Line Inclusions in Anisotropic Elastic Solids

1989 ◽  
Vol 56 (3) ◽  
pp. 556-563 ◽  
Author(s):  
Qianqian Li ◽  
T. C. T. Ting
Keyword(s):  
Integral Equation ◽  
Numerical Solutions ◽  
Form Solution ◽  
Fredholm Integral Equations ◽  
Homogeneous Field ◽  
Elastic Solids ◽  
The Matrix ◽  
The Difference ◽  
Anisotropic Elastic ◽  
Line Inclusion

A line inclusion located at x2 = 0, |x1| < 1 in the anisotropic elastic medium of infinite extent under uniform loading at infinity is considered. Stroh’s formalism is used to find the displacement and stress fields. The inclusion can be rigid or elastic. Conditions on the loading under which the line inclusion does not disturb the homogeneous field are derived. For the rigid inclusion, a real form solution is obtained for the stress and displacement along x2 = 0. When the inclusion is elastic (and anisotropic), a pair of singular Fredholm integral equations of the second kind is derived for the difference in the stress on both surfaces of the inclusion. The pair can be decoupled and asymptotic solutions of the integral equation are obtained when λ, which represents the relative rigidity of the matrix to the inclusion, is small. For the general cases, the integral equation is solved by a numerical discretization. Excellent agreements between the asymptotic and numerical solutions are observed for small λ.

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

Contact Stresses Between an Elastic Cylinder and a Layered Elastic Solid

1974 ◽  
Vol 96 (2) ◽  
pp. 250-257 ◽  
Author(s):  
P. K. Gupta ◽  
J. A. Walowit
Keyword(s):  
Integral Equation ◽  
Layer Thickness ◽  
Contact Pressure ◽  
Contact Zone ◽  
Numerical Solutions ◽  
Elastic Solid ◽  
Elastic Cylinder ◽  
Elastic Solids ◽  
Actual Contact ◽  
Wide Range

The generalized plane strain problem of the contact of layered elastic solids is reduced to an integral equation using Green’s function approach. Approximate numerical solutions are obtained by replacing the integral equation by a matrix inversion when the trapezoidal rule is used to represent the integral. Results for determining the actual contact pressure at the center of the contact zone and size of contact zone for a wide range of layer thicknesses are presented for two most practical cases, (i) when the indenter is rigid, and (ii) when the indenter is elastic having a modulus of elasticity equal to that of the substrate of the indented body. When the layer is softer than the substrate it is found that the actual contact pressure distribution is very closely determined by a weighted sum of elliptic and parabolic functions. For a substrate softer than the layer the pressures substantially deviate from an elliptical or parabolic behavior, for the cases when the layer thickness is finite. The analysis checks with the Hertzian solution in the extreme cases when the layer thickness either tends to zero or approaches infinity.

A novel technique based on Bernoulli wavelets for numerical solutions of two-dimensional Fredholm integral equation of second kind

2019 ◽  
Vol 36 (6) ◽  
pp. 1798-1819
Author(s):  
S. Saha Ray ◽  
S. Behera
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernoulli Polynomials ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Content Type ◽  
Good Convergence ◽  
Novel Technique

Purpose A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials. Design/methodology/approach To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations. Findings Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate. Originality/value The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

scholarly journals Radiation of water waves by a heaving submerged horizontal disc

1997 ◽  
Vol 337 ◽  
pp. 365-379 ◽  
Author(s):  
P. A. MARTIN ◽  
L. FARINA
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Water Waves ◽  
Numerical Solutions ◽  
Boundary Integral ◽  
Fredholm Integral Equations ◽  
Mass Coefficient ◽  
Added Mass Coefficient ◽  
Thin Rigid Plate ◽  
Hypersingular Boundary Integral Equation

A thin rigid plate is submerged beneath the free surface of deep water. The plate performs small-amplitude oscillations. The problem of calculating the radiated waves can be reduced to solving a hypersingular boundary integral equation. In the special case of a horizontal circular plate, this equation can be reduced further to one-dimensional Fredholm integral equations of the second kind. If the plate is heaving, the problem becomes axisymmetric, and the resulting integral equation has a very simple structure; it is a generalization of Love's integral equation for the electrostatic field of a parallel-plate capacitor. Numerical solutions of the new integral equation are presented. It is found that the added-mass coefficient becomes negative for a range of frequencies when the disc is sufficiently close to the free surface.

Punch Problem for Planar Anisotropic Elastic Half-Plane

2011 ◽  
Vol 27 (2) ◽  
pp. 215-226 ◽  
Author(s):  
R.-L. Lin
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Half Plane ◽  
Green's Function ◽  
Boundary Integral ◽  
Fredholm Integral Equations ◽  
Surface Traction ◽  
Rigid Punch ◽  
Anisotropic Elastic ◽  
Punch Problem

ABSTRACTThe two dimensional punch problem for planar anisotropic elastic half-plane is revisited using the Lekhnitskii's formulation with aid of the Fourier transform and boundary integral equation. Four different conditions of contact problem for the rigid punch are analyzed in this study. From the combination of surface Green's function of half-plane and Hooke's law of anisotropic material, a set of Fredholm integral equations are obtained for mixed boundary value problems. After solving the integral equation according to specified contact condition, the explicit distributions of surface traction under the punch are obtained in closed-form. From the surface traction and Green's function of anisotropic half-plane, the full-field solutions of stresses are constructed. Numerical calculations of surface traction under the rigid punch are presented base on the analysis and are discussed in detail.

