Simulation of hydroforming of steel tube made of metastable stainless steel

International Journal of Plasticity 26 (2010) 1576–1590

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International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Simulation of hydroforming of steel tube made of metastable stainless steel Lars-Erik Lindgren a,b,*, Mikael Olsson b, Per Carlsson b a b

Luleå University of Technology, SE-97187 Luleå, Sweden Dalarna University, SE-78188 Borlänge, Sweden

a r t i c l e

i n f o

Article history: Received 19 April 2009 Received in final revised form 30 January 2010 Available online 6 February 2010 Keywords: Strain-induced martensite Hydroforming Finite element simulation Plasticity model

a b s t r a c t The Olson–Cohen model for strain-induced deformation, further developed by Stringfellow and others, has been calibrated together with a flow stress model for the plastic deformation of metastable stainless steel. Special validation tests for checking one of the limitations of the model have also been carried out. The model has been implemented into a commercial finite element code using a staggered approach for integrating the stress–strain relations with the microstructure model. Results from a thermo-mechanical coupled simulation of hydroforming of a tube have been compared with corresponding experiments. The agreement between experimental results of radial expansion and martensite fraction and the corresponding computed results is good. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Metastable austenitic stainless steels are of interest for many applications. They have good ductility during the initial forming and can reach high strength after the forming as a result of strain-induced martensite formation. However both strain and temperature must be controlled carefully in the process in order to utilise these properties fully. A non-optimal forming path may either lead to fracture as the martensite formation starts too early or that the final strength of the component is lower than wanted if it starts too late in the forming process. The aim of the current work is to calibrate and validate a material model useful for simulating forming processes enabling the optimisation of the forming path. The martensite formation depends, in the general case, on the prevailing stress, strain and temperatures. The current work focuses on strain-induced martensite formation of an AISI 301 type of material, HyTensXÒ. The model has been implemented into the commercial finite element code MSC.Marc and a thermo-mechanical coupled simulation of tube hydroforming has been performed. The work consists of an experimental part with tensile tests starting at different temperatures used to calibrate the martensite and the flow stress models. The martensite formation model is the well-known model by Olson and Cohen (1975). It has been further developed by Stringfellow et al. (1992) and others (Tsuta and Cortes, 1993; Tomita and Iwamoto, 1995; Iwamoto et al., 1998, 2000, 2001; Tomita and Iwamoto, 2001; Iwamoto and Tsuta, 2002). Furthermore, additional tests were done to validate the martensite formation model. The implications of this comparison are discussed in the paper. The above model was used to simulate the Tube Hydroforming (THF) of HyTensXÒ and a comparison between computed results and the empirical THF data was performed. THF is a manufacturing process that uses a fluid medium to form a part by using a high internal pressure (Dohmann and Hartl, 1997; Koc and Altan, 2001). One of the main fields of using hydroforming is the lightweight construction of automobile parts. The advantages of hydroforming include better tolerances, decreased number of parts and an increased range of forming options. The microstructural evolution during THF has been monitored by light optical microscopy (LOM), scanning electron microscopy (SEM), electron back scattering diffraction (EBSD) and * Corresponding author. Address: Luleå University of Technology, SE-97187 Luleå, Sweden. E-mail address: [email protected] (L.-E. Lindgren). 0749-6419/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2010.01.012

