Summary: We study conditions for the \(L^p\)-dissipativity of the classical linear elasticity operator. In the two-dimensional case we show that \(L^p\)-dissipativity is equivalent to the inequality \[ \Biggl({1\over 2}-{1\over p}\Biggr)^2\leq {2(\nu- 1)(2\nu- 1)\over (3- 4\nu)^2}. \] Previously [the authors, Ric. Mat. 55, No. 2, 233--265 (2006; Zbl 1189.47043)] this result has been obtained as a consequence of general criteria for elliptic systems, but here we give a direct and simpler proof. We show that this inequality is necesasry for the \(L^p\)-dissipativity of the three-dimensional elasticity operator with variable Poisson ratio. We give also a more strict sufficient condition for the \(L^p\)-dissipativity of this operator. Finally, we find a criterion for the \(n\)-dimensional Lamé operator to be \(L^p\)-negative with respect to the weight \(|x|^{-\alpha}\) in the class of rotationally invariant vector functions.