TY - JOUR

T1 - On the dilatational wave motion in anisotropic fractal solids

AU - Joumaa, Hady

AU - Ostoja-Starzewski, Martin

N1 - Funding Information:
This research was made possible by the support from Sandia-DTRA (grant HDTRA1-08-10-BRCWMD ) and the NSF (grant CMMI-1030940 ).
Publisher Copyright:
© 2013 IMACS

PY - 2016/9/1

Y1 - 2016/9/1

N2 - This paper reports a study of wave motion in a generally anisotropic fractal medium (i.e. with different fractal dimensions in different directions), whose constitutive response is represented by an isotropic Hooke's law. First, the governing elastodynamic laws are formulated on the basis of dimensional regularization. It is discovered that the satisfaction of the angular momentum equation precludes the implementation of the classical elasticity theory which results in symmetry of the Cauchy stress tensor. Nevertheless, the classical elastic constitutive model can still be applied to explore dilatational wave propagation; in such a case, the angular momentum balance is “trivially” satisfied. The resulting problem, of eigenvalue type, is solved analytically. A computational finite element method solver is also developed to simulate the problem in its one-dimensional (1d) 1D form, it is validated by the reference solutions generated through the modal analysis. A 3D finite-difference-based solver is also developed and the obtained results match those of the 1D simulation.

AB - This paper reports a study of wave motion in a generally anisotropic fractal medium (i.e. with different fractal dimensions in different directions), whose constitutive response is represented by an isotropic Hooke's law. First, the governing elastodynamic laws are formulated on the basis of dimensional regularization. It is discovered that the satisfaction of the angular momentum equation precludes the implementation of the classical elasticity theory which results in symmetry of the Cauchy stress tensor. Nevertheless, the classical elastic constitutive model can still be applied to explore dilatational wave propagation; in such a case, the angular momentum balance is “trivially” satisfied. The resulting problem, of eigenvalue type, is solved analytically. A computational finite element method solver is also developed to simulate the problem in its one-dimensional (1d) 1D form, it is validated by the reference solutions generated through the modal analysis. A 3D finite-difference-based solver is also developed and the obtained results match those of the 1D simulation.

KW - Dilatational wave

KW - Elastodynamics

KW - Fractal solid

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U2 - 10.1016/j.matcom.2013.03.012

DO - 10.1016/j.matcom.2013.03.012

M3 - Article

AN - SCOPUS:84878696739

SN - 0378-4754

VL - 127

SP - 114

EP - 130

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

ER -