Modeling Tube-Forming of an Austenitic Stainless Steel with Exploitation of Martensite Evolution

Transcription

1 TECHNISCHE MECHANIK, 32, 2-5, (2012), submitted: November 1, 2011 Modeling Tube-Forming of an Austeniti Stainless Steel with Exploitation of Martensite Evolution T. Dally, C. Müller-Bollenhagen, H.-J. Christ, K. Weinberg Within the last years the industrial manufaturing of tubes has developed to an inreasingly omplex proess. In partiular, during the forming proedure of sheets made of austeniti stainless steel, the inrease and the ontent of strain-indued martensite needs to be ontrolled in order to ahieve the optimal strutural properties of the manufatured tube with respet to very-high-yle fatigue (VHCF). On the basis of experimental investigations this ontribution deals with the numerial simulation of the forming proess with speial onsideration of the martensite ratio as a funtion of temperature and deformation field. A onvenient approah of modeling the martensite evolution as well as the extension of this model to polyaxial states of stress and a omparison with experimental results is presented. 1 Tehnologial Bakground Within the last deade engineers beame more and more aware of the fat that a given high yle fatigue may not be suffiient to guarantee reliability of ertain mahine omponents. Strutures like engine parts, train wheels, rotors, medial devies or offshore omponents often have to sustain high frequeny vibrations and thus a number of life yles muh higher than The need for a long life expetation of suh omponents has raised interest in investigating materials in the very high yle fatigue (VHCF) regime, i.e., with loading yles of Espeially for metastable austeniti steel it appeared that there is a ertain volume fration of martensite whih optimizes the resistane to VHCF. Therefore, we present in this ontribution a strategy to model and numerially simulate the evolution of martensite in a metastable steel sheet, whih an be employed to ontrol proess parameters during industrial forming. slip bands martensite 10µm Figure 1: Needles of martensite in X5CrNi18-10 steel after after 10 9 yles The mehanial properties of austeniti stainless steel are known to signifiantly improve with growing martensite ontent. For example, the transformation indued lattie hange from fae-entered ubi γ-austenite to bodyentered ubi α -martensite inreases the disloation density in the austeniti phase whih, in turn, raises its strength. Moreover, martensite works as a barrier against mirosopi rak growth and, thus, the material resistane is inreased. Figure 1 displays the mirostruture of a speimen after yling in the VHCF regime (10 9 yles with a stress amplitude of 245 MPa); the VHCF fatigue properties are strongly influened by the formation of slip bands and martensite needles. In material siene there exist a wide range of models that desribe the martensite evolution during plasti deformation of metastable austeniti steel. However, many of these models are rather simple and do not even aount for the influene of a temperature hange while deformation and for the strain rate sensitivity of the martensite 155

2 formation, f. Mu ller-bollenhagen (2011) for a review. A lassial model was introdued by Olson and Cohen (1975) who suggest a diret dependene between the formation of martensite embryos and the quantity of shear band rossings; therefore failitating a alulation of the martensite ratio as a funtion of strain. Stringfellow et al. (1992) modified this early model in suh a manner that also the stress-state of the material was regarded as a omponent for the martensite evolution. Tomita and Iwamoto (1995) presented another modifiation based on experimental observations and stated an influene of the strain rate on the formation of shear bands. Another martensite model was developed by Tsuta and Cortes (1993) who suggested a polynomial funtion whih needs to be alibrated against tension tests. This model was modified and formulated inrementally by Heinemann (2004) and Springub (2006). Most of these marosopi models desribed the martensite evolution as a funtion of temperature and strain in an uniaxial way. However, here we onsider a stepwise forming proess whih requires a model appliable to multi-axial states. Beside temperature, stress-state and strain there are moreover additional influening fators for the evolution of martensite. Waitz et al. (2009) found out that the formation of thermally indued martensite strongly depends on the grain size of the material and an be ompletely suppressed if the grain size is too small. Of ourse the martensite evolution is ruially influened by the material s omposition; for example a high arbon onentration obstruts the martensite evolution, f. Krupp and Christ (2008). The formation of martensite orresponds to a high amount of shear bands and shear band rossings and requires a partiulary high loal shear stress in the grain. The ratios of shear stress in the loal slip plane to the applied external tension (Shmid fators) visualized by means of Eletron Baksatter Diffration (EBSD) sans are depited in Figure 2. The displayed speimen have been prestrained (ε = 0.15) in order to reah 20 vol.% of martensite. Subsequently they are loaded in two different ways. For a speimen pulled again in the same horizontal diretion the Shmid fators of different grains are shown in Figure 2 a); the white areas are martensiti. Clearly, martensite forms in areas with high Shmid fators whih orrelates to a high density of shear bands. Figure 2 b) shows the Shmid fators for the same speimen now pulled in orthogonal diretion. Again, grains with a high Shmid fator will deform plastially. However, the Shmid fator in grains with aumulated martensite is now redued. This indiates that not only the total strain but also its diretion influene the amount of martensite following from a ertain shear band volume fration. We will quantify this effet in our martensite formation model. a) b) Figure 2: Shmid fators in tensile speimens of 20 vol.% martensite in uniaxial and two-axial deformation (blue: minimum, red: maximum, white: martensite) The remaining of the paper is organized as follows: In the next setion we desribe the onept of VHCF testing and provide investigations of the VHCF properties of X5CrNi18-10 steel. Then, in Setion 3, we shortly introdue the underlying ontinuum mehanial model and present an approah to alulate the rate of martensite evolution. A disussion of the model is also given in Setion 3. The last setion summarizes our numerial results and presents the appliation of our model in a forming simulation of metastable X5CrNi18-10 steel sheets. 2 Resistane to VHCF In the literature the VHCF-behavior of metalli materials is often lassified in type I materials (dutile single phase materials without intrinsi defets) and type II materials (high strength steels with intrinsi defets). Surfae roughening aused by loal plasti deformation in favorably oriented grains is seen as the predominant damage mehanism in VHCF regime for type I materials. In ontrast, type II materials often show loal plasti deformation at inlusions that an lead to rak initiation due to debonding of the inlusion-matrix interfae or breaking of non- 156

