Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 2, pp. 377-398, Warsaw 2012

50th Anniversary of JTAM

TEMPERATURE INCREASE ASSOCIATED WITH PLASTIC
DEFORMATION UNDER DYNAMIC COMPRESSION:
APPLICATION TO ALUMINIUM ALLOY AL 6082

José-Luis Pérez-Castellanos

University Carlos III of Madrid, Department of Continuum Mechanics and Structural Analysis,

Madrid, Spain; e-mail: percaste@ing.uc3m.es

Alexis Rusinek

National Engineering School of Metz (ENIM), Laboratory of Mechanics, Biomechanics, Polymers

and Structures (LaBPS), Metz, France; e-mail: rusinek@enim.fr

The temperature increase associated with plastic deformation of a ma-
terial under loading may be measured using several techniques such as
infrared thermography (IRT). The present work investigates the tem-
perature increase at different high strain rates and initial test tempe-
ratures, using an aluminium alloy Al 6082. A Split Hopkinson Pressure
Bar (SHPB)was applied to induce high strain rates to thematerial and
an infrared camera was used to measure the temperature increase. Nu-
merical simulations of dynamic tests were performed to calculate the
temperature increase and to gain a better understanding of the process
by localmeasurements.Thus, a detailed finite-elementsmodel was deve-
loped to simulate the dynamic compression test. The fraction of plastic
work converted into heat was estimated using the Zehnder model. Nu-
merical results in terms of the strain rate and initial temperature effect
on the material temperature increase are reported and compared with
experiments.

Keywords: infraredthermography,plastic strain,adiabaticheating,Hop-
kinson bar

1. Introduction

Mechanical energy produced during plastic and elastic deformation of ametal
is partially converted into heat while the rest is stored as deformation energy
(Hodowany et al., 1999; Kappor andNemat-Nasser, 1998). This stored energy
remains in the material after loading as internal defects, phase changes and



378 J.L. Pérez-Castellanos, A. Rusinek

other permanentmicrostructural changes.Theheat generated, proportional to
a part of themechanical work, induces a temperature increase of thematerial.

The thermal-balance equation for a visco-plastic isotropicmaterial is given
by Eq. (1.1). It does not consider the effect related to phase-transformation
processes in the heat generation as discussed inRusinek andKlepaczko (2009)

λ∇2T − Ṫ =−
β

ρCP
σ : ε̇p+

α

ρCP

E

1−2v
T tr(ε̇e) (1.1)

where λ is the thermal diffusivity of thematerial, T is the absolute tempera-
ture, β is theQuinney-Taylor coefficient (Taylor andQuinney, 1934), (propor-
tion of plastic-deformation energy converted into heat), ρ is themass density,
CP is the specific heat, σ is the stress tensor, ε̇

p is the plastic strain-rate ten-
sor, ε̇e is the elastic-strain-rate tensor, α is the thermal-expansion coefficient,
ν is Poisson’s ratio, and E is Young’s modulus.

Close to 10s−1 for metals (Klepaczko et al., 2009; Oussouaddi and Kle-
paczko, 1991; Rusinek and Klepaczko, 2001; Rusinek et al., 2007) no heat is
exchanged with the surroundings, and adiabatic conditions may be assumed
λ∇2T =0. If thermoeslasticity is not considered, then the temperature incre-
ase ∆T , may be calculated as a function of the plastic strain εp

∆T(εp)=

ε
p

max
∫

0

β

ρCP
σ(εp) dεp (1.2)

where σ is the equivalent stress under uniaxial deformation and εp is the
corresponding equivalent plastic strain.

It is usual to consider β as constant in such calculations. Nevertheless, it
is accepted that β depends on the plastic strain. Different models (Aravas et
al., 1990; Rosakis et al., 2000; Zehnder, 1991) to define the Quinney-Taylor
coefficient β, have beendeveloped.Based on theZehndermodel, related to the
calculation of stored energy by unit of dislocation density, theQuinney-Taylor
coefficient β, may be defined as below

β(εp)≈ 1−
∂σ(εp)

∂εp
1

E

∣

∣

∣

∣

∣

ε̇
p

(1.3)

where E is material Young’s modulus.

