13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 4. Numerical results and discussions The present method is used to analyze the singular stresses around the corner apex A of a rectangular inclusion subjected to uniform temperature change TΔ as shown in Fig.Fig.3. Fig.4 shows a configuration for a super 8-node quadrilateral inclusion corner element. For the solution of the singular stress fields, around the square inclusion corner apex A, defined in Eqs.(8) and(9), the stress intensity factors 1K and 2K should be first determined. Generally speaking, any component of the stresses at any angle θ may be used as the compared object to determine the stress intensity factors. However, for simplicity, herein we only use the stresses at o0 θ= and o 180 θ= , that is, the stresses at points on the bisector of the vertex angle in region 2Ω (for o0 θ= ) and 1Ω (for o 180 θ= ). An example for a square inclusion (l h= ) is given herein. In the numerical analysis, To model the infinite plate, its width and height are all set to be10l ; a quarter of plate is used for element divisions due to the symmetry of its geometry and loading, and 332 4-noded stress-based element and one 8-noded super elements are utilized. The material parameters of elastic modulus 1 2 : 100 E E = and Poisson ratio 1 2 0.3 ν ν = = are employed. The singular stresses away from the apex A along the boundaries of the inclusion are analyzed with different the unknown parameters sβ . They are plotted in Fig. 5. In the Figures, for the sake of comparison, the numerical results from the commercial software ANSYS package are also shown. It is shown that the singular stresses θθ σ and r θ σ rapidly increase with the decrease of the distance away from the inclusion corner apex A along the boundary ( o 135 θ=− ) of the square inclusion, which is well recognized; When the number of sβ meets the LBB condition: greater than equal to the number of freedom degrees of nodes in the super element minus the rigid modes (=3 in plane deformation), the present numerical results are in good agreement with the solutions of the ANSYS. Fig.6 tells us a fact that the influence of thermal expansion coefficient 2αon the dimensionless stress factor 1F is limited to very small the range of 6 2 20 10 α − < × . Figs.7 and 8 give the relationships between the dimensionless stress factor 1F and material parameters. From Fig.7, it can be seen that 1F 2l 2Ω 2h 1Ω A x y TΔ 2 3 4 s o x θ 1 5 y 6 7 8 r 2Ω 1Ω p θ− A Fig.4 Configuration of a super 8-node quadrilateral inclusion corner element Fig.3 A rectangular inclusion in a infinite matrix under thermal loading
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