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magnetic measurements. The transformation of the austenite to the strain-induced a0 -martensite phase was quantified by EBSD and magnetic measurements (Carlsson and Olsson, 2006). A finite element model of the hydroforming process was created and the process was simulated. 2. Related work 2.1. Hydroforming Several authors have simulated hydroforming process and their work are not reviewed here. Instead the interested reader can consult (Ahmetoglu and Altan, 2000; Lundqvist, 2004; Mackerle, 2005). However, some recent hydroforming simulations that also have used advanced material models are briefly discussed below. Zamiri et al. (2007) validated a crystal plasticity model by comparing the results with empirical data obtained from hydroforming of an aluminium tube. The model can predict texture and gave a good agreement with measurements but requires much more computational effort than the plasticity model used in the current study. Their approach is needed when it is important to account for texture evolution during the forming process. A similar model was used by Guan et al. (2006) for THF. Korkolis and Kyriakides (2008, 2009) also studied hydroforming of aluminium tubes. They used an anisotropic plasticity model without texture evolution. Their focus was on predicting failure of the tube. The model gave a good agreement with experimental results of bursting of the tube during forming. 2.2. Martensite formation There exists a wealth of publications with experiments and models for athermal and isothermal martensite formation. The reader is referred to the reviews by Olson (2006) and Bhadeshia et al. (2001), and the book by Bhadeshia and Honeycombe (2006). Martensite formation is also a part in models for shape memory alloys, see (Fischer, 1990; Fischer et al., 1998, 2000; Jemal et al., 2009). Han et al. (2004, 2009) modified the Olson–Cohen model to accommodate the Koistinen–Marburger relation (Koistinen and Marburger, 1959). This model was applied to each martensitic variant that can be formed in a crystal. The macroscopic effect was obtained by summation over the 24 possible variants and averaging over a chosen number of grains that are assumed to be randomly oriented. This model can be combined with crystal plasticity models like in Zamiri et al. (2007). Dan et al. (2007) combined the Olson–Cohen model with an anisotropic plasticity model. Das and Tarafder (2009) performed detailed measurements on mechanically induced martensite formation in AISI 304LN. They determined different types of nucleation sites and whether the transformation was stress-assisted or strain-induced. This information is valuable for validation of the physical mechanisms of martensite formation models. The model used for the isothermal, strain-induced, martensite formation in the current study, is chosen due to its relative simplicity. The first author of this paper found it to be very accurate in an earlier evaluation on a more restricted test set (Lindgren, 2004). Its background and history of development is given below. 2.3. Macroscopic properties of multiphase materials The current work uses a simple approach for obtaining macroscopic properties for a two-phase material that can be applicable for large-scale simulations. It does only consider the fraction of the phases and does not account for their morphology. There are other more advanced models for this purpose that attempt this. The interested reader is directed to the references of the recent work by Delannay et al. (2008). They used a mean-field approach for a multiphase material with less than 20% austenite that may form martensite with specific assumptions about the phase morphology and the distribution of the macroscopic strain to them. Perdahcioglu (2008) evaluated in his thesis several homogenization approaches for a metastable austenitic stainless steel. Leblond and Devaux (1988) and Mahnken et al. (2009) used micromechanical finite element models to study the macroscopic behaviour of a representative volume element with different volume fractions of martensite. 3. Martensite formation in austenitic stainless steels Slip, twinning and martensite formation are three mechanisms that can accommodate the deformation of metals at low temperatures (Tsakiris and Edmonds, 1999). Usually, the metals deform by dislocation slip, which is the main source of deformation in classical plasticity. Higher deformation rates, lower temperatures and lower stacking fault energies promote twinning and eventually mechanically induced martensite formation. Metastable austenitic stainless steel can form martensite under certain conditions. Spontaneous martensite formation occurs when the temperature sinks below the Ms-temperature. The Koistinen–Marburger equation (Koistinen and Marburger, 1959) is usually applied for this case. The transformation is called stress-assisted for temperatures above Ms and starts on the same sites responsible for spontaneous transformation during cooling. This is sometimes modelled by including the effect of stress on the Ms-temperature (Patel and Cohen, 1953; Olson and Cohen, 1975; Tanaka and Sato, 1985; Inoue and Wang,

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1985; Fischer, 1990). The strain-induced martensite transformation is active at temperatures above Ms. The martensite then nucleates at new sites at intersection of shear bands created by the plastic deformation (Olson and Cohen, 1982) but also mechanical twins can act as nucleation sites (Tsakiris and Edmonds, 1999). It is observed (Ganesh Sundara Raman and Padmanabhan, 1994; Lebedev and Kosarchuk, 2000; Spencer et al., 2004) that e-martensite first forms at stacking faults within the c phase. Thereafter a0 -martensite is formed at shear band intersections. It is also possible that e-martensite transforms into a0 -martensite at larger strains. The latter is magnetic and is measured during the hydroforming tests performed in the current study. The e-martensite has smaller specific volume as compared with the c phase whereas the a0 -martensite expands. Lebedev and Kosarchuk (2000) observed that tensile loading promotes the latter transformation. Diani and Parks (1998) also evaluated the effect of the stress state on the martensite formation. Geijselaers and PerdahcIoglu (2009) and PerdahcIoglu et al. (2008) investigated the effect of the stress state on the mechanically induced martensite formation on an austenitic stainless steel (12Cr–9Ni–4Mo). They did not find that the plastic straining had any decisive role and wanted to attribute the martensite formation solely to be stress driven. No martensite can form at all if the temperature is above Md. The three ranges of martensite formation are summarised in Fig. 1. Details of the used model for strain-induced martensite are given in next chapter. A summation of the deformation over the different martensite crystallographic variants that can be formed leads to a volume and shape change (shear) in the material. The interaction between the transformation and the applied stress gives a macroscopic straining, transformation induced plasticity (TRIP). This can be defined as ‘‘a significantly increased plasticity during a phase change at a stress lower than normal yield stress” (Fischer et al., 2000). It requires an applied stress simultaneously with the phase change. The TRIP-effect consists of two parts; an accommodation effect due to the volume increase, the Greenwood-Johnson mechanism, and an orientation effect due to the shear, Magee mechanism. The contribution from the shear effect is often ignored in thermal driven martensite formation as it is assumed to be self-accommodating (Bhadeshia et al., 2001; Oddy et al., 1992) provided the applied stress is not too large. The model by Leblond (1989) and Leblond et al. (1989) does not include the latter effect. Oddy et al. (1992) also assumed that the shear may be self-accommodating, which was motivated by the agreement with measurements in their work. The current work follows the proposal by Leblond (1989). He proposed that transformation induced plasticity can be excluded as a separate term beside the classical plasticity when the stress state is above the yield limit, i.e. during plastic straining. His motivation was that it is difficult to differentiate between classical plasticity and transformation induced plasticity when calibrating a flow stress model. 4. Model for strain-induced martensite formation The current work focuses on strain-induced martensite formation (SIMT). Olson and Cohen (1975) proposed a physical based model for SIMT where shear band intersections has a probability to be nuclei for martensite formation. Stringfellow et al. (1992) extended this model by including stress into the probability function. The model was further developed (Tsuta and Cortes, 1993; Tomita and Iwamoto, 1995; Iwamoto et al., 1998, 2000, 2001; Tomita and Iwamoto, 2001; Iwamoto and Tsuta, 2002). The paper by Garion et al. (2007) gives the reader a good review about the modelling of strain-induced martensite formation in stainless steels. The model has also been used for modelling the behaviour of low-alloy TRIP steels (Bouquerel et al., 2006). It is assumed that only two phases coexist in the material. They are austenite (c) and a0 -martensite (M). Thus it is only necessary to compute the martensite fraction and use

