scholarly journals Eshelby's problem of polygonal inclusions with polynomial eigenstrains in an anisotropic magneto-electro-elastic full plane

2015 ◽  
Vol 471 (2179) ◽  
pp. 20140827 ◽  
Author(s):  
Y.-G. Lee ◽  
W.-N. Zou ◽  
E. Pan
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Inclusion Problem ◽  
Linear Quadratic ◽  
Boundary Integrals ◽  
General Terms ◽  
Present Solution ◽  
Full Plane ◽  
Basic Functions ◽  
Inhomogeneity Problem

This paper presents a closed-form solution for the arbitrary polygonal inclusion problem with polynomial eigenstrains of arbitrary order in an anisotropic magneto-electro-elastic full plane. The additional displacements or eigendisplacements, instead of the eigenstrains, are assumed to be a polynomial with general terms of order M + N . By virtue of the extended Stroh formulism, the induced fields are expressed in terms of a group of basic functions which involve boundary integrals of the inclusion domain. For the special case of polygonal inclusions, the boundary integrals are carried out explicitly, and their averages over the inclusion are also obtained. The induced fields under quadratic eigenstrains are mostly analysed in terms of figures and tables, as well as those under the linear and cubic eigenstrains. The connection between the present solution and the solution via the Green's function method is established and numerically verified. The singularity at the vertices of the arbitrary polygon is further analysed via the basic functions. The general solution and the numerical results for the constant, linear, quadratic and cubic eigenstrains presented in this paper enable us to investigate the features of the inclusion and inhomogeneity problem concerning polynomial eigenstrains in semiconductors and advanced composites, while the results can further serve as benchmarks for future analyses of Eshelby's inclusion problem.

Closed Form Solution for Outflow Between Corotating Disks

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Closed Form ◽  
Stokes Equations ◽  
Closed Form Solution ◽  
Tangential Velocity ◽  
Form Solution ◽  
Navier Stokes ◽  
Published Data ◽  
Rotational Inertia ◽  
Viscous Losses ◽  
Present Solution

Mathematical expressions are derived for flow velocities and pressure distributions for a laminar flow in the gap between two rotating concentric disks. Fluid enters the gap between disks at the center and diverges to the outer periphery. The Navier–Stokes equations are linearized in order to get closed-form solution. The present solution is applicable to the flow between corotating as well as contrarotating disks. The present results are in agreement with the published data of other investigators. The tangential velocity is less for contrarotating disks than for corotating disks in core region of the radial channel. The flow is influenced by rotational inertia and convective inertia both. Dominance of rotational inertia over convective inertia causes backflow. Pressure depends on viscous losses, convective inertia, and rotational inertias. Effect of viscous losses on pressure is high at small throughflow Reynolds number. The convective and rotational inertia influence pressure significantly at high throughflow and rotational Reynolds numbers. Both favorable and unfavorable pressure gradients can be found simultaneously depending on a combination of parameters.

DELEGATED MANAGEMENT IN DYNAMIC DUOPOLIES

2010 ◽  
Vol 12 (02) ◽  
pp. 93-114
Author(s):  
VLADIMIR P. PETKOV
Keyword(s):  
Functional Form ◽  
Closed Form Solution ◽  
Industry Structure ◽  
Form Solution ◽  
Linear Quadratic ◽  
Current Effort ◽  
First Mover Advantage ◽  
And Control ◽  
Direct Management ◽  
First Mover

This paper studies the commitment value of delegation in a model of dynamic competition. We argue that separating ownership and control delivers an instantaneous first-mover advantage. Thus, delegation would enable an oligopolistic firm to increase its equilibrium profit relative to direct management. We focus on remuneration strategies that provide managers with intertemporal production incentives: future wages depend on current effort. Their composition and functional form are endogenously determined by the requirement for Markov perfection. For the case of linear-quadratic payoffs, we obtain a closed-form solution for the equilibrium wage strategies which is independent of industry structure.

