Abstract: Under normal condition,and hydrostatic pressure and uniaxial stress conditions,the different modes of longitudinal and shear wave velocities of YAG:single crystal hadbeen measured by ultrasonic pulse echo overlape technique. Based on those data,the secondorder and third-order elastic constants were derived,which are:C₁₁=-334. 78 GPa,C₁₂=109. 95 GPa,C₄₄=115. 25 GPa and C₁₁₁=-3377. 66 GPa,C₁₁₂=-841. 16 GPa, C₁₂₃=-32. 86 GPa, C₁₄₄=-37. 66 GPa, C₁₆₆=-579. 38 GPa, C₄₅₄₆=-114. 60 GPa.From those SOEC and TOEC, the Debye temperature,Grruneisen constant γ,and equation of state of YAG:YAG:N_d^3＋ are obtained