Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 3, pp. 645-665, Warsaw 2009 RELATION BETWEEN STRAIN HARDENING OF STEEL AND CRITICAL IMPACT VELOCITY IN TENSION This paper is dedicated to our friend, Prof. Janusz Roman Klepaczko who passed away in August 15, 2008, for his pioneer contributions to the understanding of the Critical Impact Velocity phenomenon José A. Rodŕıguez-Mart́ınez University Carlos III of Madrid, Department of Continuum Mechanics and Structural Analysis, Madrid, Spain; e-mail: jarmarti@ing.uc3m.es Alexis Rusinek Engineering School of Metz (ENIM), Laboratory of Mechanics, Biomechanics, Polymers and Structures(LaBPS), Metz, France; e-mail: rusinek@enim.fr Angel Arias University Carlos III of Madrid, Department of Continuum Mechanics and Structural Analysis, Madrid, Spain In the present paper, a numerical study on the influence of strain harde- ning on the Critical Impact Velocity (CIV) in tension is conducted. Finite element code ABAQUS/Explicit is used to carry out numerical simulations of dynamic tension tests in a wide range of impact velocities up to that corresponding to the CIV. The constitutive relation due to Rusinek and Klepaczko (2001) has been used to define the material behaviour. Strain hardening parameters of the RK model were varied during the simulations. Numerical results are comparedwith those obtained from the analytical de- scription of CIV proposed by Klepaczko (2005). Satisfactory agreement is found between numerical and analytical approaches. The analysis allows for a better understanding of the causes responsible of the CIV appearance. Key words: critical impact velocity, RKmodel, dynamic tension 1. Introduction The study of materials subjected to extreme loading conditions like crash, impact or explosion, is of considerable interest in different industrial fields. A relevant amount of publications can be found in the international literatu- re dealing with high strain rate behaviour of metallic materials related with 646 J.A. Rodŕıguez-Mart́ınez et al. different engineering applications (Arias et al., 2008; Borvik et al., 2002; For- restal and Piekutowski, 2000; Klepaczko, 2006; Klepaczko and Klosak, 1999; Klepaczko et al., 2009; Nemat-Nasser andGuo, 2003; Rusinek andKlepaczko, 2003; Rusinek et al., 2005). Whenmetals are subjected to impulsive loads, the effects of strain harde- ning, strain rate and temperature sensitivity play the main role in the beha- viour of amaterial. Moreover, the thermal coupling cannot be ignored at high strain rates (Klepaczko, 2005). The heat energy due to plastic deformation cannot be transmitted and thematerial behaves under an adiabatic condition of deformation. Such a condition induces localization of deformation which is a precursor of failure. In addition, in dynamic problems, the propagation of elastic and plastic waves that, depending on the initial boundary value pro- blem, could totally govern the response of the material is observed (Rusinek et al., 2005, 2008). An example of an initial boundary value problem which is ruled by the plastic wave effect is the phenomenon called Critical Impact Velocity (CIV). This phenomenon takes place when the speed of plastic waves reaches zero due to localization of plastic deformation in adiabatic conditions. Thus, the existence ofCIV formetals imposes theupper limit to thedynamic tension test for determination of material properties. Loading conditions corresponding to CIV could be reached in some industrial processes like fast cutting, high speed machining or ballistic impact. TheCIV is considered as amaterial property (Clark andWood, 1950; Kle- paczko, 2005;Mann, 1936). Suchaconclusionwas reported for thefirst timeby Mann (1936). In that study, tension impact tests revealed that the maximum energy absorbed by a specimen was well defined for a certain impact veloci- ty independently of length of the specimen. Later, Clark and Wood (1950) confirmed experimentally the existence of CIV in tension. Different specimen lengths were tested in a wide range of impact velocities. The conclusion was analogous to that previously achieved byMann (1936). However, the value of CIV in tension may suffer considerable variations depending on the material considered. Such a conclusion was drawn by Hu andDaehn (1996) estimating analytically CIV in tension for severalmaterials, Fig.1. The normalizedmaterial density ρn introduced in Fig.1 is the ratio of the density to the constant ρn = ρ/K where σ=K · (ε p)n. There are not many studies dealing with the influence of thermo- viscoplastic behaviour of the material on the CIV value. There are not many materials with an identified CIV in tension. Moreover, up to now, the cau- Relation between strain hardening of steel... 