Strain Rate Compression Research Articles

Micro-laminated CoCrFeMnNi−TiNp/CoCrFeMnNi high-entropy alloy matrix composite with bimodal grain structure via multi-material selective laser melting (MM-SLM) additive manufacturing

Deviating from most studies on selective laser melting (SLM) additively manufactured single metals or homogeneous metal matrix composites (MMCs), this work engaged in a multi-material SLM (MM-SLM) route for micro-laminated MMCs with bimodal grain structures. Structurally hierarchical micro-laminate of CoCrFeMnNi high-entropy alloy (HEA)-based HEA−TiNp/HEA MMC was built by the MM-SLM to overcome the strength-ductility trade-off and improve the anti-impact performance under high strain-rate compression deformation. The dynamic impacts were performed on the SLM-built HEA and MM-SLM-built HEA−TiNp/HEA using a split Hopkinson pressure bar (SHPB) system, with the strain rates of 1000 s−1, 2000 s−1, and 3000 s−1. The structurally hierarchical HEA−TiNp/HEA exhibited higher strength and deformability than the HEA. The strain-rate-dependent work hardening capacity and macroscopic plasticity of the HEA−TiNp/HEA indicated a better resistance to dynamic shocks. The promising results pave a road for achieving high-performance HEA-based MMCs of the MM-SLM-built hierarchical structures with a relatively low proportion of nano-ceramic reinforcements and bimodal-grained HEA matrix.

The evolution of adiabatic shear band in high Co–Ni steel during high strain-rate compression

The microstructural evolution and formation mechanism of adiabatic shear band (ASB) in M54 steel were investigated using split Hopkinson pressure bar (SHPB). Specially, the strain response to ASB was analyzed systematically by strain controlling dynamic compression tests at high strain-rates. The results demonstrated that the ASBs started to form at 23% strain and shearing failure strain was increased to 28% at strain-rate of 4000 s−1. The ASBs in the M54 steel was consist of a transformed adiabatic shear bands (T-ASB) and deformed adiabatic shear bands (D-ASB). The evolution of the ASBs was revealed and the generation of fine equiaxed nano-grains within ASBs was attributed to the rotational dynamic recrystallization mechanism. The twinning process was actually the reverse transformation of ω→bcc and the driving force was the lattice instability of martensite matrix caused by carbon redistribution.

Strain rate effect of frozen ballast aggregates under compression: Experiments and discrete element simulations

Ballast railway tracks in cold regions are typically affected by severe freezing damage. To further investigate the mechanical properties of the ballast layer under freezing conditions, ice–bonded ballast specimens are prepared, and uniaxial compression tests are performed in a low-temperature environment. The effect of strain rate on the stress–strain curve, failure mode, and compressive strength are discussed comprehensively under two ice content ranges. Meanwhile, a dilated polyhedron discrete element method based on bonded particle model (DP-DEM-BPM) was developed to establish the discrete element model of ice–bonded ballast under uniaxial compression. The results show that the strain rate significantly affects the deformation characteristics of the frozen specimens. When the strain rate is low, the stress–strain curves exhibit clear strain-softening characteristics. As the strain rate increases, the post-peak stress decreases more rapidly, and the brittleness becomes more prominent. Meanwhile, the strain rate can be categorised into ductile, ductile–brittle transition, and brittle zones based on the stress–strain curve and failure mode. The specimens in the ductile zone primarily present axial splitting failure, whereas those in the brittle zone present brittle fracture with multistage local crushing. The corresponding discrete element simulation reveals the internal mechanism of failure modes in the specimen from the point of view of the bond. In addition, the uniaxial compressive strength and effective elasticity modulus increase in the ductile zone and then decrease in the brittle zone as the strain rate increases. The relationship between compressive strength and strain rate can be expressed using a piecewise function.

