13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Stress analysis by the EFGM and analysis of order of singularity by the FEM The characteristics of the EFGM is that the interpolation function at evaluation point is given by the information in the domain influence (See Fig 2.). Example of the interpolation function is written as Eq. (1). ( ) ( ) ( ) ( ) ( ) { ( )} { } q u q u u q u q u q u T n n x x x x x x = + + + + = Λ 3 3 2 2 1 1 , (1) where q indicates shape function, and n indicates number of referred nodes in the domain influence. In addition, forth order spline function is employed as the weighting function in the EFGM. Procedure of discretization is same as the traditional FEM. Detail of discretization is shown in reference [3]. Figure 2. Domain influence In addition, if the order of singularity is expressed by λ, the stress distribution is written as 1 1 1/ 1/ − − = ∝ = p p ij r r r λ σ . Here, p indicates characteristic root. Because the stress is expressed as gradient of the displacements iu , the relationship the displacements and distance r is given p iu r ∝ by the integration of 1− ∝ p ij r σ . If displacements for each element are expressed by interpolation function shown in Eq. (2) and the interpolation function is substituted to equation of the principle of virtual work, a characteristic equation is finally derived as shown in Eq. (3). ( ) ∑ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 8 1 0 , , i i ri p r hu r r u r φ θ , (2) [ ] [ ] [ ] ( ){ } { } A B C x 0 = + + p p2 , (3) where iu is expressed by o iu u− , and iu and ou represent spherical displacements at an arbitrary point in the spherical surface. In addition, ih indicates the shape function. In the Eq. (3), p indicates characteristic root and vector { }x denotes superposed displacement vector in entire domain, and matrices [ ]A , [ ]B and [ ]C represent the coefficient matrices derived by finite element procedure. Detail of this formulation is shown in reference [4].
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