We have carried out numerical ab initio calculations of the elastic constants for several cubic ordered structures modeling titanium carbonitride (TiC x N 1Àx ) alloys. The calculations were performed using the full-potential linear augmented plane-wave method (FPLAPW) to calculate the total energy as functions of volume and strain, after which the data were fit to the traditional Murnaghan equation of state and to a polynomial function of strain to determine the formation energy; the bulk modulus; and the elastic constants C 11 , C 12 , and C 44 . The predicted equilibrium lattice parameters are slightly higher than those found experimentally (on average by 0.2 pct). The computed formation energy indicates that the alloys are stable in the entire range of the carbon concentration x and the maximum stability is obtained for 0.5 # x # 0.75. The computed bulk modulus, the shear modulus G, and the Young's modulus E are within approximately 2, 1, and 2 pct of the experimentally measured characteristics, respectively. The maximum deviation is observed for TiC and TiN. The moduli G, E, and Poisson's ratio reach a maximum value at approximately the middle of the concentration range, which is due to the fact that the shear modulus C 44 shows a maximum value for a valence electron concentration (VEC) in the range of 8.25 to 8.5. The other shear modulus (C 11 -C 12 )/2 does not exhibit any maximum overall concentration range and instead has a flat dependence in the range mentioned previously. Such a concentration behavior of the elastic constants is related to specific changes in the band structure of TiC x N 1Àx alloys caused by the orthorhombic and monoclinic strains that determine the shear moduli (C 11 -C 12 )/2 and C 44 , respectively.

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