PSI - Issue 28

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

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transverse shear (mode II) — by specifying on these lateral faces the tangential displacements which directed parallel to the x -axis in opposite directions relative to each other. The stress singularity exponents given by Pageau et al. (1995a) and obtained using the proposed numerical algo rithm based on the extraction of stress asymptotics by the finite element method in accordance with relations (5) are presented in Table 2 for modes I and II. Table 2. Stress singularity exponents for a crack whose plane is perpendicular to the surface of an elastic half-space. λ 1 (mode II) λ 2 (mode I) Material Pageau et al. (1995a) Proposed algorithm Pageau et al. (1995a) Proposed algorithm

Isotropic (steel)

0.3929 0.4543

0.40 0.46

0.5483 0.5227

0.55 0.52

Orthotropic (composite)

In this case, the di ff erence in the singularity exponents obtained by these two methods is no more than 1.5%.

4. Numerical analysis of stresses near common vertices of several radial cracks

The algorithm for the numerical analysis of stress singularities considered in this work and other similar algorithms (Becker et al. (1974); Raju et al. (1981)) are applicable for variants with one singular term of representation (1). For other variants, including for the vertex of several radial cracks, the stress behavior in the vicinity of singular points is determined by several singular terms (Korepanova et al. (2013)). For these options, the possibilities of numerical methods are limited to constructing a reliable pattern of stress field in the vicinity of singular points. The validity test of the obtained stresses is the comparison of the numerical results obtained with di ff erent discretizations near singular points. Let us investigate the common vertex of two spatial radial cracks located inside an elastic isotropic half-space. Each of the cracks is a wedge-shaped crack with a right angle and is a quarter plane. To analyze the stress behavior in the vicinity of common vertex of two radial cracks, a cylinder of radius R and height h is considered. The common vertex of two spatial radial cracks is located on the cylinder axis at a depth R from its upper base and coincides with the origin of the rectangular coordinate system xyz . The y -axis is oriented along the axis of the cylinder. Crack faces are perpendicular to the upper base. The common edge of the cracks is located on the cylinder axis. The angle between the cracks is ϕ . The distributed normal tensile load σ 0 is applied to the lateral surface of the cylinder. Sliding conditions are specified on the bottom base of the cylinder (displacements along the y -axis are equal to zero). Upper base of the cylinder and crack faces are stress-free. The design scheme is shown in Fig. 4. The calculations were performed at various angles ϕ between cracks. The following parameter values were used in the calculations here and after: h / R = 6, σ 0 = 1 kPa, the elastic modulus and Poisson’s ratio of the cylinder E = 1 GPa, ν = 0 . 3.

a

b

O

h





R

Fig. 4. (a) Design scheme for two radial cracks; (b) top view of the upper base of the cylinder.