This paper is devoted to the derivation of trace bounds for elastic moment tensors. Starting from the integral equation formulation of the elastic moment tensor, we establish that its trace can be obtained as a sum of minimal energies. We then recover the so-called Hashin-Shtrikman bounds, and show that these bounds can be tightened for inclusions which have some local thickness. As an application, we show that the volume of the inclusion can be estimated by the elastic moment tensor.