On the modelling of strain ageing in a metastable austenitic stainless steel

Journal of Materials Processing Technology 212 (2012) 46–58

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On the modelling of strain ageing in a metastable austenitic stainless steel Rikard Larsson ∗ , Larsgunnar Nilsson Division of Solid Mechanics, Linköping University, SE-58183 Linköping, Sweden

a r t i c l e

i n f o

Article history: Received 6 April 2011 Received in revised form 30 June 2011 Accepted 3 August 2011 Available online 10 August 2011 Keywords: Plastic anisotropy Isotropic-distortional hardening Portevin–Le Châtelier effect Dynamic strain ageing Static strain ageing Martensitic transformation

a b s t r a c t The plastic hardening of metastable austenitic stainless steel is partly governed by martensitic transformation, the occurrence of serrated plastic flow, and plastic strain ageing phenomena. In this paper an elasto-viscoplastic material model with isotropic distortional plastic hardening is developed. The model accounts for static and dynamic strain ageing as well as the martensitic transformation. An experimental programme has been conducted in order to fit the model parameters to an austenitic stainless steel within the EN 1.4310 standard. The identification of the dynamic strain ageing was based on so called jump tests, where a sudden strain rate increase was shown to result in an instantaneous positive strain rate sensitivity followed by negative steady state strain rate sensitivity. Furthermore, the static strain ageing was identified by unloading tensile test specimens at specified plastic strains and then reloading these specimens after different periods of time. The observed material behaviour in the test situations can be predicted by the developed model. Lastly, the model was validated by predicting the force–displacement relation of the material in a shear test; the prediction agrees well with experimental results. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Demands on light weight structures, often with complex geometries, have lead to an increased interest in high strength steels with good formability. The austenitic stainless steels have a great potential in these applications due to their excellent formability and high strength. The formability is related to strain localisation, which is associated with plastic hardening, but which also can be affected by the occurrence of serrated plastic flow, cf. Hopperstad et al. (2007), and martensitic transformation. Serrated plastic flow is referred to as the Portevin–Le Châtelier, PLC, effect. Furthermore, rolled sheet metal often show an orthotropic plastic behaviour, where the axes of orthotropy lie along the rolling direction (RD), transversal direction (TD) and normal direction (ND), respectively. The high strength and deformation hardening in austenitic stainless steels are partly due to phase transformations during plastic deformation. Martensitic transformation can be spontaneous, stress-assisted, or strain induced, see Seetharaman (1984). The transformations also depend on the strain rate, cf. Hecker et al. (1982). Ramírez et al. (1992) suggested a model for martensitic transformation based on an energy assumption, where the martensitic fraction was determined by the temperature and plastic strain. They also presented a non-linear mixture rule based on the strains of the two phases and their individual hardening. In a later work, Tsuta and Cortes (1993) reformulated this model

∗ Corresponding author. Tel.: +46 13 281195; fax: +46 13 282717. E-mail address: [email protected] (R. Larsson). 0924-0136/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.08.003

into an incremental form for use in multiaxial plasticity problems. The model was further developed by Hänsel et al. (1998). The Hänsel model does not explicitly depend on strain rate, but high strain rates will affect the temperature due to adiabatic heating, and thus also affect the transformation rate. Hänsel used a linear mixture rule for the two phases and assumed a constant difference between the yield stresses of the two phases. Serrated plastic flow is often observed in austenitic stainless steels, cf. Meng et al. (2009). Seetharaman (1984) analysed both stress assisted and strain induced martensitic transformation in an AISI 316 austenitic stainless steel at various temperatures. It was found that stress assisted martensitic transformation gave birth to serrated yielding. Serrated yielding is generally associated with a negative strain rate sensitivity, SRS, of the yield stress. Strain rate sensitivity can be identified as an instantaneous SRS, which corresponds to the immediate stress response at strain rate changes, and a steady state SRS, which is present at constant strain rates. The difference between the two can be explained by the ageing of pinned dislocations during deformation, i.e. dynamic strain ageing, DSA. There is a competition between the DSA and instantaneous strain rate sensitivity, cf. Mesarovic (1995), which may lead to a negative or a positive SRS, depending on the strain rate and on the temperature. The strength of pinned dislocations increases with time due to the diffusion of solute atoms. Thus, the yield stress increases with the ageing time at the obstacles. When the stress reaches a critical yield stress, the dislocations move to another obstacle. The time for this motion is considered short relative to the ageing time, which is inversely proportional to the strain rate, cf. McCormick (1988). Thus, a strain rate increase leads to a shorter

