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Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel
Accepted Manuscript
Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel Selcuk Hazar, Bo Alfredsson, Junbiao Lai PII: DOI: Reference:
S0167-6636(17)30149-7 10.1016/j.mechmat.2017.10.001 MECMAT 2802
To appear in:
Mechanics of Materials
Received date: Revised date: Accepted date:
24 February 2017 14 September 2017 2 October 2017
Please cite this article as: Selcuk Hazar, Bo Alfredsson, Junbiao Lai, Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel, Mechanics of Materials (2017), doi: 10.1016/j.mechmat.2017.10.001
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Highlihgts • Both stress and strain induced phase change are taken into account in the definition of constitutive behavior of high strength steels.
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• The phase transformation enhances and dominates the strength differential effect over plasticity effects.
• Strength differential effect simulated successfully by employing the phase transformation kinetics together with a pressure sensitive surface for plastic yielding.
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• The nonlinearity of elastic strains are also taken into account in the material model.
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Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel
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Selcuk Hazara,∗, Bo Alfredssona , Junbiao Laib a Department
of Solid Mechanics, Royal institute of technology (KTH), 100 44 Stockholm, Sweden b SKF Engineering & Research Centre, P.O. Box 2350, 3430 DT Nieuwegein, The Netherlands
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Abstract
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The stress and strain induced solid to solid phase transformation of retained austenite in a martensitic high strength bearing steel has been studied. Monotonic tension experiments that were carried out at different temperatures using this high strength steel showed that not only the strain induced but also the stress induced phase change plays a crucial role in the phase transformation of retained austenite to martensite. In the material model, plastic deformation was defined using the Drucker Prager yield surface through a nonassociated flow rule accompanied by nonlinear kinematic and isotropic hardening. The hardening was coupled with stress and strain induced phase transformations. A nonlinear elastic effect based on elastic dilation was included in the constitutive model by extending the bulk modulus with a second order term. For the finite element analysis, the material model was written as a user defined material subroutine (UMAT). The numerical simulations were done using ABAQUS and compared to monotonic tension, compression and cyclic experiments. The results showed that the strength differential effect and the volumetric change under loading are closely related to the transformation of retained austenite to martensite. At low temperatures the effect of stress induced phase transformation on yield strength was noticeable. It was concluded that at certain temperatures both strain and stress induced phase transformations significantly affect mechanical behavior of the high strength steel.
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Keywords: Transformation induced plasticity, Stress induced phase transformation, Strain induced phase transformation, Strength differential effect, Nonassociated flow rule
∗ Corresponding
author Email address: [email protected] (Selcuk Hazar)
Preprint submitted to Mechanics of Materials
October 10, 2017
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1. Introduction
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At equilibrium conditions, materials tend to stay in their crystallographically more stable phase i.e. the atomic configuration having the lowest energy level. In high strength bearing steels and other materials undergoing phase change upon cooling, solid to solid phase transformation occurs from the parent phase (austenite), which is stable at high temperatures, to the product phase (martensite) that is stable at low temperatures. During quenching of the high strength steel, when the temperature is above the martensite finish temperature and the energy level required to complete the transformation is not reached, austenite cannot transform completely to martensite. As a result, a metastable phase named retained austenite forms. The amount of austenite retained after heat treatment is affected by carbon content, quenching temperature and alloy composition. Aside from temperature change, application of load and shear band intersection triggers the phase change of retained austenite to martensite. This mechanically induced phase change enhances ductility, formability, toughness and strength of such steels and makes them attractive for automotive, aerospace and variety of other engineering applications. The martensitic transformation of retained austenite upon loading can be either stress or strain induced (assisted). When the temperature is below transformation start temperature (Msd ), stress induced displacive martensitic transformation of the retained austenite to martensite takes place at nucleation sites in the parent phase. Up to Msd stress induced transformation takes place below the yield stress (σ Y ) of the material. In this temperature range, the transformation start stress increases linearly with the temperature (Bolling and Richman,
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1970a). Kinetics of stress induced transformation is described using either Gibbs or Helmholtz free energy potentials by considering the thermodynamic effect of the applied stress (Abeyaratne and Knowles, 1993; Bhattacharyya and Weng, 1994; Chan et al., 2012; Entchev and Lagoudas, 2004). Using Helmholtz free energy, Moumni et al. (2011) proposed a model for stress induced phase transformation based on the formalism of principle of generalized standard materials with internal constraints (Halphen and Nguyen, 1974; Moumni, 1995). The simulations that they performed using the thermomechanical model had good correlations with experimental results. In the literature, the constitutive models of stress induced transformation are studied not only for high strength steels but also for other alloys like Nitinol (Arghavani et al., 2010; Auricchio et al., 2007; Chan et al., 2012; Levitas and Ozsoy, 2009a; Moumni et al., 2011; Popov and Lagoudas, 2007; Saleeb et al., 2011; Zaki and Moumni, 2007b; Zhang et al., 2007). Strain induced (assisted) phase change initiates through formation of highly potent new sites via shear band interactions when the temperature is above Msd . In this case, depending on temperature, the transformation start stress can be higher than the yield stress. In previous studies, the strain induced transformation, that is commonly investigated in TRIP steels, was found to undergo irreversible martensitic transformation (Cherkaoui et al., 2000, 1998; Goel et al., 1985; Idesman et al., 2000; Iwamoto and Tsuta, 2000; Leblond et al., 1986; Olson and Azrin, 1978; Olson and Cohen, 1975; Sangal et al., 1985; Stringfellow et al., 1992; Taleb and Sidoroff, 2003; Tomita and Iwamoto, 1995). Olson and Cohen (1975) proposed a model that outlines the strain induced phase transformation behavior in austenite steel through interaction of shear bands. Their model was only able to capture uniaxial stress effects under isothermal conditions. Stringfellow et al. (1992) enhanced the model of Olson and Cohen (1975) by considering the stress state sensitivity, since the evolution of martensite at a material point is effected not only by the plastic strain and temperature but also by the stress state history. The model proposed by Stringfellow et al. (1992) was improved by taking into account of the influence of strain rate on toughness and ductility (Tomita and Iwamoto, 1995), thermocoupled effects (Iwamoto, 2002, 2004), deformation mode dependence (Iwamoto et al., 1998) and the effect of grain size (Iwamoto and Tsuta, 2000). Perlade et al. (2003) and Papatriantafillou et al. (2006) used the model of Stringfellow et al. (1992) to study the multiphase TRIP steels having different retained austenite levels. In a later study Sierra and Nemes (2008) modified the same model to study the interaction between embedded retained austenite islands and the surrounding matrix. In the early attempts, formulation of transformation kinetics in TRIP steels is based on either large scale orientation of the transformation strain in the martensite phase (Magee, 1966) or the micromechanical plastic strain accommodation that emerges from martensitic phase transformation and arises in the parent phase (Greenwood and Johnson, 1965). Based on these two principles, most of the constitutive models for TRIP steels are derived using either macro scale (Cherkaoui et al., 2000; Fischer et al., 1998, 2000; Leblond et al., 1989;
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Levitas, 1998; Moumni et al., 2011; Perlade et al., 2003; Tanaka et al., 1996) or micro scale approaches (Levitas and Ozsoy, 2009a,b; Ma and Hartmaier, 2015). In the vast majority of constitutive models developed for materials undergoing ˙ is prescribed as phase transformation, the rate of martensite volume fraction (ζ) the main internal state variable. Previous researches have documented that the rate of martensite transformation is affected by Ms temperature (Inoue et al., 1984; Neu and Sehitoglu, 1992), hydrostatic stress (Jacques et al., 2001; Neu and Sehitoglu, 1992), carbon content (Neu and Sehitoglu, 1992; Seong et al., 2004), quenching temperature and rate, plastic strains (Seong et al., 2004) and percentage of retained austenite (Neu and Sehitoglu, 1992). The study by Young (1988) showed that evolution of martensite fraction with respect to plastic strain during stress induced transformation is more linear compared to the case of strain induced phase change. Not only the transformation kinetics, but also the morphology of the martensite microstructure formed via stress or strain assisted transformation differs. Strain induced phase change results in fine lath martensite phase formed at the shear band intersections, whereas plate martensite formation is observed when the transformation is induced by stress (Young, 1988). Phase transformation in TRIP steels results in a strength differential effect (SDE), which is known as the asymmetry between tensile and compressive yield stresses, because the retained austenite is more susceptible to transform to martensite under tension than in compression. The stress induced transformation is affected by temperature, for this reason SDE is found to be higher when the temperature is decreased (Singh et al., 2000). It is observed that SDE depends not only on temperature but also on the hydrostatic stress, therefore Drucker Prager yield surface has to be used to define the plastic flow (Casey and Sullivan, 1985; Spitzig and Richmond, 1984). In addition, in high strength steels the associated flow rule would result in a plasticity induced volume change which is different from the experimental findings, for this reason a nonassociated flow rule has to be considered in the material model (Linares Arregui and Alfredsson, 2010; Stoughton and Yoon, 2004). Not only the phase transformation but also microcracking, residual stresses, volume expansion and solute dislocation interaction are some of the factors that can give rise to SDE. For instance Linares Arregui and Alfredsson (2010) observed SDE in high strength bainitic steel which did not undergo any mechanically induced phase transformation. Previously, the stress dependency of elastic modulus for high strength steels is discussed by some researchers (Cleveland and Ghosh, 2002; Lee et al., 2013; Linares Arregui and Alfredsson, 2013; Sun and Wagoner, 2011; Yoshida et al., 2002; Zavattieri et al., 2009). According to the results of push–pull cyclic experiments by Sommer et al. (1991) on SAE 52100 high strength steel, when the linear elastic part is subtracted, a sickle shaped asymmetric hysteresis loop is obtained in stress vs. plastic strain graphs. In a recent work, Linares Arregui and Alfredsson (2013) showed that the nonlinear elastic material behavior should be considered in the material model of the high strength bainitic roller bearing steels. Moreover, Linares Arregui and Alfredsson (2010) used combined nonlinear kinematic and nonlinear isotropic behaviors to precisely define the
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Msd
Stress Induced Transformation Coupled Transformation
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Figure 1: The effect of temperature on the start of strain and stress induced transformations. Yellow region represents where the austenite is stable, blue and gray zones represents the stress induced and coupled transformation zones respectively.
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plastic material response in their experimental measurements. Figure 1 illustrates the temperature effect on the start of strain and stress induced transformations. At a certain critical temperature stress induced transformation starts at a certain stress level, σs , which can be below yield stress (Bolling and Richman, 1970a; Olson and Cohen , 1972). Therefore coupled transformation behavior takes place after plastic yielding. Strain induced transformation on the other hand starts at the onset of yielding. Above Msd the shape of yield stress curve representing the case of coupled transformation depends on material properties but increases up to a certain temperature above which no phase change is observed (Young, 1988). Therefore, coupled transformation behaviour takes place after plastic yielding. According to Neu and Sehitoglu (1991) it is difficult to distinguish stress induced from strain induced transformation when the deformation increases beyond the yield point. Recently Ma and Hartmaier (2015) proposed a constitutive model considering strain induced martensite nucleation and stress assisted martensite growth using Helmholtz free energy potential. Although they stated that their numerical calculations fit to the experimental results of Lebedev and Kosarchuk (2000), no direct comparison were presented between the numerical results and the corresponding measurements. In the present research, stress and strain induced phase transformation mechanisms were coupled with plastic deformation that was described using a Drucker Prager yield surface with a nonassociated flow rule accompanied with combining nonlinear kinematic and isotropic hardening. For this purpose the model represented by Stringfellow
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2. Material model
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This section presents the constitutive model of the investigated martensitic high strength bearing steel (DIN 100CrMnMoSi8), with the chemical composition given in Table 1. Figure 2 shows the microstructure of the quenched and tempered martensitic steel with retained austenite content.
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et al. (1992) was extended by including the stress induced transformation effect using Nguyen's formalism of standard materials, which was applied to TRIP steels by Moumni et al. (2011). The temperature dependence of the onset of nonlinear stress strain behavior was the main motivation for implementing coupled stress and strain induced phase change. Compared to the models existing the literature this model couples stress and strain induced phase changes with a nonassociated flow rule and nonlinear elasticity. The predictions were in a good agreement with the experimental findings. Since the morphology and mechanisms of the stress and strain induced martensite phases differ, the proposed model includes the contribution of each phase separately and in combination to the other mechanical properties of the high strength steel, such as nonlinear elasticity, nonlinear hardening and pressure dependence of the yield surface. The paper is structured in the following way: First the material model is described in section two. The first subsection introduces the free energy potential and the effect of nonlinear elasticity. The second subsection describes the nonlinear kinematic and isotropic hardening rules together with nonassociated flow rule. The third subsection presents the strain induced phase transformation mechanism and combines it with the stress induced transformation mechanism. In section three experiments and determination material parameters are discussed. The section 4 describes the method and procedure of numerical simulations. Finally, conclusions are drawn.
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Table 1: Chemical composition (wt %) of 100CrMnMoSi8 (Lai et al., 2016).
Cr 1.86
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2.1. Definition of free energy potential and nonlinear elasticity Patel and Cohen (1953) were among the first researchers to investigate the contribution of the applied loading to phase transformation through a free energy formulation. Using a similar approach, Olson and Cohen (1982) represented the stress state sensitivity of the transformation and the toughening effect of stress assisted transformation. In this part, the thermodynamic force conjugate to evolution of martensite fraction was computed using the formalism of generalized standard materials with the internal constraints that were derived by
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Figure 2: Microstructure of martensitic high strength steel DIN 100CrMnMoSi8 (Lai et al., 2016), in this work some of the retained austenite islands are marked inside red circles.
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Halphen and Nguyen (1974). A similar aproach was implemented by Moumni et al. (2011). In this study Reuss scheme was used to define the macroscopic strain tensor which is an average over the representative volume element and given as (1 −
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By considering the contribution of internal variables of plastic deformation, the free energy potential can be written as 1 m m m 1 a a a εij Eijkl εkl + ζ εij Eijkl εkl + C(T ) W = (1 − ζ) 2 2 (5) 3 X 1 1 1 pl m m m el 3 pl + C χij χij + K1 εkk + Q εeq + exp(−bεeq ) . 3 m=1 3 b 8
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where Λij , ν1 and ν2 are the Lagrange multipliers. ν1 and ν2 are associated with unilateral constraints and
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In Eq. (5), E a and E m are the linear elasticity tensors for martensite and austenite phases. C(T ) is the phase change heat density that is assumed to be a linear function of temperature (Zaki and Moumni, 2007b) C(T ) = ρ(T − Ms ) + κ.
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In Eq. (5), C is the slope between the back stress Xij , where X˙ ij = 32 C ε˙pl ij − pl ˙ γXij λ, and the plastic strain rate ε˙ij when γ = 0, C/γ defines the saturated value of evolution of the back stress. χij is the conjugate of back stress Xij and defined as 3 X 3 m m χ˙ ij = C Xij (7) 2 m=1
In Eq. (5), Q is the limiting value of the change in isotropic yield surface size 3 and b is the rate of hardening before reaching the stress limit. The term K1 εel kk represents the nonlinear elastic potential with the same nonlinear bulk modulus parameter K1 for both phases. εel ij is the elastic strain tensor that is composed of the elastic strains of the martensite and austenite phases, a m εel ij = (1 − ζ)εij + ζεij .
