Solution for elliptic inclusion in an infinite plate with remote loading and Eshelby’s eigenstrain of polynomial type

2021 ◽  
pp. 108128652110600
Author(s):  
YZ Chen
Keyword(s):  
Stress Field ◽  
Complex Variable ◽  
Infinite Plate ◽  
Inclusion Problem ◽  
Elliptic Inclusion ◽  
Inherent Difficulty ◽  
Polynomial Type ◽  
Stress Components ◽  
The Matrix ◽  
Remote Loading

In this paper, a particular inhomogeneous inclusion problem is studied. In the problem, Eshelby’s eigenstrain takes the type [Formula: see text], where m+ n = 2, and the remote loadings [Formula: see text], [Formula: see text] are applied. In the solution, the complex variable method is used. The continuity conditions along the interface of the matrix and the inclusion are formulated exactly. Because the stress field is no longer uniform in inclusion in this case, the studied problem has an inherent difficulty. After some manipulation, the final result for stress components [Formula: see text], [Formula: see text] and [Formula: see text] in inclusion are obtainable. In the present study, [Formula: see text], [Formula: see text] and [Formula: see text] are no longer uniform.

Elliptic inclusions in a stressed matrix

1963 ◽  
Vol 59 (4) ◽  
pp. 821-832 ◽  
Author(s):  
R. D. Bhargava ◽  
H. C. Radhakrishna
Keyword(s):  
Minimum Energy ◽  
Elastic Field ◽  
Theory Of Elasticity ◽  
Inclusion Problem ◽  
Elliptic Inclusion ◽  
Linear Elasticity Theory ◽  
Energy Principle ◽  
Uniform Tension ◽  
Minimum Energy Principle ◽  
The Matrix

AbstractThis paper treats an extension of the problem considered by the authors in a recent paper (1). The minimum energy principle of the classical theory of elasticity was used in the above paper for evaluating the elastic field when an elliptic region (the inclusion, which could be of a material different from the rest) undergoes spontaneous dimensional change in an otherwise unstrained infinite medium (the matrix). By modification of this method, it has been possible to deal with the case when the inclusion is spherical or circular and the matrix is under uniform tension at infinity (2). The present paper deals with the much more general case when the matrix is under tension, at infinity, inclined at any angle to the major axis of the elliptic inclusion. The solution has been possible by the combination of the complex variable method coupled with minimum energy principle and superposition methods of linear elasticity theory. As a consequence we immediately derive almost without further calculation many particular cases, viz. (i) the inclusion problem in a matrix under axial tension parallel to either of the axes, (ii) under all round uniform tension (or pressure) etc. It is obvious that the results for the respective cases of a circular inclusion can be deduced from these results.It also solves the problem of composite sections under external forces at infinity because of the complete freedom in choosing the elastic constant of the inclusion which can be different from that of the matrix. As a corollary, it solves the problem of a cavity under stress at infinity.

Cavitation at the Ends of an Elliptic Inclusion Inside a Plate Under Tension

1968 ◽  
Vol 35 (3) ◽  
pp. 505-509 ◽  
Author(s):  
M. A. Hussain ◽  
S. L. Pu ◽  
M. A. Sadowsky
Keyword(s):  
Mixed Boundary ◽  
Contact Region ◽  
Infinite Plate ◽  
Major Axis ◽  
Uniaxial Stress ◽  
Theory Of Elasticity ◽  
Transcendental Equation ◽  
Elliptic Inclusion ◽  
Unstressed State ◽  
The Matrix

An oblong elliptic inclusion is perfectly filled in a hole in an infinite plate in the unstressed state. Cavities at the ends of the inclusion will appear as a result of the application of uniaxial stress at infinity in the direction of the major axis of the ellipse. Analytical formulation of the problem leads to a mixed boundary-value problem of the mathematical theory of elasticity. A Fredholm integral equation of the first kind is derived for the normal stress with the range of integration being unknown (corresponding to the unknown region of contact). Applying the theorem which has recently been established based on a variational principle, a transcendental equation is obtained for determining the contact region. Numerical results are given for various values of the elastic constants of both the matrix and the inclusion. Application of the results to fiber-reinforced composite materials is discussed.

