Theoretical and experimental investigations are being carried out on Cu based alloys due to their technologically important shape memory properties and pseudo-elasticity, which are intimately associated with the martensitic transformation. The transition between the two phases, martensite to austenite, is of continued interest in academics and in industry. The shape memory effect, superelastic properties and biocompatibility are being applied in a variety of fields. Cu based SMA system has large vibrational entropy, high damping capacity and good economic viability. All these make it a potential candidate in the field of sensors and actuators. The concurrent knowledge of the second order elastic constants (SOEC) and third order elastic constants (TOEC) enables a better understanding of the nonlinear elasticity exhibited by these alloys. We have used a model based on deformation theory and Keating’s potential scheme to obtain the expressions for TOEC of the above alloys. In this paper we have calculated the complete sets of six non-vanishing TOEC of Cu-Al-Ni, Cu-Al-Zn, Cu-Al-Be and Cu-Al-Pd and are presented along with the available experimental data. It is remarkable that all the third order elastic constants are negative, indicating an increase in the vibrational frequencies under stress, giving rise to an increase in the strain-free energy. The absolute values of the TOEC are large. This means that the bcc phase observed is considerably anharmonic. The TOEC C144 representing the shear mode has a smaller value than C111. Hence, the effect of pressure is much greater on longitudinal wave velocity than on the shear wave velocity in the above Cu based SMA. The mode Grüneisen parameters of the acoustic waves are determined based on the quasi-harmonic approximation method. The low temperature limit of the lattice thermal expansion and the Anderson– Grüneisen parameter of these alloys are also obtained.