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.     

Numerical modeling of brittle fracture in porous CFC-materials

Both energy and stress criteria are necessary conditions for fracture but neither the one nor the other is sufficient. A combination of these criteria is proposed in [1]. This combined criterion is used for numerical simulation of crack propagation by the 4-point bending test in porous materials. Examples of such materials are carbon-carbon composites (CFC) [2, 3]. Micrographs of the cross-sections of these materials are used for FEM modeling of the crack propagation on the basis of the proposed criterion. Results of the numerical modeling are compared with experimental results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Existence and convergence for quasi‐static evolution in brittle fracture

This paper investigates the mathematical well‐posedness of the variational model of quasi‐static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford and Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time‐continuous quasi‐static growth is to pass to the limit as the time‐discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {u_n} of SBV‐functions to its BV‐limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u_n}. In particular, it is shown that the notion of minimizer of a Mumford and Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time‐continuous quasi‐static evolution. © 2003 Wiley Periodicals, Inc.     

Epsilon‐stable quasi‐static brittle fracture evolution

We introduce a new definition of stability, ε‐stability, that implies local minimality and is robust enough for passing from discrete‐time to continuous‐time quasi‐static evolutions, even with very irregular energies. We use this to give the first existence result for quasi‐static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time, but not necessarily global minimizers. The key ingredient in our model is the physically reasonable property, absent in global minimization models, that whenever there is a jump in time from one state to another, there must be a continuous path from the earlier state to the later along which the energy is almost decreasing. It follows that these evolutions are much closer to satisfying Griffith's criterion for crack growth than are solutions based on global minimization, and initiation is more physical than in global minimization models. © 2009 Wiley Periodicals, Inc.     

The fracture of brittle and ductile crystals. Force and deformation criteria

The growth and branching of sharp cracks in ideal single crystals are investigated. Neuber-Novozhilov force and deformation criteria are proposed for the branching of sharp cracks; these criteria describe the brittle, quasibrittle, quasiductile and ductile behaviour of materials on fracture. For internal cracks, simple relations are obtained that describe the branching of cracks when the Coulomb-Mohr single-crystal theoretical strength curves are known for a generalized stress state. The possibility of multiple branching of cracks is found, which is linked to the multiplicity of the eigenvalues on loss of stability of the system. It is established that, for ideal single crystals, the principle of local symmetry is satisfied in the vicinity of the crack tip if the axis of symmetry of the crystal coincides with the axis of the crack. When there are asymmetrical disturbances of the atomic lattice in the vicinity of the crack tip, or when the axis of symmetry of the single crystal does not coincide with the crack axis, the principle of local symmetry is not satisfied.     

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.     

Tuning of mass dampers for preventing brittle fracture by employing teaching learning based optimization

In design of reinforced concrete structures, a ductile design is needed, because the brittle fracture is sudden. Also, the energy absorbed by the structure is effective for ductile structures during earthquake excitations. In design regulations, especially for vertical supporters such as columns, the shear force and axial force capacity is limited, although the strength of the members is not at the limits. For that reason, the shear forces of structures excited by earthquakes can be reduced by adding a tuned mass damper (TMD), but TMD must be optimized and it must not be heavy. If the mass of the TMD is not optimized, the ductile behaviour constraint about the axial force capacity cannot be provided. The compressive strength of the concrete is a measure for the effective application of a TMD to a reinforced concrete structure. Since the mass is limited by the design constraint, the other design variables of TMD such as period and damping ratio are optimized. Using the formulas for frequency and damping ratio related to a preselected mass may not be sufficient and a precise optimum solution for preventing brittle fracture. Metaheuristic methods can be used by using a specific limit. In this study, Teaching Learning Based Optimization (TLBO) is employed to find optimum TMD parameters. Several earthquake excitations were used in the optimization process. A three storey RC frame structures with different compressive strength of concrete is investigated. Three stopping criteria have been used in the proposed methodology. The first criterion is the reduction of the maxi-mum shear force value for the most critical excitation to the ductile behaviour limit. The second criterion is the reduction of the ratio of maximum first storey displacements of structure with and without TMD. This value is reduced under a user defined value, but the user defined value can be entered as zero. This value can be iteratively increased for minimization. The last criterion is the reduction of the acceleration transfer function of the structure. The proposed method is feasible on finding optimum TMD ensuring ductility conditions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

On a variational,free-discontinuity formulation of brittle fracture: mean-field relaxation and numerical solution

In this paper we address the resolution of two important issues arising in the context of the relaxed variational formulation of the incremental free–boundary value problem of brittle fracture. First issue, is how by recasting the formulation into a discrete, minimum–maximum problem one can avoid the undesirable scale effects expressed in terms of the characteristic size and domain–shape dependence of the calculated minimum; second, how by a remeshing procedure in combination with a domain–shape update for tracking the propagating 0–th level set one can reconstruct the crack surface. We finally illustrate our approach by a geometrically linear 2–dimensional example for crack propagation in an initially isotropic brittle solid. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)