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Exp Tech
https://doi.org/10.1007/s40799-017-0215-0
Accurate Data Reduction for the Uniaxial Compression Test
J. A. Gruber 1
Received: 26 June 2017 / Accepted: 4 October 2017
# The Society for Experimental Mechanics, Inc 2017
Abstract Raw load-displacement data from the uniaxial
compression test are shown to be contaminated by both machine deflection and seating effects at the specimen surface.
Machine displacement is found to be a nonlinear and reversible function of applied load, insensitive to the displacement
or loading rate, but sensitive to environmental conditions such
as temperature. An empirical function is proposed which provides an accurate fit to machine displacement data. Specimen
seating effects alter the measured load-displacement data by
introducing additional nonlinearity, reducing the elastic loading slope, and shifting the load data to higher displacement.
These effects are shown to originate from non-ideality of the
specimen geometry and are independent of friction. A simple
analytical model to predict specimen seating effects was proposed. The model was then used to determine a robust method
for determining the proper correction to compensate for specimen non-ideality. A new method for compression test data
reduction is proposed that accounts for both machine compliance and specimen seating effects.
Keywords Compression . Mechanical testing . Test
machines . Data reduction . Plasticity
Introduction
Uniaxial compression testing is a simple and effective way to
characterize the mechanical behavior of most engineering
* J. A. Gruber
jason.gruber@unnpp.gov
1
Core Structural Materials Technology, Bettis Atomic Power
Laboratory, West Mifflin, PA 15122, USA
materials. While compression testing has not been as universally employed as other mechanical test methods such as tensile testing, there are significant advantages gained by testing
materials in compression. A typical compression test involves
a relatively small specimen with particularly simple geometry,
usually a right circular cylinder. In comparison, tensile test
specimens typically require far more material; often, the grip
section of a tensile specimen alone is several times larger than
the size required for one or more comparable compression test
specimens. Compression test specimens may be fabricated
from thin plate material along the through-thickness direction,
a test orientation that is usually not viable for other test
methods. The simple test geometry also makes the fabrication
of compression specimens less complex and consequently less
time-consuming than most other test methods.
Uniaxial compression testing involves testing a material
under a particularly simple stress state, matched only by tensile testing in this respect. A tensile test, however, produces a
limited range of data under this condition, as plastic instability
(necking) inevitably forms at low to moderate strains.
Compression testing does not suffer from this problem, and
in fact allows reliable flow stress data to be measured to very
high strains. For this reason, it is especially useful in testing
and constitutive model development for metal forming
processes.
In contrast to other methods, the geometry of a compression test often makes the use of instrumentation to sense displacement (measuring devices directly attached to the specimen) impractical. Specimens are typically small, and the open
space around the specimen becomes increasingly restricted
during the test. Compression testing at moderate to high
strains therefore usually relies on displacement measurements
determined by the machine. For servohydraulic systems, this
is typically provided by a linear variable displacement transducer (LVDT) embedded in the hydraulic actuator, or for
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electromechanical machines, through a rotary encoder attached to a leadscrew.
While such displacement data may be both highly accurate
and precise, they invariably include contributions from both
the displacement of the specimen and from the machine itself.
The latter contribution is due to the machine deflecting under
load, and leads to the concept of Bmachine compliance^ or its
inverse, Bmachine stiffness^ [1–4]. Because only the specimen displacement is characteristic of the piece being tested,
the machine displacement is an unwanted contribution that
must be removed. Attempting to correct raw compression test
data for machine deflection has become common practice,
although not all commonly used methods produce valid results. In particular, the machine response to applied loads is
nonlinear, and an accurate correction must account for this.
This work presents experimental results and discussion
pertaining to machine displacement first.
Correcting for machine compliance has been the main focus of previous data reduction methods. However, another
aspect of the problem has been largely ignored. Seating effects
at the test specimen may also contribute to discrepancies between the expected and the Bcompliance corrected^ specimen
displacement. This is because imperfections in the specimen
itself – non-parallelism or non-planarity of the specimen ends,
surface roughness induced by machining or grinding – may
result in a load-displacement characteristic that is measurably
different than that expected for a perfect specimen of the same
nominal geometry. Regardless of how well the machine compliance is characterized, imperfections in the specimen geometry can lead to three additional artefacts in the compliance
corrected data, namely an initial nonlinearity in the displacement, a reduced loading slope in the elastic region, and
a shift in the displacement. These effects will be demonstrated through detailed numerical modeling, experimental data, and theoretical calculations. We will show
that removing the machine displacement component
from the raw displacement measurement, while a necessary step, does not fully correct all unwanted features
found in the data.
The purpose of this report is to critically assess the various
methods used for reducing raw uniaxial compression test data,
and to develop an improved method based on new observations and theory. The goal of data reduction is to transform the
raw data into a form that most resembles what would be measured if the test machine was perfectly stiff, the specimen had
an ideal cylindrical geometry, and the interaction between the
specimen and platens was frictionless. In this ideal case, the
constitutive relationship for the material may be inferred from
the experimental data exactly. The methods described here
provide a solution to the first two aspects of the problem,
but do not address friction effects. We will show, however,
that corrections for machine compliance and specimen irregularities are largely independent of friction.
This work is relevant to compression testing where it is
impossible or impractical to independently measure specimen
displacement. Standardized test methods [5, 6] exist for cases
where instrumentation, such as an extensometer, are used to
measure specimen displacement. Those methods advocate
specimen geometries that are of limited use in measuring mechanical behavior at high strain because their large cross sectional area would lead to excessively high loads at large displacement, their high aspect ratio would lead to buckling instability, and/or the closure of the compression platens on an
extensometer would limit the total strain that may be achieved.
The examples and discussion presented here are limited to
behavior typical of metals, but may be applicable to other
material systems as well.
