Acoustic radiation force generated by two counterpropagating acoustic waves in a thin layer of soft material can induce large deformation, and hence can be applied to design acoustomechanical actuators. Owing to the sensitivity of wave propagation to material geometry, the change of layer thickness may enhance wave propagation and acoustic radiation force, causing a jumping larger deformation, i.e., snap-through instability. Built upon the basis of strong elliptic condition, we develop a generalized theoretical method to evaluate the acoustomechanical stability of soft material actuators. We demonstrate that acoustomechanical instability occurs when the true tangential stiffness matrix ceases to be positive definite. Our results show that prestresses can not only enhance significantly the acoustomechanical stability of the soft material layer but also amplify its actuation stretch in thickness direction.