I don't know exactly, but one may seek justification for this type of observation by investigating the microstructure at the vicinity of central Torsion line which can experience a substantial amount of unbalanced dislocation movement during torsion test.

Many thanks Prof. Arabi!

There are already experimental evidence of grain refinement and dislocation movement in the central zone but these contradict the common believe about the zero deformation in the zone based on the existing mechanical theory. Seems a new theory needs to explain the observations.

Perhaps it may add to what Professor Arabi says to remember that the grain structure is very unlikely to be axisymmetric and of course the structure on an atomic scale cannot be centrally symmetric.

Many Thanks Peter!

What you said explains why a simple idealized velocity field can not explain the complex situation. Perhaps this is a gap where people can contribute by developing new theories to quantify the processing parameters.

First of all, usually there is not very much grain refininement at the rotation axis of a HPT disk, provided the geometry of the torsion test is well maintained; a coarser structure at the disk center is often well visible even after very large deformations. However, there are several factors explaining why plastic deformation or structural changes occur at the centre of a HPT sample:

- the ideal geometry is exactly that: ideal, i.e., is never perfectly attained and the disk axis of rotation will suffer some wandering during the straining; even a very samll deviation from the original position leads to large strains when the number of HPT revolutions applied to the disk is big

- there is a non-negligible plastic deformation by compression superposed to the shear deformation, more intense at the beginning of the test

- there is a shear strain gradient in the deformed HPT sample; it requieres the storage of geometrically necessary dislocations, their contribution being more evident near the sample centre, where the shear strain is small.

 I believe that maintaining the GND is the most important factor.

A recent publication (Rathmayr et al, Mater. Sci Eng , A 560 ( 2013) 224-231) presents the  grain structures at diiferent tangential and radial directions. Not much variation in radial direction.

Just to add something to this interesting discussion, the strain tensor is really null at the center line, but the infinitesimal rotation tensor in maximal at this point, if you consider the subgrain rotation mechanism suggested to take place in dynamic recrystallization, it is very likely that the infinitasimal rotation tensor may trigger grain refinement close to the center line, but I agree with Prof. Gil Sevillano, the usual situation woul be to have coarse grains closer to the center line.

I forgot to mention another contribution to deformation at the very centre: the material is not uniform, it is heterogeneous (clearly in the polycrystalline case); then, the stress-strain fields are heterogeneous too and the state of the disk axis is non-locally conditionned.

Many thanks for all the nice contributions specially those by Prof Gil Sevillano and Prof. Geraldo Schön.

Regarding to the scale of the work hardening near the center, please see for examples the following research:

Article Strain gradient plasticity modelling of high-pressure torsion

Article Microstructure evolution in commercial purity aluminum durin...

One should not forget that the zero deformation is miss leading terminology used  by engineers. The heterogeneous deformation tensor has two parts, the symmetric part which is called pure strain (self-conjugate) tensor and the anti-symmetric part that is pure rotational type (similar to screw dislocations).  Therefore one should expect very large plastic rotational deformation there, which may relieved partially  by the formation of screw dislocations and  disinclinations.  The formation of small grains at the central region due to the fragmentation of large grains  due to large torsional stresses (zero dilation or divergence) is an another mechanism for the deformation energy relief.

Unfortunately,  the treatment of the rotational part of the strain tensor  has been  almost forgotten in the theory of elasticity with few exceptions, which may be very imported in imperfect solids.  Since then one can't presume  that the anti-symmetric part of the  stress tensor is negligible.  Actually the stress tensor can only be symmetric if the body is in complete mechanical equilibrium condition not only for the vector sum of forces but also for the moments. One can also prove that the strain energy associated with the anti-symmetric part of the strain tensor involve the double inner product with the anti-symmetric part of the stress tensor. If  we use  the  engineering language; the part of the body, which  exposed to pure torsion or micro rotation  still carries excess strain energy with respect to the unstrained body. That is the energy needed  as the driving force at the central  region of the body  for the formation of small grains out of the  big ones in order to compensate  the extra interfacial free energy necessary for the grain refinement. 

Yes, according to the torsional theory, the point of zero torsional strain is at the axis of rotation. However, in physical point of view, a point has a volume. Therefore, the center volumetric material cannot have zero volume and zero deformation. In addition, in strain calculation at a point we are using relative displacement of this point to other nearest points. So, in theoretical point of view, we need a volumetric space to calculate strain at a point.

Dear Rahim, you are perfectly right. But center here doesn' t imply the geometric center of a  bulk  material but  rather a  region where affects of the tortional distortion can be observed and calculable,  and the asscoiated distorion energy  may computed like a   stored energy induced by ordinary strain. 

Thanks for everybody's contribution, specially those recently made by Tarik. All these explanations together with the fact that a perfect antisymmetric process is not practical makes quantitative description of the process very difficult.

This leds us to another question: why the same problem doesn't exist for the case of torsion of a shaft: an antisymmetric description seems to be a quite reasonable quantitative description!!

Your new comments in the light of this analogy is most welcomed.