Analytical elastic-plastic solutions for the torsion of a round bar with a heterogeneous cross section

Hello,

I am trying to derive/find an analytical expression
for the torsion of a round bar made of an heterogeneous material. But I just do
not seem to find any solution available in the literature other than for a
homogeneous material, which can be find in any theory of plasticity book (e.g.
Theory of Plasticity, 3rd Ed., J. Chakarabarty, pp. 132-136).

The heterogeneous round bar shown below consists of a
hardened outer layer and a core denoted material 1 and 2, respectively. The
elastic-plastic behavior of the two materials are assumed to be governed by the
linear hardening stress-strain curve shown in figure 2(b). Material 1 and 2
have initial yield strength in shear τy1 and τy2 respectively where τy1 > τy2. The plastic hardening modulii for the
two materials are H1 and H2, where H1 < H2. Both materials
have identical elastic properties given by the shear modulus G and Poisson’s ratio. Assume the bar has length equal to unity. Determine an explicit
relation between the end twist qx and the applied torque Tx.

Could anyone recommend an article or a textbook that
might give some hints on how to solve this problem. Any help in this regard
would be greatly appreciated.

Thanks and best regards,

Imme