Methods
This work includes both experimental test data and the results
of numerical simulations. A general summary of our test system and numerical methods is presented here.
Experimental All experimental results were obtained through
testing on a 100 kN load capacity Instron 8821S servohydraulic
test machine. This machine is fitted with a high-vacuum environmental chamber capable of temperatures in excess of
1000 °C and vacuum better than 1 × 10−5 Torr. Room temperature testing is performed at atmospheric pressure in air.
Elevated temperatures are achieved by resistive heating of tungsten elements in vacuum.
Below the upper crosshead is a 100 kN Instron AlignPRO
alignment fixture, followed by a 100 kN Instron Dynacell load
cell. The load cell is calibrated to meet the requirements of
ASTM E4 [7]. Typical loading in our tests does not exceed
about 40 kN in compression. Below the load cell is a custom
imperial-metric coupling (7.62 cm length, 6.35 cm diameter,
4140 steel) and a hardened steel pushrod (20.32 cm length,
3.175 cm diameter) with cooling channels that extends into
the upper opening of the environmental chamber. This actively cooled pushrod prevents heating of the load cell and other
components external to the environmental chamber. Attached
to this is a Nimonic 100 high-temperature nickel based alloy
collet (Nimonic 100), holding a cemented tungsten carbide
pushrod (15.24 cm length, 2.438 cm diameter, Vista Metals
grade VM-17). All connections are threaded, with the exception of the tungsten carbide pushrod, which is held within the
collet by eight set screws. The tungsten carbide pushrod is the
only component of this train that is directly exposed to the
heated zone of the environmental chamber.
Attached to the lower crosshead is a hydraulic actuator,
which contains an integral LVDT from which total displacement is measured. The actuator has a total travel of 12.7 cm,
and is capable of a maximum displacement rate of
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approximately 4 cm/s. The output resolution of this LVDT is
0.0001 mm, and from calibration is found to achieve better
than 1% absolute accuracy over its entire range. From the
hydraulic actuator is a hardened steel pushrod with cooling
channels (20.32 cm length, 3.175 cm diameter) that extends
into the lower opening of the environmental chamber. Within
the environmental chamber and attached to the end of this
pushrod is a high-temperature nickel-based alloy collet
(Nimonic 100), holding a cylindrical, cemented tungsten carbide pushrod (16.51 cm length, 4.76 cm diameter, Vista
Metals grade VM-17). As with the upper portion of the load
train, all connections are threaded with the exception of the
tungsten carbide pushrod, which is held within the collet by
eight set screws, and similarly, the tungsten carbide pushrod is
the only component of this train that is directly exposed to the
heated zone of the environmental chamber.
Finally, between the upper and lower tungsten carbide
pushrods are two reaction-bonded silicon nitride discs
(0.5 cm length, 1.27 cm diameter) that act as upper and lower
platens, contacting the test specimen. Silicon nitride has an
exceptionally high modulus (>300 GPa from room temperature to 1000 °C), and provides a machine/specimen interface
that is a stiff and highly polished surface. It also prevents
diffusion bonding between metallic test specimens and the
cemented tungsten carbide pushrods, or between the pushrods
themselves when no specimen is present.
Some compression test data has been measured specifically
for this study. In all cases, the material used is beta-quenched
Zircaloy-4. This material is isotropic and exhibits typical
power-law type strain hardening behavior at room temperature. All compression specimens have nominal dimensions of
0.5 cm length and 0.5 cm diameter.
Simulation Various test cases were simulated using the finite
element method (FEM). These calculations were performed
using the Abaqus simulation package from Dassault Systèmes
Simulia. The compression test is modeled using an elasticplastic, nominally cylindrical test specimen compressed between two rigid, parallel planar surfaces, Fig. 1. The model
specimen has the same dimensions as our real compression
specimens, with 0.5 cm length and 0.5 cm diameter. The model uses second-order, reduced integration hexahedral elements
and nonlinear geometry correction. In some simulations, the
specimen geometry is made to deviate slightly from a perfect
cylinder. The mesh topology shown in Fig. 1 was used in all
simulations; this minimizes possible effects due to different
mesh density or element type when comparing different simulation results. Contact between the specimen and the platen
surfaces is modeled as standard isotropic Coulomb friction
with various friction coefficients. Because the compression
platens are modeled as rigid bodies, the test machine itself is
effectively modeled as perfectly rigid and free of compliance.
All deviations from ideal behavior in our simulations are
Fig. 1 Finite element mesh of the uniaxial compression test. Elements
are colored according to computed von Mises stress. This simulation used
an elastic-plastic model for Alloy 690 with a total displacement of
0.125 mm and a displacement rate of 0.125 mm/s, with Coulomb friction
coefficient μ = 0.1
therefore due to the deformation behavior of the test specimen,
as well as interactions between the specimen and the platens.
Simulated compression tests are performed with a constant
displacement rate.
A constitutive model for the test specimen was chosen that
represents typical metallic behavior at room temperature. The
model assumes an elastic modulus E = 216.6 GPa and
Poisson’s ratio ν = 0.28. The plastic model assumes an isotropic, von Mises flow rule, with a rate-dependent, JohnsonCook model form for hardening,
0:52 1 þ 0:0085lnε˙ p ð1Þ
σ ¼ 209:30 þ 1109:27 εp
where σ is the von Mises flow stress (MPa), εp is the plastic
contribution to the von Mises strain (mm/mm), and ε̇ p is the
rate of change of the plastic von Mises strain (mm/mm/s). This
model is based on measured elastic and plastic deformation of
the nickel-based Alloy 690 at room temperature.
Results and Discussion
Machine Displacement – Observations It is useful to first
consider the behavior of a test machine with an applied compressive load, but with no specimen. At a minimum, universal
test machines used in compression testing include upper and
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lower crossheads, and between them, a load cell, compression
platens, and various other linkages. These components are
appropriately sized to withstand a range of applied loads without plastically deforming. The term Bmachine compliance^
invokes the idea of linear elastic response, but as discussed
by Kalidindi et al. [2], typical machine response is nonlinear.
