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 τ_y1 and τ_y2 respectively where τ_y1 > τ_y2. 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 q_x and the applied torque T_x.

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