This study examines the suitability of the functions phi _m(r)=-Ar^-n+B ln(1+pr^-m) for describing the mechanical properties of FCC and BCC metals. It is shown that the conditions for stability of the crystals can be expressed uniquely as a function of the parameter q(=p(2/a₀)^m). Here A, n, B, p and m are potential parameters and a₀ is the lattice parameter of the crystal. The potential parameters for the functions phi _m(r) have been evaluated for eight FCC and eleven BCC metals using experimental values of the elastic moduli C₁₁/C₁₂ for only those members of the family of functions phi _m(r) which can satisfy the Born stability criteria of the metals in their stress-free state. The functions phi _m(r) have been used to predict the pressure-volume behaviour of FCC (Ni and Th) and BCC (V, Mo and W- alpha ) metals at pressures large enough to produce considerable anharmonicity; the results are then compared with the results obtained using generalised Morse functions as well as experimental shock-wave data. The present potential model predicts the pressure-volume behaviour much more accurately than the generalised Morse functions particularly at high pressures for FCC metals. The function phi _m(r) has been used to locate the stress-free FCC phase of iron with a cell length a₀=3.6444 AA in its FCC phase and transition enthalpy (BCC to FCC) of 1.1 kJ mol^-1, which are in good agreement with the experimental data. Numerical calculations are carried out for the mechanical stability of nickel and thorium subjected to hydrostatic compressive and tensile stresses according to four different criteria of stability. Values of the bulk modulus, shear modulus, Green moduli, Milstein moduli and stretch moduli have been computed for nickel and thorium as a function of lattice deformation following hydrostatic compressive and tensile stresses.