Mechanical Behavior of a Structural Steel at Different Rates of Loading-mild steel

Mechanical Behavior of a Structural Steelat Different Rates of LoadingN.K.Singh,E.Cadoni,M.K.Singha,and N.K.GuptaAbstract The purpose of this chapter is to investigate the mechanical properties of a structural steel under quasi-static and dynamic loads.Specimens of as-received low-carbon mild steel are tested on universal testing machine to study their stress-strain behavior under quasi-static tension(0.001sÀ1)and compression (À0.001sÀ1).Then,the specimens are tested under split Hopkinson pressure bar (SHPB)and modified Hopkinson bar(MHB)to study their material properties under dynamic compressive(À550,À800sÀ1)and tensile(250,500sÀ1)loading, respectively.The material parameters of the existing Johnson-Cook model are determined.Finally,the applicability of the existing Johnson-Cook material model to represent the mechanical behavior of mild steel in plastic zone is examined.Keywords Mild steel•Hopkinson pressure bar•Strain rate sensitivity•Tension •Compression•Johnson-Cook material model1IntroductionLow-carbon steels such as mild steels are widely used as a major structural material in severalfields of engineering.It is found that the stress-strain behavior of mild steels depends on the loading rate[1,9,14].Hence,the knowledge of the mechani-cal behavior of such steels at different strain rates is crucial in severalfields of N.K.Singh•M.K.Singha(*)•N.K.GuptaDepartment of Applied Mechanics,Indian Institute of Technology Delhi,New Delhi110016,Indiae-mail:nilambersingh@;maloy@am.iitd.ac.in;nkgupta@am.iitd.ac.inE.CadoniDynaMat Laboratory,University of Applied Sciences of Southern Switzerland,6952Canobbio,Switzerlande-mail:ezio.cadoni@supsi.chS.Chakraborty and G.Bhattacharya(eds.),Proceedings of the International Symposium859 on Engineering under Uncertainty:Safety Assessment and Management(ISEUSAM-2012), DOI10.1007/978-81-322-0757-3_57,#Springer India2013860N.K.Singh et al. engineering in order to improve the safety against crash,impacts,and blast loads. For example,the crashworthiness characteristics of automobile,train,and airplane are the primary concerns for the transportation industry.The penetration of body armors and sheltering structures by bullets is a challenging problem for the terrorist-hit areas.The safety of socially sensitive structures such as tunnels,bridges,and offshore platforms is of primary national importance.Therefore,several experi-mental studies are reported in the literature over the last few decades to understand the influence of loading rate on the yield stress andflow stress of different grades of mild steel.The study of the mechanical properties at high strain rate needs special experi-mental techniques to record the stress wave propagation in the materials.The Hopkinson pressure bar is one of the widely used experimental techniques for measurement of the mechanical properties of materials at high strain rates[3]. Initially,Campbell and Duby[1]and Marsh and Campbell[9]studied the strain rate sensitivity of EN2A mild steel under dynamic compression;Santosham and Ramsey[14]compared the dynamic elastic behavior of mild steel(type1020), aluminum,and copper.In general,it is observed that theflow stress of mild steel increases with the increase of strain rate.The compressive response of copper,steel, and tungsten alloy steel was studied by Couque and Walker[4]with a split Hopkinson pressure bar technique at strain rates varying from3,000to7,000sÀ1. Lee and Liu[8]investigated theflow stresses of S15C mild steel and S50C medium alloy heat-treatable steel under different compressive strain rates and wide range of test temperatures.Subsequently,Hopkinson pressure bar has been modified by several investigators to study the constitutive laws of different low-carbon steels under dynamic tensile or shear ngseth et al.[7]studied the stress-strain behavior (yield stress,flow stress,and strain hardening)of St52-3N mild steel under tension and shear at different strain rates(0.001–1,000sÀ1).It is observed that the lower yield stress is approximately twice as strain rate sensitive as stresses in the harden-ing region and that the ultimate tensile strength marginally reduces at high strain rates.Vedantam et al.[18]analyzed the tension stress-strain data for mild and DP590steels and observed that mild steel is more strain rate sensitive compared to DP590.High strain rate tension tests have been performed on AISI1018CR steel specimens by Sasso et al.[15].Rusinek et al.[11–13]studied the stress-strain behavior of ES steel at different strain rates and temperatures.Huh et al.