If a potential hardening law is used (which allows an isotropic behaviour to
be defined while considering just hardening), an analytical expression for the



Temperature increase associated with plastic deformation... 379

Quinney-Taylor coefficient β(εp), Eq. (1.4), may be defined (Zehnder, 1991)
from Eq. (1.3)

β(εp)= 1−n
(εp

ε0

)n−1

(1.4)

where ε0 is the yield strain and n is the hardening exponent of the material.
The temperature increase in a specimen deformed during plastic deformation
may be calculated using Eq. (1.2) and Eq. (1.4).
There are at least two procedures for measuring the temperature incre-

ase ∆T associated with the plastic strain εp. The first one is based on the
use of a rapid-response thermocouple connected to the specimen tested. The
second one is based on the measurement of the thermal radiation of the spe-
cimen emitted during deformation. It corresponds to the temperature field
observed on the surface of the material.
The IRT technique has been widely used to correlate plastic strain to

temperature increase ∆T(εp) in metal and polymeric materials. This techni-
que is steadily more used due to the development of new high-speed infrared
cameras. During the last decade, several authors were using the IRT tech-
nique to estimate the temperature increase of materials during elastic and
plastic deformation. For example, Kappor and Nemat-Nasser (1998), Man-
son et al. (1994), Rabin and Rittel (2000) measured the temperature incre-
ase under dynamic compression tests using a SHPB device. Chrysochoos and
Louche (2000) applied thermographic analysis to the deformation process of
Polymethyl-methacrylate (PMMA). Trojanowski et al. (1998) used IRT to in-
vestigate theheat generation inanepoxy resinand inmetal specimens, coupled
to a SHPB.Titanium-alloy specimenswere tested byMacdougall andHarding
(1998), using a split Hopkinson torsion and tensile bars and by using an in-
frared radiometer to measure the temperature increase during the test. The
IRT technique to investigate impact processes of steels at high strain rateswas
used by Guduru et al. (2001), Rusinek et al. (2003), and Rodŕıguez-Mart́ınez
et al. (2010).
The objective of the present study is to investigate the temperature in-

crease ∆T associated with the plastic strain εp at high strain rates ε̇
p
using

an aluminium alloy Al 6082. The technique used to measure the temperature
increase is based on the IRT. Concerning the mechanical loading, a SHPB
is used to reach high strain rates under dynamic compression. Coupling the
SHPBand IRTtechnique, it allows the temperature increase ∆T to bedefined
and estimated at high strain rates during plastic deformation ∆T(εp, ε̇

p
).

For the test simulation, a detailed numerical model of the SHPB test was
developed using a FE code.With this 3Dmodel, extensive numerical analyses
weremade to calculate the temperature increase and other local quantities to



380 J.L. Pérez-Castellanos, A. Rusinek

compare with the experiments. This kind of technique using coupling experi-
ments and simulations, steadily more used, is called the “InverseMethod”. It
allows one to reach local quantities to complement measurements of macro-
scopic behaviour made during experiments, for a better understanding of the
material behaviour (Rusinek et al., 2008; Miguélez et al., 2009).

2. Test description

Guzmán et al. (2009), and Pérez-Castellanos et al. (2010) have developed an
original set-up and a SHPB testing procedure with a simultaneous measure-
ment of the temperature increase ∆T(εp, ε̇

p
), using an infrared camera (IRC)

and the dynamic mechanical behaviour of the material σ(εp, ε̇
p
,T). The next

part of the paper provides a detailed description of the technique used.

2.1. Test set-up

Fordynamic testing, aSHPBwasused.Thisdevice, developedoriginallyby
Kolsky (1949) consists of two coaxial barsbetweenwhich the specimen is fixed.
A cylindrical striker of the same material and with the same cross-sectional
area as the bars is launched, using a pneumatic gas gun, to impact against
the incident bar. As a result of the impact, a compressive square-shaped pulse
travels through the incident bar until it reaches the specimen; the magnitude
of the incident wave is σi = ρCV0/2 with a duration of tp = 2Lp/C. In the
previous formulas, ρ is the bar density of the material; C is the elastic wave
speed, which depends on Young’s modulus E; with C =

√

E/ρ; V0 is the
velocity of the striker just before impact, and Lp is the striker length. When
the incidentwave εi reaches the incident bar-specimen interface, it is split into
reflected εr, and transmittedwaves εt. Two strain gauges, on full bridge, fixed
on the incident bar and the transmitter bar, respectively, allow recording of
the elastic strain waves. The strains are related to the stress quantities using
Hooke’s law.

