PSI - Issue 28

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

2246

2

Among these are points on the body surface at which the surface smoothness condition is violated, the type of bound ary conditions changes, various materials come into a contact, or internal points at which the smoothness condition of the contact surface of di ff erent materials is violated. Singular points of di ff erent types are of frequent occurrence in computational models for various applied problems of elasticity theory. In the presence of a singular solution, the vicinities of the singular points, as a rule, are zones of high stress concentration in practice. It is here that the fracture process begins and the structure integrity is violated. Many researchers are studying the stress behavior in the vicinity of singular points. For two-dimensional and three dimensional problems of the linear elasticity theory, the various variants of singular points was considered. The results achieved in this area are presented in su ffi cient detail in a few review papers, for example, of Sinclair (2004); Paggi et al. (2008). One of the ways of studying the stress singularity is associated with the analysis of solutions in the vicinity of singular point, which, according to the works of Williams (1952); Kondrat’ev et al. (1967), have the following form σ ∼ k = 1 A k ξ k r λ k − 1 , r → 0 , ω < λ 1 < λ 2 < . . . < λ k < . . . (1) Here, r is the distance to the singular point, A k are the intensity coe ffi cients, and ξ k are the functions of the angular distribution of the stress field σ in the vicinity of a singular point. In the plane case, ξ k depends on one polar angular variable ϕ , in which case ω = 0, and in the spatial case on two spherical variables ϕ , θ , then ω = − 0 . 5. The limits on the values of λ k determined by the value of ω follow the finite condition of the potential strain energy in the vicinity of singular points. For solutions of the form (1), in the presence of values λ k which are satisfying the condition Re λ k < 1, the stresses tend to infinity as r tending to zero. It is these values that will determine the singular behavior of stresses in the vicinity of the singular points. Within this most common approach to research of stress singularity for two-dimensional problems, the object of study is the vicinities of the vertices of wedge-shaped regions: homogeneous or composite plane wedges (Fig. 1), on the edges of which the boundary conditions are set (in stresses or displacements). Similarly, for three-dimensional problems, the objects of study are the points at the edge of the spatial wedge (which can also be both homogeneous and composite) and the vertices of homogeneous and composite conical regions, such as the vertices of circular and noncircular cones, trihedral and polyhedral wedges. A number of researchers, in particular Mihailov (1979) and Huang et al. (2007), have shown that the solutions to the problems of plane strain state and antiplane strain of wedges obtained in the planes perpendicular to the edge of the spatial wedge determine the occurrence of singular solutions at the points of the edge of a spatial wedge, through which the appropriate plane is drawn (Fig. 2). Thus, the stress behavior near the points at the edge of the spatial wedge is determined by solving the two-dimensional problems of plane strain state and antiplane strain. Among the problems of constructing singular solutions in the framework of the linear theory of elasticity, most of the works are devoted to the study of the stress singularity in the vicinity of singular points of isotropic bodies. Analytical approaches dominated for solving two-dimensional problems. The history of genesis and development of these approaches is described in the review paper of Paggi et al. (2008). Analytical approaches can be easily adapted to the case of cylindrically-orthotropic bodies; however, the case of general rectilinear anisotropy brings about some

 



r

r

 





 

b

a

Fig. 2. Schematic representation of the spatial wedge and one of the plane wedges in the cross-section perpendicular to the edge.

Fig. 1. (a) Homogeneous and (b) composite plane wedges.