Xc ¼ 1  XM :

ð1Þ

The amount of shear band formation is related to plastic strain in austenite by

X_ sb ¼ Asb ð1  X sb Þe_ c ;

ð2Þ

σ

Ms

σ Yγ Yield limit of austenite

M σs

Plastic deformation of austenite, no martensite

- as sist ed ess Str

Spontanous martensite thermal formation

Strain-induced nucleation

Md

T

Fig. 1. Martensite formation in different stress and temperature regimes. The stress needed to trigger the formation is indicated together with the yield limit of the austenitic phase.

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where Xsb is the volume fraction of shear bands and ec is the effective plastic strain in austenite. Asb is a function according to Eq. (5) that has to be calibrated. The definition of the effective plastic strain rate for von Mises plasticity is

e_ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p e_ e_ ; 3 ij ij

ð3Þ

This definition is also used for effective plastic strain rates of each individual phase. The von Mises associated flow rule is assumed and applied on the macroscopic level to give its effective plastic strain rate. Eqs. (27) and (28), described later, give the plastic strain rates of the individual phases from the macroscopic one. The factor Fsb is written as

e_ c _eref

Asb ¼ ðAðTÞ  Ar K r Þ

!M ð4Þ

;

where, T, is the temperature, e_ ref is a reference strain rate and the triaxiality factor Kr is defined as

Kr ¼

rkk ;  3r

ð5Þ

 is the von Mises effective stress. Iwamoto et al. (1998) and Iwamoto (2004) where rkk is the sum of the normal stresses and r used a second order polynomial for A(T) whereas we use only a linear temperature dependency in the current study. The fraction of shear bands is related to the number of shear band intersections per unit volume, NI, by a power law,

NI ¼ gX nsb ;

ð6Þ

where g is a geometric constant and the exponent n typically has a value around 4–5. The number of operational martensite nucleation sites, NM, is assumed to be equal to NI multiplied by a probability function, n, (Stringfellow et al., 1992) as

NM ¼ nNI :

ð7Þ

The probability function is defined as

1 nðgÞ ¼ pffiffiffiffiffiffiffi 2prg

Z

g

12

e



g 0 g 0 rg

2 0

dg ;

ð8Þ

1

where the driving force is

g ¼ T þ g 1 K r :

ð9Þ

The increase in martensite fraction is assumed to be proportional to N_ M giving

d X_ M ¼ ð1  X M ÞN_ M ¼ ð1  X M Þ ½nðgÞgX nsb : dt

ð10Þ

This together with Eq. (2) gives

  dn n _ c þ HðgÞ  _ _ X_ M ¼ ð1  X M Þ ngnX n1 g A ð1  X Þ e g X sb sb sb sb : dg

ð11Þ

_ is included to reflect the irreversibility of the martensite formation process. Thus the evolution The Heaviside function, HðgÞ, equation for the martensite formation becomes

_ X_ M ¼ ð1  X M ÞðAM e_ c þ BM gÞ;

ð12Þ

AM ¼ Asb ngnX n1 sb ð1  X sb Þ;

ð13Þ

with

dn _ BM ¼ gX nsb HðgÞ; dg g_ ¼ T_ þ g 1 K_ r :

ð14Þ ð15Þ

_ is the effect of the change in probability in Eq. (8) and is ignored in (Iwamoto et al., 1998, 2001). The last term in Eq. (12), BM g, The martensite model parameters were determined as described later in the paper. 5. Mechanical and thermal properties The macroscopic properties of the austenite–martensite mixture is obtained from simple linear mixture rules and properties assigned to the individual phases. Bouquerel et al. (2006) used a non-linear mixture rule proposed by Balliger and Gladman (1981). A mixture rule, that includes same strain, iso-strain, or same stress, iso-stress, assumptions regarding

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the deformation of the phases as special cases, was used by (Kuang et al., 2009). Perlade et al. (2003) assumed no plastic strain at all in the martensite phase. Non-linear mixture rules will be able to represent the macroscopic flow stress better when the fraction of one phase is small. It can be noted that the martensite formation model would also need improvements if processes with small final martensite fractions are of interest as e-martensite is present initially as discussed in Section 3. These mixture rules imply that also the thermo-elastic properties are dependent on the plastic strain via the strain-induced martensite formation described earlier. The material properties of interest are elastic modulus, thermal expansion and yield limit. The elastic and thermal properties are taken from the literature whereas the plastic parameters are calibrated. The macroscopic Young’s modulus is computed as

