Fractal dimension of brittle fracture

Summary Using a worldwide catalog of earthquakes we analyze the distribution of distances between pairs of earthquake hypocenters to determine the spatial fractal dimension of an earthquake fracture. As the time span of the catalog increases, asymptotically reaches the value 2.1–2.2 for shallow earthquakes. Approximately the same asymptotic value of dimension is obtained for a catalog of earthquakes with aftershocks removed. This value is compared with other known determinations of for brittle fractures. The fractal dimension declines to 1.8–1.9 for intermediate events (depth interval 71–280 km) and to about 1.5–1.6 for deeper events. Taking into account various possible errors and biases, we conclude that the fractal dimension of brittle shear fracture in rocks is 2.20 ± 0.05.     

Kinetic theory of brittle fracture of polymeric glasses

Conclusions 1. The stress distribution over the individual bonds at the end of a crack is taken into account on the basis of fracture mechanics using the Bartenev-Razumovskaya-Rebinder kinetic model for polymeric glasses.2. The rate of growth of a transverse edge crack in a strip subjected to tensile stresses is obtained as a function of the parameter l; this dependence is similar to that observed experimentally.3. The time characteristics of the strength are calculated numerically with reference to polymethyl methacrylate.4. An approximate analytic expression is obtained for the lifetime of a strip of polymeric glass; at _o<<_k this expression coincides with the exact result calculated on a computer.Institute of Physical Chemistry, Academy of Sciences of the USSR, Moscow. Moscow Engineering Physics Institute. Translated from Mekhanika Polimerov, No. 1, pp. 3–11, January–February, 1977.     

Electron-fractographic investigation of the brittle fracture of polyethylene

The results are presented of an electron-microscope investigation of the brittle fracture of linear polyethylene at the temperature of liquid nitrogen. It is shown that the physical nature of the structural details of the microrelief of the fracture surface is conditioned by local plastic deformation of the stratified lamellar structures in a thin layer adjacent to the fracture surface preceding and accompanying fracture. The possibility of local self-heating is discussed and a possible mechanism of plastic deformation of the crystal platelets at low temperatures is proposed.Mekhanika Polimerov, Vol. 3, No. 2, pp. 286–290, 1967     

Modeling of fracture of fibrous composites with brittle components

Conclusions A numerical analysis of the stress field at the tip of a crack showed high localization, increasing with increase of anisotropy and normal and shear stresses at the crack tip, which can cause various types of fracture. The use of the J-integral for estimates of the flow of elastic energy to the crack tip, simulated by a greatly elongated elliptical notch, is apparently possible in approximate calculations, taking into account localization of the zone of nonlinear behavior of the material. A scheme of estimating the crack resistance of a fibrous material with brittle components, taking into account the contribution of various forms of microfracture to energy dissipation, is proposed. The calculated value of G^* differs from the experimental data by about 20%, which, taking into account the simplicity of the calculation method, can be considered satisfactory.Report presented at the Fourth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, October, 1980).Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 615–619, July–August, 1981.The author thanks S. T. Mileiko for formulating the problem, advice, and discussion of the work.     

Relation between fracture and deformation in oriented polymers

The development of submicroscopic cracking under load has been studied (by a small-angle X-ray scattering technique) in such oriented polymers as Kapron (Nylon 6), polypropylene, etc. The increase in crack density (N_cr) is compared with the variation of the tensile strain (). The observed correlation between and N_cr indicates that the processes of fracture and deformation of oriented polymers are closely connected. The presence of a similar correlation in loading-relaxation-repeat loading experiments establishes fracture as the primary process.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 43–47, January–February, 1970.     

Concerning the mechanism of fracture of oriented glass-reinforced plastics in uniaxial compression

The mechanism of fracture of an oriented glass-reinforced plastic in uniaxial compression in various directions in the plane of the sheet is considered on the basis of experimental observations of the microstructure and data on the variation of the linear and volume strains with increasing load. It is shown that increases in load are accompanied by a steady increase in crack development and damage accumulation.Leningrad Structural Engineering Institute. Translated from Mekhanika Polimerov, No. 2, pp. 238–242, March–April, 1971.     