The temperature increase during the test is measured using an IRC with
a temperature resolution of 20mK. The frequency acquisition of the camera
maybe increased by reducing the frame size (subwindowmode).A subwindow
of 64x16 pixels was used, thus achieving 3290 frames per second, with an
acquisition time of 40µs (TACQ). Before starting the tests, the camera was
calibrated using a black body from 273K to 373K.



Temperature increase associated with plastic deformation... 381

Thespecimen isplaced ina test chamber to isolate it fromspuriousexternal
radiation. Thus the IRCmeasures only thermal radiation. Inside the chamber,
there are electrical resistances controlled by thermocouples with which the
specimen may be heated until reaching the defined test temperature. The
maximum temperature allowable is 573K.

A dark chamber is placed between the test chamber and the IRC. This
chamber was designed taking into account the working focal distance of the
IRC.

Figure 1a shows a picture of the test chamber and the dark chamber con-
nected to theHopkinsonbar.Figure 1b showsapicture inside the test chamber
with the electrical resistances to enable the test to be performed at different
initial temperatures.

Fig. 1. (a) Coupled SHPB, test chamber, and dark chamber, (b) internal
test-chamber description

Afundamental aspect of the testing system is the synchronismbetween the
SHPBandthe IRC.This synchronismis achievedbymeansof a trigger system.
The synchronization system consists of two photoelectric sensors, two optical
fibres, and a time counter. During experiments, the photo-sensors detect the
striker arrival and generate a trigger signal. By the use of this synchronization
system, it is possible to associate the temperature measured by the IRCwith
a value of the strain in the specimen during the deformation process.

2.2. Test procedure

The test procedure is based on the classical SHPB testing methodology.
The main difference is that several mechanical and electronic time intervals
must be controlled to ensure synchronism between the SHPB and the IRC,
allowing a better analysis on time of the signals.



382 J.L. Pérez-Castellanos, A. Rusinek

The strain rate ε̇ imposed on the specimen depends on the striker veloci-
ty V0 and is proportional to the air pressure stored in the gas gun chamber and
to the specimen length Lo. Thenominal strain εN, the strain rate ε̇N, and the
nominal stress σN, imposed on thematerialmaybe calculatedwith knowledge
of the incident wave εi, the reflected wave εr, and the transmitted wave εt,
recorded, using the following expressions, Eq. (2.1). This is based on the the-
ory of one-dimensional elastic-wave analysis while assuming mechanical-force
equilibrium of the specimen

εN(t)=
2C

Lo

t
∫

0

εr(τ) dτ ε̇N(t)=
2C

Lo
εr(t)

σN(t)=E
A

AS
εt(t)

(2.1)

where C is the velocity of the elastic waves, AS is the initial cross section
of the specimen, A is the cross section of the bar, Lo is the initial length
of the specimen, and E is the Young modulus of the bar material. The true
stress σ, and true strain ε,maybe calculated, respectively, using the following
expressions, Eq. (2.2), and assuming plasticity with constant volume ν =0.5

σ=σ(1−εN) ε=− ln(1−εN) (2.2)

The physical process of temperature measurement by the IRC is described
as follows. The IRC receives and measures the outcoming radiance from the
object Rout. This radiance has two components: the emitted Remi, and the
reflected Rref . The emitted radiance may be calculated as Remi = ǫR0(Temi)
where R0(Temi) is the radiance emitted by the black body for the same spe-
cimen temperature Temi, and ǫ is the specimen-surface emissivity factor. The
reflected radiance may be calculated as Rref = rR0(Tamb) where R0(Tamb) is
the radiance emitted by the black body at room temperature Tamb, and r the
specimen-surface reflectance (r=1−ǫ). The radiance received andmeasured
by the IRC is related to its emitted and reflected components as

Rout = ǫR0(Temi)+(1− ǫ)R0(Tamb) (2.3)

Rout is measured using the IRC, R0(Tamb) may be calculated using the well-
known Plank’s law expression, and ǫ must be firstly estimated. From these
parameters, R0(Temi) may be calculated using Eq. (2.3). Finally, the tempe-
rature of the specimen T may be deduced as the value at which the integral
of the Plank law expression R0(λ,T), over the spectral range of the camera,
equals to R0(Temi).



