Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Xiaoqing Jin ◽  
Ding Lyu ◽  
Xiangning Zhang ◽  
Qinghua Zhou ◽  
Qian Wang ◽  
...  
Keyword(s):  
Deformation Gradient ◽  
Elastic Field ◽  
Analytical Form ◽  
Ellipsoidal Inclusion ◽  
Normal Vector ◽  
Full Field ◽  
Practical Applications ◽  
Starting Point ◽  
Complete Set ◽  
Outward Unit

The celebrated solution of the Eshelby ellipsoidal inclusion has laid the cornerstone for many fundamental aspects of micromechanics. A well-known difficulty of this classical solution is to determine the elastic field outside the ellipsoidal inclusion. In this paper, we first analytically present the full displacement field of an ellipsoidal inclusion subjected to uniform eigenstrain. It is demonstrated that the displacements inside inclusion are linearly related to the coordinates and continuous across the interface of inclusion and matrix. The exterior displacement, which is less detailed in existing literatures, may be expressed in a more compact, explicit, and simpler form through utilizing the outward unit normal vector of an auxiliary confocal ellipsoid. Other than many practical applications in geological engineering, the displacement solution can be a convenient starting point to derive the deformation gradient, and subsequently in a straightforward manner to accomplish the full-field solutions of the strain and stress. Following Eshelby's definition, a complete set of the Eshelby tensors corresponding to the displacement, deformation gradient, strain, and stress are expressed in explicit analytical form. Furthermore, the jump conditions to quantify the discontinuities across the interface are discussed and a benchmark problem is provided to validate the present formulation.

Steklov eigenvalue problem with a-harmonic solutions and variable exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Belhadj Karim ◽  
Abdellah Zerouali ◽  
Omar Chakrone
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Boundary ◽  
Normal Vector ◽  
Variable Exponents ◽  
Variational Technique ◽  
Unit Normal Vector ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Outward Unit ◽  
Outward Unit Normal Vector

AbstractUsing the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences:\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}}{(N\geq 2)} is a bounded domain of smooth boundary {\partial\Omega} and ν is the outward unit normal vector on {\partial\Omega}. The functions {m\in L^{\infty}(\partial\Omega)}, {p\colon\overline{\Omega}\mapsto\mathbb{R}} and {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

The elastic field outside an ellipsoidal inclusion

1959 ◽  
Vol 252 (1271) ◽  
pp. 561-569 ◽  
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Elastic Constants ◽  
Strain Energy ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Harmonic Potential ◽  
Surface Distribution ◽  
Shear Transformation ◽  
General Method

The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.

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.

scholarly journals CFD Analysis of Urban Canopy Flows Employing the V2F Model: Impact of Different Aspect Ratios and Relative Heights

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Fabio Nardecchia ◽  
Annalisa Di Bernardino ◽  
Francesca Pagliaro ◽  
Paolo Monti ◽  
Giovanni Leuzzi ◽  
...  
Keyword(s):  
Flow Field ◽  
Water Channel ◽  
Flow Regimes ◽  
Two Dimensional ◽  
Mean Flow ◽  
Ansys Fluent ◽  
Practical Applications ◽  
Aspect Ratios ◽  
Starting Point ◽  
Step Down

Computational fluid dynamics (CFD) is currently used in the environmental field to simulate flow and dispersion of pollutants around buildings. However, the closure assumptions of the turbulence usually employed in CFD codes are not always physically based and adequate for all the flow regimes relating to practical applications. The starting point of this work is the performance assessment of the V2F (i.e., v2¯ − f) model implemented in Ansys Fluent for simulating the flow field in an idealized array of two-dimensional canyons. The V2F model has been used in the past to predict low-speed and wall-bounded flows, but it has never been used to simulate airflows in urban street canyons. The numerical results are validated against experimental data collected in the water channel and compared with other turbulence models incorporated in Ansys Fluent (i.e., variations of both k-ε and k-ω models and the Reynolds stress model). The results show that the V2F model provides the best prediction of the flow field for two flow regimes commonly found in urban canopies. The V2F model is also employed to quantify the air-exchange rate (ACH) for a series of two-dimensional building arrangements, such as step-up and step-down configurations, having different aspect ratios and relative heights of the buildings. The results show a clear dependence of the ACH on the latter two parameters and highlight the role played by the turbulence in the exchange of air mass, particularly important for the step-down configurations, when the ventilation associated with the mean flow is generally poor.

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

2001 ◽  
Vol 440 ◽  
pp. 269-291 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Curvature Tensor ◽  
Deformation Gradient ◽  
Bending Stiffness ◽  
Boundary Element Methods ◽  
Simple Shear Flow ◽  
Normal Vector ◽  
Membrane Thickness ◽  
Bending Moments ◽  
Balance Equations ◽  
Capsule Deformation

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

Spaces and Operators

2012 ◽  
Author(s):  
G. F. Roach ◽  
I. G. Stratis ◽  
A. N. Yannacopoulos
Keyword(s):  
Normal Vector ◽  
Connected Component ◽  
Unit Normal Vector ◽  
Open Set ◽  
Outward Unit ◽  
Outward Unit Normal Vector ◽  
Unit Normal

This chapter consists mainly of definitions and various properties (without proofs) of spaces and operators used in this book. It defines O as an open set in Rᶰ such that it is locally on one side of its boundary Γ‎ := δ‎O, which is supposed to be bounded and Lipschitz. The chapter is mainly focused on the case of N = 3. Further, without loss of generality, the chapter supposes that Γ‎ is connected (for otherwise, one could work separately at each connected component). Such a set O is referred to as ‘regular’ in what follows. Let n denote the outward unit normal vector to Γ‎. In addition, let Oₑ := Rᶰ∖Ō: By N₀ we denote the set N ∪ {0}.

scholarly journals Further Considerations of the Elastic Inclusion Problem

1964 ◽  
Vol 14 (1) ◽  
pp. 61-70 ◽  
Author(s):  
R. J. Knops
Keyword(s):  
Elastic Inclusion ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Inclusion Problem ◽  
Infinite Matrix ◽  
Mean Values ◽  
Shear Moduli ◽  
Weakly Interacting

SummaryAn equation is derived for the strains of an arbitrary elastic field in an infinite matrix perturbed by several inclusions. The equation is solved exactly when the shear moduli of the inclusions and matrix are identical, and also when only a single ellipsoidal inclusion perturbs a field uniform at infinity. Mean-values of the strains are then calculated for non-uniform fields perturbed either by an ellipsoid or by a system of weakly-interacting spheres.

scholarly journals Matched filter in low-number count Poisson noise regime: Efficient and effective implementation

2018 ◽  
Vol 616 ◽  
pp. A25 ◽  
Author(s):  
R. Vio ◽  
P. Andreani
Keyword(s):  
Numerical Simulations ◽  
Matched Filter ◽  
Analytical Form ◽  
Analytical Framework ◽  
Poisson Noise ◽  
Saddle Point Approximation ◽  
Number Count ◽  
Practical Applications ◽  
Effective Implementation ◽  
Non Gaussian

The matched filter (MF) is widely used to detect signals hidden within the noise. If the noise is Gaussian, its performances are well-known and can be described in an elegant analytical form. The treatment of non-Gaussian noises is often cumbersome as in most cases there is no analytical framework. This is true also for Poisson noise which, especially in the low-number count regime, presents the additional difficulty to be discrete. For this reason in the past methods have been proposed based on heuristic or semi-heuristic arguments. Recently, an analytical form of the MF has been introduced but the computation of the probability of false detection or false alarm (PFA) is based on numerical simulations. To overcome this inefficient and time-consuming approach, we propose here an effective method to compute the PFA based on the saddle point approximation (SA). We provide the theoretical framework and support our findings by means of numerical simulations. We also discuss the limitations of the MF in practical applications.

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