They demonstrated this fact by loading a test machine in compression with no specimen present.
This conclusion is supported by our data as well. Figure 2
shows several examples of the measured load-displacement
response of our Instron 8821S servohydraulic test machine
when no specimen is present. The tests shown here all begin
with a finite separation between the compression platens and,
consequently, zero load, followed by motion of the actuator to
bring the platens into contact. Actuator motion, and the associated displacement as measured by the integral LVDT, then
increases while an increasing load is measured by the load
cell. However, by design no specimen displacement occurs
between the platens after initial contact, and therefore the test
provides a measurement of the machine displacement as a
function of applied load. After reaching some predetermined
load, the actuator motion is reversed and unloading exhibits a
similar load-displacement relationship, although some hysteresis is common. This process is repeatable and follows the
same load-displacement path regardless of the maximum load
attained. These results demonstrate that the machine response
is in general nonlinear, reversible, and path-independent.
The reversibility and path independence of the machine
displacement suggest that the observed behavior is dominated
by elastic deformation. However, the simple view of the load
train as a series of elastic springs implies a linear response.
Kalidindi et al. noted this discrepancy and suggested that the
machine displacement consists of two regimes, with one dominated by Bnonlinearity associated with the many connections
and/or linkages typically present in the loading system^ [2],
Fig. 2 Measured machine displacement (no specimen present) as a
function of applied load for displacement rates of 0.0025, 0.025, 0.25,
and 2.5 mm/s. A linear fit to the data at high loads provides the parameters
K and dns of the model given in (equations (2-3))
and the other dominated by linear elastic behavior. We agree
with this assessment, but further propose that the nonlinear
behavior is specifically a result of the seating of various contact surfaces in the load train. This might occur, for example,
at threaded connections, or between flat mating surfaces that
are nonparallel in the unloaded condition. Even between
contacting surfaces with good geometrical fit, surface roughness effects may provide a measurable contribution. We will
demonstrate analogous effects of seating or contact of the test
specimen in the following section and show that a nonlinear
elastic response is one result. We make this distinction here
because it is clear that the observed behavior is essentially a
characteristic of the machine (as no specimen is present),
whereas Kalidindi et al. state that Bthe magnitude of this [nonlinear] effect is very sensitive to sample geometry, sample
placement and the type of lubrication^ [2]. This statement
confounds the effects of seating within the load train, a characteristic behavior of the machine, with the same effects at the
test specimen ends, which will be shown to depend on irregularities in the specimen geometry. Similarly, some authors
have attempted to study the characteristic behavior of test
machines, i.e. the Bmachine stiffness,^ by analysis of test data
with compression specimens present, and have invariably
found the results to have poor repeatability. Hockett and
Gillis [3, 4] used this method and computed machine stiffness
coefficients that varied significantly with material type, test
temperature, and load at yield. Kalidindi et al. [2] describe a
method for determining machine displacement directly by
comparing final specimen height to measured displacement
(their BMethod III^), but found greater scatter in the resulting
data than when compared to measurements with no specimen
present. In each case, the authors have analyzed their test data
without accounting for the additional effects due to specimen
seating.
Compression testing is often performed at a wide range of
strain rates and temperatures, and so it is useful to understand
how these variables impact the characteristic machine displacement. Figure 2 shows results of room temperature machine displacement tests performed at different displacement
rates, which are nearly indistinguishable. As expected based
on the assumption of elastic, path-independent behavior, the
characteristic machine displacement is found to be independent of displacement rate. There are cases, however, where the
observed machine displacement may appear to depend on the
displacement rate, such as if the programmed displacement
data acquisition rate is too slow for the test conditions, or if
data processing techniques, for example noise filtering, are
chosen to be too aggressive. Machine displacement tests
should be performed over a range of displacement rates to
ensure that these effects are negligible.
Figure 3 shows results of machine displacement tests performed at various temperatures. Test temperature is shown to
have a measureable effect on characteristic machine
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Machine Displacement – Model Our model for the machine
displacement, dm, assumes that the load train consists of a series
of bulk material regions that behave like perfect linear elastic
springs, interspersed with regions where nonlinear effects due to
seating and redistribution of stresses occur. The contributions to
dm from individual regions are assumed to be additive, and so
d m ¼ d linear þ d nonlinear ¼ K P þ d nonlinear
Fig. 3 Measured machine displacement (no specimen present) as a
function of applied load at temperatures as indicated. All tests were
performed with a displacement rate of 0.25 mm/s and 1 h soak times at
temperature
displacement for our machine. This is expected, as several
inches of each pushrod are within the heated zone in our
environmental chamber. This behavior will be strongly machine dependent, and will also depend on the particular
heating schedule, as the temperature profile of larger machine
components may take longer to equilibrate than the temperature of test specimens. Therefore, the machine displacement
should be determined at every test temperature of interest, and
should not be assumed from testing at room temperature. For
high temperature testing, characterizing the nonlinear machine
behavior at low loads becomes even more important. This is
because, for a given specimen size, the loads required to deform the specimen will decrease with increasing temperature,
and therefore more of the measured load-displacement data
will include nonlinear effects from the machine. Where practical, machine displacement tests should also reproduce the
detailed heating schedule and soak times that will be applied
during compression testing.
Even with a carefully executed machine displacement test,
the resulting load-displacement is only an approximation to
the true machine characteristic. This is because one pair of
mating surfaces – the upper and lower platen surfaces – is
not present in an actual compression test. It is instead replaced
by two contact areas: one between the top of the specimen and
the upper platen, and one between the bottom of the specimen
and the lower platen. It is therefore worthwhile to ensure that
the platen surfaces are as flat and parallel as possible, so that
seating of these surfaces during a machine displacement test
has as small a contribution as possible. If the actual surfaces
that will contact the specimen can be removed (as with our
silicon nitride discs) and a machine displacement test can be
performed without them, a comparison of the two results provides an indication of the size of this contribution. For our
system, the difference is less than the measurement resolution.
