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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di ffi culties due to the fact that in the polar coordinate system the coe ffi cients of the di ff erential equations become variable. These di ffi culties were overcome mainly by numerical approaches. The known numerical and numerical analytical approaches allow us to obtain the eigenvalues for the vertex of any wedge with anisotropic properties, and, in a particular case, with isotropic properties. Much fewer works have been devoted to studying the stress singularity in the vicinity of the singular points in bodies made of functionally graded materials. Functionally graded materials (FGMs or simply graded materials) are characterized by changes in material properties in space. There are a few papers in the literature dealing with isotropic functionally graded bodies, in which the stress singularity exponents are determined based on the above-mentioned analytical methods. Examples of such works can be found in the review part of the paper of Fedorov et al. (2018). No studies devoted to evaluating the stress singularity exponents in the vicinity of the singular points of anisotropic functional-gradient bodies have been found in the literature. In solving three-dimensional problems of the stress singularity, the majority of studies have used various versions of numerical methods, mainly finite element method and boundary element method. The known variants of numerical methods such as proposed by Pageau et al. (1995a), allow solving the problem of stress singularity in the vicinity of the vertices of any homogeneous and composite conical domains with isotropic properties and any boundary conditions; however, the literature contains the results only for simple configurations and several boundary conditions. There are few results for complex objects, such as the vertex of intercrossing spatial cracks. Korepanova et al. (2013) developed the numerical method, which was applied to analyze the stress singularity near the vertices of a single and two intercrossing wedge-shaped cracks. The developed numerical method can be used for di ff erent boundary conditions imposed on the crack faces; however the results were obtained only for stress free conditions on the crack faces for the case of two intercrossing cracks. Some of these numerical methods can be adapted for the case of a special type of anisotropy. Despite the fact that some of the methods, for example, proposed by Dimitrov et al. (2001), allow us to obtain eigenvalues for the vertices of homogeneous and composite conical regions with general anisotropic properties; the results are presented for some simple configurations (and some boundary conditions): the vertex of a plane crack in an elastic half-space, the plane of which is perpendicular to the surface of the half-space (Pageau et al. (1995a)); common vertex of three di ff erent materials on the surface of a composite half-space (Pageau et al. (1995a,b)). There are no results for the vertices of homogeneous and composite conical regions with functionally graded properties. Of great practical interest are those singular points that define the vertices of plane or spatial cracks, considered in the framework of two-dimensional or three-dimensional statements of the problems of elasticity theory, respectively. As can be seen from the review, there are few results on the stress singularity of such complex objects as the inter section point of the planes of three-dimensional cracks. This paper presents the results of studying the asymptotic behavior of stresses in the vicinity of the common vertex of several radial spatial cracks in an isotropic material, the plane of which is perpendicular to the surface of the elastic half-space. The idea of an algorithm considered in this work for determining the power law dependence of stresses in the vicinity of singular points was first proposed by Raju et al. (1981). The algorithm is based on the assumption that, in a su ffi ciently small vicinity of the singular point, the stress behavior is described by one term of the representation (1). Then, according to Williams (1952) and Kondrat’ev et al. (1967), stress distribution along a radial line outgoing from a singular point can be expressed as σ = A 1 r λ − 1 + O r λ , (2) where r is the distance from the singular point, A 1 is the constant, λ is the parameter, characterizing the degree of stress singularity, and O r λ represents the terms of the order r λ and higher. For small distances r , the singular term dominates and Eq. (2) can be approximated by σ A 1 r λ − 1 (3) or log σ = log A 1 + ( λ − 1) log r . (4) 2. Description of the algorithm for the numerical analysis of stresses in the vicinity of singular points