[5]studied the mechanical behavior of different high-strength steels used in automobile indus-try at different strain rate(0.001–1,000sÀ1)and observed that strain rate hardening is strongly coupled with strain hardening.Campbell and Ferguson[2]studied the sensitivity of shearflow stress of mild steel EN3B(British standard)that was measured at temperatures from195to713 K and strain rates(10À3to 4Â104sÀ1).Klepaczko[6]studied the behavior of low-alloy steel XC18(French standard)under double shear at different strain rates(0.001–10,000sÀ1).An experimental study on the high strain rate tensile behavior of aluminum alloy 7075T651and IS2062mild steel is reported by Pothnis et al.[10].Mechanical Behavior of a Structural Steel at Different Rates of Loading861 The purpose of this chapter is to study the mechanical behavior of commercially available mild steel at Delhi under quasi-static,dynamic tensile,and dynamic compressive loads.The quasi-static experiments are conducted on a universal testing machine,while high strain rate experiments are conducted on traditional and modified Hopkinson bar apparatus.Further,the applicability of the existing Johnson-Cook material model to represent the strain hardening and strain rate hardening of mild steels under tensile and compressive loads is examined.2Experimental SetupsThree different experimental techniques are employed to study the mechanical properties of mild steel at different rates of loading.The stress-strain behavior of mild steel specimens under quasi-static tension and compression is obtained from the universal testing machine.High strain rate experiments under dynamic com-pressive and tensile loads are carried out in the split Hopkinson pressure bar and modified Hopkinson bar apparatus,respectively,whose working principles are described here.The schematic diagram of the split Hopkinson pressure bar,available at the Impact Mechanics laboratory of IIT Delhi,is shown in Fig.1.It consists of an incident bar(also known as input bar),one output bar,one striker,and instrumen-tation for recording strain data.The input and output bars(each of20-mm diameter and1,500-mm length)are coaxially aligned and have free axial horizontal move-ment.Cylindrical specimen is sandwiched between the input and output bars.The striker of400-mm length and mass1.0kg strikes the input bar at a specified velocity,which creates a trapezoidal compressive stress wave at the striking end of the incident bar.The compressive stress wave(e i)propagates through the input bar and reaches the specimen,where one part of the incident wave is reflected back (e r)to the incident bar and another part(e t)is transmitted to the output(transmission) bar.The strain pulses are measured at the strain gauge stations on the bars(Fig.1). The crushing of the specimen is subjected to the load equilibrium as the signals e i þe r and e t are equal.A pulse shaper is used before the input bar,which deforms plastically as the striker strikes on it.The diameter of the pulse shaper is marginally more than the diameter of the Hopkinson bars,whereas the thickness is1mm.The plastic deformation of the pulse shaper increases the rise time of the compressivepulse in the input bar.This is also useful to modify the shape of the pulse and to minimize thefluctuation in the wave pulse.Experiments under dynamic tensile loads have been performed on the modified Hopkinson bar(Fig.2)available at the DynaMat laboratory of the University of Applied Sciences of Southern Switzerland.It consists of a prestressed bar,incident bar,output bar,and instrumentation for recording strain data.The specimen is connected between the input bar and the output bar.Here,tension in the prestressed bar is suddenly released to create a tensile pulse(e i)in the input bar.The tensile pulse propagates to the specimen with speed C0,where one part of the pulse reflects(e r )back in the input bar due to the impedance mismatch and the other part gets transmitted (e t )through the output bar.The fracture process of the specimen is subjected to the load equilibrium as the signals e i þe r and e t are equal.The strain rate,obtained during the experiment,depends on the prestress value (based on assumed strain rate)of the prestressed bar.The details of the working principle of modified Hopkinson bar are reported elsewhere [16,17]and not repeated here for the sake of brevity.3Calculation of StressesThe wave propagation in both the split Hopkinson pressure bar and modified Hopkinson bar is assumed to be governed with one-dimension elastic wave propa-gation theory by neglecting wave dispersion.Then,the engineering stress (s s ),engineering strain (e s ),and strain rate (_es )on the tested specimens are expressed as [3]s s ¼E A A s e t ;e s ¼À2C 0L Z t 0e r dt ;and _e s ¼À2C 0L e r ;(1)where E is the modulus of elasticity for the bar material,A is the cross-sectional area of the bar,A s is the cross-sectional area of the specimen,L is the gauge length of the specimen,and C 0is the speed of the stress wave in the pressure bar.e r and e t are