Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 4, pp. 1027-1042, Warsaw 2010 EXPERIMENTAL CHARACTERISATION AND MODELLING OF THE THERMO-VISCOPLASTIC BEHAVIOUR OF STEEL AISI 304 WITHIN WIDE RANGES OF STRAIN RATE AT ROOM TEMPERATURE Alexis Rusinek National Engineering School of Metz (ENIM), Laboratory of Mechanics, Biomechanics, Polymers and Structures, Metz cedex, France; e-mail: rusinek@enim.fr José A. Rodŕıguez-Mart́ınez University Carlos III of Madrid, Department of Continuum Mechanics and Structural Analysis, Madrid, Spain Raphaël Pesci ENSAM–Laboratory of Physics and Mechanics ofMaterials, FRECNRS 3236, Metz cedex, France Julien Capelle National Engineering School of Metz (ENIM), Laboratory of Mechanics, Biomechanics, Polymers and Structures, Metz cedex, France This paper is dedicated to our friend and colleague Prof. W.K. Nowacki who passed away on August 14, 2009. Thank you for your kindness, humility, advice and our long French-Polish collaboration. A thought to Alicja Prof. W.K. Nowacki, Solmech/2006 In this investigation, thermo-viscoplastic behaviour of austenitic steel AISI 304 has been characterised in tension underwide ranges of strain rate at room temperature. This metal possesses an elevated strain hardening rate and ductility which enhance its capability for absorbing energy under mechanical 1028 A. Rusinek et al. loading. It has been observed that the rate sensitivity of the material is inde- pendent of plastic strain. Moreover, it has been noticed that beyond a certain level of loading rate the flow stress of the material sharply increases. In agre- ement with experimental evidences reported in the literature, this behaviour is assumed to be caused by the drag deformation mode taking place at high strain rates. Based on such considerations, the thermo-viscoplastic behaviour of the material has been macroscopically modelled by means of the extended Rusinek-Klepaczko model to viscous drag effects. Satisfactory matching has been found between the experiments and analytical predictions provided by the constitutive relation. Key words: steel AISI 304, extended RK model, strain rate sensitivity, consti- tutive relation, thermal imaging 1. Introduction Understanding dynamic behaviour of metallic alloys has gathered the efforts ofmany researchers during the last decades (Kumar et al., 1968; Clifton et al., 1984; Follansbee, 1986; Regazzoni et al., 1987; Follansbee and Kocks, 1988; Zerilli andArmstrong, 1992; Huang et al., 2009). Development of testing tech- niques which enable one to determine the deformation modes taking place in metallic materials at high strain rates has been a challenge approached by the most relevant experimentalists over the years (Mann, 1936; Clark and Wood, 1957; Klepaczko, 1994; Nemat-Nasser et al., 2001; Nguyen andNowac- ki, 1997). Thus, supported by those experimental investigations, important advances on constitutive modelling of metallic alloys have been carried out (Johnson andCook, 1983; Zerilli andArmstrong, 1987; Nemat-Nasser and Li, 1998; Molinari and Ravichandran, 2005; Voyiadjis and Almasri, 2008; Dur- renberger et al., 2007-2008; Rusinek et al., 2008). Derivation of constitutive descriptions capable of predicting the thermo-viscoplastic response of mate- rials is of fundamental interest to analyse the loading processes. Important industrial sectors like automotive and naval industries are nowadays beco- ming more interested in replacing the traditionally used phenomenological hardening laws (Cowper andSymonds, 1952; Johnson andCook, 1983) bymo- re advanced physical-based constitutive descriptions (Zerilli and Armstrong, 1987; Nemat-Nasser and Li, 1998; Rusinek and Klepaczko, 2001). The lat- ter provides a more accurate description of the material behaviour under wide ranges of loading conditions. Such an advantage allows optimization of materials used for building mechanical elements with elevated structural responsibility. Experimental characterisation and modelling... 