Gauss-Chebyshev Quadrature Formulas Research Articles

Investigation on the 2D Contact of Multilayer Functionally Graded Piezoelectric Material Coating Under Conducting Indenters

Considering the contact problem for functionally graded piezoelectric material (FGPM) coating under the different kinds of conducting indenter. Effects of gradient index on the contact mechanical behavior of functionally graded piezoelectric coating were investigated. In this paper, a model of multilayer functionally graded piezoelectric coatings is established. The contact problem of multilayer functionally graded piezoelectric coatings was transformed into singular integral equation(s) by using the Fourier integral transform technology and transfer matrix method. Gauss-Chebyshev quadrature formula is used to obtain the surface stress distribution and charge distribution in multilayer functionally graded piezoelectric coating-substrate system under a rigid conducting flat indenters and a conducting cylindrical indenters. According to the numerical results, the effects of different variations of material parameters of FGPM coating on the indentation and electrical potential are analyzed. The distribution of stress and electrical displacement in functionally graded piezoelectric coating are analyzed. The results show that the different variations of material parameters of functionally graded piezoelectric material coating have an important influence on the contact behavior in the structures.

Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

A hypersingular integral equations (HSIEs) of the first kind on the interval [ 1 ; 1 ] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.

Cylindrical indentation of an elastic bonded layer with surface tension

Surface tension is a prominent factor for the deformation of solids at micro-/nano-scale. This paper investigates the effects of surface tension on the two-dimensional contact problems of an elastic layer bonded to the rigid substrate. Under the plane strain assumption, the elastic field induced by a uniformly distributed pressure within a finite width is formulated by applying the Fourier integral transform, and the limiting process leading to the solutions for a line force brings the requisite surface Greens function. For the indentation of an elastic layer by a rigid cylinder, the corresponding singular integral equation is derived, and subsequently solved by using an effective numerical method based on Gauss-Chebyshev quadrature formula. It is found from the theoretical and numerical results that the existence of surface tension strongly enhances the hardness of the elastic layer and significantly affects the distribution of contact pressure, when the size of contact region is comparable to the elastocapillary length. In addition, an approximated relationship between external load and half-width of contact is generalized in an explicit and concise form, which is useful and convenient for practical applications.

Indentation load–depth relation for an elastic layer with surface tension

The load–depth relation is a fundamental requisite in nanoindentation tests for thin layers; however, the effects of surface tension are seldom included. This paper concerns micro-/nano-sized indentation by a rigid sphere of a bonded elastic layer. The surface Green’s function with the incorporation of surface tension is first derived by applying the Hankel integral transform, and subsequently used to formulate the governing integral equation for the axisymmetric contact problem. By using a numerical method based on the Gauss–Chebyshev quadrature formula, the singular integral equation is solved efficiently. Several numerical results are presented to investigate the influences of surface tension and layer thickness on contact pressure, surface deformation, and bulk stress. It is found that when the size of contact is comparable to the ratio of surface tension to elastic modulus, the contribution of surface tension to the load–depth relation becomes quite prominent. With the help of a parametric study, explicit general expressions for the indentation load–depth relation as well as the load–contact radius relation are summarized, which provide the groundwork for practical applications.

A dimension-wise method and its improvement for multidisciplinary interval uncertainty analysis

• The Chebyshev polynomial approximation is used in multidisciplinary interval uncertainty analysis. • The dimension-wise method is improved through an iterative procedure to address systematic coupling effects. • The high accuracy of the proposed iterative dimension-wise method is demonstrated by numerical examples. Considering that uncertain factors widely exist in practical engineering, this study develops an improved dimension-wise method for multidisciplinary interval uncertainty analysis, in which the extremums of each interval variable for determining systematical response bounds are solved using the Chebyshev polynomial approximation and iterative criterion. First, Chebyshev basis functions are involved to construct the approximate relation between the system output variables and initial interval parameters dimension by dimension in multidisciplinary frameworks. The Gauss–Chebyshev quadrature formulas are then utilized to confirm coefficients of the fitting function. Due to the weakness of traditional dimension-wise method, that is, it ignores the coupling effects of uncertain variables, a new iterative dimension-wise method (IDWM) where the nominal states of intervals can be updated in each iteration, is proposed. Discussions on efficiency and accuracy are further expounded. Both numerical and engineering examples are eventually given to demonstrate the usage and validity of the developed methodology, and results indicate that the presented IDWM has a superiority in uncertainty propagation problems of multidisciplinary issues.

