Gauss Chebyshev Integration Formula Research Articles

On the receding contact between an inhomogeneously coated elastic layer and a homogeneous half-plane

The general double contact problem of an inhomogeneously coated elastic layer indented against a homogeneous half-plane by a rigid punch is investigated. The problem is solved under the assumptions that the contact at both interfaces is frictionless, the three materials possess different shear modulus, and the shear modulus of the functionally graded coating varies exponentially. By the standard method of Fourier integral transforms, the problem is reduced to a system of singular integral equations, which are subsequently transformed into algebraic ones by Gauss–Chebyshev integration formulas. Massive numerical experiments are performed to quantify the influence of stiffness distribution in coating, material and geometric properties of the compound structure, and indentation load on the contact pressure and contact length at both surfaces of contact. The results suggest that in the case of a hard and thick coating large pressure reduction can be achieved for the receding contact at the layer/substrate interface.

Saint-Venant torsion of a circular bar with radial cracks incorporating arbitrarily varied surface elasticity

ABSTRACTWe analytically investigate the contribution of arbitrarily varied surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The varied surface elasticity is incorporated by using a modified version of the continuum-based surface/interface model of Gurtin and Murdoch. In our discussion, the surface shear modulus is assumed to be arbitrarily varied along the crack surfaces. Both internal and edge cracks are studied. By using Green's function method, the boundary value problem is reduced to the Cauchy singular integro-differential equation of first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials, and the collocation method. The torsion problem of a cylinder containing two symmetric collinear radial cracks of equal length with symmetrically varied surface elasticity is also solved by using a similar method. Our numerical results indicate that the variation of the surface elasticity exerts a significant influence on the strengths of the logarithmic stress singularity at the crack tips, the torsional rigidity, and the jump in warping function.

Continuous contact problem of a functionally graded layer resting on an elastic half-plane

In this study, the continuous contact problem of a functionally graded layer resting on an elastic half-plane and loaded by a rigid rectangular stamp is examined. The problem is solved assuming that the functionally graded (FG) layer is isotropic and the shear modulus and mass density vary exponentially throughout the layer’s thickness. However, the body force of the elastic half-plane is neglected. In addition, it is assumed that all surfaces are frictionless and only compressive stress is transferred along the contact surfaces. The mathematical problem is reduced to a singular integral equation in which the contact stress under the rigid stamp is unknown using the Fourier integral transform and boundary conditions related to the problem. This singular integral equation is solved numerically using the Gauss–Chebyshev integration formula. The dimensionless contact stress under the rigid stamp, the initial separation loads and the initial separation distances between the FG layer and the elastic half-plane are obtained for various dimensionless quantities.

Surface strain gradient effects in the torsion of a circular bar with radial cracks

We study the contribution of surface strain gradient elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The surface strain gradient elasticity is incorporated by using an enriched version of the continuum-based surface/interface model of Gurtin and Murdoch. By using Green’s function method, the original boundary value problem is reduced to a Cauchy singular integro-differential equation which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the presence of surface strain gradient elasticity on the crack faces, the stresses are bounded at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface strain gradient elasticity is also solved by using a similar method. Numerical results indicate that the surface strain gradient effect exerts a significant influence on the torsional rigidity and the jump in warping function. In particular, the jump in warping function forms a cusp shape with zero enclosed angle at the crack tips.

Saint-Venant torsion of a circular bar with a bridged radial crack incorporating surface elasticity

We examine the contribution of crack bridging and surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The surface effect on the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. The bridging force is assumed proportional to the crack opening displacement, while the bridging stiffness is allowed to vary arbitrarily along the crack. Both an internal crack and an edge crack are studied. By using the Green’s function method, the boundary value problem is reduced to a first-order Cauchy singular integro-differential equation. After the utilization of the Gauss–Chebyshev integration formula and the expansion of both the unknown dislocation density and the known variable bridging stiffness into Chebyshev polynomials, the singular integro-differential equation is solved numerically by using the collocation method. The stresses exhibit a weak logarithmic singularity rather than a strong square root singularity at the crack tips due to the surface effect. The strengths of the logarithmic singularity at the crack tips, the reduction in torsional rigidity and the jump in warping function across the crack faces are calculated. Our results also show that both surface elasticity and crack bridging will stiffen the cracked cylinder and that the stiffening effect of crack bridging is remarkable in the case when the crack is very long or is extremely close to the lateral surface of the cylinder.

Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.