R. Larsson, L. Nilsson / Journal of Materials Processing Technology 212 (2012) 46–58

ageing time for arrested dislocations and thus to a lower yield stress. Static strain ageing, SSA, is another phenomenon which refers to the increase in yield stress observed when re-loading a specimen which has been unloaded some time after pre-straining, cf. Kubin et al. (1992). Static strain ageing is related to pinning of dislocations by solute atoms and pinning of new dislocation sources at the grain boundaries, cf. Leslie and Keh (1962). Static strain ageing is distinguished from DSA since the plastic strain rate vanishes, and there is no repetitive pinning-unpinning of dislocations. Thus, the ageing time is prescribed in the case of SSA, whereas it depends on the plastic strain rate in the case of DSA. The increment in increased stress at yielding depends both on the level of pre-strain as well as on the ageing time, cf. Kubin et al. (1992). Dynamic strain ageing has been the subject of several research projects. McCormick (1988) developed a model, henceforth denoted the MC model, based on the concentration of solute atoms at pinned dislocations which depends on the ageing time, ta , of such dislocations. The evolution of the ageing time depends on the plastic strain rate, where the steady state value, ta,ss , is strain rate dependent and acts as a target value for ta . An instantaneous strain rate increase is followed by a transient period where ta decreases towards the new steady state value. The model was further developed by Mesarovic (1995), and has gained some popularity. Zhang et al. (2001) used the MC model to investigate the morphology of the PLC bands by Finite Element, FE, analyses. Mazière et al. (2010) used the MC model and found that the band width (and thus also the maximum plastic strain rate within the band) is mesh dependent, whereas the band propagation speed and the plastic strain carried by the band are not. They suggested a non-local approach in order to overcome the mesh dependency. Serrated yielding has been widely studied in the literature for a variety of materials. Clausen et al. (2004) presented an experimental programme on the AA5083 aluminium alloy including tests in various directions, temperatures and strain rates. The negative SRS was found to give serrated plastic flow. In a subsequent work, Hopperstad et al. (2007) used the MC model in order to investigate the influence of the PLC effect on plastic instability. They found that the PLC effect reduces strain to necking under both uniaxial and biaxial stress states. This work was extended by Benallal et al. (2008), where Digital Image Correlation was used in order to detect the PLC bands, study their propagation and validate the material model. Zavattieri et al. (2009) investigated the PLC effect in an austenitic TWIP steel. Meng et al. (2009) analysed tensile tests in an AL6XN austenitic stainless steel. The start of serration and the serration itself were found to depend both on temperature and strain rate. In order to model the static strain ageing, SSA, Ballarin et al. (2009b) used a Voce (1948) hardening rule, with two terms. The first term was positive, corresponding to the plastic hardening, while the second term was negative modelling the transient behaviour of the stress–strain relation due to the SSA. Ballarin et al. (2009a) analysed the bake hardening effect, a special case of SSA, during complex loading paths. An accurate prediction of the material characteristics is essential in order to control the mechanical behaviour of the material in its forming processes and in its subsequent usage. Phenomenological models have gained popularity due to their simplicity and low computational cost. Plastic anisotropy is generally taken into account by an anisotropic yield criterion. If the model obeys an associated flow rule, the effective stress, , accounts for the anisotropy both in yield stress and in the direction of plastic flow. Several anisotropic effective stress functions have been proposed over the years. Barlat and Lian (1989) proposed a three parameter effective stress function for plane stress based on a linear transformation of the stresses. Further development along this line has lead to the eight

47

parameter function proposed by Barlat et al. (2003). Aretz (2004) independently developed an eight parameter effective stress based on the work of Barlat and Lian (1989). Barlat et al. (2007) showed that there is a one to one correspondence between these two eight parameter functions. Aretz (2008) allowed the effective stress parameters to vary with plastic strain in order to achieve an isotropic-distortional hardening. The isotropic hardening is often taken directly from experimental data. However, e.g. in the case of a serrated stress–strain relation, these data generally need to be filtered. Thus, analytical functions are popular in order to model the plastic hardening. Such models are also used to extrapolate experimental data. Numerous functions for isotropic hardening have been proposed, e.g. by Voce (1948). The Voce function predicts a saturation stress which is not always found from experimental results, cf. Lademo et al. (2009). Larsson et al. (2011) used an extended Voce function for the hardening until the onset of diffuse necking. For subsequent plastic strain levels the extended Voce function was replaced by a powerlaw function. Thus, the extrapolation was determined from the stress and plastic hardening at diffuse necking, and from one additional parameter corresponding to the stress at 100% plastic strain, which was determined from inverse modelling. In this project an experimental programme has been performed on HyTens 1000, an austenitic stainless steel within the EN 1.4310/ASTM 301 standard. The HyTens 1000 steel is cold rolled to achieve improved mechanical properties. Uniaxial tensile tests in various directions with respect to the RD and a biaxial bulge test were performed in order to evaluate the plastic anisotropy and its evolution. The anisotropy was considered to be independent of strain rate, whereas the effective stress exponent depends on the martensitic fraction. An additional experimental programme, including both various strain rates and so called jump tests, have been performed in order to investigate the instantaneous and steady state SRS caused by DSA. Moreover, the static strain ageing, SSA, at room temperature was evaluated by reloading interrupted tensile tests after various periods of time. A constitutive model accounting for these effects was implemented for shell elements, with the stress in the normal direction included, i.e. through thickness stress, into the FE-code LS-DYNA, Hallquist (2009). The plastic anisotropy was accounted for by using the isotropic-distortional hardening model proposed by Aretz (2008), as an extension to his eight parameter effective stress, Aretz (2004). Isothermal conditions have been assumed in the investigated applications, since the plastic strain rates are considered to be small. In order to make the model useful for future coupled thermo-mechanical analyses, the martensitic transformation follows the Hänsel model. The DSA effect follows the McCormick (MC) model, while the SSA effect was modelled according to Ballarin et al. (2009b) which was extended with time dependency. The resulting constitutive model is presented in Section 3, and its calibration procedures in Section 4. Results from FE analyses of shear tests have been used in order to validate the model.

2. Experimental work Uniaxial tension, with dogbone and notched specimens, and simple shear tests in  = 0◦ , 45◦ and 90◦ directions with respect to the RD, and a biaxial bulge tests were performed in order to investigate the plastic anisotropy. The notched tensile test specimen was designed to give a plane strain state, in which the transversal strain, εT , is negligible in comparison to the longitudinal strain, εL , and the normal, i.e. through thickness, strain, εN . This test is henceforth denoted plane strain test. The deformation was applied by prescribed crosshead velocities in the tests. Apart from tensile tests with different crosshead velocities, tests with sudden increases,

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Fig. 1. Geometry of the tensile test specimen. Dimensions in mm.