Application of machine learning in predicting the rate-dependent compressive strength of rocks

Accurate prediction of compressive strength of rocks relies on the rate-dependent behaviors of rocks, and correlation among the geometrical, physical, and mechanical properties of rocks. However, these properties may not be easy to control in laboratory experiments, particularly in dynamic compression experiments. By training three machine learning models based on the support vector machine (SVM), back-propagation neural network (BPNN), and random forest (RF) algorithms, we isolated different input parameters, such as static compressive strength, P-wave velocity, specimen dimension, grain size, bulk density, and strain rate, to identify their importance in the strength prediction. Our results demonstrated that the RF algorithm shows a better performance than the other two algorithms. The strain rate is a key input parameter influencing the performance of these models, while the others (e.g. static compressive strength and P-wave velocity) are less important as their roles can be compensated by alternative parameters. The results also revealed that the effect of specimen dimension on the rock strength can be overshadowed at high strain rates, while the effect on the dynamic increase factor (i.e. the ratio of dynamic to static compressive strength) becomes significant. The dynamic increase factors for different specimen dimensions bifurcate when the strain rate reaches a relatively high value, a clue to improve our understanding of the transitional behaviors of rocks from low to high strain rates.

Deformation behavior and processing map of ATI 425 with initial lamellar microstructure

The effect of hot compression temperatures and strain rates on deformation behavior and resultant microstructure of ATI 425 alloy with initial lamellar microstructure was investigated in this study. The temperature and strain rate of the hot compression test were chosen to be in the 700-1100 °C and 0.001-1 s-1 ranges, respectively. The stress-strain curve and microstructure evaluation show that the alloy's flow softening is associated with globularization and dynamic recrystallization mechanisms. The constitutive equation calculates the activation energy for the α/β and β regions to be 348 kJ/mol and 201 kJ/mol, respectively. Dynamic recovery and partial recrystallization are the dominant structure modification mechanisms in the beta single-phase region. Bending and fragmentation of alpha plates is the dominant mechanism of microstructure promotion in the α/β region at low temperatures and low strain rates, less than 0.1s-1. Local shear and alpha plate break-up are the main factors in structural modification at high strain rates, greater than 0.1s-1. The extracted process map at 0.5 strain revealed three zones: instability, safe zone, and peak zone, with power dissipation efficiencies of 0 -0.25%, 30-40%, and above 40%.

High strain-rate compression behavior of woven CF/PEEK thermoplastic composites at the glassy state and high-elastic state

A reliable hot air-flow heating approach is creatively carried out for high strain rate tests without environment tank. The out-of-plane and in-plane compression of carbon fiber reinforced polyetheretherketone (CF/PEEK) woven composites are investigated from 400 to 4000/s at the glassy state and high-elastic state. The real-time and in-situ deformation and failure process of the composite specimens under the extreme thermo-mechanical loads are successfully captured, which can reflect the impact dynamic behavior of the material near the ultimate service temperature. Empirical laws for mechanical index of the woven composites are given. The results reveal that the thermal softening effect outperforms the strain-rate strengthening effect. For out-of-plane impact at low strain rates, damage mode transitions from the interface and matrix cracking at glassy state to the “parallel” shear and fiber extrusion at high-elastic state. Under high strain rates, it reveals the “fragmented” shear mode at glassy state. For in-plane impact, the temperature effect leads to the failure mode changing from the delamination and local shear to the mixed feature of kinking bands, delamination, and multiple-band shear with “bulge” behavior. And the multiple-band shear and delamination are more likely to occur along the in-plane direction as the strain rate increased.

Analysis of the Wall Force on the Bubble Collapse of the Benzamide Based on Molecular Dynamics Simulation

The cavitation damage of the pollutant wall caused by shock wave and microjet in the process of bubble collapse has attracted widespread attention from scholars. However, many researches focus on bubble collapse from the macroscopic experimental point of view, the dynamics of which is still not clear. In this paper, it constructs bubbles with different radii, runs different models under different compressive strain rates, takes the pressure on the upper and lower walls of pollutants as the research target, and summarizes the influence of compressive strain rates, initial radius of bubbles and temperatures on the release pressure of bubble collapse on the pollutant wall. The results show that: as the initial radius of bubbles increase, the maximum pressure on the upper and lower walls of benzamide increases first and then decreases. The pressure release rate of bubble increases with the increase of the compressive strain rate. Based on the pressure on the upper and lower wall surfaces of the benzamide, the temperature and initial radius of the bubble have a large influence, and the influence of the compressive strain rate is small. Under the simulation conditions of the paper, when bubble collapse is used to release energy to treat pollutants, the recommended bubble radius is 10-15 Å. The paper not only reveals the influence of the initial radius of the bubble, the compressive strain rate and the temperature on the pressure released by the bubble collapse on the pollutant wall, but also provides theoretical guidance for the application of cavitation.