647 Fig. 1. Estimation of CIV in tension for different materials (Hu andDaehn, 1996); CIV=A+Bρ−Cn ,A=15.175,B=342.2,C =0.65102 ses which are behind the CIV value exhibited by each particular material are hardly known. Such lack of information is due to different causes: • The experiments required to identify CIV are sophisticated and need expensive technical resources. • The analytical estimations of CIV may be subjected to strong assump- tions. Such assumptions may considerably modify the results obtained from the analytical description of the process. This problemwill be exa- mined ahead in the present paper. Numericalmethods have recently become of relevance in analising theCIV problem (Klosak et al., 2001; Rusinek et al., 2005). In the present paper, the FE code ABAQUS/Explicit is used to conduct numerical simulations of fast tension tests. The application of FE analysis allows to determine the relevance ofdifferent aspects of thematerial behaviouron theCIVvalue.Using FE simulations, the time and cost required to obtain results for a particular problem are reduced in comparison with experiments. In the present paper, the constitutive relation due to Rusinek and Klepaczko (2001) (RK model) is used to define the material behaviour. Strain hardening parameters of the RK model are varied during simulations. Their influence on the CIV value is evaluated. The analysis is conducted for a wide range of impact velocities up to that corresponding to the CIV. Numerical results are compared with those obtained from the analytical description proposed by Klepaczko (2005). The analysis allows for a better understanding of the causes responsible of theCIV appearance. 648 J.A. Rodŕıguez-Mart́ınez et al. 2. The Rusinek-Klepaczko model The RK is a physical-based model founded on the additive decomposition of stress σ (Klepaczko, 1975; Kocks et al., 1975; Seeger, 1957). Thus, the total stress is an addition of two terms σµ and σ ∗, which define the strain harde- ning and thermal activation processes, respectively, Eq. (2.1)1. The first one is called the internal stress and the second one, the effective stress. Themultipli- cative factor E(T)/E0 defines Young’s modulus evolution with temperature, Eq. (2.1)2 (Klepaczko, 1998a) σ(εp, ε̇p,T)= E(T) E0 [σµ(εp, ε̇p,T)+σ ∗(ε̇p,T)] (2.1) E(T)=E0 { 1− T Tm exp [ θ∗ ( 1− Tm T )]} T > 0 where E0, Tm and θ ∗ denote Young’s modulus at T = 0K, the melting temperature and the characteristic homologous temperature, respectively. The constant θ∗ defines thermal softening depending on the crystal lattice of the material (Rusinek et al., 2009). The effective stress is defined as follows σ∗(ε̇ p ,T)=σ∗0 〈 1−D1 ( T Tm ) log ε̇max ε̇ p 〉m∗ (2.2) where σ∗0 is the effective stressat T =0K, D1 is thematerial constant, ε̇ max is the maximum strain rate accepted for a particular analysis and m∗ is a con- stant allowing one define the strain rate-temperature dependency (Klepaczko, 1987). The internal stress is defined by the plasticity modulus B(ε̇ p ,T) and the strain hardening exponent n(ε̇ p ,T) which are dependent on the strain rate and temperature σµ(ε p, ε̇ p ,T)=B(ε̇ p ,T)(ε0+ε p)n(ε̇ p ,T) (2.3) The explicit formulation describing themodulus of plasticity is given by B(ε̇p,T)=B0 ( T Tm log ε̇max ε̇p ) −ν (2.4) where B0 is a material constant, ν describes temperature sensitivity and ε̇max is the maximum strain rate validated for this model. Relation between strain hardening of steel... 