1029 For example, experimental testing and theoreticalmodelling of themetallic alloys applied for construction of crash-box structures has focussed the efforts of many investigators. Among thematerials specifically developed for such an application field, the TRIP (Transformation Induced Plasticity) steels have particularly attracted the interest of the scientific community (Fischer et al., 2000; Delannay et al., 2008; Rodŕıguez-Mart́ınez et al., 2010). Thus, in this work, themechanical behaviour of the high-alloy TRIP ste- el AISI 304 has been investigated. The material has been characterised in tension underwide ranges of strain rate at room temperature. Its deformation modes have been determined. Based on such understanding of the material behaviour, AISI 304 has been macroscopically modelled using the extended Rusinek-Klepaczko model to viscous drag effects (Rusinek and Rodŕıguez- Mart́ınez, 2009). It has been proven that this physical-based constitutive re- lation provides an accurate description of the plastic response of the material for all the loading conditions considered. 2. Experimental characterisation of the tensile behaviour of steel AISI 304 at room temperature Steel AISI 304 is the most versatile and most widely used stainless steel. It has excellent formingandwelding characteristics. Extensively used in avariety of industries, its typical applications include pipelines, heat exchanger railings, springs or threaded fasteners. AISI 304 is an austenitic steel (in the undeformed state, themicrostructu- re of AISI 304 is constituted by 100% of austenite) containing large amount of alloying elements asCr andNi improving pitting and corrosion resistance, Table 1. Table 1.Chemical composition of steelAISI 304 [%weight] (De et al., 2006) C Mn Cr Ni Mo Cu Si Nb 0.06 1.54 18.47 8.3 0.30 0.37 0.48 0.027 Next, the thermo-viscoplastic behaviour of this material is examined. Tensile experiments have been performed within wide ranges of strain rate, 10−4s−1 ¬ ε̇ p ¬ 103s−1, at room temperature. The geometry and dimensions of the tensile specimens used in the characterisation are depicted in Fig.1. The thickness of the samples is t=1mm. 1030 A. Rusinek et al. Fig. 1. Geometry and dimensions of the tensile specimens [mm] (Rusinek et al., 2008) It has to be highlighted that this specimen design allows for avoiding geo- metrical disturbances that frequently take place when small samples are used to reach very high strain rates during testing (Rusinek et al., 2005). 2.1. Thermo-viscoplastic characterisation at low strain rates In this section, results from the low strain rate tests performed are pre- sented; the macroscopic behaviour of the material is illustrated in different graphs, see Figs. 2 to 3. Large ductility and an elevated strain hardening rate first characterise the mechanical behaviour of this steel, Fig.2. Let us ana- lyse the case of testing at ε̇ p = 0.002s−1; AISI 304 reaches plasticity for σ ≈ 310MPa, from this point on the flow stress level starts to increase until the saturation condition is reached, σ ∣ ∣ ∣ dσ/dε=0 ≈ 975MPa. Fig. 2. (a) Flow stress evolution as a function of strain and (b) strain hardening evolution as a function of strain and stress at low strain rate and room temperature Moreover, the low strain rate tests were filmed using a high speed infra- red camera. That procedure allowed measuring the temperature increase ∆T Experimental characterisation and modelling... 