HJC Model Research Articles

About the strain-rate effect in HJC model

The strain-rate effect in HJC model is evaluated in this paper. Based on a comparison of strain-rate effects between the HJC and KC models, it shows that the determination and implementation procedures in the HJC model lead to smaller strain-rate effect. Some problems on the determination of strain-rate effect in the HJC model, i.e. the utilisations of the ‘leftward’ pressure correction, over-simplified strength curve and the cross-domain stress data, are discussed, which cause the underestimation of the strain-rate constant C. A more logical and practically-applicable ‘rightward’ pressure correction method is proposed to determine the strain-rate effect in the HJC model. The strain-rate effect determined from the high-pressure data is larger than that determined from the uniaxial compressive data, which facilitate further study on the pressure-dependent strain-rate effect.

Triaxial compressive behaviour of ultra-high performance geopolymer concrete (UHPGC) and its applications in contact explosion and projectile impact analysis

Ultra-high performance geopolymer concrete (UHPGC) is increasingly recognised as a promising construction material in protective engineering owing to its exceptional material strength and eco-friendly characteristics. However, there is currently limited research data on the behaviour of UHPGC under complex stress states, and the applicability of some common concrete dynamic constitutive models to UHPGC requires calibration and evaluation. This paper presents an experimental study on the triaxial compression (TXC) behaviour of UHPGC under varying confining pressures with a range of 0–40 MPa. The stress-strain curves under various confinement levels and failure patterns of UHPGC are discussed. The results indicate that the peak axial stress and toughness of UHPGC increase with higher confining pressure levels, but it still exhibits noticeable softening behaviour. Under uniaxial compression, UHPGC typically fails with inclined shear and axial tensile cracks, while under TXC, particularly at confining pressures above 20 MPa, the failure mode transitions to shear failure with prominent oblique shear cracks. Additionally, existing failure criteria are calibrated to match the strength envelop of UHPGC, with the Power-law failure criterion providing the closest fitting results. Subsequently, utilising both current and pre-existing TXC data, three commonly used concrete constitutive models, including HJC, K&C and CSC, are calibrated with respect to the dominant strength parameters for UHPGC in the finite element (FE) hydrocode ANSYS/LS-DYNA. Further, numerical simulations are conducted to evaluate the effectiveness of such three calibrated concrete models in replicating the response of UHPGC panels exposed to the contact explosion or projectile impact. The simulation results demonstrate that the K&C and CSC models effectively replicate the localised damage of UHPGC panels under contact explosions, while the HJC model more accurately simulates the depth of penetration (DOP) for thick UHPGC panels under projectile impact.

Investigation of dynamic compression properties of basalt fibre reactive powder concrete

Basalt fibre reactive powder concrete (BFRPC) was prepared by incorporating basalt fibres (BFs) instead of steel fibre into reactive powder concrete. In this study, the impact resistance of BFRPC was experimentally and numerically analysed at different strain rates (101–102 s−1). The 75 mm dia. split-Hopkinson pressure bar was used for impact compression tests and the finite-element software LS- Dyna was used for numerical simulation analysis. The results showed that the high-strength BFRPC has an obvious strain rate effect and the dynamic increase factor (DIF) of compressive strength increases logarithmically with strain rate. Meanwhile, the parameters of the Comité Euro-international du Béton (CEB) model were refitted and the relationship between strain rate and DIF was established. By using the Holmquist−Johnson−Cook material constitutive model (HJC model), the stress−strain curves and failure patterns obtained were consistent with the experimental results. The incorporation of BFs significantly improves the deformation properties of BFRPC.

A modified HJC model for geological materials subjected to blasting loadings

In this paper, a modified HJC model is proposed to address the problem that the HJC model cannot simulate tensile damage cracks. The modified HJC model modifies the strength surface, strain rate effect and damage model of the original HJC model, respectively. The modified model is embedded in LS-DYNA software for single and multiple finite element validation. The effects of different parameters on the modified HJC model stress–strain curves are analyzed in the single finite element validation. In multiple finite element verification, uniaxial and triaxial compression experiments, Brazilian disc splitting experiments, SHPB dynamic compression experiments, and SHPB dynamic splitting experiments are performed. The results of single and multiple finite element numerical validation illustrate that the modified HJC model can well simulate the pre-peak and post-peak nonlinear mechanical behaviors of geological materials such as rocks and concrete. Compared with the HJC model, the damage pattern of the modified HJC model is closer to the real damage pattern. For the strain rate effect, the modified HJC model can reflect the real compressive strain rate effect and improve the sensitivity of tensile strength to strain rate. The results of the numerical simulation of the blasting funnel illustrate that: the modified HJC model can simulate explosive tensile cracking well and can explain the formation mechanism of blasting funnel due to the distinction between tensile and compressive meridian surfaces and the addition of tensile damage. The modified HJC model is more applicable to the numerical simulation of engineering blasting.