E ¼ ð1  X M ÞEc þ X M EM ;

ð16Þ

with data from Iwamoto (2004), in GPa,

Ec ¼ 215:7  0:0692ðT þ 273:15Þ

ð17Þ

EM ¼ 237:3  0:00692ðT þ 273:15Þ:

ð18Þ

and

It is assumed that Poisson’s ratio for both austenite and martensite is constant, m = 0.33, for both martensite and austenite. The split into deviatoric and volumetric stress states in the incremental stress–strain algorithm requires the shear modulus,

G ¼ ð1  X M ÞGc þ X M GM ;

ð19Þ

and the bulk modulus,

K ¼ ð1  X M ÞK c þ X M K M :

ð20Þ

Shear and bulk moduli are determined by Young’s modulus and, in this case, the constant value for Poisson’s ratio. The macroscopic thermal dilatation is computed as

eth ¼ ð1  X M Þethc þ X M ethM ;

ð21Þ

with data from Iwamoto (2004),

ethc ¼ 0:015 þ 17:3  106 ðT þ 273Þ

ð22Þ

ethM ¼ 11  106 ðT þ 273Þ:

ð23Þ

and

The shift between the curves in Eqs. (22) and (23) is one third of the difference in specific volume between the two phases. The macro-yield limit is computed as

rY ¼ ð1  X M ÞrYc þ X M rYM ;

ð24Þ

with the model for the yield limit of each phase taken from Tomita and Iwamoto (1995) and Iwamoto and Tsuta (2000) and other studies by the same group. It is written as c

c

c

rYc ¼ C c4 eC5 T þ C c1 ½1  eC2 ec C3

ð25Þ

for austenite, and M

M

M

rYM ¼ C M4 eC5 T þ C M1 ½1  eC2 eM C3

ð26Þ

for martensite. The parameters in Eqs. (25) and (26) for the material are determined as described later. The distribution of the plastic strain between the two phases is obtained, as in (Perlade et al., 2003), by use of the iso-work principle

rYc e_ c ¼ rYM e_ M

ð27Þ

e_ p ¼ ð1  X M Þe_ c þ X M e_ M :

ð28Þ

and

Further discussions about the iso-work principle can be found in (Bouaziz and Buessler, 2004). Eq. (27) gives less plastic straining of the phase with higher yield limit. The high strength of the martensite can also be used to motivate that there should be no plastic straining of the martensite at all. Iwamoto (2004) used a multiscale model of austenite–martensite and found that the plastic strain in the martensite was considerably smaller than in the austenite. The current used approach, Eqs. (27) and (28), was compared with measurements using high energy X-ray diffraction measurements in Hedström et al. (2009) where the plastic strain evolution of the two phases in HyTensXÒ was monitored during tensile

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testing. The agreement was found to be good. It can be noted that the derivation above implies that the formed martensite has the same effective plastic strain as the existing martensite. Martensite only forms during plastic straining according to the model. Therefore, the transformation induced plasticity is included in the classical plasticity as proposed by Leblond (1989) (Section 2.1.b) as stated at the end of Section 3. The heat conductivity was taken from Iwamoto (2004) as

k ¼ ð1  X M Þkc þ X M kM ;

ð29Þ

with kc =16.3 [W/(m C)] and kM =24.2 [W/(m C)]. The macroscopic heat capacity was taken as 460 [J/(kg C)]. The latent heat for the martensite formation was 625 [J/(kg)]. 6. Stress–strain algorithm The stress–strain algorithm has two tasks. Firstly, it will compute the stress from a given strain and secondly it will compute the consistent constitutive matrix. The latter is required when the 2nd order convergence of the Newton-Raphson method should be preserved and is needed in implicit finite element formulations. The book by Simo and Hughes (1997) describes plasticity and viscoplasticity models and their numerical implementation. The book by Crisfield (1991) describes basic numerical methods in Section 6 and extensions of these models are given in Crisfield (1997). Belytschko et al. (2000) also devote some chapters to constitutive models and stress–strain algorithms. Application to viscoplasticity can be found in chapter 15.12 in (Crisfield, 1997) and chapter 5.9.8 in (Belytschko et al., 2000) where a flow strength equation is solved to obtain the effective plastic strain increment. The flow strength equation is an equation that defines the viscoplastic strain rate in terms of other variables. This approach is also used in Frost and Ashby (1982), Raboin (1993), and Marin and McDowell (1997). Ponthot (2002) describes the application of the radial-return method to viscoplasticity. The return mapping method is an operator-split approach where an elastic predictor is followed by a plastic corrector. The elastic predictor step is the computation of a trial state assuming that no plastic strain increment occurs. If the effective trial stress is larger than the yield limit, then a plastic corrector step must be performed. There are different options for the return direction of the stress state towards the yield surface. The backward Euler or implicit method uses the return direction at the end of the increment. Ortiz and Martin (1989) show that only the implicit approach for the return mapping gives a symmetric consistent tangent for von Mises plasticity and the associated flow rule. They also note (Ortiz and Popov, 1985) that this method may also be a better choice when large strain increments are expected. Different approaches for implementing the stress–strain algorithm were evaluated in Domkin (2005). A set of coupled microstructure evolution equations and the consistency condition has to be solved to determine the plastic straining and martensite formation. Domkin (2005) formulated the following system of equations non-linear for increment in microstructure variables Dq and effective plastic strain Dep h