649 The strain hardening exponent is defined as follows n(ε̇p,T)=n0 〈 1−D2 ( T Tm ) log ε̇p ε̇min 〉 (2.5) where n0 is the strain hardening exponent at T = 0K, D2 is the material constant and ε̇min is the minimum strain rate validated for this model. In the case of adiabatic conditions of deformation, the approximation of thermal softening of the material via adiabatic heating is given by ∆Tad = β ρCp εp ∫ εe σ(ξ, ε̇ p ,T) dξ (2.6) where β is theTaylor-Quinney coefficient, ρ is thematerial density and Cp is the specific heat at a constant pressure. Transition from isothermal to adia- batic conditions is assumed at ε̇p = 10s −1, in agreement with experimental observations and numerical estimations (Berbenni et al., 2004; Oussouaddi and Klepaczko, 1991; Rusinek et al., 2007). On the basis of model calibration for DH-36 steel reported in Klepaczko et al. (2009), twomaterial constants of the RKmodel are varied, n0 and B0, see Table 1. The range of variation of these parameters is given in Table 1. The material parameters remained constant during simulations are listed in Table 2. Table 1.Parameters of theRKmodel varied during analytical and numerical analysis n0 [–] B0 [MPa] 0.1, 0.2, 0.3, 0.4 750, 1250, 1750, 2250 Table 2.Parameters of the RKmodel assumed constant D2 σ ∗ 0 m ∗ ν D1 E0 θ ∗ Tm Cp β ρ [–] [MPa] [–] [–] [–] [GPa] [–] [K] [J/kg·K] [–] [kg/m3] 0.05 500 2 0.02 0.5 200 0.7 1600 470 0.9 7800 The constitutive relation has been implemented in ABAQUS/Explicit via a user subroutine using the implicit consistent algorithm proposed by Zaera and Fernandez-Saez (2006). In the following section, the configuration used to conduct numerical si- mulations is described. 650 J.A. Rodŕıguez-Mart́ınez et al. 3. Numerical configuration and validation Geometry and dimensions of the specimen used are based on a previous work (Rusinek et al., 2005). Such geometry of the specimen allows for observing well developed necking (Rusinek et al., 2005). A scheme of the specimen is shown in Fig.2. The thickness of the sample is ts = 1.65mm. Its impacted side is subjected to a constant velocity during the simulation. Themovements are restricted to the axial direction. The opposite impact side is embedded. Such configuration idealizes boundary conditions required for the test. Itmust be noted that during experiments it might be difficult to obtain such an ar- rangement (the applied velocity may not be constant during the whole test, transversal displacements of the specimen may occur). However, this numeri- cal configuration is suitable to impose a constant level of deformation rate on the active part of the specimen during the simulations. Fig. 2. Geometry and dimensions [mm] of the specimen used during simulations; Lg =36mm,Lr =37mm,Lt =20mm,W0 =10mm,W1 =20mm The active part of the specimen has been meshed using hexahedral ele- ments whose aspect ratio was close to 1 : 1 : 1 (≈ 0.5× 0.5× 0.5mm3). This definition is in agreement with the considerations reported by Zukas and Scheffler (2000). Beside the active part of the specimen two transition zones are defined. These zones are meshed with tetrahedral elements, Fig.3. Such transition zones allow for increasing the number of elements along the 3 xis of the specimen, Fig.3. This technique is used to get hexahedral elements in the outer sides of the sample maintaining the desired aspect ratio 1 : 1 : 1. Theboundary conditions applied to simulationsmust guarantee the tensile state in the active part of the specimen. In Fig.3a, triaxiality contours during the numerical simulation are shown. It can be observed that the triaxiality Relation between strain hardening of steel... 651 value in the active part of the specimen is that corresponding to the tension state σtriaxiality =0.33. Fig. 3. (a) Mesh configuration used during numerical simulations. (b) Numerical estimation of the triaxiality contours For validation of the numerical approach, a comparison between the analy- tical predictions of themodel and the numerical results is conducted in terms of true stress alongwithplastic strain, Fig.4. It can be seen that the numerical results fit the analytical predictions of the model. It validates the numerical configuration. The oscillation obtained in the numerical values is caused by the elastic wave propagation. It is dissipated along the loading time due to spread of plasticity in the active part of the specimen, Fig.4. Fig. 4. Comparison of the analytical predictions with numerical results (elastic wave propagation: C0 = √ E(T)/ρ=5200m/s, 15µs→ 78mm=length specimen) In the following section, the influence of the main strain hardening para- meters on CIV in tension is analysed. 