“jumps”, in the crosshead velocity have been performed. Moreover, static strain ageing, SSA, tests have been carried out, in which the material dependency on ageing time in pre-strained, unloaded and reloaded tests have been evaluated. In general three tests of each kind have been conducted in order to evaluate variations in the results. For the readers convenience, the result from just one specimen is presented in some figures, which is visually and subjectively chosen to be representative at this test condition. In the following sections, the test procedures will be described and selected results will be presented. 2.1. Tensile tests The geometry of the tensile test specimen is shown in Fig. 1. The longitudinal strain, εL = ln (L/L0 ), and the transversal strain, εT , were both measured with mechanical extensometers. The gauge length used for the longitudinal strain was L0 = 12.5 mm, whereas specimen width was used for the width strain. The true stress is  = F/A, where A is the actual cross section area calculated from the extensometer strain, i.e. A = A0 L0 /L, where A0 is the undeformed cross section area. 2.1.1. Monotonic tensile tests Monotonic tensile tests with various constant crosshead velocities were conducted. First, tensile tests in the  = 0, 45◦ and 90◦ -directions relative to the RD were conducted. The applied crosshead velocity was 0.5 mm/min, resulting in a strain rate of ε˙ L ≈ 10−4 s−1 . These results were used to evaluate the plastic anisotropy. The stress–strain relations are shown in Fig. 2. Notice that the stress–strain relations from the monotonic tests in the  = 45◦ - and  = 90◦ -directions are similar, whereas the relation corresponding to the RD lies significantly below the others for plastic strains εL < 0.15. However, at larger strain these relations are close, indicating a distortional hardening. Foremost in the RD, distinct jumps in stress levels can be identified. The stresses and strains, respectively, are plotted versus crosshead displacement in Fig. 3. Notice that the width strain is measured at one longitudinal

Fig. 2. The stress–plastic strain relations obtained from monotonic tensile tests.

Fig. 3. True stresses and longitudinal and transversal logarithmic strains plotted versus the crosshead displacement.

position located at the centre of the specimen. Both the longitudinal and transversal strains undergo jumps. It is concluded that a longitudinal strain increment is accompanied by a negative transversal strain increment. Moreover, the true stress–crosshead displacement, i.e.  − ı, relations appear to be more smooth, see Fig. 3. Tensile tests, both at higher and lower crosshead velocities, were conducted. Since the hardening in the RD differs significantly from the hardening in the other two directions, the study of SRS was conducted in the TD. Fig. 4 shows the relation between the stress and the strain evolution as function of the crosshead displacement. It is noted that the strain level at which the serration starts increases with increasing strain rate. All relations are cut at the maximum force, where diffuse necking is assumed to occur. Thus, it is also concluded that the longitudinal strain at diffuse necking decreases with strain rate. This effect will not be further investigated in this work. 2.1.2. Jump tests In general, the deformation in the tensile tests was applied by a constant crosshead velocity. However, in a series of tensile tests,

Fig. 4. True stresses and longitudinal logarithmic strains plotted versus the crosshead displacement for various crosshead velocities. The tests were conducted in the TD.

R. Larsson, L. Nilsson / Journal of Materials Processing Technology 212 (2012) 46–58

49

Fig. 5. Schematic evolution of ageing stress, ta , viscous stress, v , and ageing time, ta , as a function of plastic strain in the case of an instantaneous increase in plastic strain rate.

Fig. 7. Influence of ageing time. For a clearer picture, only one typical curve for each ageing time is represented in the magnified part of the graph.

this crosshead velocity was suddenly increased after an amount of plastic straining, resulting in a strain rate jump. Henceforth, these tests are denoted “jump” tests. The strain rate jump results in an instantaneous stress increase, followed by a transient period with plastic softening. The principal stress response at a plastic p p∗ strain rate jump at ε = ε is shown in Fig. 5, where the plastic hardening is removed and, thus, only the part of the yield stress which depends on plastic strain rate is included. McCormick (1988) defined two stress measures that can be identified from the jump tests. The instantaneous stress increase,  i , is defined as the difference between the stress levels just before and just after the strain rate increase. The other measure,  ss , is the difference relative to the reference stress level after the transient period. The jump tests were conducted in the TD, and the strain rate sensitivity was assumed to be isotropic. The experimental results are presented in Fig. 6 together with a reference stress–strain relation. All stress–strain relations are translated (<15 MPa) in the stress direction, and thus the stress levels just before the strain rate jumps coincide with the reference data. The strain rate jumps were applied at two different strain levels, see Fig. 6. Due to serrated yielding seen in the reference data, cf. Fig. 6(b), the steady state SRS, i.e.  ss , was evaluated from only the strain rate jumps at the lower strain level. However, the instantaneous SRS, i.e.  i , was evaluated from all of the test data.

in time between the loadings. The specimens were unloaded at εL,0 ≈ 0.085, based on the extensometer. Thus, the strain levels in other regions of the specimen differ. The results are shown in Fig. 7. Three specimens were tested for each ageing time. However, in order to more clearly illustrate the difference between the different ageing times, only one representative curve for each ageing time is presented in the magnified part of Fig. 7. Obviously, the ageing time has a considerable effect on the yield stress at the second loading. Contrary to the strain rate tests and jump tests, these tests were conducted in the RD.

2.1.3. Ageing tests In order to evaluate the influence of time on the SSA, tensile specimens were loaded, unloaded and reloaded with a varying lag

2.2. Plane strain test The geometry of the test specimen is shown in Fig. 8. The longitudinal deformation, ı, was measured with an extensometer with an initial length L0 = 23 mm, as indicated in the figure. The deformation was applied by a constant crosshead velocity, v = 0.06 mm/min. Three repetitions in each direction were done. The experimental results, i.e. nominal stress–displacement relations, from the plane strain tests are shown in Fig. 9. 2.3. Shear test The geometry of the shear test specimen is shown in Fig. 10, also used by Larsson et al. (2011). The test specimen was not clamped, but mounted with clevis pins through holes in the specimen, which were lubricated to minimise friction and to permit possible

Fig. 6. The stress–strain relations from the jump tests. (a) ε∗L ≈ 0.04 and (b) ε∗L ≈ 0.084.

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R. Larsson, L. Nilsson / Journal of Materials Processing Technology 212 (2012) 46–58

in each direction were performed. The results are presented in Fig. 9. 2.4. Balanced biaxial test

Fig. 8. Geometry of the plane strain test specimen. Dimensions in mm. The deformation ı was measured with an extensometer placed symmetrically over both notches as indicated in the figure.