ð2Þ
The term dlinear is the combined contribution of all regions
that behave like linear elastic springs, with K an effective
linear machine compliance, and P the applied load. The contributions to machine displacement from all regions behaving
in a nonlinear way are combined into the term dnonlinear. In our
machine displacement tests at room temperature, we observe a
total nonlinear contribution of about 0.025 cm over at least 11
distinct contact regions. Clearly, dnonlinear results from very
complex stress and contact relations, on length scales that
make accurate characterization of the relevant geometries,
and subsequent detailed analyses impractical. The term dnonlinear must therefore be fit using test data like that shown
in Fig. 2. Because dnonlinear tends to increase rapidly upon
initial loading but saturate at high loads, we find that the following expression provides a reasonably good fit to measured
data in most cases:
d nonlinear ¼ d ns ½1−expð−k Pm Þ
ð3Þ
The term dns is the Bnonlinear saturation^ value, k and m
are fitting constants, and P is again the applied load.
The model described here has the property that at high
loads, dm is an approximately linear function of load, while
at low loads, dm is dominated by nonlinear effects, and at zero
load, dm is zero. Although the model is purely empirical, the
expression given here provides a reasonable balance between
reproducing the observed behavior, while still being simple
enough to fit with linear methods. In particular, because dnonlinear → dns as load increases, approximate values of K
and dns may be taken as the slope and intercept of the tangent
to dm(P) at high load, Fig. 2. After estimates of K and dns are
determined, if the quantity
K P−d m
ln −ln
þ1
d ns
ð4Þ
is plotted as a function of lnP, the result may be fit with a linear
approximation. The slope provides the parameter m and the
intercept gives lnk. This approximate fit of dm(P) may then be
fine-tuned to fit the raw machine displacement data. It should
be noted that the measured displacement and load data in a
machine displacement test may require small shifts depending
on how accurately the zero points of each are established prior
to the test. A benefit of fitting an analytical expression like that
provided here, rather than an arbitrary form dictated by the test
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data, is that this zero crossing is strictly enforced.
Figure 4 demonstrates the quality of fit achieved when
using (eqs. (2–3)). Over the range of applied loads, the fit
approximates the measured displacement to within about
±0.001 mm. This is nearly within the limit of precision of
the LVDT, and should be adequate for most applications. If
a smaller range of loads is sufficient to cover all test cases, the
machine displacement may be fit over a reduced dataset and
this will generally improve the quality of fit even more.
Seating Effects at the Test Specimen – Observations A
machine displacement test (compression with no test specimen present) provides an accurate estimate of machine deflection under load, and therefore can be used to estimate actual
specimen displacement from the total measured displacement.
Figure 5 shows an example of load-displacement data measured in a compression test, compared with the same data
when corrected for machine displacement. Subtracting
machine displacement has resulted in visibly less initial
nonlinearity, an increased loading slope, and has shifted
the test data such that higher loads occur at lower displacement. However, not all nonlinearity has been removed. Likewise, the loading slope during elastic loading is lower than that estimated by linear elasticity theory. A higher linear contribution to the machine displacement would be needed to account for this discrepancy, but based on the observed reproducibility of the
characteristic machine displacement, it’s unlikely that
such a change is justified.
As noted above, the addition of a compression test specimen introduces two new pairs of mating surfaces, which may
lead to additional nonlinearity in the initial load-displacement
response. This is why applying a correction for machine displacement, as in Fig. 5, rarely eliminates all of the nonlinearity
present in the test data. A less intuitive consequence is
that the loading slope (elastic region) of the test data is
Fig. 5 Initial load-displacement data from a uniaxial compression test of
Zr-4 at 0.25 mm/s. The dashed line shows the expected load-displacement
behavior based on specimen geometry and known elastic modulus
typically lower than what would be expected based on a
known elastic modulus. In addition, the nonlinearity has
the effect of adding a shift to the measured displacement. If not removed, this shift could lead to errors in
the computed stress-strain relation at all strains, not just
at low strains where the effect is most apparent.
As a simple demonstration of the seating effect, we have
performed finite element method (FEM) simulations of the
uniaxial compression test with both perfect and imperfect
specimen geometries. The compression platens are modeled
as rigid bodies, and therefore dm = 0 by construction. All deviations from ideal behavior are therefore due to the deformation behavior of the test specimen, as well as interactions between the specimen and the platens. Figure 6
shows verification that with a perfect specimen geometry and no friction, the FEM model produces a loaddisplacement response that closely matches that calculated explicitly from the chosen constitutive model. Also,
as expected under such conditions, there is no initial
nonlinearity, and the initial loading slope satisfies the
following approximation to linear elastic response,
P ðd Þ ¼
Fig. 4 Difference between measured and fit (equation (2)) machine
displacement at room temperature
A0 E
d
L0
ð5Þ
where P is the applied load, A0 is the initial cross sectional area
of the specimen, d is the specimen displacement, and L0 is the
initial specimen length. The elastic modulus that might be
inferred from the simulated load-displacement data agrees
with the model modulus to within less than 0.01% error.
Also shown in this figure is the result of simulations with
identical test conditions, except with nonzero friction
coefficients. The effect of friction is negligible in the
elastic loading region, but becomes significant after the
onset of plastic strain.