652 J.A. Rodŕıguez-Mart́ınez et al. 4. Analysis and results The first results reported are those corresponding to the variation of the pa- rameter n0. 4.1. Effect of the strain hardening exponent n0 Analytical predictions of the RKmodel in terms of flow stress along with strain for several values of n0 are shown in Fig.5. It can be observed that the strain hardening dσ/dεp strongly increases with n0, Fig.5b. However, the yield stress level is considerably diminished, Fig.5a. The condition of instabi- lity dσ/dεneck = σ (Considere, 1885) is revealed as highly dependent on the strain hardening exponent n0. The augmentation of strain hardening delays the appearance of instabilities, increasing ductility of the material, Fig.5c. In Fig.5d it is shown that at a high rate of deformation the instability stra- in εneck remains constant. Such a conclusion is in agreement with the ob- servations reported in Rusinek and Zaera (2007). The condition of trapping of plastic deformation dσ/dεp = 0 → Cp = 0, is analysed in Fig.5e. Since the strain hardening increases with n0, the plastic wave speed also does it. Notable differences in the value of the strain corresponding to Cp = 0 con- dition are predicted for different values of n0. Due to these considerations, great influence of the strain hardening exponent n0 on the CIV value can be expected. These expectations are fulfilled in sight of the numerical results shown in Fig.6, where equivalent plastic strain contours are shown for two different impact velocities (V0 =120m/s and V0 =100m/s) and several values of n0. For both impact velocities, in the case of n0 =0.1, thedeformation is localised close to the impact end,Fig.6.TheCIV is reached.Onthe contrary, in the case of n0 =0.4, the necking takes place in themiddle, in one case (V0 =120m/s), and in the opposite impact side, in another case (V0 =100m/s), Fig.6e-h. In those last cases, the plastic deformation is spread along the whole active part of the specimen. In Fig. 8, the equivalent strain rate contours estimated by numerical simu- lations is shown. In the case of n0 = 0.1, the strain rate level is not uniform along the active part of the specimen, Fig.8a-c. A high level of the deforma- tion rate is instantaneously reached after the impact in the zone where the necking takes place, Fig.8a. In the case of n0 = 0.4, once inertia effects are dissipated, the strain rate level along the active part of the specimen remains constant until the necking appears, Fig.8d. Relation between strain hardening of steel... 653 Fig. 5. Analytical predictions using RK model of (a) flow stress and (b) strain hardening along with plastic deformation for different values of n0 at T =300K and 5000s−1. (c) Elongation of the active part of the specimen with n0 at T =300K for V0 =100m/s and V0 =120m/s. (d) Evolution of strain of instability along with strain rate. (e) Analytical predictions using RKmodel of the plastic wave speed with plastic strain for different values of n0 at T =300K and 3000s −1 The trapping of plastic deformation when the CIV is reached induces the loss of equilibrium in the specimen behaviour. In Fig.9, the Input (measu- red on the impacted end) and the Output (measured on the clamped end) forces predicted by the numerical simulations for strain hardening exponents 654 J.A. Rodŕıguez-Mart́ınez et al. Fig. 6. Numerical estimation of the equivalent plastic strain contours for two impact velocities V0 =120m/s (a)-(d) and V0 =100m/s (e)-(h) and different strain hardening coefficients n0 =0.1, 0.2, 0.3, 0.4 Fig. 7. Numerical estimation of the transversal displacement of the active part of the specimen for several values of n0; (a) V0=120m/s, (b) V0 =100m/s Fig. 8. Numerical estimation of the strain rate contours using different values of the strain hardening exponent n0 in the case of V0 =120m/s, 6000s −1, (a)-(b) t=28µs, (c)-(d) t=52µs Relation between strain hardening of steel... 