A new material model for concrete subjected to high rate of loading

This study presents a modified HJC model for simulating the constitutive behaviour of concrete under large strains, high strain rate and high pressure. The original HJC model has been modified by introducing the shear cracking, pore crushing, tensile cracking, and hydrostatic expansion damage models to the yield surface for reproducing the crushing, spalling (crater formed at the front surface), scabbing (crater formed at the rear surface), and radial cracking in the concrete target. The Lode-Angle model has been introduced to describe the yield surface to account for the substantial difference in the shear strength between the tensile and compressive meridians. The strain rate effect has been considered to suitably model the dynamic compressive and tensile strengths. Single element simulations have been performed to substantiate the effect of modifications introduced in the constitutive model. The results of the single element simulation obtained by using the proposed model (MHJC_K&I) have been compared with the results of the two latest modified (MHJC_Kong [1] and MHJC_Liu [2]) models available in the literature. The MHJC_K&I model has described more realistic strength softening under uniaxial compression when compared with the experiment results. Further, the MHJC_K&I model has described accurately the magnitude of tensile strength under uniaxial tension in comparison to the other two models. Under hydrostatic tension, the MHJC_K&I model described the exponential strength softening and the MHJC_Kong model described the linear softening while the MHJC_Liu model had limitation in describing the softening behaviour. The ballistic experiments have also been performed on 100 and 60 mm thick plain concrete targets against 0.4 kg steel projectile at incidence velocities in the range 103–220 m/s. The damage induced in the concrete targets as well as the residual projectile velocities have been obtained. The MHJC_K&I model has been found to effectively reproduce the damage with respect to the size and depth of the crater as well as the radial cracking at the front and rear surfaces of the plain concrete target. The MHJC_Kong and MHJC_Liu models underestimated the size and depth of the crater and described limitations in reproducing the radial cracking. The residual velocities obtained by using the proposed model (MHJC_K&I) have been found to have a good agreement with the actual results.

Assessment of the ballistic impact response of Cor-Tuf UHPC concrete using the HJC constitutive model

Concrete offers superior strength in compressive loadings and is implemented for many applications. The high compressive strengths enable concrete to resist high strain rate loading scenarios such as ballistic impacts. A variety of concrete denoted as Cor-Tuf, which is classified as ultra-high-performance concrete with a compressive strength of 210 MPa, is evaluated in this study. The response of this concrete is assessed through a finite element analysis under the high strain rate loadings of ballistic impacts. To capture the response of the concrete, a plasticity and damage constitutive model denoted as the HJC model is implemented. The parameters of this model are calibrated to the Cor-Tuf concrete using confined compression experiments, unconfined compression experiments, and shock experiments. The concrete target is impacted at speeds between 610 m/s through 1112 m/s, and the results are compared to existing experimental data. Our results show that the HJC model can predict the response of this impact to the Cor-Tuf concrete targets as an average error of 5.85% is found. The results of this study present parameters which can be implemented with the HJC concrete model for future studies to model the response of the Cor-Tuf UHPC.

Accuracy Assessment of “Step-by-Step” Simulation Modeling Method for Rock Breaking by TBM Disc Cutters Assisted with Laser

