Anti-plane Problem Research Articles

Closed-form approximations to borehole stresses for weak transverse isotropic elastic media

We have derived new closed-form approximations to borehole stresses for weak tilted-transverse-isotropic media with boreholes oriented in the plane of isotropy but not necessarily along a principal stress direction. By introducing Green's three anisotropic parameters, ω1,ω2 and ω3, into Lekhnitskii's formalism for anisotropic elastic media to calculate stresses around boreholes subjected to internal pressure and in situ stress, we first have recasted the general borehole stresses into more compact expressions that are useful to highlight explicitly the impact of the elastic anisotropy. We have shown the equivalence between Lekhnitskii's and Green's equations for the in-plane components and extended Green's expressions to the anti-plane problem. By linearizing the expressions for weak anisotropy degree, i.e. ∣ωi∣≪1, we have shown that the borehole stress expressions are the sum of the isotropic “Kirsch” solution and additional terms that depend on the anisotropic parameters ω1+ω2 and ω3. The additional terms have azimuthal/radial functional dependence of higher orders (cos4θ,sin4θ) and (1/r2,1/r4,1/r6) for in-plane components, and, (cos3θ,sin3θ) and (1/r2,1/r4) for anti-plane components. Our verification examples show that the new borehole stress expressions provide good approximations not only for weak anisotropy but also moderate and strong anisotropy in certain cases.

Diffraction of elastic waves from object in layer with slightly corrugated surface

This work is concerned with the influence of corrugated surfaces on waves diffracted from an object in an elastic layer. A boundary value problem is formulated to simulate an anti-plane problem for a harmonic load acting on the upper surface of the layer. By using the boundary integral equation method and the perturbation technique, the considered problem is reduced to a pair of integral equations. By constructing the Green’s function, the scattering problem in a one-mode frequency range is solved. To check the validity of the proposed technique, several numerical examples for different geometrical shapes of the corrugated bottom are presented.

Fracture analysis of one-dimensional hexagonal quasicrystals: Researches of a finite dimension rectangular plate by boundary collocation method

As an important supplement and development to crystallography, the applications about quasicrystal materials have played a core role in many fields, such as manufacturing and the space industry. Due to the sensitivity of quasicrystals to defects, the research on the fracture problem of quasicrystals has attracted a great deal of attention. We present a boundary collocation method to research fracture problems for a finite dimension rectangular one-dimensional hexagonal quasicrystal plate. Because mode I and mode II problems for onedimensional hexagonal quasicrystals are like that for the classical elastic materials, only the anti-plane problem is discussed in this paper. The correctness of the present numerical method is verified through a comparison of the present results and the existing results. And then, the size effects on stress field, stress intensity factor and energy release rate are discussed in detail. The obtained results can provide valuable references for the fracture behavior of quasicrystals.

Soil and topographic effects on ground motion of a surficially inhomogeneous semi-cylindrical canyon under oblique incident SH waves

To elucidate the ground motion amplification due to combined soil and topographic effects, an analytical formulation in the framework of classical elastodynamics is derived for the scattering of oblique incident plane SH waves by a semi-cylindrical canyon covered by a local inhomogeneous soil layer with a radially-varying modulus. Allowing the shear modulus of the finite soil layer covering the circular surface of the canyon as a power of the radial distance, the governing equation of motion for the anti-plane shear problem is derived and solved analytically by the method of wave function expansion. The ground motions for both of the homogeneous and inhomogeneous canyons under oblique incident waves can be computed efficiently and a comprehensive set of numerical examples are presented as illustrations. The degree of inhomogeneity of the soil layer and its thickness are found to affect the magnitude and the pattern of ground motion amplification of the cylindrical canyon surface depending on the frequency content, the irregular topography and obliquity of the wave incidence.

Dynamic response of an icosahedral quasi-crystalline medium with a Griffith crack under mechanical loadings

Elastodynamic response of three-dimensional icosahedral quasi-crystals under mechanical loadings is presented in this article. Two kinds of cases are considered here, in which the plane problem and anti-plane problem are included. The phonon and phason elastic fundamental fields along with their coupling effect in crack analysis are explicitly presented in terms of the theoretical and numerical analyses. Using the finite difference method, some numerical results of stresses, displacements, and normalized dynamic stress intensity factor are obtained. By comparing the results of quasi-crystals, this article reveals the influence of phonon and phason elastic fundamental fields, which occupy an important position in dynamic deformation behavior of quasi-crystals. Of course, phonon–phason coupling effect also should not be neglected.