1031 in the material duringmechanical loading. Knowledge of the temperature in- crease ∆T leads to better understanding of dissipative effects taking place duringmaterial straining. These are, for example, adiabatic heating, marten- sitic transformation or damagemechanics as discussed inRusinek et al. (2002, 2003), Nowacki et al. (2004). In the following graphs, Fig.3, themaximum increase of temperature ∆T during plastic deformation for a strain rate of ε̇ p =0.01s−1 is reported. The material temperature is continuously increasingwith stress and strain, Fig.3a. It has to be noted that a non-linear relation between the temperature increase and plastic deformation is found, Fig.3. Fig. 3. (a)Maximum temperature increase and flow stress versus strain at low strain rate; (b) maximum temperature increase versus time at low strain rate It is detected that this non-linearity is coming from two different sources: • Dependence of the inelastic heat fraction β with plastic strain • Dissipative effects of the martensitic transformation which takes place in this material during straining at low strain rates (Rusinek and Kle- paczko, 2009). The maximum temperature recorded during the test is close to ∆Tmax ≈ 110K, Fig.3b. The drastic increase of temperature taking place close to the saturation stress condition has to be analysed, Fig.3b. Such a sharp augment of the temperature occurs when necking is formed. The heat generated in the instability is hardly spread to the rest of the sample. The strong elevation of the strain rate level in the necking zone leads to local adiabatic conditions of deformation. Next, the results obtained from the high strain rate tests are shown in order to offer a comprehensive analysis of the material rate sensitivity. 1032 A. Rusinek et al. 2.2. Thermo-viscoplastic characterisation at high strain rates Flow stress evolution versus plastic strain for high loading rates is depicted in Fig.4. Under such loading conditions the material keeps the elevated work hardening rate enhancing its ductility. Fig. 4. Flow stress evolution as a function of strain for different high strain rate levels at room temperature It has been noticed that the material strain hardening is kept rather con- stant with the loading rate variations, Fig.4. Thus, the Volume Thermally Activated (VTA) of the material is assumed independent of the plastic strain (Taylor, 1992), Eq. (2.1) V ∗ ≈ kT ∂ ln(ε̇ p ) ∂σ∗ ∣ ∣ ∣ ∣ T,εp (2.1) In the previous expression, Eq. (2.1), k is the Boltzmann constant and T is the absolute temperature. It has to be noticed that beyond a certain level of loading rate, ε̇ p ­ 500s−1, the flow stress of thematerial sharply increases, Fig. 5. Such ob- servation is in agreementwith the experimental results reported byFollansbee (1986), Fig.5. According to several authors (Campbell and Fergusson, 1970; Nemat-Nasser et al., 2001; Rusinek and Rodŕıguez-Mart́ınez, 2009) let us as- sume this behaviour caused by drag deformation mechanisms taking place at high strain rates (Kapoor and Nemat-Nasser, 1999). Based on the previous considerations, the macroscopic behaviour of the AISI 304 is modelled using the extendedRK (Rusinek-Klepaczko) model to viscousdrageffects (RusinekandRodŕıguez-Mart́ınez, 2009).This constitutive description takes into account the independenceof thematerial rate sensitivity with the plastic strain. Experimental characterisation and modelling... 1033 Fig. 5. Flow stress evolution as a function of the strain rate at room temperature for a strain level imposed of εp =0.1. Comparison with the results reported by Follansbee (1986) 3. Macroscopic modelling of the thermo-viscoplastic behaviour of steel AISI 304 under wide ranges of strain rate Next, the formulation of the extended RK model to viscous drag effects and the comparison of its analytical predictions with experimental results are re- ported. 3.1. The extended RK model to viscous drag effects The constitutive description is based on the additive decomposition of the equivalent stress (Seeger, 1957; Zerilli and Armstrong, 1987; Nemat-Nasser and Guo, 2003; Abed and Voyiadjis, 2005; Kocks, 2001), Eq. (3.1) σ(εp, ε̇ p ,T)= E(T) E0 [σµ(ε p, ε̇ p ,T)+σ∗(ε̇ p ,T)]+σvs(ε̇ p ) (3.1) where each term of the previous expression, Eq. (3.1), is defined below. Themultiplicative factor E(T)/E0 definesYoung’smodulusevolutionwith temperature (Klepaczko, 1998), Eq. (3.2) E(T)=E0 { 1− T Tm exp [ θ∗ ( 1− Tm T )]} ∀T > 0 (3.2) where E0, Tm and θ ∗ denote respectively Young’s modulus at T = 0K, the melting temperature and the characteristic homologous temperature. This expression allows for defining the material thermal softening depending on its crystal lattice (Rusinek et al., 2009). In the case of FCC metals like the steelAISI 304, θ∗ ≈ 0.9, as discussed in Rusinek et al. (2009). 