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Fig. 6 Computed load-displacement response of Alloy 690 at 0.125 mm/
s displacement rate. The compression specimen is a perfect, right circular
cylinder, and the Coulomb friction coefficient takes values of μ = 0.0, 0.1,
0.2, 0.4, and 0.8. Solid line is the load-displacement behavior computed
analytically based on the assumed constitutive model, and matches the
FEM result with μ = 0.0. Dashed line is the expected elastic response
based on specimen geometry and elastic modulus
Nonlinearity can be shown to be the result of seating effects, which is essentially a result of non-ideal specimen geometry. A simple case that is amenable to FEM modeling is to
assume one end of a cylindrical test specimen is planar, but its
normal is not coincident with the cylindrical axis, Fig. 7. The
other specimen end is perfectly flat and parallel to the
contacting platen. The parameter h shown in Fig. 7 is
varied from zero (no tilt) to 0.00635 mm, and simulations are performed under otherwise identical conditions.
Figure 8 shows the result of assuming increasing Btilt,^
as characterized by the parameter h. The consequences
of this particular geometry are an initial nonlinear loading region, followed by a loading slope that is significantly lower than that observed for the ideal geometry,
and a shift in the load-displacement response that extends throughout the entire dataset, including the plastic region. Figure 9 shows similar results with a single tilt parameter
h, but with various friction coefficients. Friction is seen to
have no measurable effect on the initial nonlinearity or the
loading slope.
Fig. 7 BTilted^ (left) and
Brough^ (right) specimen geometry definitions
Fig. 8 Computed load-displacement response of a Btilted^ Alloy 690
compression specimen, with h parameter as indicated. All simulations
used a displacement rate of 0.125 mm/s and a Coulomb friction coefficient μ = 0.1. Dashed line is the expected elastic response based on
specimen geometry and elastic modulus
Another form of non-ideality is surface roughness. Even if
contacting surfaces are macroscopically flat and parallel, surface roughness creates a distribution of surface heights that
together may result in effects similar to the Btilted^ specimen
described above. This idea was explored through the characterization and testing of Zircaloy-4 compression specimens
with various surface preparations. Figure 10 compares images
of the ends of two Zircaloy-4 specimens, each produced by
facing on a lathe. The difference in surface quality is due to
differences in speed and depth of the cut. The height distribution of each surface was determined using a Keyence VKX200 laser scanning microscope, Fig. 11. The roughness averages for the two surfaces were determined to be Ra =
2.00 μm and Ra = 3.98 μm, respectively, which are typical
of turned surfaces [8, 9]. Figure 12 shows a comparison of
measured load-displacement data (corrected for machine displacement) for these two specimens. There is a measurable
difference in the extent of the nonlinear loading region and
the loading slope. The loading slope of the low roughness
specimen almost matches the expected slope (based on an
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Fig. 9 Computed load-displacement response of a Btilted^ Alloy 690
compression specimen with h = 0.0635 mm and friction coefficient as
indicated. Dashed line is the expected elastic response based on specimen
geometry and elastic modulus
Fig. 11 Surface height probability densities of Zr-4 compression specimen ends. The average and median of each distribution are shown by
solid and dashed vertical lines, respectively
assumed modulus E = 99.3 GPa), while the slope of the high
roughness specimen is considerably lower.
Assuming that the qualitative features of these results apply
to other geometries as well, what is the larger significance?
First, the elastic modulus itself, as inferred from a noninstrumented compression test, is likely to be an underestimate
of its true value. The slope of the load-displacement curve depends on both the nominal specimen geometry, and as shown
here, the magnitude of any deviations from this geometry.
Because these deviations are small, they are difficult to characterize accurately for any particular specimen prior to testing,
and it is usually impractical to model the test explicitly, as done
here. An accurate estimate of the elastic modulus under compression may only be possible in cases where the specimen
geometry and precision in manufacturing are taken into account, or when instrumentation is used. The next section proposes criteria for selecting the specimen geometry when measuring the elastic modulus in compression.
Another consequence of a reduced loading slope is that the
method of measuring machine compliance (usually assumed
to be a constant) by matching the loading curve to the expected linear elastic response will generally result in a machine
compliance that is too high. This is because the raw data
contains artefacts from both the machine deflection and seating at the specimen surfaces, while this method for measuring
machine compliance ignores the latter. Similarly, attempting
to apply a Bcompliance correction^ to raw data based on
matching the loading curve to the linear elastic response will
typically result in an overcorrection everywhere beyond the
linear elastic region.
If the elastic modulus of the material is already known, the
load-displacement relationship in the linear elastic region may
be assumed to follow (equation (5)). The effect of a shift in the
load-displacement response appears to be the most problematic, as it leads to the load being underestimated throughout
the entire test. Fortunately, it appears that this problem can be
Fig. 10 Zr-4 compression test
specimen end surface finishes.
Low roughness (a) and high
roughness (b) surfaces, with Ra =
2.00 μm and Ra=3.98 μm,
respectively
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behavior as the bulk. Taken together, they approximate the
true height profile of the surface. However, because there is
no interaction between separate columns, the deformation behavior of each is essentially that of an ideal compression specimen and may be computed exactly. As the platen above the
tilted surface moves into the specimen, it contacts an increasing number of these columns, and is therefore subjected to an
increasing load. The measured load P on the compression
platen as a function of total platen displacement d is then an
integral over the loads due to each column. Assuming the
deformation is purely elastic, this gives
Pðd Þ ¼ ∫ E εlocal ðx; y; d Þ dA
A
Fig. 12 Corrected load-displacement data for Zr-4 compression specimens of different surface roughness. Dashed lines are computed loaddisplacement response based on the surface height distributions shown
in Fig. 11 and the elastic seating model given by (equation (13))
corrected by simply shifting the displacement data by a fixed
constant, Fig. 13. For the titled specimen geometry studied
above, the correct shift is equal to the geometric parameter h. After shifting each curve by this amount, the
load-displacement behavior at displacements greater than
about 2h are in excellent agreement with the ideal case.