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Given the expressions of the free energy density W and of the constraints potential Wl in (3) and Eqs (5), the Lagrangian L = W + Wl was constructed to define the generalized forces (Halphen and Nguyen, 1974; Zaki and Moumni, 2007b) 1 a a a 1 m m m L(εij , T, εaij , εm , ζ) = (1 − ζ) ε E ε + ζ ε E ε + C(T ) ij 2 ij ijkl kl 2 ij ijkl kl 3 1 1 X m m m 1 3 pl pl + K1 εel + C χ χ + Q ε + exp(−bε ) kk ij ij eq eq 3 3 m=1 b h i pl tr − Λij (1 − ζ)(εaij ) + ζ(εm ij ) + εij + εij − εij
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in the equations above Cζ is the only nonzero thermodynamic force. Using the Eqs (10), (11), (12) and (14) the constitutive equation became 2
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In the work of Linares Arregui and Alfredsson (2013) a small but significant nonlinear elastic behavior was detected during uniaxial cyclic push pull experiments on high strength bainitic steel. The nonlinear elastic behavior was traced to the volumetric bulk behavior of the material and defined using the bulk modulus as a function of elastic strains, K(εel kk ) and a constant shear modulus, G. Using Eq. (15) eq el K(εel (20) kk ) = K0 + K1 εkk .
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The nonlinearity of Young's modulus and Poisson's ratio could be written in terms of their relation to K(εel kk ) as (21)
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Appendix A shows the derivation of uniaxial elastic strain, εel 11 , at tension or compression using Eqs (21) and (23). 10
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Figure 3: W L is the linear elastic strain energy and W N L is the nonlinear elastic strain energy.
The strain energy function of the nonlinear elastic material became 1 2 a NL a a 2 a a a W =(1 − ζ) K0 − G0 (εkk ) + G0 εij εij + 2 3 1 2 1 el 3 m 2 m m m ζ K0m − Gm (ε ) + G ε ε kk 0 ij ij + K1 εkk , 2 3 0 3
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L which can be stated alternatively as W N L = W L + 13 K1 εel repkk , where W resents sum of the linear elastic strain energies of both phases. The change of W N L / W L with respect to hydrostatic strains is plotted in Fig. 3 for ζ = 0.60, 0.80, 0.99 to see the effect of phase change on nonlinear elasticity. Fig. 3 shows that strain energy is effected by nonlinearity when the strains are high and the nonlinear elasticity is more noticeable at high ζ when the retained austenite level is low. The stress strain graph of a push pull experiment under a cyclic load with a compressive mean load, Pm = −20.8 kN and amplitude of Pa = 48.5 kN performed by Linares Arregui and Alfredsson (2010) (test no: 13832) is given in Fig. 4(a). The figure shows the 1st and 10th cycles of the experiment. The effect of nonlinear elasticity on the inelastic stress strain cycle is given in Fig. 4(b). The inelastic curves are derived either by subtracting linear elastic (L), εin 11 = el = ε − ε and Eq. (24) ε11 − σ/E eq , or nonlinear elastic (NL) strains, using εin 11 11 11 from the experimental results in Fig. 4(a). When the two cyclic responses shown in Fig. 4(b) are compared, it is observed that nonlinear elastic strains exist during unloading, since steeper response is obtained for the NL curves. The vertical unloading response from both tension and compression that was derived
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2.2. Nonlinear kinematic and isotropic hardening
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By virtue of the SDE observed in the push pull experiments conducted by Linares Arregui and Alfredsson (2010), it was found that a Drucker Prager type (Drucker and Prager, 1952) yield surface with a linear dependency on the hydrostatic stress had to be used to define the plastic behavior. Hence, the yield surface was defined as f = σeq + 3aσ h − σ Y , (26) where a is the slope of the offset yield stress with respect to hydrostatic stress σ h = σii /3. σ Y defines the yield stress which could be expressed using nonlinear isotropic hardening behavior (Chaboche, 1991)
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σ Y = σ Y 0 + R.
˙ R˙ = b (Q − R) λ.
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using NL illustrates the presence of nonlinear elastic strains. Hence, the cyclic response must be studied in order to separate nonlinear elasticity from plasticity. The martensite fraction ζ at start of the first cycle was taken as ≈ 0.78 and at 10th cycle it was ≈ 0.85. These values, ζ = 0.78, 0.85, were measured using X-ray diffraction on an undeformed reference configuration and on the specimen midsection where it had broken after 10 cycles.
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The equivalent stress, σeq , used in Eq. (26) was defined as
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d d where σij is the deviatoric stress, σij = σij −σh δij , and δij is the Kronecker delta. In Eq. (30), the nonlinear kinematic hardening is introduced using the back stress tensor, described by Chaboche (1986, 2008); Frederick and Armstrong (2007), 2 ˙ − γXij λ. (31) X˙ ij = C ε˙pl 3 ij In the definition by Chaboche (1986) three sets of kinematic hardening parameters are used 2 m m m˙ X˙ ij = C m ε˙pl (32) ij − γ Xij λ, 3 with 3 X m Xij = Xij . (33)
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d 3 d σij − Xij σij − Xij 2
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Experimental measurements of the volume change due to the dislocation motion in austenitic steels undergoing phase change was found to be different from the predictions of the associated flow rule (Garofalo and Wriedt, 1962; Linares Arregui and Alfredsson, 2010; Rauch et al., 1975; Spitzig and Richmond, 1984), for this reason a nonassociated flow rule was used,
M
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267 268
˙ ε˙pl ij = λ
270 271
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273
274 275 276
δg . δσij
R−
3 P
m=1
m 3 γ m m 2 C m αij αij
≤
σY 0 + R
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266 267 268
(35)
a∗ ≤ 1, a
269 270 271 272
(36)
in order to satisfy the dissipation equation. The derivation is stated in Appendix B. Using the same procedure as (Linares Arregui and Alfredsson, 2010) the value of a∗ was continuously checked during the simulations to satisfy the condition in Eq. (36) when the plastic strains developed. 13
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(34)
The nonassociated flow ruled used to define plastic material behavior might result in a physically instabil solution and violence of principal of maximum dissipation. For this reason, the bounds on the pressure dependence in nonassociated flow rule were defined by the
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3 X 3 γm m m X X = constant, 4 C m ij ij m=1
where a∗ was used to predict plastic dilatation ∆V /V0 , which is known to be very small compared to volume expansion due to phase change (Neu and Sehitoglu, 1992). Using the nonassociated flow rule the plastic strain was defined as
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g = σeq + 3a∗ σ h − σ Y +
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2.3. Strain induced transformation coupled with stress induced phase transformation The strain induced phase change, that occurs after plastic yielding, see Fig. 1, was introduced by implementing the extended version of Olson and Cohen's model (Olson and Cohen, 1975) which was proposed by Stringfellow et al. (1992). The evolution of strain induced martensite fraction, ζ˙strain , was formulated as ζ˙strain = (1 −
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˙ + Bζ Σ),
Aζ given in Eq. (37) was defined as
287
Aζ = αβ(1 − fsb )(fsb )r−1 P ,
291 292 293 294 295 296
where εeq is the equivalent plastic strain of the austenite phase. It was calculated using Eq. (35) assuming the same hardening properties for both phases (Zaki et al., 2010). P in Eq. (39) represents the probability function for transformation start, which was determined by assuming a Gaussian distribution of the potency of the shear bands to activate the nucleation sites that initiate transformation (Stringfellow et al., 1992). The probability function was defined as " 2 # ZΓ 1 1 Γ0 − g¯ P =√ exp − dΓ0 , (41) 2 sg 2π
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where α, β and r are the material parameters. The volume fraction of shear bands, fsb , is a function of equivalent plastic strain. The equation that describes fsb was written as fsb = 1 − exp −αεpl(a) , (40) eq pl(a)
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281 282
283
284 285 286
288 289 290
291 292 293 294 295 296 297
−∞
where g¯ is the mean value of the distribution and sg is the standard deviation. Γ depends on temperature and stress triaxiality through a dimensionless driving force function for martensite nucleation
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(39)
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(37)
where ζ is the overall martensite volume fraction and Σ represents the stress ii , which has an effect not only on the nucleation of voids triaxiality, Σ = σσeq (Rice and Tracey, 1969) but also on nucleation of martensite sites (Jacques et al., 2001; McClintock, 1968). Σ˙ was calculated as σ˙ ii σ˙ eq ˙ Σ=Σ − . (38) σii σeq
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ζ)(Aζ ε˙pl eq
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Γ = g0 − g1 Θ + g2 Σ,
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(42)
in which g0 is a material parameter, g1 and g2 are the temperature and the linear triaxiality dependency coefficients. Θ is the normalized temperature defined as ˙ = 0, P˙ could be derived as Θ = MTs . When Θ " 2 # g2 1 Γ − g¯ ˙ ˙ P =√ exp − Σ. (43) 2 sg 2πsg 14
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In Eq (45) ∆v is the relative volume change due to martensite transformation (∆v ≈ 0.02 − 0.05 (Leal, 1984)). N is defined as N = using σeq A = A0 + A1 , sa
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1 ˙ (Cζ )2 ξ1 (ξ2 − ζ) + Cζ (Aζ ε˙pl eq + Bζ Σ). 2
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307
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(47)
where ξ1 and ξ2 are material parameters and Cζ is given by Eq. (13). In Eq. (13) the term multiplied by Cζ is responsible for stress induced phase transformation and determines the value of evolution of stress induced martensite fraction, ζstress . In order to satisfy the dissipation inequality the requirement of Cζ > 0 is discussed in the Appendix C. 3. Experiments and Material parameters To examine stress and strain induced transformations and determine the parameters related to the stress and strain induced transformation mechanisms, monotonic tests were performed at different temperatures. For this purpose, tension experiments were carried out at 22◦ C (RT), 100◦ C and 150◦ C using servo-hydraulic test machines and the specimens shown in Fig. 5. The thin steel specimens with rectangular cross section used in this study were wire electro-discharged from forged rod specimens which had been hardened together with the specimens used in the experiments conducted by Linares Arregui and Alfredsson (2010). Thus, the material and heat treatment were 15
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(46)
Therefore, the evolution of martensite fraction can be found using the function ∂Φ ˙ ζ˙ = = Cζ ξ1 (ξ2 − ζ) + Aζ ε˙pl (48) eq + Bζ Σ, ∂Cζ
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where A0 and A1 are material parameters and sa is the reference austenite hardness (Stringfellow et al., 1992). Finally, the stress and strain transformations were united through the pseudo potential for dissipation of the phase transformation proposed by Moumni et al. (2011). It was here modified as
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and A was calculated
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d 3σij 2σeq
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(44)
where H() is the Heaviside function. Using the evolution of martensite in Eq. (37), the transformation strain became ˙ √1 AN + 1 ∆v δij . ε˙tr = ζ (45) ij 3 2
Φ= 315
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The parameter Bζ in Eq. (37) was (Stringfellow et al., 1992) " 2 # g2 1 Γ − g¯ r ˙ Bζ = √ β(fsb ) exp − H(Σ), 2 sg 2πsg
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Figure 5: Tensile test specimen, thickness 0.70 – 0.76 mm. Scale is cm with mm divisions. 4000 3500
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σ (MPa)
2500
σ (MPa)
1800 1600 1400 1200 1000 800 600 400 200 0 0.0
2000
Exp. RT Exp. 75 ◦ C Exp. 100 ◦ C Exp. 150 ◦ C 0.2
0.4
0.6
ε
0.8
1.0
1000
Exp. Tension Exp. Compression
500
0 0.0
1.4 1e−2
0.5
1.0
1.5 2.0 ε
2.5
3.0
3.5 4.0 1e−2
(b)
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(a)
1.2
1500
333 334 335 336 337
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identical for all specimens used in the present and earlier experiments by Linares Arregui and Alfredsson (2010). The specimens with the aforementioned steel grade DIN 100CrMnMoSi8 had been heat treated to martensitic microstructure with a retained austenite content. The monotonic test setup was placed in an oven to maintain constant temperature during the experiment and the axial strains were measured using extensometers. To avoid assembly stresses, the fixtures were carefully aligned with the load line using metal shims. In addition to these experiments, the results of tension, compression and cyclic experiments performed at 75◦ C were taken from the study of Linares Arregui and Alfredsson (2010). The stress strain curves in monotonic tension, compression and cyclic experiments are presented in Figs 6 and 7(a). In this work, the plastic and nonlinear elastic material parameters given by Linares Arregui and Alfredsson (2010) and Linares Arregui and Alfredsson (2013) were used as starting values to determine the material parameters of the corresponding model that considers phase transformation. In their work the phase transformation was not explicitly modeled, instead all the inelastic strains were included in the plastic description. The material properties for nonlinear kinematic and isotropic hardening given in Table 2 are based on the
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Figure 6: a Tension experiments to specimen rupture b. Tension and compression experiments at 75◦
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0.95 1000
−1000 −2000
0.85 0.80
Exp. 75 ◦ C Cyclic
−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015
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σ (MPa)
0.90 0
FE 75 ◦ C Tension FE 75 ◦ C Cyclic Exp. 75 ◦ C Tension Exp. 75 ◦ C Cyclic
−0.010 −0.005
ε
(a)
0.000 ε
0.005
0.010
(b)
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studies by Linares Arregui and Alfredsson (2010). Since the present investigation focused on and modeled the phase change taking into acount both stress and strain induced transformation, some parameters in the plastic description that are given in Table 2 were adjusted from the values in the reference (σ Y 0 , a, a∗ , b, Q, C1 , C2 , C3 , γ1 , γ2 , γ3 ). The plasticity parameters were updated iteratively using inverse modeling of the stress strain curves with respect to the experimental results in Figs 6 and 7(a). The parameters quantify the effects of the different mechanical phenomena described in Eqs (1) to (48). They are grouped accordingly in Table 2. In previous studies it is reported that the elastic modulus of martensite is less than that of austenite (Liu and Shan, 2011; Mendiguren et al., 2015) therefore different elastic moduli have to be defined for each phase. The equivalent elasticity tensor should vary during the transformation with respect to the change of martensite fraction as given in Eq. (16). Here the Poisson's ratio and nonlinear bulk modulus K(εel kk ) were taken from Linares Arregui and Alfredsson (2013). The same values were used for both phases. The Young’s moduli for each phase were then determined by inverse modeling of the monotonic experiments presented in Fig. 6, the curves describing the nonlinear elastic behavior given in Fig. 4(b) and using Eqs (16) and (20). The X–ray diffraction (XRD) measurements were performed using a Bruker D8 Discover instrument and Co K–alpha radiation. A 1D silicon strip detector with 192 strips were used and a programmable divergence slit maintained the sample illumination at 5 mm throughout the measurements. The instrument was operated at 40 kV and 35 mA, and the step size used was 0.03 deg. The counting time was varied depending on the cross-section area of the sample to assure proper counting statistics. The accuracy of 2.2-3 % was maintained in the date analysis of single martensite fraction values. The measured X-ray diffraction values at different temperatures are shown in Fig. 8. The parameters affecting the kinetics of the phase transformation were set from the retained austenite levels measured after deformation. The parameter
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Figure 7: a. Cyclic loading, experimental data taken from Linares Arregui and Alfredsson (2010) (test no: 13838). b. Uniaxial strain vs. martensite fraction.