Temperature increase associated with plastic deformation... 383

The IRC records the radiation during an exposure interval [tFR, tFR +
TACQ] where tFR is the timewhen the frame recording starts and TACQ is the
acquisition time. From the recorded radiation, a temperature value is determi-
ned. This temperature is associated with themiddle point, tFR+TACQ/2, of
the exposure interval to relate itwith the plastic strain, using the synchronism
system.
Ineach test, a set of successive IR imageswasalso recordedby the IRC.The

images were processed to draw a temperature map and a temperature profile
along a generatrix line (∆T vs. pixel). This temperature profile includes the
specimenandone segment of each bar (incident and transmitter) onboth sides
of the specimen with a total length equal to the horizontal dimension of the
window used (64×16 pixels), Fig.1b and Fig.3.
The frame rate of the camera used does not allow more than two or three

temperature measurements during the complete loading time. Therefore, to
have more values of the temperature increase, several tests must be perfor-
med for the same initial conditions. Thus, the temperature-increase value
∆T(εp, ε̇

p
) for a set of initial conditions, was calculated using an average qu-

antity ∆T =N−1
∑N
1 ∆Ttest where N is the number of tests performed.

3. Results and discussion

3.1. Material

The material used in this study is an aluminium alloy Al 6082 (Mrówka-
Nowotnik andSieniawski, 2005;Mannet al., 2007;Manpinget al., 2008;Agena,
2009;Dadbakhsh et al., 2010). This alloy is amedium-strength structural alloy
that includes in its chemical composition 0.7-1.3 Si, 0.4-1.0Mn, 0.6-1.2Mgand
0.4-0.15 Cr; the manganese controls the grain structure and size. Depending
of the manganese added, this alloy has high values of mechanical properties
and is used for industrial application where both lightness and high strength
are required. In Table 1, the mechanical static parameters characterizing our
alloy Al 6082 are reported.

Table 1.Mechanical parameters of alloy Al 6082

Tensile strength Yield stress Elongation Young’s modulus
[MPa] [MPa] [%] [GPa]

300-320 280 13 69.5

The material was supplied in the form of extruded bars and machined
to the final geometry: 7.0mm long and 14mm in diameter inducing a ratio



384 J.L. Pérez-Castellanos, A. Rusinek

s0 =h0/d0 =0.5. The specimens previously underwent a process of polishing
and a lubricant (MoS2) was used to reduce the friction effect µ between the
specimen and the bars. The friction effect is related to the parameter s0 as
reported inMalinowski andKlepaczko (1986), andJankowiak et al. (2011). All
the specimens before the tests were coveredwith soot to increase thematerial-
emissivity factor ǫ.

The value of the material-emissivity factor ǫ was calculated experimen-
tally. A specimen was heated on the heating plate from room temperature to
393K.During the heating process, the temperature time coursewasmeasured
simultaneously by the IRC and by a thermocouple fixed on the specimen. The
two temperature measurements were compared to determine the emissivity-
factor value ǫ of the material considered.With the use of this technique, the
average of the emissivity factor was estimated to ǫ=0.8.

3.2. Experimental results

Several compression tests were performed at room temperature, 293K, for
different strain rates.The reproducibilityof the testswasverifiedbycomparing
several results under the same experimental conditions. Thus the results in
terms of σ−εp and ∆T−εp curves for the same set of experimental conditions
were consistent with amaximumdifference of 1% for an imposed condition in
terms of the strain rate and initial temperature.

Figure 2 shows, as an example, the true stress/true plastic-strain relation-
ship that resulted at room temperature and different strain rates varying from
300s−1 ¬ ε̇

p
¬ 1500s−1.

From the true-stress/true-strain curves, the logarithmic strain-rate-
sensitivity coefficient was calculated, m = (∂ logσ/∂ log ε̇)|ε,T = 0.03, which
may be considered as a moderate value in the interval of the strain rates te-
sted. Nevertheless, remarkable temperature sensitivity of this alloy is noted,
ν =(∂ logσ/∂ logT)|ε,ε̇ =−0.4.

Figure 3 shows, as an example, several profiles of temperature increase that
resulted during the tests at room temperature for a strain rate of ε̇=600s−1.
Each profile shows a relatively constant temperature value with two peaks
corresponding to the perimeter of the circular bases of the specimens. These
peaks aredue to the friction effect inducingan additional temperature increase
and, therefore, not considered to represent the temperature increases in the
specimen.