Rock breaking by laser-assisted disc cutters is a novel high-efficiency rock breaking mode that combines mechanical stress induced by the disc cutters with thermal cracking by laser. This paper presented a “step-by-step” simulation modeling concept, and conducted an in-depth study on potential influencing factors of simulation accuracy at each key step. First, the prediction accuracy of laser holes in laser drilling simulation was discussed. Second, the SHPB simulation and experiment were carried out to evaluate the accuracy of the selected material constitutive model in simulating the dynamic fracture damage of rock. Then, taking the laser-assisted rock-penetrating process of the scaled disc cutter as an example, the simulation prediction accuracy of rock-breaking by the disc cutter was analyzed. Finally, the simulation and experiment of laser-assisted disc cutter penetration into rock was carried out, and then the feasibility of the “step-by-step” concept was analyzed. The results show that: (1) in the laser drilling simulation, the predicted accuracy of laser hole size is higher when the power is low; with the laser power increases, the large amount of glass glaze will affect the subsequent modeling accuracy; (2) the HJC model can be used to simulate the transient nonlinear fracture damage behavior of granite; (3) the damage morphology of the granite obtained by the penetration simulation is highly similar to the experimental results, and the load curve should be corrected by the peak point fitting method. The results show the application prospects of the proposed numerical modeling method in future laser-assisted TBM tunnelling.

Study on Dynamic Splitting Properties of S-PP Hybrid Fiber Concrete after High Temperatures

To study the dynamic tensile mechanical properties of steel polypropylene hybrid fiber reinforced concrete (SP-HFRC) after high temperature, split Hopkinson pressure bar (SHPB) dynamic splitting tests were carried out, and the optimal fiber content combination was obtained. With the plain concrete (PC) as the control, the effects of fiber addition on energy dissipation and failure forms of concrete specimens after high temperatures were analyzed. LS-DYNA software was used to simulate the dynamic splitting test. The results show that the splitting strength of specimens increases first and then deteriorates with the increase of temperature. After high temperatures, HFRC has a positive and negative fiber hybrid effect. Among the studied fiber mixture combinations, S1PP0.2 (1 vol% steel fiber + 0.2 vol% polypropylene fiber) concrete has the best splitting resistance. Compared with PC, the splitting strength increases by 106.8% at 25 °C and 128.2% at 800 °C. From the perspective of energy, we can conclude that adding hybrid fiber can significantly improve the dynamic splitting and tensile toughness of concrete after high temperatures, and defining damage variables can better characterize the damage degree of concrete. PC cracks seriously after high temperatures, while S1PP0.2 concrete cracks but does not disperse at 800 °C, showing ductile failure characteristics. By modifying some parameters of the HJC model, the state of high-temperature concrete mechanical properties can be better characterized after deterioration. The simulated failure process shows an excellent agreement with the experimental results.

A Dynamic Coupled Elastoplastic Damage Model for Rock-like Materials Considering Tension-Compression Damage and Pressure-Dependent Behavior

A dynamic coupled elastoplastic damage model for rock-like materials is proposed. The model takes unified strength theory as the strength criterion. To characterize the different damage between compression and tension, two damage variables both in compression and in tension are introduced into the model. The former is represented by the generalized shear plastic strain and volumetric plastic strain and the latter is expressed with the generalized shear plastic strain. Furthermore, the model takes the strain rate effect into account to reflect the strength enhancement under dynamic loading. Because of the difference in plastic hardening between compression and tension, a modified hardening function is adopted. At the same time, the volume strain from HJC model is modified to be consistent with one of continuum mechanics. The developed model is numerically implemented into LS-DYNA with a semi-implicit algorithm through a user-defined material interface (UMAT). The reliability and accuracy of the developed model are verified by the simulation of four basic experiments with different loading conditions. The proposed model was found to be applicable to different mechanical behaviors of rock-like materials under static and dynamic loading conditions.

A dynamic bounding surface plasticity damage model for rocks subjected to high strain rates and confinements

A dynamic bounding surface plasticity damage model is developed for rocks with the consideration of high strain rates and confinements. This is achieved through adopting the strength surface of Holmquist-Johnson-Cook (HJC model)/ Johnson-Holmquist-ceramic (JH2 model) model as bounding surface, and employing the critical state concept for brittle-ductile transition. In addition, a simple continuum damage evolution law is taken for strength degradation and a simple vertical mapping rule is adopted to link current stress state with its corresponding strength state. A series of hardening rules that account for both plasticity and strain rate, and damage evolution law, etc., are proposed within the framework of consistent condition of dynamic model. A ‘S’ shape dynamic increase factor (DIF) function is adopted to consider the strain-rate effect of dynamic loadings, which aims to adjust the size of bounding surface with the consideration of radial enhancement. The proposed model is validated through modeling conventional triaxial compression tests under different confining pressures, dynamic triaxial tests, and Split Hopkinson Pressure Bar (SHPB) tests with various strain rates. Good agreement between modeling results and experimental data indicates that dynamic characteristics of rock, including rate and confinements dependency, are captured well by the proposed model. The proposed model is then implemented into a commercial software, GDEM, to simulate the stress wave propagation problem. Simulation results by the proposed model and two benchmark models are compared, which further validate the capability and applied potential of the proposed model.