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Table 2: Material parameters used to fit the numerical predictions to experimental data.
Value
Parameter
920 MPa
a
76 × 10−6 Q
135 GPa 700 GPa 50 GPa
γ1 γ2 γ3
5.2 57 220◦ C 3.05 420 MPa 350
Stress Induced Transformation κ ξ1
0.054 MPa 0.010 (MPa·s)−1
ρ ξ2
7.8 MPa/◦ C 1.02
195 GPa 0.27
E0a K1
220 GPa -4000 GPa
M
g2 g¯ Ms β sa sg
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382
383
384
PT
β in Eqs (39) and (44) depends on ratio of the average volume of martensite and shear band intersection. In this study, the start value for β was based on values in the literature (Papatriantafillou, 2005; Stringfellow et al., 1992). The value of β = 3.05 that fits the transformation kinetics given in Figs 8(a) and 8(b) was finally determined again using inverse modeling with the FEM subroutine. The exponent r in Eqs (39) and (44) is commonly taken as 4 − 4.5 for martensitic TRIP steels (Iwamoto, 2002; Papatriantafillou, 2005; Stringfellow et al., 1992). Here r = 4 was found to give the best fit to the experimental data represented in Figs 8(a) and 8(b). In general α in Eqs (40) and (39), which controls the rate of shear band formation based on a stacking fault energy, is temperature dependent (Sierra and Nemes, 2008; Tomita and Iwamoto, 1995). Since the temperature degree of freedom was satisfied through the functions Γ(Θ, σ) and C(T ) in Eqs (42), (6), α and β were here kept temperature in-
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387 388
389 390 391
392
950 500 50
0.015 0.00623 800 4 0.0371 60 176000
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180 MPa
Strain Induced Transformation A0 A1 α r ∆v g0 g1
Nonlinear elasticity E0m ν0
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0.04
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Value
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Parameter Yield Surface σY 0 Flow potential a∗ Isotropic hardening b Kinematic hardening C1 C2 C3
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0.86
ζ
0.84 0.82 0.80
0.792 0.790 0.788 0.786 0.784 0.782
FE 75 ◦ C Exp. 75 ◦ C
0.780
0.78 0.000 0.002 0.004 0.006 0.008 0.010 0.012
0.778 0.00
ε
(a)
CR IP T
0.88
0.794
FE RT FE 75 ◦ C FE 100 ◦ C FE 150 ◦ C Exp. RT Exp. 75 ◦ C Exp. 100 ◦ C Exp. 150 ◦ C
ζ
0.90
0.01
0.02
ε
0.03
0.04
0.05
(b)
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dependent. The parameters responsible for temperature dependence (g1 , ρ, κ) were obtained using the results of X-ray diffraction measurements presented in Figs 7(b) and 8(b). The g1 value was fitted to the measurements representing the change of ζ at high temperatures (75◦ C, 100◦ C and 150◦ C) where strain induced transformation was dominant. The parameters ρ and κ on the other hand changes the transformation start stresses in Eq. 6. Therefore they were determined considering the yield strength at RT to 75◦ C, where stress induced transformation starts before yielding, see Fig. 8(a). Parameters g0 , g2 , g¯ and sg in the probability function are determined through their fit to the X–ray diffraction measurements given in Fig. 8(a), especially at high temperatures where the strain induced phase transformation is dominant. The dimensionless parameters A0 , A1 and sa given in Eq. (46) were used to calculate A, which account for the strains related to shape change. These parameters were determined from the stress strain curves in Fig. 6. The starting values were based on the values by Iwamoto (2002); Papatriantafillou (2005); Stringfellow et al. (1992). According to Stringfellow et al. (1992) the difference in kinetics and morphology of stress and strain induced phase transformation affect the magnitude of transformation strains. However, in this study the same A was used to define the evolution of all transformation strains. Figure 2 shows the retained austenite as islands at the prior austenite grain boundary intersections. Some islands of retained austenite are circled in red in Fig. 2 which represent small inclusions constrained in martensite matrix that effect phase change and redistributes the stress field in the transformation zone. Therefore the volume expansion of retained austenite is not free and it is mainly constricted by elastic deformation of the martensite matrix. Using the empirical equation given by Moyer and Ansell (1975) the relative volume expansion, ∆v in Eq. (45), during martensitic transformation of the iron-carbon alloy with 0.91 wt % C was calculated as 0.0371. The determined value was in the range of 0.2 % to 0.4 %, that is reported by Leal (1984) and commonly used by previous
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Figure 8: a. X-ray measurements of transformation of retained austenite to martensite in tension specimens are compared to model predictions. b. Simulated transformation kinetics during compression and X-ray measurement at experiment termination.
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40 mm
1.5 mm 4 mm
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r = 4 mm
Figure 9: FE model, thickness 0.70 – 0.76 mm. The triangles represents the symmetry conditions.
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researchers. The relation between the parameters which were only related to the kinetics of the stress induced phase change, ξ1 and ξ2 , was determined using Eq. (48) (Moumni et al., 2011)
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∂ζ ∂T ζ˙stress = = Cζ ξ1 (ξ2 − ζ). ∂T ∂t 426
Therefore
ζ1
T Rf
M
ξ1 = T˙
Rζ2
dζ (ξ2 −ζ)
422
423 424 425
(49) 426
(50)
Cζ dT
Ms
429 430 431 432 433 434
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428
where Tf = 165◦ C representing the temperature at the end of heat treatment when ζ2 = 0.78 and Ms = 220◦ C (Ovako, 2016) at the martensite start where ζ1 = 0. Using ξ2 = 1, which is defined as the maximum martensite fraction that can be obtained as a result of phase transformation as stated by Moumni et al. (2011), and Eq. (50) ξ1 , was calculated. Table 3 lists the minimum number of experiments required to determine the model parameters, the number of experiments used here for parameter determination, the additional number of experiments used for model verification and which parameters are fitted to the corresponding curves. Some of the calculated parameters indicated in Table 3 were fitted independently (σ Y 0 , a, a∗ , b, Q, Ms , g1 , κ, ρ, ξ1 , ξ2 ) since they define a unique characteristic. Some parameters (C1 , C2 , C3 , γ1 , γ2 , γ3 , A0 , A1 , g2 , g¯, α, β, sa , sg , g0 , E0m , E0a , ν0 , K1 ) required number of iterations. The other parameters are either calculated through expressions existing in the literature or taken directly from the literature (∆v , ξ1 , ξ2 ,r). The experiments were monotonic tension to specimen rupture or a predefined strain, monotonic compression to a predefined strain and cyclic pushpull to rupture. Also, the tension experiments must be performed at different temperatures in the transition temperature range, see Fig. 1.
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438
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440 441
442 443 444
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Table 3: Number of experiments used
Used
Verification
Tensile
3
6
2
Compression
1
1
1
Cyclic
1
1
1
X-ray diff.
8
8
2
4. Methods of Numerical simulations
Parameters σ Y 0 , a, b, Q, C1 , C2 , C3 , γ1 , γ2 , γ3 A0 , A1 , g2 , α, sa , E0m , E0a , ν0 , K1 A0 , A1 , g2 b, Q, C1 , C2 , C3 , γ1 , γ2 , γ3 α, sa , E0m , E0a , ν0 , K1 A0 , A1 , g2 , g¯, r, β, sg , g0 , g1 , κ, ρ
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Test
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5. Results
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As seen from true stress true strain curves in Fig. 10, the finite element results were in good agreement with the tension experiments performed at different temperatures. Figure 6(a) indicates that the yield stress increases with the increase of temperature from RT to 75◦ C, but over 75◦ C only a small difference is observed. According to Neu and Sehitoglu (1992) during stress induced transformation, the yield strength in tension increases considerably with temperature. Therefore for TRIP steels, the change of yield point can be used to identify whether stress induced transformation takes place before strain induced transformation or not. To see the temperature dependence of stress induced transformation and the change in transformation start stress, stress strain curves representing the data obtained from the tensile experiments to rupture and those calculated using FE are plotted separately in Figs 6(a) and 11. In the FE computations
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FE calculations were done using ABAQUS through a user defined material subroutine (UMAT) that followed Eqs (26) to (48) with the material properties given in Table 2. Figure 9 shows the FE model used in the analysis. A 3D mesh with 67872 C3D8 elements was used after a mesh size and convergence study. The total CPU time was 13.1 × 103 s and the ratio of CPU time with and without considering phase transformation was 1.2. In the UMAT, stress and strain induced phase transformations were coupled with nonlinear elasticity, nonlinear kinematic and isotropic hardening behaviors and formulated using fully implicit integration of the constitutive relations. In the numerical solution, initially stress induced transformation was considered, the evolution of martensite fraction and transformation strains were calculated from Eqs (48) and (45). During dislocation movement, strain induced transformation takes place at shear band intersections along with stress induced phase change. Together they yield the evolution of martensite fraction. The flowchart of the UMAT subroutine is summarized in Algorithm 1.
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1400
1200
1200 1000
σ (MPa)
σ (MPa)
1600
800 600 400
FE RT Exp. RT
200
200
500
ε
(b)
1500
σ (MPa)
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1000
FE 75 ◦ C Exp. 75 ◦ C
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
M
σ (MPa)
1500
600
400
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 ε
(a)
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500
FE 100 ◦ C Exp. 100 ◦ C
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 ε
PT
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 ε
FE 150 ◦ C Exp. 150 ◦ C
(d)
CE
(c)
AC
Figure 10: Tensile experiments and finite element results at a. RT b. 75◦ C conducted by Linares Arregui and Alfredsson (2010) (test no: 13823) c. 100◦ C d. 150◦ C
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Algorithm 1 UMAT subroutine for plastic deformation, stress and strain induced phase change. Initial Conditions pl 0 0 ζ0 = 0.78, [εtr ij ]0 = 0, [εij ]0 = 0, k:iteration number, n:increment number, [∆A] = [A]n+1 − [A]n . 1. Trial Stress pl 0 pl 0 0 0 tr 0 Set : k = 0, ζn+1 = ζn0 , [εtr ij ]n+1 = [εij ]n , [εij ]n+1 = [εij ]n pl
k [∆εpl ij ] =
d k m k 3 [σij ]n+1 −[Xij ]n+1 k Y 2 [σ ]n+1
AN US
Calculate trial stress: [σij ]0n+1 = [Eijkl ]0n+1 ([εkl ]0n+1 −[εkl ]0n+1 −[εkl tr ]0n+1 )+ 2 0 K1 [(εel kk ) ]n+1 δij ) 2. Check stress induced transformation and yield function k if (( Φ > 0 & ζ˙ > 0 ⇒ Cζ > 0) & fn+1 < 0) then only stress induced transformation occurs [∆ζ]k+1 = Cζ ξ1 (ξ2 − [ζ]k+1 n+1 ) k+1 [εtr ] calculated using Eq. (45) ij n+1 update stresses using Eq. (15) and go to step 6. k else if (Cζ < 0 and fn+1 > 0) then yielding occurs before stress induced transformation [f ]kn+1 = 0 gives ∆λ plastic strains are calculated using: + 3a∗ [σ h ]kn+1 [∆λ]k
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k k k [∆ζstrain ]k = (1 − ζn+1 )(Aζ [∆εpl eq ] + Bζ [∆Σ] ) k else if Cζ > 0 and ζ < 1 and fn+1 > 0 then yielding, strain and stress induced phase changes occur k k k k [∆ζ]k = (1 − ζn+1 )(Aζ [∆εpl eq ] + Bζ [∆Σ] ) + Cζ ξ1 (ξ2 − ζn+1 ) k [εtr ij ]n+1 calculated using Eq. (45) else Elastic deformation and go to step 6. end if pl k+1 k+1 tr k+1 3. Update [ζ]k+1 n+1 ,[εij ]n+1 , [εij ]n+1 and [Eijkl ]n+1 . 4. Calculate stress using : pl k+1 k+1 k tr k+1 el 2 k+1 [σij ]k+1 n+1 = [Eijkl ]]n+1 ([εkl ]n+1 − [εkl ]n+1 − [εkl ]n+1 ) + K1 [(εkk ) ]n+1 δij ) 5. Set k=k+1 and go to step (2). 6. Set n = n + 1 and go to step (1). 7. When the calculation finishes at all the integration points ABAQUS checks the force equilibrium.
