The Eshelby Tensors in a Finite Spherical Domain—Part I: Theoretical Formulations

2006 ◽  
Vol 74 (4) ◽  
pp. 770-783 ◽  
Author(s):  
Shaofan Li ◽  
Roger A. Sauer ◽  
Gang Wang
Keyword(s):  
Boundary Condition ◽  
Volume Fraction ◽  
Spherical Inclusion ◽  
Finite Size ◽  
Transversely Isotropic ◽  
Solution Procedure ◽  
Eshelby Tensor ◽  
Homogenization Theory ◽  
Fredholm Type ◽  
Spherical Domain

This work is concerned with the precise characterization of the elastic fields due to a spherical inclusion embedded within a spherical representative volume element (RVE). The RVE is considered having finite size, with either a prescribed uniform displacement or a prescribed uniform traction boundary condition. Based on symmetry and group theoretic arguments, we identify that the Eshelby tensor for a spherical inclusion admits a unique decomposition, which we coin the “radial transversely isotropic tensor.” Based on this notion, a novel solution procedure is presented to solve the resulting Fredholm type integral equations. By using this technique, exact and closed form solutions have been obtained for the elastic disturbance fields. In the solution two new tensors appear, which are termed the Dirichlet–Eshelby tensor and the Neumann–Eshelby tensor. In contrast to the classical Eshelby tensor they both are position dependent and contain information about the boundary condition of the RVE as well as the volume fraction of the inclusion. The new finite Eshelby tensors have far-reaching consequences in applications such as nanotechnology, homogenization theory of composite materials, and defects mechanics.

Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers

2016 ◽  
Vol 83 (7) ◽  
Author(s):  
Guannan Wang ◽  
Marek-Jerzy Pindera
Keyword(s):  
Volume Fraction ◽  
Local Stress ◽  
Transversely Isotropic ◽  
Local Fields ◽  
Homogenization Theory ◽  
Classical Solutions ◽  
Equilibrium Equations ◽  
Fiber Volume ◽  
Unidirectional Composites ◽  
Cylindrically Orthotropic

The elasticity-based, locally exact homogenization theory for unidirectional composites with hexagonal and tetragonal symmetries and transversely isotropic phases is further extended to accommodate cylindrically orthotropic reinforcement. The theory employs Fourier series representations of the fiber and matrix displacement fields in cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. Satisfaction of periodicity conditions for the inseparable exterior problem is efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement solution convergence with relatively few harmonic terms. As demonstrated in this contribution, this also applies to cylindrically orthotropic reinforcement for which the eigenvalues depend on both the orthotropic elastic moduli and harmonic number. The solution's demonstrated stability facilitates rapid identification of cylindrical orthotropy's impact on homogenized moduli and local fields in wide ranges of fiber volume fraction and orthotropy ratios. The developed theory provides a unified approach that accounts for cylindrical orthotropy explicitly in both the homogenization process and local stress field calculations previously treated separately through a fiber replacement scheme. Comparison of the locally exact solution with classical solutions based on an idealized microstructural representation and fiber moduli replacement with equivalent transversely isotropic properties delineates their applicability and limitations.

scholarly journals Eshelby's problem of a spherical inclusion eccentrically embedded in a finite spherical body

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160808 ◽  
Author(s):  
W.-N. Zou ◽  
Q.-C. He
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Superposition Principle ◽  
Auxiliary Problem ◽  
Spherical Inclusion ◽  
Spherical Body ◽  
Physical Parameters ◽  
Tensor Fields ◽  
Spherical Domain

Resorting to the superposition principle, the solution of Eshelby's problem of a spherical inclusion located eccentrically inside a finite spherical domain is obtained in two steps: (i) the solution to the problem of a spherical inclusion in an infinite space; (ii) the solution to the auxiliary problem of the corresponding finite spherical domain subjected to appropriate boundary conditions. Moreover, a set of functions called the sectional and harmonic deviators are proposed and developed to work out the auxiliary solution in a series form, including the displacement and Eshelby tensor fields. The analytical solutions are explicitly obtained and illustrated when the geometric and physical parameters and the boundary condition are specified.

scholarly journals Evolution of Shape and Volume Fraction of Superconducting Domains with Temperature and Anion Disorder in (TMTSF)2ClO4

Crystals ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 72
Author(s):  
Kaushal K. Kesharpu ◽  
Vladislav D. Kochev ◽  
Pavel D. Grigoriev
Keyword(s):  
Cooling Rate ◽  
Annealing Time ◽  
Volume Fraction ◽  
Density Wave ◽  
Spin Density Wave ◽  
Finite Size ◽  
Organic Superconductors ◽  
Organic Superconductor ◽  
Sample Length ◽  
Sample Shape