3 fish-eye frature Inlusion 10µm 500µm Figure 3: Modes of VHCF failure: fish-eye frature along inlusions in a speimen of 27 vol.% of martensite (left) and failure by fine granular area in a speimen of 54 vol.% of martensite (right) metalli inlusions. Beause of their high dutility austeniti stainless steels may be ategorized as type I materials regarding the VHCF-behavior. However, metastable steels like X5CrNi18-10 undergo a deformation-indued phase transformation from austenite to ε-martensite and to the hard α0 -martensite phase. This phenomenon of transformation indued plastiity enables exellent strength and dutility and has been investigated intensively, for example by Fisher et al. (2000) and Dimatteo et al. (2006). Moreover, they show a pronouned strain hardening and an ontain different types of inlusions. These properties lassify X5CrNi18-10 for the type II ategory. However, our experimental observations showed that in partiular in lusters of martensite the dominant failure mode in the VHCF regime hanges from surfae roughening (Figure 1) to internal rak initiation leading to fisheye frature. A very slow rak propagation starting from inlusions is the reason for a Fine Granular Area (FGA), see Figure 3. This Fine Granular Area frature was found to be partiulary dominant at a high martensite ontent. In order to develop a quantitative relationship between the volume fration of martensite and the VHCF behavior several speimens were monotonially prestrained in two different ways and ompared to virgin austeniti speimen. Firstly, a one-step plasti deformation at different temperatures resulting in two martensite volume ontents (27% and 54%) and seondly, a two-step deformation by onstant temperature but different amounts of deformation in order to reah the 27% and the 54% of martensite volume were performed. The latter deformation leads to a higher disloation density. The results of the VHCF tests are displayed in Figure 4 and 5. Figure 4: S-N urves for one-step deformed speimens with different martensite ontent Figure 5: S-N urves for two-step predeformed speimens with different martensite ontent In the fully austeniti ondition a true VHCF fatigue limit exists. The onstant fatigue limit is mainly aused by a martensite-assisted yli hardening proess. A higher martensite ontent of 27 vol % martensite enhanes the HCF properties and the fatigue limit remains independent of the number of yles even in the VHCF regime. At 54 vol.% martensite the more brittle behavior of the martensite phase enables rak initiation at inlusions and the formation of FGAs leading to failure in the VHCF regime. Figure 5 shows the VHCF fatigue limit of the two-step predeformed speimens whih has the same gradient as the one-step limit for 54 vol.%. The two-step deformed speimens with 27 vol.% martensite also showed a behavior similar to the one-step deformed ounterparts, even though the yield strength and disloation density of these speimen were onsiderably higher. The VHCF properties are not determined by disloation density and diretion or amount of predeformation but by the martensite ontent instead. These and further experimental observations 157

