Thermomechanical analysis of the yield behaviour of a semicristalline polymer
J.-M. Muracciole*,**, B. Wattrisse*, A. Chrysochoos*,**
* Mechanics and Civil Engineering Laboratory, Montpellier II University, France
** Polytech’Montpellier , Montpellier II University, France
ABSTRACT
This paper first presents the characteristics of a new experimental set-up using digital image correlation and infrared
thermography. The kinematical data are used to track the temperature variations of material surface elements. They are then
combined to construct local energy balance. To illustrate the interest of such an approach, the paper then describes the
calorimetric effects accompanying the propagation of necking in a plasticized PolyAmide 11. A thermodynamic analysis of
cyclic loading finally aims to show the existence of an entropic elastic effect generally associated with rubber-like materials.
Introduction
The cold drawing of semi crystalline polymer has been observed in the early 1930s by Carothers and Hill [1]. Ever since,
various attempts were made to explain this phenomenon in terms of crystallinity modification. This behaviour, usually related to
a rearrangement of the crystallites, depends on the nature of the stretched materials. In some polymers, during necking, parts
of chains in crystalline blocks are unfolded and transfer into amorphous phase with more or less orientation (see Peterlin and
Olf [2], Gent et al. [3]). In others, Strauch and Schara [4] exhibit crystallite realignment. Other authors, Waddon and J.,
Karttunen [5], emphasize solid-solid phase transition. Nevertheless, everyone agrees that a local rise of temperature occurs in
the neck shoulders.
To perform energy balance during such local phenomena, we have developed an experimental approach to the material
behaviour characterization which combines digital image correlation (DIC) with infrared thermography (IRT). The DIC
techniques give access to the measurement of displacement and deformation fields while the thermal images enable us, under
certain conditions, to estimate the distribution of heat sources induced by the deformation process. Those thermal images
were recorded using an infrared camera and a specific pixel calibration method described in [6]. A specialized electronic
system was performed to get a simultaneous recording of speckle and infrared images. A combined treatment of kinematics
and thermal data then allowed us to track the temperature of material surface elements (MSEs) and to construct local energy
balances. This processing is well adapted for studying the thermomechanical behaviour of material when localization
mechanisms occur at the observation scale fixed by the optical lenses. Applications to steels and polymers were already
carried out for monotonic quasi-static loadings [7, 8].
In what follows, after having recalled the main characteristics of the experimental arrangement, we present the protocol
defined to compute the heat sources taking into account the existence of localization zones. To illustrate this we consider
tension tests on plasticized Polyamide 11 (PA11) samples. For this kind of material the necking zone spreads out during
tension stages. Using thermal and kinematical data, we analyze the thermomechanical coupling effects that appear in the
necking zone during a cyclic loading around a given tension state.
Energy balance
We used the Generalized Standard Materials formalism to derive the local heat equation [9]. Within such a framework, the
equilibrium state of each volume material element is characterized by a set of n state variables. The chosen state variables
are: T, the absolute temperature, ε, a strain tensor, and (α1, …, αn-2), the n-2 scalar components of the vector α of “internal”
variables that sum up the microstructural state of the material. By construction, the thermodynamic potential is the Helmholtz
free energy ψ. Combining both first and second principles of thermodynamics, the local heat equation reads:
ρC (
∂T
+ v .gradT ) − div( k.gradT ) =
∂t
∂ 2ψ
∂ 2ψ
∂ψ ⎞
∂ψ
⎛
: ε& + ρT
ρT
⋅ α& + ⎜ σ − ρ
⎟ : ε& − ρ ∂α ⋅ α& + re
T
T
∂
∂ε
∂
∂α
∂ε
⎝
⎠
144444244444
3 1444424444
3
•
w •the +w thc
(1)
d1
where
ρ=
C=
v =
k=
mass density
specific heat
velocity vector
heat conduction tensor
The thermomechanical coupling sources and the intrinsic dissipation d1 have been grouped in the right hand side of Eq.(1).
The symbol
sources is
( )
•
•
w the
means that the variation of ( ) is path-dependent. For the polyamide considered here, one of the coupling
, the Lord Kelvin’s source associated with the thermoelastic effect due to the thermo-dilatability of the polymer.
•
may exist and represent possible interactions between temperature and
We stress that other coupling sources w thc
microstructure. For instance, in the case of PA 11 another coupling source will be evidenced in what follows. This coupling will
be in phase with the mechanical cyclic loading, in contrast with the classical thermoelastic effect which is in phase opposition
with the loading.
