Rusinek-Klepaczko Model Research Articles

Comparative Study on High Strain Rate Fracture Modelling Using the Application of Explosively Driven Cylinder Rings.

The effect of different constitutive modelling choices is crucial under a high strain rate as encountered in ballistic applications. Natural fragmentation of explosively driven cylinder rings is chosen as a simplified example to describe the ability of numerical simulations to describe fractures. The main research interests are the importance of (i) material imperfections, (ii) the accuracy of fracture models vs. damage models, (iii) the plasticity algorithm (stress update), (iv) the introduction of a triaxiality cutoff criterion to the damage models, and (v) different constitutive models (plasticity and damage). Due to the complexity of the propagation and coalescense of multiple cracks in classical methods, smoothed-particle hydrodynamics (SPH) is used as a tailor-made method to discretise the model. An elasto-plasticity model, a damage model and an equation of state describe the material behaviour. The required material parameters are determined based on stress–strain curves from quasi-static and dynamic tests. The Johnson–Cook model, with and without a modification of the strain rate term, and the Rusinek–Klepaczko model are used to describe plasticity. These plasticity models are combined either with the Johnson–Cook, the Lemaitre, or the Dolinski–Rittel damage model and the Mie–Grüneisen equation of state. The numerical results show that (i) a random distribution of initial damage increases irregularity of cracks, and gives more realistic fragment shapes, (ii) a coupling of plasticity model and fracture criterion has only a small effect on the fracture behaviour, (iii) using an iterative plasticity solver has a positive effect on the fracture behaviour, although this effect is marginal, (iv) adding a triaxiality cutoff criterion to the damage models improves the predicted fragment masses in the numerical simulations significantly, and (v) good accordance between experiments and numerical simulations are found for the Dolinski–Rittel and Lemaitre damage model with both plasticity models.

Thermo-viscoplastic behavior of 304 austenitic stainless steel at various strain rates and temperatures: Testing, modeling and validation

This paper presents a systematic study of the thermo-viscoplastic behavior of a 304 austenitic stainless steel (ASS). The experiments were conducted over a wide range of strain rates (10−3 s−1 to 3270 s−1) and temperatures (-163°C to 172°C), for which the deformation behavior of 304 ASS becomes more complex due to the strain-induced martensitic transformation (SIMT) effect. Dynamic tests at low/elevated temperatures were conducted using the Hopkinson technique coupled with a cooling device/heating furnace, and temperature distribution within the specimen was verified to be uniform. Experimental results showed that the strain hardening rate of 304 ASS was strongly affected by SIMT effect. For quasi-static tests (10−3 s−1 to 1 s−1) at low temperatures (-163°C to -20°C), the stress-strain relations exhibited an S-shape and a second strain hardening phenomenon. The strain rate sensitivity and temperature sensitivity of 304 ASS were also different from metallic materials deformed by dislocation glide. Several unexpected phenomena including the negative strain rate sensitivity and the changing temperature sensitivity from quasi-static to dynamic tests were observed. Based on experimental results, an extension of the Rusinek-Klepaczko (RK) model considering SIMT effect was used to simulate the deformation behavior of 304 ASS: it predicted flow stress curves of 304 ASS above -60°C correctly. In addition, to validate the extended RK model and the identified model parameters, numerical simulations of ballistic impact tests of 304 ASS plates at various temperatures were carried out, showing a good agreement with experiments.

Comparative Analysis of Dynamic Plasticity Models

Abstract In the paper, a new phenomenological interpretation of some of the principal temporal effects of high-rate plastic deformation of metals from a united viewpoint is represented. A comparative analysis of some of the well-known dynamic plastic deformation models is given. Influence of strain rate on the stress-strain relation in a wide range of strain rates for different types of aluminum alloys and steels is described by the relaxation model of plasticity, by original and improved empirical Johnson-Cook models, and by the phenomenological Rusinek-Klepaczko model. It is shown that the structural-time model (“the relaxation model of plasticity”) is capable to effectively predict a wide spectrum of materials responses to fast and slow dynamic loading.

Mechanical behaviour modelling under dynamic conditions: Application to structural and high strength steels

Current needs in the design and optimization of complex ballistic protection structures lead to the development of more and more accurate numerical modelling for high impact velocity. The aim of developing such a tool is to be able to predict the protection performance of structures using few experiments. Considering only numerical approach, most important issue to have a reliable simulation is to focus on material behaviour description in term of constitutive relation and failure model for high strain rates, large field of temperatures and complex stress states. In this context, the study deals with the behaviour of two steels including a high strength steel and a structural steel. For this application, the materials can undergo both quasi-static and dynamic loading. Thus the strain rate range studied is varying from 10−3 to more than 103 s−1. Although the high strength steels do not exhibit high strain rate sensitivity, the temperature increases during dynamic loading is inducing thermal softening. Thus, temperature sensitivity is defined up to 500 K under quasi-static and dynamic conditions. Then, experiments are used to define the parameters of several constitutive relations like the Johnson-Cook model or the Rusinek Klepaczko model.