Three-dimensional mesoscopic investigation on equation of state for dry sand under shock compression

The dynamic behavior of dry sand subjected to intensively dynamic loads such as blasts and projectile penetration is a topic of current research for military protection engineering. The equation of state (EoS) plays a crucial rule under such high pressure and high strain-rate. However, experimental investigation on the EoS of dry sand with a comprehensive considerations of sand configuration, such as the particle shape, grain composition and porosity is a complex and expensive task. In the present study, a well-established three-dimensional mesoscopic model for dry sand is presented, in which the dynamic behavior of sand particle (quartz) are described by the HJC model with systematically calibrated parameters. The mesoscopic model is then used to investigate the dynamic response and EoS for dry sand subjected to shock compression and is validated against existing fly-plate-impact test data, in which two homogenization methods used to connect the mesoscopic variables and macroscopic EoS are adopted. A thorough analysis of the numerical shock compression simulation demonstrates that the shock compression of dry sand system can be divided into three stages, i.e., the initial rearrangement of particles, the following formation of stable force chains and the final translation motion. Finally, the effect of sand configuration on the EoS is systematically conducted. It is found that the particle shape and grain composition have limited influence on EoS, the friction between sand particles moderately influences the EoS and porosity strongly influences the EoS.

A novel dynamic cavity expansion model to predict the resistance of reactive powder concrete (RPC) against projectile impact

The dynamic cavity expansion (DCE) model is a commonly used analytical model to predict the resistance of concrete materials subjected to projectile impact. In this paper, a modified Griffith strength criterion was proposed for reactive powder concrete (RPC), which has a form easy to solve and can well capture the tensile, compressive and triaxial strength of RPC. Combined with the Murnaghan equation of state, the proposed strength criterion was used as the material model for RPC. The cavity expansion resistance was calculated numerically and analytically. It indicates that the numerical and analytical solutions are almost consistent under the same material response. Then, the strain softening behavior that cannot be resolved analytically was further introduced into the material model, and the numerical solution of the DCE model was obtained using LS-DYNA. To verify the present method, the 122.6 MPa RPC targets were tested using 14.5 mm diameter ogival nose projectiles at impact velocities of from 286 to 942 m/s. Combined with previous penetration model, the penetration depth of the present test was predicted. In addition, numerical simulation of this penetration test adopting the HJC model was also performed. It shows predictions of the present model are in good agreement with test and simulation results. Furthermore, the present model was used to predict the existing test date of RPCs with compressive strength of between 67.5 and 140 MPa, and the effectiveness of the present method was verified by comparing with predictions of previous empirical formulas.

Simulation and Research on Dynamic Crushing of Rod Concrete

The dynamic mechanical properties of waste glass powder concrete are studied, and the dynamic compressive strength, peak strain, stress-strain curve and dynamic enhancement factor change with strain rate and waste glass powder content are analyzed. Based on the HJC model of concrete, the SHPB test was simulated by ANSYS/LS-DYNA software. With the increase of impact pressure, the phenomenon from block spalling to aggregate crushing appears. When the impact velocity is low, several radial through cracks appear in the test block, and the test block is separated into several blocks with larger volume. With the increase of impact velocity, multiple cracks develop simultaneously.

Numerical simulations on cutting of frozen soil using HJC Model

Numerical simulations on cutting of frozen soil using HJC Model

Experimental and numerical investigation on penetration of clay masonry by small high-speed projectile

