Constitutive equations for the critical resolved shear stresses of Monocrystalline Al-Cu and Al-Mg Solid Solutions

Abstract

Constitutive equations that govern the critical resolved shear stresses (CRSS), τc, as a function of temperature, T, and concentration, c, for the dilute binary Al-Cu and Al-Mg alloys were derived in this study. Single crystals of these alloys, together with pure aluminum, were grown by the Bridgman method. The flow stresses of the crystals were measured in the [358] orientation at various temperatures from liquid nitrogen to boiling water to obtain the CRSS. The derivations were based on the expressions for the Friedel model and Labusch model that account for the contributions of solutes to the strength of the crystals. The CRSS of pure aluminum at various temperatures, τc(0, T), were also included in the expressions for these models to give the general form of the constitutive equations as follows: τc(c,T) = τc(0, T) + C1cn[1 - C2T2/3]. In this expression, n is the concentration dependence of τc, C1 and C2 are two experimentally-determined parameters. By plotting \( \frac{{\partial {\tau _c}\left( {c,T} \right)}}{{\partial {c^n}}} \) vs. T2/3 from the experimental data for Al-Cu and AlMg alloys one can readily obtain the slopes and intercepts from the straight-line fit. These slopes and intercepts are directly related to the constants C1 and C2, from which the governing constitutive equations for the Al-Cu and Al-Mg alloys can be obtained. The derived equations were used to plot the curves of τc against c or T for both alloys and it was found that the agreement between the experimental data and predicted curves is reasonably good. The two parameters, C1 and C2, were also theoretically estimated and compared with the experimentally-determined values. The theoretically calculated values of C2 are within the range of the measured values for both models whereas the great discrepancy between the theoretically-calculated and measured C1 will be discussed.