A balanced biaxial bulge test was performed where the pressure was applied with a punch made of silicon, and the in-plane strains and the radius of the bulge were measured using Digital Image Correlation. The biaxial stress–strain relation can be found from the in-plane strains, the hydrostatic pressure from the punch and the current bulge radius, see Fig. 11(a). The strain rate was not controlled in the bulge test. Instead the velocity of the punch was controlled, resulting in a varying strain rate in the test specimen, p in the range 0.01 < |ε˙ ND | < 0.1. It should be noted that the stress levels are not to be trusted for small strains, and that the test was interrupted before strain localisation due to limitations in the test equipment. Furthermore, the plastic strains in the RD, εRD , and in the TD, εTD , were measured such that the relation between them could be evaluated, see Fig. 11(b). 3. Constitutive model A constitutive model, based on previously described material behaviour, was implemented in LS-DYNA, cf. Hallquist (2009), as a user-defined material model for shells. In this section, the governing equations of the model are presented. A corotated material ˆ frame is used, where the corotated rate of deformation tensor, D, ˆ are written as and the corotated Cauchy stress tensor, ,

Fig. 9. Nominal stress–displacement relations from the plane strain tensile and shear tests.

ˆ = R T DR, D

rotations, cf. Larsson et al. (2011). The deformation was applied by a constant crosshead velocity v = 0.03 mm/min, which resulted in p a strain rate of ε˙ ≈ 10−4 s−1 . The displacement was measured on

where R and U are the rotation tensor, respectively, and the right stretch tensor in the polar decomposition F = RU of the deformation gradient, F. Henceforth, the corotational superscript (ˆ) will be excluded and all subsequent constitutive relations will be related to the corotated configuration. Since the elastic deformations are

clamps in which the clevis pins were mounted. Three repetitions

ˆ = R T R

Fig. 10. (a) Geometry of the shear test specimen. Dimensions in mm.

(1)

R. Larsson, L. Nilsson / Journal of Materials Processing Technology 212 (2012) 46–58

51

p

p

Fig. 11. Results from the balanced biaxial test: (a) Stress–normal plastic strain relation. (b) Relation between εRD and εTD .

considered small, an additive decomposition of the rate of deformation tensor is introduced, D = De + Dp De

(2) Dp

and are the elastic and plastic parts, respectively. The where update of the corotated Cauchy stress is assumed to be hypoelastic ˙ = C : De = C : (D − Dp )

(3)

where C is the elastic material stiffness tensor and the superscript dot denotes the material time derivative. In this work elastic isotropy is assumed. The plastic rate of deformation is given by an associated flow rule

∂f D = ˙ ∂ p



t

p

∗ ∗ ∂ ∂ ∂11 ∂ ∂22 = + ∗ ∗ ∂33 ∂11 ∂33 ∂22 ∂33

=−

∂ ∂ ∂ ∂ − =− − ∗ ∗ ∂11 ∂22 ∂11 ∂22

(9)

(4)

where ˙ and f are the plastic multiplier and the yield function, respectively. The accumulated equivalent plastic strain is evaluated from ˙ dt

ε =

∗ = 22 22 − 33 , as proposed by Borrvall and Nilsson (2003). The 11-direction corresponds to the RD, the 22-direction corresponds to the TD and the 33-direction corresponds to the normal direction, ND. Since the effective stress is independent of hydrostatic pressure, i.e. ( + I) = (), it follows that plastic incompressibility is preserved, since

implying that tr Dp = 0. In this work, a mixed isotropic-distortional hardening was introp duced by allowing the effective stress parameters Ai = Ai (ε ) and the exponent a = a(VM ) to be functions of the equivalent plastic strain and the martensitic fraction, respectively.

(5)

0

3.2. Yield stress

3.1. Yield function As the temperature, T, and the martensitic fraction, VM , influence the yield stress, a linear mixture rule is assumed for the yield stress

The following yield function is assumed



f =  − 0 − R − SSA − ta =

≤0

p

p

= v (ε˙ )

forε˙ = 0 p forε˙ > 0

(6)

where  is the effective stress,  0 is the initial yield stress and R is the plastic hardening, including the contribution from the martensitic fraction, VM . The contributions from the static and dynamic strain ageing are governed by  SSA and ta , respectively, and v is a viscous stress. The plastic anisotropy was accounted for by using an effective stress function, (), based on the work by Aretz (2004), 2 () = |1 |a + |2 |a + |  1 −   2 |a a

where 1 2



p

=

  1   2



p

∗ + A (ε ) ∗ A8 (ε )11 1 22



2 =

∗ + ∗ 11 22

2

±

±



p

p

p

p

2

p

2

2 p

∗ − A (ε ) ∗ A5 (ε )11 6 22

2

(10)

where R and R˛ are the yield stresses for the parent austenite and martensite phases, respectively, and RT is a contribution from the temperature influence on the yield stress. Due to the complexity of evaluating the hardening in the austenite and martensite separately, the following simplification was made p

p

p

p

R(ε , VM ) = R (ε ) + VM (ε )RM (ε ) + RT T

 

(11)

=R˛ −R

(7)

∗ − A (ε ) ∗ A2 (ε )11 3 22

p

R(ε , VM ) = (1 − VM )R (ε ) + VM R˛ (ε ) + RT T

p

+ A4 (ε )2 12 21 (8) p

+ A7 (ε )2 12 21

The yield criterion, i.e. f = 0, with  according to Eqs. (7) and (8), is convex when a ≥ 1 according to Aretz (2004). The effective stress ∗ ∗ = was originally written for plane stress state, i.e. 11 11 and 22 = 22 . However, in order to account for the normal stress,  33 , it was ∗ = subtracted from the in plane normal stress, i.e. 11 11 − 33 and

where the difference, RM , in yield stress for the martensite and austenite was assumed to be a constant. The evolution of the plastic hardening was obtained from the tensile tests. For plastic strains below a transition strain, εt1 , an extended Voce law was fitted to the