Application of optical field analysis of tensile tests for calibration of the Rusinek–Klepaczko constitutive relation of Ti6Al4V titanium alloy

In this paper, the application of the Rusinek–Klepaczko relation to describe the constitutive behaviour of Ti6Al4V titanium alloy with an HCP crystalline structure was proposed. The calibration of model coefficients was carried out on the basis of tensile tests. To obtain true stress–strain curves at quasi-static and dynamic loading conditions, the optical field measurement method was applied to determine the history of specimen cross-sections at the necking point. The outline of the specimen was tracked by virtual strain gauges implemented in TEMA Motion software. Adiabatic characteristics obtained at high strain rates using a pre-tension Hopkinson bar were corrected into quasi-isothermal using an analytical approach. Subsequently, a visco-plastic model calibrated using introduced methodology was validated using the finite element method. Engineering stress–strain curves, calculated using ABAQUS software incorporating the Rusinek–Klepaczko model, showed good agreement with experimental data at both quasi-static and dynamic deformation regimes. Moreover, numerical analysis of tensile tests shows that the strain, temperature and stress triaxiality distribution is non-homogenous in specimen cross-sections perpendicular to the loading direction. The value of the strain, temperature and stress triaxiality also depends on the strain rate.

An exponential strain dependent Rusinek–Klepaczko model for flow stress prediction in austenitic stainless steel 304 at elevated temperatures

In this paper, to predict flow stress of Austenitic Stainless Steel (ASS) 304 at elevated temperatures the extended Rusinek–Klepaczko (RK) model has been modified using an exponential strain dependent term for dynamic strain aging (DSA) region. Isothermal tensile tests are conducted on ASS 304 for a temperature range of 323–923K with an interval of 50K and at strain rates of 0.0001s−1, 0.001s−1, 0.01s−1 and 0.1s−1. DSA phenomenon is observed from 623 to 923K at 0.0001s−1, 0.001s−1 and 0.01s−1. Material constants are calculated using data obtained from these tensile tests for non-DSA and DSA region separately. The predicted results from the RK model are compared with the experimental data to check the accuracy of the constitutive relation. It is observed that to find out the constants of this model, some initial assumptions are required, and these initial values affect the predicted values. Hence, Genetic Algorithm (GA) is used to optimize the constants for RK model. Statistical measures such as the correlation coefficient, the average absolute error and standard deviation are used to measure the accuracy of the model. The resulting values of the correlation coefficient for ASS 304 for non-DSA and DSA region using modified extended RK model are 0.9828 and 0.9701. This modified, extended RK model is compared with Johnson–Cook (JC), Zerilli–Armstrong (ZA) and Arrhenius models and it is observed that specifically in DSA region, the modified extended RK model gives highly accurate predictions.

Establishment and comparison of four constitutive models of 5A02 aluminium alloy in high-velocity forming process

In high-velocity forming process, material constitutive relationships are remarkably changed, which has a significant effect on forming process and deformation behaviors. Based on the quasi-static tensile experiments and compression experiments at different temperatures, and dynamic tensile experiments and compression experiments at room temperature, the constitutive behaviors of 5A02 aluminium alloy and the mechanism of high strain rate influencing material dynamic responses are investigated in this paper. According to the above experimental results, Johnson–Cook model (J–C model), modified Cowper–Symonds model (C–S model), modified Rusinek–Klepaczko model (R–K model) and artificial neural network model (ANN model) have been proposed to describe the stress–strain relationship of 5A02 aluminium alloy in high-velocity forming process, and their predictability are compared by employing several statistical evaluation measures. The results show that the prediction precision of J–C model is limited, for the calculated coefficient of strain rate is too little to reflect the increasing trend of stress as strain rate is elevated; modified C–S model takes into consideration the effect of strain in the description of strain rate effect, thus it has a more precise correlation between predicted stress and measured stress; based on the influences of strain and strain rate on the dislocation evolution mechanism, the prediction precision of modified R–K model is higher when the strain rate is in the range of 10−3–3000s−1, while the precision decreases as the strain rate exceeds 3000s−1 due to the changed micromechanism of dislocation evolution; ANN model exhibits most powerful predictability due to its self-organizing ability, self-learning ability and high capacity of error tolerance.