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rupture of the specimens was not modeled. Instead all simulations continued to the total strain when the last experiment ruptured. During the experiments it was observed that the stress was homogeneously distributed and the specimens deformed without any necking. Therefore the localization of inhomogeneous stresses was not considered in the numerical simulations. Figure 6(a) shows that the start of stress induced phase change increases with the temperature up to a certain value above which no phase change occurs prior to yielding. Thus 23
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σ (MPa)
1800 1600 1400 1200 1000 800 600 400 200 0 0.0
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FE RT FE 75 ◦ C FE 100 ◦ C FE 150 ◦ C 0.4
0.6
ε
0.8
1.0
1.2
1.4 1e−2
AN US
0.2
Figure 11: FE calculations
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the results obtained were compatible with the findings of Neu and Sehitoglu (1992). The results by Neu and Sehitoglu (1992) indicated that it is difficult to separate the stress and strain induce transformations by only analyzing monotonic loading experiments at one specific temperature, especially if the material yields before transformation. To illustrate the transformation kinetics distinctly, the conditions considering only stress induced or strain induced transformations were modeled and are plotted in Fig. 12 at two different temperatures. At RT in Fig. 12(a), the dashed green curve, representing the case where only stress induced transformation is active, coincides with black curve, including both transformation mechanics, till the start of strain induced transformation. Therefore, it is seen that for low temperature the stress induced transformation initiates before yielding and is a major part of the total transformation. At 75◦ C strain induced phase change becomes more dominant and the stress induced phase change takes place after yielding as is represented in Fig. 12(b). For visual representation of the dependence of yield strength on transformation behavior, the uniaxial stress versus martensite fraction is plotted in Fig. 13(a). By comparing the onset of nonlinearity in Fig. 6(a) with transformation start stress in Fig. 13(a) it can be clearly seen that the yield strength decreases with decreasing temperature as the result of early transformation start at low temperatures of stress induced transformation. On the other hand at higher temperatures the effect of temperature on yield strength is less because no stress induced transformation occurs before yielding. Moreover, Fig. 13(b) supports the claim that at RT phase change starts before yielding since εtr 11 starts evolving before εpl . It was concluded that by performing a series of different eq monotonic tension experiments at different temperatures it was possible to separate between stress and strain induced transformation. Figure 1 illustrates
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PT
482
498 499 500
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for low temperatures that the decrease of inelastic deformation with decreasing temperature shows that stress induced transformation is present and predominating. It was then detected separately from strain induced transformation, see the schematic start stress below yielding for stress induced transformation in Fig. 1. Strain induced transformation starts after yielding which in the present temperature range is relatively constant. It was then found that after yielding, strain induced transformation was more pronounced than stress induced. The XRD measurements are shown in Fig. 8(a) together with the ζ curves obtained from FE using the calibrated material properties. When all data points were considered together with the model prediction in the figure, the measurements supports each other and deviation from the model is less than 0.5 % for each retained austenite value. Hence, the overall accuracy of multiple measurements were much better than the single point estimate suggests. Figures 14(a) and 14(b) represent the contour plot of martensite fraction at
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the early transformation stage ( ε11 = 0.00275 in the test part with homogeneous strain and maximum ε11 = 0.0095 in the notch root) and at the specimen rupture under tensile loading at 75◦ C, see Fig. 6(a). During phase change that occurs more predominantly under tension than compression, the volume increase induces an increase in strain, resulting in a nonlinear stress-strain behavior in advance and/or during dislocation motion. For this reason, in high strength steels with retained austenite undergoing phase change, the asymmetry between compressive and tensile yield stresses seen in Fig. 6(b) is largely a result of phase transformation. The triaxiality parameter Σ in Eq. (38) affects the probability of martensite formation at the dislocation sites during strain induced phase change through the parameter Γ in Eq. (42) in the probability function P , Eqs (37), (41) and (44). The strain induced model of Stringfellow et al. (1992) that fitted to the martensite fraction evolution data of Young (1988) showed that martensite formation decreases when the triaxility is negative as in the case of compression. Additionally, Young (1988) reported that transformation curves for stress induced transformation exhibit linear behavior with respect to axial strains, whereas the curves for strain induced transformation was found to be nonlinear. The comparison of tension–compression tests and the calculated FE results together with X-ray
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diffraction measurements given in Fig. 8(b) pointed out that under compression a small amount of transformation occurs. The simulated behavior of ζ versus axial strain plotted in Fig. 8(b) differs quite markedly to the tensile loading case in Fig. 8(a). In compression, the transformation is dominated by the stress induced transformation that develops at the early stages of the deformation. This part of the transformation is furthermore linear with the strain, which agree with the findings of Young (1988). The last part of the curve includes small strain induced transformation but at this stage the rate of transformation ζ˙ is much lower than the early part. The strength differential effect was also examined for cyclic load and compared to the experiments performed by Linares Arregui and Alfredsson (2010) (test no:13889) in Fig. 7. The XRD results represented in Fig. 7(b) are in agreement with the simulations where the dotted the black line represents the martensite fraction under monotonic tensile loading and dashed blue curve is the evolution of martensite in the first cycle. The results confirm that the phase transformation predominantly develops during tension. Note that the retained austenite measurement for the cyclic curve in Fig. 7(b) was not directly used in determining the phase transformation parameters. Hence, it gave an indication on the model prediction. For high strength steels, the slope of the yield surface in the meridian plane is a ≈ 0.02 (Allen and Wilson, 2004; Guo et al., 2008; Spitzig and Richmond, 1979), which is smaller than the value determined by Linares Arregui and Alfredsson (2010), a ≈ 0.25. However, this value included all nonelastic effects. The effect of the pressure sensitive plasticity parameter, a on yield strength and
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phase transformation was investigated by using four different a values in tensile simulations. The parameter sensitivity study represented in Fig. 16(a) shows that the decrease in yield stress for increasing a is more significant at 75◦ C than RT. This result is mainly due to the stress induced transformation that occurs at lower temperatures before yielding. Therefore, the dependency of hydrostatic stress is influenced more by stress induced phase change, where the transformation is initiated by elastic stresses, compared to strain induced phase transformation. To determine the value of parameter a, not only the stress strain response but also the effect of a on phase transformation kinetics was studied. From Fig. 16(b) it is seen that decreasing a from a = 0.12 to a = 0.04 decreases the predicted martensite fraction at a certain strain value. Experimental measurements on martensite fraction ζ in Fig. 16(b), yield stress in Fig. 16(a) and the tensile and compressive stress strain responses in Fig. 6(b) were used to establish a = 0.04 as the best yield stress slope for the present material. During strain induced phase change, which dominates for elevated temperatures, the evolution of martensite fraction occurs at dislocation sites and the effect on the yield surface slope is illustrated in Fig. 16(b). Therefore the experimented evolution of ζ at elevated temperatures were also considered. The effect of phase change on SDE was analyzed for two different temperatures through simulations of monotonic loading where the different parts of phase changes were executed. The evolution of such simulated SDE is plotted against inelastic strains in Fig. 17. Figure 17(a) shows that for increased temperatures, here exemplified with 75◦ C, strain induced transformation together with Drucker Prager plasticity coincides with the curve representing the full model and they yield closer results to the experimental measurements of Linares Arregui and Alfredsson (2010) performed at 75◦ C. The difference in SDE between experimental measurements and full model predictions in Fig. 17(a) is explained in Fig. 6(b) by the corresponding difference in the compression curves at moderate strain. Figure 6(b) also illustrates that in absolute stress values the model deviation is small. Figures 17(a) and 17(b) includes simulated curves that only contains pressure sensitive plasticity, using Drucker Prager, i.e. the phase transformation was turned off. Comparing these curves with the full model prediction clearly illustrates that phase transformation dominates SDE effect for the current material structure. The result also show that stress induced phase change has only a small effect on SDE for high temperatures. Besides, when the temperature decreases to a level where the stress induced phase transformation is more effective on yield strength, e.g. RT in this case, the stress induced transformation is mainly responsible for SDE especially at low inelastic strains, see Fig. 17(b). Figure 18 compares the calculated volume changes during tensile and compressive loading to experimental measurements of Linares Arregui and Alfredsson (2010). It was observed that the computed volume change during compression is less than that during tensile loading which agree with the results by Neu and Sehitoglu (1992).
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The present study couples the plastic deformation obeying nonlinear hardening rules and nonassociated flow rule with stress and strain induced phase change. Good agreement was obtained with the experimental results. In addition to phase transformation kinetics, the nonlinearity of elastic strains were taken into account in the material model while scrutiny of the plastic response cycles from cyclic experiments clearly showed that nonlinear elastic effects do have influences for the high elastic strains that can develop in this type of high strength bearing steel. Although the full material model includes a relatively large amount of material parameters, a limited number of well planned and well instrumented experiments were sufficient to calibrate the model. The required series include monotonic tensile experiments to some different strains and at some different temperatures together with at least one compression and one cyclic experiment. The data captured should include retained austenite levels and densities at test termination as well as the stress strain curves. The experiments showed that the solid to solid phase transformation of retained austenite to martensite affects the material behavior before and after plastic deformation through stress and strain induced phase transformations and changes in the yield strength of the material. Results indicated that the change of the yield strength with temperature is the main indicator of stress induced transformation that takes place before plasticity and defines start of the measured yielding at RT. At low temperatures, approximately below 75◦ C for the present material, stress induced transformation results in a reversed tem-
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perature relation with decreasing yield point for decreasing temperature. Also at these temperatures phase transformation was only stress induced, i.e. no strain induced transformation existed. At 75◦ C and for higher temperatures up to the martensite start temperature plastic yielding develop before phase change. Here strain induced transformation dominates the transformation, although some stress induced transformation continues to develop. By utilizing the understanding on how the yield point changes with temperature and the corresponding measured data points, stress and strain induced phase transformation were successfully separated from each other, their magnitude measured and the model parameter values were determined. Thus, the proposed model summarized in Eq. (48) separates stress and strain induced transformation. It was concluded that the stress and strain induced phase transformation was active in the present material and that the processes can be separated by coupled modeling with a series of different monotonic experiments at different temperatures where both the stress strain curves were captured and the retained austenite levels after experiment termination was determined. Using the proposed model not only the coupled behavior but also distinctly stress and strain induced phases can be identified. Therefore, using this modeling technique, for complex loading condition like the stress distribution around a crack tip or at contact surfaces, contour plot of strain and stress induced martensite transformation zones can be plotted separately and their contribution to the mechanical behavior can be calculated independently. According to the monotonic experiments and X-ray diffraction measurements it was concluded that the amount of retained austenite that transforms to martensite is higher under tension than compression. The calculations for compressive loads at 75◦ C showed that a small stress induced phase transformation at the early stage of compression accompanied by strain induced phase change which is very small in total compared to the change under tensile loading. The analysis showed that phase transformation enhances and dominates the SDE over plasticity effects and that SDE can be simulated successfully by employing the coupled stress and strain induced phase transformation kinetics together with a pressure sensitive surface for plastic yielding and flow as is proposed in the current study. In the light of the aforementioned reasons it is claimed that both stress and strain induced phase change should be taken into account in the definition of constitutive behavior of the present class of high strength steels. Therefore, the material model that was capable of capturing nonlinear hardening behaviors, pressure sensitive plastic yielding and nonlinear elasticity together with stress and strain induced phase change was required to simulate the mechanical material behavior including the SDE in the high strength martensitic bearing steel accurately.
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Leblond, J.B., Devaux, J., Devaux, J.C., 1989. Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases. Int. J. Plast. 5, 551–572. Lee, J., Lee, J.Y., Barlat, F., Wagoner, R., Chung, K., Lee, M.G., 2013. Extension of quasi-plastic-elastic approach to incorporate complex plastic flow behavior - application to springback of advanced high-strength steels. Int. J. Plast. 45, 140–159. 35
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Appendix A. Elastic strain for monotonic tension-compression
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The equation
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el 9Geq 0 K(εkk ) E(εel eq , kk ) = 3K(εel kk ) + G0
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el and σ h = K0eq εel kk + K1 εkk was evaluated for uniaxial tension and compression. The uniaxial stress was then calculated through
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el Equation (22) together with εel kk = (1 − 2v)ε11 yields 3(K0eq + K1 εel el kk ) − 2G εel εkk = 1 − eq 11 . 3(K0eq + K1 εel kk ) + G0
el Using Eq. (53), εel kk can be defined in terms of ε11 as p eq el 2 3(K0eq + Geq (3(K0eq + Geq el 0 ) 0 ) + 36K1 G0 ε11 εkk = − + . 6K1 6K1
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The dissipation potential of the system defined through free energy potential by Eq. (5) was
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∂g κ˙ R = −λ˙ (58) ∂R When a nonassociative flow rule is employed in order to compute the bounds for the influence of pressure that satisfies the dissipation equation Eq. (56) the following analysis has to be done. Equations (57), (58) and (35) were used in Eq. (56) and the stresses were written in deviatoric and dillatation components so that ! 3 X 3 γm m m ∗ h D = σeq + 3a σ + X Xij − R + Cζ ζ˙ ≥ 0 (59) m ij 2 C m=1 41
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Appendix B. Criterion for pressure dependence in the nonassociated flow rule
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The positivity of Cζ ζ˙ is discussed in appendix C. Therefore for the rest of the analysis the method used by Linares Arregui and Alfredsson (2010) is followed. Since λ˙ > 0 and f = 0 during plastic deformation, σY 0 − 3σ h (a − a∗ ) +
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Linares Arregui and Alfredsson (2010) states that in the material model that considers the nonassociated flow rule with pressure dependent plastic flow rule, the chosen parameters a∗ and a has to be controlled throughout the simulations to verify the stability of the solution and satisfaction of dissipation equation.
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Appendix C. Dissipation Potential
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The pseudopotential of dissipation was
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where both terms in this Cζ = 0. This yields the condition Cζ ≥ division are positive and numerator is zero when there is no plastic deformation. It is known that stress induced phase change starts before strain induced phase change below a certain temperature and the evolution of stress induced martensite fraction defined in Eq (49), ζstress ≥ 0. In addition to that the term ˙ Aζ ε˙pl eq + Bζ Σ is positive during plastic deformation and zero otherwise. Therefore here, as a transformation function the condition of Cζ ≥ 0 should be used. This conditon satisfies positivity of the term responsible from the dissipation during phase change Cζ ζ˙ in dissipation equation.
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˙ Aζ ε˙pl eq +Bζ Σ , − ξ1 (ξ 2 −ζ)
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Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel Selcuk Hazar, Bo Alfredsson, Junbiao Lai PII: DOI: Reference:
S0167-6636(17)30149-7 10.1016/j.mechmat.2017.10.001 MECMAT 2802
To appear in:
Mechanics of Materials
Received date: Revised date: Accepted date:
24 February 2017 14 September 2017 2 October 2017
Please cite this article as: Selcuk Hazar, Bo Alfredsson, Junbiao Lai, Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel, Mechanics of Materials (2017), doi: 10.1016/j.mechmat.2017.10.001
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Highlihgts • Both stress and strain induced phase change are taken into account in the definition of constitutive behavior of high strength steels.
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• The phase transformation enhances and dominates the strength differential effect over plasticity effects.
• Strength differential effect simulated successfully by employing the phase transformation kinetics together with a pressure sensitive surface for plastic yielding.
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• The nonlinearity of elastic strains are also taken into account in the material model.
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Mechanical modeling of coupled plasticity and phase transformation effects in a martensitic high strength bearing steel
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Selcuk Hazara,∗, Bo Alfredssona , Junbiao Laib a Department
of Solid Mechanics, Royal institute of technology (KTH), 100 44 Stockholm, Sweden b SKF Engineering & Research Centre, P.O. Box 2350, 3430 DT Nieuwegein, The Netherlands
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Abstract
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The stress and strain induced solid to solid phase transformation of retained austenite in a martensitic high strength bearing steel has been studied. Monotonic tension experiments that were carried out at different temperatures using this high strength steel showed that not only the strain induced but also the stress induced phase change plays a crucial role in the phase transformation of retained austenite to martensite. In the material model, plastic deformation was defined using the Drucker Prager yield surface through a nonassociated flow rule accompanied by nonlinear kinematic and isotropic hardening. The hardening was coupled with stress and strain induced phase transformations. A nonlinear elastic effect based on elastic dilation was included in the constitutive model by extending the bulk modulus with a second order term. For the finite element analysis, the material model was written as a user defined material subroutine (UMAT). The numerical simulations were done using ABAQUS and compared to monotonic tension, compression and cyclic experiments. The results showed that the strength differential effect and the volumetric change under loading are closely related to the transformation of retained austenite to martensite. At low temperatures the effect of stress induced phase transformation on yield strength was noticeable. It was concluded that at certain temperatures both strain and stress induced phase transformations significantly affect mechanical behavior of the high strength steel.