Þ ¼ 0 H ¼ HðDq; Dep ; h q; h rkk ; h r

nþ1

 ðDep ; nþ1 qÞ  rY ðDep ; nþ1 qÞ ¼ 0 f ¼r

ð30Þ ð31Þ

In the above equations isotropic hardening is assumed and therefore no back stress is included. The system of equations in hH are the microstructure evolution equations that are solved at time ht = nt = hDt where nt is the time at beginning of the increment and Dt = n+1t  nt is the length of the time step and h e [0, 1]. All other variables are also linearly interpolated between nt and n+1t values. q is the vector with the microstructure variables

qT ¼ ðec ; eM ; X M ; X sb Þ

ð32Þ

The yield condition, Eq. (31), is fulfilled at the end of the increment corresponding to the approach in the radial-return method. hH denotes the four scalar equations h

H1 ¼ h rYc Dec  h rYM DeM ¼ 0

H2 ¼ ð1  h X M ÞDec þ h X M DeM  Dep ¼ 0 h H3 ¼ DX M  ð1  h X M Þðh AM Dec þ h BM DgÞ ¼ 0 h

h

H4 ¼ DX sb  h Asb ðh K r ; h TÞ  ð1  h X sb ÞDec ¼ 0

ð33Þ ð34Þ ð35Þ ð36Þ

The first two corresponds to the assumption about plastic strain distribution, Eqs. (27) and (28). The third is the martensite formation according to Eq. (12). The final equation is the shear band evolution according to Eq. (2). The best variant is to update the martensite simultaneously with the plastic strain. Then the consistent constitutive matrix will be quite complex and unsymmetric as the thermo-elastic properties of the material depend on the plastic strain via the martensite fraction (Domkin, 2005). This is due to the fact that plastic strain changes the amount of martensite. This in turn changes the thermal dilatation as well as the elastic and plastic properties via the used mixture rules for the material properties.

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A simpler approach, resembling the case of h = 0 above, is used in the current work. The microstructure is frozen when updating the stress state. Then the standard consistent constitutive matrix is the same as for the standard von Mises plasticity without strain driven phase changes. Then the plastic straining is determined from nþ1

 ðDep ; n q; nþ1 TÞ  rY ðn q; nþ1 TÞ ¼ 0; f ¼r

ð37Þ

Thereafter the martensite fraction is updated according to its evolution equations. This staggered approach limits the size of the increments in the stress updating if the error should be kept small or that small time steps have to be taken in the finite element simulation. The latter problem can be circumvented by sub-stepping in the stress updating algorithm if the martensite increase becomes too large in a time step. 7. Experimental setup 7.1. Calibration and validation tests for material modelling Two sets of tests have been performed, one used for calibration and one for validation of the model. The first set, consisting of seven tests, was used to calibrate the martensite formation model, Eq. (12), as well as the plastic flow model, Eqs. (24)–(26). Uniaxial tension tests were performed at room temperature. The test samples were given initial temperatures ranging from 47 to 90 °C and the temperature of the test sample was recorded during the test, see Fig. 2. The strain rate was 0.0001 s1 in the tests. Measured forces and displacements were used to calculate the true stress and strain exhibited by the test samples. Magnetic measurements were used to determine the martensite fraction of the deformed microstructure (Talonen et al., 2004; Carlsson and Olsson, 2006). The non-magnetic e-martensite is thus not included. Additional validation tests were performed with the same measurements as for the calibration set. The test pieces were first strained at high temperature, >85 °C, to 5%, 10% and 15%, respectively. Then they were unloaded and cooled to room temperature (25–30 °C). Thereafter, tensile testing was performed in the same manner as for the calibration tests. The temperature history versus strain is shown in Fig. 3. The motivation for the choice of validation tests is as follows. The first term in the used Olson–Cohen martensite formation model, Eq. (12), is proportional to the plastic strain rate of austenite. However, when this deformation occurs at high temperatures, then the probability of martensite formation is low, Eq. (8). Therefore, we wanted to see if high temperature deformation, prestraining, affects martensite formation during subsequent deformation at lower temperatures and if the model can account for this. Taleb and Petit (2006) evaluated this for the bainite transformation and found that prestraining of austenite affected subsequent bainite formation. The measured stresses and martensite fractions, together with computed values for the calibration as well as for the validation tests are shown in the results section later in the paper. 7.2. Tube hydroforming Hydroforming of cylindrical welded tubes made of HyTensXÒ, Table 1 gives their dimensions, were performed. The tests were done at room temperature with different maximum pressures ranging from 350 to 700 bar, see Table 2. The ambient temperature was about 20–22 °C during the tests. Details and further measurements are given in (Carlsson and Olsson

Fig. 2. Measured temperature versus strain for the seven calibration tests.

