Kolsky Bar Research Articles

Dynamic Strength and Fragmentation of Highly Oriented Ti3SiC2 under Multiaxial Compression

MAX phases are distinguished by their unique kink band formation, a distinct deformation mechanism in layered materials. This study explores the influence of global grain orientation c-axis, strain rate, and stress state on the compressive response of highly oriented Ti3SiC2 through experimental methods. A Kolsky (or split-Hopkinson) bar is employed to evaluate the dynamic compressive response under uniaxial and biaxial (planar confinement) conditions under 102s-1 strain rate. Macroscopic ultra-high-speed visualization during loading and microscopic post-mortem fractography reveal that confinement states significantly impact both macroscopic failure patterns and microscopic fracture mechanisms. Notably, biaxial loading with dynamic load edge-on to the grains and 80MPa planar confinement along the layers resulted in the highest dynamic compressive strength observed (1636±136MPa), a 66% increase compared to the unconfined uniaxial dynamic condition. The planar confinement appears to delay crack propagation and enhance inelastic deformation.

A digital-twin driven Split Hopkinson bar layout for the tensile characterization of thin, low impedance, sheet-like materials

The Split Hopkinson or Kolsky bar is one of the most popular devices when it comes to the mechanical characterization of material samples under high strain-rates. While testing of high impedance materials, such as metal alloys, is relatively straight forward, samples with low impedance pose certain challenges. The present work focuses on the detailed implementation of a high strain-rate tensile testing method for thin, low impedance, sheet-like materials by using the Split Hopkinson test principle. In order to find a suitable Split Hopkinson setup a digital twin was created using explicit finite element methods. With the help of the digital twin, the design of the transmission bar and the sample holders including the friction liners were explored. The numerical model indicated, that a hollow transmission bar with a moderate tapering (hollow bar 1.0) is suited for the characterization of low impedance materials over a wide strain-rate range. Furthermore, this setup has to be combined with an asymmetrical sample holder configuration (heavier on the incident side and lighter on the transmission side) and aluminum friction liners to return accurate results. This numerically derived setup was validated against experimental tests on paper, representative of low impedance, sheet-like materials.

On the Dynamic Mechanical Behavior of Wrought and AM 17-4 Precipitation-Hardenable Stainless Steels Under Rapid Heating

Dynamic large-strain plasticity problems in metals can produce temperatures high enough to alter the microstructure, but the limited time-at-temperature prevents complete transformation, thereby making the material strength time-dependent. Precipitation reactions (age-hardening) are an important class of transformations that can create time-dependent dynamic plasticity under rapid heating and loading. This work explores the dynamic behavior of a precipitation-hardenable stainless steel (17-4) produced by wrought and Additive Manufacturing (AM) methods with a rapidly-heated Kolsky bar technique. Wrought 17-4, a martensitic stainless steel, is examined in three common heat treatments (solution-treated, peak-aged and over-aged) at temperatures up to 1000 °C and heating times limited to about three seconds. Solution-treated wrought 17-4 is observed to thermally-harden at aging temperatures (> 400 °C) due to rapid precipitate growth. Peak-aged precipitation strengthening becomes ineffective above 550 °C, as peak-aged material becomes indistinguishable from the solution treated-condition. Over-aged wrought 17-4 does not behave like either of the other conditions, owing to the effect of the extended heat treatment on the precipitates and on the martensite matrix. Stress-relieved AM 17-4 exhibits high dynamic strength and strain hardening at room-temperature due to its meta-stable austenite content and partial age-hardening during the build or stress-relief treatment. A plasticity model is developed for solution-treated wrought 17-4 that captures time-dependent aging effects that are derived from separate aging kinetics experiments. A separate model is developed for over-aged wrought 17-4 that contains no time-dependence as the precipitate population in this material appears to be more stable under rapid heating.

Dynamic Behavior of Twisted UHMWPE Yarns

Twisting is an effective method to reduce defects during fabric weaving and improve the impact resistance of the armor material. This study examined the dynamic behavior of twisted ultra-high molecular weight polyethylene (UHMWPE) yarns. Material characterization was conducted using three-dimensional X-ray tomography and scanning electron microscopy (SEM). Dynamic mechanical experiments were carried out on the twisted yarns using a Kolsky bar, with the yarn's failure process captured in real time through high-speed imaging. Additionally, the digital imaging correlation (DIC) technique was used to calibrate the strain measured by the Kolsky bar. Post-fracture analysis was performed using SEM. The effects of twisting extent, strain rate, and gauge length on the dynamic behavior of twisted yarns were studied. It was observed that all twisted yarns underwent elongation, shrinkage, fiber breakage, untwisting, and yarn rupture. Furthermore, as the twisting degree and strain rate increased, and the gauge length decreased, the fracture area became more concentrated. The tensile strength peaked at a twisting degree of 250 t/m and decreased thereafter. Moreover, twisting was found to alter the sensitivity of the yarns' stress–strain curves and tensile properties to the strain rate and gauge length.

High-throughput quantification of quasistatic, dynamic and spall strength of materials across 10 orders of strain rates

Abstract The response of metals and their microstructures under extreme dynamic conditions can be markedly different from that under quasistatic conditions. Traditionally, high strain rates and shock stresses are achieved using cumbersome and expensive methods such as the Kolsky bar or large spall experiments. These methods are low throughput and do not facilitate high-fidelity microstructure–property linkages. In this work, we combine two powerful small-scale testing methods, custom nanoindentation, and laser-driven microflyer (LDMF) shock, to measure the dynamic and spall strength of metals. The nanoindentation system is configured to test samples from quasistatic to dynamic strain-rate regimes. The LDMF shock system can test samples through impact loading, triggering spall failure. The model material used for testing is magnesium alloys, which are lightweight, possess high-specific strengths, and have historically been challenging to design and strengthen due to their mechanical anisotropy. We adopt two distinct microstructures, solutionized (no precipitates) and peak-aged (with precipitates) to demonstrate interesting upticks in strain-rate sensitivity and evolution of dynamic strength. At high shock-loading rates, we unravel an interesting paradigm where the spall strength vs. strain rate of these materials converges, but the failure mechanisms are markedly different. Peak aging, considered to be a standard method to strengthen metallic alloys, causes catastrophic failure, faring much worse than solutionized alloys. Our high-throughput testing framework not only quantifies strength but also teases out unexplored failure mechanisms at extreme strain rates, providing valuable insights for the rapid design and improvement of materials for extreme environments.

Influence of Sampling Rate on Reproducibility and Accuracy of Miniature Kolsky Bar Experiments

Abstract The purpose of this work is to develop and verify a method for quantitatively analyzing data collected from Kolsky bar experiments and to confirm its validity by comparing it to a finite element (FE) model. This study also aims to demonstrate the need for higher sample rate capture in miniature Kolsky bars, 3.16 mm diameter used in this work, by comparing results from two different data acquisition setups on identically sized experimental setups. We identified that the sample capture rate needed to accurately depict experimental results on small-scale systems is at least 400 kHz, which is far greater than what is typically assumed for lager systems. Finally, a statistical method for evaluating results is presented and expanded upon, which removes the dependence on the knowledge and experience of the experimentalist to interpret the data. Using this analysis technique on the two different systems examined in this study, we find upwards of 3.5 times better loading condition reproducibility and up to a 20 MPa reduction in the standard deviation of the sample stress profile, confirming the need for higher quality frequency capture rates.

Impact of Imperfect Kolsky Bar Experiments Across Different Scales Assessed Using Finite Elements

Abstract Typical Kolsky bars are 10–20 mm in diameter with lengths of each main bar being on the scale of meters. To push 104+ strain rates, smaller systems are needed. As the diameter and mass decrease, the precision of the alignment must increase to maintain the same relative tolerance, and the potential impacts of gravity and friction change. Finite element models are typically generated assuming a perfect experiment with exact alignment and no gravity. Additionally, these simulations tend to take advantage of the radial symmetry of an ideal experiment, which removes any potential for modeling nonsymmetric effects, but has the benefit of reducing computational load. In this work, we discuss results from these fast-running symmetry models to establish a baseline and demonstrate their first-order use case. We then take advantage of high-performance computing techniques to generate half symmetry simulations using Abaqus® to model gravity and misalignment. The imperfection is initially modeled using a static general step followed by a dynamic explicit step to simulate the impact events. This multistep simulation structure can properly investigate the impact of these real-world, nonaxis symmetric effects. These simulations explore the impacts of these experimental realities and are described in detail to allow other researchers to implement a similar finite element (FE) modeling structure to aid in experimentation and diagnostic efforts. It is shown that of the two sizes evaluated, the smaller 3.16-mm system is more sensitive than the larger 12.7 mm system to such imperfections.

Two-dimensional evolution of temperature and deformation fields during dynamic shear banding: In-situ experiments and modeling

Adiabatic shear band (ASB) is a significant failure mechanism observed in metals and alloys under impact loading. Though ASB formation has been widely assumed to be a one-dimensional thermo-mechanically-coupled instability problem, it is crucial to recognize that adiabatic shear banding is essentially a two-dimensional propagating event in space. However, it is challenging to perform in-situ characterization of temperature-deformation fields during ASB formation due to the extremely small spatial and temporal scales involved. To obtain the two-dimensional features of ASB evolution, a newly developed plane-array infrared imaging system and microspeckle-based digital image correlation (DIC) technique are synchronized with the Kolsky bar system. By incorporating interrupted tests, “quasi-synchronous” characterization of temperature-deformation-microstructure evolution during ASB formation in hat-shaped specimens of Ti–6Al–4V is achieved. A phase-field model incorporating energy-based shear banding criteria and independently calibrated model parameters is established to simulate the dynamic shear failure process, which is demonstrated to be able to well reproduce experimentally observed temperature and deformation evolution. Based on experimental characterization and simulation results, the two-dimensional features and thermo-mechanical aspects of ASB formation are presented. Energy dissipation of shear banding is estimated based on the measured temperature field, demonstrating good agreement with the calibrated values in the phase-field model. The “propagation” and “percolation” modes along the band are analyzed, which can be predicted by the introduction of a shear band process zone. The influences of thermal and microstructural softening on shear failure are also clarified through a comprehensive analysis of temperature and microstructure evolution.

A Kolsky Bar for High-Rate Indentation

A Kolsky Bar for High-Rate Indentation

Dynamic compressive failure of coal at different burial depths

Dynamic disasters frequently occur in deep coal mines, where coals and rocks are under high in situ stresses. To reveal the failure mechanism of deep burying coals, the specimens were made of buried coals at a depth of about 1 km. An improved Kolsky bar was applied to conduct the dynamic compression experiments subjected to different hydrostatic pressure i.e., 0, 5, 10, 15, and 20 MPa for simulating in situ stresses corresponding to depths of 0, 200, 400, 600, and 800 m. The findings demonstrate that the dynamic compressive strength of coal at a specified depth is rate-dependent. And the dynamic strength is positively correlated with the burial depth at a given loading rate. Moreover, a Mohr–Coulomb strength criterion for coal under dynamic loadings that considers the effect of depth is established, and the model describes the experimental results well.

The effects of carbonaceous inclusions and their distributions on dynamic failure processes in boron carbide ceramics

AbstractCommercially available boron carbide ceramics typically have heterogeneous microstructures that contain distributions of processing‐induced inclusions. The inclusions that are rich in carbon (i.e., carbonaceous) govern the underlying mechanisms of brittle fracture through wing crack formation, and thus dictate the mechanical response of the ceramic. In this study, we investigate the dynamic failure of five boron carbide ceramic materials with different inclusion populations. All of the materials were prepared by hot‐pressing; four of these boron carbides contained different sizes and concentrations of carbonaceous inclusions, while one contained no carbonaceous inclusions. The heterogeneity distributions were characterized in some detail for statistical analysis using scanning electron microscopy and quantitative image analysis. A modified compression Kolsky bar setup with in situ ultra‐high‐speed microscopic imaging (10 million frames per second) was then used to study the influence of the inclusion distributions on the dynamic failure processes in these materials, at nominal high strain rates of 102–103 s−1. The in situ ultra‐high‐speed microscopy highlighted the link between micro and macroscale failure processes and demonstrated that the carbonaceous inclusions are indeed the preferential sites for nucleation of wing cracks, as previously hypothesized based on post‐mortem observations. The relative orientation of an inclusion with respect to the compression axis was shown to affect the likelihood that it would participate in crack nucleation. All of the ceramics were also found to have orientation‐dependent peak compressive stress, regardless of the presence of carbonaceous inclusions, suggesting that grain orientation distributions are also important.

Simulation and dimensional analysis of instrumented dynamic spherical indentation of ductile metals

Finite element (FE) modeling of instrumented dynamic indentation experiments in a miniature Kolsky bar is undertaken. Geometry, including an output bar with machined spherical indenter tip, and velocity history boundary conditions are extracted directly from experimental diagnostics. The test material (i.e., substrate) is polycrystalline aluminum alloy Al 6061-T6. The constitutive model used in simulations accounts for isotropic elasticity and isotropic plasticity with strain hardening, strain-rate hardening, and thermal softening under adiabatic conditions. The FE model, with representative material parameters culled from the literature, accurately reproduces the curvature of the experimental load versus depth data for three different experimental indentation velocity histories. A framework for dimensional analysis of instrumented dynamic spherical indentation is set forth, improving upon prior work. Parametric FE simulations reveal sensitivity, or lack thereof, of the predicted response to variations in the proposed independent dimensionless variables encompassing material properties. For the indenter size, maximum depth, and maximum strain rate imposed experimentally on the order of 103/s, force-depth predictions are nearly unaffected by realistic variations in mass density, melting temperature, and thermal softening parameters when the sample is initially at room temperature. Predictions are affected by elastic constants (the elastic modulus and to a lesser extent, Poisson’s ratio), initial yield strength, two strain hardening parameters, and strain rate sensitivity. Predictions are also notably affected by initial temperature, with thermal softening prominent at high enough initial temperature or much higher loading rates. Based on the dimensional analysis, static indentation and elevated temperature indentation experiments are proposed for extraction of quasi-static and thermal material properties from previously uncharacterized metals, and dynamic indentation is proposed for extraction of rate sensitivity that cannot be obtained from static tests. Rate sensitivity obtained in this way from the novel instrumented dynamic spherical indentation experiments and FE simulations produces a parameterized stress–strain response for Al 6061-T6 reasonably validated by external studies for strain rates up to the order of 103/s.

Direct comparison between experiments and dislocation dynamics simulations of high rate deformation of single crystal copper

A long standing challenge in computational materials science is to establish a quantitative connection between the macroscopic properties of plastic deformation with the microscopic mechanisms of dislocations in crystalline materials. Although the discrete dislocation dynamics (DDD) simulation method has been developed for several decades with the goal of addressing this challenge, a one-to-one comparison between the DDD predictions on single crystal stress–strain curves and experimental measurements under identical conditions has not been possible to date. Such a comparison is an essential step towards establishing a dislocation-physics based theory of plasticity and a multiscale framework of the plastic behaviors of crystalline materials. Here we provide direct comparisons between the stress–strain curves of Cu single crystals under high strain rate loading in the [001] and [011] directions obtained from miniaturized desktop Kolsky bar experiments and those from DDD simulations under identical loading conditions. With an appropriate set of parameters, DDD simulations can produce stress–strain curves that are in reasonable agreement with the experimental results. However, the dislocation mobility values needed to achieve this agreement are an order of magnitude lower than expected based on previous measurements and atomistic simulations. We hypothesize that this discrepancy could be caused by drag forces from jogs and point defects produced during the plastic deformation. Cross-slip of screw dislocations is also found to be necessary to capture the experimental stress–strain behavior, especially for the [011] loading direction. This work provides an example of how direct comparisons between DDD simulations and experimental measurements can provide new insight into the fundamental mechanisms of plastic deformation.

Toward Material Property Extraction from Dynamic Spherical Indentation Experiments on Hardening Polycrystalline Metals

Static indentation and dynamic indentation are reviewed, with a focus on extraction of material properties of isotropic strain-hardening polycrystalline metals that may be rate- and temperature-sensitive. Static indentation is reviewed first, followed by dynamic indentation, since the former is regarded as a specialization of the latter with inertia, rate dependence, and adiabatic heating excluded. Extending concepts from the literature review, a treatment of dynamic indentation using dimensional analysis is forwarded, and a general framework for extraction of material property information (i.e., constitutive model parameters) from instrumented dynamic spherical indentation experiments is set forth. In an example application of the methodology, experimental data obtained from instrumented spherical indentation in a miniature Kolsky bar apparatus are evaluated via dimensional analysis. The substrate material is aluminum alloy Al 6061-T6. Several definitions of indentation strain proposed for static indentation are assessed for dynamic indentation, as are indentation strain rates. While the fidelity of the experimental method and inertial effects could inhibit extraction of elastic properties, extraction of certain plastic constitutive properties may be feasible. Current data are insufficient to enable determination of a complete and unique set of all physical properties. Motivated by the present review and analysis, new experiments and simulations are proposed that would identify influences of material properties, facilitating their extraction from data.

Hydraulic-based testing and material modelling to investigate uniaxial compression of thermoset and thermoplastic polymers in quasistatic-to-dynamic regime

The strain rate dependence of thermoplastics and thermosets in the range 10-3–102 s−1 is still unclear due to experimental difficulties. This interval cannot be fully treated using a quasistatic-based electromechanical machine nor a high-speed test relying on the Kolsky bar method. This paper presents a uniaxial compression test set-up to cover the aforementioned range providing robust stress–strain measurements up to 100 s−1. This set-up is assessed at room temperature with a semi-crystalline polymer (SCP) polyether-ether-ketone (PEEK) KT880-NT and a toughened thermoset epoxy PR520. High-speed 3D stereo digital image correlation (DIC) and an infra-red camera are used for data acquisition. The macroscopical true stress–strain curves demonstrate the importance of the thermal softening effect taking place in this range, accompanied with a temperature rise. In terms of characterization, an advanced unified SCP (USCP) model is employed to quantify the rate- and temperature-dependence for both polymers. A straightforward FEM-based parameter identification procedure is proposed requiring only two compressive stress–strain curves. The USCP model is further applied to PEEK 450G from literature to highlight the features of SCPs. The comparison of PEEK and PR520 is discussed pointing out the characterization challenges due to the transitional quasistatic-to-dynamic testing regime.

Miniature Beryllium Split-Hopkinson Pressure Bars for Extending the Range of Achievable Strain-Rates

Conventional Split Hopkinson Pressure Bars (SHPB) or “Kolsky” bars are often used for determining the high-rate compressive yield and failure strength of materials. However, for experiments generating very high strain-rates (>103/s) miniaturization of the setup is often required for minimizing the effects of elastic wave dispersion in order to enable the inference of decreasingly short loading events from the data. Miniature aluminum and steel bars are often sufficient for meeting these requirements. However, for high enough strain-rates, miniaturization of steel or aluminum Kolsky bars may require prohibitively small diameter bars and test specimens that could become inappropriate for inferring representative properties of materials with large grain size relative to the test specimen size. The use of a beryllium Kolsky bar setup is expected to enable high rates to be accessible with larger diameter bars/specimen combinations due to the inherent physical properties of beryllium, which are expected to minimize the effects of elastic wave dispersion. For this reason, a series of beryllium Kolsky bars have been developed, and, in this paper, the dispersion characteristics of these bars are measured and compare the data with those of similarly sized 7075-T6 aluminum and C350 maraging steel. The results, which agree well with the theory, show no appreciable frequency dependence of the elastic wavespeed in the data from the beryllium bars, demonstrating its advantage over aluminum and steel in application to Kolsky bars.

On the Mechanical Response of Aluminum Alloy 6061-T6 under Extreme Strains and Strain Rates, and Rapid Heating

The flow stress of aluminum alloy 6061-T6 produced by conventional thermomechanical methods has been investigated under the application of strains up to 80% at a strain rate of the order of 50001/s, and temperatures ranging from ambient to near melting. The high strain and strain rate deformation was imposed using a Kolsky bar compression apparatus equipped with a fast pulse-current heating system to reach the target temperatures at high heating rates. The flow stress was measured under strains, strain rates, temperatures and heating rates that match thermomechanical conditions developed in the primary shear zone of a specially designed machining with chip pulling test method. Flow stress error is estimated by statistics obtained from experimental replicates. It is expected that these measurements will enable the formulation of realistic machining models for Al6061-T6. Temperature measurements were obtained using non-contact infrared (IR) full field imaging together with embedded micro-thermocouple (TC) point probing. The TC measurements were used to estimate the emissivity of the aluminum specimen via separate, in situ calibration tests, and to track temperature evolution during the compression, also via separate testing, not compromising the flow stress measurements. Type-K TC measurements were made using the separated junction principle by embedding the TC wires into two micromachined holes in the specimen. The temperature measurement technique is discussed in detail, and temperature uncertainties are estimated. The Kolsky bar and machining test with chip pulling will be used going forward to study the effect of heating time on the dynamic thermal softening behavior of Al6061-T6, which has been shown to be time-sensitive under quasi-static loading when temperatures exceed 200°C, due to Mg-Si precipitate growth.

Effect of Grain Size on the Properties of Aluminum Matrix Composites with Graphene

The structure and mechanical properties of composites consisting of a metal matrix based on aluminum and its alloys of different compositions (AA-3003 and AA-5154) and graphene synthes sized in situ under a layer of molten salts were investigated depending on the chemical composition and grain size of the matrix. Aluminum matrix composites of three compositions were studied in as-cast coarse-grained, deformed fine-grained (grain size < 1 mm), and deformed sub microcrystalline (grain size < 1 μm) states in order to compare the structural characteristics of composites with different grain sizes. The composites were subjected to deformation with a split Hopkinson (Kolsky) bar and to dynamic-channel angular pressing. The hardness and dynamic mechanical properties of the composites were measured at strain rates ε˙ from 1.8 − 4.7 × 103 to 1.6 − 2.4 × 105 s−1. It was found that grain refinement induced a sharp increase in the hardness of composites with various compositions (by a factor of 1.6–2.6). A correlation of the elastic-plastic properties of the aluminum matrix composites with the grain sizes and chemical compositions of the matrices was established. A transition from coarse-grained to sub microcrystalline structure was shown to improve the elastic-plastic properties on average by a factor of 1.5. It was proved that the reinforcing effect of graphene increased with the decreasing grain size of the matrix. Mechanisms of reinforcement of the aluminum matrix composites using graphene were proposed.

Adiabatic Shear Failure of Ultrafine Grained Titanium Under Impact Loading: An in-situ Experimental Study

Background: Ultrafine-grained (UFG) titanium, which is with high yield strength, biocompatibility and corrosion resistance, is widely used in biomedical and industrial applications. However, adiabatic shear localization (ASL) is often observed in UFG materials due to their worse deformation stability under impact loading. This instability will easily result in formation of adiabatic shear bands (ASB), a narrow band located in the ASL zoom, and finally cause the fracture of material. The main objective of this work is to study the adiabatic shear behavior of UFG titanium under impact loading, including macro- and micro-properties, temperature rise, ASB failure, etc. Methods: A synchronization apparatus, which consisted of Kolsky bar system, high-speed camera system and high-speed infrared temperature measuring system, was set up to carry out the in-situ study of the mechanical properties, temperature rise, and adiabatic shear failure process of UFG pure titanium. Microstructure of the material was also analyzed in this work. Results: The critical strain of UFG pure titanium for adiabatic shear localization is about 0.37 and 0.69 for UFG Ti and CG Ti, respectively. The peak shear stress of UFG Ti is 500MPa. The propagation velocity of ASB in UFG titanium is 533~800m/s, and 160~320m/s for CG Ti. The temperature rsie within ASB of UFG titanium is 307~732℃, and 212-556℃ for CG Ti. The intense temperature rise is after the peak stress and the initiation of ASB most of the time. Conclusion: UFG Ti has good mechanical properties, however, it is easier to form ASB and cause adiabatic shear failure under impact loading when compared with CG Ti. Temperature rise may not play a major role in the formation of ASB in UFG Ti, but may be the consequence of ASB. Results of this work will help researchers better understand the failure of UFG metals under impact loading.

Apparatus for measuring strength in biaxial compression.

Most measurements of compressive strength of ductile materials have involved Hopkinson-Kolsky bars or Taylor anvils placing samples in uniaxial compression. In these geometries, strain is limited by the tendency of the sample to petal, in analogy to necking in uniaxial tension. Estimation of strength for any other form of the stress tensor requires assuming a shape of the yield surface; because data exist only for uniaxial compression, these assumptions are untested. In an imploding spherical shell, compression is biaxial, the plastic strain may not be small, and the material behavior may be nonlinear as a result of work hardening and heating by plastic work. We outline a method of measuring the strengths of materials in biaxial compression, both quasistatically and dynamically, using the compression of thin spherical shells. We suggest surrounding the shell with an annulus filled with a mixture of H2 and Cl2 gases whose homogeneous ignition is initiated by a flash of blue and near-ultraviolet light. Less promising approaches are described in Appendixes A-C.