On the Imperfectly Bonded Spherical Inclusion Problem

1999 ◽  
Vol 66 (4) ◽  
pp. 839-846 ◽  
Author(s):  
Z. Zhong ◽  
S. A. Meguid
Keyword(s):  
Superposition Principle ◽  
Spherical Inclusion ◽  
Elastic Strain Energy ◽  
Elastic Field ◽  
Displacement Discontinuity ◽  
Inclusion Problem ◽  
Interfacial Condition ◽  
Interfacial Sliding ◽  
Equivalent Inclusion Method ◽  
The Matrix

An exact solution is developed for the problem of a spherical inclusion with an imperfectly bonded interface. The inclusion is assumed to have a uniform eigenstrain and a different elastic modulus tensor from that of the matrix. The displacement discontinuity at the interface is considered and a linear interfacial condition, which assumes that the displacement jump is proportional to the interfacial traction, is adopted. The elastic field induced by the uniform eigenstrain given in the imperfectly bonded inclusion is decomposed into three parts. The first part is prescribed by a uniform eigenstrain in a perfectly bonded spherical inclusion. The second part is formulated in terms of an equivalent nonuniform eigenstrain distributed over a perfectly bonded spherical inclusion which models the material mismatch between the inclusion and the matrix, while the third part is obtained in terms of an imaginary Somigliana dislocation field which models the interfacial sliding and normal separation. The exact form of the equivalent nonuniform eigenstrain and the imaginary Somigliana dislocation are fully determined using the equivalent inclusion method and the associated interfacial condition. The elastic fields are then obtained explicitly by means of the superposition principle. The resulting solution is then used to evaluate the average Eshelby tensor and the elastic strain energy.

The Stress Field and Intensity Factor due to Crazes Formed at the Poles of a Spherical Inhomogeneity

1994 ◽  
Vol 61 (4) ◽  
pp. 803-808 ◽  
Author(s):  
Z. M. Xiao ◽  
K. D. Pae
Keyword(s):  
Stress Intensity Factor ◽  
Intensity Factor ◽  
Stress Field ◽  
Elastic Moduli ◽  
Superposition Principle ◽  
Numerical Examples ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix ◽  
Good Agreement

The problem of two penny-shaped crazes formed at the top and the bottom poles of a spherical inhomogeneity has been investigated. The inhomogeneity is embedded in an infinitely extended elastic body which is under uniaxial tension. Both the inhomogeneity and the matrix are isotropic but have different elastic moduli. The analysis is based on the superposition principle of the elasticity theory and Eshelby’s equivalent inclusion method. The stress field inside the inhomogeneity and the stress intensity factor on the boundary of the craze are evaluated in the form of a series which involves the ratio of the radius of the penny-shaped craze to the radius of the spherical inhomogeneity. Numerical examples show the interaction between the craze and the inhomogeneity is strongly affected by the elastic properties of the inhomogeneity and the matrix. The conclusion deduced from the numerical results is in good agreement with experimental results given in the literature.

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.

On the Eigenstrain Problem of a Spherical Inclusion With an Imperfectly Bonded Interface

1996 ◽  
Vol 63 (4) ◽  
pp. 877-883 ◽  
Author(s):  
Z. Zhong ◽  
S. A. Meguid
Keyword(s):  
Spherical Inclusion ◽  
Normal Displacement ◽  
Theoretical Treatment ◽  
Interfacial Condition ◽  
Isotropic Materials ◽  
Interfacial Sliding ◽  
Bonded Interface ◽  
Special Cases ◽  
Normal Separation ◽  
Displacement Jumps

This article provides a comprehensive theoretical treatment of the eigenstrain problem of a spherical inclusion with an imperfectly bonded interface. Both tangential and normal discontinuities at the interface are considered and a linear interfacial condition, which assumes that the tangential and the normal displacement jumps are proportional to the associated tractions, is adopted. The solution to the corresponding eigenstrain problem is obtained by combining Eshelby’s solution for a perfectly bonded inclusion with Volterra’s solution for an equivalent Somigliana dislocation field which models the interfacial sliding and normal separation. For isotropic materials, the Burger’s vector of the equivalent Somigliana dislocation is exactly determined; the solution is explicitly presented and its uniqueness demonstrated. It is found that the stresses inside the inclusion are not uniform, except for some special cases.

