0
Research Papers

# Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

[+] Author and Article Information
Lixiang Yang

The Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210yang.1130@osu.edu

Yung-Yu Chen

The Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210chen.1352@osu.edu

S.-T. John Yu1

The Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210yu.274@osu.edu

1

Corresponding author.

J. Appl. Mech 77(6), 061003 (Aug 16, 2010) (15 pages) doi:10.1115/1.4001545 History: Received May 01, 2009; Revised January 19, 2010; Posted April 02, 2010; Published August 16, 2010; Online August 16, 2010

## Abstract

This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three $9×9$ coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

<>

## Figures

Figure 1

Cartesian coordinates for the elastic relations of quartz, a solid of trigonal 32 symmetry. The x3 axis is in the orthotropic direction.

Figure 2

Wave polarization for waves propagating along three Cartesian axes

Figure 3

Rotation of two-dimensional coordinates in the x2-x3 plane

Figure 4

Two-dimensional slowness curves of quartz for waves propagating in the x2-x3 plane

Figure 5

Wave polarization of quartz in the x2-x3 plane. The compression directions on the x2-x3 plane are at ϕ=−72 deg, −17.8 deg, 40.8 deg, and 90 deg with respect to the x2 axis. In addition, we add ϕ=0 deg, a noncompression direction, as a reference.

Figure 6

The slowness curves of quartz in the x1-x3 plane

Figure 7

Polarization vectors for waves propagating on the x1-x3 plane. Polarization vectors show that pure longitudinal wave along x1 and x3 directions. The compression directions are at φ=0 deg and 90 deg. In addition, we add φ=30 deg as an arbitrary direction for reference.

Figure 8

Two-dimensional slowness curves of quartz in the x1-x2 plane

Figure 9

Polarization vectors for waves propagating on x1-x2 plane in the directions of θ=0 deg, 60 deg, 120 deg, 180 deg, 240 deg, and 300 deg. All above directions are compression direction. As a reference, θ=30 deg, a noncompression direction is also considered.

Figure 10

Rotation of three-dimensional Cartesian coordinates

Figure 11

Three-dimensional outer slowness surfaces of quartz. (a) The inner slowness surface plotted based on the fastest wave speed. (b) The combined outer surface for two slower wave speeds. (c) The combined slowness surfaces cut by using an x1-x2 plane. (d) The combined slowness surfaces cut by using an x2-x3 plane.

Figure 12

Wave polarization vectors of waves propagating in the nine compression directions inside a quartz crystal, a solid of trigonal 32 symmetry

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections