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 Exp Tech 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 Exp Tech 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 Exp Tech 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 Exp Tech 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 Exp Tech 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. Exp Tech 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 Exp Tech 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 Exp Tech 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 Exp Tech & & & 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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