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Keywords: Transformation induced plasticity, Stress induced phase transformation, Strain induced phase transformation, Strength differential effect, Nonassociated flow rule
∗ Corresponding
author Email address: [email protected] (Selcuk Hazar)
Preprint submitted to Mechanics of Materials
October 10, 2017
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1. Introduction
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At equilibrium conditions, materials tend to stay in their crystallographically more stable phase i.e. the atomic configuration having the lowest energy level. In high strength bearing steels and other materials undergoing phase change upon cooling, solid to solid phase transformation occurs from the parent phase (austenite), which is stable at high temperatures, to the product phase (martensite) that is stable at low temperatures. During quenching of the high strength steel, when the temperature is above the martensite finish temperature and the energy level required to complete the transformation is not reached, austenite cannot transform completely to martensite. As a result, a metastable phase named retained austenite forms. The amount of austenite retained after heat treatment is affected by carbon content, quenching temperature and alloy composition. Aside from temperature change, application of load and shear band intersection triggers the phase change of retained austenite to martensite. This mechanically induced phase change enhances ductility, formability, toughness and strength of such steels and makes them attractive for automotive, aerospace and variety of other engineering applications. The martensitic transformation of retained austenite upon loading can be either stress or strain induced (assisted). When the temperature is below transformation start temperature (Msd ), stress induced displacive martensitic transformation of the retained austenite to martensite takes place at nucleation sites in the parent phase. Up to Msd stress induced transformation takes place below the yield stress (σ Y ) of the material. In this temperature range, the transformation start stress increases linearly with the temperature (Bolling and Richman,
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1970a). Kinetics of stress induced transformation is described using either Gibbs or Helmholtz free energy potentials by considering the thermodynamic effect of the applied stress (Abeyaratne and Knowles, 1993; Bhattacharyya and Weng, 1994; Chan et al., 2012; Entchev and Lagoudas, 2004). Using Helmholtz free energy, Moumni et al. (2011) proposed a model for stress induced phase transformation based on the formalism of principle of generalized standard materials with internal constraints (Halphen and Nguyen, 1974; Moumni, 1995). The simulations that they performed using the thermomechanical model had good correlations with experimental results. In the literature, the constitutive models of stress induced transformation are studied not only for high strength steels but also for other alloys like Nitinol (Arghavani et al., 2010; Auricchio et al., 2007; Chan et al., 2012; Levitas and Ozsoy, 2009a; Moumni et al., 2011; Popov and Lagoudas, 2007; Saleeb et al., 2011; Zaki and Moumni, 2007b; Zhang et al., 2007). Strain induced (assisted) phase change initiates through formation of highly potent new sites via shear band interactions when the temperature is above Msd . In this case, depending on temperature, the transformation start stress can be higher than the yield stress. In previous studies, the strain induced transformation, that is commonly investigated in TRIP steels, was found to undergo irreversible martensitic transformation (Cherkaoui et al., 2000, 1998; Goel et al., 1985; Idesman et al., 2000; Iwamoto and Tsuta, 2000; Leblond et al., 1986; Olson and Azrin, 1978; Olson and Cohen, 1975; Sangal et al., 1985; Stringfellow et al., 1992; Taleb and Sidoroff, 2003; Tomita and Iwamoto, 1995). Olson and Cohen (1975) proposed a model that outlines the strain induced phase transformation behavior in austenite steel through interaction of shear bands. Their model was only able to capture uniaxial stress effects under isothermal conditions. Stringfellow et al. (1992) enhanced the model of Olson and Cohen (1975) by considering the stress state sensitivity, since the evolution of martensite at a material point is effected not only by the plastic strain and temperature but also by the stress state history. The model proposed by Stringfellow et al. (1992) was improved by taking into account of the influence of strain rate on toughness and ductility (Tomita and Iwamoto, 1995), thermocoupled effects (Iwamoto, 2002, 2004), deformation mode dependence (Iwamoto et al., 1998) and the effect of grain size (Iwamoto and Tsuta, 2000). Perlade et al. (2003) and Papatriantafillou et al. (2006) used the model of Stringfellow et al. (1992) to study the multiphase TRIP steels having different retained austenite levels. In a later study Sierra and Nemes (2008) modified the same model to study the interaction between embedded retained austenite islands and the surrounding matrix. In the early attempts, formulation of transformation kinetics in TRIP steels is based on either large scale orientation of the transformation strain in the martensite phase (Magee, 1966) or the micromechanical plastic strain accommodation that emerges from martensitic phase transformation and arises in the parent phase (Greenwood and Johnson, 1965). Based on these two principles, most of the constitutive models for TRIP steels are derived using either macro scale (Cherkaoui et al., 2000; Fischer et al., 1998, 2000; Leblond et al., 1989;
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Levitas, 1998; Moumni et al., 2011; Perlade et al., 2003; Tanaka et al., 1996) or micro scale approaches (Levitas and Ozsoy, 2009a,b; Ma and Hartmaier, 2015). In the vast majority of constitutive models developed for materials undergoing ˙ is prescribed as phase transformation, the rate of martensite volume fraction (ζ) the main internal state variable. Previous researches have documented that the rate of martensite transformation is affected by Ms temperature (Inoue et al., 1984; Neu and Sehitoglu, 1992), hydrostatic stress (Jacques et al., 2001; Neu and Sehitoglu, 1992), carbon content (Neu and Sehitoglu, 1992; Seong et al., 2004), quenching temperature and rate, plastic strains (Seong et al., 2004) and percentage of retained austenite (Neu and Sehitoglu, 1992). The study by Young (1988) showed that evolution of martensite fraction with respect to plastic strain during stress induced transformation is more linear compared to the case of strain induced phase change. Not only the transformation kinetics, but also the morphology of the martensite microstructure formed via stress or strain assisted transformation differs. Strain induced phase change results in fine lath martensite phase formed at the shear band intersections, whereas plate martensite formation is observed when the transformation is induced by stress (Young, 1988). Phase transformation in TRIP steels results in a strength differential effect (SDE), which is known as the asymmetry between tensile and compressive yield stresses, because the retained austenite is more susceptible to transform to martensite under tension than in compression. The stress induced transformation is affected by temperature, for this reason SDE is found to be higher when the temperature is decreased (Singh et al., 2000). It is observed that SDE depends not only on temperature but also on the hydrostatic stress, therefore Drucker Prager yield surface has to be used to define the plastic flow (Casey and Sullivan, 1985; Spitzig and Richmond, 1984). In addition, in high strength steels the associated flow rule would result in a plasticity induced volume change which is different from the experimental findings, for this reason a nonassociated flow rule has to be considered in the material model (Linares Arregui and Alfredsson, 2010; Stoughton and Yoon, 2004). Not only the phase transformation but also microcracking, residual stresses, volume expansion and solute dislocation interaction are some of the factors that can give rise to SDE. For instance Linares Arregui and Alfredsson (2010) observed SDE in high strength bainitic steel which did not undergo any mechanically induced phase transformation. Previously, the stress dependency of elastic modulus for high strength steels is discussed by some researchers (Cleveland and Ghosh, 2002; Lee et al., 2013; Linares Arregui and Alfredsson, 2013; Sun and Wagoner, 2011; Yoshida et al., 2002; Zavattieri et al., 2009). According to the results of push–pull cyclic experiments by Sommer et al. (1991) on SAE 52100 high strength steel, when the linear elastic part is subtracted, a sickle shaped asymmetric hysteresis loop is obtained in stress vs. plastic strain graphs. In a recent work, Linares Arregui and Alfredsson (2013) showed that the nonlinear elastic material behavior should be considered in the material model of the high strength bainitic roller bearing steels. Moreover, Linares Arregui and Alfredsson (2010) used combined nonlinear kinematic and nonlinear isotropic behaviors to precisely define the
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Stress Induced Transformation Coupled Transformation
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Figure 1: The effect of temperature on the start of strain and stress induced transformations. Yellow region represents where the austenite is stable, blue and gray zones represents the stress induced and coupled transformation zones respectively.
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plastic material response in their experimental measurements. Figure 1 illustrates the temperature effect on the start of strain and stress induced transformations. At a certain critical temperature stress induced transformation starts at a certain stress level, σs , which can be below yield stress (Bolling and Richman, 1970a; Olson and Cohen , 1972). Therefore coupled transformation behavior takes place after plastic yielding. Strain induced transformation on the other hand starts at the onset of yielding. Above Msd the shape of yield stress curve representing the case of coupled transformation depends on material properties but increases up to a certain temperature above which no phase change is observed (Young, 1988). Therefore, coupled transformation behaviour takes place after plastic yielding. According to Neu and Sehitoglu (1991) it is difficult to distinguish stress induced from strain induced transformation when the deformation increases beyond the yield point. Recently Ma and Hartmaier (2015) proposed a constitutive model considering strain induced martensite nucleation and stress assisted martensite growth using Helmholtz free energy potential. Although they stated that their numerical calculations fit to the experimental results of Lebedev and Kosarchuk (2000), no direct comparison were presented between the numerical results and the corresponding measurements. In the present research, stress and strain induced phase transformation mechanisms were coupled with plastic deformation that was described using a Drucker Prager yield surface with a nonassociated flow rule accompanied with combining nonlinear kinematic and isotropic hardening. For this purpose the model represented by Stringfellow
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2. Material model
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This section presents the constitutive model of the investigated martensitic high strength bearing steel (DIN 100CrMnMoSi8), with the chemical composition given in Table 1. Figure 2 shows the microstructure of the quenched and tempered martensitic steel with retained austenite content.
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et al. (1992) was extended by including the stress induced transformation effect using Nguyen's formalism of standard materials, which was applied to TRIP steels by Moumni et al. (2011). The temperature dependence of the onset of nonlinear stress strain behavior was the main motivation for implementing coupled stress and strain induced phase change. Compared to the models existing the literature this model couples stress and strain induced phase changes with a nonassociated flow rule and nonlinear elasticity. The predictions were in a good agreement with the experimental findings. Since the morphology and mechanisms of the stress and strain induced martensite phases differ, the proposed model includes the contribution of each phase separately and in combination to the other mechanical properties of the high strength steel, such as nonlinear elasticity, nonlinear hardening and pressure dependence of the yield surface. The paper is structured in the following way: First the material model is described in section two. The first subsection introduces the free energy potential and the effect of nonlinear elasticity. The second subsection describes the nonlinear kinematic and isotropic hardening rules together with nonassociated flow rule. The third subsection presents the strain induced phase transformation mechanism and combines it with the stress induced transformation mechanism. In section three experiments and determination material parameters are discussed. The section 4 describes the method and procedure of numerical simulations. Finally, conclusions are drawn.
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Table 1: Chemical composition (wt %) of 100CrMnMoSi8 (Lai et al., 2016).
Cr 1.86
Mn 0.81
Mo 0.59
Si 0.21
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Fe balance
2.1. Definition of free energy potential and nonlinear elasticity Patel and Cohen (1953) were among the first researchers to investigate the contribution of the applied loading to phase transformation through a free energy formulation. Using a similar approach, Olson and Cohen (1982) represented the stress state sensitivity of the transformation and the toughening effect of stress assisted transformation. In this part, the thermodynamic force conjugate to evolution of martensite fraction was computed using the formalism of generalized standard materials with the internal constraints that were derived by
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Figure 2: Microstructure of martensitic high strength steel DIN 100CrMnMoSi8 (Lai et al., 2016), in this work some of the retained austenite islands are marked inside red circles.
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Halphen and Nguyen (1974). A similar aproach was implemented by Moumni et al. (2011). In this study Reuss scheme was used to define the macroscopic strain tensor which is an average over the representative volume element and given as (1 −
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− εij = 0,
(1 − ζ) > 0.
and
ν1 > 0,
ν1 ζ = 0, ν2 > 0,
ν2 (1 − ζ) = 0.
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(4)
By considering the contribution of internal variables of plastic deformation, the free energy potential can be written as 1 m m m 1 a a a εij Eijkl εkl + ζ εij Eijkl εkl + C(T ) W = (1 − ζ) 2 2 (5) 3 X 1 1 1 pl m m m el 3 pl + C χij χij + K1 εkk + Q εeq + exp(−bεeq ) . 3 m=1 3 b 8
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where Λij , ν1 and ν2 are the Lagrange multipliers. ν1 and ν2 are associated with unilateral constraints and
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εpl ij
Using Eqs (1) and (2), the constraints potential, Wl , in the form of a Lagrangian formulation, was given by h i pl tr Wl = −Λij (1 − ζ)εaij + ζεm (3) ij + εij + εij − εij − ν1 ζ − ν2 (1 − ζ),
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where the terms εaij and εm ij are elastic strains of austenite and martensite phases, respectively. Martensite volume fraction, ζ, that is bounded in the interval [0, 1] can be stated in inequality form as
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In Eq. (5), E a and E m are the linear elasticity tensors for martensite and austenite phases. C(T ) is the phase change heat density that is assumed to be a linear function of temperature (Zaki and Moumni, 2007b) C(T ) = ρ(T − Ms ) + κ.
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In Eq. (5), C is the slope between the back stress Xij , where X˙ ij = 32 C ε˙pl ij − pl ˙ γXij λ, and the plastic strain rate ε˙ij when γ = 0, C/γ defines the saturated value of evolution of the back stress. χij is the conjugate of back stress Xij and defined as 3 X 3 m m χ˙ ij = C Xij (7) 2 m=1
In Eq. (5), Q is the limiting value of the change in isotropic yield surface size 3 and b is the rate of hardening before reaching the stress limit. The term K1 εel kk represents the nonlinear elastic potential with the same nonlinear bulk modulus parameter K1 for both phases. εel ij is the elastic strain tensor that is composed of the elastic strains of the martensite and austenite phases, a m εel ij = (1 − ζ)εij + ζεij .
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Given the expressions of the free energy density W and of the constraints potential Wl in (3) and Eqs (5), the Lagrangian L = W + Wl was constructed to define the generalized forces (Halphen and Nguyen, 1974; Zaki and Moumni, 2007b) 1 a a a 1 m m m L(εij , T, εaij , εm , ζ) = (1 − ζ) ε E ε + ζ ε E ε + C(T ) ij 2 ij ijkl kl 2 ij ijkl kl 3 1 1 X m m m 1 3 pl pl + K1 εel + C χ χ + Q ε + exp(−bε ) kk ij ij eq eq 3 3 m=1 b h i pl tr − Λij (1 − ζ)(εaij ) + ζ(εm ij ) + εij + εij − εij
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− ν1 ζ − ν2 (1 − ζ),
(9)
where Eq. (4) must be verified. The state equations can be written as
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∂L,εij = σij
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⇒
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Λij − σij = 0, 2 K1 εel kk δij
(10)
a + εakl Eijkl − Λij = 0,
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m + εm kl Eijkl − Λij = 0, 1 a a a m m εkl Eijkl εkl − εm kl Eijkl εkl 2
(12)
pl tr (1 − ζ)(εaij ) + ζ(εm ij ) + εij + εij − εij = 0.
(14)
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in the equations above Cζ is the only nonzero thermodynamic force. Using the Eqs (10), (11), (12) and (14) the constitutive equation became 2
tr el σij = Eijkl (εkl − εpl kl − εkl ) + K1 εkk δij ,
where Eijkl is
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(16)
At undeformed state, assuming the same Poisson's ratio ν0 for both phases, the equivalent Young's, E0eq , bulk, K0eq , and shear, Geq 0 moduli were expressed as
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(18) (19)
In the work of Linares Arregui and Alfredsson (2013) a small but significant nonlinear elastic behavior was detected during uniaxial cyclic push pull experiments on high strength bainitic steel. The nonlinear elastic behavior was traced to the volumetric bulk behavior of the material and defined using the bulk modulus as a function of elastic strains, K(εel kk ) and a constant shear modulus, G. Using Eq. (15) eq el K(εel (20) kk ) = K0 + K1 εkk .
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−1 1−ζ ζ + , E0a E0m E0eq , K0eq = 3(1 − 2ν0 ) E0eq Geq . 0 = 2(1 + ν0 )
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The nonlinearity of Young's modulus and Poisson's ratio could be written in terms of their relation to K(εel kk ) as (21)
ν(εel kk ) =
eq 3K(εel kk ) − 2G0 eq . 2(3K(εel kk ) + G0 )
(22)
Using Eq. (15) the hydrostatic stress, σ h = σii /3, was stated as 2
eq el el el σ h = K(εel kk )εkk = K0 εkk + K1 εkk .