A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion

2011 ◽  
Vol 78 (3) ◽  
Author(s):  
Xiaoqing Jin ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Elastic Field ◽  
Form Solution ◽  
Eshelby Tensor ◽  
Cylindrical Inclusion ◽  
Closed Form Expression ◽  
Equivalent Inclusion Method ◽  
Explicit Analytical Solution ◽  
Present Solution

From the analytical formulation developed by Ju and Sun [1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574], it is seen that the exterior point Eshelby tensor for an ellipsoid inclusion possesses a minor symmetry. The solution to an elliptic cylindrical inclusion may be obtained as a special case of Ju and Sun’s solution. It is noted that the closed-form expression for the exterior-point Eshelby tensor by Kim and Lee [2010, “Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion,” ASME Trans. J. Appl. Mech., 77, p. 024503] violates the minor symmetry. Due to the importance of the solution in micromechanics-based analysis and plane-elasticity-related problems, in this work, the explicit analytical solution is rederived. Furthermore, the exterior-point Eshelby tensor is used to derive the explicit closed-form solution for the elastic field outside the inclusion, as well as to quantify the elastic field discontinuity across the interface. A benchmark problem is used to demonstrate a valuable application of the present solution in implementing the equivalent inclusion method.

Efficient Numerical Modeling of Hertzian Line Contact for Material With Inhomogeneities

2012 ◽  
Author(s):  
Xiaoqing Jin ◽  
Zhanjiang Wang ◽  
Qinghua Zhou ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Half Plane ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
General Purpose ◽  
Form Solution ◽  
Inclusion Problem ◽  
Computational Domain ◽  
Line Contact ◽  
Equivalent Inclusion Method ◽  
Parametric Studies

The present work proposes an efficient and general-purpose numerical approach for handling two-dimensional inhomogeneities in an elastic half plane. The inhomogeneities can be of any shape, at any location, with arbitrary material properties (which can also be non-homogeneous). To perform the numerical analysis, we first derive an explicit closed-form solution for a rectangular inclusion with uniform eigenstrain components, where the inclusion is aligned with the surface of the half plane. In view of the equivalent inclusion method, an inhomogeneity problem can be converted to a corresponding inclusion problem. In order to determine the distribution of the equivalent eigenstrain, the computational domain is meshed into rectangular elements whose resultant contributions can be efficiently computed using an efficient algorithm based on fast Fourier transform (FFT). In principle, there is no specific limitation on the type of the external load, although our major concern is the contact analysis. Parametric studies are performed and typical results highlighting the deviation of the current solution from the classical Hertzian line contact theory are presented.

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.

Vibration of porous functionally graded plates with geometric discontinuities and partial supports

2020 ◽  
Vol 234 (21) ◽  
pp. 4149-4170
Author(s):  
Gaurav Bansal ◽  
Ankit Gupta ◽  
Varun Katiyar
Keyword(s):  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Functionally Graded Plate ◽  
Stress Variation ◽  
Boundary Constraints ◽  
Present Solution ◽  
Effective Material Properties ◽  
Geometric Discontinuities

Vibrational study of the porous functionally graded plate with geometric discontinuities and partial supports has been presented in the present paper. The kinematics of functionally graded plate is based on the refined exponential shear deformation theory. The displacement field has been refined by dividing the in-plane and out of the plane displacements into bending and shear components. The theory accounts for the nonlinear transverse shear stress variation along with the thickness with only four unknowns. The closed-form solution (Navier’s solution), as well as FEM-based solution, have been used for the vibration analysis of functionally graded plate. The geometric discontinuities have been incorporated in terms of a circular cut-out of different sizes at the center of the plate. Modified rule of mixtures, modified sigmoid law, and trigonometric law have been used to compute the effective material properties of the functionally graded plate. A C0 continuous iso-parametric FEM formulation has been used to attain the results in the case of FEM solution, and the efficacy of the present solution is demonstrated by comparing the results with the available literature. The results reflect that the porosity inclusion, circular cut-out, and position of the boundary constraints have a notable influence on the fundamental frequency of the functionally graded plate. It is also concluded that after a specific radius of circular cut-out, the vibration response of functionally graded plate exhibits nonlinearity in nature.

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