PSI - Issue 28

Andrey Fedorov et al. / Procedia Structural Integrity 28 (2020) 2245–2252 Andrey Fedorov / Structural Integrity Procedia 00 (2020) 000–000

2248

4

In this paper, the parameter λ is evaluated using the FEM procedure, in which the finite element mesh has gradual refinements near singular point. To obtain relationship (4) through numerical experiments, it is necessary to find the discretization such that in the vicinity of the singular point for a number of nodal points r 1 , r 2 , . . . , r n on the radial line issuing from the singular point, the following relations are fulfilled with su ffi cient accuracy λ − 1 ≈ log σ 1 σ 2 log r 1 r 2 ≈ log σ 2 σ 3 log r 2 r 3 ≈ . . . ≈ log σ n − 1 σ n log r n − 1 r n . (5) Where r 1 , r 2 , . . . , r n are the distances from the singular point; σ 1 , σ 2 , . . . , σ n are stresses at the corresponding nodal points r 1 , r 2 , . . . , r n ; λ is required stress singularity exponent. Determination of relationship (5) allows us to calculate the value of λ , that specifies the character of stress behavior in the vicinity of the singular point. This algorithm has been verified for various types of singular points of two-dimensional problems, where the small est eigenvalue is a real number (Korepanov et al. (2013)). In the considered two-dimensional numerical experiments, the di ff erence between the singularity exponents found using this numerical algorithm from the values obtained from analytical solutions is no more than 0.08%. To illustrate additional arguments for the reliability of the results obtained on the basis of this numerical algorithm, let us consider a three-dimensional example related to the estimation of the stress behavior at the vertex of crack, the plane of which is perpendicular to the surface xOy (Fig. 3a). The stress singularity is estimated at the point O with coordinates x = y = z = 0. The contour of the sphere in Fig. 3a is shown only for visualizing the crack in the elastic half-space. For this problem the numerical results on the stress singularity exponents for isotropic and orthotropic materials, the elastic characteristics of which are presented in Table 1, are given by Pageau et al. (1995a). 3. Verification of the algorithm for the numerical analysis of stresses in the vicinity of singular points

Table 1. Elastic characteristics of the elastic half-space. Material

E i ( MPa )

G i j ( MPa )

ν i j

Isotropic (steel)

210000

0.3

Orthotropic (carbon-fiber composite)

E x = 130300, E y = E z = 9377

G xy = G xz = 4502, G yz = 2865

ν xy = nu xz = nu yz = 0 . 33

When using the numerical algorithm, instead of an elastic half-space, a cube is used as a design scheme, the size of the faces of which does not matter in this problem (Fig. 3b). In this case, the conditions of opening mode (mode I) are realized by specifying on the lateral faces the normal displacements parallel to the xOz plane, and the conditions of

z

z

y

2 a

O

x

90°

360°

O

2 a

y

x

a b Fig. 3. (a) The crack whose plane is perpendicular to the surface of the elastic half-space; (b) its design scheme.