Asymptotic Solutions of Penny-Shaped Inhomogeneities in Global Eshelby’s Tensor

2001 ◽  
Vol 68 (5) ◽  
pp. 740-750 ◽  
Author(s):  
Q. Yang ◽  
W. Y. Zhou ◽  
G. Swoboda
Keyword(s):  
Stress Field ◽  
Three Dimensional ◽  
Uniform Stress ◽  
Isotropic Matrix ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
Asymptotic Expressions ◽  
Eshelby’S Tensor ◽  
Uniform Stress Field

In this paper, a three-dimensional penny-shaped isotropic inhomogeneity surrounded by unbounded isotropic matrix in a uniform stress field is studied based on Eshelby’s equivalent inclusion method. The solution including the deduced equivalent eigenstrain and its asymptotic expressions is presented in tensorial form. The so-called energy-based equivalent inclusion method is introduced to remove the singularities of the size and eigenstrain of the Eshelby’s equivalent inclusion of the penny-shaped inhomogeneity, and yield the same energy disturbance. The size of the energy-based equivalent inclusion can be used as a generic damage measurement.

Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization

2021 ◽  
pp. 1-35
Author(s):  
Chunlin Wu ◽  
Liangliang Zhang ◽  
Huiming Yin
Keyword(s):  
Taylor Series ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Elastic Field ◽  
Analytical Form ◽  
Tetrahedral Elements ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
Eshelby’S Tensor ◽  
Domain Discretization

Abstract The paper extends the recent work (JAM, 88, 061002, 2021) of the Eshelby's tensors for polynomial eigenstrains from a two dimensional (2D) to three dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity by using Eshelby's equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby's tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the 𝐶0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of the Eshelby's tensor, the elastic analysis is robust, stable and efficient.

Atomic transport via point defects in crystals III. Migration in a stress field

1985 ◽  
Vol 398 (1814) ◽  
pp. 203-208 ◽  
Keyword(s):  
Stress Field ◽  
Point Defects ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Molecular Volume ◽  
Uniform Stress ◽  
Atomic Transport ◽  
Elastic Dipole ◽  
Uniform Stress Field ◽  
Defect Species

The kinetic theory of isothermal atomic transport via point defects that was presented in two previous papers (Franklin, A. D. & Lidiard, A. B. Proc. R. Soc. Lond . A 389, 405–431 (1983) and Franklin, A. D. & Lidiard, A. B. Proc. R. Soc. Lond . A 392, 457–473 (1984)) has been expanded into a three-dimensional formulation to analyse transport in an applied non-uniform stress field. The fluxes of the various defect species take the general form familiar from non-equilibrium thermodynamics, while the contribution to the force on defect species Y arising from the stress σ αβ is confirmed to be v ∇(λ (Y) αβ σ αβ ), where v is the molecular volume of the solid and λ (Y) αβ is the elastic-dipole strain tensor of the defect species Y (summation over repeated Cartesian indices α, β is here assumed). Full details of these calculations are presented in Lidiard, A. B. A. E. R. E. Rep . no R. 11367 (1984).

Two-Ellipsoidal Inhomogeneities by the Equivalent Inclusion Method

1975 ◽  
Vol 42 (4) ◽  
pp. 847-852 ◽  
Author(s):  
Z. A. Moschovidis ◽  
T. Mura
Keyword(s):  
Computer Program ◽  
Strain Field ◽  
Applied Strain ◽  
Stress And Strain ◽  
Isotropic Matrix ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix ◽  
Final Stress

The problem of two ellipsoidal inhomogeneities in an infinitely extended isotropic matrix is treated by the equivalent inclusion method. The matrix is subjected to an applied strain field in the form of a polynomial of degree M in the position coordinates xi. The final stress and strain states are calculated for two isotropic ellipsoidal inhomogeneities both in the interior and the exterior (in the matrix) by using a computer program developed. The method can be extended to more than two inhomogeneities.

Effective Stiffness of a Debonded Particle in Particulate Composites

2007 ◽  
Vol 334-335 ◽  
pp. 33-36 ◽  
Author(s):  
Akihiro Wada ◽  
Yusuke Nagata ◽  
Shi Nya Motogi
Keyword(s):  
Three Dimensional ◽  
Particulate Composite ◽  
Spherical Particles ◽  
Particulate Composites ◽  
Particle Volume ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
Load Carrying ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

In this study, partially debonded spherical particles in a particulate composite are analyzed by three-dimensional finite element method to investigate their load carrying capacities, and the way to replace a debonded particle with an equivalent inclusion is examined. The variation in Young’s modulus and Poisson’s ratio of a composite with the debonded angle was evaluated for different particle arrangements and particle volume fractions, which in turn compared with the results derived from the equivalent inclusion method. Consequently, it was found that by replacing a debonded particle with an equivalent orthotropic one, the macroscopic behavior of the damaged composite could be reproduced so long as the interaction between neighboring particles is negligible.

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.

Elastic Fields in Double Inhomogeneity by the Equivalent Inclusion Method

2000 ◽  
Vol 68 (1) ◽  
pp. 3-10 ◽  
Author(s):  
H. M. Shodja ◽  
A. S. Sarvestani
Keyword(s):  
Elastic Constants ◽  
Present Method ◽  
Present Theory ◽  
Actual Distribution ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Ellipsoidal Inhomogeneity ◽  
Equivalent Inclusion ◽  
Comparison Results ◽  
Special Cases

Consider a double-inhomogeneity system whose microstructural configuration is composed of an ellipsoidal inhomogeneity of arbitrary elastic constants, size, and orientation encapsulated in another ellipsoidal inhomogeneity, which in turn is surrounded by an infinite medium. Each of these three constituents in general possesses elastic constants different from one another. The double-inhomogeneity system under consideration is subjected to far-field strain (stress). Using the equivalent inclusion method (EIM), the double inhomogeneity is replaced by an equivalent double-inclusion (EDI) problem with proper polynomial eigenstrains. The double inclusion is subsequently broken down to single-inclusion problems by means of superposition. The present theory is the first to obtain the actual distribution rather than the averages of the field quantities over the double inhomogeneity using Eshelby’s EIM. The present method is precise and is valid for thin as well as thick layers of coatings, and accommodates eccentric heterogeneity of arbitrary size and orientation. To establish the accuracy and robustness of the present method and for the sake of comparison, results on some of the previously reported problems, which are special cases encompassed by the present theory, will be re-examined. The formulations are easily extended to treat multi-inhomogeneity cases, where an inhomogeneity is surrounded by many layers of coatings. Employing an averaging scheme to the present theory, the average consistency conditions reported by Hori and Nemat-Nasser for the evaluation of average strains and stresses are recovered.

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