On the influence of the distance between edges of cracks on brittle fracture

We study the influence of the shape of a hole (cavity or crack) in an elastic plane on brittle fracture. We use a modified method of solving macrocrack problems. The results obtained are generalizations of known problems. One figure. Two tables. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 26–30, 1991.     

Accumulation of submicroscopic defects in fluoroplastics during their viscous fracture

The character of the accumulation of submicrocracks in fluoroplastics 4, 4D, and 4DM during a moderate-duration cold stretching was studied by x-ray diffraction patterns and mechanical methods. It was shown that during viscous fracture of fluoroplastics, an active accumulation of submicrodefects takes place, which substantially depends on the previous thermal history of the samples. Irreversible submicrofractures accumulate in annealed samples not only at the opening of the main crack, but also throughout the entire volume of the sample. Qualitative correlations have been established between the crystallinity and the concentration of the submicrocracks, on the one hand, and the yield point and the ultimate strength of the polymers, on the other. The degree of damage before the fracture of the fluoroplastic samples was evaluated phenomenologically.     

Critical behavior and the limit distribution for long-range oriented percolation. I

We consider oriented percolation on ${\mathbb{Z}}^d\times{\mathbb{Z}}_+$ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on ${\mathbb{Z}}^d$ . Suppose that D(x) decays as |x|^?d?α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension $d_c=2(\alpha\wedge2)$ . We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to $e^{-c|k|^{\alpha\wedge2}}$ for some c > 0.     

Variation of the work of brittle fracture in the impact bending of the prestressed polymer material 40K60V

Tensile stresses in a polymer in the viscous-flow state affect the work of brittle fracture in subsequent impact bending, with the formation of a maximum on the A- curves. The effect of crystallinity, preliminary molecular orientation, and the magnitude of the stresses on the position of this maximum has been studied in relation to possible conformational changes in the molecules. The experiments were performed on a polymer material consisting of an alloy of wax and rosin.Mekhanika Polimerov, Vol. 3, No. 4, pp. 696–702, 1967     

Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation

We prove that the Fourier transform of the properly scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α > 0 converges to ${e^{-C|k|^{\alpha\wedge2}}}$ for some ${C\in(0,\infty)}$ above the upper-critical dimension ${{{dc \equiv 2(\alpha \wedge 2)}}}$ . This answers the open question remained in the previous paper (Chen and Sakai in Probab Theory Relat Fields 142:151–188, 2008). Moreover, we show that the constant C exhibits crossover at α = 2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.     

Bases in oriented matroids

Let M be an oriented matroid. One can define exactly two assignments of +1 and ?1 to permutations of bases of M canonically associated with the orientation of M.     

Convexity in oriented matroids

We generalize to oriented matroids classical notions of Convexity Theory: faces of convex polytopes, convex hull, etc., and prove some basic properties. We relate the number of acyclic orientations of an orientable matroid to an evaluation of its Tutte polynomial.     

Convexity in oriented graphs

For vertices u and v in an oriented graph D, the closed interval I[u,v] consists of u and v together with all vertices lying in a u−v geodesic or v−u geodesic in D. For S⊆V(D), I[S] is the union of all closed intervals I[u,v] with u,v∈S. A set S is convex if I[S]=S. The convexity number con(D) is the maximum cardinality of a proper convex set of V(D). The nontrivial connected oriented graphs of order n with convexity number n−1 are characterized. It is shown that there is no connected oriented graph of order at least 4 with convexity number 2 and that every pair k, n of integers with 1⩽k⩽n−1 and k≠2 is realizable as the convexity number and order, respectively, of some connected oriented graph. For a nontrivial connected graph G, the lower orientable convexity number con⁻(G) is the minimum convexity number among all orientations of G and the upper orientable convexity number con⁺(G) is the maximum such convexity number. It is shown that con⁺(G)=n−1 for every graph G of order n⩾2. The lower orientable convexity numbers of some well-known graphs are determined, with special attention given to outerplanar graphs.     

Accessibility in oriented networks

The aim of this work consists of allocating a value that allows us to emphasize the importance of each player in a cooperative game when the cooperation possibilities are limited according to the links of an oriented network. The proposed concept of accessibility tries to conjugate the marginal contributions of each node as a game player with the cooperation geometry imposed by the digraph that models the network. We study general properties of this concept and particularly with respect to oriented paths. Concrete applications are proposed.