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Experimentally simulating high-rate behaviour: Rate and temperature effects in polycarbonate and PMMA


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This paper presents results from applying a recently developed technique for experimentally simulating the high-rate deformation response of polymers. The technique, which uses low strain rate experiments with temperature profiles to replicate high-rate behaviour, is here applied to two amorphous polymers, polymethylmethacrylate (PMMA) and polycarbonate, thereby complementing previously obtained data from plasticized polyvinyl chloride. The paper presents comparisons of the mechanical data obtained in the simulation, as opposed to those observed under high-rate loading. Discussion of these data, and the temperature profile required to produce them, gives important information about yield and post-yield behaviour in these materials.
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Cite this article: Kendall MJ, Siviour CR. 2014
Experimentally simulating high-rate
behaviour: rate and temperature eects in
polycarbonate and PMMA. Phil.Trans.R.Soc.A
372: 20130202.
One contribution of 12 to a Theme Issue ‘Shock
and blast: celebrating the centenary of
Bertram Hopkinson’s seminal paper
of 1914 (Part 1)’.
Subject Areas:
mechanical engineering, materials science
polymer, PVC, PMMA, polycarbonate,
rate dependence, adiabatic
Author for correspondence:
C. R. Siviour
Electronic supplementary material is available
at or
Experimentally simulating
high-rate behaviour: rate
and temperature eects in
polycarbonate and PMMA
M. J. Kendall and C. R. Siviour
Department of Engineering Science, University of Oxford,
Parks Road, Oxford OX1 3PJ, UK
This paper presents results from applying a recently
developed technique for experimentally simulating
the high-rate deformation response of polymers. The
technique, which uses low strain rate experiments
with temperature profiles to replicate high-rate
behaviour, is here applied to two amorphous
polymers, polymethylmethacrylate (PMMA) and
polycarbonate, thereby complementing previously
obtained data from plasticized polyvinyl chloride.
The paper presents comparisons of the mechani-
cal data obtained in the simulation, as opposed to
those observed under high-rate loading. Discussion
of these data, and the temperature profile required
to produce them, gives important information about
yield and post-yield behaviour in these materials.
1. Introduction
Since the pioneering experiments of Kolsky [1], the
split Hopkinson bar has been used to measure the
mechanical properties of polymers under high strain
rate loading and to explore rate dependence in these
materials. One of the earliest studies was by Chou
et al. [2], who examined four polymers between 104
and 103s1. A more comprehensive investigation was
conducted by Walley & Field [3], who investigated
17 materials, and extended the range of strain rates
studied up to ca 20 000 s1using a miniaturized direct
impact Hopkinson bar system. A key observation in
this research was that the strain rate dependence of
many, but not all, polymers was much greater in the
high-rate (Hopkinson bar) experiments than in the
low-rate tests. Because of the difficulty in testing at
strain rates (˙ε) between 10 and 500 s1, a bilinear
dependence on log(˙ε) has often been reported, although
2014 The Author(s) Published by the Royal Society. Allrights reser ved.
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it was appreciated that the true behaviour was more likely to be continuous. This has been
confirmed previously in studies [47] that produced data over a more complete set of rates.
As a result of these observations, there was discussion in the literature about the cause of the
observed nonlinearity in rate dependence—in particular considering the difficulty in separating
material behaviour from artefacts of the experimental techniques used. It is important in high-rate
testing to ensure that specimens are correctly designed to minimize the effect of inertia [8,9]and
the time taken to achieve stress equilibrium [10], which is particularly difficult when testing low-
modulus materials like polymers, and these were cited by some authors as possibly contributing
to the observed response. However, it is now well established that the apparent bilinearity is the
result of a lower order (i.e. low temperature) relaxation process whose frequency dependence
brings them into the testing regime at high strain rates [11,12]. Experiments can be performed
at different strain rates and temperatures, and the relationship between the low-rate transitions
and high-rate behaviour can be described using time–temperature equivalence. If the range
of rates and temperatures is sufficiently small, which is usually the case in Hopkinson bar
studies [11,13], this can be done by using a linear relationship between log(˙ε) and temperature.
As expected, in materials which have no lower order transition, or if the transitions occur at
sufficiently low temperatures, a linear dependence on log(˙ε) is observed [14]. Time–temperature
superposition has been applied in this context to polymers [15,16], composites [17,18]and
polymer-based foams [19]. Furthermore, authors have successfully applied two process models to
describe rate and temperature dependence, and this forms the basis of recent constitutive model
development [12,15,16,2025].
Having used time–temperature equivalence to better understand and model the high-rate
behaviour of polymers, a logical further step is to reproduce the high-rate behaviour by
performing low-rate experiments at reduced temperatures; such experiments were first published
by Kendall & Siviour [26]. There are a number of motivations for doing this:
there are many more available diagnostics for low-rate experiments, allowing
microstructural phenomena in, for example, polymer-based composites or foams to be
better observed during deformation;
a reliable method for experimentally simulating high-rate behaviour would be of benefit
for low-modulus polymers, for which dynamic stress equilibrium is difficult to achieve;
such experiments will give a better understanding of the internal processes that govern
the behaviour of polymers under high-rate and impact loads.
In order to fully reproduce high-rate stress–strain curves under quasi-static loading, not only must
the modulus and initial yield be simulated by reducing the temperature at which the experiments
are performed, but the adiabatic nature of high-rate loading must also be taken into account.