Application of an extended Rusinek–Klepaczko constitutive model to predict the mechanical behavior of 6082-T6 aluminum under Taylor impact test conditions

The application of an extended Rusinek–Klepaczko constitutive equation to predict the mechanical response of 6082-T6 aluminum under the Taylor impact test conditions was presented in this article. The numerical results obtained in the study were verified through a comparison with the experimental data extracted from the Taylor anvil-on-rod impact experiments. It was concluded that the extended Rusinek–Klepaczko constitutive model predicts the behavior of the tested aluminum alloy under applied loading conditions with satisfactory accuracy. Moreover, it was found that the plastic wave phenomenon in this material is very limited and that there was no region of constant plastic wave velocity. Strain rates up to 1.6 × 104 s−1 were recorded during the Taylor impact experiments; therefore, this value may be set as the upper limit of the extended Rusinek–Klepaczko model for the alloy, which was validated with the anvil-on-rod experiment.

Results of Strain Rate and Temperature on Mechanical Properties of Selected Structural Steels

The article presents results of application of Rusinek-Klepaczko (RK) constitutive equation for the purposes of modelling of mechanical behaviour of four grades of structural steels. In order to calibrate the RK model a series of compression tests at wide range of strain rates were carried out using two testing stands: servo-hydraulic machine and split Hopkinson bar. The results were analysed to determine influence of temperature and strain rate on visco-plastic response of tested steels. It was concluded that at low temperatures the thermally activated mechanisms of plastic deformation results in the strong strain rate hardening effects, whereas at high temperatures they are limited.

Compressive Viscoplastic Response of 6082‐T6 and 7075‐T6 Aluminium Alloys Under Wide Range of Strain Rate at Room Temperature: Experiments and Modelling

Abstract: In this work, the viscoplastic behaviour of 6082‐T6 and 7075‐T6 aluminium alloys is examined over a wide range of strain rates. Three different testing techniques were applied to this investigation: low‐rate experiments were performed using a regular servo‐hydraulic testing machine, high‐rate tests were conducted using a split Hopkinson bar apparatus and very‐high‐rate experiments were carried out using a miniaturised direct impact test arrangement. The latter testing set‐up allowed for the characterisation of material flow at strain rates up to . These experimental results showed a sharp increase in the rate sensitivity of the materials once a threshold loading rate of is exceeded. This behaviour may be attributed to the presence of viscous drag on high‐velocity dislocation motion. In addition, the thermo‐viscoplastic behaviour of the 6082‐T6 and 7075‐T6 aluminium alloys was analytically described using the extended Rusinek–Klepaczko model of viscous drag effects. Satisfactory correlation was observed between the experiments and the constitutive model results over the entire range of strain rates studied,

Study of thermo-viscoplastic behaviour of TA6V titanium alloy

In this paper, the thermo–viscoplastic behaviour of TA6V under compression loading is analysed. Experiments using two different setups have been performed. Tests at low strain rates were conducted using hydraulic machine. Dynamic tests were carried out for high strain rates using a modified Hopkinson bar which is based on direct–impact technique. Based on these experimental results and to describe thermo–viscoplastic behaviour of titanium alloy studied here, RK (Rusinek–Klepaczko) constitutive relation has been used, this model was proposed in 2001 and takes into account strain hardening, strain rate, and temperature sensitivity. In the present paper, extended RK model is used to approximate TA6V behaviour. A satisfactory agreement has been found between the model predictions and the experimental results for strain rates lower than 5000 s−1.

Thermo-viscoplastic constitutive relation for aluminium alloys, modeling of negative strain rate sensitivity and viscous drag effects

In this paper are presented two extensions of the Rusinek–Klepaczko constitutive relation [Rusinek A, Klepaczko JR. Shear testing of sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress. Int J Plasticity 2001;17:87–115] to define the behaviour of aluminium alloys at wide ranges of strain rate and temperature. The formulations reported extend the validity of the Rusinek–Klepaczko model for the definition of particular aspects of the behaviour of aluminium alloys, namely the negative strain rate sensitivity and the viscous drag. Such formulations are applied to describe the thermo-viscoplastic behaviour of two commercial aluminium alloys ( AA 5083-H116 and AA 7075) using experimental data available in the literature [Clausen AH, Børvik T, Hopperstad OS, Benallal A. Flow and fracture characteristics of aluminium alloy AA5083-H116 as function of strain rate, temperature and triaxiality. Mater Sci Eng A 2004;364:260–72; El-Magd E, Abouridouane M. Characterization, modelling and simulation of deformation and fracture behaviour of the light-weight wrought alloys under high strain rate loading. Int J Impact Eng 2006;32:741–58]. Their analytical results are compared with those obtained from the phenomenological constitutive relations reported in Clausen et al. (2004) and El-Magd and Abouridouane (2006). The best agreement with experiments is achieved by the predictions provided by the extended Rusinek–Klepaczko models. Moreover, the formulations presented may be implemented into FE code using for example the algorithm reported in [Zaera R, Fernández-Sáez J. An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations. Int J Solids Struct 2006;43:1594–612].

