Irregular Inhomogeneities in an Anisotropic Piezoelectric Plane

2012 ◽  
Vol 79 (2) ◽  
Author(s):  
L. G. Sun ◽  
K. Y. Xu ◽  
E. Pan
Keyword(s):  
Quantum Wire ◽  
Body Force ◽  
Numerical Solutions ◽  
Elementary Functions ◽  
Line Integral ◽  
Induced Field ◽  
Full Plane ◽  
Piezoelectric Fields ◽  
Force Concept ◽  
Eshelby Problem

This paper presents an analytical solution for the Eshelby problem of polygonal inhomogeneity in an anisotropic piezoelectric plane. By virtue of the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields in the corresponding inclusion are first expressed in terms of the line integral along its boundary with the integrand being the Green’s functions, which is carried out analytically. The Eshelby inhomogeneity relation for the elliptical shape is then extended to the polygonal inhomogeneity, with the final induced field involving only elementary functions with small steps of iteration. Numerical solutions are compared to the results obtained from other methods, which verified the accuracy of the proposed method. Finally, the solution is applied to a triangular and a rectangular quantum wire made of InAs within the semiconductor GaAs full-plane substrate.

scholarly journals Mixed-convection laminar film condensation on a semi-infinite vertical plate

1995 ◽  
Vol 300 ◽  
pp. 207-229 ◽  
Author(s):  
Jian-Jun Shu ◽  
Graham Wilks
Keyword(s):  
Vertical Plate ◽  
Body Force ◽  
Numerical Solutions ◽  
Force Convection ◽  
Film Condensation ◽  
Approximate Methods ◽  
Saturated Vapour ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Infinite Vertical Plate

The flow of a uniform stream of pure saturated vapour past a cold, semi-infinite vertical plate is examined. The formulation incorporates the limits of both pure forcedconvection and pure body-force-convection laminar film condensation. Detailed asymptotic and exact numerical solutions are obtained and comparisons drawn with approximate methods and experimental results reported in the literature.

Singular Integral Equation Method for Interaction Between Elliptical Inclusions

1998 ◽  
Vol 65 (2) ◽  
pp. 310-319 ◽  
Author(s):  
Nao-Aki Noda ◽  
Tadatoshi Matsuo
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Singular Integral ◽  
Body Force ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Integral Equation Method ◽  
The Body ◽  
Force Density ◽  
Cauchy Type

This paper deals with numerical solutions of singular integral equations in interaction problems of elliptical inclusions under general loading conditions. The stress and displacement fields due to a point force in infinite plates are used as fundamental solutions. Then, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns are the body force densities distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. To determine the unknown body force densities to satisfy the boundary conditions, four auxiliary unknown functions are derived from each body force density. It is found that determining these four auxiliary functions in the range 0≦φk≦π/2 is equivalent to determining an original unknown density in the range 0≦φk≦2π. Then, these auxiliary unknowns are approximated by using fundamental densities and polynomials. Initially, the convergence of the results such as unknown densities and interface stresses are confirmed with increasing collocation points. Also, the accuracy is verified by examining the boundary conditions and relations between interface stresses and displacements. Randomly or regularly distributed elliptical inclusions can be treated by combining both solutions for remote tension and shear shown in this study.

scholarly journals Mechanics-Based Solution Verification for Porous Media Models

2016 ◽  
Vol 20 (5) ◽  
pp. 1127-1162 ◽  
Author(s):  
M. Shabouei ◽  
K. B. Nakshatrala
Keyword(s):  
Vector Fields ◽  
Body Force ◽  
Numerical Solutions ◽  
Mechanical Power ◽  
Reciprocal Relation ◽  
A Posteriori ◽  
Numerical Examples ◽  
Brinkman Equations ◽  
Mathematical Properties ◽  
Solution Verification

AbstractThis paper presents a new approach to verify the accuracy of computational simulations. We develop mathematical theorems which can serve as robusta posteriorierror estimation techniques to identify numerical pollution, check the performance of adaptive meshes, and verify numerical solutions. We demonstrate performance of this methodology on problems from flow thorough porous media. However, one can extend it to other models. We construct mathematical properties such that the solutions to Darcy and Darcy-Brinkman equations satisfy them. The mathematical properties include the total minimum mechanical power, minimum dissipation theorem, reciprocal relation, and maximum principle for the vorticity. All the developed theorems have firm mechanical bases and are independent of numerical methods. So, these can be utilized for solution verification of finite element, finite volume, finite difference, lattice Boltzmann methods and so forth. In particular, we show that, for a given set of boundary conditions, Darcy velocity has the minimum total mechanical power of all the kinematically admissible vector fields. We also show that a similar result holds for Darcy-Brinkman velocity. We then show for a conservative body force, the Darcy and Darcy-Brinkman velocities have the minimum total dissipation among their respective kinematically admissible vector fields. Using numerical examples, we show that the minimum dissipation and total mechanical power theorems can be utilized to identify pollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman equations are shown to satisfy a reciprocal relation, which has the potential to identify errors in the numerical implementation of boundary conditions. It is also shown that the vorticity under both steady and transient Darcy-Brinkman equations satisfy maximum principles if the body force is conservative and the permeability is homogeneous and isotropic. A discussion on the nature of vorticity under steady and transient Darcy equations is also presented. Using several numerical examples, we will demonstrate the predictive capabilities of the proposeda posterioritechniques in assessing the accuracy of numerical solutions for a general class of problems, which could involve complex domains and general computational grids.