Elastic inclusions and inhomogeneities in transversely isotropic solids

1994 ◽  
Vol 444 (1920) ◽  
pp. 239-252 ◽  
Keyword(s):  
Closed Form ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Closed Form Expression ◽  
Inclusion Problem ◽  
Stress Vector ◽  
Equivalent Inclusion Method ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

A method that introduces a new stress vector function ( the hexagonal stress vector ) is applied to obtain, in closed form, the elastic fields due to an inclusion in transversely isotropic solids. The solution is an extension of Eshelby’s solution for an ellipsoidal inclusion in isotropic solids. The Green’s functions for double forces and double forces with moment are derived and these are then used to solve the inclusion problem. The elastic field inside the inclusion is expressed in terms of the newtonian and biharmonic potential functions, which are similar to those needed for the solution in isotropic solids. Two more harmonic potential functions are introduced to express the solution outside the inclusion. The constrained strain inside the inclusion is uniform and the relation between the constrained strain and the misfit strain has the same characteristics as those of the Eshelby tensor for isotropic solids, namely, the coefficients coupling an extension to a shear or one shear to another are zero. The explicit closed form expression of this tensor is given. The inhomogeneity problem is then solved by using Eshelby’s equivalent inclusion method. The solution for the thermoelastic displacements due to thermal inhomogeneities is also presented.

Elastohydrodynamic Lubrication Analysis of Point Contacts With Consideration of Material Inhomogeneity

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Zhang Shengguang ◽  
Wang Wenzhong ◽  
Zhao Ziqiang
Keyword(s):  
Film Thickness ◽  
Numerical Algorithms ◽  
Elastohydrodynamic Lubrication ◽  
Rolling Bearing ◽  
Inclusion Problem ◽  
Material Inhomogeneity ◽  
Equivalent Inclusion Method ◽  
Lubrication Performance ◽  
The Matrix ◽  
Lubricated Contact

Inhomogeneities in matrix may significantly affect the performance of mechanical elements, such as possible fatigue life reduction for rolling bearing due to stress concentration induced by inhomogeneities; on the other hand, most components operate under lubrication environment. So far the numerical algorithms to solve lubrication problems without the consideration of inhomogeneities or inclusions are well developed. In this paper, the combination of elastohydrodynamic lubrication (EHL) and inclusion problem is realized to consider the effect of material inhomogeneity on the lubrication performance and subsurface stress distribution, etc. The matrix inhomogeneity will induce disturbed displacement, which will modify the film thickness and consequently result in the change of lubricated contact pressure distribution, etc. The matrix inhomogeneity is treated as the homogeneous inclusion with equivalent eigenstrain according to equivalent inclusion method (EIM), and the disturbed displacement is calculated by semi-analytical method (SAM). While the pressure and film thickness distributions are obtained by solving Reynolds equation. The iterative process is realized to consider the interaction between lubrication behavior and material response. The results show the inhomogeneity in contacting body will greatly influence the lubricated contact performance. The influences are different between compliant and stiff inhomogeneity. It is also found that different sizes and positions of inhomogeneity can significantly affect the contact characteristic parameters.

Elastic Field Due to a Dynamically Transforming Spherical Inclusion

1990 ◽  
Vol 57 (4) ◽  
pp. 845-849 ◽  
Author(s):  
Y. Mikata ◽  
S. Nemat-Nasser
Keyword(s):  
Phase Transformation ◽  
Systematic Study ◽  
Spherical Inclusion ◽  
Closed Form Solution ◽  
Elastic Field ◽  
Form Solution ◽  
Eshelby Tensor ◽  
Inclusion Problem ◽  
Static Limit ◽  
Stress Components

As a first step towards a systematic study of the interaction between a stress-pulse traveling in transformation toughened ceramics and possible phase transformation of zirconia particles, a dynamic inclusion problem is investigated. An exact closed-form solution is obtained for the case of a spherical inclusion. With this result, the dynamic Eshelby tensors for the inside and outside fields of the spherical inclusion are defined and determined. It is confirmed that the static Eshelby tensor is obtained as a static limit of the dynamic Eshelby tensor. It is found in the numerical results that the frequency of the dynamic inclusion has a relatively large influence on the amplitudes of the stress components inside and outside the inclusion.

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.

Stress Concentrations in Three-Dimensional Electrostriction

1967 ◽  
Vol 34 (2) ◽  
pp. 431-436 ◽  
Author(s):  
T. E. Smith
Keyword(s):  
Displacement Vector ◽  
Spherical Inclusion ◽  
Crack Problem ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Uniform Electric Field ◽  
Inclusion Problem ◽  
Stress Concentrations ◽  
Displacement Potential ◽  
Penny Shaped Crack

Using the techniques employed in developing a Papkovich-Neuber representation for the displacement vector in classical elasticity, a particular integral of the kinematical equations of equilibrium for the uncoupled theory of electrostriction is developed. The particular integral is utilized in conjunction with the displacement potential function approach to problems of the theory of elasticity to obtain closed-form solutions of several stress concentration problems for elastic dielectrics. Under a prescribed uniform electric field at infinity, the problems of an infinite elastic dielectric having first a spherical cavity and then a rigid spherical inclusion are solved. The rigid spheroidal inclusion problem and the penny-shaped crack problem are also solved for the case where the prescribed field is parallel to their axes of revolution.

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.

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