  In a recent work (Dick et al, arXiv:2310.06187), we considered a linear
stochastic elasticity equation with random Lam\'e parameters which are
parameterized by a countably infinite number of terms in separate expansions.
We estimated the expected values over the infinite dimensional parametric space
of linear functionals ${\mathcal L}$ acting on the continuous solution $\vu$ of
the elasticity equation. This was achieved by truncating the expansions of the
random parameters, then using a high-order quasi-Monte Carlo (QMC) method to
approximate the high dimensional integral combined with the conforming Galerkin
finite element method (FEM) to approximate the displacement over the physical
domain $\Omega.$ In this work, as a further development of aforementioned
article, we focus on the case of a nearly incompressible linear stochastic
elasticity equation. To serve this purpose, in the presence of stochastic
inhomogeneous (variable Lam\'e parameters) nearly compressible material, we
develop a new locking-free symmetric nonconforming Galerkin FEM that handles
the inhomogeneity. In the case of nearly incompressible material, one known
important advantage of nonconforming approximations is that they yield optimal
order convergence rates that are uniform in the Poisson coefficient. Proving
the convergence of the nonconforming FEM leads to another challenge that is
summed up in showing the needed regularity properties of $\vu$. For the error
estimates from the high-order QMC method, which is needed to estimate the
expected value over the infinite dimensional parametric space of ${\mathcal
L}\vu,$ we %rely on (Dick et al. 2022). We are required here to show certain
regularity properties of $\vu$ with respect to the random coefficients. Some
numerical results are delivered at the end.
