Pair Of Dual Integral Equations Research Articles

Interaction of anti‐plane shear waves with two collinear cracks in 1D hexagonal piezoelectric quasicrystals

AbstractThe present article presents the interaction of anti‐plane shear waves by two collinear cracks in one dimensional (1D) hexagonal piezoelectric quasicrystals (PQCs). With the aid of Fourier transform, the mixed boundary value problem (MBVP) is transformed into three pairs of dual integral equations, which are solved analytically by Hilbert transform. The explicit expressions for dynamic stress intensity factors (DSIFs) of phonon and phason fields, crack opening displacement (COD) and electric displacement (ED) are derived in closed form and some special cases are studied. Numerical values of DSIFs of phonon and phason fields, COD and ED are plotted to show the effect of crack length and electric boundary condition. Moreover, the DSIFs of phonon field and COD are represented graphically for single crack.

Vertical vibration of rigid strip footings on poroelastic soil layer of finite thickness

This paper presents a numerical solution that describes the response to forced vertical vibrations of rigid strip footings founded on the surface of a poroelastic soil layer of finite thickness. To mathematically describe this mixed boundary value problem, we relax the boundary conditions and cast it as a pair of dual integral equations, which are transformed to a system of linear equations by means of Jacobi orthogonal polynomials. The system is solved numerically to provide the frequency-dependent footing compliance, and the associated nearby soil displacement. Results obtained with the model at hand underline the important contribution of standing waves to the response of the soil-foundation system, and draw the limits of existing solutions that consider the foundation soil as infinite half-space, where standing waves cannot form.

Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

Scattering of SH wave by a crack in a functionally graded one dimensional hexagonal piezoelectric quasicrystals

AbstractIn this paper, the dynamic interaction between SH wave and a crack in functionally graded one‐dimensional hexagonal piezoelectric quasicrystals (PQCs) is studied by using integral transform technique. To make the analysis executable, the material properties are assumed to vary partially in the direction perpendicular to the crack face following an exponential function. The problem was transformed through Fourier transform into three pairs of dual integral equations, which are solved numerically by the Copson method. The explicit expressions of phonon field, phason field and electric field on crack surface are determined. The dynamic field intensity factors of crack tip are given and some special cases are discussed. Numerical examples are then conducted to reveal the influences of amplitude, incident angle and wave number of SH wave and crack length, gradient parameter on the normalized dynamic stress intensity factors (DSIFs). The results of this paper lay a theoretical foundation for the design and nondestructive testing of QCs structures.

Vertical vibration of a rigid strip footing on a half-space of unsaturated porous soil

This study presents an analytical solution to the compliance coefficient of rigid strip footings lying on an unsaturated porous soil half-space when subjected to vertical vibration. Using the Fourier transform techniques, the complex interaction between unsaturated soils and rigid strip foundations is transformed into a mixed boundary-value problem based on the developed framework of Biot’s model. The wavenumber domain auxiliary function and boundary conditions are used to convert the governing equations to a pair of dual integral equations. To yield the compliance function, the integral equations are transformed into a system of inhomogeneous linear equations by utilizing hypergeometric polynomials and Bessel functions. The proposed analytical solution is verified against existing solutions to interactions between foundations and saturated soil or single-phase soil by adjusting the parameters in the unsaturated soil model. Additionally, the impacts of soil skeleton saturation, intrinsic permeability, and Poisson’s ratio of unsaturated soil on the dynamic compliance coefficient of rigid strip footings are examined. It is shown that the Poisson’s ratio of the soil skeleton and the soil saturation at critical saturation have a significant effect on the dynamic compliance coefficient, but have a negligible effect on the intrinsic permeability.

Horizontal vibration of rigid strip footings on poroelastic half-space

This paper presents an analytical method to determine the response of rigid strip footings to harmonic horizontal loads, while considering the foundation soil as poroelastic, two-phase medium. The derived solution is the first of its kind that allows considering in a robust manner the coupled effects of soil deformation and flow on the dynamic response of strip footings to horizontal loads. The strategy for treating this complex problem analytically comprises transforming the mixed boundary conditions to a pair of dual integral equations that provide the contact stresses at the soil-footing interface, which are solved by means of Jacobi orthogonal polynomials. Presentation of the solution is followed by a discussion on how soil flow parameters affect the dynamic response of strip footings to horizontal loads, and the importance of considering the foundation soil as two-phase medium.

Non‐local stress and electric displacement solution of a 3D semi‐permeable rectangular crack in infinite orthotropic piezoelectric materials

AbstractThe paper presents the non‐local stress and electric displacement solution of a 3D semi‐permeable rectangular crack in infinite orthotropic piezoelectric materials (OPMs). With the help of the generalized Almansi's theorem and 2D Fourier transform, the boundary problem is formulated by three pairs of dual integral equations, and the displacement jumps across the crack surfaces are defined. The Schmidt method is used to solve the dual integral equations. The non‐local stress field (NSF) and the non‐local electric displacement field (NEDF) along the crack edges are deduced. Numerical results are reported to explain the influence of the size of the rectangular crack, the lattice parameter and the electric permittivity of the air inside the crack on the NSF and the NEDF at the crack edges in OPMs in detail. The present non‐local solutions exhibit no stress and electric displacement singularities along the crack edges, and may be benefit future works.