Masonry is one of the most common building materials, which is used due to low cost, good sound and heat insulation properties, easy availability and locally available material [1]. Masonry walls are consisted of bricks and mortar which interconnect regularly, the bricks are jointed together by the mortar through various conventional patterns. Since its extensive application it would certainly be subjected to a variety of strikes from human activities such as wars and military conflicts apart from erosion of natural forces wind, rain and so on. The knowledge of masonry will be useful if on urban battlefield because of the inability of conventional forces to manoeuvre troops and vehicles with ease. Many reviewer hold that urban warfare is on the increase, with recent events in Iraq, Libya and Syria indicating that the battleground is shifting inexorably to large and small conurbations. Thus it is necessary to work out how to operate in the future urban environment and maximise our capabilities at a time of diminishing military strength [2]. As pointed out by C. S. Meyer, the U.S. Army’s concern in urban operations has brought about efforts at the U.S. Army Research Laboratory to develop an initial set of material model parameters to enable physics-based penetration simulations of high-fidelity brick and mortar masonry wall models. A search of publicly released literature turned up not much concerning modeling and simulation of weapon effects against brick and mortar targets. The characteristic of concrete under dynamics load has been studied in detail however the investigation of dynamic material properties of brick and mortar is in early stages [3].What has been historically employed to investigate the dynamic characteristic of concrete includes theoretical method, experimental method and the numerical simulation. It has been shown that numerical stimulation is a cost-effective method and can provide more information of process history for sophisticated analysis than experimental method. Nevertheless, the data from experiments were crucial because on the one hand any modern analysis and stimulation computation should necessarily be validated by those come from real range tests, on the other hand the material parameters which are of vital importance to obtain a reasonable and compelling result must be furnished by or refer to experimental data [4]. The experimental exploration indicates that brick and mortar joint act similarly as concrete. Many material models are proposed to describe the behaviour of concrete, which are able to apply to solution of simulating behaviour of masonry reasonably. Li et al. [5] investigated the response mode of masonry subjected to vented gas explosion through experiment and simulation, in which the anisotropic material model *MAT_96 (MAT_BRITTLE_DAMAGE) has been adopted. The material model has been developed by Govindjee et al.[6]. Meyer explored profoundly the potential of the HJC model to apply in simulation of penetrations, which was initialled by Holmquist and Johnson in 1993 [3].

Numerical Study of Dynamic Behaviors of Concrete Under Various Strain Rates

AbstractAs the dynamic behavior of the concrete is different from that under static load, this research focuses on the study of dynamic responses of concrete by simulating the split Hopkinson pressure bar (SHPB) test. Finite element code LS-DYNA is used for modeling the dynamic behaviors of concrete. Three continuous models are reviewed and the Holmquist-Johnson-Cook model (HJC) is introduced in detail. The HJC model which has been implemented in LS-DYNA is used to represent the concrete properties. The SHPB test model is established and a few stress waves are applied to the incident bar to simulate the dynamic concrete behaviors. The stress-strain curves are obtained. The stress distributions are analyzed. The crack initiation and propagation process are described. It is concluded that: the HJC model can modeling the entire process of the fracture initiation and fragmentation; the compressive of the concrete is significantly influenced by the strain rates.

Theoretical considerations on cavity diameters and penetration depths of concrete materials generated by shaped charge jets using the targets response modes described by a modified HJC model

The traditional dynamic cavity expansion theory generally deals with the problem of penetrator penetration into targets at lower velocity, which is considered that the materials near the penetrator/target interface are in the plastic state. For shaped charge jets penetration, the concrete materials in a certain distance away from the jet/target interface are in the fluid state. Thus, the radial stress at the cavity surface derived from the dynamic cavity expansion theory cannot be directly applied as the target resistance. The current paper aims to establish a theoretical model for shaped charge jets penetration into concrete targets based on the cavity expansion idea, assuming that the cavity expansion stress is the stress at the flow/fully dense interface. Firstly, the hydrostatic pressure-volume strain relationship described by the HJC concrete constitutive model was analyzed; result indicated that four response regions were produced when concrete subjected to load: the elastic region, the cracked region, the plastic-like (the “plastic” behavior of concrete is named as “plastic-like” since the behavior has nothing to do with the dislocation movement of metals that reflects the plastic behavior) transitional region and the fully dense region. The HJC yield surface equation was then modified by discussing the elastic-cracked-transitional-fully dense response in front of the flow/fully dense interface. With the changing particle velocity at the flow/fully dense interface, it showed that another three response modes were produced in front of the flow/fully dense interface, i.e., the elastic-transitional-fully dense response, the elastic-fully dense response, and the fully dense response. The applications of the equations of state in different response modes were analyzed. Secondly, the calculation model of the radial stress at the flow/fully dense interface was proposed based on the modified HJC model. Thirdly, the shaped charge jets penetration theories were analyzed. In the end, the depth of penetration (DOP) experiments for three different shaped charges into C200 reactive powder concrete (RPC) targets were carried out to verify the theoretical model. Results showed that the cavity diameters and penetration depths predicted by the elastic-cracked-transitional-fully dense response, the elastic-transitional-fully dense response and the elastic-fully dense response exhibited good agreement with the experimental data, whereas that by the fully dense response existed relatively large errors comparing with the other three response modes.