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el 9Geq 0 K(εkk ) eq , 3K(εel kk ) + G0
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The uniaxial elastic strain can be calculated using q eq 2 eq 4 σ 3 σ11 K1 + (K0 ) − K0 11 el ε11 = + . 3Geq 6K1 0
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Appendix A shows the derivation of uniaxial elastic strain, εel 11 , at tension or compression using Eqs (21) and (23). 10
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Figure 3: W L is the linear elastic strain energy and W N L is the nonlinear elastic strain energy.
The strain energy function of the nonlinear elastic material became 1 2 a NL a a 2 a a a W =(1 − ζ) K0 − G0 (εkk ) + G0 εij εij + 2 3 1 2 1 el 3 m 2 m m m ζ K0m − Gm (ε ) + G ε ε kk 0 ij ij + K1 εkk , 2 3 0 3
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L which can be stated alternatively as W N L = W L + 13 K1 εel repkk , where W resents sum of the linear elastic strain energies of both phases. The change of W N L / W L with respect to hydrostatic strains is plotted in Fig. 3 for ζ = 0.60, 0.80, 0.99 to see the effect of phase change on nonlinear elasticity. Fig. 3 shows that strain energy is effected by nonlinearity when the strains are high and the nonlinear elasticity is more noticeable at high ζ when the retained austenite level is low. The stress strain graph of a push pull experiment under a cyclic load with a compressive mean load, Pm = −20.8 kN and amplitude of Pa = 48.5 kN performed by Linares Arregui and Alfredsson (2010) (test no: 13832) is given in Fig. 4(a). The figure shows the 1st and 10th cycles of the experiment. The effect of nonlinear elasticity on the inelastic stress strain cycle is given in Fig. 4(b). The inelastic curves are derived either by subtracting linear elastic (L), εin 11 = el = ε − ε and Eq. (24) ε11 − σ/E eq , or nonlinear elastic (NL) strains, using εin 11 11 11 from the experimental results in Fig. 4(a). When the two cyclic responses shown in Fig. 4(b) are compared, it is observed that nonlinear elastic strains exist during unloading, since steeper response is obtained for the NL curves. The vertical unloading response from both tension and compression that was derived
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2.2. Nonlinear kinematic and isotropic hardening
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By virtue of the SDE observed in the push pull experiments conducted by Linares Arregui and Alfredsson (2010), it was found that a Drucker Prager type (Drucker and Prager, 1952) yield surface with a linear dependency on the hydrostatic stress had to be used to define the plastic behavior. Hence, the yield surface was defined as f = σeq + 3aσ h − σ Y , (26) where a is the slope of the offset yield stress with respect to hydrostatic stress σ h = σii /3. σ Y defines the yield stress which could be expressed using nonlinear isotropic hardening behavior (Chaboche, 1991)
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σ Y = σ Y 0 + R.
˙ R˙ = b (Q − R) λ.
where λ˙ represents the plastic multiplier, λ˙ = loading, R in Eq. (28) can be integrated to
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(28) q
2 pl pl 3 ε˙ij ε˙ij
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In the equation above, σ Y 0 represents the virgin yield stress at zero hydrostatic ˙ was defined as stress. The evolution of the yield surface radius, R,
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using NL illustrates the presence of nonlinear elastic strains. Hence, the cyclic response must be studied in order to separate nonlinear elasticity from plasticity. The martensite fraction ζ at start of the first cycle was taken as ≈ 0.78 and at 10th cycle it was ≈ 0.85. These values, ζ = 0.78, 0.85, were measured using X-ray diffraction on an undeformed reference configuration and on the specimen midsection where it had broken after 10 cycles.
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The equivalent stress, σeq , used in Eq. (26) was defined as
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(30)
d d where σij is the deviatoric stress, σij = σij −σh δij , and δij is the Kronecker delta. In Eq. (30), the nonlinear kinematic hardening is introduced using the back stress tensor, described by Chaboche (1986, 2008); Frederick and Armstrong (2007), 2 ˙ − γXij λ. (31) X˙ ij = C ε˙pl 3 ij In the definition by Chaboche (1986) three sets of kinematic hardening parameters are used 2 m m m˙ X˙ ij = C m ε˙pl (32) ij − γ Xij λ, 3 with 3 X m Xij = Xij . (33)
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Experimental measurements of the volume change due to the dislocation motion in austenitic steels undergoing phase change was found to be different from the predictions of the associated flow rule (Garofalo and Wriedt, 1962; Linares Arregui and Alfredsson, 2010; Rauch et al., 1975; Spitzig and Richmond, 1984), for this reason a nonassociated flow rule was used,
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(35)
a∗ ≤ 1, a
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(36)
in order to satisfy the dissipation equation. The derivation is stated in Appendix B. Using the same procedure as (Linares Arregui and Alfredsson, 2010) the value of a∗ was continuously checked during the simulations to satisfy the condition in Eq. (36) when the plastic strains developed. 13
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The nonassociated flow ruled used to define plastic material behavior might result in a physically instabil solution and violence of principal of maximum dissipation. For this reason, the bounds on the pressure dependence in nonassociated flow rule were defined by the
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where a∗ was used to predict plastic dilatation ∆V /V0 , which is known to be very small compared to volume expansion due to phase change (Neu and Sehitoglu, 1992). Using the nonassociated flow rule the plastic strain was defined as
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2.3. Strain induced transformation coupled with stress induced phase transformation The strain induced phase change, that occurs after plastic yielding, see Fig. 1, was introduced by implementing the extended version of Olson and Cohen's model (Olson and Cohen, 1975) which was proposed by Stringfellow et al. (1992). The evolution of strain induced martensite fraction, ζ˙strain , was formulated as ζ˙strain = (1 −
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˙ + Bζ Σ),
Aζ given in Eq. (37) was defined as
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where εeq is the equivalent plastic strain of the austenite phase. It was calculated using Eq. (35) assuming the same hardening properties for both phases (Zaki et al., 2010). P in Eq. (39) represents the probability function for transformation start, which was determined by assuming a Gaussian distribution of the potency of the shear bands to activate the nucleation sites that initiate transformation (Stringfellow et al., 1992). The probability function was defined as " 2 # ZΓ 1 1 Γ0 − g¯ P =√ exp − dΓ0 , (41) 2 sg 2π
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where α, β and r are the material parameters. The volume fraction of shear bands, fsb , is a function of equivalent plastic strain. The equation that describes fsb was written as fsb = 1 − exp −αεpl(a) , (40) eq pl(a)
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where g¯ is the mean value of the distribution and sg is the standard deviation. Γ depends on temperature and stress triaxiality through a dimensionless driving force function for martensite nucleation
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where ζ is the overall martensite volume fraction and Σ represents the stress ii , which has an effect not only on the nucleation of voids triaxiality, Σ = σσeq (Rice and Tracey, 1969) but also on nucleation of martensite sites (Jacques et al., 2001; McClintock, 1968). Σ˙ was calculated as σ˙ ii σ˙ eq ˙ Σ=Σ − . (38) σii σeq
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in which g0 is a material parameter, g1 and g2 are the temperature and the linear triaxiality dependency coefficients. Θ is the normalized temperature defined as ˙ = 0, P˙ could be derived as Θ = MTs . When Θ " 2 # g2 1 Γ − g¯ ˙ ˙ P =√ exp − Σ. (43) 2 sg 2πsg 14
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In Eq (45) ∆v is the relative volume change due to martensite transformation (∆v ≈ 0.02 − 0.05 (Leal, 1984)). N is defined as N = using σeq A = A0 + A1 , sa
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(47)
where ξ1 and ξ2 are material parameters and Cζ is given by Eq. (13). In Eq. (13) the term multiplied by Cζ is responsible for stress induced phase transformation and determines the value of evolution of stress induced martensite fraction, ζstress . In order to satisfy the dissipation inequality the requirement of Cζ > 0 is discussed in the Appendix C. 3. Experiments and Material parameters To examine stress and strain induced transformations and determine the parameters related to the stress and strain induced transformation mechanisms, monotonic tests were performed at different temperatures. For this purpose, tension experiments were carried out at 22◦ C (RT), 100◦ C and 150◦ C using servo-hydraulic test machines and the specimens shown in Fig. 5. The thin steel specimens with rectangular cross section used in this study were wire electro-discharged from forged rod specimens which had been hardened together with the specimens used in the experiments conducted by Linares Arregui and Alfredsson (2010). Thus, the material and heat treatment were 15
306
(46)
Therefore, the evolution of martensite fraction can be found using the function ∂Φ ˙ ζ˙ = = Cζ ξ1 (ξ2 − ζ) + Aζ ε˙pl (48) eq + Bζ Σ, ∂Cζ
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where A0 and A1 are material parameters and sa is the reference austenite hardness (Stringfellow et al., 1992). Finally, the stress and strain transformations were united through the pseudo potential for dissipation of the phase transformation proposed by Moumni et al. (2011). It was here modified as
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and A was calculated
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d 3σij 2σeq
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(44)
where H() is the Heaviside function. Using the evolution of martensite in Eq. (37), the transformation strain became ˙ √1 AN + 1 ∆v δij . ε˙tr = ζ (45) ij 3 2
Φ= 315
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The parameter Bζ in Eq. (37) was (Stringfellow et al., 1992) " 2 # g2 1 Γ − g¯ r ˙ Bζ = √ β(fsb ) exp − H(Σ), 2 sg 2πsg
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Figure 5: Tensile test specimen, thickness 0.70 – 0.76 mm. Scale is cm with mm divisions. 4000 3500
AN US
3000
σ (MPa)
2500
σ (MPa)
1800 1600 1400 1200 1000 800 600 400 200 0 0.0
2000
Exp. RT Exp. 75 ◦ C Exp. 100 ◦ C Exp. 150 ◦ C 0.2
0.4
0.6
ε
0.8
1.0
1000
Exp. Tension Exp. Compression
500
0 0.0
1.4 1e−2
0.5
1.0
1.5 2.0 ε
2.5
3.0
3.5 4.0 1e−2
(b)
M
(a)
1.2
1500
333 334 335 336 337
CE
338
identical for all specimens used in the present and earlier experiments by Linares Arregui and Alfredsson (2010). The specimens with the aforementioned steel grade DIN 100CrMnMoSi8 had been heat treated to martensitic microstructure with a retained austenite content. The monotonic test setup was placed in an oven to maintain constant temperature during the experiment and the axial strains were measured using extensometers. To avoid assembly stresses, the fixtures were carefully aligned with the load line using metal shims. In addition to these experiments, the results of tension, compression and cyclic experiments performed at 75◦ C were taken from the study of Linares Arregui and Alfredsson (2010). The stress strain curves in monotonic tension, compression and cyclic experiments are presented in Figs 6 and 7(a). In this work, the plastic and nonlinear elastic material parameters given by Linares Arregui and Alfredsson (2010) and Linares Arregui and Alfredsson (2013) were used as starting values to determine the material parameters of the corresponding model that considers phase transformation. In their work the phase transformation was not explicitly modeled, instead all the inelastic strains were included in the plastic description. The material properties for nonlinear kinematic and isotropic hardening given in Table 2 are based on the
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Figure 6: a Tension experiments to specimen rupture b. Tension and compression experiments at 75◦
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16
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0.95 1000
−1000 −2000
0.85 0.80
Exp. 75 ◦ C Cyclic
−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015
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ζ
σ (MPa)
0.90 0
FE 75 ◦ C Tension FE 75 ◦ C Cyclic Exp. 75 ◦ C Tension Exp. 75 ◦ C Cyclic
−0.010 −0.005
ε
(a)
0.000 ε
0.005
0.010
(b)
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M
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studies by Linares Arregui and Alfredsson (2010). Since the present investigation focused on and modeled the phase change taking into acount both stress and strain induced transformation, some parameters in the plastic description that are given in Table 2 were adjusted from the values in the reference (σ Y 0 , a, a∗ , b, Q, C1 , C2 , C3 , γ1 , γ2 , γ3 ). The plasticity parameters were updated iteratively using inverse modeling of the stress strain curves with respect to the experimental results in Figs 6 and 7(a). The parameters quantify the effects of the different mechanical phenomena described in Eqs (1) to (48). They are grouped accordingly in Table 2. In previous studies it is reported that the elastic modulus of martensite is less than that of austenite (Liu and Shan, 2011; Mendiguren et al., 2015) therefore different elastic moduli have to be defined for each phase. The equivalent elasticity tensor should vary during the transformation with respect to the change of martensite fraction as given in Eq. (16). Here the Poisson's ratio and nonlinear bulk modulus K(εel kk ) were taken from Linares Arregui and Alfredsson (2013). The same values were used for both phases. The Young’s moduli for each phase were then determined by inverse modeling of the monotonic experiments presented in Fig. 6, the curves describing the nonlinear elastic behavior given in Fig. 4(b) and using Eqs (16) and (20). The X–ray diffraction (XRD) measurements were performed using a Bruker D8 Discover instrument and Co K–alpha radiation. A 1D silicon strip detector with 192 strips were used and a programmable divergence slit maintained the sample illumination at 5 mm throughout the measurements. The instrument was operated at 40 kV and 35 mA, and the step size used was 0.03 deg. The counting time was varied depending on the cross-section area of the sample to assure proper counting statistics. The accuracy of 2.2-3 % was maintained in the date analysis of single martensite fraction values. The measured X-ray diffraction values at different temperatures are shown in Fig. 8. The parameters affecting the kinetics of the phase transformation were set from the retained austenite levels measured after deformation. The parameter
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Figure 7: a. Cyclic loading, experimental data taken from Linares Arregui and Alfredsson (2010) (test no: 13838). b. Uniaxial strain vs. martensite fraction.
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370 371
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378 379
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Table 2: Material parameters used to fit the numerical predictions to experimental data.
