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.

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.

An Elastic Strip Pressed Against an Elastic Half Plane by a Steadily Moving Force

1978 ◽  
Vol 45 (1) ◽  
pp. 89-94 ◽  
Author(s):  
G. G. Adams
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Mixed Boundary ◽  
Contact Region ◽  
Theory Of Elasticity ◽  
Strain Theory ◽  
Fredholm Integral Equations ◽  
Elastic Strip ◽  
Concentrated Force ◽  
Moving Force

An infinite elastic strip is pressed against an elastic half plane of a different material by a steadily moving concentrated force. Using the plane strain theory of elasticity, it is shown that the problem can be decomposed into its symmetric and antisymmetric parts. These mixed boundary-value problems are then solved by reduction to Fredholm integral equations subject to certain other conditions. For various material combinations, and a range of speed, the extent and location of the contact region as well as the contact pressure will be computed and illustrated graphically.

Two-dimensional elliptic inclusions

1963 ◽  
Vol 59 (4) ◽  
pp. 811-820 ◽  
Author(s):  
R. D. Bhargava ◽  
H. C. Radhakrishna
Keyword(s):  
Potential Energy ◽  
Elastic Field ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Minimum Potential Energy ◽  
Elliptic Inclusion ◽  
Inclusion Problems ◽  
Equilibrium Size ◽  
The Matrix ◽  
Minimum Potential

AbstractThe simple concept of minimum potential energy of the classical theory of elasticity, first applied to solve inclusion problems (1) by one of the authors (R. D. B.), who considered spherical and circular inclusions, has now been extended to solve elliptic inclusion problems. The complex-variable method of determining the elastic field, first enunciated by A. C. Stevenson in the U.K. and N. I. Muskhelishvili in the U.S.S.R., has been used to determine the elastic field in the infinite material (the matrix) around the inclusion. Strain energies are calculated. The equilibrium size of an elliptic inclusion of elastic (Lamé's) constants λ1 and μ1, differing from those of matrix, for which the constants are λ and μ, has been determined.An independent check on the calculations has been made by testing the continuity of normal and shearing stresses. The results also agree with the known results for the much simpler case when inclusion and matrix are of the same material.

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.

Contact Stress Analysis of a Layered Transversely Isotropic Half-Space

1992 ◽  
Vol 114 (2) ◽  
pp. 253-261 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Region ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Stress Components ◽  
Contact Failure

The three-dimensional problem of contact between a spherical indenter and a multi-layered structure bonded to an elastic half-space is investigated. The layers and half-space are assumed to be composed of transversely isotropic materials. By the use of Hankel transforms, the mixed boundary value problem is reduced to an integral equation, which is solved numerically to determine the contact stresses and contact region. The interior displacement and stress fields in both the layer and half-space can be calculated from the inverse Hankel transform used with the solved contact stresses prescribed over the contact region. The stress components, which may be related to the contact failure of coatings, are discussed for various coating thicknesses.

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.

scholarly journals Improved Generalized Single-Source Tangential Equivalence Principle Algorithm with Contact-Region Modeling Method for Array Structures

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yao Han ◽  
Hanru Shao ◽  
Jianfeng Dong
Keyword(s):  
Equivalence Principle ◽  
Contact Region ◽  
Near Field ◽  
Single Source ◽  
Matrix Vector Multiplication ◽  
The Matrix ◽  
Array Structures ◽  
Fast Multipole Algorithm ◽  
Matrix Vector ◽  
Equivalence Principle Algorithm

An improved generalized single-source tangential equivalence principle algorithm (GSST-EPA) is proposed for analyzing array structures with connected elements. In order to use the advantages of GSST-EPA, the connected array elements are decomposed and computed by a contact-region modeling (CRM) method, which makes that each element has the same meshes. The unknowns of elements can be transferred onto the equivalence surfaces by GSST-EPA. The scattering matrix in GSST-EPA needs to be solved and stored only once due to the same meshes for each element. The shift invariant of translation matrices is also used to reduce the computation of near-field interaction. Furthermore, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix-vector multiplication in the GSST-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

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