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.

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.

Closed Form Solution for Outflow Between Corotating Disks

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Closed Form ◽  
Stokes Equations ◽  
Closed Form Solution ◽  
Tangential Velocity ◽  
Form Solution ◽  
Navier Stokes ◽  
Published Data ◽  
Rotational Inertia ◽  
Viscous Losses ◽  
Present Solution

Mathematical expressions are derived for flow velocities and pressure distributions for a laminar flow in the gap between two rotating concentric disks. Fluid enters the gap between disks at the center and diverges to the outer periphery. The Navier–Stokes equations are linearized in order to get closed-form solution. The present solution is applicable to the flow between corotating as well as contrarotating disks. The present results are in agreement with the published data of other investigators. The tangential velocity is less for contrarotating disks than for corotating disks in core region of the radial channel. The flow is influenced by rotational inertia and convective inertia both. Dominance of rotational inertia over convective inertia causes backflow. Pressure depends on viscous losses, convective inertia, and rotational inertias. Effect of viscous losses on pressure is high at small throughflow Reynolds number. The convective and rotational inertia influence pressure significantly at high throughflow and rotational Reynolds numbers. Both favorable and unfavorable pressure gradients can be found simultaneously depending on a combination of parameters.

A Closed-Form Solution for the Global Quadratic Hedging of Options under Geometric Gaussian Random Walks

2019 ◽  
Keyword(s):  
Closed Form ◽  
Recursive Algorithm ◽  
Closed Form Solution ◽  
Form Solution ◽  
Closed Form Expression ◽  
Problem Solution ◽  
Underlying Asset ◽  
Quadratic Hedging ◽  
Hedging Problem ◽  
The Impact

This study obtains a closed-form solution for the discrete-time global quadratic hedging problem of Schweizer (1995) applied to vanilla European options under the geometric Gaussian random walk model for the underlying asset. This extends the work of Rémillard and Rubenthaler (2013), who obtained closed-form formulas for some components of the hedging problem solution. Coefficients embedded in the closed-form expression can be computed either directly or through a recursive algorithm. The author also presents a brief sensitivity analysis to determine the impact of the underlying asset drift and the hedging portfolio rebalancing frequency on the optimal hedging capital and the initial hedge ratio.

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.

EXACT PRICING WITH STOCHASTIC VOLATILITY AND JUMPS

2010 ◽  
Vol 13 (06) ◽  
pp. 901-929 ◽  
Author(s):  
FERNANDA D'IPPOLITI ◽  
ENRICO MORETTO ◽  
SARA PASQUALI ◽  
BARBARA TRIVELLATO
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Exact Algorithm ◽  
Form Solution ◽  
Closed Form Expression ◽  
Barrier Options ◽  
Form Expression ◽  
Variance Swaps ◽  
Jump Diffusion Model

A stochastic volatility jump-diffusion model for pricing derivatives with jumps in both spot return and volatility underlying dynamics is presented. This model admits, in the spirit of Heston, a closed-form solution for European-style options. The structure of the model is also suitable to explicitly obtain the fair delivery price for variance swaps. To evaluate derivatives whose value does not admit a closed-form expression, a methodology based on an "exact algorithm", in the sense that no discretization of equations is required, is developed and applied to barrier options. Goodness of pricing algorithm is tested using DJ Euro Stoxx 50 market data for European options. Finally, the algorithm is applied to compute prices and Greeks for barrier options and to determine the fair delivery prices for variance swaps.

scholarly journals Enhancing the electrical and thermal conductivities of polymer composites via curvilinear fibers: An analytical study

2019 ◽  
Vol 24 (10) ◽  
pp. 3231-3253 ◽  
Author(s):  
Marco Salviato ◽  
Sean E Phenisee
Keyword(s):  
Closed Form ◽  
Electric Conductivity ◽  
Electrostatic Potential ◽  
Mechanical Performance ◽  
Transverse Isotropy ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Form Solution ◽  
Closed Form Expression ◽  
Element Analysis