SpecimenTension waveTension waveInput bar (3m)Output bar (6m)Pre-stressed Bar (6m)Fig.2Schematic diagram ofthe modified Hopkinson bar availableat University of Applied Sciences of SouthernSwitzerlandAir gun Fig.1Schematic diagram of the split Hopkinson pressure bar available at IIT Delhi862N.K.Singh et al.reflected and transmitted pulse strain.The corresponding true stress (s T )and the true strain (e T )on the specimen may be estimated ass T ¼s s 1þe s ðÞand e T ¼ln 1þe s ðÞ:(2)4Materials and Specimen DesignLow-carbon mild steel is obtained from a Delhi-based supplier,whose chemical composition in terms of weight %is C:0.16%,Si:0.27%,Mn:0.77%,S:0.016%,and P:0.024%.Flat specimens (Fig.3a )of the material having gauge length 10mm,width 4mm,and thickness 2mm are prepared for tensile testing.The variation in width and thickness of the specimens is less than Æ2and Æ3%,respectively.The specimen is inserted in the specimen holders,and a steel dowel is inserted to prevent slippage.The specimen and supports are threaded jointly for 8mm,so that the specimen can be gripped with the bars rigidly at its both ends.Cylindrical specimens (Fig.3b )of thickness 10mm and diameter 10mm are employed for compression testing.5Results and DiscussionsStress-strain behavior of as-received mild steel at different strain rates under tensile and compressive loads is discussed here.The engineering stress versus engineering strain curves are compared at various rates of loading in Fig.4,whereas the corresponding true stress versus true strain curves are compared in Fig.5.The yield stresses in these curves are measured at 0.2%offset strain.Three samples at each strain rate are tested at room temperature 25 C,and the average value of the mechanical properties isconsidered.Fig.3Photograph of the mild steel specimens,(a )Tensile specimens (gauge length 10mm,width 4mm,and thickness 2mm),(b )Compression specimens (diameter 10mm,thickness 10mm)Mechanical Behavior of a Structural Steel at Different Rates of Loading 863In case of tensile loading,the engineering yield stresses at strain rates 0.001,250,and 500s À1are 361,688,and 815MPa,respectively,whereas the engineering ultimate tensile strengths are,respectively,483,574,and 576MPa.It has been observed that yield stress of mild steel is more sensitive to strain rate than ultimate tensile strength.The ultimate tensile strength is almost maintained at high strain rates.The flow stress in the strain hardening zone increases marginally compared to the yield stress of the material as the strain rate increases.The flat and transverse sections of the fractured samples are shown in Fig.4at different tensile loading.The reduction in cross-sectional area at strain rates 0.001,250,and 500s À1are 65.36,66.72,and 68.16%,respectively.Therefore,the ductility of the material slightly increases at high strain rate compared to the quasi-static condition.The true yield stresses at strain rates 0.001,250,and 500s À1are 363,691,and 820MPa,-0.001s -1-250s -1-500s -1-1200-800-40004008001200E n g i n e e r i n g S t r e s s (M P a )Engineering Strain (%)Fig.4Engineering stress versus engineering strain curves along with the photographs of the deformed and fracturedspecimens-1200-60006001200T r u e S t r e s s (M P a )True Strain (%)Fig.5True stress versus truestrain curves of mild steel atdifferent rates of loading 864N.K.Singh et al.respectively,whereas the true ultimate tensile strengths are,respectively,596,702,and 704MPa.After the ultimate tensile strength,the necking in the material starts and it is very difficult to analyze the complex stresses that arise during necking.Therefore,true stress-strain curves are considered up to the point of instability at different strain rates under tensile loads.The fracture energy and toughness of the material at high strain rate is higher in comparison to the quasi-static condition because the area under the stress-strain curve at high strain rates is relatively more.In case of compressive loading,the engineering yield stresses at strain rates 0.001,550,and 800s À1are 541,749,and 761MPa,respectively,whereas the corresponding true yield stresses are,respectively,546,758,and 769MPa.As the compressive load increases,the flow stress in the strain hardening zone of the material also increases in the above range of strain rate.This grade of mild steel is stronger in compression compared to tension.The yield stress increases with the strain rate under both tensile and compressive loads.The yielding instability with upper and lower peaks has been found lower in compressive loads compared to the tensile loads.The strain rate remains approximately constant during plastic deforma-tion of the material.In case of dynamic compression,the maximum deformation is shown in Figs.4and 5as the flow stress increases continuously during a particular loading after