The first step in such a study is to distinguish between the strain rates in which the thermal
conditions under loading are adiabatic or isothermal. Although the transition between these
conditions is not sharp, the identification is aided by the fact that changes in polymer behaviour
are typically dependent on log(˙ε), so the time scales available for heat conduction out of the
specimen vary by many orders of magnitude, and approximate calculations are appropriate.
Knowing the thermal diffusivity (κ=kc,wherekis the thermal conductivity, ρis the density
and Cis the heat capacity) of the specimen material allows calculation of a characteristic time
scale, t,for heat to conduct out of the specimen,
4κ, (1.1)
where lis the specimen length. This can be compared with the time scale of the experiment,
which goes as 1/(˙ε), in order to calculate the transition from essentially isothermal to essentially
adiabatic conditions. Hence, if low-rate experiments are used to simulate those at high strain
rates, the temperature must gradually be increased during the experiment to compensate for
the mechanical work which is converted to heat during the loading, and which appears as a
temperature rise in the high-rate experiments.
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0 0.2 0.4 0.6 0.8 1.0
true strain
15 s–1
simulating 15 s–1
3300 s–1
simulating 3300 s–1
true stress (MPa)
true stress (MPa)
0 0.2 0.4 0.6 0.8 1.0
true strain
85 s–1
simulating 85 s–1
15 s–1
simulating 15 s–1
1.5 s–1
simulating 1.5 s–1
0.001 s–1
simulating 0.001 s–1
Figure 1. Results from simulating high strain rate deformation in plasticized (20 wt%) PVC. (a) Simulating initial yields and
adiabatic conditions at a wide range of strain rates; (b) comparison of theoretical (assuming β=1) and applied temperature
rise for the experiment simulating 1.5 s1. (Reproduced with permission from [26]. Copyright c
2013 Elsevier.)
The second step is to identify the proportion of mechanical work converted to heat, often known
as the βfactor, which can be identified by comparing the temperature rise in the specimen
with that which would be expected if all the applied work is converted to heat; currently, it
is assumed that the heat capacity is independent of strain. Measuring temperate rises under
high-rate loading is difficult; however, important studies have been conducted by a number of
authors. A notable achievement was by Chou et al. [2], who showed that the temperature rise in
specimens increases significantly after yield. Arruda et al. [25] presented visible increases in strain
softening with increases in strain rate, coupled with corresponding temperature measurements
using infrared detectors. Regev & Rittel [27,28] used thermocouples to measure the temperature
rise in polycarbonate (PC) specimens undergoing high-rate deformation and later confirmed
the accuracy of this method via infrared techniques. Hillmansen and co-workers [29,30] studied
plastic work being converted to heat by studying high-density polyethylene (HDPE) and found
that conversion was approximately 100%, and Garg et al. [31] obtained the same result in PC.
However, Li & Lambros [32], using high-speed infrared detector array at rates around 103s1,
found that the amount of plastic work converted to heat for PC ranged from 50% to 100%
depending upon the applied strain, and data from Mulliken [33] indicate a value of about 80%.
Kendall & Siviour [26] first simulated the high-rate behaviour of a polymer (polyvinyl chloride
(PVC) with 20 wt% plasticizer) through low-rate experiments with temperature profiles, and the
methodology used in that research forms the basis of this paper. Experiments at rates from 1.5
to 3300 s1were simulated at 0.001 s1using a commercial load frame with an environmental
chamber. An initial set of quasi-static experiments over a wide range of temperatures, and a
further set of room temperature experiments over a wide range of strain rates, allowed initial
temperatures to be chosen for each of the simulation experiments, to match the yield stress of
the ‘real’ high-rate tests. Temperature rises were then calculated by assuming a βfactor of 1,
and that the heat capacity of the material is independent of strain rate, strain and temperature.
This assumption requires further investigation; however, the use of this calculation is supported
by the experimental temperature rises discussed above. The rate-dependent response of the
material was successfully simulated in the low-rate experiments up to strains of ca 0.5. In
particular, the significant strain hardening observed in low-rate, low-temperature deformation
was removed by the temperature profiling to give a large strain response which was very
similar to that at high rates. Figure 1 gives an overview of the technique: figure 1ashows a
full set of data from simulating stress–strain relationships over a wide range of strain rates,
while figure 1bshows room temperature medium-rate and low-temperature quasi-static stress–
strain curves with the same yield stress, and the modified low-rate curve which is achieved by
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0 0.1 0.2 0.3
temperature rise (K)
true stress (MPa)
true strain
1450 s–1 drilled
1250 s–1 thermocouple
1470 s–1 thermocouple
1200 s–1
1250 s–1 theoretical
1470 s–1 theoretical
1470 s–1 experimental
1250 s–1 experimental
Figure 2. Measurements of temperature rise in plasticized (20 wt%) PVC. Data from four experiments are shown: one at
1200 s1with an undrilled specimen, one at 1450 s1using a drilled specimen with no thermocouple, and two specimens with
embedded thermocouples, one at 1250 s1and one at 1470 s1.
increasing the temperature to simulate adiabatic conditions. Figure 2 presents a temperature
profile measured using thermocouples during high-rate deformation. Data for all figures are
shown in the electronic supplementary material.