4 lead us to onlude that a volume fration of vol.% of martensite is required for optimal VHCF resistane of this steel. 3 Mehanial Modeling of Martensite Formulation 3.1 Continuummehanial Model of Metastable CrNi Steel The state of motion of a solid is fully desribed by the deformation gradient F whih an be divided into an elasti and a plasti omponent, F = F e F p. This multipliative split leads to the following representation of the spatial veloity gradient, l = Ḟ F 1 = Ḟ e (F e ) 1 +F e Ḟ p (F p ) 1 (F e ) 1, }{{}}{{} =:l e =:l p whih will be linearized as indiated. The onsidered material is now haraterized by a free Helmholtz-energydensity A = A(F e, F p, ε p, T ) with effetive plasti strain ε p and temperature T. The rate of plasti deformation is subjeted to the von Misesflow rule l p = ε p M with tr (M) = 0, M M = 3 2, εp 0, where tensor M desribes the diretion of plasti flow. Now the free Helmholtz-energy-density an be splitted into an elasti and a plasti part whih again is additively deomposed into work hardening and phase transforming omponents (a more detailed desription is given in Dally and Weinberg (2011)): A = W e (ε e, T ) + W p (ε p, T ) = W e (ε e, T ) + W γ (ε p, T ) + W α (ε p, T ). In an time-inremental approah with Δt = t n+1 t n this energy will be optimized with respet to the plasti variables. The used onstitutive update algorithm starts with the assumption that the material s state at time t n and the state of deformation and temperature at time t n+1 is given. The problem is to determine the state of the material at time t n+1. For this purpose an inremental deformation energy funtion f n is formulated: f n (ε n+1, T n+1, ε p n+1, M) = W e (ε e n+1, T n+1 ) + W γ (ε p n+1, T n+1) + W α (ε p n+1, T n+1). Minimization of f n with respet to the inner variables leads to the effetive inremental strain-energy density: W eff (ε n+1, T n+1 ) = min f n (ε n+1, T n+1, ε p ε p n+1,m n+1, M). The definition of an elasti preditor strain ε trial n+1 = ε n+1 ε p n and the aording trial stress σ trial n+1 failitates the following restatement of f n : f n (ε n+1, T n+1, ε p n+1, M) = W e (ε trial n+1 Δε p M n+1, T n+1 ) + W p (ε p n+1, T n+1). The effetive plasti strain ε p n+1 then an be evaluated by solving σtrial n+1 σ trial n+1 n+1 3μΔε p σtrial σ 0 (1 + εp n + Δε p ε 0 ) 1 m σ0 ( ε p ε p 0 = 2μεtrial, dev n+1 ) 1 k = 0 (1) using a Newton-Raphson iteration. Here σ 0 = σ 0 (T ) is the initial yield stress and m the hardening exponent. Aside of the last term whih follows from martensite hardening, Equation (1) orresponds to lassial von Mises plastiity. 3.2 Evolution of Martensite The assumption that the formation of martensite during forming proesses strongly depends on the temperature is supported by speifi experiments: The left piture in Figure 6 shows the martensite formation in a speimen whih is strained uniaxially with ε = 0.14 at a starting temperature of T (0) = 70 C. In the middle piture the tension test is progressed with ε = 0.34 at ambient temperature T (0) = 23 C. Both experiments lead to a similar martensite ratio of 0.55 in the speimens - however the arrangements of the martensiti setions differ from 158

5 Figure 6: Formation of martensite (dark areas) in austeniti steel; speimen uniaxially strained with ɛ = 0.15 at T = 70 o C (left) and ɛ = 0.34 at T = 70 o C (middle), both resulting in approximately 54 vol.% martensite. At right the martensite fration is 25 vol.%, here shear band rossings (A) and shear bands (B) are marked. eah other. In the right piture the martensite formation is illustrated for a speimen whih is strained uniaxially with ε = 0.15 at a starting temperature of T (0) = 20 C. The shear bands and shear band rossings whih work as seeding points for martensite evolution are indiated. Experimental observations suggest a diret dependene between the formation of martensite embryos and the quantity of shear band rossings whih an be determined by the shear band volume. Therefore, our point of departure to model the martensite evolution is the lassial model of Olson and Cohen (1975). In inremental form it reads ε = α n β sn 1 (1 s)(1 ), (2) s = α (1 s), ε with martensite ratio and strain ε, whereas α denotes the rate of shear band evolution, β is the probability of martensite formation at a shear band rossing, s the ratio of shear band volume and n is a material speifi onstant. In the original model α and β are assumed to be onstant during the simulation with the result, that the model is only appliable for isothermal proesses. However, due to the fat that the temperature T hanges rapidly during the regarded proesses like tension tests or sheet forming, a non-isothermal evolution equation for martensite formation is required. For this purpose we extend the model of Olson and Cohen (1975) in suh a manner that we set the onstants α and β as funtions of temperature and urrent martensite ratio: α(t ) = α 1 T α 2, β(t, ) = β 1 exp (β 2 T + β 3 ), with onstants α 1, α 2, β 1, β 2 and β 3, whih need to be adjusted to the speifi material. This orresponding adjustment takes into aount that the martensite evolution depends on the grain size (see Waitz et al. (2009)). An extension of this non-isothermal but uniaxial model to polyaxial states of stress is outlined in the following. Let us first onsider a sheet whih is deformed in one diretion x and afterwards in an orthogonal diretion y with the amounts ε x and ε y respetively, see Figure 7. From experimental observations it is known that only a ertain ratio Ψ of the omposed shear bands in x-diretion is signifiant onerning the martensite evolution in y-diretion. In the following x and y denote the martensite rates and s x, s y the rates of shear band volumes whih are indued by the strains ε x and ε y. In order to alulate the martensite ratio in suh a biaxial tension test one has to alulate s x, s y and x using equation (2). At the alulation of y, however, one has to add the relevant part Ψ s x of the shear band volume ratio omposed in x-diretion additionally to s y : y ε y = α n β (s y + Ψ s x ) n 1 (1 (s y + Ψ s x ))(1 y ). Finally the martensite rates x and y based on the strains ε x and ε y get ombined, see Figure 9, so that the total martensite ratio of = x + (1 x ) y = x + y x y 159