the lower yield point.It has also been observed that the rate of strain hardening is more under compression as compared to tension.6Material ModelThe Johnson-Cook model considers three independent phenomena,which are isotropic hardening,strain rate hardening,and thermal softening,and these can be isolated from each other.Here,only two phenomena,isotropic hardening and strain rate hardening,have been considered due to the unavailability of temperature data.The model can be expressed ass ¼A þB e p n ÂÃÂ1þC ln _e ý ;(3)where e p is the equivalent plastic strain,_e is the strain rate,_e 0is the reference strain rate,and _eü_e _e 0=is the dimensionless plastic strain rate for _e 0¼0.001s À1.The constants A ;B ,C ;and exponent n are the four material parameters,which are determined using experimentally obtained true stress versus true strain curves.A is the true yield stress of the material corresponding to 0.2%offset true strain at the strain rate of 0.001s À1.The material parameters B and n represent the strain hardening effects of the material at the strain rate 0.001s À1,whereas C represents the strain rate sensitivity of the material.The parameters B ,C ;and n are determined by curve-fitting method in plastic zone from the true yield stress corresponding to 0.2%offset true strain to the point of instability in case of tension.These parameters have been determined up to 8%deformation of the material under compression by Mechanical Behavior of a Structural Steel at Different Rates of Loading 865the same procedure.The estimated material parameters (A ,B ,C ,and n )for the Johnson-Cook model are presented in Table 1for both tension and compression,and hence,the models may be expressed asTension:s ¼363þð779Þe p 0:743ðÞh i Â1þC :ln _e ý (4)Compression:s ¼546þð2692Þe p 0:928ðÞh i Â1þC :ln _eý (5)Here,the strain rate parameter,C ,can be used at a particular strain rate from Table 1.The true stress versus plastic strain curves,obtained from the Johnson-Cook model (Eqs.4and 5),are compared with the experimental results in Fig.6at different strain rates under tensile and compressive loadings.It is observed that the Johnson-Cook model with the estimated material constants (A ,B ,C ,and n )has good agreement with the experimental results.Table 1Johnson-Cook parameters for mild steel Mode A (MPa)B (MPa)n CTension 3637790.7430(0.001s )0.01744(250s )0.01438(500s )Compression 54626920.9280(0.001s À1)0.01824(550s À1)0.02067(800s À1)The value in the bracket indicates strain rate (s-1)for the strain rate sensitivity,CStrain Rate = -800 1/sT r u eS t r e s s(M P a )Plastic Strain (%)Strain Rate = -550 1/sT r u e S t r e s s (M P a )Plastic Strain (%)Strain Rate = -0.001 1/s T r u e S t r e s s (M P a )Plastic Strain (%)Strain Rate = 500 1/sT r u e S t r e s s (M P a )Plastic Strain (%)Strain Rate = 250 1/s T r u e S t r e s s (M P a )Plastic Strain (%)Strain Rate = 0.001 1/s T r u e S t r e s s (M P a )Plastic Strain (%)Fig.6Comparison between predicted and measured true plastic stress versus true plastic strain curves at different strain rates (a )0.001s À1,(b )250s À1,(c )500s À1,(d )À0.001s À1,(e )À550s À1,and (f )À800s À1866N.K.Singh et al.Mechanical Behavior of a Structural Steel at Different Rates of Loading867 7ConclusionsThe isotropic hardening and strain rate hardening of as-received mild steel under tension and compression loads are investigated here.Experiments are conducted at quasi-static,dynamic compressive and tensile strain rates(À800to500sÀ1).The parameters for the Johnson-Cook material model are determined.The present investigation reveals that:•Mild steel is strain rate sensitive.The yield stress increases sharply with the strain rate under both dynamic tensile and compressive loads.However,flow stresses in the strain hardening zone of the material change marginally.•Mild steel exhibits ductile fracture.•Theflow stress obtained from the Johnson-Cook material model with estimated material parameters is in good agreement with the experimental results.•The fracture energy and toughness of the material at high strain rate is higher compared to that at quasi-static condition.Acknowledgments Authors thank D.Forni and M.Dotta of the DynaMat Laboratory,University of Applied Sciences of Southern Switzerland for the help in some experiments.The research was financially supported by the Indo-Swiss Joint Research Program sponsored by the“Department of Science and Technology”of India and“Swiss State Secretariat of Education and Research”of Switzerland.References1.Campbell JD,Duby J(1956)The yield behaviour of mild steel in dynamic compression.Proc R Soc Lond A236(1204):24–402.Campbell JD,Ferguson WG(1970)The temperature and strain-rate dependence of the shearstrength of mild steel.Philos Mag21(169):63–823.Chen W,Song B(2011)Split 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