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Fig. 3. Measured temperature versus strain for the three validation tests. The fall in temperature corresponds to the strain where the prestraining phase was finished and the sample was cooled and unloaded. Thereafter the temperature increases during final loading.

Table 1 Cylinder tube data. Material

HyTensXÒ

Thickness, t0 [mm] Average diameter, D0 [mm] Length [mm] Length of formed area [mm]

1.01 25.0 500 120

Table 2 Tube samples investigated. Samples

Pressure [bar]

#0 #1 #2 #3 #4 #5 #6 #7 #8

0 350 400 450 500 550 600 675 700

(2006). The amount of formed a-martensite during the forming process was measured by a Ferroscope (Nagy et al., 2004; O’Sullivan et al., 2004; Vertesy et al., 2005). Furthermore, the dimensions of the deformed part of the tube were measured after each of the forming tests in Table 2. The results for the martensite fraction given in (Carlsson and Olsson, 2006) were recalibrated using a more correct method based on (Talonen et al., 2004).

8. Computational model An axisymmetric finite element model of the setup described above was made. The mesh is shown in Fig. 4. Half of the length of the pipe need only be modelled due to symmetry conditions. The outer layer of elements in the left part of the mesh corresponds to the grip of the hydroforming machine that holds the pipe. A quasistatic, coupled thermo-mechanical simulation using the staggered approach was performed accounting for large deformations and strains. Quasistatic means that inertia forces were ignored. The volumetric strain in the element was underintegrated in order to avoid locking due to the incompressible plasticity. The temperature transferred from the thermal analysis was uniform over the volume of each element in the mechanical analysis to avoid locking due to the mismatch between the degree of interpolated strain in the mechanical element and temperature field in the thermal element (Lindgren, 2001a–c, 2006, 2007; Oddy et al., 1990). The inside of the pipe is subjected to pressure loading varying in time. It increases linearly up to 700 bar at 30 s. Thereafter, it was held for 90 s and finally reduced to zero again during 30s. Thus only one unloaded configuration of Table 2, the

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Fig. 4. Axisymmetric finite element model.

700 bar case, was evaluated. All other results are evaluated on the model with an applied pressure, i.e. without unloading. A total of 991 time steps were used in the analysis. A user routine, WKSLP, with user defined internal state variables was used to compute the plastic properties. User routines were also needed to compute Young’s modulus, Eq. (16), thermal dilatation, Eq. (21), and heat conductivity, Eq. (29). The material properties are given earlier in the paper with exception of the calibrated flow stress properties. The latter are given in the next chapter. A low thermal contact resistance was assumed between pipe and the holder as well as toward the pressure medium. The convective heat transfer between the pipe and the air was taken as 10 [W/(m2°C)]. The friction coefficient between holder and the pipe was set to 0.3. These values are estimates and therefore the sensitivity of the results with respect to them was evaluated and found to be small. 9. Results 9.1. Calibration of martensite formation and flow stress models An in-house Matlab™ tool has been used to calibrate the material parameters using a least square approach (Lindgren et al., 2003; Lindgren, 2004) to find the parameters that give measured martensite fraction as well as flow stress. The results for the martensite formation model are given in Table 3 and the flow stress model in Table 4. Note that as the tests are only uniaxial, then the triaxality factor is the same for all tests and the coefficient Ar cannot be calibrated but is taken from Iwamoto and Tsuta (2002) for a similar steel (AISI 304 type). Furthermore, only one strain rate was used and thus the exponent for the rate dependency, M, in Eq. (4) is set to zero. The contribution from the stress state to the driving force in Eq. (8) was ignored by setting g1 to 0. The function A(T) in Eq. (4) is taken as a linear function. Thus the number of model parameters is reduced compared with, for example, the work in (Iwamoto et al., 1998; Iwamoto, 2004).

AðTÞ ¼ Alow

ðT high  TÞ ðT  T low Þ þ Ahigh ðT high  T low Þ ðT high  T low Þ

ð38Þ

where T low was chosen to 40 °C and Thigh to 60 °C. The computed martensite fraction for the calibration tests is compared with the measured in Fig. 5 and the computed and measured axial stress is shown in Figs. 6 and 7 using the parameters in Tables 3 and 4. 9.2. Validation of martensite formation model The computed martensite fraction is compared with the measured in Fig. 8 and the computed and measured axial stress is shown in Fig. 9 using the parameters in Tables 1 and 2. The measured temperature and strain from the validation tests, prestraining and final tension test, were used to drive the model. Thus the initial prestraining is included in the figures below. It should be noted that the test specimen was completely unloaded in the experiment whereas it was only cooled down in the modelling. Therefore, the stress does not fall during this stage in the model.