6 is obtained. A similar proedure is appliable in the ase of a sheet that is pulled in three spatial diretions by the eigenstrains ε 1, ε 2 and ε 3 : For i, I = 1, 2, 3 (no summation) the martensite ratios i indued by ε I is alulated by and their ombination gives i ε I = α n β (s i + Ψ 3 j=1,j i s j ) n 1 (1 (s i + Ψ 3 j=1,j i s j ))(1 i ) = 1 + (1 1 ) 2 + (1 1 (1 1 ) 2 ) 3 = F x F y x y F x, ε x F y, ε y x = 1 2, y = 1 3, = 2 3 = x + y Figure 7: Two-diretional tension test. The speimen is first deformed in x-diretion, afterwards in orthogonal y-diretion Figure 8: Combination of martensite ratios x and y to the total martensite ratio 4 Simulation Results 4.1 Tensile Tests At first we study the martensite evolution at uni-diretional and two-diretional tension tests. Figure 9 shows the inrease of the martensite ratio for different initial temperatures T (0) and exposes the support of martensite evolution by low temperatures. In Figure 10 the martensite ratio depending on the strain rate ε is presented. It beomes obvious that the martensite evolution is inhibited due to low heat emission if a high strain rate is applied. The nie orrelation between the simulations and the experimental results marked by dashed lines in both Figures beomes obvious T (0) = 23 C T (0) = 10 C T (0) = 0 C T (0) = 23 C T (0) = 10 C T (0) = 0 C ε = s 1 ε = 0.005s 1 ε = 0.02s 1 ε = s 1 ε = 0.005s 1 ε = 0.02s ε ε Figure 9: Martensite volume fration as a funtion of ε and T (0); ε = 0.005s 1 Figure 10: Martensite volume fration as a funtion of ε and ε; T (0) = 23 C 160

7 Now we onsider a speimen that is deformed in one diretion x with ε x = 1 4 in the first stage followed by a deformation in an orthogonal diretion y with ε y = 1 4. In Figure 11 the martensite evolution is illustrated by using a strain rate of ε = s 1 during the first, but different strain rates during the seond stage of the forming proess. The best orrelation between experimental data (again marked by dashed lines) and simulation results is performed with Ψ 3/4. In Figure 12 the martensite evolution is presented for different initial temperatures. The progression of is similar but not idential to the one during the uniaxial tension test presented in Figure 9. (In both Figures the first derivative of is disontinuous at ε = 0.25 due to the hange of loading diretion.) ε = s 1 ε = s 1 ε = 0.005s 1 ε = 0.005s T (0) = 23 C T (0) = 10 C T (0) = 0 C ε ε Figure 11: Martensite volume fration as a funtion of ε and ε; T (0) = 23 C, Ψ = 0.75, ε = s 1, first stage of forming Figure 12: Martensite volume fration as a funtion of ε and T (0); ε = 0.005s 1, Ψ = Forming Simulation of an X5CrNi18-10 Sheet Finally we have a loser look on the loal martensite evolution during a proess that forms a X5CrNi18-10-sheet to a tube. One possibility of forming suh a tube is presented in Figure 13: At first a sheet made of austeniti stainless steel is transformed by orresponding tools and welded to a onventional tube. Afterwards the tube may be bent in order to get the final shape. Figure 13: Forming proess of a sheet to a tube The main goal of simulation is the optimization of the strutural properties of the manufatured tube with respet to very-high-yle fatigue. Therefore, the ontrol of the strain-indued martensite and thus an aurate omputation of the martensite evolution is required. At this point we fous on the first stage of the forming-proess, the U- forming. Here exist several possibilities to ontrol the martensite evolution: Apart from the strain rate and the initial temperature one an also vary the retention fore F that appears at the holders while the sheet is pressed in the given form by a speial tool. F F F F F F Figure 14: Shematis of the forming proess presented in Figure 13 The evolution of martensite is modeled as presented in Setion 3, all variables like plasti strain, temperature and martensite ratio, whih are of entral interest during the proess, are alulated by means of a UMAT-subroutine for Abaqus. Our results are displayed in Figures 15 and 16. The martensite volume fration is presented for retention 161