Nonlocal fracture analysis of an interface crack between a functionally graded coating and a homogeneous substrate under thermal loading

In this paper, the thermoelastic response of an interface crack between a functionally graded coating and a homogeneous substrate under thermal loading is investigated using the nonlocal continuum theory to eliminate the stress singularities at the crack tips and address the size effects. The differential form of nonlocal theory is utilized to obtain the governing equations of the thermal stress field, whereas the integral form is employed for the boundary conditions. Finally, the mixed, boundary-value problems are reduced to two pairs of dual integral equations using Fourier transform. The Schmidt method is adopted to solve the dual integral equations numerically. The numerical results confirmed the finite values of thermal stresses at the crack tips, which enable us to establish a straightforward maximum stress-based criterion instead of the traditional, stress intensity factor- or energy-based fracture criterion. In addition, the location of maximum stresses is found to be slightly away from the crack tip, which is determined by the nonlocal characteristic length. Parametric studies are conducted to investigate the influence of nonhomogeneous properties on the thermal fracture behavior of the interface crack.

Fundamental solutions of two 3D rectangular semi-permeable cracks in transversely isotropic piezoelectric media based on the non-local theory

PurposeThe paper aims to present the non-local theory solution of two three-dimensional (3D) rectangular semi-permeable cracks in transversely isotropic piezoelectric media under a normal stress loading.Design/methodology/approachThe fracture problem is solved by using the non-local theory, the generalized Almansi's theorem and the Schmidt method. By Fourier transform, this problem is formulated as three pairs of dual integral equations, in which the elastic and electric displacements jump across the crack surfaces. Finally, the non-local stress and the non-local electric displacement fields near the crack edges in piezoelectric media are derived.FindingsDifferent from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack edges in piezoelectric media.Originality/valueAccording to the literature survey, the electro-elastic behavior of two 3D rectangular cracks in piezoelectric media under the semi-permeable boundary conditions has not been reported by means of the non-local theory so far.

Stress analysis of a mode‐I crack in a 1D hexagonal quasicrystal based on the non‐local theory

AbstractThis paper studies the non‐local theory solution of a Mode‐I crack in a 1D hexagonal quasicrystal (QC) by using the generalized Almansi's theorem and the Schmidt method. Based on the Fourier transform, the boundary value problem is reduced to two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. The dual integral equations are derived. Finally, the expression of the phonon stress field and the phason stress field at the crack tip are obtained. Numerical examples are provided to show the effects of the lattice parameter and the crack length on the phonon stress field and the phason stress field near the crack tip. Different from the classical solution, that the present solution exhibits no stress singularity at the crack tips in a 1D hexagonal QC.

Fracture analysis of a Griffith crack under thermal-mechanical loading using nonlocal theory

In this paper, the nonlocal theory is utilized to deal with a Griffith crack problem under thermal-mechanical loading for the first time to eliminate stress singularity at the crack tip. A partially, thermally insulated Griffith crack subjected to a remote heat flux and mechanical loading is investigated. Fourier transform is employed to convert the mixed, boundary-value problem to two pairs of dual integral equations, which are then solved using the Schmidt method. The unknown jumps of displacement across crack faces are expressed by a series of Jacobi polynomials. The numerical results show that the stress at the crack tip is finite, and the maximum value occurs at a short distance from the crack tip. The results can be employed to build a maximum stress-based criterion instead of the traditional, stress intensity factor- or energy-based fracture criterion.

Vertical vibration of a rigid strip footing on viscoelastic half‐space

SummaryThis paper presents an analytical method for modeling the dynamic response of a rigid strip footing subjected to vertical‐only loads. The footing is assumed to rest on the surface of a viscoelastic half‐space; therefore, effects of hysteretic soil damping on the impedance of the foundation and the generated ground vibrations are considered in the solution. To solve the mixed boundary value problem, we use the Fourier transform to cast a pair of dual integral equations providing contact stresses, which are solved by means of Jacobi orthogonal polynomials. The resulting soil and footing displacements and stresses are obtained by means of the Fourier inverse transform. The solution provides more realistic estimates of footing impedance, compared to existing solutions for elastic soil, as well as of the attenuation of ground vibrations with distance of the footing. The latter is important for the estimation of machine vibration effects on nearby structures and installations.

Dynamic intensity factors for multiple permeable cracks in piezoelectric materials

AbstractThis article presents the dynamic intensity factors for multiple permeable Mode‐I cracks in piezoelectric materials plane by means of the generalized Almansi's theorem. The Fourier transform is used and the present problem is formulated into two pairs of dual integral equations. The dual integral equations are solved by the Schmidt method. The dynamic stress intensity factors (DSIFs) and the dynamic electric displacement intensity factor (DEDIF) at the crack tips are given. It is shown that the effects of the crack length, the distance between two parallel cracks and the circular frequency of incident wave near the crack tips on the DSIFs and DEDIF.