In highly anisotropic organic superconductor (TMTSF)2ClO4, superconducting (SC) phase coexists with metallic and spin-density wave phases in the form of domains. Using the Maxwell-Garnett approximation (MGA), we calculate the volume ratio and estimate the shape of these embedded SC domains from resistivity data at various temperature and anion disorder, controlled by the cooling rate or annealing time of (TMTSF)2ClO4 samples. We found that the variation of cooling rate and of annealing time affect differently the shape of SC domains. In all cases the SC domains have oblate shape, being the shortest along the interlayer z-axis. This contradicts the widely assumed filamentary superconductivity along the z-axis, used to explain the anisotropic superconductivity onset. We show that anisotropic resistivity drop at the SC onset can be described by the analytical MGA theory with anisotropic background resistance, while the anisotropic Tc can be explained by considering a finite size and flat shape of the samples. Due to a flat/needle sample shape, the probability of percolation via SC domains is the highest along the shortest sample dimension (z-axis), and the lowest along the sample length (x-axis). Our theory can be applied to other heterogeneous superconductors, where the size d of SC domains is much larger than the SC coherence length ξ, e.g., cuprates, iron-based or organic superconductors. It is also applicable when the spin/charge-density wave domains are embedded inside a metallic background, or vice versa.

scholarly journals An effective thermal conductivity and thermomechanical homogenization scheme for a multiscale Nb3Sn filaments

2021 ◽  
Vol 10 (1) ◽  
pp. 187-200
Author(s):  
Xiaoyu Zhao ◽  
Guannan Wang ◽  
Qiang Chen ◽  
Libin Duan ◽  
Wenqiong Tu
Keyword(s):  
Volume Fraction ◽  
Material Design ◽  
Axial Direction ◽  
High Heat ◽  
Thermomechanical Properties ◽  
Homogenization Theory ◽  
High Heat Flux ◽  
Element Analysis ◽  
Hierarchical Microstructure ◽  
Thermal Conductivities

Abstract A comprehensive study of the multiscale homogenized thermal conductivities and thermomechanical properties is conducted towards the filament groups of European Advanced Superconductors (EAS) strand via the recently proposed Multiphysics Locally Exact Homogenization Theory (LEHT). The filament groups have a distinctive two-level hierarchical microstructure with a repeating pattern perpendicular to the axial direction of Nb3Sn filament. The Nb3Sn filaments are processed in a very high temperature between 600 and 700°C, while its operation temperature is extremely low, −269°C. Meanwhile, Nb3Sn may experience high heat flux due to low resistivity of Nb3Sn in the normal state. The intrinsic hierarchical microstructure of Nb3Sn filament groups and Multiphysics loading conditions make LEHT an ideal candidate to conduct the homogenized thermal conductivities and thermomechanical analysis. First, a comparison with a finite element analysis is conducted to validate effectiveness of Multiphysics LEHT and good agreement is obtained for the homogenized thermal conductivities and mechanical and thermal expansion properties. Then, the Multiphysics LEHT is applied to systematically investigate the effects of volume fraction and temperature on homogenized thermal conductivities and thermomechanical properties of Nb3Sn filaments at the microscale and mesoscale. Those homogenized properties provide a full picture for researchers or engineers to understand the Nb3Sn homogenized properties and will further facilitate the material design and application.

A Semi-Analytical Approach to Time Dependent Squeezing Flow of Cu and Ag Water-Based Nanofluids

2019 ◽  
Vol 393 ◽  
pp. 121-137 ◽  
Author(s):  
S.R. Mishra ◽  
Debi P. Bhatta ◽  
J.K. Dash ◽  
Oluwole Daniel Makinde
Keyword(s):  
Volume Fraction ◽  
Similarity Transformation ◽  
Velocity Slip ◽  
Solution Procedure ◽  
Vital Role ◽  
Unknown Coefficient ◽  
Squeezing Flow ◽  
Physical Parameters ◽  
Parallel Plates ◽  
Nonlinear Odes

Study reveals the axisymmetric squeezing flow of nanofluids through two parallel plates. Both Copper (Cu) and Silver (Ag) nanoparticles along with water treated as base fluid have been taken into consideration. Viscous dissipation effect and velocity slip both enhance the present study. The non-dimensional form of governing nonlinear ODEs is obtained with the suitable choice of similarity transformation. The complex ODEs are solved analytically imposing Adomain Decomposition Method (ADM). The influence of emerging parameters such as nanoparticle volume fraction, unsteadiness parameter, Eckert number, etc. have been described by visualizing graphically and the tabular values represent the unknown coefficient and computation is made for various values of physical parameters. The present result is compatible with the earlier which confirms the accuracy of the solution procedure. It reveals that point of inflection is marked in the velocity profiles of both Ag and Cu water nanofluids for the effects of various physical parameters. Squeezing number play a vital role in the velocity profile and it is observed that near the lower plate Ag nanoparticle dominates over Cu nanoparticles and further, after the middle of the channel the effect is reversed. 2010 Mathematics Subject Classification: 76D05, 76D10, 76M60, 76S05. *Corresponding Author’s Email: HYPERLINK "mailto:[email protected]" [email protected] Mobile No.: (+91)-9937169245

scholarly journals Homogenization of a Conductive and Radiative Heat Transfer Problem, Simulation With CAST3M

