Materials Science and Engineering A 518 (2009) 56–60

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Materials Science and Engineering A

journal homepage: www.elsevier.com/locate/msea

Determination of strain-hardening exponent using double compression test

R. Ebrahimi ∗ , N. Pardis
Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran

a r t i c l e i n f o a b s t r a c t

Article history: In this investigation a new method for determination of strain-hardening exponent (n) is introduced. The
Received 9 February 2009 presented method is named “Double Compression Test” in which two specimens with the same compo-
Received in revised form 5 April 2009 sition and geometrical dimensions but different processing background are compressed simultaneously.
Accepted 24 April 2009
One of the two specimens is in annealed condition while the other has experienced a predetermined
amount of pre-strain. This difference leads to different final length of the specimens which can be used
Keywords:
in the theoretical relation presented to calculate the strain-hardening exponent. The major advantage of
Double compression test
this method is its independency to the stress–strain data which is essential in the conventional method
Strain-hardening exponent
Pre-strain
for determination of strain-hardening exponent. The test was performed experimentally and the results
Finite element analysis were compared with those obtained by the conventional method. Finally, the test was simulated using
the commercial finite element code, ABAQUS/Explicit, to investigate it in more details. Experimental and
simulation results showed that this test is capable of determining strain-hardening exponent with good
accuracy.
© 2009 Elsevier B.V. All rights reserved.

1. Introduction applied loads, indenter displacement, flow stress and strain-

hardening exponent of steels. The model they presented yields
The value of the strain-hardening exponent (n) is of major impor- to the steel mechanical parameters,  y and n from the inden-
tance in forming operations since it controls the amount of uniform tation displacement–load curve. Kim et al. [6] evaluated plastic
plastic strain the material can undergo during a tensile test before flow properties by characterizing indentation size effect using
strain localization, or necking, sets in leading to failure. It is easily a sharp indenter and found a linear relationship between the
shown that the maximum amount of uniform plastic deformation strain-hardening exponent and the log of the indentation size
in tensile straining is given by the strain-hardening exponent (n) effect characteristic length for Ni and structural steel samples with
which is known as the Considère criterion. As a result, a high coef- different plastic pre-strain values.
ficient facilitates complex-forming operations without premature Antoine et al. [7] showed that there is a linear relationship
failure [1]. between the value of the strain-hardening exponent and the
Therefore, strain-hardening exponent is an important parame- yield strength, and presented a model giving the value of the
ter reflecting a material’s hardening property and its determination strain-hardening exponent for Ti-IF steel. Nagarjuna et al. [8] inves-
is of great importance. A standard method to perform this task is tigated the relationship between strain-hardening exponent (n)
based on using stress–strain data obtained from uniaxial tensile and grain size of Cu–26Ni–17Zn alloy by analysis of constants in
test. Stress–strain curves are usually represented by the Hollomon Hollomon equation as a function of grain size. They found that
equation. Therefore, by plotting stress–strain data on logarithmic strain-hardening exponent (n) is independent of grain size in the
coordinates, it can be shown that the slope of the line in the fully range of 15–120 ␮m. Narayanasamya et al. [9] performed a study
plastic region defines the strain-hardening exponent (n) [2,3]. on the instantaneous strain-hardening behavior of an aluminum
Shinohara [4] investigated relationship between strain- powder metallurgy composite with various percent of iron con-
hardening exponent and load dependence of Vickers hardness in tents and for the various stress state conditions with two different
copper and showed that the slope of the load dependence of the aspect ratios. They calculated the instantaneous strain-hardening
hardness was a good measure for correlating with strain-hardening exponent (ni ) and the strength coefficient (ki ) using mathematical
exponent (n). By using the instrumented spherical-indentation expressions. Zhang et al. [10] used two equations correlating the
technique, Nayebi et al. [5] presented a relationship between strain-hardening exponent and the strength coefficient with the
yield stress–strain behavior, and also the fracture strength with the
fracture ductility and presented a simple theoretical method of cal-
∗ Corresponding author. Tel.: +98 711 2307293; fax: +98 711 2307293. culating the strain-hardening exponent and the strength coefficient
E-mail address: ebrahimy@shirazu.ac.ir (R. Ebrahimi). of metallic materials.

0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2009.04.050
 R. Ebrahimi, N. Pardis / Materials Science and Engineering A 518 (2009) 56–60 57

Table 1
Nomenclature The initial dimensions of the aluminum specimens.

Specimen H0 (mm) D0 (mm) Aspect ratio

k strength coefficient
I 45 30 1.5
n strain-hardening exponent
II 15 10 1.5
H0 initial height of cylinder
Hp height of pre-strained cylinder after deformation
Ha height of annealed cylinder after deformation where, subscripts “a” and “p” apply to annealed and pre-strained
H total reduction in height conditions respectively. Thus:
D0 initial diameter of cylinder  n
Dp diameter of pre-strained cylinder after deformation ε0 + ε̄p Aa
= (2)
Da diameter of annealed cylinder after deformation ε̄a Ap
A0 initial cross section area of cylinder
Ap cross section area of pre-strained cylinder after where A0 , Ap and Aa are related to one another according to the
deformation following equations:
Aa cross section area of annealed cylinder after defor-
Aa = A0 · exp(ε̄a ) (3)
mation
ε0 amount of pre-strain Ap = A0 · exp(ε̄p ) (4)

The strain-hardening exponent (n) is calculated using equation:

ε̄a − ε̄p
n=   (5)
In the present study, a new approach is introduced which is capa- ε0 +ε̄p
ln ε̄a
ble of determining strain-hardening exponent without any need to
know the stress–strain data. This method just deals with geomet- By this method, strain-hardening exponent can be simply cal-
rical dimensions of the work piece before and after the test. culated by just using geometrical measurements, without needing
to know the load values.
2. Theory
3. Experimental procedure
Double compression test is based on simultaneous compression
of two specimens having the same composition and geometrical Two sets of cylindrical specimens with aspect ratio 1.5 and
dimensions, one in the annealed condition, while the other under- dimensions mentioned in Table 1 were machined from an alu-
gone a predetermined amount of pre-strain. The scheme of the test minum alloy rod with unknown mechanical properties. These
is illustrated in Fig. 1. specimens were then annealed at 430 ◦ C for 2 h. The specimens
In the absence of internal effects, the same axial force (F) is in set (I) were compressed to strain values of 0.2, 0.4, 0.6 and 0.8
transmitted through both cylindrical work pieces. Considering a as illustrated in Fig. 2. Compression tests were carried out using a
frictionless condition and using the Hollomon equation, this can be screw press with the compression rate of 0.1 mm/s at room tem-
expressed by the following equation: perature.
Then, new specimens were machined out of these compressed
k(ε0 + ε̄p )n · Ap = k(ε̄a )n · Aa (1) cylinders. The final dimensions of these specimens were exactly the
same as those in set (II). A pair of specimens including an annealed
and a strained specimen with pre-strain value ε0 = 0.2, were com-
pressed simultaneously to the amount of H = 10 mm. The test
set-up is illustrated in Fig. 3. The same procedure was repeated for
specimens with pre-strain values ε0 = 0.4, ε0 = 0.6 and ε0 = 0.8. The
specimens’ new heights were measured and are listed in Table 2.
Using these dimensions and Eq. (5), it is possible to obtain the
value of strain-hardening exponent without considering any load
parameter.

Fig. 2. Aluminum specimens compressed to strain values of 0.2, 0.4, 0.6 and 0.8 from
Fig. 1. A scheme of double compression test. left to right.
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