Following the success of this approach, a further programme of research, presented here, has
been performed to investigate polymethylmethacrylate (PMMA) and PC. These two materials
were chosen because of a number of interesting features in their behaviour. For PMMA, the very
low thermal diffusivity means that, for specimens of moderate size, even experiments at 0.1 s1
are adiabatic, and these were simulated using experiments at 0.001 s1. This gives the opportunity
to compare ‘real’ and ‘simulated’ data on a single testing apparatus, without the additional
complications of high-rate testing. Subsequently, a method for measuring temperature rises in
these specimens was developed [34], and the data obtained were used to refine the simulation. For
PC, the effect of different βfactors was investigated, motivated by observations in the literature
that the rate dependence of strain hardening in this material is less than might be expected from
the measured temperature rises. Interesting results were obtained on both materials, and these
are discussed in the context of current literature. As well as the results themselves, this paper
shows how the simulation technique can be used to better elucidate and understand the interplay
between temperature and strain rate in governing the mechanical response of polymers under
large strain deformation.
2. Experimental techniques and results
(a) Experimental techniques
Two testing apparatuses were used in these studies. Quasi-static experiments (up to 0.1 s1)
were performed on a commercial load frame (Instron) with an environmental chamber, using
a standard liquid nitrogen attachment to achieve low temperatures and a clip gauge attached
to the loading anvils to measure the specimen displacement, from which strain was calculated.
The chamber’s thermocouple, which was used to control the temperature, was supplemented by
an additional thermocouple attached to the loading anvil near to the specimen. During constant
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temperature tests, the chamber was allowed to equilibrate to the set point for between 5 and
10 min, by which time the two thermocouple readings differed by less than 1C. For the adiabatic
simulation experiments, the temperature was increased during the experiment by increasing the
set point on the environmental chamber according to the plastic work done on the specimen,
using heat capacities obtained from the literature. Temperatures reported later are from the
thermocouple placed near the specimen.
High-rate experiments (1000 s1and above) were performed in a split Hopkinson bar with
12.7 mm diameter titanium alloy (Ti6Al4V) bars instrumented with foil strain gauges. The input
bar was 1000 mm long, and instrumented halfway along its length; the output bar was 500 mm
long, and instrumented 50 mm from the specimen–bar interface.
(b) PMMA
Experiments were performed on an extruded PMMA rod (rate and temperature dependence
given in [35]). Specimens were right cylinders of length and diameter 8 mm. The data in the
thesis show that PMMA specimens of this size exhibit enhanced strain softening at rates as low
as 0.01 s1, and the implication that the experiments become close to adiabatic at these rates is
confirmed by equation (1.1), which gives a characteristic time of 150 s (κ=0.11 mm2s1)forheat
diffusion out of the specimen. PMMA also presents interesting time- and temperature-dependent
behaviour in that the lower order beta transition occurs at room temperature even at quasi-static
strain rates. The rate dependence of this transition is significantly larger than that of the glass (α)
transition, 25 K/decade strain rate instead of 5 K/decade, and the two transitions begin to merge
at approximately 1 s1(between 10 and 100 Hz) [35].
The first simulation experiment was performed at a rate of 0.001 s1and an initial temperature
of 7.25C, to produce the same yield stress as a room temperature experiment at 0.1s1.The
temperature was increased during the experiment assuming a βfactor of 1 and fully adiabatic
conditions. The stress–strain curve obtained (figure 3) agreed extremely closely with the curve
it was attempting to simulate at 0.1 s1, the differences between the curves being similar to the
variability one would normally expect when testing multiple specimens.
Following this very encouraging result, a number of methods were investigated to measure
the real temperature rise in the specimen at 0.1 s1. Because of the semi-brittle nature of PMMA,
it was not possible to do this using thermocouples inserted into holes in the specimen. Instead
a new thermocouple system was designed which can be sandwiched between two half-length
specimens [34]; this was shown not to interfere with the measured mechanical response of
the specimen and was able to accurately measure the temperature up to true strains of at
least 1. The temperature rise measured was less than that assumed in the initial simulation
experiment, which may be attributed to the βfactor being less than 1 or the conditions not
being fully adiabatic. Further investigations were performed using a thermal camera to look for
temperature gradients in the specimen, none were observed, and replacing the steel anvils with
more insulating, plastic, ones, which made no difference to the measured stress–strain response.
Although these experiments are not wholly conclusive, they indicate that the experimental
conditions are adiabatic, and therefore that either the βfactor is less than 1 or the heat capacity
changes when the specimen yields.
The simulation experiment was therefore repeated using the measured temperature rise, the
result of which is also shown in figure 3. It is clear that using this temperature rise gives less
strain softening and more strain hardening in the simulation experiment compared with the ‘real’
data and the other simulation curve. This indicates that, although not all the work done on the
specimens at the higher strain rate is being converted to heat, it is all contributing to increased
molecular mobility within the material, which requires higher temperature rises to simulate using
heat alone. These observations will be discussed further later.1
1It is noted that in this second simulation experiment more work is done on the specimen, indicating that a slightly higher
temperature rise is justified; however, the difference in work is only approximately 2.5%.
