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.

A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion

2011 ◽  
Vol 78 (3) ◽  
Author(s):  
Xiaoqing Jin ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Elastic Field ◽  
Form Solution ◽  
Eshelby Tensor ◽  
Cylindrical Inclusion ◽  
Closed Form Expression ◽  
Equivalent Inclusion Method ◽  
Explicit Analytical Solution ◽  
Present Solution

From the analytical formulation developed by Ju and Sun [1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574], it is seen that the exterior point Eshelby tensor for an ellipsoid inclusion possesses a minor symmetry. The solution to an elliptic cylindrical inclusion may be obtained as a special case of Ju and Sun’s solution. It is noted that the closed-form expression for the exterior-point Eshelby tensor by Kim and Lee [2010, “Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion,” ASME Trans. J. Appl. Mech., 77, p. 024503] violates the minor symmetry. Due to the importance of the solution in micromechanics-based analysis and plane-elasticity-related problems, in this work, the explicit analytical solution is rederived. Furthermore, the exterior-point Eshelby tensor is used to derive the explicit closed-form solution for the elastic field outside the inclusion, as well as to quantify the elastic field discontinuity across the interface. A benchmark problem is used to demonstrate a valuable application of the present solution in implementing the equivalent inclusion method.

scholarly journals The dynamic generalization of the Eshelby inclusion problem and its static limit

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160256 ◽  
Author(s):  
Luqun Ni ◽  
Xanthippi Markenscoff
Keyword(s):  
Integral Form ◽  
Transformation Strain ◽  
The Self ◽  
Eshelby Tensor ◽  
Inclusion Problem ◽  
Static Limit ◽  
Governing System ◽  
Eshelby Inclusion ◽  
Interior Domain ◽  
Eshelby Problem

The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids ( doi:10.1016/j.jmps.2016.02.025 )) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.

Axially Symmetric Radial Flow of Rigid/Linear-Hardening Materials

1979 ◽  
Vol 46 (2) ◽  
pp. 322-328 ◽  
Author(s):  
D. Durban
Keyword(s):  
Radial Flow ◽  
Closed Form Solution ◽  
Wire Drawing ◽  
Form Solution ◽  
Flow Rule ◽  
Axially Symmetric ◽  
Linear Hardening ◽  
Stress Components ◽  
Von Mises ◽  
Good Agreement

A closed-form solution has been discovered for axially symmetric radial flow of rigid/linear-hardening materials. It is assumed that the materials obey the von Mises flow rule and that the flow field is in steady state. Explicit expressions for the stress components and the radial velocity are given. The applicability of the solution to wire drawing or extrusion is discussed. Some approximate formulas are derived and shown to be in good agreement, within their range of validity, with experimental results for drawing.

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.

scholarly journals Best theory diagrams for multilayered plates considering multifield analysis

2017 ◽  
Vol 28 (16) ◽  
pp. 2184-2205 ◽  
Author(s):  
M Cinefra ◽  
E Carrera ◽  
A Lamberti ◽  
M Petrolo
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Equilibrium Equations ◽  
Material Parameters ◽  
Stress Components ◽  
Minimum Number ◽  
Multilayered Plates ◽  
Carrera Unified Formulation ◽  
Definition Of

This work presents the best theory diagrams (BTDs) for multilayered plates involved in multifield problems (mechanical, thermal and electrical). A BTD is a curve that reports the minimum number of terms of a refined model for a given accuracy. The axiomatic/asymptotic technique is employed in order to detect the relevant terms, and the error is computed with respect to an exact or quasi-exact solution. The models that belong to the BTDs are constructed by means of a genetic algorithm and the Carrera Unified Formulation (CUF). The CUF defines the displacement field as an expansion of the thickness coordinate. The governing equations are obtained in terms of few fundamental nuclei, whose form does not depend on the particular expansion order that is employed. The Navier closed-form solution has been adopted to solve the equilibrium equations. The analyses herein reported are related to plates subjected to multifield loads: mechanical, thermal and electrical. The aim of this study is to evaluate the influence of the type of the load in the definition of the BTDs. In addition, the influence of geometry, material parameters and displacement/stress components are considered. The results suggest that the BTD and the CUF can be considered as tools to evaluate any structural theory against a reference solution. In addition, it has been found that the BTD definition is influenced to a great extent by the type of load.

Efficient Numerical Modeling of Hertzian Line Contact for Material With Inhomogeneities

2012 ◽  
Author(s):  
Xiaoqing Jin ◽  
Zhanjiang Wang ◽  
Qinghua Zhou ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Half Plane ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
General Purpose ◽  
Form Solution ◽  
Inclusion Problem ◽  
Computational Domain ◽  
Line Contact ◽  
Equivalent Inclusion Method ◽  
Parametric Studies

The present work proposes an efficient and general-purpose numerical approach for handling two-dimensional inhomogeneities in an elastic half plane. The inhomogeneities can be of any shape, at any location, with arbitrary material properties (which can also be non-homogeneous). To perform the numerical analysis, we first derive an explicit closed-form solution for a rectangular inclusion with uniform eigenstrain components, where the inclusion is aligned with the surface of the half plane. In view of the equivalent inclusion method, an inhomogeneity problem can be converted to a corresponding inclusion problem. In order to determine the distribution of the equivalent eigenstrain, the computational domain is meshed into rectangular elements whose resultant contributions can be efficiently computed using an efficient algorithm based on fast Fourier transform (FFT). In principle, there is no specific limitation on the type of the external load, although our major concern is the contact analysis. Parametric studies are performed and typical results highlighting the deviation of the current solution from the classical Hertzian line contact theory are presented.

Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
B. R. Kim ◽  
H. K. Lee
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Eshelby Tensor ◽  
Cylindrical Inclusion ◽  
Normal Vector ◽  
Eshelby’S Tensor ◽  
Cylindrical Inclusions ◽  
Outward Unit ◽  
Outward Unit Normal Vector

With the help of the I-integrals expressed by Mura (1987, Micromechanics of Defects in Solids, 2nd ed., Martinus Nijhoff, Dordrecht) and the outward unit normal vector introduced by Ju and Sun (1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574), the closed form solution of the exterior-point Eshelby tensor for an elliptic cylindrical inclusion is derived in this work. The proposed closed form of the Eshelby tensor for an elliptic cylindrical inclusion is more explicit than that given by Mura, which is rough and unfinished. The Eshelby tensor for an elliptic cylindrical inclusion can be reduced to the Eshelby tensor for a circular cylindrical inclusion by letting the aspect ratio of the inclusion α=1. The closed form Eshelby tensor presented in this study can contribute to micromechanics-based analysis of composites with elliptic cylindrical inclusions.

Interaction Between an Edge Dislocation and a Rigid Elliptical Inclusion

1986 ◽  
Vol 53 (2) ◽  
pp. 382-385 ◽  
Author(s):  
M. H. Santare ◽  
L. M. Keer
Keyword(s):  
Rigid Body ◽  
Edge Dislocation ◽  
Complex Potential ◽  
Closed Form Solution ◽  
Elastic Field ◽  
Form Solution ◽  
Two Dimensional ◽  
Elliptical Inclusion ◽  
Contour Plots ◽  
Rigid Elliptical Inclusion

A solution is presented for the two-dimensional elastic field created by the interaction of an edge dislocation with a rigid elliptical inclusion. The complex potential approach by Muskhelishvili is used and a closed-form solution is obtained. Contour plots for the glide component of the Peach–Koehler forces are presented. Particular attention is paid to the rigid body of the inclusion with respect to the dislocation.

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