Conditional analysis of temperature and strain rate effects on dissipation structure in turbulent non-premixed jet flames

This work presents results from simultaneous high-resolution temperature and velocity measurements in a series of turbulent non-premixed jet flames. The filtered Rayleigh scattering (FRS)-based temperature measurements demonstrate sufficient signal-to-noise (SNR) and spatial resolution to estimate the smallest scalar length scales and accurately determine dissipation rate fields. A comprehensive set of conditional statistics are used to characterize the small-scale structure, including the dependence of dissipation layer widths on Reynolds number, temperature, and dissipation magnitude. In general, the dissipation layer thickness decrease with increasing Reynolds number and increase with increasing temperature. However, dissipation layer widths show two distinct behaviors with respect to dissipation magnitude. For small dissipation values, increases in magnitude results in broadening of the dissipation layer, while for larger magnitude values of dissipation, the layer widths are thinned, highlighting the complexity of small-scale turbulent mixing. Additionally, measured ratios of the dissipation layer width to the Batchelor length scale are consistent across all Reynolds numbers and agree with previous studies in non-reacting flows. The unique aspect about the current set of measurements is the ability to examine the interaction of dissipation structure with turbulent flow parameters for the first time in turbulent non-premixed flames. Particularly, the strain rate/dissipation relationship is examined and compared to previous studies in non-reacting flows. It is found that the dissipation layers tend to align normal to the principal compressive strain axis and this tendency increases with increasing Reynolds number. For the lowest Reynolds number case, no dependence of the dissipation layer width nor dissipation rate magnitude on strain rate is found. However, for higher Reynolds numbers, a strong dependence of the dissipation layer width and dissipation rate magnitude on the principal compressive strain rate is observed. These results indicate the direct role of the compressive strain rate field on small-scale mixing structure in reacting flows.

Microstructural evolution and kinetic characteristics during metadynamic recrystallization in a Ni-Cr-Mo alloy

The metadynamic recrystallization (MDRX) characteristics of a Ni-Cr-Mo alloy are investigated using the double-pass thermal compression tests. The impacts of double-pass compression parameters on the kinetic features and microstructural changes are clarified. Results signify that the MDRX softening behaviors are remarkably accelerated with increasing of compression temperature, first-pass strain or strain rate. The bulging of the MDRX grain boundary and the annihilation/interaction of substructures become weaken at low compression temperature or small first-pass strain, enhancing the grain refining during MDRX. However, the MDRX grains coarsen rapidly with decrease of strain rate. Based on the comprehensive impact of double-pass compression parameters on softening fraction and grain size, the MDRX kinetic model and MDRX grain size model are developed. The predicted results are well matched the experimental data, which suggests that the established models can precisely characterize the changes of softening characteristics and grain size of the researched alloy during MDRX.

On the formation of thrust fault-related landforms in Mercury’s Northern Smooth Plains: A new mechanical model of the lithosphere

There are numerous tectonic shortening structures distributed across the planet, Mercury. As Mercury’s largest single volcanic deposit, the northern smooth plains (NSP) is dominated by thrust fault-related landforms, showing particularity in their tectonic patterns compared with their counterparts in other geological terrains on Mercury. Geomorphic interpretations of these landforms assume an internal layering lithosphere to account for the deformation accommodating superficial units, implying the deformation in the NSP is thin-rooted dominated. However, the commonly used lithospheric mechanical model is an oversimplification that only allows for the sharp transition from brittle to ductile deformation, failing to explain the thin-rooted deformation well. In this work, we propose a new mechanical model incorporating the semi-brittle deformation in the lithosphere to account for an equivalent weak layer at shallow depth, filling the gap between brittle and ductile deformation. In addition, we implement 2-D numerical simulations to simulate the formation of thrust fault-related landforms in the NSP of 3.8 billion years ago. As a result, we obtain surface topographies roughly consistent with lobate scarps. Our results also support that most thrust fault-related landforms were likely formed over a period with a gradually decreased background compressive strain rate, and these landforms can retain their basic geomorphic features on this planet with little to no erosion. Although the physical properties of semi-brittle deformation are not fully understood, considering such a deformation model in planetary science is still promising, especially when studying the thermodynamic processes of a planet.

