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.

Eigenstresses in Anisotropic Films

1991 ◽  
Vol 239 ◽  
Author(s):  
Ferdinando Auricchio ◽  
Mauro Ferrari
Keyword(s):  
Stress Field ◽  
Experimental Determination ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Ceramic Film ◽  
Stress States ◽  
Resultant Stress ◽  
Structural Theories

ABSTRACTA closed-form solution for a macroscopically homogeneous, fully anisotropie layer subject to non-uniform through-thickness eigenstrain is presented, and employed in determining the three-dimensional deformation and stress states of a thermally loaded ceramic film with microstructure-induced macroscopic anisotropy. The resultant stress field is compared with those that could be deduced by experimental determination of the curvature and the classical structural theories.

The Interior Stress Field Caused by Tangential Loading of a Rectangular Patch on an Elastic Half Space

1987 ◽  
Vol 109 (4) ◽  
pp. 627-629 ◽  
Author(s):  
N. Ahmadi ◽  
L. M. Keer ◽  
T. Mura ◽  
V. Vithoontien
Keyword(s):  
Stress Field ◽  
Internal Stress ◽  
Half Space ◽  
Elastic Half Space ◽  
Vertical Load ◽  
Elementary Functions ◽  
Rectangular Patch ◽  
Internal Stress Field ◽  
Interior Stress

A solution is obtained for the tangential loading on a rectangular patch. The solution gives the internal stress field in terms of elementary functions and is a form analogous to the solution for vertical load developed by Love.

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.

Exact closed-form solutions for Lamb’s problem—III: the case for buried source and receiver

2020 ◽  
Vol 224 (1) ◽  
pp. 517-532
Author(s):  
Xi Feng ◽  
Haiming Zhang
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Closed Form Solution ◽  
Time History ◽  
Natural Generalization ◽  
Elastic Half Space ◽  
Form Solution ◽  
Elementary Functions ◽  
Heaviside Step Function ◽  
Lamb’S Problem

SUMMARY In this paper, we derive the exact closed-form solution for the displacement in the interior of an elastic half-space due to a buried point force with Heaviside step function time history. It is referred to as the tensor Green’s function for the elastic wave equation in a uniform half-space, also a natural generalization of the classical 3-D Lamb’s problem, for which previous solutions have been restricted to the cases of either the source or the receiver or both are located on the free surface. Starting from the complex integral solutions of Johnson, we follow the similar procedures presented by Feng and Zhang to obtain the closed-form expressions in terms of elementary functions as well as elliptic integrals. Numerical results obtained from our closed-form expressions agree perfectly with those of Johnson, which validates our explicit formulae conclusively.

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.

Closed-form solution of Cattaneo equation including volumetric source in relation to laser short-pulse heating

2011 ◽  
Vol 89 (7) ◽  
pp. 761-767 ◽  
Author(s):  
H. Al-Qahtani ◽  
B.S. Yilbas
Keyword(s):  
Stress Field ◽  
Closed Form ◽  
Short Pulse ◽  
Closed Form Solution ◽  
Transformation Method ◽  
Substrate Material ◽  
Form Solution ◽  
Cooling Period ◽  
Wave Nature ◽  
Heating Model

The wave nature of the heating model is considered, incorporating the Cattaneo equation with the presence of a volumetric heat source. The volumetric heat generation resembles the step input laser short-pulse intensity. The governing of the heat equation is solved analytically using the Laplace transformation method. The stress field generated due to thermal contraction and expansion of the substrate material is formulated and the closed-form solution is presented. It is found that the wave nature of the heating is dominant during the period of the irradiated short-pulse; however, in the late cooling period, the wave nature of heating is replaced by diffusional heat conduction, governed by Fourier’s law. The stress field during the heating cycle is compressive and becomes tensile in the cooling cycle.

Elastic-Plastic Stress Field in a Coated Continuous Fibrous Composite Subjected to Thermomechanical Loading

1999 ◽  
Vol 66 (3) ◽  
pp. 750-757 ◽  
Author(s):  
L. You ◽  
S. Long ◽  
L. Rohr
Keyword(s):  
Stress Field ◽  
Governing Equation ◽  
Yield Criterion ◽  
Fibrous Composite ◽  
Closed Form Solution ◽  
Metallic Matrix ◽  
Form Solution ◽  
Concentric Cylinders ◽  
The Matrix ◽  
Strain Relationship

A micromechanics investigation was performed in the present work to analyze the stress field in a coated continuous fibrous composite subjected to thermal and mechanical loading based on a four-concentric-cylinders model. A temperature-independent stress-plastic strain relationship for the metallic matrix and coating layer with linear strain-hardening behaviour were introduced. Tresca’s yield criterion and the associated flow law were employed to derive the governing equation of the coating and matrix. The closed-form solution of the governing equation was obtained. Some numerical examples were given. The numerical results indicate that the plasticity of the coating greatly decreases the circumferential and axial stresses in the coating itself, but has very limited influence on the stresses in other constituents of the composite. The plasticity of the matrix imposes no significant influence on all the stresses in the composite.

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