1034 A. Rusinek et al. The athermal stress reads as follows, Eq. (3.3) σµ(ε p, ε̇ p ,T)=B(ε̇ p ,T)(ε0+ε p)n(ε̇ p ,T) (3.3) It macroscopically defines dislocation interactions with long-range obstac- les which determine the strain hardening rate of the material. The explicit formulas describing the modulus of plasticity B(ε̇ p ,T) and the strain harde- ning exponent n(ε̇ p ,T) are given by Eqs. (3.4) n(ε̇ p ,T)=n0 〈 1−D2 ( T Tm ) log ( ε̇ p ε̇min )〉 (3.4) B(ε̇ p ,T)=B0 〈( T Tm ) log (ε̇max ε̇ p )〉 −ν ∀ T > 0 where B0 is thematerial constant, ν is proportional to temperature sensitivity, n0 is the strain hardening exponent at T =0K, D2 is the material constant, ε̇min is the lower strain rate limit of the model and ε̇max is the maximum strain rate level accepted for a particular material. TheMcCauley operator is defined as follows 〈•〉= { • if 〈•〉­ 0 0 if 〈•〉< 0 In the following graphs, the evolution of both the modulus of plasticity B(ε̇ p ,T) and the strain hardening coeficient n(ε̇ p ,T) with temperature and strain rate, Fig.6 is depicted. It can be observed that increasing temperature leads to diminution of thematerial flow stress level and strain hardening rate. It allows aproperdefinitionof the characteristic thermal softening takingplace in metallic materials subjected to high loading rates. The thermal stress is the flow stress component defining macroscopically the rate dependent interactions with short range obstacles for the dislocation motion. It denotes the rate controlling the deformationmechanism from ther- mal activation. Based on the theory of thermodynamics and kinetics of slip (Kocks et al., 1975), Rusinek and Klepaczko derived the following expression (Rusinek and Klepaczko, 2001), Eq. (3.5) σ∗(ε̇ p ,T)=σ∗0 〈 1−D1 ( T Tm ) log (ε̇max ε̇ p )〉m∗ (3.5) where σ∗0 is the effective stress at T = 0K, D1 is the material constant and m∗ is the constantdefining the reciprocity strain rate-temperature (Klepaczko, 1987). Experimental characterisation and modelling... 1035 Fig. 6. Evolution of (a) modulus of plasticity and (b) strain hardening coefficient with strain rate and temperature The formulation used for defining the viscous drag stress component is co- ming from the investigations due to Kapoor andNemat-Nasser (1999). Based on theoretical considerations and supported by experimental evidences for a certain number of metals, they proposed the following relation, Eq. (3.6) σvs(ε̇ p )=χ[1− exp(−αε̇ p )] (3.6) where χ is a material constant and α represents an effective damping coeffi- cient affecting the dislocation motion (Nemat-Nasser et al., 2001). In the following graphs, the evolution of the viscous drag stress termwithin wide ranges of strain rate as a function of the material constants χ and α, Fig.7 is depicted. For a strain rate ε̇ p ¬ 100s−1, the viscous drag component is negligible nomatter the value of previousmaterial parameters.Moreover, it can be observed that this formulation enables one to define the two stages of the drag regime experimentally observed (Kapoor and Nemat-Nasser, 2000); the first stage of flow stress linearly increasing with the strain rate and the subsequent stage of the rate sensitivity no longer active. In the case of adiabatic conditions of deformation, the constitutive relation is combinedwith the energybalance principle,Eq. (3.7). Sucha relation allows for approximation of the thermal softening of the material by means of the adiabatic heating ∆T(εp,σ)= β ρCp ε p max ∫ 0 σ(εp, ε̇ p ,T)dεp (3.7) 1036 A. Rusinek et al. Fig. 7. Evolution of the viscous drag stress component as a function of strain rate (a) as a function of χ and (b) as a function of α where β is theTaylor-Quinney coefficient assumed as constant, ρ is themate- rial density and Cp is the specific heat at constant pressure. Transition from isothermal to adiabatic conditions of deformation is assumedat ε̇ p =10s−1 in agreementwith experimental observations reported for example inOussouaddi and Klepaczko (1991), Berbenni et al. (2004). Next, the extendedRKmodel to viscous drag effects is applied to describe the thermo-viscoplastic behaviour of steelAISI 304. 