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0 0.2 0.4 0.6 0.8
temperature rise (K)
true stress (MPa)
true strain
0.1 s–1
simulating 0.1 s–1 experimental rise
0.1 s–1 theoretical rise
0.1 s–1 temperature input
0.1 s–1 experimental rise
0.1 s–1 temperature input
simulating 0.1 s–1 theoretical rise
Figure 3. Simulation experiments on PMMA. Comparison of room temperature stress–strain curves at 0.1 s1and two
dierent simulations performed at 0.001 s1, one using the theoretical temperature rise expected for assuming β=1and
(c) Polycarbonate
In previous work, Mulliken & Boyce [12,33] applied a model for the rate-dependent mechanical
response of polymers to a number of different materials. The model [33] included a term to
simulate adiabatic conditions, which could be switched on and off as required. Using this term,
where appropriate, the model was able to capture strain softening, and the difference between
adiabatic and isothermal conditions, in a number of polymers, but not PC. A number of other
researchers [3,11,16,20,27,31,32,36] have studied the rate dependence of PC, and the available
data indicate that, unlike the PMMA and plasticized PVC discussed above, the amount of strain
softening after yield does not increase with strain rate, nor does the amount of strain hardening
decrease: instead, there is a decrease in strain softening, and an increase in strain hardening. These
observations are unexpected: although available measurements of temperature rises in PC under
high-rate deformation show somewhat differing results (§1), the data at rates of around 2000s1
point towards 80% conversion of work to heat in high-rate loading.
These seemingly contradictory observations make PC an ideal candidate for further
investigation using the simulation method.
Experiments were performed on a commercial, extruded PC rod, the loading axis of the
specimens is aligned with that of the rod, and specimens were right cylinders 2.5 mm long and
5 mm in diameter. From equation (1.1), we expect the characteristic time scale for heat diffusion
out of the specimen to be about 10 s (κ=0.11 mm2s1). Quasi-static and high-rate room
temperature data were obtained, supplemented by enough temperature-dependent data to
identify the initial temperature required to simulate the high-rate loading. A comparison of the
quasi-static and high-rate data (figure 4) shows clearly the increase in yield stress owing to
the rate dependence of the lower order beta transition, which in PC is very distinct from the
α, with up to a 100 K temperature spacing between the two transitions [11,33]. Figure 4 also
shows the unusual (compared with PVC and PMMA) softening and hardening behaviour of this
material. It is possible, therefore, that this behaviour is due to a more significant contribution from
the entropic network back stress at high rates [37].
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true stress (MPa)
6400 s–1
4480 s–1
2545 s–1
1175 s–1
0.1 s–1
0.01 s–1
0.001 s–1
yield stress (MPa)
log(strain rate (s–1))
0 0.2 0.4 0.6 0.8 1.0
true strain
Figure 4. Room temperature stress–strain curves for PC, with rate dependence of yield stress inset.
b= 0.1
b= 0.05
b= 0.2
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
temperature rise (K) temperature rise (K)
true stress (MPa)true stress (MPa)
true strain true strain
Figure 5. Results from simulating the PC stress–strain relationship at 2550 s1in experiments at 0.001 s1using four dierent
βfactors. The starting temperature was 7.25C in all cases.
A number of experiments, all at 0.001 s1with an initial temperature of 46C, were
performed to simulate the high-rate response at 2550 s1; the results of these experiments
are shown in figure 5. Initially, a βfactor of 1 (and fully adiabatic loading) was assumed, but the
simulation experiment showed completely different post-yield behaviour to the ‘real’ high-rate
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
energy of deformation (kJ kg–1)
true strain
Figure 6. Energies of deformation in PMMA: W, mechanical work of deformation; Q, heat of deformation; U,internalor
‘stored energy.
data. The assumed βfactor was then decreased a number of times, until eventually, at β=0.05,
the simulated stress–strain curve matched the high-rate data. It can be seen from figure 5 that
this results from a very small temperature rise. This implies that, despite the fact that in reality
nearly all the applied work is converted to temperature rise in high-rate loading of PC, the
material behaves as if the temperature rise is very small, and hence further work is required to
fully understand the mechanisms that govern post-yield response and PC’s lack of sensitivity
to thermal changes owing to the heat created during high strain rate loading.
3. Discussion
The adiabatic simulation technique has now been applied to three polymers—PVC, PMMA and
PC—with very different results. In all three cases, experiments at 0.001s1were used to simulate
higher rate loading by increasing the starting temperature of the experiment and increasing
the temperature during the experiment to simulate the adiabatic conditions in the materials.
Although the simulated rates were different—103to 3300 s1for 20 wt% plasticized PVC,
2550 s1for PC and 0.1 s1for PMMA—in all three cases the beta transition plays a role in the
mechanical response. In fact, for plasticized PVC, the simulations encompass both higher rates at
which the beta transition does affect the response, and lower rates in which it does not.
However, it is clear that there is a difference in the outcomes of the simulation experiments,
which may indicate underlying differences in the mechanics of post-yield behaviour in these
In PVC, a βfactor of 1 was assumed, which is supported by measurements at high rates.
This assumption led to good simulation of high-rate behaviour.
In PMMA, a βfactor of 1 led to good simulation, but was in fact too high; using the real
temperature rise in the specimen led to a simulated stress that was above the real value.
—InPC,aβfactor of 1, although consistent with data from the literature, led to a simulated
stress below the real value. The βfactor had to be reduced to 0.05 for an accurate
simulation; this is below any measured values reported in the literature.
From the PMMA temperature rise data, the energies of deformation can be calculated at a
strain rate of 0.1 s1. The work of deformation is calculated from the stress–strain curve, while
the heat of deformation comes from the temperature rise in the specimen. The remaining energy
is called the internal, or stored, energy [33,38]. Figure 6 shows these three values as functions
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of strain. Approximately 75% of the work is converted to heat; the stored energy increases
rapidly until a strain of about 0.3 and then more slowly for the remainder of the experiment;
this is consistent with Rudnev’s data on PC. It is well known that in quasi-static deformation
of amorphous polymers there is a structural evolution towards a less compact higher energy
structure with lower flow stress, which causes strain softening in the observed response [37].