Effect of fibre addition on the static and dynamic compressive properties of ambient-cured geopolymer concrete

This paper presents an experimental study on the static and dynamic compressive properties of ambient-cured fibre reinforced geopolymer concrete (FRGPC) containing mono steel, mono polypropylene (PP) or hybrid steel-PP fibres. A Split Hopkinson Pressure Bar (SHPB) was used to carry out dynamic compressive tests under different strain rates up to 130.5 s−1. The effects of fibre addition on static compressive strength, splitting tensile strength, dynamic failure process and damage mode, dynamic compressive strength, specific energy absorption and dynamic increase factor (DIF) were investigated and analysed. The results revealed mono steel fibre as the most effective in improving static compressive and splitting tensile strengths, which increased by 21.92% and 21.05% respectively, compared to plain geopolymer concrete (GPC). The dynamic compressive strength of both plain GPC and FRGPC increased with the increase of strain rate. There is a positive relationship between DIFs of compressive strengths and strain rates for all GPC and FRGPC samples. The samples containing mono PP fibres showed the most strain rate sensitivity with respect to compressive strengths and specific energy absorption within the tested strain rates. Furthermore, empirical formulae are proposed accordingly to predict the DIFs and specific energy absorption of ambient-cured GPC and FRGPC.

Stretched vortex layer flamelet

A new sub-grid flamelet model is developed based on a steady-state, viscous vortex layer with an imposed compressive strain orthogonal to both the vorticity and the shear force. The imposed normal strain results in the stretching of the vorticity in the direction orthogonal to the plane of the shear strain. The model has certain special features. (i) Non-premixed flames, premixed flames, or multi-branched flame structures are determined rather than prescribed. (ii) Three components of velocity exist although the flamelet model is two-dimensional. (iii) The effect of variable density is addressed in the flamelet model. The vortex-layer configuration distinguishes this model from recent models by this author that address flames in other practical and ubiquitous vortical structures: tube-like stretched vortices or vortices stretched in two directions. Thereby, the model collection available for application to vortical configurations found in turbulent combustion is increased. Solutions to the Navier-Stokes equations and the associated scalar equations governing the flamelet model are obtained without the boundary-layer approximation and demonstrate the impact of the new features of the model. The shear strain and vorticity are found not to affect the scalar profiles, mixing and burning rates, or the two velocity components orthogonal to the shear force; only the velocity parallel to the shear force is affected. The compressive strain rate is the influential constraint affecting the scalars and the other two velocity components. However, the vorticity is strongly affected by the scalar profiles. The relation to classical concepts for incompressible flow is discussed.

Revealing the Geometrically Necessary Dislocation Density Evolution during Hot Compression of AA7050

Dislocations play a critical role in metal forming processes, and accurate values of dislocation density are important in modelling these processes. However, direct determination of the dislocation density is challenging. In this study, electron backscatter diffraction is used to estimate the evolution of geometrically necessary dislocation density as a function of plastic strain, strain rate and temperature in hot compression of AA7050 alloy. The geometrically necessary dislocation density was found to increase at a higher strain rate and lower temperature; the higher dislocation density in these samples promoted continuous dynamic recrystallisation leading to grain refinement. At lower strain rates and higher temperatures, the dislocation densities were lower and dislocations formed into walls, channels and cells. These observations agree with accepted theories of dislocation evolution and demonstrate the capability of electron backscatter diffraction to provide representative dislocation density values as well as comprehensive information linking plastic flow with microstructural evolution.Keywords: electron backscatter diffraction (EBSD), geometrically necessary dislocations (GNDs), hot deformation, AA7050

Combined effects of cryogenic temperature and strain rates on compressive behavior of concrete

Concrete structures in extremely cryogenic environments may be subject to dynamic loadings as a result of occasional accidents or terrorist attacks during the service life. Considering the cryogenic temperature effect and strain rate effect of each phase for concrete, a meso-scale numerical model of the concrete subjected to cryogenic temperature and dynamic loadings was presented in this study, which serves as a preliminary foundation for calculating and evaluating the performances of concrete structures at cryogenic temperature. Taking concrete cube as an example, the dynamic compressive behavior and strain rate effect of concrete specimens at cryogenic temperature from 20 °C to –160 °C were modelled and investigated. The numerical results indicate that the failure appearances and compressive strength are highly related to the cryogenic temperature and dynamic loadings. The compressive strength at cryogenic temperature is greater than that at ambient temperature, and the sensitivity of dynamic compressive strength to strain rate is enhanced at cryogenic temperature. Moreover, the absorbed energy and dynamic elastic modulus tend to increase with the decrease of temperature and increase of strain rate. According to the numerical results, an empirical formula on dynamic increase factor of compressive strength at cryogenic temperatures (CDIF T) was established.

