# 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

AttachmentSize hetero_bar.png6.58 KB mat1&2.png3.1 KB

### try this paper

J.  Inst.  Maths.  Applics  (1979)  24,  353-378
Elastic-Plastic  Torsion  of  Heterogeneous  Cylindrical  Bars
D.   CIORANESCU,  J.   SAINT  JEAN  PAULIN
AND
H.   LANCHON
Following the study by Lanchon  (1974) on elastic-plastic torsion of cylindrical bars
with  simple  or  multiple  connected  cross  sections,  we  give  here  a  generalized
formulation  including, in particular, the case of bars reinforced by longitudinal  fibres.

and search for literature using this one as a reference

Good luck

Frank

------------------------------------------
Ruhr-University
Bochum
Germany 