•
Per unit volume, the intrinsic dissipation d1 is the difference between the rate of deformation energy w def
= σ : D (where σ is
the Cauchy stress tensor and D the strain rate tensor) and the sum of elastic and stored energy rates
w e• + w s• = ρ ψ,ε . ε& + ρ ψ,α .α& . At finite strain, the chosen strain variable ε is the Hencky strain tensor.
Finally, considering Fourier’s law of heat conduction, (q = -k gradT), the left hand side of Eq.(1) becomes a partial derivative
operator applied to temperature. Its estimate then leads to a local determination of the overall heat source
•
•
w h• = w the
+ w thc
+ d1 .
Experimental design and image processing
The experimental set-up involved a MTS hydraulic testing machine (frame 100 kN, load cell ±25 kN), an IRFPA camera
(Cedip, Jade 3) and a visible CCD camera (Camelia, 8M). The lens axes of the cameras were kept fixed and perpendicular to
both sides of the sample surface. We developed a special electronic master-slave system to ensure a perfect synchronization
of the frame grabbing of the two cameras. Moreover, the SYNCHROCAM system enabled us to record simultaneously various
analogical signals (force, crosshead displacement, etc.) provided by the testing machine. It also sets the data capture
frequency. The performances of the system are the following: the rate of the master camera can vary –insofar as the camera
-4
allows it – from 1 frame per hour to 500 frames per second. Finally the acquisition time accuracy is better than 10 s which is
sufficient for classical mechanical tests. The main characteristics of the two cameras are given in Table 1.
Table 1. Main characteristics of both cameras
Camelia 8M
Cedip Jade 3
Image size
(pixels)
3500×2300
240×320
Pixel size
2
(µm )
10×10
30×30
Frame rate
(Hz)
2.7
250
Kinematical fields
Speckle image processing gives the time course of various kinematical variables. We worked with white light and we used
artificial speckle using white and black paints. The camera must be carefully set so that the CCD detector remains parallel to
the sample surface. Indeed, each out-of-plane movement (translation or rotation) produces a parallax error which distorts the
-4
-4
images. A typical error on the strain measurements due to the setting up of the camera, was about 1 10 to 2 10 .
X
X
e0
l0
Y, y
L0
Y
Z
O0
X, x
U
O0
V
W
Z, z
Y
O0, O
Z
Figure 1. Frames, coordinates and specimen geometry
e0=4 mm, l0=10mm , L0= 60 mm
The image processing was systematically realized in two steps after the test:
- first step: the displacement field was estimated. Generally, the displacement of each point located on the surface of the
sample, has three components: 2 in-plane components, say U and V, and 1 out-of-plane component, say W. In figure 1, (L0, l0,
e0) define the initial geometry of the sample test section, while (X, Y, Z) and (x, y, z) denote respectively the Lagrangian
coordinates and Eulerian coordinates. The components U and V were directly computed by a digital image correlation [10].
- second step: the strains (or the strain-rates) were then derived from the displacements by space (and time) differentiation.
Each computational step used a particular numerical processing. The reader interested in this subject is referred to [10].
To achieve accurate strain measurements, the displacement field must be filtered before any differentiation. We chose a fitting
technique based on a local least-squares (linear or quadratic) approximation of the discrete displacement data which also
allows the computation of the derivatives. The choice of the approximation zone (AZ) is very important in the differentiation
process. The optimized AZ depends on the signal-to-noise ratio and the amplitude of the sought derivatives.
Using standard parameters, the accuracy on the displacement calculation was about 5 10
-4
strain measurement was 5 10 .
-2
pixel and the accuracy on the
Temperature and heat source fields
In the framework of our experiments, several hypotheses were made. First, we checked that the external heat supply remained
time-independent so that: re = -k ∆T0 where T0 is the equilibrium temperature field of the sample. Besides, for tests performed
on thin flat specimen, [11] showed that the temperature measured at the sample surface remains very close to the depth-wise
averaged temperature.
Taking into account all these hypotheses and integrating Eq.(1) throughout the thickness of the specimen, the heat equation
can be markedly simplified into a two-dimensional partial derivative equation
⎛ ∂θ
θ ⎞
•
ρC ⎜ + v .gradθ +
⎟ − k ∆θ = wh
∂
τ
t
th ⎠
⎝
(2)
where
θ = T − T0 = difference between T and T0, averaged according to thickness
τth
= a time constant characterizing heat losses perpendicular to the
plane of the specimen
The heat conduction in the plane is taken into account by the two-dimensional Laplacian operator ∆.
