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.

Multiple ellipsoidal/elliptical inhomogeneities embedded in infinite matrix by equivalent inhomogeneous inclusion method

2021 ◽  
pp. 108128652110071
Author(s):  
Xiu-wei Yu ◽  
Zhong-wei Wang ◽  
Hao Wang
Keyword(s):  
Separation Distance ◽  
Stress And Strain ◽  
Infinite Matrix ◽  
The Finite Element Method ◽  
Stress Concentrations ◽  
Strain Fields ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
Small Separation

Traditional equivalent inclusion method provides unreliable predictions of the stress concentrations of two spherical inhomogeneities with small separation distance. This paper determines the stress and strain fields of multiple ellipsoidal/elliptical inhomogeneities by equivalent inhomogeneous inclusion method. Equivalent inhomogeneous inclusion method is an inverse of equivalent inclusion method and substitutes the subdomains of matrix with known strains by equivalent inhomogeneous inclusions. The stress and strain fields of multiple inhomogeneities are decomposed into the superposition of matrix under applied load and each solitary inhomogeneous inclusion with polynomial eigenstrains by the iteration of equivalent inhomogeneous inclusion method. Multiple circular and spherical inhomogeneities are respectively used as examples and examined by the finite element method. The stress concentrations of multiple inhomogeneities with small separation distances are well predicted by equivalent inhomogeneous inclusion method and the accuracies improve with the increase of eigenstrain orders. Equivalent inhomogeneous inclusion method gives more accurate stress predictions than equivalent inclusion method in the problem of two spherical inhomogeneities.

Numerical methods for solving the equivalent inclusion equation in semi-analytical models

2021 ◽  
pp. 135065012110001
Author(s):  
Zhiqiang Yan ◽  
Mengqi Zhang ◽  
Shulan Jiang
Keyword(s):  
Matrix Material ◽  
Current Method ◽  
Equation System ◽  
Analytical Models ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix ◽  
Convergence Mechanism ◽  
Computing Accuracy

Equivalent inclusion method is the basis for semi-analytical models in tackling inhomogeneity problems. Equivalent eigenstrains are obtained by solving the consistency equation system of the equivalent inclusion method and then stress disturbances caused by inhomogeneities are determined. The equivalent inclusion method equation system can only be solved numerically, but the current fixed-point iteration method may not be able to achieve deep convergence when the Young's modulus of inhomogeneity is lower than that of the matrix material. The most significant innovation of this paper is to reveal the non-convergence mechanism of the current method. Considering the limitation, the Jacobian-free Newton Krylov algorithm is selected to solve the equivalent inclusion method equation. Results indicate that the new algorithm has significant advantages of computing accuracy and efficiency compared with the classic method.

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.

Equivalent Inclusion Method for the Stokes Flow of Drops Moving in a Viscous Fluid

2014 ◽  
Vol 81 (7) ◽  
Author(s):  
H. M. Yin ◽  
P.-H. Lee ◽  
Y. J. Liu
Keyword(s):  
Viscous Fluid ◽  
Stokes Flow ◽  
The Body ◽  
Function Technique ◽  
Elongation Ratio ◽  
Spherical Drop ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix

The equivalent inclusion method is presented to derive the Stokes flow of multiple drops moving in a viscous fluid at a small Reynolds number. The drops are replaced by inclusions with the same viscosity as the fluid, but an eigenstrain rate field that is a fictitious nonmechanical strain rate field is introduced to represent the viscosity mismatch between each drop and the matrix fluid. The velocity and pressure fields can be solved by considering the body force and eigenstrain rate on the inclusions with the Green's function technique. When one spherical drop is considered, the solution recovers the closed-form classic solution. This method is versatile and can be used in the simulation of a many-body system with different drop size, elongation ratio, and viscosity. Numerical examples demonstrate the capability and accuracy of the proposed formulation and illustrate particles' rotation and motion caused by particle interactions.

Numerical Simulation on Nanoparticles Integrated Laser Shock Peening of Aluminum Alloy

2009 ◽  
Author(s):  
Chang Ye ◽  
Gary J. Cheng
Keyword(s):  
Numerical Simulation ◽  
Particle Size ◽  
Laser Shock Peening ◽  
Particle Orientation ◽  
Stress Wave Propagation ◽  
Stress And Strain ◽  
Laser Shock ◽  
The Matrix ◽  
Final Stress ◽  
Shock Peening

In this paper, numerical simulation of nanoparticle integrated laser shock peening of aluminum alloys was carried out. A “tied constraint” was used to connect the matrix and nanoparticle assembly in ABAQUS package. Different particle size and particle volumes fraction (PVF) were studied. It was found that there is significant stress concentration around the nanoparticles. The existence of nanoparticle will influence the stress wave propagation and thus the final stress and strain state of the material after LSP. In addition, particle size, PVF and particle orientation all influence the strain rate, static residual stress, static plastic strain and energy absorption during the LSP process.

scholarly journals Numerical Modelling of Stress/Strain Field Arising in Diamond-Impregnated Cobalt

2014 ◽  
Vol 59 (2) ◽  
pp. 443-446 ◽  
Author(s):  
J. Borowiecka-Jamrozek ◽  
J. Lachowski
Keyword(s):  
Strain Field ◽  
Diamond Crystal ◽  
Metallic Matrix ◽  
Stress And Strain ◽  
Stress Strain ◽  
Strain Fields ◽  
Matrix Potential ◽  
The Matrix ◽  
Saw Blade ◽  
Diamond Retention

Abstract The paper presents results of computer simulations of the stress/strain field built up in a cobalt matrix diamond impregnated saw blade segment during its fabrication and after loading the protruding diamond with an external force. The main objective of this work was to create better understanding of the factors affecting retention of diamond particles in a metallic matrix of saw blade segments, which are produced by means of the powder metallurgy technology. The effective use of diamond impregnated tools strongly depends on mechanical and tribological properties of the matrix, which has to hold the diamond grits firmly. The diamond retention capability of the matrix is affected in a complex manner by chemical or mechanical interactions between the diamond crystal and the matrix during the segment manufacture. Due to the difference between the thermal expansion coefficients of the diamond and metallic matrix, a complex stress/strain field is generated in the matrix surrounding each diamond crystal. It is assumed that the matrix potential for diamond retention can be associated with the amount of the elastic and plastic deformation energy and the size of the deformation zone occurring in the matrix around diamonds. The stress and strain fields generated in the matrix were calculated using the Abaqus software. It was found that the stress and strain fields generated during segment fabrication change to a large extent as the diamond crystal emerges from the cobalt matrix to reach its working height of protrusion.

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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