Contact problem for a functionally graded layer indented by a moving punch

In this study moving contact problem for a rigid cylindrical punch and a functionally graded layer is considered. The punch subjected to concentrated normal force, and moves steadily with a constant subsonic velocity on the boundary. Poisson׳s ratio is taken as constant, and both the elasticity modulus and the mass density are assumed to vary exponentially in depth direction. By using Fourier transform and boundary conditions, the governing equations are reduced to a Cauchy singular integral equation. The numerical solution of the singular integral equation is obtained by using Gauss–Chebyshev integration formulas. Numerical results for the contact area, the contact stress and the normal stresses are given. This study is limited in that the elasticity modulus and the mass density vary with the same function. However, it is the first attempt to investigate the moving contact problem with FGMs.

Analysis of the Antiplane Problem of Functionally Graded Coating with Generally Oriented Crack

Investigated is the anti-plane problem of functionally graded strip with an inclined crack sandwiched between a half infinite substrate and an elastic strip. Fourier transforms and integral equation are used to reduce the crack problem to a system of singular integral equation with Cauchy kernel, which is solved numerically by application of the Gauss-Chebyshev integration formula. Numerical results show the effects of the material gradient parameter and crack configuration on the field intensity factors of the crack.

Receding contact problem for a coated layer and a half-plane loaded by a rigid cylindrical stamp

In this study, the frictionless contact problem between a coated layer and a half-plane is considered when they are pressed by a rigid cylindrical stamp. Upon loading, two unknown contact widths and contact pressures occur on the contact areas the advanced contact area between the stamp and the coating; and the receding contact area between the layer and the half-plane. This problem is reduced to two singular integral equations by using the Fourier transform and applying the boundary conditions of the problem. The numerical solution of the system of singular integral equations is obtained by applying the Gauss–Chebyshev integration formulas.

Contact problem of a functionally graded layer resting on a Winkler foundation

The contact problem for a functionally graded layer supported by a Winkler foundation is considered using linear elasticity theory in this study. The layer is loaded by means of a rigid cylindrical punch that applies a concentrated force in the normal direction. Poisson’s ratio is taken as constant, and the elasticity modulus is assumed to vary exponentially through the thickness of the layer. The problem is reduced to a Cauchy-type singular integral equation with the use of Fourier integral transform technique and the boundary conditions of the problem. The numerical solution of the integral equation is performed by using Gauss–Chebyshev integration formulas. The effect of the material inhomogeneity, stiffness of the Winkler foundation and punch radius on the contact stress, the contact area and the normal stresses are given.

Mode III fracture problem of an FGPM strip bonded to an FGPM half-plane with embedded arbitrarily oriented cracks

This article studies the anti-plane crack problem of a functionally graded piezoelectric material (FGPM) cracked strip bonded to an FGPM-cracked half-plate. Each crack is arbitrarily oriented. The material properties are assumed to be exponential forms varied in the direction normal to the interface. Using Fourier transforms, the problem can be reduced to a system of singular integral equations, which are solved numerically by applying the Gauss–Chebyshev integration formula. Numerical calculations are carried out to obtain the crack driving forces, such as the stress intensity factors (K), energy density factors (S), and the energy release rates (G), at the crack tips. The influences of the non-homogeneous parameter, crack orientation, electrical displacement, and the crack length on the normalized factors and rates are graphically discussed.

Crack-contact problem for an elastic layer with rigid stamps

The plane crack-contact problem for an infinite elastic layer with two symmetric rectangular rigid stamps on its upper and lower surfaces is considered. The elastic layer having an internal crack parallel to its surfaces is subjected to two concentrated loads p on its upper and lower surfaces trough the rigid rectangular stamps and a pair of uniform compressive stress <TEX>$p_0$</TEX> along the crack surface. It is assumed that the contact between the elastic layer and the rigid stamps is frictionless and the effect of the gravity force is neglected. The problem is reduced to a system of singular integral equations in which the derivative of the crack surface displacement and the contact pressures are unknown functions. The system of singular integral equations is solved numerically by making use of an appropriate Gauss-Chebyshev integration formula. Numerical results for stress-intensity factor, critical load factor, <TEX>$\mathcal{Q}_c$</TEX>, causing initial closure of the crack tip, the crack surface displacements and the contact stress distribution are presented and shown graphically for various dimensionless quantities.

Mode III fracture problem of two arbitrarily oriented cracks located within two bonded functionally graded material strips

This paper shows the anti-plane crack problem of two bonded functionally graded material (FGM) strips. Each strip contains an arbitrarily oriented crack. The material properties of the strips are assumed in exponential forms varied in the direction normal to the interface. After employing the Fourier transforms, the unknowns are solved from the interface conditions, boundary conditions and the condition on the crack surfaces. The problem can then be reduced to a system of singular integral equations, which are solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. In the discussions, several degenerated problems are considered to demonstrate the influence of the non-homogeneous parameters, crack orientations, edge effects and the crack interactions on the normalized intensity factors. In general, the factors are larger when crack tips are located in stronger material. Also, the factors increase as the crack is oriented in the direction normal to the interface. The conclusions made in this research can be used to evaluate the safety of two bonded strips once the cracks exist inside the structure.