Analytic relations for two-dimensional indentations with surface tension

Surface tension plays a significant role in micro- and nanoindentation tests. Based on the solution of a concentrated force acting on an elastic half-plane with surface tension, the two-dimensional indentations of an elastic half-plane by a cylindrical indenter, a wedge indenter and a flat-ended indenter are formulated, and by employing the Gauss–Chebyshev quadrature formula, the corresponding singular integral equations are numerically solved. For each indentation, the analytic solutions of two limit cases considering only bulk elasticity or surface tension are derived. Then through simple combinations of the results of two limit cases, the analytic relations between load and contact half-width or indent depth are formulated for the general case, and compared with the numerical results. The analytic relations established in this paper facilitate better characterization of mechanical properties of materials by micro/nanoindentation tests.

Effects of surface tension on the nanoindentation with a conical indenter

In the present paper, the nanoindentation of an elastic half-space by a conical indenter is investigated with the influence of surface tension. Based on the solution of a point force acting on an elastic half-space with surface tension, the singular integral equation of this problem is formulated and is then solved numerically by using the Gauss–Chebyshev quadrature formula. Surface tension flattens the pressure distribution in the contact region. Compared to the result of the classical elasticity model, surface tension evidently decreases the normal displacement on the surface of the half-space. The explicit relations between load and contact radius, and between load and indent depth are derived, which are helpful to characterize the mechanical properties of the materials in nanoindentation.

General Relations of Indentations on Solids With Surface Tension

The conventional contact mechanics does not account for surface tension; however, it is important for micro- or nanosized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half-space by a rigid sphere, cone, and flat-ended cylinder are investigated, and the corresponding singular integral equations are formulated. Due to the complicated structure of the integral kernel, it is difficult to obtain their analytical solutions. By using the Gauss–Chebyshev quadrature formula, the integral equations are solved numerically first. Then, for each indenter, the analytical solutions of two limit cases considering only the bulk elasticity or surface tension are presented. It is interesting to find that, through a simple combination of the solutions of two limit cases and fitting the direct numerical results, the dependence of load on contact radius or indent depth for general case can be given explicitly. The results incorporate the contribution of surface tension in contact mechanics and are helpful to understand contact phenomena at micro- and nanoscale.

Micro/Nanocontact Between a Rigid Ellipsoid and an Elastic Substrate With Surface Tension

For micro/nanosized contact problems, the influence of surface tension becomes prominent. Based on the solution of a point force acting on an elastic half space with surface tension, we formulate the contact between a rigid ellipsoid and an elastic substrate. The corresponding singular integral equation is solved numerically by using the Gauss–Chebyshev quadrature formula. When the size of contact region is comparable with the elastocapillary length, surface tension significantly alters the distribution of contact pressure and decreases the contact area and indent depth, compared to the classical Hertzian prediction. We generalize the explicit expression of the equivalent contact radius, the indent depth, and the eccentricity of contact ellipse with respect to the external load, which provides the fundament for analyzing nanoindentation tests and contact of rough surfaces.

Computing rational Gauss–Chebyshev quadrature formulas with complex poles: The algorithm

We provide an algorithm to compute arbitrarily many nodes and weights for rational Gauss–Chebyshev quadrature formulas integrating exactly in spaces of rational functions with complex poles outside [−1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O ( n ) . This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions on [−1, 1] with arbitrary complex poles outside this interval.

Analysis of mode III crack perpendicular to the interface between two dissimilar strips

The present work is concerned with the problem of mode III crack perpendicular to the interface of a bi-strip composite. One of these strips is made of a functionally graded material and the other of an isotropic material, which contains an edge crack perpendicular to and terminating at the interface. Fourier transforms and asymptotic analysis are employed to reduce the problem to a singular integral equation which is numerically solved using Gauss–Chebyshev quadrature formulae. Furthermore, a parametric study is carried out to investigate the effects of elastic and geometric characteristics of the composite on the values of stress intensity factor.

