On the elastic symmetries of growing mixed-mode cracks

Many continuum damage mechanics models for quasi-brittle materials are based on the reduction of stiffness due to elliptical crack or penny-shaped microcracks in the material. Because of this a numerical study of growing elliptical cracks in a unit cube is undertaken with the help of an FEM simulation.The propagation of the crack is governed by the principle of maximum driving force [1]. For each propagation step the tensor of elasticity is calculated and its symmetries are analyzed. It will be shown that the elastic symmetry in each step is close to orthotropy and can be approximated by an elliptical crack. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New damage evolution model of rock material

As a kind of natural engineering material with original defects, there are distinctly nonlinear and anisotropic mechanical behaviors for rock materials. Nevertheless, the rock damage mechanics can solve this problem well. However, for the complexity of mechanical property of rock material, the mature and applicable model to describe the rock failure process and the method to determine the maximum damage value have not been established very well. To solve this problem, one new damage evolution model for rock material has been proposed. In this model, the least energy consumption principle proposed to describe the fracture process of materials is used. Using the experimental data of granite sample under uniaxial compression and the results of numerical tests under uniaxial tension and uniaxial compression, this model is verified. Moreover, the results of the new model have been compared with the results of the tests (numerical test and real test) and the traditional damage model. The comparison shows that the new model has the higher accuracy and better reflects for the fracture process of the granite sample. Moreover, the released damage energies of the new model and Mazars model are different, and the released damage energy of the new model is slightly less than that of the Mazars model.     

On the topological derivative due to kink of a crack with non-penetration. Anti-plane model

A topological derivative is defined, which is caused by kinking of a crack, thus, representing the topological change. Using variational methods, the anti-plane model of a solid subject to a non-penetration condition imposed at the kinked crack is considered. The objective function of the potential energy is expanded with respect to the diminishing branch of the incipient crack. The respective sensitivity analysis is provided by a Saint-Venant principle and a local decomposition of the solution of the variational problem in the Fourier series.     

On the topological derivative due to kink of a crack with non-penetration. Anti-plane model

A topological derivative is defined, which is caused by kinking of a crack, thus, representing the topological change. Using variational methods, the anti-plane model of a solid subject to a non-penetration condition imposed at the kinked crack is considered. The objective function of the potential energy is expanded with respect to the diminishing branch of the incipient crack. The respective sensitivity analysis is provided by a Saint-Venant principle and a local decomposition of the solution of the variational problem in the Fourier series.     

On the peculiarities of a crack growth path in anisotropic solid

One of the possibilities to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. In this connection the influence of the elastic moduli of an anisotropic material on the possibilities of crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are supposed to be isotropic. For studying a direction of a crack growth path several crack models are considered. It is shown that a thin elongated elliptical hole as a crack model leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for some class of orthotropic materials a crack deviates from the straight path just after it starts to grow even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loading is also considered. This problem is reduced to the problem of the fracture direction determination for thin elongated elliptical cavity slightly inclined to the initial direction. In the frame of the proposed approach the conditions of instability are obtained. It is shown that for some class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is wider than one for which a crack deviates from the straight crack path just after its start. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

On the solvability of the Cauchy problem with growing initial data for a class of anisotropic parabolic equations

We study the Cauchy problem for a doubly nonlinear parabolic equation with anisotropic degeneration in the case where the initial data are locally finite Radon measures growing, generally saying, at infinity. The weak solution of the problem is obtained as the limit of regular solutions with smoothed initial data.     

Long-term failure of a layered viscoelastic composite material with a crack under a time-dependent load

The long-term failure of a layered viscoelastic composite caused by precritical propagation of a coin-shaped crack is studied. It is assumed that the crack is located inside a viscoelastic layer (the layer of binder) parallel to the layer orientation. The crack development due to stretching of the composite massive by uniformly distributed external forces increasing with time is described. It is assumed that these forces act perpendicularly to the plane of crack propagation. The investigation is carried out within the framework of Boltzmann-Volterra linear theory for resolving integral operators with difference kernels describing the deformation of a material with time-dependent rheological properties. An irrational function of the viscoelastic integral operator is presented in the form of a proper continued fraction and transformed using the method of operator continued fractions. Numerical solutions are obtained for resolving integral operators with the kernel in the form of Rabotnov exponential-fractional function. The kinetics of crack growth with a prefailure zone commensurable with the crack length is described. A comparison with the results obtained in terms of the concept of thin structure of the crak tip is given.Timoshenko Institute of Mechanics, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 545–558, July–August, 2000.     

