Third-Order Elastic Constants ofα-Phase Cu-Al Alloys

Abstract

Complete sets of the six third-order elastic (TOE) constants have been determined for single-crystal $\ensuremath{\alpha}$-phase Cu-Al alloys with compositions of 3.1-, 7.4-, and 10.8-at.% Al. The TOE constants were determined experimentally using a modulated ultrasonic-pulse-superposition technique to measure changes in natural sound velocities with applied hydrostatic pressure and uniaxial tensile stress. The relative magnitudes among the alloy TOE constants are basically the same as those of pure Cu. The magnitudes of the TOE constants ${C}_{111}$, ${C}_{112}$, and ${C}_{166}$ decrease monotonically with increasing Al concentration. The greatest fractional change among these is approximately 0.7% per at.% Al for ${C}_{166}$. ${C}_{123}$, ${C}_{144}$, and ${C}_{456}$ are smaller than the above three constants, and it is not possible to detect systematic changes upon alloying. The hydrostatic-pressure derivatives of the second-order elastic constants have also been obtained. The pressure derivatives of ${C}_{44}$, ${C}^{\ensuremath{'}}=(\frac{1}{2})({C}_{11}\ensuremath{-}{C}_{12})$, and $B=(\frac{1}{3})({C}_{11}+2{C}_{12})$ all decrease with increasing Al concentration. The largest fractional change is in the pressure derivative of ${C}^{\ensuremath{'}}$. The change in the pressure derivative of $B$ appears negligible. The TOE constants have been used to calculate the high-temperature thermodynamic Gr\"uneisen parameter in the anisotropic continuum model. The composition dependence of the Gr\"uneisen parameter is then used to calculate the composition dependence of the thermal expansion. The change in the thermal expansion upon alloying Cu with Al is found to be - 0.19 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{} ${\mathrm{C}}^{\ensuremath{-}1}$ per at.% Al.