Modelling of thermo-viscoplastic behaviour of DH-36 and Weldox 460-E structural steels at wide ranges of strain rates and temperatures, comparison of constitutive relations for impact problems

In this paper, the thermo-viscoplastic behaviour of DH-36 and Weldox-460-E steels is analyzed at wide ranges of strain rates and temperatures. These materials are commonly used for naval applications. Thus, they may be subjected to a wide range of exploitation temperatures and at the same time to high strain rates due to accidental impact or explosion. The thermo-viscoplastic behaviour of these materials has been modeled by application of RK ( Rusinek–Klepaczko) constitutive relation. The predictions obtained using RK constitutive relation have been compared with JC ( Johnson–Cook) and PB ( Physical Base) constitutive relations with use of the experimental results reported in the works of Nemat-Nasser and Guo [Nemat-Nasser, S., Guo, W.G., 2003. Thermomechanical response of DH-36 structural steel over a wide range of strain rates and temperatures. Mech. Mat. 35, 1023–1047] and Borvik et al. [Børvik, T., Hopperstad, O.S., Berstad, T., Langseth, M., 2001. A computational model of viscoplasticity and ductile damage for impact and penetration. Eur. J. Solid. Mech. A. 20, 685–712]. For both metals, a satisfactory agreement is reported between the experimental results and the analytical predictions using RK model at wide ranges of strain rates and temperatures (10 −3 s −1 to 10 4 s −1, and 77 K to about 1000 K). Especially for high strain rate level, the predictions of RK model are notably more precise than those predictions obtained using PB and JC models. This proof converts RK model in suitable for modeling impact problems. Finally, numerical simulations of perforation process of DH-36 and Weldox 460-E steel plates impacted by conical non-deformable projectiles have been carried out using RK and JC models. Numerical results using FE simulations have revealed substantial influence of the constitutive relation concerning the ballistic limit, residual velocity and failure time predictions for the same initial and boundary conditions.

Extension of R– K constitutive relation to phase transformation phenomena

In this paper an extension of the Rusinek–Klepaczko (RK) constitutive relation is presented. The new formulation proposed, allows defining the phase transformation effect observed on macroscopic scale using a phenomenological approach. The key point is to introduce in the original formulation of RK model a new stress component based on evolution of martensite, which takes into account strain, strain rate and temperature effects. Analytical predictions of the extended constitutive relation are compared with experimental results for 301Ln2B steel. This material is chosen since the phase transformation is well observed during quasi-static loading inducing a strong increase of strain hardening rate during plastic deformation. Satisfactory agreement between analytical and experimental observations has been obtained. The phenomenological extension of RK model allows to reduce substantially the computational time in comparison with models based on physical background, for example [Papatriantafillou I, Agoras M, Aravas N, Haidemenopoulos G. Constitutive modeling and finite element methods for TRIP steels. Comput Methods Appl Mech Eng 2006;195:5094–114]. At the same time, the number of material constants defining the extended RK model is reduced. Seven constants are needed to identify the RK model in its original formulation and five are added for the description of phase transformation process.

Constitutive equations for multiphase TRIP steels at high rates of strain

Multiphase TRansformation Induced Plasticity (TRIP) steels show an excellent combination of high strength and high strain values, making them ideally suited for use in vehicle body structures. A complex synergy of three different phases (ferrite, bainite and austenite) on the one hand, and the metastable character of the austenite on the other hand, give the material indeed a high energy absorption potential. The knowledge and understanding of the dynamic behaviour of these sheet steels is essential to investigate the impact-dynamic characteristics of the structures. Therefore split Hopkinson tensile tests are performed in a strain rate range of 500 to 2000 s -1 . Three TRIP steel grades with a different Al and Si content were studied. The experimental results show that these steels preserve their excellent shock-absorbing properties in dynamic conditions. The typical high strain rate loading conditions and the complex behaviour of TRIP steels offer a unique investigation opportunity. This behaviour can be described with phenomenological material models that can be used for numerical simulations of car crashes. The Johnson-Cook model, a frequently used model in finite element codes, is well-suited to describe the dynamic behaviour of the investigated TRIP steels. This model is compared to the Rusinek-Klepaczko model.