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Ultra-sonic Measurement and Computation of Elastic Constants

Abstract

We have developed a laboratory exercise using an ultrasonic test system to measure elastic constants in copper single crystals, as well as moduli in polycrystalline samples. This laboratory is coordinated with the Introduction to Physics of Materials class taught to junior-level Materials Science and Engineering majors. Students in the class have usually taken a course in Mechanical Behavior of Solids, and are familiar with macroscopic (and destructive) testing of elastic moduli. This exercise provides an excellent basis for a discussion of the propagation of phonons in solids, the relationship between microscopic and macroscopic properties, destructive vs. non-destructive test methods, and an application of matrix algebra. We have recently added a computational component to the lab: use of density functional theory (using the program Plane Wave Self- Consistent Field, PWSCF) to calculate the elastic constants and compare to the measured values. The free software is part of the Quantum-ESPRESSO package.1 The use of DFT also allows the demonstration of the nature of the approximations that must be made to solve the many-body Schrödinger equation for electrons in a condensed phase. This follows discussions in the first half of the course about the hydrogen atom and how extending such calculations require approximations due to the many-body problem.

Introduction and Motivation

Materials Scientists and Engineers are increasingly using computational tools to predict material properties, in addition to using analytical tools to measure them. We feel it is essential to incorporate some of these tools, as well as knowledge of the theories on which they are based, into our undergraduate curriculum. One opportunity to do so is in our junior-level Introduction to Physics of Materials course. This is a two-quarter sequence, required by MSE majors, that introduces quantum mechanics and applications to materials and engineering, band structures, cohesive energy, thermal behavior, electrical conduction, etc. The first quarter emphasizes an introduction to quantum mechanics; the second quarter emphasizes applications, and is taught with a weekly two-hour laboratory. One of the lab exercises we have used for several years is the measurement of elastic constants using ultrasonic testing. Most of the students in the class have already taken a course in Mechanical Behavior of Solids, in which they use destructive testing (compression or tension), to measure elastic properties. They have familiarity with moduli, stiffness and compliance from this course, but they have not previously used non- destructive testing (ultrasound). This lab offers an excellent opportunity to expose students to a second method that yields some of the same information they obtained with conventional mechanical testing. We have recently written a follow-up exercise which uses computational free-ware to calculate these same constants. These computations rely heavily on approximations, and students see the limitations of the program when calculating their results that often differ from “book values” by 10-20 percent. This is underscored by the comparison with their measured values, which are much closer to the literature.

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Stair, K., & Liu, J. Z., & Asta, M. (2006, June), Ultra Sonic Measurement And Computation Of Elastic Constants Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--1286