Table 2presents the results of themeasured temperature increase including
the initial temperature T0 the average strain rate reached during the test ε̇

p
,



Temperature increase associated with plastic deformation... 385

Fig. 2. σ−εp curves for Al 6082 alloy at room temperature; (a) ε̇
p

=300s−1,
(b) ε̇

p

=600s−1, (c) ε̇
p

=1000s−1 and (d) ε̇
p

=1500s−1

Fig. 3. Profiles of temperature increase corresponding to different strain levels at
room temperature and strain rate ε̇

p

=600s−1; (a) εp =0.027 and (b) εp =0.046

the plastic strain level εp, and the associated recorded temperature increase
under dynamic compression ∆T .

From the previous values, Table 2, it is observed that ∆T(εp) did not
depend on the strain rate for an imposed strain level and initial temperature;
this is due to a reduced strain-rate sensitivity of the alloyAl 6082, as discussed
previously.



386 J.L. Pérez-Castellanos, A. Rusinek

Table 2. Tests results: test temperature T0, plastic strain rate reached ε̇
p
,

plastic strain εp and the associated temperature increment ∆T

T0 [K] ε̇
p
[s−1] εp [–] ∆T [K]

600 0.014 1.4
296 600 0.027 3.0

600 0.046 5.8

296

1000 0.026 3.1
1000 0.047 6.1
1000 0.07 9.7
1000 0.092 13

1500 0.038 3.5
373 1500 0.127 15

1500 0.195 20

Immediately after the process of plastic deformation ended, a small tem-
perature decrease was detected. This temperature decrease was due to the
elastic recovery as observed precisely under a quasi-static compression test.
The load applied to the specimen consisted of a periodic compression force
with an amplitude of 32.3kN and a frequency of 0.185Hz. This load amplitu-
de corresponds to a material-stress level of σloading = 210MPa, which is less
than the material yield stress. The frequency value was chosen to be able to
reproduce it on the test machine.

Figure 4 shows the loading force vs. time and the temperature increase vs.
time relations.

Fig. 4. Loading force vs. time and temperature vs. time relationsmeasured during
the experiment



Temperature increase associated with plastic deformation... 387

The maximum temperature increase was found to be close to 0.5K. It
was also observed that the decreasing slope defines a temperature decrease
of the same magnitude as the thermal expansion parameter of the material.
The temperature increase, ∆T(εp, ε̇

p
), corresponds to a rate of 0.42K/s. This

rate corresponds exactly to the rate calculated using the analytical expression
derivedbyThomson(1853),which is the last termofEq. (1.1) andrepresenting
the temperature variation associated with the elastic strain, Eq. (3.1).

The temperature increase, as elastic recovery, proved to be of the same
order of magnitude, close 0.5K. Thus, the temperature decrease observed at
the end of the test may be considered to correspond to the elastic recovery
process.

3.3. Numerical analyses

A three-dimensional finite-element model was used to simulate the SHPB
tests. The commercial finite-element code ABAQUS/Explicit, 2006, was used
to investigate the precise thermo-mechanical process in the specimen under
dynamic loading and the phenomenon of elastic-wave propagation. The FEM
model developed is described in the following sections.

3.3.1. Model description

The model includes the two elastic bars, the striker, and the specimen
using real dimensions (incident and transmitter bars 1m longwith a diameter
of 22mm and for the striker a length of 330mmwith the same bar diameter;
Fig.5.

Fig. 5. 3D finite elementmodel of SHPB test

The specimen and bars are free of movement along the bars axis. The
radial movement of the nodes on the model axis is restricted.

The mesh of the incident bar, the transmitter bar, the specimen, and
the striker were made using hexahedral elements with 8 nodes, C3D8R
(ABAQUS/Explicit, 2006), reproducing the total geometry of the device. The



388 J.L. Pérez-Castellanos, A. Rusinek

density of themeshdirectly influences two fundamental factors of the analysis:
the accuracy of the solution reached and the time of calculations. Therefore,
it is necessary to achieve a balance between these two variables. The study
of the sensitivity of the mesh was carried out before beginning the numerical
analyses considering just the incident and the transmitter bars.
Several numerical models with FE meshes with 18320 to 34508 elements

were considered. On the first time, a comparison wasmade of the incident εi,
the reflected εr, and the transmitted εt waves calculated using these diffe-
rent numerical models. It was found that there is no strong variation between
the wave profiles and level calculated (Fig.6). On the second time, the tem-
perature increase on the specimen surface calculated using these numerical
models were also compared; as a result, minor influence of the mesh density
on the temperature increase (in the interval of elements number considered)
was deduced. Thus, the density of 18320 elements was considered optimal.

