Multiscale modeling of elastic wave propagation in heterogeneous materials

• Author / Creator

  Wang, Chen

• Heterogeneous materials have been used extensively due to their desirable properties achieved by combining various constituents and tailored local structures. More recently, the metamaterials have attracted extensive attention because of their exotic dynamic properties, which are caused by either or both of the periodic and resonant local structures. Since the separation of scales exists between the fine-scale features and the macroscopic structures, the multiscale modeling and simulation of the elastodynamic behavior of the heterogeneous materials are essential for their design, optimization and application. This thesis aims to develop computational methods for the multiscale modeling of elastic wave propagation in heterogeneous materials.

  Firstly, an analytical-numerical method is developed for the multiple scattering problem of elastic media with interacting inhomogeneities under time-harmonic antiplane incident waves. The main focus is on the detailed evaluation of the effectiveness and accuracy of the method in the determination of the local dynamic behavior of such heterogeneous media with significant numbers of inhomogeneities. The method is based on the eigenfunction expansion and the use of a pseudo-incident wave technique. Then, by introducing the Helmholtz decomposition, the method is extended to in-plane problems. The accuracy and effectiveness of the method for dealing with multiple interaction problems are discussed in detail.

  Then, for the periodic materials, an analytical-numerical method is developed for determining the eigenstate of the unit cell under designated frequency and propagation direction. Based on the eigenfunction expansion and Floquet-Bloch theory, the nonlinear eigenvalue problem is established. The Newton's method is employed for computing the expansion coefficients as the eigenvector by which the eigenstate of the unit cell can be determined. The method is validated by the comparison with the finite element method.

  Based on the explicitly solved multiple scattering wave fields and the eigenstates of periodic unit cell, two kinds of computational homogenization methods are developed. The first kind is based on the domain averaging, and two methods are developed. The first method is based on the volume averages of the field variables with considering the effective wave form, which is iteratively adjusted using the self-consistent scheme. The second method is developed for periodic materials based on the kinetic energy equivalence. The homogenization results are verified by comparing the direct numerical simulations of the original heterogeneous material and the homogeneous substitution with the obtained effective properties.

  Another computational homogenization method is developed based on the boundary matching technique. The effective material properties are obtained by being adjusted so that the boundary response of the representative volume element has the minimum mismatch with that of a congruent piece of homogeneous material. According to different frequency ranges and materials, different RVE models are established. The validity of the homogenization is also verified by the comparison of direct numerical simulations. The homogenization results obtained by using boundary matching method and the domain averaging method are in a good consistency.

  At last, the multiscale modeling method is summarized by combining the developed methods. The method for recovering the local response from the homogenized model is developed. The effectiveness and accuracy of the method are shown by general examples of elastic wave propagation in heterogeneous materials with fine-scale local structures.

• Subjects / Keywords

  • Elastic wave
  • Homogenization
  • Multiscale modeling

• Graduation date

  Fall 2019

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-fss7-e841

• License

  Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.