Algorithm 882

We present a numerical procedure to compute the nodes and weights in rational Gauss-Chebyshev quadrature formulas. Under certain conditions on the poles, these nodes are near best for rational interpolation with prescribed poles (in the same sense that Chebyshev points are near best for polynomial interpolation). As an illustration, we use these interpolation points to solve a differential equation with an interior boundary layer using a rational spectral method. The algorithm to compute the interpolation points (and, if required, the quadrature weights) is implemented as a Matlab program.

Performance evaluation of high rate space–time trellis-coded modulation using Gauss–Chebyshev quadrature technique

The performance analysis of high rate space-time trellis-coded modulation (HR-STTCM) using the Gauss-Chebyshev quadrature technique is presented. HR-STTCM is an example of space-time codes that combine the idea used in trellis coded modulation (TCM) design that is signal set expansion and set partitioning into its construction. HR-STTCM construction is based on the concatenation of an outer TCM encoder and inner space-time block code. This paper evaluates the exact pairwise error probability of HR- STTCM based on the Gauss-Chebyshev quadrature formula. Comparison of numerical and simulation results shows that the proposed method is accurate. The method used is shown to be computationally simpler than those in the literature.

Rational Gauss-Chebyshev quadrature formulas for complex poles outside $[-1,1

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [-1,1] to arbitrary complex poles outside [-1,1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [-1,1].

On computing rational Gauss-Chebyshev quadrature formulas

We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside [ − 1 , 1 ] [-1,1] . Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O ( n ) O(n) . This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on [ − 1 , 1 ] [-1,1] with arbitrary real poles outside this interval.

Analysis of vertical cracking phenomena in tensile-strained epitaxial film on a substrate: Part I. Mathematical formulation

This paper presents an analysis of a single vertical crack and periodically distributed vertical cracks in an epitaxial film on a semi-infinite substrate where the cracks penetrate into the substrate. The film and substrate materials have different anisotropic elastic constants, necessitating Stroh formalism in the analysis. The misfit strain due to the lattice mismatch between the film and the substrate serves as the driving force for crack formation. The solution for a dislocation in an anisotropic trimaterial is used as a Green function, so that the cracks are modeled as the continuous distributions of dislocations to yield the singular integral equations of Cauchy-type. The Gauss–Chebyshev quadrature formula is adopted to solve the singular integral equations numerically. Energy arguments provide the critical condition for crack formation, at which the cracks are energetically favorable configurations, in terms of the ratio of the penetration depth into the substrate to the film thickness, the ratio of the spacing of the periodic cracks to the film thickness, and the generalized Dundurs parameters between the film and substrate materials.

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n − 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi-infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.

Simple algorithm to calculate T-F electron emission current density

The Gauss-Chebyshev quadrature formula is suggested to calculate the transparency of the potential barrier at the metal-vacuum boundary. It is shown that in the most interesting cases ONE node is enough to obtain accuracy of a few percent when calculating thermo-field electron emission current density.

Dynamic crack propagation between two bonded orthotropic plates

The problem of crack propagation along the interface of two bonded dissimilar orthotropic plates is considered. Using Galilean transformation, the problem is reduced to a quasistatic one. Then, using Fourier transforms and asymptotic analysis, the problem is reduced to a pair of singular integral equations with Cauchy‐type singularity. These equations are solved using Gauss‐Chebyshev quadrature formulae. The dynamic stress intensity factors are obtained in closed form expressions. Furthermore, a parametric study is introduced to investigate the effect of crack growth rate and geometric and elastic characteristics of the plates on values of dynamic stress intensity factors.

Convergence Acceleration of Gauss–Chebyshev Quadrature Formulae

The aim of this paper is to accelerate, via extrapolation methods, the convergence of the sequences generated by the Gauss–Chebyshev quadrature formula applied to functions holomorphic in ]−1,1[ and possessing, in the neighborhood of 1 or −1, an asymptotic expansion with log (1±x)(1±x)α, (1±x)α, α>−1, as elementary elements.