Tem analysis of multiple-energy arsenic implanted and Rta'd silicon

1987 ◽  
Vol 45 ◽  
pp. 246-247
Author(s):  
R.A. Herring
Keyword(s):  
Device Fabrication ◽  
High Concentration ◽  
Tem Analysis ◽  
Defect Clusters ◽  
High Fluences ◽  
Small Defect ◽  
Before And After ◽  
The Matrix ◽  
The Difference ◽  
Factor Type

Rapid thermal annealing (RTA) of ion-implanted Si is important for device fabrication. The defect structures of 2.5, 4.0, and 6.0 MeV As-implanted silicon irradiated to fluences of 2E14, 4E14, and 6E14, respectively, have been analyzed by electron diffraction both before and after RTA at 1100°C for 10 seconds. At such high fluences and energies the implanted As ions change the Si from crystalline to amorphous. Three distinct amorphous regions emerge due to the three implantation energies used (Fig. 1). The amorphous regions are separated from each other by crystalline Si (marked L1, L2, and L3 in Fig. 1) which contains a high concentration of small defect clusters. The small defect clusters were similar to what had been determined earlier as being amorphous zones since their contrast was principally of the structure-factor type that arises due to the difference in extinction distance between the matrix and damage regions.

Development of Prolonged Delivery of Tramadol and Dissolution Translation by Statistical Data Treatment

2011 ◽  
Vol 4 (1) ◽  
pp. 1331-1337
Author(s):  
P B Parejiya ◽  
B S Barot ◽  
P K Shelat
Keyword(s):  
Drug Release ◽  
Study Data ◽  
Stability Study ◽  
Dissolution Time ◽  
Matrix Tablet ◽  
Burst Effect ◽  
The Matrix ◽  
The Mean ◽  
The Difference ◽  
Dissolution Efficiency

The present study was carried out to fabricate a prolonged design for tramadol using Kollidon SR (Polyvinyl acetate and povidone based matrix retarding polymer). Matrix tablet formulations were prepared by direct compression of Kollidon SR of a varying proportion with a fixed percentage of tramadol. Tablets containing a 1:0.5 (Drug: Kollidon SR) ratio exhibited a rapid rate of drug release with an initial burst effect. Incorporation of more Kollidon SR in the matrix tablet extended the release of drug with subsequent minimization of the burst effect as confirmed by the mean dissolution time, dissolution efficiency and f2 value. Among the formulation batches, a direct relationship was obtained between release rate and the percentage of Kollidon SR used. The formulation showed close resemblance to the commercial product Contramal and compliance with USP specification. The results were explored and explained by the difference of micromeritic characteristics of the polymers and blend of drug with excipients. Insignificant effects of various factors, e.g. pH of dissolution media, ionic strength, speed of paddle were found on the drug release from Kollidon-SR matrix. The formulation followed the Higuchi kinetic model of drug release. Stability study data indicated stable character of Batch T6 after short-term stability study.

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Plastic failure zone characteristics and stability control technology of roadway in the fault area under non-uniformly high geostress: A case study from Yuandian Coal Mine in Northern Anhui Province, China

2020 ◽  
Vol 12 (1) ◽  
pp. 406-424 ◽  
Author(s):  
Yaoguang Huang ◽  
Aining Zhao ◽  
Tianjun Zhang ◽  
Weibin Guo
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Stress Test ◽  
Pressure Coefficient ◽  
Stability Control ◽  
Failure Zone ◽  
Plastic Failure ◽  
Comparison Results ◽  
High Geostress ◽  
The Difference

AbstractIn order to explore the effective support method for deep broken roadway, based on the in situ stress test results, the analytical and numerical solutions of the stress and the range of plastic failure zone (PFZ) in a circular roadway subjected to non-uniform loads were obtained using analytical and finite difference numerical methods based on the elastoplastic theory, respectively. Their comparison results show that the analytical and numerical methods are correct and reasonable. Furthermore, the high geostress causes the stress and range of PFZ in roadway roof and floor to increase sharply while those in roadway ribs decrease. Moreover, the greater the difference of horizontal geostress in different horizontal directions is, the larger the range of PFZ in roadway roof and floor is. The shape of PFZ in roadway varies with the ratio of horizontal lateral pressure coefficient in x-direction and y-direction. Finally, according to the distribution characteristics of PFZ and range of PFZ under the non-uniformly high geostress, this paper has proposed a combined support scheme, and refined and optimized supporting parameters. The field monitoring results prove that the roadway deformation and fracture have been effectively controlled. The research results of this paper can provide theoretical foundation as well as technical reference for the stability control of deep broken roadway under non-uniformly high geostress.

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