  In the region between close-to-touching hard inclusions, the stress may be
arbitrarily large as the inclusions get closer. The stress is represented by
the gradient of a solution to the Lam\'e system of linear elasticity. We
consider the problem of characterizing the gradient blow-up of the solution in
the narrow region between two inclusions and estimating its magnitude. We
introduce singular functions which are constructed in terms of nuclei of strain
and hence are solutions of the Lam\'{e} system, and then show that the singular
behavior of the gradient in the narrow region can be precisely captured by
singular functions. As a consequence of the characterization, we are able to
regain the existing upper bound on the blow-up rate of the gradient, namely,
$\epsilon^{-1/2}$ where $\epsilon$ is the distance between two inclusions. We
then show that it is in fact an optimal bound by showing that there are cases
where $\epsilon^{-1/2}$ is also a lower bound. This work is the first to
completely reveal the singular nature of the gradient blow-up in the context of
the Lam\'{e} system with hard inclusions. The singular functions introduced in
this paper play essential roles to overcome the difficulties in applying the
methods of previous works. Main tools of this paper are the layer potential
techniques and the variational principle. The variational principle can be
applied because the singular functions of this paper are solutions of the
Lam\'{e} system.