The thermal data being discrete and noisy, the differential operator was traditionally estimated by low-pass filtering and
according to the properties of discrete Fourier transform and Fourier series [12]. To facilitate the coupling of numerical
methods developed for kinematical and thermal data processing, we chose to use local least-squares fitting techniques to
calculate the left hand side of Eq.(3). The thermal data were spatially approximated by a parabolic function, corresponding to
locally constant conduction heat losses in the neighbourhood of each pixel of the thermal image. As already mentioned, the
dimensions of AZ affect the data fitting efficiency, and thus may influence the operator estimate.
In this work, the displacement data were also used for the processing of thermal images to track the same material zone as
the one observed by the CCD camera. This allowed us to take into account the convective terms of the material time derivative
of the temperature throughout the test.
Tests and results
Tensile tests were performed at room temperature on ISO R527 PA11 samples (see figure 1). PA11 is a thermoplastic semicrystalline polymer with a triclinic phase and, at room temperature, a glassy amorphous phase. This raw material has
undergone an ageing processing. The glass transition associated with the transition of the amorphous phase occurs at around
45°C.
Figure 2 illustrates the macroscopic mechanical response of the material during the test (load-displacement curve). The tests
-3 -1
consisted in a velocity-controlled loading (imposed strain rate of around 8 10 s ) until the displacement reached 82.5 mm.
Then the control mode was switched over to the load canal, and 9 sinusoidal cycles were completed (force amplitude = 600 N,
th
time period = 5 s). The sample fracture occurred at the beginning of the 10 cycle during the loading phase.
1500
load (N)
neck propagation
fracture
0
load-controlled cycles
0
50
displacement (mm)
100
Figure 2. Mechanical response of PA11 sample (T0 = 25°C)
The plateau in figure 2 indicates that this material develops a localized neck in tension. We chose to perform the load cycles
after the development of the neck in order to characterize the thermomechanical behaviour of the material at large strains,
when the macromolecular chains are highly stretched [13-15]. As the oriented chain is much stiffer than the un-oriented one,
the orientation mechanism leads to a neck propagation phenomenon. The increase of material stiffness limits the narrowing of
the necked zone. To extend further the specimen elongation, the neck must move, drawing material from the un-necked
region: the strain-rate is then localized in the necking shoulders, while it remains approximately equal to zero within and
outside these necking zones.
The necking is consequently associated with a heterogeneous response of the material. The concentration of heat sources in
the necking shoulders and the low diffusivity of the studied polymer generate important temperature gradients that must be
taken into account in the computation of the heat sources [8]. The measurement fields being relatively homogeneous
according to the specimen width, we chose, for concision’s sake, the following representation of our measurement fields: we
plot the time course of the data profiles captured along the longitudinal axis of the specimen.
The space-time charts (figures 3, 4, 5) are constructed with the abscissa axis representing the time and the ordinate axis being
the longitudinal sample axis. The mechanical response σ 0 ( t ) = F ( t ) ( l 0e0 ) , where F (t ) stands for the applied load as a
function of time, was superimposed to give the reader a familiar landmark to link the local pattern of the measurements to the
loading state of the sample.
θ(X,t), (K)
θ(x,t), (K)
0
A
20
20
20 A
B
80
C
D
10
120
0
160
0
50
100
t (s)
X (mm)
x (mm)
40
15
B C
40
10
D
60
5
80
0
150
0
50
(a)
100
t (s)
150
(b)
Figure 3. Time course of: (a) θ(x ,t), (b) θ(X, t)
Figure 3a presents the time evolution of a crude thermo-profile directly extracted from infrared data files (Eulerian
configuration). The room temperature was around 300 K. The darker horizontal lines are related to bad pixels in the IRFPA
detector for which no thermal data is available. As the camera is fixed, each pixel no longer corresponds to a fixed MSE as
soon as the strain becomes important. The paths of four MSEs named A, B, C and D were plotted to illustrate this. Note that
the maximum amplitude of the temperature variations reached more than 20°C at point A, which fully justified the choice of an
anisothermal thermomechanical framework. Furthermore, this temperature increase was sufficient to warm up the material to
the glass transition temperature in the neighbourhood of the neck.