Unfortunately, in a real compression test the appropriate shift
would be unknown without detailed characterization of the
specimen geometry. Moreover, other geometries may require
a different approach, and so some other method must be developed to determine the correct shift. This point is addressed
in the following section.
Seating Effects at the Test Specimen – Model It is useful to
develop a simpler model description to predict more general
behavior, and in doing so, we will arrive at a practical method
for estimating the appropriate shift. We apply it first to the
Btilted^ specimen geometry shown in Fig. 7. The main assumption in our model is that the entire specimen behaves like
a collection of independent columns, each with perfectly flat
and parallel ends and infinitesimally small cross sectional area
dA. Each column is assumed to have the same constitutive
A0 E
P ðd Þ ¼
L0
2h
π
A0 E
ðd−hÞ
L0
ð11Þ
ð6Þ
where dlocal(x, y, d) is the local displacement of the material at point (x, y) on the surface when the platen displacement is d. Because the nominal specimen length L0
is much greater than local deviations from L0, the local
column length L0, local is approximated everywhere as
L0, making the integral easier to solve explicitly. The
integration area A may be taken to be that portion of
the surface that is in contact with the platen at the given
displacement, as dlocal is zero elsewhere and does not
contribute to the load. For the Btilted^ geometry shown in
Fig. 7,
h
d local ðx; y; d Þ ¼
x−ðh−d Þ
ð7Þ
R0
for any point in contact with the platen. Here, R0 is the initial
specimen radius. For a platen displacement of d, the points in
contact
with
the platen satisfy
d
ð8Þ
R0 1−
≤ x ≤ R0
h
qffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffi
− R20 −x2 ≤ y≤ þ R20 −x2
ð9Þ
and provide the appropriate integration limits. The final result
of the integration may be simplified to
" 3=2 " 1=2 ##
1 d
d 3=2 1
d π
d
d
d 1=2
d
2−
−
2−
−asin 1−
1−
− 1−
3 h
h
2
h 2
h
h
h
h
This expression provides meaningful values when d ≤ 2h.
When d > 2h, i.e. when the tilted surface is in full contact with
the platen, the appropriate expression is
P ðd Þ ¼
d local ðx; y; d Þ
E
dA≈ ∫ d local ðx; y; d Þ dA
L
L
A
0;local
0A
¼E∫
ð10Þ
showing that the load-displacement curve eventually achieves
the slope predicted for an ideal specimen (equation (5)), although shifted in displacement by h. The result of this model
is compared to results from FEM simulations in Fig. 14. This
simple analytical model provides a reasonable estimate of the
FEM result, especially when the deviation from ideality is
Exp Tech
Fig. 13 Load-displacement data of Btilted^ compression test data of Fig.
8. Each curve has been shifted by the geometric parameter h (see Fig. 7)
small. In fact, even better agreement can be achieved if we
relax the assumption of a purely elastic response and include
plasticity. In this case, we determine the relationship between
the local stress and displacement from the elastic-plastic constitutive model, and numerically integrate
Pðd Þ ¼ ∫ σlocal ðx; y; d Þ dA
A
ð12Þ
The results of the elastic-plastic seating model are also
shown in Fig. 14. This model provides a surprisingly accurate
estimate of the load-displacement behavior of the tilted compression specimen, even beyond yield.
A comparison of the elastic and elastic-plastic models
and their relation to the FEM data in Fig. 14 helps to
explain the observed behavior in the FEM data. First,
the good agreement of both elastic and elastic-plastic
models at very small displacements suggests that the
Fig. 14 Load-displacement response of Btilted^ Alloy 690 test cases
from FEM simulation (open points), elastic seating model (dashed line),
and elastic-plastic seating model (solid line)
initial nonlinearity is due to a purely elastic seating
response. For large deviations, local plastic deformation
during seating may become significant at relatively
small displacements, and consequently the loading slope
is lower than that predicted by the purely elastic model.
Conversely, if the deviation from an ideal geometry is
relatively small, the entire loading curve may be dominated by nonlinear elastic behavior. It may even be possible to extract the correct elastic modulus, assuming
that sufficient care has been taken in correcting for machine displacement. This is possible when the magnitude of the deviation is small compared to the displacement at yield.
The successful application of our simple model in this
case suggests that the treatment of the specimen as a collection of independently deforming columns may be a
reasonable approximation. In fact, because each column
is assumed to behave independently, we can divorce the
model from any particular surface geometry and instead
work with height distributions. This allows us to extend
the model to treat surface roughness, which is perhaps the
most likely cause of deviation from an ideal compression
specimen geometry. For this model, we assume the same
height distribution p(z) on each surface, Fig. 11. The function p(z) is a probability density and is nonzero between
the limits −h/2 and +h/2. For a purely elastic model, the
load is calculated as
2A0 E h=2
∫
d local ðz; d Þ pðzÞdz
L0 ðh−d Þ=2
2A0 E h=2
h−d
¼
pðzÞdz
∫ðh−d Þ=2 z −
L0
2
P ðd Þ ¼
ð13Þ
The factors of 2 occur throughout this expression because the model now includes both top and bottom specimen surfaces. This expression can be numerically integrated using the height distributions shown in Fig. 11. The
resulting load-displacement predictions for the Zircaloy-4
specimens are shown in Fig. 12 along with the experimental data. Much like the elastic seating model predictions for
the tilted specimen in Fig. 14, there is good agreement
between the model and the experimental measurement at
low loads. Deviations occur at higher loads, especially for
the high roughness specimen, presumably due to the onset
of local plastic deformation which is not addressed by the
elastic model.
Now, we wish to use the model to develop a method
for estimating the appropriate amount to shift the data.
The main difficulty is that we assume no prior knowledge
of p(z). Surprisingly, this is possible with only one additional assumption, namely that the mean and median of
p(z) are approximately the same. This is true for any symmetric distribution, such as the Btilted^ geometry, but also
Exp Tech
approximately true for our test specimens (Fig. 11) and
for surface roughness profiles in general [9].