Fig. 6. Analysis of the mesh-sensitivity of the SHPBmodel using only the
transmitter bar

The contact conditions between the surfaces of the striker-incident bar,
incident-bar specimen, and specimen-transmitter barwere taken into account.
The tangential interaction between the contact surfaces can be established by
means of Coulomb’s friction law.
Based on the Klepaczko andMalinowski model (Malinowski andKlepacz-

ko, 1986), additional energy for a deformation range is observed depending on
the specimen ratio used during the tests. This energy, related to friction, in-
duces an extra stress increase; the stress level will change, increasing with the
friction coefficient. Thus, the stress levelmeasured includesmaterial behaviour
and friction effects. To correct the latter, the model proposes the expression

µ(ε)=
σ−σ0
σ
3s0[exp(ε)]

3

2 (3.1)



Temperature increase associated with plastic deformation... 389

where σ is the compressive true stress, σ0 is the compressive true stress deter-
mined for an ideal test without friction specimen/bars, s0 =h0/d0 is the ratio
between the initial specimen height and diameter, and ε is the true strain.
TheKlepaczko andMalinowskimodel has some limits andmay be used as

an average range of µ varying from 0 to 0.25. A recent complete analysis of
this model is reported in Jankowiak et al. (2011).
Anexperimental characterization of the friction coefficient µ(ε)was carried

out using the Klepaczko and Malinowski model. Figures7a and 7b show the
relation between µ and ε calculated for a strain rate of 300s−1 and 600s−1,
respectively. It can be seen that µ remains more or less constant and equal
to 0.09. This value of friction coefficient was the one used during numerical
calculations (the same for all test conditions) and implies no contact.

Fig. 7. Relations µ−ε for different strain rates under dynamic compression,
Eq. (3.1)

During the experiments, the striker velocity V0 justbefore impact,wasme-
asuredwith twofibre-opticphotoelectric sensors.Thisvelocitywasusedduring
numerical simulations as the initial boundary condition. The wave transmis-
sion through the bars occurs after the impact, and the incident, reflected, and
transmittedwaves can be calculated along themesh, andmore precisely, for a
distance corresponding to it where the strain gauges are glued along the bars.
The material of the incident and transmitter bars is an Inconel steel:

Young’smodulus equal to 211GPaandPoisson’s ratio equal to 0.3; thedensity
considered is equal to 8190kg/m3. Thematerial of the striker is steel: Young’s
modulus equal to 207GPa andPoisson’s ratio of 0.3; the density considered is
7800kg/m3.
To describe behaviour of the isotropic alloy Al 6082, the J2 theory was

assumed to define the flow stress. The elastic constants were defined from
the corresponding experimental test. The hardening function considered was



390 J.L. Pérez-Castellanos, A. Rusinek

defined using the Johnson-Cook model, Eq. (3.2), since the material exhibits
a low strain-rate dependency (frequently reported for aluminium 6082). This
model is well defined to describe this kind of material behaviour (linear strain
rate sensitivity)

σ(εp, ε̇
p
,T)= [A+B(εp)n]

(

1+C ln
ε̇
p

ε̇0

)

(1−T∗m) → ε̇
p
­ ε̇0

σ(εp,T)= [A+B(εp)n](1−T∗m) → ε̇
p
< ε̇0

(3.2)

with

T∗ =
T −Troom
Tmelt −Troom

where σ is the equivalent stress, εp is the equivalent plastic strain, ε̇
p
is

the equivalent plastic strain rate, ε̇0 is the reference strain rate, Tmelt is the
melting temperature, and Troom is the room temperature. As material para-
meters, A is the initial yield stress, B is the hardening modulus, n is the
hardening exponent, C is the coefficient of strain rate sensitivity, and m is
the temperature sensitivity exponent.
The parameters A,B,C, n, and mwere calculated by applying the root-

mean-square method coupled to Eq. (3.2) and based on experimental true
stress/plastic strain curves obtained at 296K for ε̇

p
= 300s−1, at 296K for

ε̇
p
=1000s−1 and at 373K for ε̇

p
=1500s−1. Figure 8 shows the experimental

true stress/plastic strain curve obtained at 296K for ε̇
p
= 600s−1 and the

corresponding true stress/plastic strain curve using the values calculated for
parameters A, B, C, n, and m. A good agreement was found between the
experiments and numerical results.

Fig. 8. Experimental and J−C true stress-plastic strain curves for a test at 296K
and ε̇

p

=600s−1

Non-constant values of the Quinney-Taylor coefficient β were considered.
The previous relation proposed by Zehnder, Eq. (1.4), is valid for materials



Temperature increase associated with plastic deformation... 391

showing a strain/energy transformed into heat dependency as alloy Al 6082 in
this experiment. The temperature increase ∆T , measured during the tests at
room temperature T0 =296K, and for a plastic-strain value ε

p =0.027, can
be determined from Table 2, these values being ∆T = 3K for ε̇

p
= 600s−1

and ∆T = 3.2K (extrapolated from εp = 0.026) for ε̇
p
= 1000s−1. Both

temperature-increase values are practically equal, and thus it can be deduced
that the proportion of plastic work converted into heat β does not depend on
the strain rate for alloy Al 6082.
Nevertheless, no conclusions can be drawnwith respect to the dependency

of β on temperature because the mechanical behaviour of alloy Al 6082 de-
pends heavily on temperature. The variation of the temperature increase ∆T
associatedwith plastic strain is due tomainly the aforementioned dependency.
These results lead to the conclusion that the use of the Zehnder model is

appropriate in this case, since the testedmaterial is not strain-rate dependent
as reported before.
Thetemporal evolution of the stress-strain state of the systemwasanalysed

bydirect integration of the equations ofmotion and the constitutive equations.
The implicit algorithm proposed by Zaera and Fernández-Sáez (2006) was
used to integrate the Johnson-Cook constitutive equation. This algorithmwas
modified to include the dependency of β on εp, Eq. (1.4).
For the validation of the numerical model proposed, the experimental and

the analytical elastic waves were compared. Figure 9 shows the experimental
and numerical incident, reflected and transmitted nominal stress waves; the
three waves verify the relation

εI +εR = εT (3.3)

A good agreement between the experiments and numerical results was found.
The radial-nominal stress component is also plotted in Fig.9. It can be

seen that the values of this wave are practically negligible, allowing for the
fulfilment of the hypothesis of negligible radial inertia during the SHPB test
due to the initial shape geometry (Malinowski and Klepaczko, 1986).

3.3.2. Results

Figure 10 shows the relations σ−εp foundduring thenumerical analyses at
room temperature and a strain rate of 300s−1, 600s−1, and 1000s−1, together
with the β−εp relations calculated using Eq. (1.4). It may be observed that
the usual hypothesis taking β as constant is, in this case, available for high
plastic-strain values. The medium value calculated for β was 0.9, which is
similar to that currently used for metal alloys.



392 J.L. Pérez-Castellanos, A. Rusinek

Fig. 9. Incident, reflected and transmitted longitudinal-nominal stress waves and
radial-nominal stress wave

Fig. 10. Relations σ−εp and β−εp for different dynamic strain rates at room
temperature

Figures 11a and 11b show, respectively, a map of the plastic deformation
and the temperature increase for calculation time of t = 120µs, initial tem-
perature of 296K and a strain rate of ε̇

p
=1000s−1.

The cylindrical surface of the specimen shows uniform fields of plastic de-
formation (Fig.11a and temperature Fig.11b). It is observed using numerical
simulations that on the specimen bases, local increaments of plastic defor-
mation and temperature take place as observed during experiments using IR
temperaturemeasurements. These local increaments are related to friction ef-
fects between the specimen and the bars. Localized small non-symmetrical
irregularities are also noticed; these irregularities are due to the non-exact
cylindrical symmetry of the mesh. A temperature increase on the specimen



Temperature increase associated with plastic deformation... 393

Fig. 11. (a) Plastic strain and (b) temperature maps resulting from numerical
simulation at room temperature for a strain rate ε̇

p

=1000s−1 and corresponding
to a plastic strain value εp =0.07

surface (initial test temperature equal to 296K) between 7K and 13K can be
seen. This value is of the same order as the one measured during the experi-
ments (Table 2).