Table 3 Martensite formation model parameters for HyTensX. Alow

Ahigh

Ar

rg

g0

g1

g

n

M

32.2

2.0

7.4

168.3

115.5

0

8.36

4.5

0

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L.-E. Lindgren et al. / International Journal of Plasticity 26 (2010) 1576–1590 Table 4 Flow stress model parameters for HyTensX. c

c

c

c

c

C1

C2

C3

C4

C5

CM 1

CM 2

CM 3

CM 4

CM 5

1215

0.96

0.635

1093

0.0046

1682

19.9

4.1

7287

0.0175

Fig. 5. Computed and measured martensite fraction versus strain as obtained in the calibration tests.

Fig. 6. Computed and measured axial stress versus strain as obtained in the calibration tests.

9.3. Simulation of hydroforming of pipe The computed martensite fraction of the hydroformed tube is compared with the measured in Fig. 10 for a location at centre of tube thickness. It should be noted that the measurements were done for interrupted, i.e. unloaded pipes, whereas the computed values are for the loaded pipe. The final unloading at the end of loading to maximum pressure of 700 bar was elastic and therefore does not influence the computed martensite fraction in Fig. 10. The computed and measured radial expansion is shown in Fig. 11. The computed result is from one simulation with continuous increasing pressure. Thus the decrease in radius due to the unloading is not included in the computed radial expansion in contrast to the measured results. The unloading phase, results not shown, gave a decrease in radius less than 0.2 mm when unloading occurred from the final pressure of 700 bar. Therefore the computed values used in the comparison in Fig. 11 are larger than what would have been obtained if the values after unloading at each pressure had been used. 10. Discussions As revealed by Fig. 8, the used model does not capture the effect of prestraining. A plot of martensite fraction versus stress, not shown, reveals that our measurements do not agree with the findings of Perdahcioglu et al. (2008). They proposed

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Fig. 7. Magnification of upper part of curves in Fig. 6.

Fig. 8. Measured and computed martensite versus strain for validation tests, including the prestraining phase. The true strain on the horizontal axis includes the pre-tension phase.

Fig. 9. Measured and computed axial stress versus strain for validation tests. The true strain on the horizontal axis includes the pre-tension phase.

a purely stress based model for the martensite formation following the approach by Patel and Cohen (1953). They attributed the increase in martensite fraction for prestrained specimens to higher stress caused by the additional deformation

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Fig. 10. Measured (symbols) and computed (line) martensite fraction versus pressure in hydroformed pipe.

Fig. 11. Measured (symbols) and computed (line) radial expansion of pipe versus pressure in hydroformed pipe.

hardening in the prestraining phase. Guimarães (2008) determined the influence of prestraining on the martensite start temperature for thermal driven formation and found that it influences Ms. He added internal (micro) stresses due to the plastic straining to the Patel criterion in order to accommodate this observation into the model. Spencer et al. (2004) observes that pre-existing dislocations accelerates the transformation. Therefore, it seems that additional sites due to plastic straining matters for the transformation and not only the higher stress due to increased hardening. The only possibility for the Olson–Cohen model to include the effect of the prestraining is via the change in driving forces. This is the second term, BM, in Eq. (12). However, the shear band formation, Xsb, is also small at high temperatures making the contribution of this term during cooling after prestraining small. The authors have preferred not to stretch the model and try to modify the shear band formation by modifying Eqs. (2) and (4) as we do not have any measurements of the shear band fractions to calibrate and validate this part. We do observe from our data, the plots are not shown in the paper, that the martensite formation seems to start later than the plastic deformation. But it should be noted that used measurement technique during the tensile testing cannot register the non-magnetic e-martensite that is believed to form initially. Lebedev and Kosarchuk (2000) found that the formation of deformation induced martensite starts with the creation of e-martensite, which reaches its maximum around 6% strain and is reduced when a0 -martensite is formed later. Otherwise the combined martensite and flow stress model is valid for temperatures from 40 to +60 °C. The highest temperature starting with 90 °C and finishing below 50 °C does not show equal good agreement. However, the martensite model forms very little martensite at 90 °C, as should be the case. The rate dependency of the model has not been calibrated. However, Iwamoto and Tsuta (2000) and Iwamoto (2004) found for a similar material

e_ c e_ ref

!M ¼

e_ c

!0:01

0:0005

ð39Þ

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for this term in Eq. (4). Thus the rate dependency is probably low. An increase in rate will increase the martensite formation but at the same time the heating due to the plastic deformation counteracts. This was noted by Hedström et al. (2009) when comparing formation at different rates. The model of the hydroforming case shows good agreement with measurements. This indicates that the triaxiality factor taken from (Iwamoto and Tsuta, 2002) can be used also for HyTensXÒ. There are uncertainties about the conditions during the hydroforming as the temperature of the tube was not logged. The sensitivities of reasonable variations in the room temperature as well as heat transfer between tool and tube were evaluated and found small. Fixing the outer edges of the pipes or, like in the presented model, or using free edges and friction between tool and tube gives very similar results. The numerical implementation of the coupling between the microstructure and plasticity models using a staggered approach requires small increment in the finite element simulation. The effect of these and more accurate approaches were evaluated in (Domkin, 2005). This did not matter for the current case, as the non-linearity of the problem anyhow required the use of small time steps for convergence in the Newton-Rapshon procedure. Finally, it should be noted that the flow stress model accommodates some of the TRIP effects as a consequence of following Leblond (1989). He proposed that the TRIP strain should be include into the classical plasticity when martensite if formed during plastic straining. An alternative is to add the complete, stress state dependent TRIP term, as a separate term in the inelastic strains, see Stringfellow et al. (1992). 11. Conclusions

