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 closedform domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansionmore »
Green's function for anisotropic dispersive poroelastic media based on the Radon transform and eigenvector diagonalization
A compact Green's function for general dispersive anisotropic poroelastic media in a fullfrequency regime is presented for the first time. First, starting in a frequency domain, the anisotropic dispersion is exactly incorporated into the constitutive relationship, thus avoiding fractional derivatives in a time domain. Then, based on the Radon transform, the original threedimensional differential equation is effectively reduced to a onedimensional system in space. Furthermore, inspired by the strategy adopted in the characteristic analysis of hyperbolic equations, the eigenvector diagonalization method is applied to decouple the onedimensional vector problem into several independent scalar equations. Consequently, the fundamental solutions are easily obtained. A further derivation shows that Green's function can be decomposed into circumferential and spherical integrals, corresponding to static and transient responses, respectively. The procedures shown in this study are also compatible with other pertinent multiphysics coupling problems, such as piezoelectric, magnetoelectroelastic and thermoelastic materials. Finally, the verifications and validations with existing analytical solutions and numerical solvers corroborate the correctness of the proposed Green's function.
 Award ID(s):
 1812573
 Publication Date:
 NSFPAR ID:
 10199183
 Journal Name:
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
 Volume:
 475
 Issue:
 2221
 Page Range or eLocationID:
 20180610
 ISSN:
 13645021
 Sponsoring Org:
 National Science Foundation
More Like this


Abstract We outline and interpret a recently developed theory of impedance matching or reflectionless excitation of arbitrary finite photonic structures in any dimension. The theory includes both the case of guided wave and freespace excitation. It describes the necessary and sufficient conditions for perfectly reflectionless excitation to be possible and specifies how many physical parameters must be tuned to achieve this. In the absence of geometric symmetries, such as parity and timereversal, the product of parity and timereversal, or rotational symmetry, the tuning of at least one structural parameter will be necessary to achieve reflectionless excitation. The theory employs a recently identified set of complex frequency solutions of the Maxwell equations as a starting point, which are defined by having zero reflection into a chosen set of input channels, and which are referred to as Rzeros. Tuning is generically necessary in order to move an Rzero to the real frequency axis, where it becomes a physical steadystate impedancematched solution, which we refer to as a reflectionless scattering mode (RSM). In addition, except in singlechannel systems, the RSM corresponds to a particular input wavefront, and any other wavefront will generally not be reflectionless. It is useful to consider the theory asmore »

Consider the elastic scattering of a timeharmonic wave by multiple wellseparated rigid particles with smooth boundaries in two dimensions. Instead of using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of the Helmholtz equation. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The wellposedness of the new integral equations is established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving elastic scattering problems with multiple particles.

The chief objective of this paper is to explore energy transfer mechanism between the subsystems that are coupled by a nonlinear elastic path. In the proposed model (via a minimal order, two degree of freedom system), both subsystems are defined as damped harmonic oscillators with linear springs and dampers. The first subsystem is attached to the ground on one side but connected to the second subsystem on the other side. In addition, linear elastic and dissipative characteristics of both oscillators are assumed to be identical, and a harmonic force excitation is applied only on the mass element of second oscillator. The nonlinear spring (placed in between the two subsystems) is assumed to exhibit cubic, hardening type nonlinearity. First, the governing equations of the two degree of freedom system with a nonlinear elastic path are obtained. Second, the nonlinear differential equations are solved with a semianalytical (multiterm harmonic balance) method, and nonlinear frequency responses of the system are calculated for different path coupling cases. As such, the nonlinear path stiffness is gradually increased so that the stiffness ratio of nonlinear element to the linear element is 0.01, 0.05, 0.1, 0.5 and 1.0 while the absolute value of linear spring stiffness ismore »

We investigate aspects of the spherical squirmer model employing both largescale numerical simulations and asymptotic methods when the squirmer is placed in weakly elastic fluids. The fluids are modelled by differential equations, including the upperconvected Maxwell (UCM)/OldroydB, finiteextensibility nonlinear elastic model with Peterlin approximation (FENEP) and Giesekus models. The squirmer model we examine is characterized by two dimensionless parameters related to the fluid velocity at the surface of the microswimmer: the slip parameter $\xi $ and the swirl parameter $\zeta $ . We show that, for swimming in UCM/OldroydB fluids, the elastic stress becomes singular at a critical Weissenberg number, Wi , that depends only on $\xi$ . This singularity for the UCM/OldroydB models is independent of the domain exterior to the swimmer, or any other forces considered in the momentum balance for the fluid – we believe that this is the first time such a singularity has been explicitly demonstrated. Moreover, we show that the behaviour of the solution at the poles is purely extensional in character and is the primary reason for the singularity in the OldroydB model. When the Giesekus or the FENEP models are utilized, the singularity is removed. We also investigate the mechanism behind themore »