Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Elastic and Anelastic Properties of a Copper-Precipited Alloy Steel



H M. Ledbetter, Sung Kim, H Ogi


We studied the elastic and anelastic properties of an alloy steel containing 1.1 mass % copper subjected to various isothermal agings. The steel's hardness increases with aging time; after reaching a maximum, it decreases. Peak hardness was 17% higher than in the nonaged material. Using acoustic-resonance spectroscopy, we determined the complete elastic-stiffness tensor Cij and the associated internal friction Q-1ij. In the shear stiffnesses Cij, the materials show an elastic anisotropy of about 3 %. With increasing annealing time, all the usual elastic stiffnesses (B, E, G, C1) increase and then decrease slightly after reaching optimum aging (maximum hardness). Within measurement error, the Poisson ratio v remains unchanged. All the elastic stiffnesses correlate linearly with hardness. Mass-density measurements suggest a three-stage aging process, proposed originally by Hornbogen and confirmed by the Young-modulus measurements of Lahiri and coworkers. Internal friction Q-1 decreases continuously with increasing annealing time. This C-Q-1 signature suggests that most of Q-1 arises from dislocations, which we verified by measuring strain-amplitude effects. As dislocations are pinned, stiffness increases and Q-1 decreases, following the Koehler-Granato-Lucke stretched-vibrating-string theory. The well-known equation of motion for this theory is Ay + By + Ky = bςoexp(iωt). From our measurement results, we deduced A, B, K, and other properties such as the relaxation time τ, the dislocation-segment length l, and the maximum frequency in the Q-1 frequency curve fm. The striking correlation among the B(t), K(t), τ (t), l(t), and hardness (t) curves leaves little doubt that the internal friction reflects the hardening mechanism: diffusion of copper atoms from b.c.c. lattice sites to form b.c.c. copper-rich clusters, which become noncoherent and transform to an f.c.c. phase after reaching a certain size.
Metallurgical and Materials Transactions B-Process Metallurgy and Materials Processing Science


clusters, coefficients, elastic-stiffness, ferritic steel, hardness, high-strength low-alloy steel, internal friction, Koehler-Granato-Lucke therory, precipitates


Ledbetter, H. , Kim, S. and Ogi, H. (2008), Elastic and Anelastic Properties of a Copper-Precipited Alloy Steel, Metallurgical and Materials Transactions B-Process Metallurgy and Materials Processing Science (Accessed May 30, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created October 16, 2008