Research on dynamic constitutive model of ultra-high performance fiber-reinforced concrete

A dynamic constitutive model for ultra-high performance fiber-reinforced concrete (UHPFRC) was investigated in this paper. In view of the particularity of UHPFRC material, two improvements including the modifications of yield surface and the change of strain rate effect due to the adjunction of steel fibers were made based on the Holmquist-Johnson-Cook model (HJC model). Further, on the purpose of model parameter determination, a series of mechanical tests for UHPFRC material, including uniaxial compression test, bending test, uniaxial cyclic compression test and SHPB test were carried out. And the 22 calculation parameters required in the improved HJC model such as strength parameters, damage parameters and pressure parameters were obtained through data analysis and fitting. Finally, the numerical simulation of the SHPB test performed in this study and the drop hammer impact test in the literature were carried out in order to verify the rationality and practicability of the improved HJC model. The numerical simulation results of SHPB test show that the failure morphology and stress-strain curves of UHPFRC specimens are in good agreement with the test results. When the improved HJC model and the appropriate corrosion strain value (1.0% in this work) were selected, the mid-span deflection-time curve obtained by numerical simulation is consistent with the result of the drop hammer impact test.

On the penetration and perforation of concrete targets struck transversely by ogival-nosed projectiles - a numerical study

This paper carries out a numerical study on the penetration and perforation of (reinforced) concrete targets subjected to impact by ogival-nosed projectiles at normal incidence. A recently developed computational constitutive model for concrete subjected to dynamic loadings, which takes account of pressure dependency, strain hardening and softening, strain rate effects, Lode effect and failures in both high and low confining pressures, is employed in the numerical simulations. Parametric studies are performed numerically to examine the influences of impact velocity on the load-time histories in the case of semi-infinite concrete targets and the effects of impact velocity and concrete target thickness on the projectile residual velocities and the target cracking patterns in the case of finite reinforced concrete slabs. It transpires that the present model predictions are found to be in good agreement with experimental observations in terms of penetration depth, load-time history, residual velocity, impact cratering and scabbing size and pattern. It also transpires that the present model predicts better the sizes of both impact and scabbing craters than the modified K&C model and the modified HJC model even though these three models produce similar results for residual velocities.

The 3D meso-scale model and numerical tests of split Hopkinson pressure bar of concrete specimen

In the present study, a 3D meso-scale model of plain concrete has been developed with consideration of the aggregates, mortar and interface transition zone (ITZ) between them. Firstly, the generation methods of the random polyhedron aggregate and ITZ are introduced, as well as the process how to create the meso-scale finite element (FE) model of the concrete specimen through grid mapping method. Then, based on the experiment data and repeated numerical simulations, the appropriate material parameters of HJC model with respect to each ingredient in concrete are determined. Based on the waveform data and the specimen deformation and failure process in the split Hopkinson pressure bar (SHPB) test of the concrete specimen, the validation of the numerical model is well verified. Subsequently, the numerical SHPB tests of concrete specimen with the experiment size of Ф120 × 100 mm under various strain rates are performed. The formation processes of four typical failure patterns of the concrete specimen in the experiment are well reproduced, i.e. “slight-spalling”, “breaking”, “fragmentation” and “comminution”. What’s more, the cracking mechanism of concrete specimen in SHPB test is revealed. The formation mechanisms of various failure modes of the concrete specimen under different strain rates are further investigated. The research results show that the failure process of the concrete specimen in the SHPB test can be well simulated by the 3D meso-scale model developed in this paper. Furthermore, the accumulation of the tension strain damage in concrete is the main reason to initiate the cracks in the loading process. Under different loading conditions, due to the crush failure and the tension breaking failure derived from the compression expansion effect play different roles, the specimen deformation and failure processes of different modes will be exhibited.