 The calibration of the martensite and flow stress model worked well for the temperature range relevant for cold forming. It is a simple, efficient model applicable for large-scale simulations. This model can be used to optimise the THF process in order to obtain an optimal martensite fraction and strength of the formed tube.  The validation tests show that the calibrated Olson–Cohen model does not describe the effect of previous plastic straining on current martensite formation accurately. The used model can accommodate this, if the shear band formation, Eq. (2), at high temperatures is increased. This can be achieved by using higher values for Asb.  The model is not appropriate if the final reached martensite fraction is low (<10%) both due to the use of a linear mixture rule as well as the formation of the e-martensite, which is not accounted for.

Acknowledgments Outokumpu Stainless AB and HyfoTech AB are acknowledged for providing the HyTensXÒ tubes and help with the hydroforming experiments, respectively. The financial supports from Triple Steelix and the Swedish Foundation for Knowledge Competence Development (KK-stiftelsen) are gratefully acknowledged by the authors. References Ahmetoglu, M., Altan, T., 2000. Tube hydroforming: state-of-the-art and future trends. J. Mater. Process. Technol. 98, 25–33. Balliger, N., Gladman, T., 1981. Work hardening of dual-phase steels. Metal Sci. 15, 95–108. Belytschko, T., Liu, W.K., Moran, B., 2000. Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons. Bhadeshia, H.K.D.H., Honeycombe, S.R., 2006. Steels: Microstructure and Properties. Butterworth-Heinemann, Oxford. Bhadeshia, H.K.D.H., Buschow, K.H.J., Robert, W.C., Merton, C.F., Bernard, I., Edward, J.K., Subhash, M., Patrick, V., 2001. Martensitic transformation. Encyclopedia of Materials: Science and Technology, Martensitic Transformation. Elsevier, Oxford. Bouaziz, O., Buessler, P., 2004. Iso-work increment assumption for heterogeneous material behaviour modelling. Adv. Eng. Mater. 6, 79–83. Bouquerel, J., Verbeken, K., De Cooman, B.C., 2006. Microstructure-based model for the static mechanical behaviour of multiphase steels. Acta Mater. 54, 1443–1456. Carlsson, P., Olsson, M., 2006. Mechanical and microstructural characterisation of hydroformed stainless steel tubes. In: Juster, N., Rosochowski, A. (Eds.), Proceedings of ESAFORM 2006, Ninth International Conference on Material Forming, Glasgow, UK, pp. 375–379. Crisfield, M.A., 1991. Non-linear Finite Element Analysis of Solids and Structures. Essentials, vol. 1. John Wiley & Sons, Chichester. Crisfield, M.A., 1997. Non-Linear Finite Element Analysis of Solids and Structures. Advanced Topics, vol. 2. John Wiley & Sons. Dan, W.J., Zhang, W.G., Li, S.H., Lin, Z.Q., 2007. A model for strain-induced martensitic transformation of TRIP steel with strain rate. Comput. Mater. Sci. 40, 101–107. Das, A., Tarafder, S., 2009. Experimental investigation on martensitic transformation and fracture morphologies of austenitic stainless steel. Int. J. Plasticity 25, 2222–2247. Delannay, L., Jacques, P., Pardoen, T., 2008. Modelling of the plastic flow of trip-aided multiphase steel based on an incremental mean-field approach. Int. J. Solids Struct. 45, 1825–1843. Diani, J.M., Parks, D.M., 1998. Effects of strain state on the kinetics of strain-induced martensite in steels. J. Mech. Phys. Solids 46, 1613–1635. Dohmann, F., Hartl, C., 1997. Tube hydroforming-research and practical application. J. Mater. Process. Technol. 71, 174–186. Domkin, K., 2005. Constitutive models based on dislocation density. Formulation and implementation into finite element codes. Ph.D. Thesis 2003. Luleå University of Technology, Luleå, p. 35. Fischer, F.D., 1990. A micromechanical model for transformation plasticity in steels. Acta Metall. Mater. 38, 1535–1546. Fischer, F.D., Oberaigner, E.R., Tanaka, K., Nishimura, F., 1998. Transformation induced plasticity revised an updated formulation. Int. J. Plasticity 35, 2209– 2227. Fischer, F.D., Reisner, G., Werner, E., Tanaka, K., Cailletaud, G., Antretter, T., 2000. A new view on transformation induced plasticity (TRIP). Int. J. Plasticity 16, 723–748. Frost, H.J., Ashby, M.F., 1982. Deformation-Mechanism Maps – The Plasticity and Creep of Metals and Ceramics. Pergamon Press.

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