Influence of imperfect interface and fiber distribution on the antiplane effective magneto-electro-elastic properties for fiber reinforced composites

The antiplane effective coefficients of two-phase piezoelectric–piezomagnetic periodic composite materials reinforced with cylindrical, unidirectional and periodically distributed fibers are computed by means of asymptotic homogenization method (AHM). The constituents have transversely isotropic properties belonging to 6mm symmetry group and the periodic distribution of the fibers is assumed to be parallelogram-like as representative volume element (RVE). In the model, the imperfections are modeled as an idealization of spring–capacitor–inductor distributions at the interface. The antiplane local problems and the associated effective coefficients result of the AHM are explicitly described. The explicit formulae depend on the physical properties of the constituents of the phases and the constants that characterize the existence of the aforementioned imperfection. The validation of the present approach is shown by comparison with numerical results reported in the literature. The influences of the fiber spatial distributions and the imperfect fiber–matrix interface contact conditions on the effective properties are analyzed. Spatial fiber distribution induces some value changes in the magneto-electric coefficient and two possible composites property symmetry are obtained: monoclinic 2 and transversely isotropic. The effect of the imperfect contact parameter has a more pronounced value on the ME coefficient than the fiber distribution.

Inclusions inside a bounded elastic body undergoing anti-plane shear

As a problem of theoretical and practical interest, anti-plane shear is mathematically simple and prone to analytical solutions. However, almost no analytical solutions have been reached for the problem of inclusions embedded in a bounded domain and undergoing anti-plane shear eigenstrain. In this work, using some techniques of complex analysis and integral formulae derived previously, Eshelby’s anti-plane shear problem of inclusions in a finite isotropic elastic body is solved analytically. The cross-section of the inclusion with a prescribed uniform anti-plane eigenstrain can be of polygonal shape or any smooth one characterized by a Laurent polynomial, while the cross-section of the prismatic bar submitted to displacement or traction boundary conditions is required to be smooth. In particular, in the case where the cross-section of the prismatic bar and the one of the inclusion are both circular, explicit analytical expressions are derived for the strain and stress fields in different ways. Some examples are provided to illustrate how to implement the solutions based on the conformal maps.

Qualitative properties in strain gradient thermoelasticity with microtemperatures

This paper is devoted to the strain gradient theory of thermoelastic materials whose microelements possess microtemperatures. The work is motivated by an increasing use of materials which possess thermal variation at a microstructure level. In the first part of this paper we deduce the system of basic equations of the linear theory and formulate the boundary-initial-value problem. We establish existence, uniqueness, and continuous dependence results by the means of semigroup theory. Then, we study the one-dimensional problem and establish the analyticity of solutions. Exponential stability and impossibility of localization are consequences of this result. In the case of the anti-plane problem we derive uniqueness and instability results without assuming the positivity of the mechanical energy. Finally, we study equilibrium theory and investigate the effects of a concentrated heat source in an unbounded body.

Распространение упругих волн в плоском слое-волноводе с учётом упрощённой модели континуума коссера.

The problem of propagation of high-frequency and low-frequency acoustic waves in a plane elastic waveguide layer on the basis of a simplified model of the Cosserat continuum is considered. In view, the presence of micro polarity of the medium, boundary value problems for plane and antiplane deformations for different combinations of boundary conditions on the waveguide surface are formulated. In a long-wave and a short-wave approximations the obtained results are compared with the results of the classical theory of elasticity. The conditions for a possible localization of the wave energy near the surface of the waveguide are found. It is shown that in the antiplane deformation problem, the considering of micropolarity of the material is not leads to the possibility of existence of high localized forms. The frequency bands of localized and harmonic waveforms are found. In the plane deformation problem, considering of micropolarity of the material under different surface conditions may cause as a distortion of the frequency band of the localized Rayleigh wave existence so as the emergence of a new frequency band of possible localized waves.

Analysis of anti-plane problems for isotropic elastic material with two elliptical voids

Analysis of anti-plane problems for isotropic elastic material with two elliptical voids

The fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in one-dimensional hexagonal quasicrystals with piezoelectric effect

By modeling that the surfaces of the cracks and hole are in limited permeable boundary conditions, the anti-plane problem of an elliptical hole with two asymmetrical cracks in one-dimensional (1D) hexagonal quasicrystals (QCs) with piezoelectric effect is investigated by adopting the technique of conformal mapping and the Stroh-type formulism. The analytic solutions of the field intensity factors and energy release rate are presented. With the variation of the hole-size and the crack length, the proposed model can be reduced to some classic crack models. In the absence of the electric load or the phason field, the present results match with the classical results of 1D hexagonal QCs or piezoelectric materials provided in the open literature. Numerical examples are then conducted to reveal the effects of the hole-size, the crack length, the coupling coefficient and applied mechanical/electric loads on the field intensity factors and the energy release rate.

A circular inhomogeneity incorporating surface/interface strain gradient elasticity

We derive a rigorous solution to the anti-plane shear problem of a circular elastic inhomogeneity with an imperfect interface described by surface strain gradient elasticity. The loadings considered include a uniform remote stress field and a screw dislocation applied either in the matrix or in the inhomogeneity. By using analytical continuation, the problem is reduced to a third-order differential equation for a single analytic function. The analysis indicates that in the absence of the screw dislocation, the internal stress field is uniform and is reliant on two size-dependent parameters arising from surface strain gradient elasticity. The size-dependent image force acting on the screw dislocation is derived. The conditions for the existence of equilibrium positions for the dislocation in the matrix or in the inhomogeneity are obtained. Several numerical examples are presented to illustrate the solution. As an application of the solution, the stabilities of a misfit screw dislocation dipole in the matrix and a screw dislocation pair in the inhomogeneity are also discussed.

Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids

By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples are presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions.

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.

A semi-infinite crack of mode III in the bimaterial wedge

An exact solution of the antiplane problem for a semi-infinite interface crack in a piecewise-homogeneous wedge under a self-balanced load on its sides has been obtained. Three types of boundary conditions on the wedge sides were examined: the both sides being stress-free; both sides being clamped, and one side being stress-free with the second one clamped. As a result of using the Wiener–Hopf method, the solution was represented in quadratures. Green's functions were obtained for stress intensity factors; in the case of a geometrically symmetrical wedge structure simple formulae were found for these functions. The stress singularity in the apex of the wedge was studied. In contrast to the homogeneous wedge structure the asymptotic of the stresses near the apex was established to have sometimes two singular terms for some values of the composite parameters.

The weak solution of an antiplane contact problem for electro-viscoelastic materials with long-term memory

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, and the friction is modeled with Tresca’s law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field with a time-dependent variational equation for the potential field. Then we prove the existence of a unique weak solution to the model. Moreover, the proof is based on arguments of evolution equations and on the Banach fixedpoint theorem.

Longitudinal shear of a bi-material with fractional sliding contact in the interfacial crack

We construct an analytical solution to the anti-plane problem of an inhomogeneous bi--material medium with the interfacial crack considering sliding friction. The medium is exposed to an arbitrary normal and shear loading in the longitudinal direction. Using the jump function method, the problem is reduced to a solution to singular integral equations for the jumps of displacements and stresses in the areas with sliding friction. Explicit expressions for displacements, stress intensity factors and energy dissipation are obtained. Critical load values for determination of the onset of slippage are investigated. The effect of friction and loading parameters on the size of the slip zone, stress intensity factors and energy dissipation is numerically analyzed.

各種面外荷重下における２個のだ円形空孔を有する等方性弾性体問題の解析

We derive a solution for two elliptical voids that are free boundary to an elastic medium (matrix) of infinite extent under anti-plane problem. These two voids have different long and short radii and different central points. When there are two voids in the matrix, we consider the matrix as a part of poroelastic matrix. The matrix is subjected to arbitrary loading, for example, by uniform stress and concentrated force and so on. The solution is obtained as a series with an explicit general term involving the complex potential of the corresponding homogeneous problem. This procedure is referred to as heterogenization. Several numerical examples are presented with graphs.

Antiplane Axisymmetric Creep Deformation of Incompressible Medium

Model problems with simplified geometry and kinematics has a great value for developing theories of mechanical behavior of materials. Analytical solutions of such problems allow one to perform qualitative analysis of investigated process and to verify numerical solutions. Anti-plane deformation problem is one of the simplest model problems. This problem was solved for linear elastic medium, nonlinear elastic medium and elastoplastic medium. One should mention the set of papers [1–5] published by V.D.Bondar in which anti-plane deformation problem is solved in the frameworks of finite strain elasticity and elastoplasticity. In this paper anti-plane deformation problem is solved in the context of theory of large elastoplastic deformations. This theory is based on non-equilibrium thermodynamics formalism. It assumes that irreversible and reversible deformations are defined by differential transport equations. We consider flow of incompressible medium within cylindrical tube due to pressure gradient. No-slip boundary condition is set on the walls of the tube. The points of the medium are restricted to move only along lines parallel to the element of the cylinder. We suppose that medium deforms both reversibly and irreversibly. In addition, irreversible deformation accumulation is due to creeping of medium. Present work is the continuation of the research [6, 7] in which similar problem statement was used. The main distinction of works [6,7] is that irreversible deformation of medium is due to plastic flow.

Singular integral equations with two fixed singularities and applications to fractured composites

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector Riemann-Hilbert problem on a real axis with a piecewise constant matrix coefficient that has two points of discontinuity. A condition of solvability and a closed-form solution to the integral equation are derived. For the Chebyshev polynomials of the first kind in the right hand-side, the solution of the integral equation is expressed in terms of two nonorthogonal polynomials with associated weights. Based on this new generalized spectral relation for the singular operator with two fixed singularities an approximate solution to the complete singular integral equation is derived by recasting it as an infinite system of linear algebraic equations of the second kind. The method is illustrated by solving two problems of fracture mechanics, the antiplane and plane strain problems for a finite crack in a composite plane. The plane is formed by a strip and two half-planes; the elastic constants of the strip are different from those of the half-planes. The crack is orthogonal to the interfaces, and it is located in the strip with the ends lying in the interfaces. Numerical results are reported and discussed.