Solution Of Dual Integral Equations Research Articles

A Griffith Crack at the Interface of an Isotropic and Orthotropic Half Space Bonded Together

In the past 53 years, many efforts have been contributed to develop and demonstrate the properties of reinforced composite materials. The ever-increasing use of composite materials through engineering structures needs the proper analysis of the mechanical response of these structures. In the proposed work, we have an exact form of Stress components and Displacement components to a Griffith crack at the interface of an Isotropic and Orthotropic half-space bounded together. The expression was evaluated in the vicinity of crack tips by using Fourier transform method but here these components have been evaluated with the help of Fredholm integral equations and then reduce to the coupled Fredholm integral equations. In this paper, we use the problem of Lowengrub and Sneddon and reduce it to dual integral equations. Solution of these equations through the use of the method of Srivastava and Lowengrub is reduced to coupled Fredholm integral equation. Further reduces the problem to decoupled Fredholm integral equation of 2nd kind. We get the solution of dual integral equations and the problem is reduced to coupled Fredholm integral equation. We find the solution of the Fredholm integral equation and reduce it to decoupled Fredholm integral equation of 2nd kind. The Physical interest in fracture design criterion is due to Stress and crack opening Displacement components. In the end, we can easily calculate the Stress components and Displacement components in the exact form.

Boundary effect on crack kinking in a magnetoelectroelastic strip with a central crack

In this study, the boundary effect on crack kinking in a magnetoelectroelastic strip with a central crack vertical to the boundary under in-plane loadings has been investigated. By using integral transform techniques the mixed boundary value problem of the crack is reduced to the solution of dual integral equations, which can be further reduced to solving Fredholm integral equations of the second kind. Analytical solutions can be obtained when the width of the magnetoelectroelastic strip are of infinite size. The asymptotic fields around crack-tips have been obtained for the electrically and magnetically impermeable and permeable cracks, respectively. Crack kinking phenomenon is investigated by applying the criterion of maximum hoop stress intensity factor. Numerical results show that the material properties, the geometry of the cracked strip and the electric–magnetic-mechanical loading significantly affect the singular field distributions around the crack. Crack kinking is found for the permeable crack case while no kinking for the impermeable crack.

Solution of a dual integral equation arising in the contact problems of elasticity theory with the full Fourier series as the right-hand side

A class of dual integral equations is analyzed which arises in solution of a wide range of plane and antiplane contact problems of elasticity theory for a half-plane with functionally graded coating. In particular, a similar equation arises in solution of the contact problem on indentation in the presence of tangential stresses on a surface. The solution of the dual integral equation is sought in the form of a sum of even and odd functions. It makes possible to reduce the problem to independent solution of two dual integral equations over odd and even functions. Kernel transform of these equations is approximated by a product of fractional quadratic functions. The solution of dual integral equations is constructed in approximated analytical form by the bilateral asymptotic method. The expressions obtained are asymptotically exact for small and large values of a characteristic geometrical parameter.

Non-local theory behavior of multiple cracks in a functionally graded piezoelectric medium

In this paper, the non-local theory solution of multiple cracks in a functionally graded piezoelectric medium (FGPM) under the permeable electric boundary condition have been investigated. Overcoming the mathematical difficulty, a one-dimensional non-local kernel is used instead of a two-dimensional one for the fracture problem to obtain stress and electric displacement fields near the crack tips. By using Fourier transform techniques, the present problem was reduced to the solution of dual integral equations, which unknown variable is jumps of displacement across the crack surface. The present solution exhibits no stress and electric displacement singularities at the crack tips when non-local was used to investigate the problem. The non-local solution yields a finite hoop stress at the crack tips, thus allows us to use the calculated maximum stress as a fracture criterion.

P-wave interaction with a pair of rigid strips embedded in an orthotropic strip

The present paper is concerned with the problem of scattering of the P-wave by two co-planer finite rigid strips placed symmetrically in an infinitely long orthotropic strip. Using the Hilbert transform technique, the mixed boundary value problem has been reduced to the solution of dual integral equations which has finally been reduced to the solution of a Fredholm integral equation of the second kind. Solving this integral equation numerically, stress intensity factors have been calculated at the inner and outer edges of the rigid strips, and the vertical displacement outside the strips has been calculated and plotted graphically to show the effect of material orthotropy.

