  In this work, we develop a general mathematical framework on regularized
approximate cloaking of elastic waves governed by the Lam\'e system via the
approach of transformation elastodynamics. Our study is rather comprehensive.
We first provide a rigorous justification of the transformation elastodynamics.
Based on the blow-up-a-point construction, elastic material tensors for a
perfect cloak are derived and shown to possess singularities. In order to avoid
the singular structure, we propose to regularize the blow-up-a-point
construction to be the blow-up-a-small-region construction. However, it is
shown that without incorporating a suitable lossy layer, the regularized
construction would fail due to resonant inclusions. In order to defeat the
failure of the lossless construction, a properly designed lossy layer is
introduced into the regularized cloaking construction . We derive sharp
asymptotic estimates in assessing the cloaking performance. The proposed
cloaking scheme is capable of nearly cloaking an arbitrary content with a high
accuracy.
