In this short note, we consider some issues regarding the instability of some elastodynamical problems when the elasticity tensor is not positive definite. By using the so-called logarithmic convexity argument, we prove the instability of solutions when the time derivative of the elasticity tensor is semi-definite negative or it satisfies another restriction on the coefficients. The uniqueness of the solution is also concluded. Finally, a simple one-dimensional example is provided to demonstrate the numerical behaviour of the instability. •An incremental problem in elastodynamics is mathematically studied.•Instability is analyzed by using the logarithmic convexity argument.•Uniqueness of the solution is also considered.•A numerical example is performed to demonstrate the behaviour.