655 n0 =0.1 and n0 =0.4 and for the impact velocity V0 = 120m/s are compa- red. It can be observed that in the case of n0 = 0.1 the equilibrium betwe- en both forces is never reached. On the contrary, in the case of n0 = 0.4, once the inertia effects are overcome, the Input and Output forces meet for a determined force level. Plasticity acts as a filter to dissipate inertia effects. Fig. 9. Numerical estimation of the Input andOutput forces for V0 =120m/s; (a) n0 =0.1 – unstable behaviour (absence of equilibrium between Input and Output forces), (b) n0 =0.4 – stable behaviour (equilibrium between Input and Output forces) According to the experimental results published inMann (1936),Clarkand Wood (1950), Klepaczko (1998b), the CIVmay bemeasured by knowledge of the energy absorbed by the specimen during the impact. When the impact velocity is close to that corresponding to theCIV, the energy absorbed by the specimen is maximum. Then the plastic wave speed in adiabatic conditions near the impact end reaches zero dσ/dεp = 0 → Cp = 0. Once the CIV is overcome, that energy suddenly decreases. Such behaviour is well described by the numerical simulations as shown in Fig.10. 4.2. Effect of the modulus of plasticity B0 The parameter B0 rules the flow stress level of thematerial and its strain hardening. The flow stress level has an effect on the increase of temperature when the material behaves under adiabatic conditions of deformation since ∆T(σ(εp, ε̇p,T)). As the stress level increases, the material temperature does it as well. Moreover, it is known that the thermal softening accelerates the appearance of plastic instabilities and it reduces the strain hardening. Such an effect can be observed in Fig.11a. Increasing the value of B0, the nec- 656 J.A. Rodŕıguez-Mart́ınez et al. Fig. 10. Numerical estimation of the energy absorbed by the specimen along with n0 and impact velocity Fig. 11. Analytical predictions using RK model of (a) flow stress and (b) strain hardening along with plastic deformation for different values of B0 at T =300K and 4000s−1. (c) Displacement of the active part of the specimen with B0 at T =300K for V0 =100m/s and V0 =120m/s. (d) Analytical predictions using RK model of the plastic wave speed with plastic strain for different values of B0 at T =300K and 2000s−1 Relation between strain hardening of steel... 657 king condition dσ/dεneck = σ is delayed along with plastic strain. On the contrary, the condition of trapping of plastic waves dσ/dεp = 0 → Cp = 0 is moved forwards. At low values of plastic deformation the strain hardening increases with B0 (Fig.11b) increasing ductility of thematerial (Fig.11c). At high values of plastic deformation the strain hardening decreases with B0 (Fig.11d). Therefore, the parameter B0 allows for uncoupling the effect that the necking condition and trapping of plastic waves condition has on the CIV. In Fig.12, the plastic strain contours estimated by numerical simula- tions for each value of B0 considered and two different impact velocities, V0 = 120m/s and V0 = 100m/s are shown. The necking position is heavi- ly dependent on B0, Fig.12. It can be observed that the CIV is delayed with the increase of B0. For both impact velocities and B0 =750MPa the necking takes place in the impacted end of the specimen, the CIV condition is fulfilled (Figs.12-13).When B0 =2250MPa, the necking takes place in the embedded side of the specimen for V0 = 120m/s and in the middle of the sample for V0 =100m/s, see Figs.12-13. Fig. 12. Numerical estimation of the equivalent plastic strain contours using different values of the material constant B0 in the case of V0 =120m/s (a)-(d) and the case of V0 =100m/s (e)-(h) Acomparison of the strain rate contours for two values of B0 and two dif- ferent impact velocities is shown in Fig.14. In the case of B0 = 750MPa, the necking is already developed in the impacted end. In the case of 658 J.A. Rodŕıguez-Mart́ınez et al. Fig. 13. Numerical estimation of the transverse displacement of the active part of the specimen for several values of the plasticity coefficient B0; (a) V0 =120m/s, (b) V0 =100m/s Fig. 14. Numerical estimation of the strain rate contours using different values of the material constant B0 in the case of V0 =120m/s, theoretical