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R. Larsson, L. Nilsson / Journal of Materials Processing Technology 212 (2012) 46–58

experimental data, whereas for larger strains, the stress was given by two powerlaw expressions. Thus,

p

R(ε ) =

⎧ 2  ⎪ p ⎪ p ⎪ Qi (1 − e−Ci ε ) ε ≤ εt1 ⎪ ⎪ ⎪ ⎨ i=1 n

εt2 ≤ ε

 1 − V (B+1)/BeQ/T [1−tanh(C+D · T )] M

(13)

V

3.3. Dynamic strain ageing The dynamic strain ageing is modelled according to McCormick (1988). The instantaneous SRS is governed in the viscous stress, cf. Hopperstad et al. (2007),



p

1+

p ε˙



p ε˙ 0

0

ta , introduced in Eq. (6), accounts for DSA via the average ageing time, ta , cf. Mesarovic (1995)



  t ˛  −

(15)

a

where H, td and ˛ are material parameters. According to Kubin et al. (1992), td can be considered as a characteristic time constant. The evolution of the average ageing time can be written



dta =

0 dt −

p  for ε = 0 t



t p ε = dt 1 − ad a

(16)

p

for ε > 0

p where the steady state ageing time, ta,ss = /ε˙ , has been introduced and where can be considered as a strain increment corresponding to an advancement of all arrested dislocations from one obstacle to the next, cf. Zhang et al. (2001). The material is considered to be stable after its manufacturing, i.e. ageing will not take place without plastic strain. This is in contrast to the work by Hopperstad et al. (2007), where the evolution of ta is independent of the total equivalent plastic strain. Although probably is affected by plastic strain, it has been considered to be a material constant in this work. Since ta remains zero before any plastic deformation takes place, it does not affect the initial yield stress. In the case of p(1) p(2) an instantaneous change of plastic strain rate, i.e. ε˙ → ε˙ , at (2)

p∗

time t* and plastic strain ε , the new steady state value, ta,ss acts as a target value and the evolution of the ageing time ta can be found from Eq. (16), ta (t) =



(1) ta,ss

(2) − ta,ss

td [s]

˛ [-]

[-]

ε,ref [MPa] ˙

9.0 × 10−5

10.9

1.9

0.336

0.0018

119

p  i , only depends on S and ε˙ 0 . Negative SRS is obtained if  ss < 0. The total viscous stress ε˙ = v + ta is introduced, which in the case of a constant plastic strain rate, i.e. steady state condition, can be written

 

εss ˙

= S ln

p ε˙ 1+ ε˙



+ H exp

 t ˛ 



 exp

−

(t − t ∗ ) (2)

ta,ss



(2)

+ ta,ss

(17)

Thus, the transient behaviour after a strain rate change depends on

. A sketch of the viscous response of an instantaneous strain rate

(18)

d

where the superscript ss indicates steady state. 3.4. Static strain ageing Following Ballarin et al. (2009a), the static strain ageing, SSA, effect was incorporated in the model by an additional term of Voce type in the plastic hardening function. In addition, the dependency on the ageing time, , was approximated by an exponential function. Thus, the proposed contribution from the SSA to the plastic hardening yields p

(14)

where the material parameter S determines the instantaneous p strain rate sensitivity and ε˙ is a material constant. The yield stress,

ta = SH 1 − exp

H [-]

11.5

0

where A, B, C, D, b and Q are material constants.

v (ε˙ ) = S ln

ε˙ 0 [s−1 ]

increase is shown in Fig. 5. The steady state SRS depends on S, H, p ε˙ , ˛, and the ratio /td , whereas the instantaneous stress increase,

p

where  0 , Qi , Ci , Ai , Bi and Ci for i = 1, 2 are material constants. The transition strains, εt1 and εt2 , may be chosen arbitrarily. In this work, the extended Voce model was used for small strains; the first and the second transition strains were less than and close to the longitudinal plastic strain at diffuse necking, respectively. The martensitic transformation rate was evaluated using the Hänsel model, Hänsel et al. (1998),

∂VM B = (VM )b 2A ∂ε

p

S [MPa]

(12)

p

εt1 ≤ ε ≤ εt2

A1 + B1 ε ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎩ A2 + B2 ε

Table 1 Parameters for the strain rate sensitivity, SRS.

p

SSA = [sat − (sat −  ) exp(−Cε (ε − ε0 )] · · (1 − exp(−C ))

(19)

where the constants   and  sat govern the yield stress increase and the saturation level of this stress contribution, i.e. the permanent effect of the SSA, respectively. The two material constants C and Cε govern the ageing rate and the speed at which the effect vanishes with recurring plastic strain, respectively. The SSA effect probably depends both on temperature and on plastic strain, but these dependencies were excluded in this work. 4. Calibration of the material model Both direct parameter identification from experimental data and inverse FE analyses were used in order to calibrate the material model constants. In this section the calibration procedures are described. 4.1. Strain rate sensitivity and dynamic strain ageing Both the viscous stress, v (ε˙ p ), and the ageing stress, ta (ta ) contribute to the strain rate dependency. These parameters were identified in a sequence. The parameter ˛ was set equal to 0.336 p according to Hopperstad et al. (2007). Firstly, S and ε˙ were found 0

from the instantaneous SRS,  i , see Fig. 6. Secondly, the value of H and the ratio /td were found from the steady state SRS stress measure,  ss . The stress level before and after the increased strain rate was measured to evaluate  ss , with a reference curve obtained with a crosshead velocity v = 0.5 mm/min, see Fig. 6. The steady state strain rate dependency according to Eqs. (14), (15) and (18) is shown in Fig. 12. The absolute values of and td determine the plastic strain increment carried by the band. Several FE analyses with different values of and td , but with a fixed ratio, /td , were conducted in order to find these values. Details of the FE-analyses are presented in Section 5. The obtained parameter values are presented in Table 1. The reference equivalent plastic strain rate was p set to ε˙ = 10−4 s−1 , and the corresponding ε,ref was found to be ˙ 119 MPa.

