  Elasticity theory is an important component of continuum mechanics and has
had widely spread applications in science and engineering. Material interfaces
are ubiquity in nature and man-made devices, and often give rise to
discontinuous coefficients in the governing elasticity equations. In this work,
the matched interface and boundary (MIB) method is developed to address
elasticity interface problems. Linear elasticity theory for both isotropic
homogeneous and inhomogeneous media is employed. In our approach,
Lam$\acute{e}$'s parameters can have jumps across the interface and are allowed
to be position dependent in modeling isotropic inhomogeneous material. Both
strong discontinuity, i.e., discontinuous solution, and weak discontinuity,
namely, discontinuous derivatives of the solution, are considered in the
present study. In the proposed method, fictitious values are utilized so that
the standard central finite different schemes can be employed regardless of the
interface. Interface jump conditions are enforced on the interface, which in
turn, accurately determines fictitious values. We design new MIB schemes to
account for complex interface geometries. In particular, the cross derivatives
in the elasticity equations are difficult to handle for complex interface
geometries. We propose secondary fictitious values and construct geometry based
interpolation schemes to overcome this difficulty. Numerous analytical examples
are used to validate the accuracy, convergence and robustness of the present
MIB method for elasticity interface problems with both small and large
curvatures, strong and weak discontinuities, and constant and variable
coefficients. Numerical tests indicate second order accuracy in both $L_\infty$
and $L_2$ norms.