Value
Parameter
920 MPa
a
76 × 10−6 Q
135 GPa 700 GPa 50 GPa
γ1 γ2 γ3
5.2 57 220◦ C 3.05 420 MPa 350
Stress Induced Transformation κ ξ1
0.054 MPa 0.010 (MPa·s)−1
ρ ξ2
7.8 MPa/◦ C 1.02
195 GPa 0.27
E0a K1
220 GPa -4000 GPa
M
g2 g¯ Ms β sa sg
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382
383
384
PT
β in Eqs (39) and (44) depends on ratio of the average volume of martensite and shear band intersection. In this study, the start value for β was based on values in the literature (Papatriantafillou, 2005; Stringfellow et al., 1992). The value of β = 3.05 that fits the transformation kinetics given in Figs 8(a) and 8(b) was finally determined again using inverse modeling with the FEM subroutine. The exponent r in Eqs (39) and (44) is commonly taken as 4 − 4.5 for martensitic TRIP steels (Iwamoto, 2002; Papatriantafillou, 2005; Stringfellow et al., 1992). Here r = 4 was found to give the best fit to the experimental data represented in Figs 8(a) and 8(b). In general α in Eqs (40) and (39), which controls the rate of shear band formation based on a stacking fault energy, is temperature dependent (Sierra and Nemes, 2008; Tomita and Iwamoto, 1995). Since the temperature degree of freedom was satisfied through the functions Γ(Θ, σ) and C(T ) in Eqs (42), (6), α and β were here kept temperature in-
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386
387 388
389 390 391
392
950 500 50
0.015 0.00623 800 4 0.0371 60 176000
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381
180 MPa
Strain Induced Transformation A0 A1 α r ∆v g0 g1
Nonlinear elasticity E0m ν0
380
0.04
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Value
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Parameter Yield Surface σY 0 Flow potential a∗ Isotropic hardening b Kinematic hardening C1 C2 C3
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0.86
ζ
0.84 0.82 0.80
0.792 0.790 0.788 0.786 0.784 0.782
FE 75 ◦ C Exp. 75 ◦ C
0.780
0.78 0.000 0.002 0.004 0.006 0.008 0.010 0.012
0.778 0.00
ε
(a)
CR IP T
0.88
0.794
FE RT FE 75 ◦ C FE 100 ◦ C FE 150 ◦ C Exp. RT Exp. 75 ◦ C Exp. 100 ◦ C Exp. 150 ◦ C
ζ
0.90
0.01
0.02
ε
0.03
0.04
0.05
(b)
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M
395
dependent. The parameters responsible for temperature dependence (g1 , ρ, κ) were obtained using the results of X-ray diffraction measurements presented in Figs 7(b) and 8(b). The g1 value was fitted to the measurements representing the change of ζ at high temperatures (75◦ C, 100◦ C and 150◦ C) where strain induced transformation was dominant. The parameters ρ and κ on the other hand changes the transformation start stresses in Eq. 6. Therefore they were determined considering the yield strength at RT to 75◦ C, where stress induced transformation starts before yielding, see Fig. 8(a). Parameters g0 , g2 , g¯ and sg in the probability function are determined through their fit to the X–ray diffraction measurements given in Fig. 8(a), especially at high temperatures where the strain induced phase transformation is dominant. The dimensionless parameters A0 , A1 and sa given in Eq. (46) were used to calculate A, which account for the strains related to shape change. These parameters were determined from the stress strain curves in Fig. 6. The starting values were based on the values by Iwamoto (2002); Papatriantafillou (2005); Stringfellow et al. (1992). According to Stringfellow et al. (1992) the difference in kinetics and morphology of stress and strain induced phase transformation affect the magnitude of transformation strains. However, in this study the same A was used to define the evolution of all transformation strains. Figure 2 shows the retained austenite as islands at the prior austenite grain boundary intersections. Some islands of retained austenite are circled in red in Fig. 2 which represent small inclusions constrained in martensite matrix that effect phase change and redistributes the stress field in the transformation zone. Therefore the volume expansion of retained austenite is not free and it is mainly constricted by elastic deformation of the martensite matrix. Using the empirical equation given by Moyer and Ansell (1975) the relative volume expansion, ∆v in Eq. (45), during martensitic transformation of the iron-carbon alloy with 0.91 wt % C was calculated as 0.0371. The determined value was in the range of 0.2 % to 0.4 %, that is reported by Leal (1984) and commonly used by previous
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Figure 8: a. X-ray measurements of transformation of retained austenite to martensite in tension specimens are compared to model predictions. b. Simulated transformation kinetics during compression and X-ray measurement at experiment termination.
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412
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418
419 420 421
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40 mm
1.5 mm 4 mm
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P
r = 4 mm
Figure 9: FE model, thickness 0.70 – 0.76 mm. The triangles represents the symmetry conditions.
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researchers. The relation between the parameters which were only related to the kinetics of the stress induced phase change, ξ1 and ξ2 , was determined using Eq. (48) (Moumni et al., 2011)
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∂ζ ∂T ζ˙stress = = Cζ ξ1 (ξ2 − ζ). ∂T ∂t 426
Therefore
ζ1
T Rf
M
ξ1 = T˙
Rζ2
dζ (ξ2 −ζ)
422
423 424 425
(49) 426
(50)
Cζ dT
Ms
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435
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428
where Tf = 165◦ C representing the temperature at the end of heat treatment when ζ2 = 0.78 and Ms = 220◦ C (Ovako, 2016) at the martensite start where ζ1 = 0. Using ξ2 = 1, which is defined as the maximum martensite fraction that can be obtained as a result of phase transformation as stated by Moumni et al. (2011), and Eq. (50) ξ1 , was calculated. Table 3 lists the minimum number of experiments required to determine the model parameters, the number of experiments used here for parameter determination, the additional number of experiments used for model verification and which parameters are fitted to the corresponding curves. Some of the calculated parameters indicated in Table 3 were fitted independently (σ Y 0 , a, a∗ , b, Q, Ms , g1 , κ, ρ, ξ1 , ξ2 ) since they define a unique characteristic. Some parameters (C1 , C2 , C3 , γ1 , γ2 , γ3 , A0 , A1 , g2 , g¯, α, β, sa , sg , g0 , E0m , E0a , ν0 , K1 ) required number of iterations. The other parameters are either calculated through expressions existing in the literature or taken directly from the literature (∆v , ξ1 , ξ2 ,r). The experiments were monotonic tension to specimen rupture or a predefined strain, monotonic compression to a predefined strain and cyclic pushpull to rupture. Also, the tension experiments must be performed at different temperatures in the transition temperature range, see Fig. 1.
PT
427
436 437
438
AC
439
440 441
442 443 444
20
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Table 3: Number of experiments used
Used
Verification
Tensile
3
6
2
Compression
1
1
1
Cyclic
1
1
1
X-ray diff.
8
8
2
4. Methods of Numerical simulations
Parameters σ Y 0 , a, b, Q, C1 , C2 , C3 , γ1 , γ2 , γ3 A0 , A1 , g2 , α, sa , E0m , E0a , ν0 , K1 A0 , A1 , g2 b, Q, C1 , C2 , C3 , γ1 , γ2 , γ3 α, sa , E0m , E0a , ν0 , K1 A0 , A1 , g2 , g¯, r, β, sg , g0 , g1 , κ, ρ
CR IP T
Min.
AN US
445
Test
461
5. Results
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462
464
465
466
As seen from true stress true strain curves in Fig. 10, the finite element results were in good agreement with the tension experiments performed at different temperatures. Figure 6(a) indicates that the yield stress increases with the increase of temperature from RT to 75◦ C, but over 75◦ C only a small difference is observed. According to Neu and Sehitoglu (1992) during stress induced transformation, the yield strength in tension increases considerably with temperature. Therefore for TRIP steels, the change of yield point can be used to identify whether stress induced transformation takes place before strain induced transformation or not. To see the temperature dependence of stress induced transformation and the change in transformation start stress, stress strain curves representing the data obtained from the tensile experiments to rupture and those calculated using FE are plotted separately in Figs 6(a) and 11. In the FE computations
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M
460
FE calculations were done using ABAQUS through a user defined material subroutine (UMAT) that followed Eqs (26) to (48) with the material properties given in Table 2. Figure 9 shows the FE model used in the analysis. A 3D mesh with 67872 C3D8 elements was used after a mesh size and convergence study. The total CPU time was 13.1 × 103 s and the ratio of CPU time with and without considering phase transformation was 1.2. In the UMAT, stress and strain induced phase transformations were coupled with nonlinear elasticity, nonlinear kinematic and isotropic hardening behaviors and formulated using fully implicit integration of the constitutive relations. In the numerical solution, initially stress induced transformation was considered, the evolution of martensite fraction and transformation strains were calculated from Eqs (48) and (45). During dislocation movement, strain induced transformation takes place at shear band intersections along with stress induced phase change. Together they yield the evolution of martensite fraction. The flowchart of the UMAT subroutine is summarized in Algorithm 1.
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1400
1400
1200
1200 1000
σ (MPa)
σ (MPa)
1600
800 600 400
FE RT Exp. RT
200
200
500
ε
(b)
1500
σ (MPa)
ED
1000
FE 75 ◦ C Exp. 75 ◦ C
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
M
σ (MPa)
1500
600
400
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 ε
(a)
800
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1000
500
FE 100 ◦ C Exp. 100 ◦ C
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 ε
PT
0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 ε
FE 150 ◦ C Exp. 150 ◦ C
(d)
CE
(c)
AC
Figure 10: Tensile experiments and finite element results at a. RT b. 75◦ C conducted by Linares Arregui and Alfredsson (2010) (test no: 13823) c. 100◦ C d. 150◦ C
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Algorithm 1 UMAT subroutine for plastic deformation, stress and strain induced phase change. Initial Conditions pl 0 0 ζ0 = 0.78, [εtr ij ]0 = 0, [εij ]0 = 0, k:iteration number, n:increment number, [∆A] = [A]n+1 − [A]n . 1. Trial Stress pl 0 pl 0 0 0 tr 0 Set : k = 0, ζn+1 = ζn0 , [εtr ij ]n+1 = [εij ]n , [εij ]n+1 = [εij ]n pl
k [∆εpl ij ] =
d k m k 3 [σij ]n+1 −[Xij ]n+1 k Y 2 [σ ]n+1
AN US
Calculate trial stress: [σij ]0n+1 = [Eijkl ]0n+1 ([εkl ]0n+1 −[εkl ]0n+1 −[εkl tr ]0n+1 )+ 2 0 K1 [(εel kk ) ]n+1 δij ) 2. Check stress induced transformation and yield function k if (( Φ > 0 & ζ˙ > 0 ⇒ Cζ > 0) & fn+1 < 0) then only stress induced transformation occurs [∆ζ]k+1 = Cζ ξ1 (ξ2 − [ζ]k+1 n+1 ) k+1 [εtr ] calculated using Eq. (45) ij n+1 update stresses using Eq. (15) and go to step 6. k else if (Cζ < 0 and fn+1 > 0) then yielding occurs before stress induced transformation [f ]kn+1 = 0 gives ∆λ plastic strains are calculated using: + 3a∗ [σ h ]kn+1 [∆λ]k
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k k k [∆ζstrain ]k = (1 − ζn+1 )(Aζ [∆εpl eq ] + Bζ [∆Σ] ) k else if Cζ > 0 and ζ < 1 and fn+1 > 0 then yielding, strain and stress induced phase changes occur k k k k [∆ζ]k = (1 − ζn+1 )(Aζ [∆εpl eq ] + Bζ [∆Σ] ) + Cζ ξ1 (ξ2 − ζn+1 ) k [εtr ij ]n+1 calculated using Eq. (45) else Elastic deformation and go to step 6. end if pl k+1 k+1 tr k+1 3. Update [ζ]k+1 n+1 ,[εij ]n+1 , [εij ]n+1 and [Eijkl ]n+1 . 4. Calculate stress using : pl k+1 k+1 k tr k+1 el 2 k+1 [σij ]k+1 n+1 = [Eijkl ]]n+1 ([εkl ]n+1 − [εkl ]n+1 − [εkl ]n+1 ) + K1 [(εkk ) ]n+1 δij ) 5. Set k=k+1 and go to step (2). 6. Set n = n + 1 and go to step (1). 7. When the calculation finishes at all the integration points ABAQUS checks the force equilibrium.
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rupture of the specimens was not modeled. Instead all simulations continued to the total strain when the last experiment ruptured. During the experiments it was observed that the stress was homogeneously distributed and the specimens deformed without any necking. Therefore the localization of inhomogeneous stresses was not considered in the numerical simulations. Figure 6(a) shows that the start of stress induced phase change increases with the temperature up to a certain value above which no phase change occurs prior to yielding. Thus 23
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σ (MPa)
1800 1600 1400 1200 1000 800 600 400 200 0 0.0
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FE RT FE 75 ◦ C FE 100 ◦ C FE 150 ◦ C 0.4
0.6
ε
0.8
1.0
1.2
1.4 1e−2
AN US
0.2
Figure 11: FE calculations
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the results obtained were compatible with the findings of Neu and Sehitoglu (1992). The results by Neu and Sehitoglu (1992) indicated that it is difficult to separate the stress and strain induce transformations by only analyzing monotonic loading experiments at one specific temperature, especially if the material yields before transformation. To illustrate the transformation kinetics distinctly, the conditions considering only stress induced or strain induced transformations were modeled and are plotted in Fig. 12 at two different temperatures. At RT in Fig. 12(a), the dashed green curve, representing the case where only stress induced transformation is active, coincides with black curve, including both transformation mechanics, till the start of strain induced transformation. Therefore, it is seen that for low temperature the stress induced transformation initiates before yielding and is a major part of the total transformation. At 75◦ C strain induced phase change becomes more dominant and the stress induced phase change takes place after yielding as is represented in Fig. 12(b). For visual representation of the dependence of yield strength on transformation behavior, the uniaxial stress versus martensite fraction is plotted in Fig. 13(a). By comparing the onset of nonlinearity in Fig. 6(a) with transformation start stress in Fig. 13(a) it can be clearly seen that the yield strength decreases with decreasing temperature as the result of early transformation start at low temperatures of stress induced transformation. On the other hand at higher temperatures the effect of temperature on yield strength is less because no stress induced transformation occurs before yielding. Moreover, Fig. 13(b) supports the claim that at RT phase change starts before yielding since εtr 11 starts evolving before εpl . It was concluded that by performing a series of different eq monotonic tension experiments at different temperatures it was possible to separate between stress and strain induced transformation. Figure 1 illustrates
ED
483
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482
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σ (MPa)
1500
σ (MPa)
1500
1000
500 0 0.0
0.2
0.4
500
Stress & Strain induced Strain induced Stress induced 0.6 0.8 1.0 1.2 1.4 1e−2 ε
0 0.0
0.2
0.4
(a)
CR IP T
1000
Stress & Strain induced Strain induced Stress induced 0.6 0.8 1.0 1.2 1.4 1e−2 ε
(b)
AN US
Figure 12: Stress strain curve obtained from FEA, illustrating the effects and difference between stress and strain induced phase change at two different temperatures a. RT b. 75◦ C
2000
0.16
0.14
σ (MPa)
1000
FE RT FE 75 ◦ C FE 100 ◦ C FE 150 ◦ C
0.82
0.84
ζ
0.86
0.88
0.90
ED
0.80
0.10
FE RT FE 75 ◦ C FE 100 ◦ C FE 150 ◦ C
0.08
0.06
0.04
M
500 0 0.78
pl /ε11 εeq
0.12
1500
(a)
0.02 0.00 0.0
0.1
0.2
0.3
tr /ε11 ε11
0.4
0.5
(b)
509 510
for low temperatures that the decrease of inelastic deformation with decreasing temperature shows that stress induced transformation is present and predominating. It was then detected separately from strain induced transformation, see the schematic start stress below yielding for stress induced transformation in Fig. 1. Strain induced transformation starts after yielding which in the present temperature range is relatively constant. It was then found that after yielding, strain induced transformation was more pronounced than stress induced. The XRD measurements are shown in Fig. 8(a) together with the ζ curves obtained from FE using the calibrated material properties. When all data points were considered together with the model prediction in the figure, the measurements supports each other and deviation from the model is less than 0.5 % for each retained austenite value. Hence, the overall accuracy of multiple measurements were much better than the single point estimate suggests. Figures 14(a) and 14(b) represent the contour plot of martensite fraction at
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Figure 13: a. Effect of martensite fraction on strength b. evolution of nonlinear strains at monotonic tension.
