We propose a finite element method (FEM) for solving planar elasticity problems involving of heterogeneous materials using uniform grid. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart (CR) $P_1$-nonconforming basis functions so that they satisfy the jump conditions along the interface. It is well-known that the nonconforming piecewise linear FEM does not satisfy the discrete Korn’s inequality. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method. Numerical experiments for various problems show that second order convergence in $L^2$ and first order in $H^1$-norms. Moreover, the convergence order is very robust for nearly incompressible case.