A note on diffraction of normally incident P-wave by a line crack in micro-polar elastic medium

The problem of diffraction of waves due to plane harmonic P-wave incident normally on a line crack situated in an infinite micro-polar elastic medium has been studied in this paper. The problem has been solved for both low and high frequencies for small coupling parameter. The stress intensity factors (SIF) have been obtained in micro-polar elastic medium from which the corresponding stress intensity factor for classical elastic medium can be deduced.     

Wave diffraction by a line of finite crack in a saturated two-phase medium

An application of the Biot’s theory to the diffraction problem of plane harmonic dilatational waves (P-waves) of the first kind and the second kind by a line crack or geometric discontinuity of finite width embedded in a saturated two-phase medium is presented in this paper. The crack surfaces are assumed impermeable, and the integral transform method is utilized to reduce the mixed boundary-value problem to a single Fredholm integral equation. The magnitudes of the intensity of the stress fields near the crack tips measured by Mode I dynamic stress-intensity factor (dimensionless) are computed and displayed graphically against dimensionless circular frequency (ω) for several dimensionless material property values, namely, viscosity-to-permeability and mass density ratios. In the case of the normally incident P-waves of the first kind, the results in terms of stress-intensity factor are also compared with the corresponding values of dry elastic material. All the stress-intensity factor curves are shown to exhibit a similar character in that they rise to the peaks at certain frequency values and then decay with increasing frequencies. At certain frequency ranges and material property values, amplification in the dynamic stress-intensity factor can be substantially larger than those encountered in dry elastic materials. The stress-intensity factor is found to be more affected by the changes in the ratio of viscosity-to-permeability at lower mass density ratio. With fluid mass density 10% of the bulk mass density, the viscosity-to-permeability ratio of 0.01 gives the highest increase of about 32% in the magnitude of stress-intensity factor compared to the dry material counterpart value, while a decrease of about 9% is observed for the viscosity-to-permeability ratio of 100. It is also found that change in mass density ratio has significant effect upon the magnitude of stress-intensity factor at lower ratio of viscosity-to-permeability. As for the normally incident P-waves of the second kind, the presence of the pore fluid affects both the magnitude and character of the stress-intensity factor. Large variations in the magnitude of stress-intensity factor are observed as viscosity-to-permeability ratio changes from 1 to 100. At the ratio of viscosity-to-permeability of 1.0, the stress-intensity factor curves increase gradually with frequency and exhibit the peaks in curves for mass density ratio of 0.3 and higher. As the viscosity-to-permeability ratio is raised to 100, the stress-intensity factor curves increase monotonically with frequency at a much faster rate throughout the frequency range of interest (ω = 0–2), and the change in mass density ratio is shown to have little effect on the stress-intensity factor, especially within the low frequency ranges. The results obtained in this study are useful in the mechanics of fracture initiation of saturated porous materials under the fluctuating mechanical and/or pore fluid loadings that are periodic with time.     

Contact problem for a plane crack under a normally incident antiplane shear wave

The contact-interaction problem for a stationary plane crack with friction between its edges under the action of a normal (to the crack plane) harmonic shear wave is addressed. Antiplane deformation conditions are considered. The distribution of contact forces and displacement discontinuity of crack edges are studied Published in Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 138–142, May 2007.     

Rayleigh waves emitted by a propagating crack in a strain-rate dependent elastic medium

A simple model was proposed for the interpretation of the non-circular form of the Rayleigh wavefronts emitted by a fast running crack in a plate. The surface deformation around the crack tip, due to the high stress concentration there, propagated as a surface wave after fracture of this zone. On the other hand, the moving singularity of the crack tip created a dynamic stress field of varying intensity with time all over the specimen. This dynamic stress field resulted in a significant change of the mechanical properties of a strain-rate dependent material and therefore it influenced the velocity of propagation of fracture-Rayleigh wavefronts. An analysis of this varying dynamic strain field explained the non-circular form of Rayleigh waves, accompanying the propagating crack. For the experimental evaluation of the K₁-factor the method of dynamic caustics was used in conjunction with the high-speed photography technique.     