strain rate level = 6000s−1, (a)-(b) t=32µs, (c)-(d) t=76µs B0 = 2250MPa, the strain rate level remains homogeneous and uniformly spreads all along the active part of the specimen. Those differences in the sample behaviour can be observed comparing the Input andOutput forces, see Fig.15. In the case of B0 =750MPa, both forces never reach equilibrium. A different trend is reported for B0 = 2250MPa. After the inertia effects are dissipated, both forcesmeet alongwith the loading time. The estimation of energy absorbed by the specimen versus impact velocity for all the values of B0 is shown in Fig.16. It can be seen that the maximum energy absorbed by the specimen takes place for the greatest impact velocity when B0 =2250MPa. Next, the numerical estimations are compared withthe analytical results provided by the analytical model developed by Klepaczko (2005). Relation between strain hardening of steel... 659 Fig. 15. Numerical estimation of the Input and Output forces for V0 =120m/s; (a) B0 =750MPa – unstable behaviour (absence of equilibrium between Input and Output forces), (b) B0 =2250MPa – stable behaviour (equilibrium between Input andOutput forces) Fig. 16. Numerical estimation of the energy absorbed by the specimen versus B0 and V0 4.3. Analytical and numerical approach to CIV in tension According to Klepaczko (2005), CIV can be obtained by integrating the wave celerity along strain. The expression for CIV can be split into two parts CIV= εe ∫ 0 Ce(T) dε+ εpm ∫ εe Cp(εp, ε̇p,T) dεp (4.1) 660 J.A. Rodŕıguez-Mart́ınez et al. The first term of Eq. (4.1) corresponds to the elastic range. In that term, Ce(T) is the elastic wave celerity (in a general case may be dependent on tem- perature) and εe is the elastic deformation corresponding to the yield stress in a quasi-static condition. The second term corresponds to the plastic ran- ge. In that term, Cp(ε p, ε̇ p ,T) is the plastic wave celerity dependent on the strain hardening, strain rate and temperature. The upper limit of integra- tion εpm may be considered as the plastic strain value corresponding to the instability criterion dσ/dεpm =σ (Considere, 1885). Another possibility is to consider εpm as the plastic strain value corresponding to the trapping of pla- stic waves Cp → dσ/dεpm =0 (Klepaczko, 2005). However, the use of one or another possibility could stronglymodify the analytical prediction ofCIV for a determinedmaterial, see Fig.17.Moreover, the analytical solution of Eq. (4.1) depends on the constitutive relation used to define the material behaviour since Cp(ε p, ε̇ p ,T)∝σeq(ε p, ε̇ p ,T). In addition, the thermal couplingmust be taken into account (Klepaczko, 2005) and, then, the increase of temperature becomes dependent on plastic deformation dT/dεp 6=0. Fig. 17. Schematic representation of the wave speed along plastic strain for a given strain rate and temperature levels. Influence of the upper limit of integration εpm on the CIV value Next, the results of the CIV value obtained by Eq. (4.1), are compared with the values obtained from the numerical simulations. In order to get an analytical solution to Eq. (4.1), the following procedure has been followed: • The elastic contribution to CIV is calculated to obtain the stress level correspondingto εp =0 fromtheanalytical predictionsof theRKmodel. Then, by application of Hook’s law the upper limit of integration εe is obtained. Assuming a constant celerity of the plastic waves Ce ≈ 5200m/s, the elastic contribution can be obtained. Relation between strain hardening of steel... 661 • The contribution of the plastic range is calculated using the analytical predictions of the RK constitutive relation. Both conditions discussed previously – Condition 1: dσ/dεpm =σ – Condition 2: dσ/dεpm =0 are considered to calculate the upper limit of integration εpm. The analytical and numerical results obtained for the CIV are listed in Tables 3-4. Table 3. Analytical estimations of CIV and comparison with the numerical results B0 [MPa] 2250 1750 1250 750 Condition 1 144m/s 121m/s 101m/s 69m/s Condition 2 269m/s 255m/s 233m/s 190m/s Numerical 130m/s 110m/s 90m/s 70m/s Table 4. Analytical estimations of CIV and comparison with the numerical