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Fig. 13. Evolution of the effective stress parameters. Fig. 12. Strain rate dependency in the case of constant plastic strain rate. Table 2 Parameters for the plastic hardening in the TD.

4.2. Plastic hardening The two first hardening functions in Eq. (12) were fitted to p the stress–strain data in the TD for strains 0 ≤ ε ≤ εt2 = 0.25. The parameters were found from the least-square optimisation problem find

0 , Q1 , C1 , Q2 , C2 , εt1 , A1 , B1 , n1

minimise

e =

N 

p

exp

p

[ε,ref + 0 + R(εi ) − y (εi )]2 ˙

(20)

 0 [MPa]

Q1 [MPa]

C1 [-]

Q2 [MPa]

C2 [-]

655

223

484

180

41

εt1 [-]

A1 [MPa]

B1 [MPa]

n1 [-]

0.023

915

2417

0.93

εt2 [-]

A2 [MPa]

B2 [MPa]

n2 [-]

0.25

2023

-63

-1.4

i=1

where Nεp was the number of strain states at which the numerical and experimental stress–strain relations were compared in the fitp ting process. The third interval in Eq. (12), i.e. ε > εt2 , was used for extrapolation of the stress–strain relation. Continuity requirements p at ε = εt2 was used together with inverse modelling of the plane strain test in order to determine the parameters A2 , B2 and n2 . This procedure was repeated for all the material directions. The obtained parameters for the plastic hardening in the TD are presented in Table 2. 4.3. Martensitic transformation As the present experimental programme did not aim at measuring the martensitic fraction, this type of experimental data was

received from Moshfegh (2010). The data contained stress, martensitic fraction and temperature as functions of plastic strain obtained from uniaxial tensile tests with various starting temperatures. The calibration procedure to evaluate the Hänsel et al. (1998) model parameters and the related hardening coefficients, RT and RM , was based on a least-square fit. The obtained values are presented in Table 3. Regarding the experimental results presented in Fig. 2, the heating of the specimens was assumed to be negligible, due to the rather low strain rates, i.e. ε˙ ≈ 10−4 s−1 . Thus, the temperature was assumed to be equal to the room temperature, i.e. T = 293 K. Eq. (13) was integrated numerically, which gives the p martensitic fraction VM (ε ) at room temperature. Furthermore, the austenitic part of the stress–strain relation was obtained from the

Table 3 Hänsel model parameters for HyTens® 1000 and the hardening coefficients for temperature and martensitic fraction. A [-]

B [-]

C [-]

D [K− 1]

Q [K]

b [-]

VM0 [-]

RT [MPa/K]

RM [MPa]

813

213

-6.0

0.019

1184

1.6

0.13

0.75

370

Fig. 14. (a) Finite Element mesh of the tensile test. Element length Le = 1 mm. (b) Thickness distribution on the tensile test model. Fringe lines denote a thickness step of 0.001 mm.

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Fig. 15. The true stress, , and longitudinal logarithmic strain, εL , versus the applied deformation, based on the numerical extensometer for various element lengths.

Fig. 16. Experimental and numerical stress–strain relations from tensile tests. Fig. 19. Finite Element model of the plane strain tensile test.

p

A1 (εj ), . . . , A8 were calculated from the optimisation problem at p

several levels of equivalent plastic strains, εj , i.e. find

p

p

A1 (εj ), . . . , A8 (εj ) 0,45◦ ,90◦

min

eA,j =

 

0,45◦ ,90◦ p

p

(r (εj ) − r  (εj ))2 + p

p

p

 

p

p

(k (εj ) − k (εj ))2 p

+(rb (εj ) − r b (εj ))2 + (kb (εj ) − kb (εj ))2 Fig. 17. Comparisons between experimental and numerical results.

difference between the total hardening and the martensitic part, i.e. R = R − RM VM − RT T with T = 293 K.

(21) where the superscript () denotes a quantity evaluated from the p p effective stress with a set of parameters A1 (εj ), . . . , A8 (εj ) and

4.4. Plastic anisotropy In this section, the procedure for evaluating the plastic anisotropy and its evolution is briefly described. The parameters

Fig. 18. The numerical and experimental longitudinal strains as functions of crosshead displacement for uniaxial loading in the TD.

Fig. 20. Experimental and numerical results from the plane strain tests. The experimental curves are mean curves based on the three experimental results in each direction.

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55

Fig. 21. (a) Finite Element model of the shear test. (b) Details of the FE mesh in the shear zone. p

a(εj ). The yield stress ratios, r and rb , and the plastic strain ratios, k and kb , are defined as p

p

 (ε )

p

ref (ε ) p dεL 

r (ε ) = k (ε ) =

p

;

p

rb (ε ) =

p p  ; kb (ε ) = dεW  p ε

p

b (ε ) p

ref (ε )

   

p dεTD p dεRD εp

(22)

where the subscripts  and b denote a uniaxial stress state in the -direction and the balanced biaxial stress state, respectively. The reference yield stress was set to be  ref =  90 , and thus r90 = 1.

Since the transversal strain was measured locally within the gage length, the obtained relation between the longitudinal and transversal strains is serrated as well. Aretz et al. (2010) measured the transversal strain at several longitudinal locations along the tensile test specimen. The transversal strain was then evaluated as the average of these measurements in order to achieve a smooth relation between longitudinal and transversal strain components. p p However, in this work, a linear relation, εT, = k εL, + m was fitted to the experimental data in order to evaluate k , cf. Larsson et al. (2011). Despite the higher strain rate observed in the bulge test, compared to the uniaxial tensile tests, it was concluded that this effect may cause a marginal change of the shape of the yield surface.