This evolution is endothermic, and hence in the experiment at 0.1 s1the heat of deformation
is less than the work done on the specimen. When simulating the behaviour at 0.001 s1and
reduced temperature all of this energy must be replaced by thermal energy in order to replicate the
stress–strain curve.
These observations illustrate that a majority of the strain softening seen in the true stress–strain
behaviour is indeed caused by the heat of deformation, but the remainder of the strain softening
is caused by a material characteristic seen only in the higher rate of deformation (0.1 s1), and
which is not accommodated by the rate of deformation in the simulation tests (0.001 s1).
As an aside, the results from these experiments do have implications for quasi-static data in
the literature, and for the use of these data to simulate deformation of structures: even for small
specimen sizes and low strain rates, heating can play a major role in the observed mechanical
response of the specimen to deformation.
For PC, the observations are almost opposite: although the temperature increases significantly
in high-rate loading, according to the simulation technique only a small rise is required to
reproduce the post-yield behaviour observed. It appears logical to suggest that this is associated
with the beta transition in this material, although it must be noted that the very basis of the
simulation method is that both the high-rate and low-temperature data are affected by this
transition, or at least that the yield stresses are. Hence the data indicate that there is some
aspect of the strain softening/post-yield behaviour which may indeed be associated with the
beta motions, for which the interplay between rate and temperature dependence is different from
that of the yield stress, and which must be captured in order to produce a successful high-rate
constitutive model.
4. Conclusion
This paper shows that the adiabatic simulation experiments are able to successfully replicate
higher strain rate properties of three amorphous polymers. Moreover, by performing very careful
simulation experiments, the technique also allows investigation of the nature of post-yield
polymer behaviour at high rates, and its dependence on temperature rises during deformation.
The basic premise of the method is to simulate the high-rate yield by performing a low-rate
experiment at reduced temperature, and then to simulate post-yield behaviour by increasing the
temperature in the low-rate experiment. The required temperature rise can be calculated by taking
the plastic work performed on the specimen and dividing by the heat capacity. This assumes that
all of the plastic work is converted into heat, the so-called heat of deformation; in this case the
so-called β-factor is equal to 1. In practice, some of the work is converted into internal energy,
β<1, which reduces the temperature rise.
Experiments on a plasticized PVC gave results consistent with most, if not all, of the applied
work being converted to heat, and using β=1 gives very good simulation of the rate-dependent
On PMMA, using β=1 gave good simulation of the stress–strain response, but too high
a temperature rise. Using the measured value, β=0.75, does not fully capture the post-yield
deformation: the simulated stress is too high. Hence, in this material, all of the high-rate
mechanical work must be replicated by heat in order to replicate the strain softening, indicating
softening processes that occur in the higher rate response but are not present at lower rates.
In PC, the opposite effect was observed. Although literature data point to βclose to 1, only
very small temperature rises are required to simulate high-rate PC behaviour. The data therefore
indicate that there are aspects of the post-yield behaviour for which the interplay between rate
and temperature is different from that for the yield stress.
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Overall, then, the simulation method is not only able to replicate the high-rate behaviour of
a number of polymers, with clear advantages for experimental characterization of composites
produced from these polymers, but it is also sensitive enough to show important differences
between the expected and observed behaviour. Further work will be required to fully elucidate
the microscopic mechanisms responsible for the observed behaviour and to incorporate these into
appropriate high-rate constitutive models.
Acknowledgements. The authors would like to thank Dr J. L. Jordan and Dr J. R. Foley for their support of the
research, and R. Duffin and R. Froud for ongoing technical support. M.J.K. would also like to acknowledge
Dr I. Dyson for help with low-rate testing, and Dr D. R. Drodge for helpful discussions and suggestions.
The US Government is authorized to reproduce and distribute reprints for Governmental purpose
notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of
the authors and should not be interpreted as necessarily representing the official policies or endorsements,
either expressed or implied, of the Air Force Research Laboratory. The authors thank Dr M. Snyder and
Dr R. Pollak for their support.
Funding statement. Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command,
USAF, under grant no. FA8655-09-1-3088.
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... Polycarbonate (PC) is a typical glassy thermoplastic polymer with ductility, transparency and high impact resistance, and its mechanical properties have been investigated for many decades. These studies included, but were not limited to, investigation of molecular weight effects [1,2], co-monomer effects [3], rate-and temperature-dependence [4][5][6], ageing [1,2] and hydrostatic pressure dependence [7]. However, the thermal or manufacturing history of the material prior to testing can significantly affect the mechanical properties [8], and full mechanical characterisation data from consistently prepared specimens are not available in the literature. ...
... The high-rate performance of PC, from 1000 s − 1 to 10,000 s − 1 , has been extensively investigated [4][5][6]23,24]. Typically, these data again show the effect of the β-transition on the high rate response; however Safari et al. [25] also saw evidence of the γ-relaxation. ...
... An additional complication when considering rate dependence in polymers is that above strain rates of about 0.1 s − 1 , the constitutive stress-strain behaviours can be affected by adiabatic heating, particularly in the large strain regimes [32][33][34]. As the strain rate increases, the experimental conditions change from isothermal to adiabatic because the experiment occurs over a shorter timescale than is required for heat to be dissipated to the environment [35], and the heat generated from plastic work leads to a temperature rise that can cause a qualitatively different constitutive response to that observed in quasi-static tests [4,34]. ...