Alignment statistics of pressure Hessian with strain rate tensor and reactive scalar gradient in turbulent premixed flames

The relative alignment of the eigenvectors of pressure Hessian with reactive scalar gradient and strain rate eigenvectors in turbulent premixed flames have been analyzed for Karlovitz number values ranging from 0.75 to 126 using a detailed chemistry three-dimensional direct numerical simulations database of H2–air premixed flames. The reactive scalar gradient preferentially aligns with the most extensive strain rate eigendirection for large Damköhler number and small Karlovitz number values, whereas a preferential collinear alignment between the reactive scalar gradient with the most compressive strain rate eigendirection is observed in flames with small Damköhler number and large Karlovitz number. By contrast, the eigenvectors of pressure Hessian do not perfectly align with the reactive scalar gradient, and the net effect of the pressure Hessian on the evolution of the normal strain rate contribution to the scalar dissipation rate transport acts to reduce the scalar gradient in the zone of high dilatation rate. The eigenvectors of pressure Hessian and the strain rate are aligned in such a manner that the contribution of pressure Hessian to the evolution of principal strain rates tends to augment the most extensive principal strain rate for small and moderate values of Karlovitz numbers, whereas this contribution plays an important role for the evolution of the intermediate principal strain rate for large values of Karlovitz number. As the reactive scalar gradient does not align with the intermediate strain rate eigenvector, the influence of pressure Hessian contributions to the scalar–turbulence interaction remains weak for large values of Karlovitz number.

Multidirectional basketball activities load different regions of the tibia: A subject-specific muscle-driven finite element study.

The tibia is a common site for bone stress injuries, which are believed to develop from microdamage accumulation to repetitive sub-yield strains. There is a need to understand how the tibia is loaded in vivo to understand how bone stress injuries develop and design exercises to build a more robust bone. Here, we use subject-specific, muscle-driven, finite element simulations of 11 basketball players to calculate strain and strain rate distributions at the midshaft and distal tibia during six activities: walking, sprinting, lateral cut, jumping after landing, changing direction from forward-to-backward sprinting, and changing direction while side shuffling. Maximum compressive strains were at least double maximum tensile strains during the stance phase of all activities. Sprinting and lateral cut had the highest compressive (-2,862±662 με and -2,697±495 με, respectively) and tensile (973±208 με and 942±223 με, respectively) strains. These activities also had the highest strains rates (peak compressive strain rate=64,602±19,068 με/s and 37,961±14,210 με/s, respectively). Compressive strains principally occurred in the posterior tibia for all activities; however, tensile strain location varied. Activities involving a change in direction increased tensile loads in the anterior tibia. These observations may guide preventative and management strategies for tibial bone stress injuries. In terms of prevention, the strain distributions suggest individuals should perform activities involving changes in direction during growth to adapt different parts of the tibia and develop a more fatigue resistant bone. In terms of management, the greater strain and strain rates during sprinting than jumping suggests jumping activities may be commenced earlier than full pace running. The greater anterior tensile strains during changes in direction suggest introduction of these types of activities should be delayed during recovery from an anterior tibial bone stress injury, which have a high-risk of healing complications.

An experimental study and failure mechanism analysis on dynamic behaviors of plain concrete under biaxial compression-compression