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Table 4 Parameters for static strain ageing.   [MPa]

 sat [MPa]

Cε [-]

C [s−1 ]

88

16

58

2 × 10−6

According to the established plastic strain rate dependence, see Fig. 12, the stress level is predicted to deviate at maximum 20 MPa compared to the reference stress level. Thus, this effect was neglected and since the biaxial stress to normal plastic strain relation was very close to the uniaxial stress–plastic strain relation in the TD, the corresponding yield stress ratio was set constant, i.e. rb = 1. However, the plastic strain ratio for the balanced biaxial stress state, kb , varied with plastic strain according to Fig. 11(b), in p the plastic strain range 0.005 ≤ ε ≤ 0.2, whereas it was assumed to be constant for larger strain levels and for very small strain levels. The general recommendation on the choice of the yield function exponent, a, is a = 8 for FCC materials and a = 6 for BCC materials, see e.g. Hosford (1993). Thus, here the exponent is assumed to change due to phase transformations, and a linear mixture rule is used. Thus, a = (1 − VM ) · 8 + VM · 6

(23)

where the exponent a = 6 is assumed also for the BCT structure. The effective stress parameters were fitted to the experimental data such that the errors, eA,j , see Eq. (21), are negligible for all levels of p equivalent plastic strain, i.e. eA,j < 10−30 for all εj . The evolution of the effective stress parameters is shown in Fig. 13.

Fig. 22. Experimental and numerical results from the shear tests. The experimental curves are mean curves based on the three specimens in each direction.

three different meshes are shown in Fig. 15. Some small discrepancies both in the stress and in the strain responses are observed, indicating a weak mesh dependency. Fig. 16 shows the stress–strain relation from one representative tensile test in each direction together with the corresponding FE simulation results. The numerical stress–strain relation and the relation between the longitudinal and transversal plastic strains in the TD, with Le = 0.25 mm, are shown in Fig. 17 together with the corresponding experimental results. The numerical and experimental longitudinal strains as functions of crosshead displacement for uniaxial loading in the TD are shown in Fig. 18.

4.5. Static strain ageing, SSA

5.2. Plane strain tests

In order to find the parameters in Eq. (19), the stress–strain relations shown in Fig. 7 were compared to nominal stress–strain relations in the RD. The ageing time was defined as the time from the start of unloading till plastic yielding reoccurred after reloading. The parameter values were found from a least-square fit of Eq. (19) to the experimental data, and are presented in Table 4.

The FE model used for the simulation of the plane strain specimen in Fig. 8 is shown in Fig. 19. The shortest element length was 0.2 mm in the centre parts. The loading was applied by a prescribed deformation at the edge nodes, corresponding to the clamps. The deformation measure ı, see Fig. 8, was evaluated by measuring the distance between two nodes corresponding to the extensometer used in the experiments. The experimental and numerical results are shown in Fig. 20.

5. Finite Element simulations Finite Element simulations of the tensile test were used in order to find the absolute values of and td . Simulations of the planestrain tests were used in order to find the values of A2 , B2 and n2 in Eq. (12). In addition, numerical predictions of the nominal stress–displacement relations in the shear tests were used in order to validate the model. In order to avoid spurious local element thinning, a shell element with normal stress and strain components was used, i.e. shell element type 26 in LS-DYNA, Hallquist (2009). By using this element, the normal strain was required to be continuous, which corresponds to regularisation of the normal strain.

5.3. Shear test The FE model used in the analyses of the shear tests, cf. Fig. 10, is shown in Fig. 21(a). A characteristic element size in the shear zone was Le ≈ 0.06 mm, see Fig. 21(b). A few elastic elements were used to model the bolts, and the loading was applied as a prescribed motion of a bolt node. Thus, the rotation around each bolt was unconstrained. The shear tests were not included in the calibration procedure and the obtained results are used as a validation of the material model, see Fig. 22.

5.1. Tensile tests The tensile tests were analysed using FE with three different mesh densities in order to evaluate a possible mesh dependence, with characteristic element lengths of Le =1, 0.5 and 0.25 mm, cf. Fig. 14(a). A stochastic thickness distribution was applied in order to initialise the localisation, see Fig. 14(b). The loading of the test specimen was applied by a prescribed motion at the edges. In order to compare numerical and experimental results, the strain was measured between two nodes with an initial distance corresponding to the two measuring points of the mechanical extensometer, i.e. L0 = 12.5 mm. The resulting stresses and strains as functions of the edge displacement from using the

Fig. 23. Experimental and numerical results from the strain ageing tests.

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5.4. Static strain ageing, SSA One FE analysis for each ageing time was conducted, where the specimen was elastically unloaded at εL ≈ 0.084, which corresponds to the experimental procedure. The SSA model was subsequently activated in the three reloading analyses, after ageing times corresponding to the experimental procedure. The numerical results are presented in Fig. 23 together with one experimental result for each ageing time.