Eshelby tensor for a crack in an orthotropic elastic medium

In the present Note, we provide new analytical expressions of the components of Hill tensor $\mathbb{P}$ (or equivalently the Eshelby tensor $\mathbb{S}$) associated to an arbitrarily oriented crack in orthotropic elastic medium. The crack is modelled as an infinite cylinder along a symmetry axis of the matrix, with low aspect ratio. The three dimensional results obtained show explicitly the interaction between the primary (structural) anisotropy and the crack-induced anisotropy. They are validated by comparison with existing results in the case where the crack is in a symmetry plane. To cite this article: C. Gruescu et al., C. R. Mecanique 333 (2005).     

Surface contact of elliptical crack under normally incident tension–compression wave

Surface contact interaction of a plane elliptical crack under normally incident tension–compression wave is solved by the method of boundary integral equations. The contact forces and the displacement discontinuity of the crack edges are examined. The dependence of the mode I stress intensity factor on the wave number is studied. The solution is compared with the results obtained for an elliptical crack without allowance for crack edges contact interaction.     

Dynamic response of an elastic medium containing crack

A fundamental solution for an infinite elastic medium containing a penny-shaped crack subjected to dynamic torsional surface tractions is attempted. A double Laplace–Hankel integral transform with respect to time and space is applied both to motion equation and boundary conditions yielding dual integral equations. The solution of the derived dual integral equations is based on an analytic procedure using theorems of Bessel functions and ordinary differential equations. The dynamic displacements’ field is obtained by inversion of the corresponding Laplace–Hankel transformed variable. Results of a representative example for a crack subjected to pulse surface tractions are obtained and discussed.     

The stress state of an elastic orthotropic medium with a penny-shaped crack

The static-equilibrium problem for an elastic orthotropic space with a circular (penny-shaped) crack is solved. The stress state of an elastic medium is represented as a superposition of the principal and perturbed states. To solve the problem, Willis approach is used, which is based on the triple Fourier transform in spatial variables, the Fourier-transformed Greens function for an anisotropic material, and Cauchys residue theorem. The contour integrals obtained are evaluated using Gauss quadrature formulas. The results for particular cases are compared with those obtained by other authors. The influence of anisotropy on the stress intensity factors is studied.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 76–83, December 2004.     

A crack on the interface between a linear elastic medium and a stress-state dependent physically nonlinear medium

A crack on the interface between a linear elastic medium and a stress-state dependent physically nonlinear medium is studied. A numerical method to solve such problems is proposed. Some asymptotic distributions of stresses, strains, and displacements near the crack tip are obtained under the assumption that the forces and displacements are continuous on the interface.     

On safe crack shapes in elastic bodies

According to the Griffith criterion, a crack propagation occurs, provided that the derivative of the energy functional with respect to the crack length reaches some critical value. We consider a generalization of this criterion to the case of nonlinear cracks satisfying a nonpenetration condition and investigate the dependence of the shape derivative of the energy functional on the crack shape. In the paper, we find the crack shape which gives the maximal deviation of the energy functional derivative from a given critical value and, in particular, prove that this optimality problem admits a solution.     

On a single edge crack problem in an elastic strip

Summary In this paper we deal with the crack problem in an infinitely long elastic strip with a single edge crack. The crack problem is reduced to finding solutions of the integral equation of Fredholm type of second kind. Numerical calculations with high accuracy are carried out and the variation of the stress intensity factor for various crack lengths are shown in a table.

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Zum Problem des einzelnen Randrisses in einem elastischen Streifen

Übersicht Behandelt wird hier das Problem eines einzelnen Randrisses in einem unendlich langen elastischen Streifen. Das Rißproblem reduziert sich auf die Ermittlung von Lösungen der Fredholmschen Integralgleichung zweiter Art. Der durch sehr genaue numerische Rechnungen gewonnenen Spannungskonzentrationsfaktor wird für verschiedene Rißlängen tabellarisch angegeben.

     

On the strong ellipticity for orthotropic micropolar elastic bodies in a plane strain state

In the present paper we consider an orthotropic micropolar elastic material subject to a state of plane strain. In this context, we establish necessary and sufficient conditions for the strong ellipticity of constitutive coefficients. Furthermore, we study existence of progressive plane waves under the strong ellipticity conditions previously determined. Finally, we detail the results obtained for a specific class of materials related to tetragonal systems.     