In order to explore the dynamic mechanical properties of concrete under biaxial compression-compression, a true triaxial instrument was used to conduct an experimental study on the dynamic mechanical properties of plain concrete under biaxial compression-compression. Eight different lateral compressive stresses (0 MPa~14 MPa) and four strain (10−5/s, 10−4/s, 10−3/s and 10−2/s) rate were considered. From this, the effects of lateral compressive stress and loading strain rate on the biaxial compression-compression properties of concrete were compared and analyzed by obtaining the failure modes, principal compressive stress-strain curves and related mechanical characteristic parameters of plain concrete under different loading conditions. The research results show that: with the increase of lateral compressive stress, the failure modes of concrete were developed from columnar failure at low lateral compressive stress to sheet-like failure at high lateral compressive stress. With the increase of lateral stress, the development trend of concrete failure mode decreased gradually under the influence of strain rate. With the increase of lateral compressive stress, the variation range of concrete principal compressive stress decreased first and then tended to be relatively stable under the influence of strain rate. With the increase of strain rate, the increase range of concrete principal compressive stress decreased under the influence of lateral compressive stress. The influence mechanism of lateral compressive stress and strain rate on biaxial compression-compression performance of concrete was analyzed. Meanwhile, based on Kupfer biaxial compression failure criterion, a biaxial compression dynamic failure criterion model considering the effect of loading strain rate on plain concrete was proposed. The research results provided an important theoretical basis for the application and development of concrete engineering.

High Strain-Rate Through-Thickness Compression Testing of Symmetrical Inter-ply Hybrid Polymer Composites Reinforced with Carbon/Glass and Carbon/Kevlar Fibers

High Strain-Rate Through-Thickness Compression Testing of Symmetrical Inter-ply Hybrid Polymer Composites Reinforced with Carbon/Glass and Carbon/Kevlar Fibers

Ultralight and Highly Resilient Boron Nitride Nanosheet/Polyimide Foams for Energy Harvesting and Sensing

The application of highly superelastic foamlike materials has gradually expanded to various high-end areas, such as filtration purification, mechanical power generation, and energy storage, showing their considerable and expected prospects. In this paper, facile freeze-drying technology combined with a gradual thermal-imidization process was employed to prepare a series of boron nitride nanosheet (BNNS)-filled polyimide (PI) composite foams. These foams can exhibit excellent mechanical properties, including their cyclic stability for considerable cycles under long compressive loading–unloading processes, and their relatively low irreversible deformation. After 10000 cycles at the compression strain of 60%, the total strain loss of the composite filled with 12 wt % is only 14%, which is 2/3 that of pure PI foam. Based on these excellent mechanical characteristics, the foam composites exhibit well a compression-driven triboelectric performance. The effects of BNNS content, compression strain, and rate on the triboelectric properties of the composites were studied in detail. We found that a higher compression strain and compression rate will lead to stronger electrical signals. More significantly, on the basis of the one-to-one relationship between electrical signal and compressive deformation, BNNSs/PI foam will likely be used in the field of control sensing. Finally, the electrical generation device based on composites was used in an insole and keyboard to transform mechanical energy generated by human movement into electrical signals.

Fast nonlinear mechanical features decoupling to identify and predict asphalt-based composites

Nonlinear matrices employed in asphalt-based composites exhibit a prominent nonlinear elastic-viscoplastic-viscodamageable mechanical response. The constitutive model for this type of matrices requires a significant experimental effort to identify the material constants. To alleviate the characterization effort without losing reliability in the mechanical prediction, this paper presents a fast procedure decoupling the elasto-viscoplastic response from the viscodamageable one and facilitating the identification of the material constants via efficient optimization. This procedure relies on a combined experimental–numerical also applicable to other rheologically complex materials like polyurea, thermoplastics or elasto-viscoplastic polycrystals. The experimental part is deliberately designed to only conduct cost- and time-effective monotonic loads at different strain rates in tension and compression. These pure monotonic results are used in the constitutive model to predict more complex loading conditions such as multiple load–unload–reload (LUR) cycles. To do this, an elasto-viscoplastic constitutive model suitable to describe creep-like phenomena and large irreversible deformations is proposed. This model incorporates pressure and strain rate sensitivity, which is essential to capture the compression–tension asymmetry in asphalt-based composites. A rate-dependent damage model is proposed to describe the degradation rate of the elasto-viscoplastic part. The stress and the consistent tangent modulus are derived. The model implementation for user-defined finite element subroutines are given for explicit or implicit solvers. The proposed decoupling facilitates an efficient identification of the constant parameters via an in-house Nelder–Mead-based optimization method. To accelerate the identification, all the stress–strain curves, either in tension or compression, are simultaneously used with applying physically-based constrains. The framework is validated with asphalt matrix at room temperature (24 °C) and verified under LUR conditions at different rates. The identified model predicts correctly the experimental observations, proving its applicability to investigate particle-based composites with highly nonlinear matrices.