Analytical Solution for the Stress Field of Eshelby’s Inclusion of Polygonal Shape

2009 ◽  
Author(s):  
Xiaoqing Jin ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Stress Field ◽  
Closed Form Solution ◽  
Form Solution ◽  
Elementary Functions ◽  
Singular Stress ◽  
Equivalent Inclusion Method ◽  
Full Plane ◽  
Interior Stress ◽  
Line Elements ◽  
Polygonal Inclusion

Recently, we developed a closed-form solution to the stress field due to a point eigenstrain in an elastic full plane. This solution can be employed as a Green’s function to compute the stress field caused by an arbitrary-shaped Eshelby’s inclusion subjected to any distributed eigenstrain. In this study, analytical expressions are derived when uniform eigenstrain is distributed in a planar inclusion bounded by line elements. Here it is demonstrated that both the interior and exterior stress fields of a polygonal inclusion subjected to uniform eigenstrain can be represented in a unified expression, which consists of only elementary functions. Singular stress components are identified at all the vertices of the polygon. These distinctive properties contrast to the well-known Eshelby’s solution for an elliptical inclusion, where the interior stress field is uniform but the formulae for the exterior field are remarkably complicated. The elementary solution of a polygonal inclusion has valuable application in the numerical implementation of the equivalent inclusion method.

The Plane Solution for Anisotropic Elastic Wedges Under Normal and Shear Loading

1972 ◽  
Vol 39 (4) ◽  
pp. 1103-1109 ◽  
Author(s):  
D. B. Bogy
Keyword(s):  
Half Plane ◽  
Complex Function ◽  
Stress Singularity ◽  
Shear Loading ◽  
Elementary Functions ◽  
Linear Elastostatics ◽  
Full Plane ◽  
Plane Solution ◽  
Logarithmic Singularities ◽  
Anisotropic Elastic

The plane traction problem for an anisotropic wedge is solved within the theory of linear elastostatics. The technique employs the complex function representation of the plane solution in conjunction with the Mellin transform. Special attention is given to the orthotropic wedge; the uniform load solution is given in terms of elementary functions for wedge angles less than π, the logarithmic singularities in the stress field resulting from discontinuous loads on the half plane are studied, and the stress singularity at the apex is investigated for the reentrant wedges. Simplified results for the anisotropic half plane and cracked full plane are also presented.

Strain and piezoelectric fields in embedded quantum wire arrays

2006 ◽  
Vol 40 (3) ◽  
pp. 125-136 ◽  
Author(s):  
F. Han ◽  
E. Pan ◽  
J.D. Albrecht
Keyword(s):  
Quantum Wire ◽  
Wire Arrays ◽  
Piezoelectric Fields

3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. I43-I53 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Integral Method ◽  
Numerical Solutions ◽  
Vertical Component ◽  
Building Blocks ◽  
Variable Density ◽  
Rectangular Prism ◽  
Density Contrast ◽  
Gravity Potential ◽  
Line Integral

Three-dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3D mass bodies with spatially variable density contrasts. A 3D vector gravity potential is defined for a 3D rectangular prism with density contrast varying in depth and horizontally. The vertical component of the gravity anomaly equals the flux of the 3D vector gravity potential through the enclosed surface of the prism. Thus, the 3D integral for the gravity anomaly is reduced to a 2D surface integral. In turn, a 2D vector gravity potential is defined. The vertical component of the gravity anomaly equals the net circulation of the 2D vector gravity potential along the enclosed contour bounding the surfaces of the prism. The 3D integral for the gravity anomaly is reduced to 1D line integrals. Further analytical or numerical solutions can then be obtained from the line integrals, depending on the forms of the density contrast functions. If an analytical solution cannot be obtained, the line-integral method is semianalytical, requiring numerical quadratures to be carried out at the final stages. Singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them. Then the vector-potential line-integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast, varying simply in one direction and sophisticatedly in three directions. The advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast.

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