To begin, we compute the required shift for an arbitrary
distribution p(z). Given a total displacement of d, the average
displacement experienced by the specimen is
h=2
d local ¼ 2∫ðh−d Þ=2 d local ðz; d Þ pðzÞdz
h− d
h=2
¼ 2∫ðh−d Þ=2 z −
pðzÞdz
2
ð14Þ
When d = 2h, the specimen ends fully contact the platens, and
this reduces to
h=2
h=2
d local ¼ 2∫−h=2 z pðzÞdz þ h∫−h=2 pðzÞdz ¼ 2z þ h
ð15Þ
The difference between the applied displacement d and the
average displacement experienced by the specimen dlocal is
the desired shift,
d shift ¼ 2h − ð2z þ hÞ ¼ h − 2z
ð16Þ
When the height distribution p(z) is symmetric, z = 0 and
the shift is just h. This is a generalization of the result
already shown for the tilted specimen geometry. When
the height distribution is strongly skewed toward +h/2,
z → h/2 and dshift → 0; the specimen approaches the ideal geometry. Conversely, when the height distribution is
strongly skewed toward −h/2, dshift → 2h; the specimen
approaches an ideal cylinder with height L0 − 2h. Note
that the value of dshift is completely general; it does not rely on
any assumptions about the material behavior or the form of
p(z) itself.
Next, we examine the behavior of the purely elastic model
when d = dshift. In the simulated and experimental tests performed here, the elastic model provided a reasonable fit to the
measured data at small loads or displacements. Instead of
computing the load at dshift, which depends strongly on the
form of p(z), we take the derivative under the integral with
respect to d to find
is exact whenever the distribution p(z) is symmetric.
This result provides a practical way to estimate dshift. On a plot
of load-displacement data (corrected for machine displacement),
the point where the curve has slope A0E/2L0 is approximately
dshift. Equivalently, after the load-displacement data has been
shifted by the correct amount, the slope at d = 0 is A0E/2L0,
Fig. 15. More precisely, this point locates the median of the
height distribution p(z), whereas the actual value of dshift occurs
at the distribution mean. For the tilted specimen geometry, this
method provides the exact value of dshift because p(z) is symmetric. For the Zircaloy-4 test specimens examined here, the mean
and median of each distribution differ by less than one micron
(Fig. 11), and therefore this method will estimate dshift accurately
to within about a micron. This method of determining dshift is also
robust, in the sense that errors in locating the correct zero crossing
(shifts along the displacement axis) will be remedied naturally.
Assessment of Data Reduction Methods
Equipped with the observations and theory presented above,
we revisit some previously proposed methods for compression test data reduction.
Matching Elastic Loading Slope This method essentially
assumes that the test machine behaves as a linear spring. A
machine stiffness K is determined through compression testing of specimens with known elastic modulus by [1, 3, 4]
S L0 −1
K¼ ˙−
ð19Þ
P0 A0 E
where S is the displacement rate, and P˙0is the rate of loading.
The machine displacement dm is then
dm ¼ K P
ð20Þ
∂Pðd Þ
∂ 2A0 E h=2
h−d
∫ðh−d Þ=2 z −
pðzÞdz
¼
∂d
∂d L0
2
¼
A0 E h=2
∫
pðzÞdz
L0 ðh−d Þ=2
ð17Þ
Replacing d with dshift = h − 2z,
∂Pðd Þ
∂d
¼
d shift
A0 E h=2
A0 E
∫ pðzÞdz ≈
L0 hzi
2L0
ð18Þ
where the final approximation is a result of assuming that the
mean and median of p(z) are the same. Again, this assumption
Fig. 15 Initial load-displacement data of Btilted^ compression test data of
Fig. 8, shifted by the geometric parameter h. Dashed lines indicate a slope
of EA0/2L0, demonstrating the use of (equation (18))
Exp Tech
The derivation of this expression ignores nonlinearity in
the machine response, as well as seating effects at the test
specimen. However, our machine displacement test data, as
well as that of Kalidindi et al. [2], confirm that the characteristic machine displacement is a nonlinear function of load, and
therefore the value of K determined by such a method
will depend on the maximum load as well as specimen
geometry. In fact, Hockett and Gillis attempted to determine a unique K for their test system by applying this
method to hundreds of compression tests and consequently produced data with significant scatter [4].
Even if the machine response is approximately linear,
when applied to a single test, the value of K derived
in this way is likely to be an overestimate of the true
behavior due to specimen seating effects. This method
will therefore lead to compression test results that depend strongly on the stiffness of the test machine, as
well as the surface quality of the specimen ends.
Method I of Kalidindi et al. This method for measuring
machine compliance has the same essential features as that
advocated here, although seating effects at the specimen are
ignored [2]. The characteristic machine response is determined by a compression test with no specimen present. The
machine displacement is assumed to be a nonlinear function
of load, and this load-displacement data is curve-fit and applied to data reduction. Kalidindi et al. did not advocate any
particular functional form for this fit. With sufficient attention
to environmental conditions and quality of fit, this method
should accurately remove machine compliance effects from
raw test data, although seating effects will remain.
Method II of Kalidindi et al. This method measures a nonlinear machine compliance by testing the elastic deformation
of a known material [2]. In this method, a specimen of known
modulus is tested within the elastic regime, and the machine
response is inferred from the difference between the measured
and theoretical elastic load-displacement relationship. As with
their other methods, Kalidindi et al. recognize that the machine displacement is a nonlinear function of applied load,
and so the end result produces a nonlinear curve fit.