Stress Field for a Coated Triangle-Like Hole Problem in Plane Elasticity

2019 ◽  
Vol 36 (1) ◽  
pp. 55-72 ◽  
Author(s):  
S. C. Tseng ◽  
C. K. Chao ◽  
F. M. Chen
Keyword(s):  
Stress Field ◽  
Edge Dislocation ◽  
Coating Layer ◽  
Infinite Plate ◽  
Material Constant ◽  
Plane Elasticity ◽  
Analytical Continuation ◽  
Shape Factors ◽  
Trapping Mechanism ◽  
The Matrix

ABSTRACTThe stress field induced by an edge dislocation or a point force located near a coated triangle-like hole in an infinite plate is provided in this paper. Based on the method of analytical continuation and the technique of conformal mapping in conjunction with the alternation technique, a series solution for the displacement and stresses in the coating layer and the matrix is obtained analytically. Examples for the interaction between an edge dislocation and a coated triangle-like hole for various material constant combinations, coating thicknesses and shape factors are discussed. The analysis discovers that the so-called trapping mechanism of dislocations is more likely to exist near a coated triangle-like hole. The result shows that the dislocation will first be repelled by the coating layer and then attracted by a hole when the coating layer is slightly stiffer than the matrix. However, when the coating layer is sufficiently thin, the dislocation will always be attracted by a hole even the coating layer is stiffer than the matrix.

Two-Dimensional Stress Analysis of an Infinite Plate with Anisotropic Inclusions by BFM

2019 ◽  
Vol 827 ◽  
pp. 397-403
Author(s):  
Takuichiro Ino ◽  
Yohei Sonobe ◽  
Atsuhiro Koyama ◽  
Akihide Saimoto
Keyword(s):  
Mechanical Properties ◽  
Stress Field ◽  
Stress Analysis ◽  
Body Force ◽  
Infinite Plate ◽  
Inclusion Problem ◽  
Homogeneous Material ◽  
Two Dimensional ◽  
Force Method ◽  
Body Force Method

Based on the principle of a Body Force Method (BFM), any inclusion problem can besolved only by using a Kelvin solution which corresponds to a stress field caused by a point forceacting in a homogeneous infinite plate, regardless of the mechanical properties of the inclusion. Thischaracteristic is true even for an anisotropic inclusion in which the number of independent elasticconstants are larger than that of a homogeneous material. In the present study, some problems among anisotropic inclusions were analyzed numerically to demonstrate the validity.

The Elastic Field of an Elliptic Inclusion With a Slipping Interface

1986 ◽  
Vol 53 (1) ◽  
pp. 103-107 ◽  
Author(s):  
E. Tsuchida ◽  
T. Mura ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Elastic Field ◽  
Elliptic Inclusion ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Bonded Interface ◽  
The Matrix ◽  
Displacement Potentials ◽  
Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

Development of Generalized Plane-Strain Tensors for the Concentric Cylinder

1995 ◽  
Vol 62 (3) ◽  
pp. 590-594
Author(s):  
N. Chandra ◽  
Zhiyum Xie
Keyword(s):  
Fiber Volume Fraction ◽  
Volume Fraction ◽  
Transformation Strain ◽  
Inclusion Problem ◽  
Infinite Domain ◽  
Concentric Cylinder ◽  
Finite Radius ◽  
Fiber Volume ◽  
Thermal Residual Stresses ◽  
The Matrix

A pair of two new tensors called GPS tensors S and D is proposed for the concentric cylindrical inclusion problem. GPS tensor S relates the strain in the inclusion constrained by the matrix of finite radius to the uniform transformation strain (eigenstrain), whereas tensor D relates the strain in the matrix to the same eigenstrain. When the cylindrical matrix is of infinite radius, tensor S reduces to the appropriate Eshelby’s tensor. Explicit expressions to evaluate thermal residual stresses σr, σθ and σz in the matrix and the fiber using tensor D and tensor S, respectively, are developed. Since the geometry of the present problem is of finite radius, the effect of fiber volume fraction on the stress distribution can be easily studied. Results for the thermal residual stress distributions are compared with Eshelby’s infinite domain solution and finite element results for a specified fiber volume fraction.

The 〈111〉 Elliptic Inclusion in an Anisotropic Solid of Cubic Symmetry

1982 ◽  
Vol 49 (2) ◽  
pp. 353-360 ◽  
Author(s):  
H. C. Yang ◽  
Y. T. Chou
Keyword(s):  
Stress Field ◽  
Uniform Stress ◽  
Elliptic Inclusion ◽  
Cubic Symmetry ◽  
Elastic Fields ◽  
Closed Form Solutions ◽  
Strain Transformation ◽  
Anisotropic Solid ◽  
Uniform Stress Field ◽  
Free Strain

This paper deals with a generalized plane problem in which a uniform stress-free strain transformation takes place in the region of an elliptic cyclinder (the inclusion) oriented in the 〈111〉 direction in an anisotropic solid of cubic symmetry. Closed-form solutions for the elastic fields and the strain energies are presented. The perturbation of an otherwise uniform stress field due to a 〈111〉 elliptic inhomogeneity is also treated including two extreme cases, elliptic cavities and rigid inhomogeneities.

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