... The strain-rate at which this observed bi-linearity occurred coincided with the change of experimental apparatus required to perform tests at very high strain-rates, as traditionally Instron mechanical testing machines were used for low-to-intermediate strain rates and a split Hopkinson pressure bar for higher strainrates (Siviour et al., 2005). After further research, it was concluded that this observed bi-linearity is not attributed to the change of experimental apparatus, but is due to lower order relaxation processes of the polymers, which are known to occur at low temperatures, also occurring at room temperature under sufficiently high strain-rates (Kendall and Siviour, 2014). For polymers exhibiting this bi-linear relationship, the yield stress is ...
... It is evident that the PVB yield stress is sensitive to both strain-rate and temperature, with a potential time-temperature dependency similar to many other polymers. Williams et al. (1955) (Kendall and Siviour, 2014;Brown et al., 2007;Jordan et al., 2007;Williamson et al., 2008;Kendall and Siviour, 2013). The application of this approach to temperatures below the glass transition temperature has ...
Full-text available
Laminated glass panels are increasingly used to improve the blast resilience of glazed facades, as part of efforts to mitigate the threat posed to buildings and their occupants by terrorist attacks. These composite ductile panels offer superior blast resistance and result in reduced glass-related injuries, compared to the inherently brittle, monolithic glass, which has historically been used in building facades. This is due to the interlayer’s ability to both provide residual resistance, following the fracture of the glass layers, and retain glass fragments. Therefore, such panels can act as the first barrier of defence during an external explosion to prevent the blast waves from penetrating the building interior and protect occupants. The blast response of these ductile panels is still only partially understood, with an evident knowledge gap between fundamental behaviour at the material level and observations from full-scale blast tests. To enhance our understanding, and help bridge this gap, this thesis adopts a ‘first principles’ approach to investigate the effects of high strain-rate and inertia loading associated with blast loading. The former is studied by developing simplified analytical beam models, for all stages of deformation, with the focus on laminated glass with polyvinyl butyral, as this is the most commonly used interlayer in building facades. The models account for the enhanced properties of both the glass and the interlayer at high strain-rates. This enhances the residual post-fracture bending moment capacity, arising from the combined action of the glass fragments in compression and the interlayer in tension, which is considered negligible under low strain-rates. The post-fracture resistance is significantly improved by the introduction of in-plane restraint, due to the membrane action associated with panel stretching under large deflections. This is demonstrated by developing a yield condition that accounts for the relative contributions of bending and membrane action, and applying the upper bound theorem of plasticity, assuming a tearing failure of the interlayer. To validate the post-fracture capacity predicted by the derived analytical models, three- and four-point bending tests are performed at low temperature on specimens pre-fractured before testing. The pre-fracture ensures controlled and repeatable fracture patterns, and the low temperature simulates the effects of the high strain-rates that result from short-duration blast loads by taking advantage of the time–temperature dependency of the viscoelastic interlayer. A new time–temperature mapping equation is derived from experimental results available in the literature, to relate the temperatures and strain-rates that result in the same interlayer yield stress. The results of the low-temperature tests demonstrate an enhancement of the ultimate load capacity of the fractured glass by two orders of magnitude, compared to that at room temperature. Due to the time–temperature dependency of the interlayer, a similar enhancement is therefore anticipated at the high strain-rates associated with typical blast loading, as predicted by the analytical models. Additionally, comparable moment capacities were observed between the experimental results and the analytical models, and tearing failure consistently occurred in the experiments at the plastic hinge locations predicted by the analytical models. The experiments also demonstrated that the post-fracture moment capacity is unaffected by the number and size of the glass fragments. However, the moment capacity is influenced by the crack alignment between the glass layers, with significantly higher moments recorded for specimens with misaligned cracks compared to specimens with aligned cracks. To investigate the effects of inertia that are known to be significant under the accelerations experienced by a panel during a typical blast event, laminated glass specimens were impacted with polymer foam projectiles, launched from a gas gun. These tests aimed to simulate the loading from a blast pulse. The dynamic bending response was subsequently recorded with a high-speed camera. Nine different test types were performed by varying the boundary conditions, impacting both intact and pre-fractured specimens and testing glass specimens with different cross-section sizes. It was found that the collapse mechanisms formed in laminated glass specimens under short-duration pulses depend on the intensity of the loading. Under high intensity loading, the panel can resist pressures greater than the static collapse load, due to the effects of inertia. This results in a more localised collapse mechanism, compared to quasi-static loading, and explains the repeated failure pattern observed in blast tests of laminated glass panels. The incorporation of axial restraint results in a significant membrane contribution to the response, and therefore reduced deflections.
... Manual input of the temperature rise requires the measurement of the temperature rise during high rate experiments, a significant challenge due to their short duration, and does not give sufficient repeatability or precision. Furthermore, the assumption that all applied mechanical work is converted to heat is not true for all polymers [8] -including polycarbonate, which is why it was selected to be the focus of this technique paper. ...
