Nonlocal Effects In Torsional Deformation

Materials Science and Engineering A277 (2000) 148 – 153
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Nonlocal effects in torsional deformation
F. Szekely, I. Groma *, J. Lendvai
´
Department of General Physics, Eot6os Uni6ersity, Pazmany Setany 1 /A, POB 32, H -1518 Budapest, Hungary
¨¨
´´
´´
Received 22 February 1999; received in revised form 28 July 1999

Abstract
The stress distribution intorsionally deformed cylindrical polycrystalline copper samples was investigated. The strain in
torsional deformation changes with distance from the torsional axis and thus the local extent of work hardening and consequently
the local flow stress changes as well. The variation of the flow stress with radial position in the samples was measured by
microhardness tests. It was found that except for thenear-axis region the local flow stress can be correctly calculated by the Nadai
evaluation from the experimentally measured torque versus torsional angle function. Hardening is observed in the axis of the
torsionally deformed samples in spite of the fact that the shear strain is zero here. The hardening obtained in the axis depends
approximately linearly on the torsional angle, and it increaseswith grain size. This effect is interpreted by introducing nonlocal
effects in the continuum description. © 2000 Elsevier Science S.A. All rights reserved.
Keywords: Torsional deformation; Nonlocal effects; Constitutive equations

1. Introduction
Torsion tests are often used for the investigation of
plasticity at high strains, since in this deformation mode
much higher strains can beachieved than in tension or
compression tests [1 – 4]. In case of torsion, however,
the deformation is inhomogeneous inside the sample,
which makes the derivation of stress – strain curves from
the measured torque-angle relations difficult. A commonly accepted method for the determination of the
stress–strain relation for rod-shaped cylindrical samples from torsion tests is the Nadai evaluation [5]which
is based on two assumptions:
(1) The shear strain, k is proportional to the distance
from the axis of the sample, r : k = rr where r is the
angle of torsion per unit length. (This assumption
was investigated and proved experimentally by
Grewe and Kappler [6].)
(2) The shear stress (~ ) is determined only by the shear
strain: ~ = ~ (k ).

From these assumptions for the surface of thesample
the
~ (k = Rr ) = ~N(k ) =



1
dM
r
+ 3M
2yR 3
dr



(1)

relation can be derived where
M = 2y

&

R

~ (r )r 2 dr

(2)

0

is the applied torque.
At a given angle of torsion (r = const.) the stress–
strain relation ~N(k ) determined by the Nadai evaluation from the measure M (r ) curve can be converted to
~N(r ) = ~N(k = rr )

(3)

which describesthe variation of stress as a function of
r.
The aim of this work is to check the validity of the
stress distribution determined by this method in torsionally deformed samples.

2. Experimental procedure
* Corresponding author. Tel.: + 36-1-372-2802; fax: + 36-1-3722811.
E -mail address: groma@metal.elte.hu (I. Groma)

Cylindrical samples of electrolytic copper (purity
99.99%), with adiameter of 4 mm, and a gauge length

0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 5 4 0 - 7

F. Szekely et al. / Materials Science and Engineering A277 (2000) 148–153
´

of 40 mm were used for torsion and tension tests. The
samples for the stress distribution measurements were
annealed at 600°C for 1 h to get afully recrystallized
and untextured structure with an average grain size of
20 mm. Additional measurements were made on samples with grain sizes of 50 and 75 mm, which were
obtained by annealing at 800 and 900°C, respectively.
The tension tests were carried out in an INSTRON
machine at a constant crosshead velocity of 2 mm
min − 1.
For torsion tests, a free-end torsion machine was
used, in...