In this study, a central-force model was employed to compute the second- and third-order elastic constants for Nickel (Ni) and Titanium (Ti). The interatomic potential was described using the Morse potential, with parameters determined based on the lattice parameter, bulk modulus, and cohesive energy. The cutoff range for the interatomic potential was set at 176 and 168 neighbors for face-centered and body-centered cubic metals, respectively. Utilizing the calculated elastic constants, the pressure derivatives of the second-order elastic constants were further determined. The results exhibited a good agreement with experimental values. However, the comparison between experimental and theoretical values for third-order elastic constants revealed less satisfactory alignment. The theoretical predictions indicated a positive value for C_123 in all face-centered cubic metals and a negative value for body-centered cubic metals. It is noteworthy that the theoretical predictions for C_123 were consistent with a positive value for all fcc metals and a negative value for bcc metals. Contrasting this with the available experimental data on the third-order elastic constants of Ni and Ti, it was observed that C_123 was indeed positive, supporting the theoretical predictions.