Full-text available
Polymers are widely used in applications where they may be subjected to impact loading leading to high strain rate deformation. Plastic work on deformation generates heat in the material. At high strain rates, there is insufficient time for this heat to diffuse out of the material, leading to adiabatic conditions. This leads to post-yield thermal softening in the mechanical response, which modifies the material response and must be considered in Engineering design. In this paper, a novel technique is presented in which this adiabatic self-heating can be simulated experimentally at low strain rates using programmed temperature profiles. We show that, in some cases, these simulations can very accurately capture the mechanical response at higher rates, but in others, the replication is less accurate. This may give further insights into the thermodynamics of high strain rate polymer mechanics. This technique therefore enables a number of avenues of research: the work to heat conversion can be investigated systematically; diagnostic tools that are limited to low strain rates can be applied; and we can better understand material behaviour and thereby improve predictive models.
... The stress-strain relationship without confinement shows a compressive strength (peak true stress) of 130 MPa, followed by softening with gradually decreasing stress. A similar decreasing trend of compressive true stress can be found in a series of polymers and composites in the literature [37][38][39][40]. The peak true stress increases from 178 MPa with low confinement to 243 MPa with high confinement. ...
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This paper studies the uniaxial and multiaxial dynamic compressive behaviour and failure of PA66-GF30 (30 wt % glass fibre reinforced polyamide 66), a typical light weight and high strength composite increasingly used in automobiles including the electric cars in applications from ambient temperature to elevated temperatures up to 90 °C. Likewise, the constitutive relation of PA66-GF30 is characterized from quasi-static to high strain rates. Constant high strain rate loading is achieved by pulse shaping technique on a bespoke split Hopkinson bar. The stress-strain relations are pressure, strain rate and temperature dependent. Effects of strain rate and temperature are found to be decoupled on the pressure sensitivity of PA66-GF30. Beyond maximum stress, micro crack has already formed in dynamically deformed PA66-GF30, which corresponds to macro strain localization monitored by high speed photography and digital image correlation techniques. The PA66-GF30 with confinement shows adiabatic shear failure, with fibres coated by severe shear matrix facets and evenly distributed filaments. This is different from the unconfined PA66-GF30 which shows fibres pull out in the fractured matrix. A modified Drucker-Prager model is proposed to describe the pressure dependent compressive strength of PA66-GF30 over various strain rates and temperatures.
Mechanical behavior of acrylonitrile butadiene styrene (ABS) was investigated in this study with experimental and numerical studies. Specimens were fabricated by using a three-dimensional (3D) printer with horizontal and vertical deposition directions. Quasistatic tension tests at strain rates of 0.01/s and 0.1/s and dynamic compression tests at strain rates of 3,500/s, 6,000/s, and 7,500/s were performed at room temperature. Dynamic deformation behavior of test specimen was investigated via 3D Lagrangian method in LS-Dyna software with Johnson-Cook and Piecewise Linear Plasticity material models. Simulations were conducted with an adequate element size determined by a mesh dependency study. Johnson-Cook strength model parameters of ABS were determined and validated by simulation studies. Experimental results showed that increase of strain rate at tension and compression conditions led to increase of strengths for both vertically and horizontally deposited specimens. Under low strain rates, specimens were exposed to strain softening followed by strain hardening. However, under high strain rates, strain softening was observed after maximum stress was achieved. Strain softening behavior observed in dynamic compression tests was attributed to adiabatic heating in the specimen during testing. Furthermore, it has been observed that deposition direction of specimen with regard to applied load direction is significant for the mechanical strength of the material in quasistatic and dynamic tests. Simulation results for dynamic compression tests under high strain rates showed good agreement with test results in terms of final dimensions of test specimens. Results showed that Johnson-Cook material model is more accurate for modelling dynamic behavior of ABS than Piecewise Linear Plasticity model. Additionally, higher error values were observed with increasing strain rates in simulations.
Polymer nanocomposites with excellent mechanical performances have been increasingly sought after in engineering applications such as biotechnology, aerospace, and automotive areas. Through molecular dynamics (MD) simulation, this work systematically assessed the tensile performance of poly(methyl methacrylate) (PMMA) nanocomposite reinforced by randomly dispersed two-dimensional diamond - diamane. It is found that randomly dispersed diamane effectively enhances the tensile properties of PMMA with surface functionalization, and the enhancement effect can be remarkably augmented by cross-linking. Simulations reveal that the enhancement effect can be effectively tailored by the alignment of the diamane fillers. The PMMA nanocomposites exhibit much better tensile performance when the diamane fillers are uniformly aligned along the in-plane direction of the filler. Additional investigations show that larger diamane filler is preferred without cross-linking, while smaller diamane filler should be considered when cross-linking is present. Overall, the impacts of different factors on the tensile properties of PMMA nanocomposites are analysed in-depth in this work, which provides atomistic insights for the preparation of polymer nanocomposites with desired mechanical properties.
Epoxy resins are employed in many engineering applications in which mechanical properties at different strain rates and temperatures must be quantified for effective use. It is known that these properties are related through the principle of time-temperature equivalence. In this paper, the application of this equivalence is explored in detail, at both small and large strains, and used as part of a strategy for calibrating a constitutive model that can reproduce the stress-strain behaviour in different mechanical conditions. In order to achieve this, quasi-static compression experiments were performed at temperatures from −40 to +40 °C in a screw-driven load frame; room temperature rate dependent experiments were performed from 0.0005 to 6300 s⁻¹ using a screw-driven load frame, hydraulic compression system and Split Hopkinson Pressure Bar apparatus. Dynamic mechanical analysis and differential scanning calorimetry were also performed. A three-network (TN) polymer model was fit to the mechanical behaviour up to a strain of 0.5. This model, combined with time-temperature based mappings gave a good understanding of the thermomechanical behaviour of the material, and also demonstrates an approach that can be used for a wide range of polymers.
