  Analytical solutions to two axisymmetric problems of a penny-shaped crack
when an annulus-shaped (model 1) or a disc-shaped (model 2) rigid inclusion of
arbitrary profile are embedded into the crack are derived. The problems are
governed by integral equations with the Weber--Sonin kernel on two segments. By
the Mellin convolution theorem the integral equations associated with the
models 1 and 2 reduce to vector Riemann-Hilbert problems with and 3x3 and 2x2
triangular matrix coefficients whose entries consist of meromorphic and of
infinite indices exponential functions. Canonical matrices of factorization are
derived and the partial indices are computed. Exact representation formulas for
the normal stress, the stress intensity factor, and the normal displacement are
obtained and the results of numerical tests are reported.
