Microstructure Lectures

by

Ronald D. Kriz, Associate Professor
Engineering Science and Mechanics
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061



4.1 Introduction to Mechanical Behavior of Anisotropic Materials

Graphical Proof: Stiffness tensor inequalities are the necessary and sufficient
condition that defines the four possible geometries for orthorhomic symmetry.

Exact solutions of equation (43) are possible but only in the principle orthogonal planes. Solutions to equation (43) were derived by B.A. Auld, "Acoustic Fields and Waves in Solids," Vol. 1, Appendix 3, Wiley-Interscience 1973, for materials with orthorhombic symmetry.

Solutions are written interms of wave velocities with subscripts "l" for the fastest longitudinal wave, "t1" for the transverse wave that is always pure shear out-of-plane, and "t2" for the other transverse wave that is mixed mode. Only the intersection of the wave surfaces with the three principle orthogonal planes need to be considered.

--- Click on image below to view equations at a higher resolution ---

A template shown below demonstrates how these equations can be used to draw all possible geometric combinations of wave surface intersections with the principle orthogonal planes. Hence the inequalities only involve the diagonal terms of the stiffness matrix and are the necessary and sufficient conditions that define the four possible geometries. The off- diagonal terms do not initiate bifrications of the eigenvalues (graphical intersections) but they do influence the shape of the three-dimensional wave surfaces.

--- Click on image below to view equations at a higher resolution ---

Only one example is shown below that demonstrates the geometry and inequality that uniquely creates the hypothetical "Type-4" orthorhombic symmetry.

--- Click on image below to view equations at a higher resolution ---


Send comments to: rkriz@vt.edu
Ronald D. Kriz, Short Bio
Engineering Science and Mechanics
College of Engineering
Virginia Tech
Blacksburg, Virginia 24061

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Created on ESM Web-server November 1997
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Moved to Off-Campus Web-server November 2014
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