The new generation of manufacturing technologies such as additive manufacturing and automated fiber placement has enabled the development of material systems with desired functional and mechanical properties via particular designs of inhomogeneities and their mesostructural arrangement. Among these systems, particularly interesting are materials exhibiting curvilinear transverse isotropy (CTI), in which the inhomogeneities take the form of continuous fibers following curvilinear paths designed to, for example, optimize the electric and thermal conductivity, and the mechanical performance of the system. In this context, the present work proposes a general framework for the exact, closed-form solution of electrostatic problems in materials featuring CTI. First, the general equations for the fiber paths that optimize the electric conductivity are derived, leveraging a proper conformal coordinate system. Then, the continuity equation for the curvilinear transversely isotropic system is derived in terms of electrostatic potential. A general exact, closed-form expression for the electrostatic potential and electric field is derived and validated by finite element analysis. Finally, potential avenues for the development of materials with superior electric conductivity and damage sensing capabilities are discussed.

scholarly journals Joint Resource Allocation for SWIPT-Based Two-Way Relay Networks

Energies ◽  
2020 ◽  
Vol 13 (22) ◽  
pp. 6024
Author(s):  
Chunling Peng ◽  
Guozhong Wang ◽  
Fangwei Li ◽  
Huaping Liu
Keyword(s):  
Resource Allocation ◽  
Outage Probability ◽  
Closed Form ◽  
Time Allocation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Closed Form Expression ◽  
Power Splitting ◽  
Decode And Forward ◽  
Joint Resource Allocation

This paper considers simultaneous wireless information and power transfer (SWIPT) in a decode-and-forward two-way relay (DF-TWR) network, where a power splitting protocol is employed at the relay for energy harvesting. The goal is to jointly optimize power allocation (PA) at the source nodes, power splitting (PS) at the relay node, and time allocation (TA) of each duration to minimize the system outage probability. In particular, we propose a static joint resource allocation (JRA) scheme and a dynamic JRA scheme with statistical channel properties and instantaneous channel characteristics, respectively. With the derived closed-form expression of the outage probability, a successive alternating optimization algorithm is proposed to tackle the static JRA problem. For the dynamic JRA scheme, a suboptimal closed-form solution is derived based on a multistep optimization and relaxation method. We present a comprehensive set of simulation results to evaluate the proposed schemes and compare their performances with those of existing resource allocation schemes.

scholarly journals Scattering from Spheres: A New Look into an Old Problem

Electronics ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 216
Author(s):  
Giuseppe Ruello ◽  
Riccardo Lattanzi
Keyword(s):  
Closed Form ◽  
Transmission Lines ◽  
Electromagnetic Scattering ◽  
Impedance Matching ◽  
Closed Form Solution ◽  
Mie Scattering ◽  
Form Solution ◽  
Closed Form Expression ◽  
Spherical Waves ◽  
Scattered Fields

In this work, we introduce a theoretical framework to describe the scattering from spheres. In our proposed framework, the total field in the outer medium is decomposed in terms of inward and outward electromagnetic fields, rather than in terms of incident and scattered fields, as in the classical Lorenz–Mie formulation. The fields are expressed as series of spherical harmonics, whose combination weights can be interpreted as reflection and transmission coefficients, which provides an intuitive understanding of the propagation and scattering phenomena. Our formulation extends the previously proposed theory of non-uniform transmission lines by introducing an expression for impedance transfer, which yields a closed-form solution for the fields inside and outside the sphere. The power transmitted in and scattered by the sphere can be also evaluated with a simple closed-form expression and related with the modulus of the reflection coefficient. We showed that our method is fully consistent with the classical Mie scattering theory. We also showed that our method can provide an intuitive physical interpretation of electromagnetic scattering in terms of impedance matching and resonances, and that it is especially useful for the case of inward traveling spherical waves generated by sources surrounding the scatterer.

scholarly journals A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

2012 ◽  
Vol 19 (6) ◽  
pp. 1415-1426 ◽  
Author(s):  
A. Barari ◽  
A. Kimiaeifar ◽  
M.G. Nejad ◽  
M. Motevalli ◽  
M.G. Sfahani
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Analytical Techniques ◽  
Nonlinear Oscillators ◽  
Form Solution ◽  
Closed Form Expression ◽  
System Response ◽  
Classical Dynamics ◽  
Analytical Technique

Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

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.

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