  Elastic materials are ubiquitous in nature and indispensable components in
man-made devices and equipments. When a device or equipment involves composite
or multiple elastic materials, elasticity interface problems come into play.
The solution of three dimensional (3D) elasticity interface problems is
significantly more difficult than that of elliptic counterparts due to the
coupled vector components and cross derivatives in the governing elasticity
equation. This work introduces the matched interface and boundary (MIB) method
for solving 3D elasticity interface problems. The proposed MIB method utilizes
fictitious values on irregular grid points near the material interface to
replace function values in the discretization so that the elasticity equation
can be discretized using the standard finite difference schemes as if there
were no material interface. The interface jump conditions are rigorously
enforced on the intersecting points between the interface and the mesh lines.
Such an enforcement determines the fictitious values. A number of new technique
are developed to construct efficient MIB schemes for dealing with cross
derivative in coupled governing equations. The proposed method is extensively
validated over both weak and strong discontinuity of the solution, both
piecewise constant and position-dependent material parameters, both smooth and
nonsmooth interface geometries, and both small and large contrasts in the
Poisson's ratio and shear modulus across the interface. Numerical experiments
indicate that the present MIB method is of second order convergence in both
$L_\infty$ and $L_2$ error norms.