results n0 =0.4 n0 =0.3 n0 =0.2 n0 =0.1 Condition 1 146m/s 121m/s 102m/s 62m/s Condition 2 317m/s 255m/s 188m/s 118m/s Numerical 130m/s 110m/s 90m/s 80m/s It can be observed that Condition 1 provides the results which better fit the numerical estimations. Although the phenomenon of CIV is governed by Condition 2, the value ofCIV seems to be ruled by the condition of instability, Condition 1. Suchaconclusion allows for optimiziation ofmaterials usedunder dynamic applications which, eventually, may be susceptible to the appearance of instabilities. Some examples are thosematerials used for constructing balli- stic armours or crash-box structures. According to the results reported in this document, metals showing low stress level but high strain hardening seem to bemore suitable for absorbing energy instead ofmaterials showing a high flow stress but a reduced strain hardening. 662 J.A. Rodŕıguez-Mart́ınez et al. 5. Concluding and remarks In this paper, the influence of strain hardening on CIV in tension has been examined using numerical simulations. Thematerial behaviour has been defi- ned by means of the constitutive description due to Rusinek and Klepaczko. The numerical simulations have been conducted for a wide range of impact velocities up to that corresponding to the CIV. Two parameters of the strain hardening formulation of the model have been varied in order to study their influence on the CIV value. The numerical predictions of CIV have been com- paredwith the analytical results. The followingmain conclusions are obtained from the analysis: • Strain hardening shows great influence on CIV of materials. The CIV value strongly increases with strain hardening.A strain hardening incre- ase delays the appearance of plastic instabilities augmenting ductility of the material. An increase of the yield stress leads to a decrease of the energy absorbed by materials due to adiabatic heating. Thermal softe- ning is more important as the flow stress level increases, it reduces the CIV value. • Although theCIVphenomenon is founded on the trapping of plastic wa- ves, theCIV value seems to be ruled by the condition of instability. The analytical approach developed by Klepaczko (2005) allows for defining such behaviour and provides results according to numerical simulations. References 1. AriasA.,Rodriguez-Martinez J.A.,RusinekA., 2008,Numerical simula- tions of impact behaviour of thin steel to cylindrical, conical and hemispherical non-deformable projectiles,Eng. Fract. Mech., 75, 1635-1656 2. Berbenni S., Favier V., Lemoine X., Berveiller N., 2004,Micromecha- nical modelling of the elastic-viscoplastic behaviour of polycrystalline steels having different microstructures,Mat. Sci. and Eng., 372, 128-136 3. BorvikT., LangsethM.,HoperstadO.S.,MaloK.A., 2002,Perforation of 12mm thick steel plates by 20mmdiameter projectiles with flat, hemisphe- rical and conical noses. 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Solids Struct., 43, 1594-1612 30. Zukas J.A., Scheffler D.R., 2000, Practical aspects of numerical simula- tions of dynamic events: effects of meshing, Int. J. Impact Eng., 24, 925-945 Relation between strain hardening of steel... 665 Zależność między umocnieniem odkształceniowym stali a krytyczną prędkością uderzenia przy rozciąganiu Streszczenie Praca przedstawia numeryczną analizę wpływu umocnienia odkształceniowego na krytyczną prędkość uderzenia (CIV) przy rozciąganiu.W symulacjach zastosowa- no oprogramowanie ABAQUS/Explicit oparte na metodzie elementów skończonych. Obliczeń dokonano dla dynamicznych obciążeń rozciągających w szerokim zakresie prędkości uderzenia aż do osiągnięcia wartości krytycznej (CIV). Do opisu materia- łu próbki użyto równań konstytutywnychmodeli Rusinka-Klepaczki. Podczas analizy zmieniano parametry umocnienia odkształceniowego opisanego tymmodelem.Wyni- ki symulacji numerycznych porównano z analitycznymopisemCIV zaproponowanym przezKlepaczkę (2005). Uzyskano zadawalającą zgodność pomiędzy symulacją a teo- rią. Przedstawiona analiza przyczynia się do lepszego zrozumienia zjawisk odpowie- dzialnych za powstawanie krytycznej prędkości uderzenia (CIV). Manuscript received April 21, 2009; accepted for print May 23, 2009