3.2. Application of the extended Rusinek-Klepaczko model to viscous drag effects for description of the thermo-viscoplastic behaviour of steel AISI 304 Following the procedure reported by Rusinek and Rodŕıguez-Mart́ınez (2009), the following material constants for calibration of the extended RK model are obtained, see Table 2. Table 2. Material constants for calibration of the extended RK model to viscous drag effects for steel AISI 304Eqs. (3-1)-(3.7) B0 ν n0 D2 ε0 σ ∗ 0 m ∗ [MPa] [–] [–] [–] [–] [MPa] [–] 1243.6 0.001 0.36 0.035 0.0118 117.72 1.29 D1 χ α θ ∗ ε̇min ε̇max Tm [–] [MPa] [–] [–] [s−1] [s−1] [K] 0.55 200.83 0.0009774 0.9 10−5 107 1800 Experimental characterisation and modelling... 1037 Conventional physical constants of the steel can be obtained frommaterial handbooks, Table 3. Table 3.Physical constants for steel E0 [GPa] Cp [JkgK −1] β [–] ρ [kgm−3] 200 470 0.9 7800 In Fig.8, the comparison between the experiments and analytical predic- tions of the model within wide ranges of strain rate 10−3 s−1 ¬ ε̇ p ¬ 500s−1 is illustrated. It has to be highlighted that the constitutive description enables accurate determination of thematerial flow stress and strain hardening for all the loading rates considered, Fig.8. Fig. 8. Comparison between experiments and analytical predictions of the constitutive description within wide ranges of strain rate Thus, in this investigation a predictive tool which allows proper descrip- tion of the thermo-viscoplastic behaviour of steel AISI 304within wide ran- ges of loading conditions has been developed. It has to be highlighted that the constitutive description defines accurately the strain hardening and ther- mal softening effects observed in this material at high strain rates and large deformation. 1038 A. Rusinek et al. 4. Conclusions In this work, the thermo-viscoplastic behaviour of steel AISI 304 has been characterised in tensionunderwide ranges of strain rates at roomtemperature. The material shows high strain hardening and ductility for the whole range of loading conditions tested. It has been found that the rate sensitivity of the material is independent of plastic strain. Moreover, based on the experiments reported by Follansbee (1986), it has been observed that AISI 304 shows viscous drag effects at high strain rates. Thus, the material response under loading has been modelled using the extended RK model to viscous drag effects. It provides an accurate description of the material behaviour for all the loading conditions analysed. Thus, in this investigation a predictive tool for a proper definition of the thermo-mechanical response of steel AISI 304 within wide ranges of strain rates has been developed. 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Co więcej, zauważono, że powyżej pewnej pręd- kości zmian obciążenia naprężenie płynięcia materiału gwałtownie rośnie.W zgodzie z rezultatamibadańdoświadczalnychopisanymiw literaturze założono, że zachowanie takie wywołane jest pojawieniem się tłumionej postaci deformacji charakterystycznej dla wysokich prędkości odkształcenia. Na podstawie przeprowadzonychbadań zapro- ponowano makroskopowymodel termo-lepkosprężystych właściwości stali w oparciu o rozszerzonymodel Rusinka-Klepaczki dla odzwierciedlenia efektu oporu wiskotycz- nego. Otrzymano zadawalającą korelację pomiędzy wynikami uzyskanymi z konsty- tutywnegomodelu materiału oraz rezultatami badań doświadczalnych. Manuscript received April 21, 2010; accepted for print June 7, 2010