Due to the outstanding mechanical properties of carbon nanothreads, it is highly expected that carbon nanothreads will be excellent reinforcements for improving the mechanical properties of polymer composites. This chapter aims to provide significant insights into the promising role of carbon nanothreads on reinforcing polymer composites through multiscale modeling. The computational details of molecular dynamics and coarse-grained simulation are introduced. The thermal–mechanical properties of poly (methyl methacrylate) composites under tensile deformation are introduced. The effect of carbon nanothread’s topological structure on Young’s modulus, ultimate stress, interfacial shear stress, glass transition temperature, and friction factor are illustrated, and the associated interfacial interaction mechanisms are concluded. The cavitation and crazing behaviors of polymer composites subjected to tension are also investigated through mesoscale insights, and the manipulation mechanisms of the two behaviors are revealed.
Polymers are frequently used in applications in which they may be exposed to high rate or impact loading and there is growing industrial importance in understanding their mechanical behavior at different strain rates. This paper describes research that has been performed to better understand and predict the dependence on strain rate of mechanical properties such as yield and flow stress, by investigating the relationship between rate dependence and temperature dependence. In particular, the mechanical behavior of poly(vinyl chloride) with 20 wt% plasticizer has been studied at strain rates from 0.001 s−1 to 4700 s−1 and temperatures from −60 to +80 °C. In addition to the use of time-temperature superposition, this paper presents a novel experimental method in which the yield stress and the effects of adiabatic heating at high strain rates are simulated in the low rate loading.
Time–temperature equivalence is a phenomenon common to many time-dependent material systems. This is typically used to generate loci of material properties as a function of temperature and strain-rate. Past work in polyethylene has shown the yield strength to obey a simple empirical temperature/strain-rate equivalence law, with one-decade increase in strain-rate approximately equivalent to a 10 K drop in temperature. This work extends this equivalence relation to continuous stress–strain curves in polyethylene for isothermal deformation states. Large strains are accommodated by invoking adiabatic heating at elevated strain-rates, and equivalence of the isothermal response under adiabatic conditions is checked by constructing a locus of isothermal flow stresses from a series of strain-rate jump tests. Good agreement between isothermal deformation response curves was found. Finally, the system of temperature/strain-rate equivalent curves was validated for large compressive deformations at ∼104 s−1 Taylor impact, with good agreement. Thus, temperature/strain-rate equivalence methods can be used to populate a high strain-rate constitutive description directly from continuous stress–strain curves from the laboratory without extrapolating to those states using an advanced constitutive model.
The use of plastic-bonded explosives (PBXs) in weapon applications requires that they possess and maintain a level of structural/mechanical integrity. Uniaxial tension and compression experiments are typically used to characterize the mechanical response of materials over a wide range of temperatures and strain rates, providing the basis for predictive modeling in more complex geometries. After many years of data collection on a variety of PBX formulations, we have here applied the principles of time-temperature superposition to a mechanical properties database which includes PBX 9501, PBX 9502, PBXN-110, PBXN-9, and HPP (propellant). Consistencies are demonstrated between the results of quasi-static tension and compression, dynamic Split-Hopkinson Pressure Bar (SHPB) compression, and cantilever Dynamic Mechanical Analysis (DMA). Timetemperature relationships of maximum stress and corresponding strain values are analyzed, in addition to the more conventional analysis of modulus. The extensive analysis shows adherence to the principles of time-temperature superposition and correlations of mechanical response to binder glasstransition temperature (Tg) and specimen density. Direct ties exist between the time-temperature analysis and the underlying basis of a useful existing PBX mechanical model (ViscoSCRAM). Results give confidence that, with some limitations, mechanical response can be predicted at conditions not explicitly measured.
The flow stress of many metals is known to vary more strongly with strain rate epsilon above a critical value epsilon c than below it. An analysis of previous authors' measurements on copper indicates that the value of epsilon c varies systematically with specimen size. An estimate of inertial stress shows that inertia may be an important mechanism behind this observation. Qualitative arguments suggest that deformation velocity, friction and wave propagation could also contribute.
A robust physically consistent three-dimensional constitutive model is developed to describe the finite mechanical response of amorphous polymers over a wide range of temperatures and strain rates, including the rubbery region and for impact loading rates. This thermomechanical model is based on an elastic–viscoplastic rheological approach, wherein the effects of temperature, strain rate, and hydrostatic pressure are accounted for. Intramolecular, as well as intermolecular, interactions under large elastic–inelastic behavior are considered for the mechanisms of deformation and hardening. For a wide range of temperatures and strain rates, our simulated results for poly(methyl methacrylate) (PMMA) and polycarbonate (PC) are in good agreement with experimental observations.
In the present study, dynamic compression tests have been performed on polycarbonate (PC) specimen for strain rates in the range of 500 to 2000s^{-1} and on aluminum for a strain rate around 1300s^{-1} using a Split Hopkinson Pressure Bar (SHPB). Temperature measurement has been carried out using a fast response infrared optical pyrometer which is able to measure the surface temperature rise of a specimen during the tests. The temperature data shows that the mechanical work is not completely converted into heat. It implies the existence of a stored energy within the material and possibly a subsequent delay for the conversion of mechanical work into heat.