International Journal of Fracture
, Volume 10, Issue 4, pp 507523
First online:
Brittle fracture of solids with arbitrary cracks
 R. V. Gol'dsteinAffiliated withInstitute for Problems in Mechanics, Academy of Science of USSR
 , R. L. SalganikAffiliated withInstitute of Mechanics, M. V. Lomonosov Moscow State University
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One of the problems of fracture mechanics is the prediction of the propagation of cracks in solids. The present paper deals mainly with linear fracture mechanics which owes its origin to the works of A. A. Griffith [1, 2] and studies the development of cracks under sufficiently low loads when the behaviour of the material within a region sufficiently remote from the edges of cracks may be regarded as linearly elastic, At present, linear fracture mechanics [3] is restricted mainly to special kinds of loading geometry, with the crack extending rectilinearly (in a plane case) or in its plane (in a threedimensional case). The main problem here is to establish a relationship between the dimensions of cracks and the loads applied. Within the framework of linear fracture mechanics the fracture itself and other nonlinear phenomena that precede it are assumed to take place only within local regions which are small compared to the dimensions of cracks. The possibility that such a situation exists is associated with the fact that when the crack dimensions are sufficiently large the characteristic dimension of the end region is fully determined by a certain intrinsic dimension of the material structure. Therefore, if the material does not exhibit time dependency, the state of the end region at the moment of rupture becomes fully independent of the loads applied and the geometry of the solid, i.e. autonomous. The notion of autonomy [4] leads to the formulation of this theory as one of limit equilibrium.
If the conditions of rectilinear extension of the crack (or those of the crack extension in its plane) are disturbed, there arises a problem of determining not only the dimensions of the crack, but also the path of the crack extension under such conditions of loading that a slow, quasistatic crack development is possible. This problem can be actually subdivided into two: (1) Criteria for the determination of the dimensions and paths of the crack extension, and (2) Expressions for the characteristics of the stressstrain state which are constituents of these criteria through the geometry of solid with cracks and the loads applied.
As regards (1), there have been many assertions, and the connections between them are not quite clear at present. The first of the suggested criteria, namely that of local symmetry for the plane problem formulated by Barenblatt and Cherepanov [5, 6] and by Erdogan and Sih [7] can be within certain limits substantiated and generalized for the threedimensional case. The guiding principle here is the treatment of the theory of cracks from the standpoint of the method of inner and outer expansions or that of singular perturbations [8]. The concept of the stress intensity factor which is basic in linear fracture mechanics is decisive in matching inner and outer expansions to find the main term of the asymptotic solution of the complete problem. Actually the construction of the theory of equilibrium cracks [4] implicitly employs this technique for a certain specific model. More explicit indications are given in Ref. 9. In the treatment of the problem of plastic zones in the vicinity of notches, the idea of the boundary layer is employed in Ref. 10. The problem of fracture of a solid is analysed from this standpoint in Ref. 11.
As regards (2), progress has been hampered by the lack of efficient techniques for fording the stressstrain state of a solid having nonrectilinear cuts. A number of investigations have been carried out for cuts of a particular kind an arc of a circumference [12, 13], an arc of a parabola [l4], and a threelink broken line which is close to a straight line to such an extent that the boundary conditions are assumed referable to the direction of the middle portion [15, 16]. The problem of a semiinfinite curvilinear cut slightly deviating from a rectilinear one by expanding complex elastic potentials in the magnitude of deviation of the cut from the rectilinear axis tangent to the line of cut at its end is considered in Ref. 17. An exact solution of the problem of a semiinfinite cut having the form of a twolink broken line is given in Ref. 18.
The present paper is devoted to the investigation of the development of cracks under arbitrary loading conditions.
In Section 1 the criterion of local symmetry is substantiated and generalized for the threedimensional case. In Section 2 an effective procedure of finding stress intensity factors for the plane case is given, in terms of which the criterion is formulated. Closed first approximation formulas for these magnitudes are presented in the case of a slightly curved crack, numerical calculations showing the applicability of the latter with an error not exceeding 10 to 15 with the angles of deviation of the crack from the straight line coming to 20°. In Section 3 equations of extension of curvilinear cracks are derived on the basis of the first approximation formulas and criterion of local symmetry. In Section 4 some examples are considered.
 Title
 Brittle fracture of solids with arbitrary cracks
 Journal

International Journal of Fracture
Volume 10, Issue 4 , pp 507523
 Cover Date
 197412
 DOI
 10.1007/BF00155254
 Print ISSN
 03769429
 Online ISSN
 15732673
 Publisher
 Kluwer Academic Publishers
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 Authors

 R. V. Gol'dstein ^{(1)}
 R. L. Salganik ^{(2)}
 Author Affiliations

 1. Institute for Problems in Mechanics, Academy of Science of USSR, Moscow, USSR
 2. Institute of Mechanics, M. V. Lomonosov Moscow State University, Moscow, USSR