Figure 12 gives the numerical (continuous line) and experimental (symbol)
temperature profiles along a generatrix line taken on the surface of the de-
formed specimen. The numerical profile corresponds to the temperature map
using Fig,11b. The two profiles discussed above show a good agreement. The
temperature increase at the contact between the bars and specimenwas higher
than the temperature increase in the specimenmiddle zone, due to friction ef-
fects. The difference in the central zone is due to a possible variation of the
soot coating due to its ejection during experiments.

Fig. 12. Experimental and numerical temperature profiles for an imposed strain level
under dynamic compression and corresponding to a strain level of εp =0.07



394 J.L. Pérez-Castellanos, A. Rusinek

As a result of numerical simulations, the temperature increments of the
specimen surfacewere calculated and comparedwith the experimental results.
Figure 13 shows the temperature evolution during theplastic-deformation pro-
cess for two different strain rates using numerical simulations. The simulated
tests were at ε̇

p
= 600s−1 and ε̇

p
= 1000s−1, both at room temperatu-

re, and at 373K and ε̇
p
= 1500s−1. Figure 13 also plots the experimental

temperature-increase values associated with the corresponding plastic-strain
values (Table 2).

Fig. 13. Numerical and experimental T −εp results for different strain rates under
dynamic compression

A good agreement is observed between the experiments and numerical
simulations in terms of the temperature increase ∆T for the strain-rates and
test temperature considered.

4. Conclusions

By an experimental procedure developed, the temperature increase associated
withplastic deformationduringSHPBtestswasmeasured for anAl alloy 6082.

The results of the tests reveal that the temperature increases with the
strain rate and decreases when the initial test temperature is higher.

At the final deformation stage, a minor temperature decrease occurs, pre-
sumably associated with an elastic unloading process.

A three-dimensional numerical model of the SHPB test including the in-
cident bar, transmitter bar, the projectile and the specimen was developed.



Temperature increase associated with plastic deformation... 395

The Zehnder model was linked to an implicit integration algorithm and im-
plemented as a user subroutine in the commercial finite-elements code ABA-
QUS/Explicit. Thismodel offers a prediction in terms of temperature increase
associated with plastic deformation.

The results validated the applicability of Zehnder’s expression to model
the proportion of plastic-deformation energy transformed into heat β for ma-
terials that are strain dependent and strain rate non-dependent. Nevertheless,
no conclusions could be drawn with respect to the dependency of β on tem-
perature.

A good agreement was found between the experimental and numerical
results, providing a better understanding of experimental measurement using
an inverse method.

Acknowledgements

The author fromUC3MthanksDr. RolandoGuzmánLópez for his collaboration,

interest and discussions.

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Wzrost temperatury wywołany plastycznym odkształceniem przy
dynamicznym ściskaniu – analiza stopu aluminium Al 6082

Streszczenie

Wzrost temperatury w materiale związany z plastycznym odkształcaniem może
być rejestrowany różnymi metodami, w tym m.in. techniką termografii podczerwie-
ni (IRT). Prezentowana praca poświęcona jest badaniom wzrostu temperatury przy
różnym tempie odkształceń i temperatury początkowej próbek wykonanych ze stopu
aluminium Al 6082. W eksperymentach użyto zmodyfikowanego pręta Hopkinsona
do generowania szybko-zmiennych odkształceńwmateriale i jednocześnie dokonywa-
no pomiarów temperatury za pomocą kamery termowizyjnej. Przeprowadzono także
symulacje numeryczne przebiegu przyrostu temperatury pozwalające na lepsze zro-
zumienie zachodzących procesów na podstawie lokalnych pomiarowym. W tym celu
zbudowano szczegółowy model bazujący na metodzie elementów skończonych, któ-
ry przeanalizowano pod kątem dynamicznego ściskania. Część pracy odkształcenia
plastycznego zamienianego w ciepło oszacowano za pomocą modelu Zehndera. Wy-
niki obliczeń uwzględniających tempo odkształceń i temperaturę początkową na jej
przyrost w badanymmateriale zweryfikowano z rezultatami doświadczeń.

Manuscript received April 15, 2011; accepted for print July 7, 2011