6. Conclusion and discussion An experimental programme was performed on the austenitic stainless steel HyTens® 1000, including tensile tests, shear tests and plane strain tests. All tests used in the parameter identification procedure were conducted at low strain rates and isothermal conditions were assumed. It was shown that this material is sensitive both to dynamic strain ageing, DSA, and to static strain ageing, SSA. The DSA leads to a material instability and causes a discontinuous plastic flow and propagation of strain bands which results in a serrated stress–strain relation. The SSA results in a substantially higher yield stress after pre-straining and ageing. The observed serration is identified as type D, i.e. plateaus in the stress–strain relations, cf. Rodriguez (1984). However, the number of plateaus are low, and the plastic strain carried by each band is large, cf. Fig. 3. The serration starts at a lower strain level in the RD than in the other directions, an observation which is expected since the plastic hardening in the RD at low strain levels is smaller, i.e. implying that instability occurs earlier in this direction. Static strain ageing was shown to result in an increased yield stress after the specimens have been unloaded and aged in room temperature, and the SSA effect was shown to increase with ageing time. A constitutive model was implemented and calibrated to experimental data in order to predict the identified phenomena. The material model was implemented as a hypo elastic-plastic model. In order to account for plastic anisotropy, a modified version of the high exponent eight parameter effective stress by Aretz (2004) was used. By allowing the effective stress parameters to depend on plastic strain and martensitic fraction, an isotropicdistortional hardening was achieved. The Hänsel et al. (1998) model for martensitic transformation was incorporated in the material model in order to find the evolution of the martensitic fraction. The McCormick (MC) model for dynamic strain ageing, DSA, was used in order to account for the experimentally observed discontinuous yielding. So called jump tests were used to find both the instantaneous SRS and the steady state SRS, in order to calibrate the DSA model. Inverse modelling of the tensile tests was used in the final calibration of the model. A modified version of the model proposed by Ballarin et al. (2009b) was used in order to describe the SSA. Despite some discrepancies between the numerical and experimental results, the model is able to predict both the increase of yield stress with ageing time and the decrease of this effect during subsequent plastic deformation with reasonable accuracy. The anisotropic nominal stress–displacement relations from plane strain tests in various directions were very well predicted. It is to be noted, that the anisotropy in the plane strain tests at p relatively small deformations, i.e. ε ≤ εt2 and ı ≤ 2.3 mm, was successfully predicted from the evaluation of the anisotropy in the uniaxial tensile tests. For larger deformations, the plane strain tests were used in order to evaluate the extrapolation of the stress–strain relation, and cannot serve as validation of the model, however the numerical and experimental results agree well. The developed material model was able to represent the characteristic strain history as a function of the crosshead displacement

57

in the tensile tests, and the characteristics in the predicted serrated stress–strain relation were reasonable compared to experimental data. Furthermore, the start of the jerky flow was well predicted in the RD, despite the fact that the DSA model was calibrated by using experimental data from tensile tests in the TD only. This agreement was achieved by combining the DSA model with the isotropic-distortional hardening. The predicted force–displacement relations in the shear tests were used as a validation of the material model. The agreement between the numerical and experimental results are reasonable. There are two possible explanations regarding the observed discrepancies. The first is an insufficient accuracy in the evolution of the effective stress. The second is the absence of hydrostatic pressure dependency on the martensitic transformation in the Hänsel model. It is believed that the shear stress test promotes the martensitic transformation less than the uniaxial tension does. Since the Hänsel model parameters are found from tensile tests, the model may over predict the martensitic transformation in shear, and thus also, plastic hardening. The main goal of this work has been to incorporate the characteristics of observed material phenomena into a constitutive model and propose a calibration procedure to determine its parameters. The presented constitutive model requires a relative large number of constants. However, the identification of most constants is straightforward and these are evaluated sequentially and directly from experimental data. The practical impact of the included phenomena will depend on the specific application. However, it is believed to be important e.g. in sheet metal forming simulations and the prediction of failure. Acknowledgements The work presented in this paper has been carried out with funding from the SFS ProViking project “Super Light Steel Structures”. Outokumpu Stainless and Dr. Ramin Moshfegh, are greatly acknowledged for advice, material data and material supply. References Aretz, H., 2004. Applications of a new plane stress yield function to orthotropic steel and aluminium sheet metals. Model. Simul. Mater. Sci. Eng. 12, 491–509. Aretz, H., 2008. A simple isotropic-distortional hardening model and its application in elastic-plastic analysis of localized necking in orthotropic sheet metals. Int. J. Plast. 24, 1457–1480. Aretz, H., Aegerter, J., Engler, O., 2010. Analysis of earing in deep drawn cups. In: Barlat, Moon, Lee (Eds.), Proceedings of the 10th International Conference on Numerical Methods in Industrial Forming Processes. Pohang, Proc. 1252, pp. 417–424. Ballarin, V., Perlade, A., Lemoine, X., Bouaziz, O., Forest, S., 2009a. Mechanisms and modeling of bake-hardening steels. Part II. Complex loading paths. Metall. Mater. Trans. A 40 (6), 1375–1382. Ballarin, V., Soler, M., Perlade, A., Lemoine, X., Forest, S., 2009b. Mechanisms and modeling of bake-hardening steels. Part I. Uniaxial tension. Metall. Mater. Trans. A 40 (6), 1367–1374. Barlat, F., Brem, J.C., Yoon, J.-W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminium alloy sheets. Part. Theory. Int. J. Plast. 19, 1297–1319. Barlat, F., Lian, J., 1989. Plastic behaviour and stretchability of sheet metals. Part I. A yield function for orthotropic sheets under plane stress conditions. Int. J. Plast. 5, 51–66. Barlat, F., Yoon, J.W., Cazacu, O., 2007. On linear transformations of stress tensors for the description of plastic anisotropy. Int. J. Plast. 23 (5), 876–896. Benallal, A., Berstad, T., Børvik, T., Hopperstad, O., Koutiri, I., de Codes, R.N., 2008. An experimental and numerical investigation of the behaviour of AA5083 aluminium alloy in presence of the Portevin Le Châtelier. Int. J. Plast. 24, 1916–1945. Borrvall, T., Nilsson, L., 2003. Revision of the implementation of material 36 in LSDYNA. Engineering Research AB, Linköping, private communication. Clausen, A., Børvik, T., Hopperstad, O., Benallal, A., 2004. Flow and fracture characteristics of aluminium alloy AA5083-H116 as function of strain rate, temperature and triaxiality. Mater. Sci. Eng. A 364 (1–2), 260–272. Hallquist, J., 2009. LS-DYNA Theory Manual. Livermore Software Technology Corporation, Livermore. Hänsel, A.H.C., Hora, P., Reissner, J., 1998. Model for the kinetics of straininduced martensitic phase transformation at non-isothermal conditions for the

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