Note on volume integrals of the elastic field around an ellipsoidal inclusion

1972 ◽  
Vol 2 (3) ◽  
pp. 199-200 ◽  
Author(s):  
K. Tanaka ◽  
T. Mori
Keyword(s):  
Elastic Field ◽  
Ellipsoidal Inclusion

The elastic field outside an ellipsoidal inclusion

1959 ◽  
Vol 252 (1271) ◽  
pp. 561-569 ◽  
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Elastic Constants ◽  
Strain Energy ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Harmonic Potential ◽  
Surface Distribution ◽  
Shear Transformation ◽  
General Method

The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.

scholarly journals Further Considerations of the Elastic Inclusion Problem

1964 ◽  
Vol 14 (1) ◽  
pp. 61-70 ◽  
Author(s):  
R. J. Knops
Keyword(s):  
Elastic Inclusion ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Inclusion Problem ◽  
Infinite Matrix ◽  
Mean Values ◽  
Shear Moduli ◽  
Weakly Interacting

SummaryAn equation is derived for the strains of an arbitrary elastic field in an infinite matrix perturbed by several inclusions. The equation is solved exactly when the shear moduli of the inclusions and matrix are identical, and also when only a single ellipsoidal inclusion perturbs a field uniform at infinity. Mean-values of the strains are then calculated for non-uniform fields perturbed either by an ellipsoid or by a system of weakly-interacting spheres.

Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Xiaoqing Jin ◽  
Ding Lyu ◽  
Xiangning Zhang ◽  
Qinghua Zhou ◽  
Qian Wang ◽  
...  
Keyword(s):  
Deformation Gradient ◽  
Elastic Field ◽  
Analytical Form ◽  
Ellipsoidal Inclusion ◽  
Normal Vector ◽  
Full Field ◽  
Practical Applications ◽  
Starting Point ◽  
Complete Set ◽  
Outward Unit

The celebrated solution of the Eshelby ellipsoidal inclusion has laid the cornerstone for many fundamental aspects of micromechanics. A well-known difficulty of this classical solution is to determine the elastic field outside the ellipsoidal inclusion. In this paper, we first analytically present the full displacement field of an ellipsoidal inclusion subjected to uniform eigenstrain. It is demonstrated that the displacements inside inclusion are linearly related to the coordinates and continuous across the interface of inclusion and matrix. The exterior displacement, which is less detailed in existing literatures, may be expressed in a more compact, explicit, and simpler form through utilizing the outward unit normal vector of an auxiliary confocal ellipsoid. Other than many practical applications in geological engineering, the displacement solution can be a convenient starting point to derive the deformation gradient, and subsequently in a straightforward manner to accomplish the full-field solutions of the strain and stress. Following Eshelby's definition, a complete set of the Eshelby tensors corresponding to the displacement, deformation gradient, strain, and stress are expressed in explicit analytical form. Furthermore, the jump conditions to quantify the discontinuities across the interface are discussed and a benchmark problem is provided to validate the present formulation.

A translated rigid ellipsoidal inclusion in an elastic medium

1991 ◽  
Vol 434 (1892) ◽  
pp. 571-585 ◽  
Keyword(s):  
Elastic Medium ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Ellipsoidal Inclusion ◽  
Arbitrary Orientation ◽  
Isotropic Matrix ◽  
The Matrix ◽  
Ellipsoidal Surface ◽  
Axis Of Symmetry ◽  
Anisotropic Elastic

A rigid ellipsoidal inclusion is embedded at arbitrary orientation in a homogeneous, arbitrarily anisotropic, elastic matrix and is translated infinitesimally by an externally imposed force. We find directly the relation between the force and translation vectors, and the stress, strain and rotation concentrations over the ellipsoidal surface, without having to solve the equations of equilibrium in the matrix, or the fundamental ones of a point force. We refer particularly then to a spheroid aligned along the axis of symmetry of a transversely isotropic matrix, and subsequently to the full elastic field of a general ellipsoid in an isotropic matrix.

A rotated rigid ellipsoidal inclusion in an elastic medium

1991 ◽  
Vol 433 (1887) ◽  
pp. 179-207 ◽  
Keyword(s):  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Arbitrary Orientation ◽  
Governing Equations ◽  
Simple Singularity ◽  
The Matrix ◽  
Ellipsoidal Surface ◽  
Rotation Vectors ◽  
Anisotropic Elastic ◽  
Full Solution

A rigid ellipsoidal inclusion is embedded at arbitrary orientation in a homogeneous, arbitrarily anisotropic, elastic matrix and is rotated infinitesimally by means of an imposed couple. Far away the matrix remains either unstrained or at a prescribed uniform strain. A simple ‘singularity’ representation of the elastic field is proposed. It yields directly the relation between the couple and rotation vectors, and the stress, strain and rotation concentrations over the ellipsoidal surface, without having to solve either the governing equations of equilibrium in the matrix, or the fundamental ones of a point force. A full solution is given for an isotropic matrix.

The Elastic Field Outside an Ellipsoidal Inclusion

1977 ◽  
Vol 44 (4) ◽  
pp. 591-594 ◽  
Author(s):  
T. Mura ◽  
P. C. Cheng
Keyword(s):  
Numerical Calculation ◽  
Unit Sphere ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Elastic Media

The elastic field outside an ellipsoidal inclusion is investigated. No particular restrictions are imposed to eigenstrain distributions throughout the inclusion and to the anisotropy of elastic media for matrices. The solution is expressed in the form of integrals defined on a subspace of a unit sphere. Since the integrands of these integrals have no singularities, the numerical calculation on a computer can easily be performed.

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