Creep damage modeling of a polycrystalline material

A polycrystalline material is investigated under creep conditions within the framework of continuum micromechanics. Geometrical 3D model of a polycrystalline microstructure is represented as a unit cell with grains of random crystallographical orientation and shape. Thickness of the plains, separating neighboring grains in the unit cell, can have non-zero value. Obtained geometry assigns a special zone in the vicinity of grain boundaries, possessing unordered crystalline structure. A mechanical behavior of this zone should allow sliding of the adjacent grains. Within the grain interior crystalline structure is ordered, what prescribes cubic symmetry of a material. The anisotropic material model with the orthotropic symmetry is implemented in ABAQUS and used to assign elastic and creep behavior of both the grain interior and grain boundary material. Appropriate parameters set allows transition from the orthotropy to the cubic symmetry for the grain interior. Material parameters for the grain interior are identified from creep tests for single crystal copper. Model parameters for the grain boundary are set from the physical considerations and numerical model validation according to the experimental data of the grain boundary sliding in a polycrystalline copper [2]. As the result of analysis representative number of grains and grain boundary thickness in the unit cell are recommended. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary layer due to the axisymmetric spreading of material on the surface of a fluid

The effect of the axisymmetric spreading of a layer of material (oil or solid particles) on the surface of a viscous fluid is studied. Assuming high Reynolds numbers, the boundary layer equation is derived and solved for general power law surface velocities. The composite streamlines show sharp turns near the surface.     

On the volume of a spherical octahedron with symmetries

The group Aff(ℚ) of affine transformations with rational coefficients acts naturally not only on the real line ℝ, but also on the p-adic fields ℚ_p. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded μ-harmonic functions for a probability measure μ on Aff(ℚ) that is supported by a finitely generated subgroup, that is, to describe the Poisson boundary. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Bifurcations of a plane crack front growing quasistatically in an elastic space

Using variational-asymptotic models of force and energy criteria, situations are found in which bifurcations of the form of the front accompanying the quasistatic propagation of a plane crack in an elastic isotropic space are possible. Two types of bifurcations are revealed for a circular crack in the case of axisymmetric loading: fluctuation of the centre of the crack while preserving its circular form and distortion of the front due to the formation of two or a larger number of “lobes”.     

On the degree evolution of a fixed vertex in some growing networks

Two preferential attachment-type graph models which allow for dynamic addition/deletion of edges/vertices are considered. The focus of this paper is on the limiting expected degree of a fixed vertex. For both models a phase transition is seen to occur, i.e. if the probability with which edges are deleted is below a model-specific threshold value, the limiting expected degree is infinite, but if the probability is higher than the threshold value, the limiting expected degree is finite. In the regime above the critical threshold probability, however, the behaviour of the two models may differ. For one of the models a non-zero (as well as zero) limiting expected degree can be obtained whilst the other only has a zero limit. Furthermore, this phase transition is seen to occur for the same critical threshold probability of removing edges as the one which determines whether the degree sequence is of power-law type or if the tails decays exponentially fast.     

Stressed state of an elastoplastic isotropic shell with a through crack with regard for material hardening

We have solved the problem of determining the stressed state of an elastoplastic isotropic shell of arbitrary curvature with a through crack with regard for material hardening. We have obtained a system of singular integral equations and solved it numerically by the method of mechanical quadratures. We have also studied the influence of material hardening on the general characteristics of the stressed state.     

Estimation of crack density due to fragmentation of brittle ellipsoidal inhomogeneities

An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ellipsoidal shape embedded in a ductile matrix under remote static loading.The energy release calculated for prolate spheroidal inhomogeneities is used in the balance of energy to determine the crack density. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)