Approximate solution of dual integral equations using Chebyshev polynomials

ABSTRACTThe aim of the present work is to introduce solution of special dual integral equations by the orthogonal polynomials. We consider a system of dual integral equations with trigonometric kernels which appear in formulation of the potential distribution of an electrified plate with mixed boundary conditions and convert them to Cauchy-type singular integral equations. We use the Chebyshev orthogonal polynomials to construct approximate solution for Cauchy-type singular integral equations which will solve the main dual integral equations. Numerical results demonstrate effectiveness of this method.

Boundary effect on crack kinking in a piezoelectric strip with a central crack

In this paper, the boundary effect on crack kinking in a piezoelectric strip with a central crack vertical to the boundary under in-plane loadings has been investigated. By using integral transform techniques the present mixed boundary value problem was reduced to the solution of dual integral equations, which can be further reduced to solving Fredholm integral equations. Analytical solutions can be obtained when the width of the piezoelectric strip are of infinite size. The asymptotic fields around crack tips have been provided for the electrically impermeable and permeable crack boundary conditions, respectively. Crack kinking phenomenon is investigated by applying the criterion of maximum hoop stress intensity factor. Numerical results show that the material properties, the geometry of the cracked strip and the electric-mechanical loading significantly affect the singular field distributions around the crack. Crack kinking is found for the impermeable crack case while no kinking for the permeable crack. Crack kinking angles for different piezoelectric materials under the impermeable crack surface condition have been predicted with certain loading and boundary conditions.

Capacitance analysis of a circular disk on a dielectric layer

Abstract Using the Hankel transform, the mixed boundary value problem of a conducting, circular disk on a dielectric layer in an electric field was reduced to the solution of dual integral equations. Analytical expressions of the capacitance of the electrical system to the second approximation were derived for the ratio of the layer thickness to the disk radius being larger than one and the ratio being less than one, respectively. Both expressions yield numerical results in accord with those reported in literature. The capacitance of a circular disk on a semi-infinite dielectric can be used for a ratio larger than or equal to 4. Using the results, the thickness effect on electrowetting of a disk-like conducting film on a dielectric substrate was discussed. There exists an upper bound of the electric voltage applied to the film above which the saturation of contact angle or local instability will occur.

Indentation of piezoelectromagneto-elastic half-space by a truncated conical punch

The indentation for piezoelectromagneto-elastic half-space by a truncated conical punch is investigated. The authors formulate the problem as the solution of dual integral equations, which are solved exactly. Analytical full-field solutions of a magnetoelectro-elastic half-space are presented. The physical quantities, which characterize the contact region and the stress singularity, are obtained. The electric displacement and magnetic induction intensity factors are presented analytically. The relations between the total force, electric and magnetic potentials under the punch and depth of penetration, radius of contact region are presented graphically. The effect of material constants in piezoelectric/magneto-strictive composite upon contact area and depth of penetration of the punch is discussed.

Mathematical model for crack arrest of a transversely cracked piezoelectromagnetic strip – Part I

A magnetic, electric and mechanical yield model is proposed for a cracked piezoelectromagnetic ceramic narrow strip. The strip is subjected to anti-plane mechanical and in-plane electric and magnetic loads, consequently the crack opens in self-similar fashion forming a magnetic, a saturation and a slide zone ahead each tip. These in turn are arrested by prescribing a magnetic, electric and mechanical load, respectively. Employing Fourier integral transform the problem reduces to the solution of three dual integral equations. The solution of dual integral equations is then expressed in terms of Fredholm integral equation of second kind. Expressions are derived for yield induction zone, slide-yield zone and saturation zone lengths, energy release rate. A case study is carried for BaTiO3–CoFe2O4 and results are presented graphically. It is shown that proposed model is capable of crack opening arrest under small-scale-yielding.

On the solution of dual integral equations

A quick method of solution of dual integral equations involving a kernel comprised of trigonometric functions is explained. Certain solvability criteria are obtained in terms of forcing functions for the unique solution of the dual integral equations.

Finite Griffith crack propagating in a non-homogeneous medium

The problem of a Griffith crack of constant length propagating at a uniform speed in a plane non-homogeneous medium under uniform load is investigated. The equilibrium equations for the non-homogeneous medium are solved by using the Fourier transforms and then the problem is reduced to the solution of dual integral equations. Solving the dual integral equations we obtain the expression for the dynamic stress intensity factor at the edge of the crack. Finally the numerical results for the stress intensity factor are obtained which are displayed graphically to show the effect of the material non-homogeneity on the stress intensity factor.