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(a)
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+8.609e−01 +8.609e−01 +8.493e−01 +8.378e−01 +8.262e−01 +8.147e−01 +8.031e−01 +7.916e−01 +7.800e−01
CR IP T
+8.609e−01 +8.493e−01 +8.378e−01 +8.262e−01 +8.147e−01 +8.031e−01 +7.916e−01 +7.800e−01
M
(b)
Figure 14: a. ζ contour plot, when a. ε11 = 0.0095 b. ε11 = 0.0112 (strains at notch root).
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the early transformation stage ( ε11 = 0.00275 in the test part with homogeneous strain and maximum ε11 = 0.0095 in the notch root) and at the specimen rupture under tensile loading at 75◦ C, see Fig. 6(a). During phase change that occurs more predominantly under tension than compression, the volume increase induces an increase in strain, resulting in a nonlinear stress-strain behavior in advance and/or during dislocation motion. For this reason, in high strength steels with retained austenite undergoing phase change, the asymmetry between compressive and tensile yield stresses seen in Fig. 6(b) is largely a result of phase transformation. The triaxiality parameter Σ in Eq. (38) affects the probability of martensite formation at the dislocation sites during strain induced phase change through the parameter Γ in Eq. (42) in the probability function P , Eqs (37), (41) and (44). The strain induced model of Stringfellow et al. (1992) that fitted to the martensite fraction evolution data of Young (1988) showed that martensite formation decreases when the triaxility is negative as in the case of compression. Additionally, Young (1988) reported that transformation curves for stress induced transformation exhibit linear behavior with respect to axial strains, whereas the curves for strain induced transformation was found to be nonlinear. The comparison of tension–compression tests and the calculated FE results together with X-ray
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σ (MPa)
1000
−1000
Exp. 75 ◦ C Cyclic FE 75 ◦ C Cyclic
−2000
AN US
−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 ε
Figure 15: Cyclic loading, experimental data taken from Linares Arregui and Alfredsson (2010). (test no: 13838)
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diffraction measurements given in Fig. 8(b) pointed out that under compression a small amount of transformation occurs. The simulated behavior of ζ versus axial strain plotted in Fig. 8(b) differs quite markedly to the tensile loading case in Fig. 8(a). In compression, the transformation is dominated by the stress induced transformation that develops at the early stages of the deformation. This part of the transformation is furthermore linear with the strain, which agree with the findings of Young (1988). The last part of the curve includes small strain induced transformation but at this stage the rate of transformation ζ˙ is much lower than the early part. The strength differential effect was also examined for cyclic load and compared to the experiments performed by Linares Arregui and Alfredsson (2010) (test no:13889) in Fig. 7. The XRD results represented in Fig. 7(b) are in agreement with the simulations where the dotted the black line represents the martensite fraction under monotonic tensile loading and dashed blue curve is the evolution of martensite in the first cycle. The results confirm that the phase transformation predominantly develops during tension. Note that the retained austenite measurement for the cyclic curve in Fig. 7(b) was not directly used in determining the phase transformation parameters. Hence, it gave an indication on the model prediction. For high strength steels, the slope of the yield surface in the meridian plane is a ≈ 0.02 (Allen and Wilson, 2004; Guo et al., 2008; Spitzig and Richmond, 1979), which is smaller than the value determined by Linares Arregui and Alfredsson (2010), a ≈ 0.25. However, this value included all nonelastic effects. The effect of the pressure sensitive plasticity parameter, a on yield strength and
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Figure 16: The effect of parameter a a. on constitutive behavior b. on martensite transformation.
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phase transformation was investigated by using four different a values in tensile simulations. The parameter sensitivity study represented in Fig. 16(a) shows that the decrease in yield stress for increasing a is more significant at 75◦ C than RT. This result is mainly due to the stress induced transformation that occurs at lower temperatures before yielding. Therefore, the dependency of hydrostatic stress is influenced more by stress induced phase change, where the transformation is initiated by elastic stresses, compared to strain induced phase transformation. To determine the value of parameter a, not only the stress strain response but also the effect of a on phase transformation kinetics was studied. From Fig. 16(b) it is seen that decreasing a from a = 0.12 to a = 0.04 decreases the predicted martensite fraction at a certain strain value. Experimental measurements on martensite fraction ζ in Fig. 16(b), yield stress in Fig. 16(a) and the tensile and compressive stress strain responses in Fig. 6(b) were used to establish a = 0.04 as the best yield stress slope for the present material. During strain induced phase change, which dominates for elevated temperatures, the evolution of martensite fraction occurs at dislocation sites and the effect on the yield surface slope is illustrated in Fig. 16(b). Therefore the experimented evolution of ζ at elevated temperatures were also considered. The effect of phase change on SDE was analyzed for two different temperatures through simulations of monotonic loading where the different parts of phase changes were executed. The evolution of such simulated SDE is plotted against inelastic strains in Fig. 17. Figure 17(a) shows that for increased temperatures, here exemplified with 75◦ C, strain induced transformation together with Drucker Prager plasticity coincides with the curve representing the full model and they yield closer results to the experimental measurements of Linares Arregui and Alfredsson (2010) performed at 75◦ C. The difference in SDE between experimental measurements and full model predictions in Fig. 17(a) is explained in Fig. 6(b) by the corresponding difference in the compression curves at moderate strain. Figure 6(b) also illustrates that in absolute stress values the model deviation is small. Figures 17(a) and 17(b) includes simulated curves that only contains pressure sensitive plasticity, using Drucker Prager, i.e. the phase transformation was turned off. Comparing these curves with the full model prediction clearly illustrates that phase transformation dominates SDE effect for the current material structure. The result also show that stress induced phase change has only a small effect on SDE for high temperatures. Besides, when the temperature decreases to a level where the stress induced phase transformation is more effective on yield strength, e.g. RT in this case, the stress induced transformation is mainly responsible for SDE especially at low inelastic strains, see Fig. 17(b). Figure 18 compares the calculated volume changes during tensile and compressive loading to experimental measurements of Linares Arregui and Alfredsson (2010). It was observed that the computed volume change during compression is less than that during tensile loading which agree with the results by Neu and Sehitoglu (1992).
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The present study couples the plastic deformation obeying nonlinear hardening rules and nonassociated flow rule with stress and strain induced phase change. Good agreement was obtained with the experimental results. In addition to phase transformation kinetics, the nonlinearity of elastic strains were taken into account in the material model while scrutiny of the plastic response cycles from cyclic experiments clearly showed that nonlinear elastic effects do have influences for the high elastic strains that can develop in this type of high strength bearing steel. Although the full material model includes a relatively large amount of material parameters, a limited number of well planned and well instrumented experiments were sufficient to calibrate the model. The required series include monotonic tensile experiments to some different strains and at some different temperatures together with at least one compression and one cyclic experiment. The data captured should include retained austenite levels and densities at test termination as well as the stress strain curves. The experiments showed that the solid to solid phase transformation of retained austenite to martensite affects the material behavior before and after plastic deformation through stress and strain induced phase transformations and changes in the yield strength of the material. Results indicated that the change of the yield strength with temperature is the main indicator of stress induced transformation that takes place before plasticity and defines start of the measured yielding at RT. At low temperatures, approximately below 75◦ C for the present material, stress induced transformation results in a reversed tem-
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perature relation with decreasing yield point for decreasing temperature. Also at these temperatures phase transformation was only stress induced, i.e. no strain induced transformation existed. At 75◦ C and for higher temperatures up to the martensite start temperature plastic yielding develop before phase change. Here strain induced transformation dominates the transformation, although some stress induced transformation continues to develop. By utilizing the understanding on how the yield point changes with temperature and the corresponding measured data points, stress and strain induced phase transformation were successfully separated from each other, their magnitude measured and the model parameter values were determined. Thus, the proposed model summarized in Eq. (48) separates stress and strain induced transformation. It was concluded that the stress and strain induced phase transformation was active in the present material and that the processes can be separated by coupled modeling with a series of different monotonic experiments at different temperatures where both the stress strain curves were captured and the retained austenite levels after experiment termination was determined. Using the proposed model not only the coupled behavior but also distinctly stress and strain induced phases can be identified. Therefore, using this modeling technique, for complex loading condition like the stress distribution around a crack tip or at contact surfaces, contour plot of strain and stress induced martensite transformation zones can be plotted separately and their contribution to the mechanical behavior can be calculated independently. According to the monotonic experiments and X-ray diffraction measurements it was concluded that the amount of retained austenite that transforms to martensite is higher under tension than compression. The calculations for compressive loads at 75◦ C showed that a small stress induced phase transformation at the early stage of compression accompanied by strain induced phase change which is very small in total compared to the change under tensile loading. The analysis showed that phase transformation enhances and dominates the SDE over plasticity effects and that SDE can be simulated successfully by employing the coupled stress and strain induced phase transformation kinetics together with a pressure sensitive surface for plastic yielding and flow as is proposed in the current study. In the light of the aforementioned reasons it is claimed that both stress and strain induced phase change should be taken into account in the definition of constitutive behavior of the present class of high strength steels. Therefore, the material model that was capable of capturing nonlinear hardening behaviors, pressure sensitive plastic yielding and nonlinear elasticity together with stress and strain induced phase change was required to simulate the mechanical material behavior including the SDE in the high strength martensitic bearing steel accurately.
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This work was conducted in cooperation with SKF ERC. The authors would ¨ like to thank Dr P. Hedstr¨ om and Mr M. Oberg for providing measuring and 31
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experimental expertise. The financial support from SKF ERC is much appreciated.
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Ottosen, N.S., Ristinma, M., 2005 The Mechanics of Constitutive Modeling . Ovako, 2016. Steel Navigator. http//www.ovako.com/steel navigator/pdf/Ovako 827 - 100CrMnMoSi8-4-6 v3.pdf, access date 2016-09-01 .
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Morin, C., Moumni, Z., Zaki, W., 2011b. Thermomechanical Coupling in Shape Memory Alloys under Cyclic Loadings: Experimental Analysis and Constitutive Modeling. Int. J. Plast. 27, 1959–1980.
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Papatriantafillou, I., 2005. TRIP Steels: Constitutive modeling and computational issues. Ph.D. thesis. University of Thessaly. Papatriantafillou, I., Agoras, M., Aravas, N., Haidemenopoulos, G., 2006. Constitutive modeling and finite element methods for TRIP steels. Comput. Methods Appl. Mech. Eng. 195, 5094–5114.
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Perlade, A., Bouaziz, O., Furn´emont, Q., 2003. A physically based model for TRIP-aided carbon steels behaviour. Mater. Sci. Eng. A 356, 145–152.
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Paul, S.K., 2013. Effect of martensite volume fraction on stress triaxiality and deformation behavior of dual phase steel. Mater. Des. 50, 782–789.
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Shan, T., Li, S., Zhang, W., Xu, Z., 2008. Prediction of martensitic transformation and deformation behavior in the TRIP steel sheet forming. Mater. Des. 29, 1810–1816.
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Patoor, E., Eberhardt, A., Berveiller, M., 1996. Micromechanical Modelling of Superelasticity in Shape Memory Alloys. Le J. Phys. IV 06, C1–277–C1–292.
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Zhang, Y., Cheng, Y.T., Grummon, D.S., 2007. Finite Element Modeling of Indentation-Induced Superelastic Effect Using a Three-dimensional Constitutive Model for Shape Memory Materials with Plasticity. J. Appl. Phys. 101, 053507–053513.
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Appendix A. Elastic strain for monotonic tension-compression
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The equation
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el 9Geq 0 K(εkk ) E(εel eq , kk ) = 3K(εel kk ) + G0
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el and σ h = K0eq εel kk + K1 εkk was evaluated for uniaxial tension and compression. The uniaxial stress was then calculated through
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el Equation (22) together with εel kk = (1 − 2v)ε11 yields 3(K0eq + K1 εel el kk ) − 2G εel εkk = 1 − eq 11 . 3(K0eq + K1 εel kk ) + G0
el Using Eq. (53), εel kk can be defined in terms of ε11 as p eq el 2 3(K0eq + Geq (3(K0eq + Geq el 0 ) 0 ) + 36K1 G0 ε11 εkk = − + . 6K1 6K1
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The dissipation potential of the system defined through free energy potential by Eq. (5) was
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m m Xij χ˙ ij − Rκ˙ R + Cζ ζ˙ > 0
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˙ ∂g = 3 sij − αij χ˙ m ij = −λ m ∂Xij 2 σeq
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(57)
∂g κ˙ R = −λ˙ (58) ∂R When a nonassociative flow rule is employed in order to compute the bounds for the influence of pressure that satisfies the dissipation equation Eq. (56) the following analysis has to be done. Equations (57), (58) and (35) were used in Eq. (56) and the stresses were written in deviatoric and dillatation components so that ! 3 X 3 γm m m ∗ h D = σeq + 3a σ + X Xij − R + Cζ ζ˙ ≥ 0 (59) m ij 2 C m=1 41
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m The corresponding work conjugates to the thermodynamic forces Xij and R can be defined as
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Appendix B. Criterion for pressure dependence in the nonassociated flow rule
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el When the Eq. (54) is used in Eq. (52), noting that εel kk = 0 for ε11 = 0, el = 0 when σ11 = 0 and solved for ε11 the following equation was obtained q eq 2 eq 4 σ11 3 σ11 K1 + (K0 ) − K0 . (55) εel = + 11 3Geq 6K1 0
D = σij ε˙pl ij −
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+ K1 εel kk ) εel 11 . K1 εel ) + G kk
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The positivity of Cζ ζ˙ is discussed in appendix C. Therefore for the rest of the analysis the method used by Linares Arregui and Alfredsson (2010) is followed. Since λ˙ > 0 and f = 0 during plastic deformation, σY 0 − 3σ h (a − a∗ ) +
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σY 0 + R
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This condition should be satisfied for large plastic deformations when the kinematic hardening is zero, Xij = 0. Hence a ≥ a∗ . For this conditions the hydrosatic tension can be formulated as 3σ h = σY 0a+R which yielded the equation
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Linares Arregui and Alfredsson (2010) states that in the material model that considers the nonassociated flow rule with pressure dependent plastic flow rule, the chosen parameters a∗ and a has to be controlled throughout the simulations to verify the stability of the solution and satisfaction of dissipation equation.
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Appendix C. Dissipation Potential
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The pseudopotential of dissipation was
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where both terms in this Cζ = 0. This yields the condition Cζ ≥ division are positive and numerator is zero when there is no plastic deformation. It is known that stress induced phase change starts before strain induced phase change below a certain temperature and the evolution of stress induced martensite fraction defined in Eq (49), ζstress ≥ 0. In addition to that the term ˙ Aζ ε˙pl eq + Bζ Σ is positive during plastic deformation and zero otherwise. Therefore here, as a transformation function the condition of Cζ ≥ 0 should be used. This conditon satisfies positivity of the term responsible from the dissipation during phase change Cζ ζ˙ in dissipation equation.
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where ( ∂Φ Cζ )0 = 0 since there is no phase change when the termodynamic force
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˙ Aζ ε˙pl eq +Bζ Σ , − ξ1 (ξ 2 −ζ)
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According to Ottosen and Ristinmaa (2005) the conditions of convex potential analysis ensures that Φ(Cζ ) − Φ(0) ∂Φ ≥ , (63) (Cζ − 0) ∂Cζ 0
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