Dynamic analysis of two collinear permeable Mode-I cracks in piezoelectric materials based on non-local piezoelectricity theory

Purpose Based on the non-local piezoelectricity theory, this paper is concerned with two collinear permeable Mode-I cracks in piezoelectric materials subjected to the harmonic stress wave. The paper aims to discuss this issue. Design/methodology/approach According to the Fourier transformation, the problem is formulated into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. Findings Finally, the dynamic non-local stress and the dynamic non-local electric displacement fields near the crack tips are obtained. Numerical results are provided to illustrate the effects of the distance between the two collinear cracks, the lattice parameter and the circular frequency of the incident waves on the entire dynamic fields near the crack tips, which play an important role in designing new structures in engineering. Originality/value Different from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack tips in piezoelectric materials. It is found that the maximum stress and maximum electric displacement can be used as a fracture criterion.

Dynamic stress intensity factor of a rectangular crack in an infinite saturated porous medium: Mode I problem

A scattering problem of a harmonic plane compressional wave disturbed by a fluid filled rectangular crack, which is embedded in an infinite saturated poro-elastic solid, is investigated. With the aid of the two-dimensional Fourier integral transform technique, the original boundary value problem is formulated by a pair of dual integral equations in terms of the crack surface displacement. By expressing the crack surface displacement into a series of Jacobi polynomials, the solution of the dual integral equations is derived, where the coefficients involved are obtained with the aid of the Schmidt method. The stress intensity factors of the Mode I crack problem are obtained. Numerical results are performed to discuss the influence of the frequency of the incident compressional wave and the geometric parameter of the rectangular crack on the stress intensity factor and crack surface displacement in detail. The present analytical solutions may benefit future simplified and numerical studies.

Fracture analysis of a 3D rectangular permeable crack and two 3D rectangular permeable cracks in the orthotropic piezoelectric media

AbstractFracture analysis of a rectangular permeable crack and two 3D rectangular permeable cracks in the orthotropic piezoelectric media (OPM) is investigated by using the generalized Almansi's theorem and the Schmidt method. Utilizing the 2D Fourier transform, the problems reduce to the solution of three pairs of dual integral equations, which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the effects of the geometric shape of the rectangular crack and the distance between two rectangular cracks on the stress intensity factors (SIFs) and electric displacement intensity factors (EDIFs) in the OPM are analyzed.

Application of Dual Integral Equations in Heat Equation for Unbounded Plate

An analytical solution to a two-dimensional nonstationary nonhomogeneous heat equation in axially symmetrical cylindrical coordinates for an unbounded plate subjected to mixed boundary conditions of the first and second kinds has been obtained. The application of the Laplace transform (L-transform) and the separation of variables result in the solution to the initial mixed boundary-value problem as the solution to a pair of dual integral equations (DIEs) with an unknown function dependent on the L-transform parameter. The DIEs solution is proposed by using the known discontinuous integrals and an infinite series method. The Green’s function is used to determine the solution to the nonhomogeneous part of the problem.

Dynamic stress intensity factors of an interfacial crack in orthotropic elastic strips under impact loading conditions

AbstractThe present article deals with the investigation of elasto‐dynamic response of a finite crack under normal and shear impact loading situated at the interface of two semi‐infinite orthotropic strips. Laplace and Fourier Integral transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane which are solved using iterations in the low frequency domain. The analytical expressions of stress intensity factors of the crack at the interface problem are found. To determine time dependence of the parameters, these equations are inverted to yield the dynamic stress intensity factors for some fiber‐reinforced orthotropic composite materials using Bellman method and the method given by H. Dubner and J. Abate. The results are depicted through graphs for different particular cases.

Dynamic non‐local theory stress analysis of two equal and collinear mode‐I cracks in the orthotropic medium

AbstractIn this study, dynamic stress analysis of two equal and collinear mode‐I cracks in the orthotropic medium is conducted using the non‐local theory, the generalized Almansi's theorem and the Schmidt method. Fourier transform is applied to solve the problem, and the problem is converted to a pair of dual integral equations. The dynamic non‐local stress fields near the crack tips are obtained. The effects of the distance between the two equal and collinear cracks, the circular frequency of the incident waves the lattice parameter on the stress field near the crack tips are discussed. The present solution exhibits no stress singularity at the crack tips in the orthotropic medium, i.e. the dynamic non‐local stress field near the crack tips is finite. The obtained results play an important role in designing new composite structures in engineering.

Impact of a torsional load on a penny-shaped crack sandwiched between two elastic layers embedded in an elastic medium

The paper is focused on the effect of a sudden impact of a torsional load on a penny-shaped crack sandwiched between two elastic layers embedded in an elastic medium. The axisymmetric mixed boundary value problem is reduced to the problem of solving a pair of dual integral equations by using Hankel and Laplace transforms. Further, the integral equations are then reduced to a Fredholm integral equation of second kind which is solved numerically. Expression for the stress intensity factor at the tip of the crack is obtained and plotted for different parameters and materials.