On generalized Kelvin solutions in a multilayered elastic medium

This paper presents fundamental singular solutions for the generalized Kelvin problems of a multilayered elastic medium of infinite extent subjected to concentrated body force vectors. Classical integral transforms and a backward transfer matrix method are utilized in the analytical formulation of solutions in both Cartesian and cylindrical coordinates. The solution in the transform domain has no functions of exponential growth and is invariant with respect to the applied forces. The convergence of the solutions in the physical domain is rigorously and analytically verified. The solutions satisfy all required constraints including the basic equations and the interfacial conditions as well as the boundary conditions. In particular, singular terms of the generalized Kelvin solutions associated with the point and ring types of concentrated body force vectors are obtained in exact closed-forms via an asymptotic analysis. Numerical results presented in the paper illustrate that numerical evaluation of the solutions can be easily achieved with very high accuracy and efficiency and that the layering material inhomogeneity has a significant effect on the elastic field.The Canadian Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Dynamic behavior of rectangular crack in three-dimensional orthotropic elastic medium by means of non-local theory

The dynamic behavior of a rectangular crack in a three-dimensional(3D)orthotropic elastic medium is investigated under a harmonic stress wave based on the non-local theory. The two-dimensional(2D) Fourier transform is applied, and the mixedboundary value problems are converted into three pairs of dual integral equations with the unknown variables being the displacement jumps across the crack surfaces. The effects of the geometric shape of the rectangular crack, the circular frequency of the incident waves,and the lattice parameter of the orthotropic elastic medium on the dynamic stress field near the crack edges are analyzed. The present solution exhibits no stress singularity at the rectangular crack edges, and the dynamic stress field near the rectangular crack edges is finite.     

The problem of interaction between a misfitting inclusion and a crack in an infinite elastic medium

The problem of interaction between a curvilinear crack (or a system of such cracks) and a misfitting inclusion of arbitrary shape (or a system of such inclusions) inside an infinite isotropic elastic medium of the same material as the inclusion was solved by using the complex potential technique and reducing the problem to a complex Cauchy type singular integral equation along the crack only (or the system of cracks).

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Résumé Le problème de l'influence mutuelle d'une fissure curviligne (ou d'un système de telles fissures) et une inclusion malajustée de forme arbitraire (ou un système de telles inclusions) dans un milieu infini élastique isotrope du même matériau que l'inclusion a été résolu en utilisant la technique des potentiels complexes et en réduisant ainsi le probléme à une équation intégrale singulière complexe du type Cauchy seulement le long de la fissure (ou du système des fissures).

     

Reflection and diffraction of SH waves due to a line source in a heterogeneous medium

The field due to a line source of harmonic SH waves embedded in a semi infinite medium whose density and rigidity vary exponentially with depth is derived in the integral form. The displacement due to diffraction at any point in the shadow zone is obtained and, by the saddle point method of evaluation of the integral, the field at any point in the illuminated region is also found. Finally, geometrical interpretation is given to the different rays arriving in the illuminated as well as in the shadow zone.Nomenclature b shear wave velocity on the free surface - C wave velocity - H _v ⁽¹⁾ (p), H _v ⁽²⁾ (p) Hankel's function of the first and second kind respectively - k Fourier transform parameter with respect to x - v the displacement - fourier transform of v with respect to x - X grazing angle - , small positive constants - positive constant, –/2 - ₀ coefficient of rigidity at the free surface - coefficient of rigidity - _is values of _i at the saddle point (i=1, 2, 3, 4) - the density of the medium - ₀ the density of the medium at the free surface - /2 frequency of vibration     

Spherical sound-wave diffraction by a sphere in a viscous medium

The problem of spherical sound-wave diffraction by a sphere in a viscous medium is discussed. Applying the method of separation of variables, we obtained exact formulas for the hydrodynamic field elements in a low-viscosity medium. Furthermore, by applying the Morse [1, 2] and Watson [3] method to the exact formulas, we determined asymptotic formulas for the same quantities in the case of long and shortwave propagation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–146, January–February, 1973.