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss–Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.

An internal crack problem for an infinite elastic layer

The elastostatic plane problem of an infinite elastic layer with an internal crack is considered. The elastic layer is subjected to two different loadings, (a) the elastic layer is loaded by a symmetric transverse pair of compressive concentrated forces P/2, (b) it is loaded by a symmetric transverse pair of tensile concentrated forces P/2. The crack is opened by an uniform internal pressure p0 along its surface and located halfway between and parallel to the surfaces of the elastic layer. It is assumed that the effect of the gravity force is neglected. Using an appropriate integral transform technique, the mixed boundary value problem is reduced to a singular integral equation. The singular integral equation is solved numerically by making use of an appropriate Gauss–Chebyshev integration formula and the stress-intensity factors and the crack opening displacements are determined according to two different loading cases for various dimensionless quantities.

Crack in a bonded functionally graded magneto-electro-elastic strip

Abstract Investigated is the anti-plane problem of functionally graded magneto-electro-elastic strip sandwiched between two functionally graded strips. It is assumed that the material properties vary exponentially with the coordinate parallel to the crack. The crack is assumed to be either magneto-electrically impermeable or permeable. Fourier transforms are used to reduce the crack problems to a system of singular integral equations, which is solved numerically by application of the Gauss–Chebyshev integration formula. Numerical results show the effects of the material gradient parameter and crack configuration on the field intensity factors of the crack.

Mode III fracture problem of an arbitrarily oriented crack in an FGPM strip bonded to a homogeneous piezoelectric half plane

This paper studies the Mode III electric-elastic field of a cracked functionally graded piezoelectric strip bonded to a homogeneous piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties of the strip vary along the strip thickness in exponential forms. By using the Fourier transform, the problem can be formulated to a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects come from the edge, crack orientations and the nonhomogeneous material parameter on intensity factors are discussed graphically.

Mode III crack problems of two bonded functionally graded strips with internal cracks

This paper deals with the anti-plane problem of two bonded functionally graded finite strips. Each strip contains an internal crack normal to the interface. The material properties of two strips are assumed to vary along the direction of the crack lines. A system of singular integral equations is derived and then solved numerically by using Gauss–Chebyshev integration formula. The influences of nonhomogeneous parameters, crack interactions and two edge conditions on the mode III stress intensity factors are investigated.

Mode III fracture problem of an arbitrarily oriented crack in a FGPM strip bonded to a FGPM half plane

This paper studies the mode III electro-elastic field of a cracked functionally graded piezoelectric strip bonded to a functionally graded piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties along x-axis vary in exponential form. By using the Fourier transform, the problem can be formulated into a system of singular integral equations and solved by applying the Gauss–Chebyshev integration formula. The effects come from the edge, crack orientation and the nonhomogeneous material parameters on intensity factors are discussed graphically.

Frictionless contact problem for a layer on an elastic half plane loaded by means of two dissimilar rigid punches

The contact problem for an elastic layer resting on an elastic half plane is considered according to the theory of elasticity with integral transformation technique. External loads P and Q are transmitted to the layer by means of two dissimilar rigid flat punches. Widths of punches are different and the thickness of the layer is h. All surfaces are frictionless and it is assumed that the layer is subjected to uniform vertical body force due to effect of gravity. The contact along the interface between elastic layer and half plane will be continuous, if the value of load factor, <TEX>${\lambda}$</TEX>, is less than a critical value, <TEX>${\lambda}_{cr}$</TEX>. However, if tensile tractions are not allowed on the interface, for <TEX>${\lambda}$</TEX> > <TEX>${\lambda}_{cr}$</TEX> the layer separates from the interface along a certain finite region. First the continuous contact problem is reduced to singular integral equations and solved numerically using appropriate Gauss-Chebyshev integration formulas. Initial separation loads, <TEX>${\lambda}_{cr}$</TEX>, initial separation points, <TEX>$x_{cr}$</TEX>, are determined. Also the required distance between the punches to avoid any separation between the punches and the layer is studied and the limit distance between punches that ends interaction of punches, is investigated. Then discontinuous contact problem is formulated in terms of singular integral equations. The numerical results for initial and end points of the separation region, displacements of the region and the contact stress distribution along the interface between elastic layer and half plane is determined for various dimensionless quantities.