W.m-3 x 106
1
20 A
E& XX ( X , t )
X (mm)
X (mm)
w •h ( X ,t )
A
20
B
0.2
D
0.1
B
C
40
0
C
40
D
60
s-1
60
-1
80
0
80
40
80
t (s)
120
160
0
0
40
(a)
80
t (s)
(b)
Figure 4. Time course of: (a)  wh• (X, t) , (b) E& XX ( X , t )
120
160
The displacement field being known, it is possible to track the temperature of any material zone. Figure 3b shows the thermal
data presented in Figure 3a, once the displacements have been taken into account. The paths of MSEs A, B, C and D
henceforth correspond to horizontal straight lines (Lagrangian configuration).
The progressive narrowing of temperature level curves is difficult to interpret because of heat diffusion. The temperature is not
indeed completely intrinsic to the material behaviour contrary to the heat sources. Figure 4a shows the time evolution of the
corresponding heat sources profile during the test. At the test beginning, distribution of w h• is rather homogeneous and
negative in accordance with classical thermoelastic behaviour during a simple tensile test. Then profiles of w h• become
positive and concentrate to give way to the localized necking. This change of sign is related to the development of positive
heat sources associated with dissipative mechanisms (viscosity, plasticity). The beginning of the plateau corresponds to the
inception of two necking lips: the upper one moves upward to the upper grip (located at the top of the figure), and the lower
one moves to the lower grip (located at the bottom of the figure). Figure 4b illustrates the space-time evolution of the
longitudinal strain-rate component E& XX ( X , t ) of the Green-Lagrange tensor. Heat sources and strain-rate distributions are very
similar: the higher the strain-rate, the larger the heat source intensity, and thus, the larger the temperature variations.
The paths of the 4 spotted MSEs A, B, C and D are plotted on Figure 4. A is associated with the inception of the necking. At
the end of the monotonic loading, A is thus located in the middle of the necked zone. B corresponds to the location reached by
the lower necking lip at the end of the monotonic loading. As shown in figure 4b, D has been chosen in the middle of a second
localization zone. Finally, C is set between B and D, in a region not yet reached by the necking.
Figures 5a,b represent the heat sources distribution during the first five cycles.
We observe that the heat sources in the selected MSEs are negative when the applied load decreases and positive when it
increases.
The oscillating part of the responses of the 4 points during the cyclic loading can be associated with coupling mechanisms.
The measured heat sources are in phase with the loading as observed in the case of materials in a rubbery state. The
amplitude difference can be interpreted as the point to point difference in the competition between two opposite coupling
effects: classical thermoelasticity (in phase opposition with loading) and rubber elasticity (in phase).
X (mm)
w h• ( X , t )
W.m-3
x 106
W.m-3 6
3 x 10
w •h ( X , t )
« σN »
A
B
C
D
2
20 A
2
B
1
C
40
0
D
60
-1
1
0
-1
80
170
175
180
t (s)
185
190
165
175
(a)
t (s)
185
195
(b)
Figure 5. (a) Time course of  wh• (X, t) during the first cycles
(b) Time evolution of sources for the different MSEs
Concluding comments
Some polymers, stretched in a rubbery state, may indeed undergo very large strains in a reversible way. This remarkable
behaviour is often called rubber (or entropic) elasticity. The fundamental hypothesis of the molecular theory of rubber elasticity
is that the variation of internal energy is proportional to the variation of the absolute temperature [16]. The analogy with ideal
gases leads to an internal energy independent of the elongation, the stress being attributed to the so-called configuration
entropy. As in classical elasticity, temperature T and strain ε are the chosen state variables. It was shown in [16-17] that for an
entropic elasticity the free energy is of the form:
ψ(T , ε ) = T ξ(ε ) + η(T )
(3)
•
. These sources are
As shown in [11], the coupling sources derived from Eq. (3) are equal to the deformation energy rate w def
then in phase with the mechanical loading contrary to the classical thermoelastic sources. This last property, the cyclic
evolution of the overall heat source and its very small asymmetry led us to claim the existence of a slightly dissipative
competition between the two coupling effects. This kind of competition was already observed in thermoplastic rubbers in the
presence of a more noticeable intrinsic dissipation [11]. A natural continuation of this work should be the local analysis of the
energy balance form within the necking shoulders using adapted optical lenses.
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