , � | � ⁄ |) for � = 0

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3

The corresponding reference stress, ��� , can then be expressed as ��� = ⎩⎨ ⎧ �

(5)

3. FE fracture analyses Finite element fracture analyses are performed to evaluate elastic and elastic-plastic J for the plates containing semi-elliptical surface cracks described in Section 2. The commercial FE package ABAQUS (2017) is used in the analyses. The dimensions of the plate are shown in Fig. 1 with W = 4 c (therefore c / W = 0.25) and L = 4 W . For all analyses, c is fixed (30 mm) and therefore also W and L . The thickness of the plate, t , is changed according to the ratio a / t for a given a / c . For the purpose of this research, the analysis matrix is selected as a / c =0.6 with a / t =0.2, 0.5 and 0.8. The FE meshes used in Lei (2004) for J and limit load analyses are used in the analyses of this current research. The cracked plate was modelled by 8-noded brick elements, the ABAQUS element type C3D8. Because of symmetry, only a quarter of the plate was modelled with proper boundary conditions applied. A typical mesh used in the analysis can be found in Lei (2004). For all cases, the crack tip was modelled using a focused mesh which enables J to be evaluated on 15 contours around the crack tip and 11 locations along the crack front. Only one quarter of the plate is modelled and symmetry boundary conditions are applied on the symmetry planes. Also, a node located at x = 0, y = t, z = 0 is fixed to prevent the rigid body movement of the model.

σ m

σ b

z

z

x y σ 2b x y σ 2m

2 L

a

2



c

2 W

t

Fig. 1 Geometry and loads of a plate containing a semi-elliptical surface crack ( φ defines the crack tip location along the crack front)

For elastic analysis, Young’s modulus E =500 MPa and Poisson’s ratio  =0.3 are adopted. For elastic-plastic analysis, the Ramberg-Osgood type stress-strain relationship is used, which can be expressed as � ⁄ = � ⁄ + ( � ⁄ ) � (6) where  and n are a material constant and the strain hardening exponent, respectively, σ 0 is a normalising stress and  0 is defined by  0 /E . Where the yield stress,  y , is required, it is taken as the 0.2% proof stress. For all elastic-plastic analyses, E = 500 MPa,  = 0.3,  = 1, E/  0 = 500 and n = 5 are used. In all analyses, small strain isotropic hardening is used with the Mises yielding criterion. The membrane stress and the through-thickness bending stress normal to the crack plane are applied at the top surface of the plane as a linearly distributed load along the plate thickness. The membrane stress and through-thickness