Diffraction of P-wave by crack in an elastic strip at asymmetric position

Scattering of P-wave by a Griffith Crack in a long elastic strip situated at an asymmetric position has been studied in this paper. Using Fourier transform, the mixed boundary value problem has been reduced to the solution of dual integral equations which finally has been reduced to the solution of a Fredholm integral equation of second kind. Solving this integral equation numerically, stress intensity factors have been calculated and illustrated graphically to show the effect of asymmetry of position of the crack.

The study of dual integral equations with generalized Legendre functions

Closed form solutions for dual integral equations involving generalized Legendre functions as kernels are obtained. Connected to these dual integral equations an exact solution for dual integral equations involving sine functions as kernels is also obtained. Properties of generalized Legendre functions and the inversion theorem for the generalized Mehler–Fock transforms are used to obtain the solution of dual integral equations

Closed-form solutions for a rectangular layer with central crack under anti-plane shear stress

We consider the problem of determining the stress distribution in a finite rectangular elastic layer containing a Griffith crack which is opened by internal shear stress acting along the length of the crack. The mode III crack is assumed to be located in the middle plane of the rectangular layer. The following two problems are considered: (A) the central crack is perpendicular to the two fixed lateral surfaces and parallel to the other two stress-free surfaces; (B) all the lateral surfaces of the rectangular layer are clamped and the central crack is parallel to the two lateral surfaces. By using Fourier transformations, we reduce the solution of each problem to the solution of dual integral equations with sine kernels and a weight function which are solved exactly. Finally, we derive closed-form expressions for the stress intensity factor at the tip of the crack and the numerical values for the stress intensity factor at the edges of the cracks are presented in the form of tables.

The breaking of a non-homogeneous fiber embedded in an infinite non-homogeneous medium

The torsion of an infinite non-homogeneous elastic cylindrical fiber, containing a penny-shaped crack embedded in an infinite non-homogeneous elastic material is considered. The cylinder and elastic medium have different shear moduli. Using integral transformation techniques the solution of the problem is reduced to the solution of dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind, which is solved numerically. Closed form expressions are obtained for the stress intensity factor and numerical values for the stress intensity factors are graphed to demonstrate the effect of non-homogeneity of the fiber and infinite medium. In the end the stress singularity is obtained when the crack touches the infinite nonhomogeneous medium (matrix).

Ian Naismith Sneddon, O.B.E. 8 December 1919 – 4 November 2000

Ian Sneddon was introduced to problems in the linear theory of elasticity involving the indentation of a surface and the extension of a crack through his work for the Ministry of Supply during World War II. He maintained a lifelong research interest in this area and also made distinguished contributions to a range of related analytical techniques, notably the application of integral transforms and the solution of dual integral equations. His many books, extensive travels and engaging personality made him very well known internationally, and he developed particularly fruitful contacts in the USAand Poland. He gave devoted service to the University of Glasgow, where the bulk of his career was spent, and played an active part in the cultural life of Scotland.

Mode I field intensity factors of infinitely long strip in piezoelectric media

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing a Griffith crack under in-plane normal loading within the framework of linear piezoelectricity. The potential theory method and Fourier transforms are used to reduce the problem to the solution of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the field intensity factors are obtained, and the influences of the electric fields for PZT-6B piezoelectric ceramic are discussed.

A Note on some Dual Integral Equations

In the present paper, appropriate solutions of dual integral equations with trigonometric function as kernel are obtained by considering the behaviour of one of the integrals of the pair at the ‘turning point’.

EFFECT OF FLUID DEPTH ON THE HYDROELASTIC VIBRATION OF FREE-EDGE CIRCULAR PLATE

This paper is concerned with the effect of water depth on the free vibration of free-edge circular plates resting on a free fluid surface. The problem addressed is formulated by using the Hankel transformation method, which leads to dual integral equations. The solution of dual integral equations is solved numerically by using Fourier–Bessel series. The fluid is assumed to be inviscid and incompressible. The Kirchhoff theory of plates is used to model the elastic thin plate. Numerical results are given in non-dimensional form for free-edge circular plates, in order to be ready-to-use in applications. To validate the theoretical results, experiments were carried out. Experimental results are in good agreement with theoretical results. It is found that the effect of the fluid depth can be neglected when the fluid depth is greater than the diameter of the circular plates but becomes significant as the water depth decreases. It is also observed in the experimental results that the fluid damping increases as the water depth decreases.