2005 ◽  
Author(s):  
K. El Ganaoui ◽  
G. Allaire
Keyword(s):  
Boundary Condition ◽  
Radiative Heat ◽  
Transfer Problem ◽  
Computer Code ◽  
Homogenization Theory ◽  
Linear Boundary ◽  
Mathematical Study ◽  
Linear Boundary Condition ◽  
The Core ◽  
Homogenous Medium

We are interested in conductive and radiative transfer of energy in the core of gas cooled reactors. Two scales characterize the problem: macroscopic and microscopic. We want to consider the domain like an equivalent homogenous medium. So we use homogenization theory to compute the effective macroscopic properties which take into account the microscopic structure. We first present a full mathematical study of a simpler conduction problem with non linear boundary condition and its simulation with the CEA’s (French Atomic Energy Commissariat) computer code CAST3M. Then we present the homogenization of the real physical problem (including radiative boundary condition).

scholarly journals Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity

2019 ◽  
Vol 16 (151) ◽  
pp. 20180911 ◽  
Author(s):  
Xiran Cai ◽  
Renald Brenner ◽  
Laura Peralta ◽  
Cécile Olivier ◽  
Pierre-Jean Gouttenoire ◽  
...  
Keyword(s):  
Cortical Bone ◽  
Volume Fraction ◽  
Transversely Isotropic ◽  
Pore Network ◽  
Anisotropic Elasticity ◽  
Homogenization Method ◽  
Femoral Diaphysis ◽  
Micro Computed Tomography ◽  
Architectural Features ◽  
Resonant Ultrasound

With ageing and various diseases, the vascular pore volume fraction (porosity) in cortical bone increases, and the morphology of the pore network is altered. Cortical bone elasticity is known to decrease with increasing porosity, but the effect of the microstructure is largely unknown, while it has been thoroughly studied for trabecular bone. Also, popular micromechanical models have disregarded several micro-architectural features, idealizing pores as cylinders aligned with the axis of the diaphysis. The aim of this paper is to quantify the relative effects on cortical bone anisotropic elasticity of porosity and other descriptors of the pore network micro-architecture associated with pore number, size and shape. The five stiffness constants of bone assumed to be a transversely isotropic material were measured with resonant ultrasound spectroscopy in 55 specimens from the femoral diaphysis of 29 donors. The pore network, imaged with synchrotron radiation X-ray micro-computed tomography, was used to derive the pore descriptors and to build a homogenization model using the fast Fourier transform (FFT) method. The model was calibrated using experimental elasticity. A detailed analysis of the computed effective elasticity revealed in particular that porosity explains most of the variations of the five stiffness constants and that the effects of other micro-architectural features are small compared to usual experimental errors. We also have evidence that modelling the pore network as an ensemble of cylinders yields biased elasticity values compared to predictions based on the real micro-architecture. The FFT homogenization method is shown to be particularly efficient to model cortical bone.

Low-frequency layer-induced kinematic parameters in transversely isotropic media and orthorhombic media

2019 ◽  
Vol 220 (2) ◽  
pp. 839-855
Author(s):  
Da Shuai ◽  
Alexey Stovas
Keyword(s):  
Volume Fraction ◽  
Velocity Ratio ◽  
Symmetry Plane ◽  
Low Frequency ◽  
Transversely Isotropic ◽  
Kinematic Parameters ◽  
Frequency Dependent ◽  
Cross Term ◽  
Transversely Isotropic Media ◽  
Isotropic Media

SUMMARY We develop a method to compute frequency-dependent kinematic parameters for an effective orthorhombic (ORT) medium. In order to investigate the influence of fracture weaknesses on the kinematic parameters, the effective ORT medium is composed based on the linear slip theory and derived by applying the limited Baker–Campbell–Hausdorff series. The frequency-dependent kinematic parameters including vertical velocity, two normal moveout velocities defined in vertical symmetry planes, and three anelliptic parameters (two of them are defined in vertical symmetry plane and one parameter is the cross-term one). We also investigate the influence of volume fraction, frequency, velocity ratio and fracture weaknesses on the effective kinematic parameters.

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