PSI - Issue 13

K. Solberg et al. / Procedia Structural Integrity 13 (2018) 1762–1767 K. Solberg / Structural Integrity Procedia 00 (2018) 000–000

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3. Analytical background The stress distribution in the different notch geometries was studied by used of the unified elastic stress field for cracks and notches proposed by Lazzarin and Tovo (1996). The stress field uses a curvilinear coordinate system in order to describe different notch geometries, the coordinate system is given by � � �� � � � � � � � � � � �� � � � . (1) Where η is a parameter used for defining the opening angle of the notch, 2α, by the relation 2α = π(2 - η). The coordinate system used to describe the notch geometries and applying the correct boundary conditions related to the notch are shown in Fig. 2 for different values of η. η = 2 corresponds to an opening angle of 0, while η = 1 corresponds to a unnotched geometry without any perturbation of the stress field. A polar coordinate system is used to describe the stress components of the stress field, the coordinate system is shown in the bottom part of Fig. 2 for sharp and blunt notch. At the notch bisector line (θ = 0) the radius of the notch, ρ , can be described by ρ = ηr 0 /(η - 1), where r 0 is the placement of the coordinate system. In the case of mode I loading, the stress field can be expressed as � �� �� �� � � √ 1 2 � � �� � �1 � � � � � � � �1 � � � � �� �1 � � � � cos�1 � � � � �� � � � � cos�1 � � � � �1 � � � � cos�1 � � � � � � � � �1 � � � � � cos�1 � � � � �cos�1 � � � � sin�1 � � � � � � � � � � � � � �� � ��� � � � � � � � �1 � � � �� � cos�1 � � � � �cos�1 � � � � sin�1 � � � � �� . (2) Where K 1 is the stress intensity factor for mode I loading, λ 1 is the Williams’ series eigenvalue for mode I loading, χ 1 , is a coefficient dependent on the opening angle and µ 1 is a parameter related to the notch radius. When describing the stress fields in the notch geometries, the σ θθ component will be plotted along the notch bisector line, and a simplified formulation of the stress field (2) formulated by Lazzarin and Filippi (2006) will be used The analytical solution of the stress field is compared with the numerical solution for each geometry obtained from the finite element software Abaqus CAE. The 3D geometry was reduced to a 2D finite element model. Plane strain was assumed and symmetry utilized, so that only one fourth of the geometry were modelled. The boundary conditions, loads and the mesh used in the notched region is shown in in Fig. 3. A unit load σ 0 is applied in tension, and linear elastic material is used with a Young’s modulus of 200 GPa and Poisson’s ratio of 0.29. Eight-node plane stress elements were used, and the mesh is refined in the region close to the notch tip in order to describe the stress gradient more accurately. �� � � √2 � � �� �1 � � � � � � � � �� � �� (3) Here � � is a parameter dependent on the opening angle of the notch. Assuming u-shaped notch and opening angle equal to zero for the unnotched and the semi-circular geometries, the parameters describing the stress fields is shown in Table 2, values obtained from Lazzarin and Filippi (2006).