However, this method is less accurate than their Method I,
because the use of a test specimen introduces additional nonlinearity due to seating effects. A careful assessment of Fig. 4
in their paper shows that, compared to their Method I data,
Method II predicts somewhat lower displacement at low loads
and higher displacement at high loads, presumably due to a
lower loading slope due to seating. Method II is also more
limited than Method I, in that data may only be obtained at
loads below yield for the particular specimen material and
geometry being used. It is and more expensive, in that one
or more test specimens are consumed.
Method III of Kalidindi et al. This method requires that a
number of compression tests are performed with the machine
to large total displacement [2]. The final length of each
specimen is then measured, and the change in length recorded. The change in specimen length is compared to the
measured displacement, and any discrepancy is attributed
to machine deflection. Like Method II, this method introduces additional error, as well as additional cost, to the
machine displacement measurement by using test specimens. An additional problem with this method is that
small differences in specimen geometry and/or friction
behavior among the various tests will lead to differences
in the load required to deform a specimen to a given final
length. This is likely the cause of the high degree of scatter reported by Kalidindi et al. for this method.
Recommended Compression Test Data Reduction
Method
We propose the following guidelines for compression test data
reduction.
Measure the characteristic machine displacement.
&
&
&
&
&
Perform one or more tests with all load train components,
platens, and other fixtures in place as they would be in a
typical compression test, but without a compression specimen present. Special care should be taken in setting up
the system to achieve good parallelism at the contacting
platens.
The machine displacement test may be performed prior to
or after compression testing. Machine displacement
should be characterized whenever a physical change
is made to the test system and also whenever load
and displacement transducers are recalibrated.
Performing such tests on a periodic basis can help
to determine if any significant changes to the system
have occurred.
The environmental conditions of the machine displacement test, in particular the test temperature
and temperature history, should match that of the
corresponding compression test as closely as
possible.
Care should be taken to properly zero load and displacement transducers prior to the test, although in
each case, offsets may be applied to the measured
data if necessary.
The test should begin at negative displacement, i.e.
no load applied to the system. The maximum positive
displacement should be chosen so as to produce a
maximum load that is at least as large as what may
be expected in a compression test. Appropriate settings may be determined using manual control prior
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&
&
&
to the test. Extreme care should be taken to not overload the system.
The machine displacement test is not particularly sensitive
to displacement rate. The chosen displacement rate
and the data sampling rate should be complementary,
so that the load-displacement curve is sampled with
good resolution. The displacement rate should also
be chosen such that the resulting test data is not
affected by noise filtering or other processing.
Machine displacement test data should be fit to an analytical form that enforces zero crossing, i.e. zero load at zero
displacement. The fit should accurately reproduce the
load-displacement behavior at all loads.
The resulting characteristic machine displacement may then
be used to correct test data from all compression
tests performed using the same machine setup and
environmental conditions.
Remove the machine displacement from raw compression
test data.
&
&
&
Raw compression test data may require offsets to properly
zero load and displacement data.
The expected machine displacement may be calculated for
each data point as a function of applied load.
The corrected specimen displacement is obtained by
subtracting the machine displacement from the measured
displacement.
be estimated from back extrapolation of the data at larger
displacements. If such approximations are made, they
should be clearly noted.
Conclusions
Raw load-displacement data from the uniaxial compression test
are contaminated by the effects of machine compliance and seating effects at the specimen surface. Machine displacement is
found to be a nonlinear and reversible function of applied load.
Machine displacement is insensitive to the displacement or loading rate, but is typically sensitive to environmental conditions,
especially temperature. An empirical function is proposed which
provides a good fit to machine displacement data. Specimen
seating affects the measured load-displacement data by introducing additional nonlinearity, reducing the elastic loading slope, and
shifting the load data to higher displacement. These effects originate from non-ideality of the specimen geometry and are independent of friction effects. A simple analytical model to predict
specimen seating effects was proposed. The model was then used
to determine a robust method for determining the proper correction (shift) to compensate for specimen non-ideality. A complete
method for compression test data reduction was proposed.
References
1.
Determine and remove the data shift due to specimen seating effects.
&
&
&
&
Plot the corrected specimen displacement (the measured
displacement minus the machine displacement). On the
same plot, determine the point where a line with slope
A0E/2L0 is tangent to the curve. The value of d at this point
is the estimate of dshift.
Remove the data shift by subtracting dshift from the
corrected specimen displacement.
All shifted data with d < dshift should be considered invalid, as the specimen is not fully seated in this region. Data
with dshift < d < ~2dshift should be used with caution, as
local plastic deformation during seating may result in
loads that are inaccurate.
For visualization purposes, the initial loading data may be
replaced by P = (A0E/L0) · d. Loads between the initial linear elastic region and displacements of about 2dshift may
2.
3.
4.
5.
6.
7.
8.
9.
House JW, Gillis PP (2000) Testing machines and strain sensors. In:
Mechanical testing and evaluation, ASM handbook, volume 8. ASM
international, Materials Park
Kalidindi SR, Abusafieh A, El-Danaf E (1997) Accurate characterization of machine compliance for simple compression testing. Exp
Mech 37:210–215
Hockett JE, Gillis PP (1971) Mechanical testing machine stiffness:
part I – theory and calculations. Int J Mech Sci 13:251–264
Hockett JE, Gillis PP (1971) Mechanical testing machine stiffness:
part II – application to data reduction. Int J Mech Sci 13:265–275
ASTM E9–09 (2009) Standard test methods of compression testing
metallic materials at room temperature. ASTM International, West
Conshohocken
ASTM E209–00 (2010) Standard practice for compression tests of
metallic materials at elevated temperatures with conventional or rapid heating rates and strain rates. ASTM International, West
Conshohocken
ASTM E4–16 (2016) Standard practices for force verification of
testing machines. ASTM International, West Conshohocken
Song JF, Vorburger TV (1992) Surface texture. In: Friction, lubrication, and wear technology, ASM handbook, volume 18. ASM
International, Materials Park
Thomas TR (1999) Rough surfaces. Imperial College Press, London
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