8 fores of 150kN and 210kN. We see that the amount of martensite strongly depends on the applied fores. This result is though not self-evident: On the one hand a higher fore F auses ultimately muh higher plasti strains and therefore leads to a support of martensite evolution. However, it also auses higher temperatures so that the formation of martensite embryos should be inhibited. During U-forming, however, it seems that the inrease of ε p after raising the retention fore is so strong that it dominates the proess of martensite evolution in this example Figure 15: Martensite in the U-formed sheet, F = 150kN, t total = 500s, T (0) = 23 C Figure 16: Martensite in the U-formed sheet, F = 210kN, t total = 500s, T (0) = 23 C Aknowledgements This material is based upon work supported by the German researh foundation (DFG) under Award Number SPP Referenes Dally, T.; Weinberg, K.: Deformation-indued martensite formation during old-working of metastable stainless steel. to be submitted to: Materials Siene and Engineering A, (2011). Dimatteo, A.; Loviu, G.; Desantis, M.; Valentini, R.; Solina, A.: Mirostrutures and properties of transformation indued plastiity steels. Assoiazione Italiana di Metallurgia, 98(11-12), (2006), Fisher, F. D.; Reisner, G.; Werner, E.; Tanaka, K.; Cailletaud, G.; Antretter, T.: A new view on transformation indued plastiity (trip). International Journal of Plastiity, 16, (2000), Heinemann, G.: Virtuelle Bestimmung des Verfestigungsverhaltens von Bändern und Blehen durh verformungsinduzierte Martensitbildung bei metastabilen rostfreien Stählen. Dissertation, ETH Zürih (2004). Krupp, U.; Christ, H. J.: Deformation-indued martensite formation during yli deformation of metastable austeniti steel: Influene of temperature and arbon ontent. Mater.Si.Eng.A, , (2008), Müller-Bollenhagen, C.: Verformungsinduzierte Martensitbildung bei mehrstufiger Umformung und deren Nutzung zur Optimierung der HCF- und VHCF-Eigenshaften von austenitishem Edelstahlbleh. Dissertation, Universität Siegen (2011). Olson, G. B.; Cohen, M.: Kinetis of strain-indued martensiti nuleation. Metall. Trans A, 6A, (1975), Springub, B.: Semi-analytishe Betrahtung des Tiefziehens rotationssymmetrisher Bauteile unter Berüksihtigung der Martensitevolution. Dissertation, PZH-Verlag, Universität Hannover (2006). Stringfellow, R. G.; Parks, D.; Olson, G. B.: A onstitutive model for transformation plastiity aompanying strain-indued martensiti transformations in metastable austeniti stainless steels. Ata Metall. Mater., 40, (1992),

9 Tomita, Y.; Iwamoto, T.: Constitutive modeling of trip steel and its appliation to the improvement of mehanial properties. Int. J. Meh. Si., 37, (1995), Tsuta, T.; Cortes, J.: Flow stress and phase transformation analyses in austeniti stainless steel under old working, part 2: Inremental theory under multiaxial stress state by the finite-element method. JSME Int J., 36, (1993), Waitz, T.; Tsuhiya, K.; Antretter, T.; Fisher, F. D.: Phase transformations of nanorystalline martensiti materials. MRS Bulletin, 34, (2009), Address: M.S. Tim Dally and Prof. Dr.-Ing. Kerstin Weinberg, Lehrstuhl für Festkörpermehanik, Universität Siegen, D Siegen Dr.-Ing. C. Müller-Bollenhagen and Prof. Dr.-Ing. H.-J. Christ, Lehrstuhl für Materialkunde und Werkstoffprüfung, Universität Siegen, D Siegen