Cyclic stress–strain behavior and load sharing in duplex stainless steels: Aspects of modeling and experiments

Acta Materialia 55 (2007) 5359–5368 www.elsevier.com/locate/actamat

Cyclic stress–strain behavior and load sharing in duplex stainless steels: Aspects of modeling and experiments R. Lillbacka a

a,c

, G. Chai

b,* ,

M. Ekh a, P. Liu b, E. Johnson c, K. Runesson

a

Department of Applied Mechanics, Chalmers University of Technology, 412 96 Go¨teborg, Sweden b Sandvik Materials Technology, R&D Centre, 811 81 Sandviken, Sweden c Swedish National Testing and Research Institute, Box 857, 501 15 Bora˚s, Sweden Received 8 January 2007; received in revised form 31 May 2007; accepted 31 May 2007 Available online 9 August 2007

Abstract From cyclic experimental tests, it has been found that the cyclic stress–strain behavior of duplex stainless steels (DSSs) is largely dependent on the applied load/strain amplitude. It is believed that the difference in the elastoplastic properties between the austenite and ferrite phases as well as the load sharing between the phases, are responsible for the dependence. In order to examine this hypothesis, three different DSSs, where the difference in the elastoplastic properties between the phases varies between them, are studied in this paper. The examinations are performed both experimentally and by simulations using multiscale material modeling. Since multiscale material modeling is used, it is possible to study in more detail the stress–strain evolution in the two phases during cycling. It is found that the difference in the elastoplastic properties between the phases affects the amount of plastic deformation occurring in the austenite and ferrite phases of the three steels. As a consequence, the cyclic stress–strain behavior is different in the three steels. A more detailed analysis of the dislocation structures formed in the two phases is also performed, and it is found that the slip band formation and the dislocation structures strongly depend on the elastoplastic properties of the individual phases as well as on the load sharing between the phases.  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Duplex stainless steels; Load sharing; Plasticity; Dislocations; Multiscale material modeling

1. Introduction Duplex stainless steels (DSSs) are used in a wide range of applications, e.g., in the power, chemical and oil industries. In these industries, materials with good corrosion resistance and mechanical properties are needed, especially since they are often subjected to cyclic loadings. In order to determine the safe operation conditions, it is crucial to understand the elastoplastic deformation and fatigue damage mechanisms in these materials. Much work has been done to understand the cyclic elastoplastic deformation mechanisms for two-phase metals [1–3], and it has been shown that they vary with the plastic strain range [2]. However, these studies, including the hardening and softening *

Corresponding author. Tel.: +46 26 263534. E-mail address: [email protected] (G. Chai).

mechanisms, are mainly performed on the macroscale, whereas, the meso/microscale elastoplastic deformation mechanisms are less studied. It is believed that the difference in the elastoplastic properties, in addition to the load sharing, between the phases is largely responsible for the varying elastoplastic deformation mechanisms with varying plastic strain range in DSSs [4]. Mateo et al. [5] observed that, although the plastic deformation in SAF 2507 starts in the austenite phase, hardness measurements after cyclic loading show that austenite is the harder of the two phases. The hardening of the austenite will lead to the transfer of plastic deformations from the austenite to the ferrite during cyclic loading. This behavior has also been observed elsewhere [6,7]. Looking further at the fatigue damage mechanisms, much work has been performed to fundamentally understand these by using simple materials such as fcc single crystals and single-phase polycrystalline metals [8–11].

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.05.056

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Ewing and Humfrey first correlated the fatigue damage with the development of slip bands [12]. Fatigue damage corresponds to formation of persistent slip bands (PSBs) on the meso/microscales and the subsequent crack initiation along these PSBs. In addition to experimental investigations, such as described above, multiscale material modeling can be used for investigating how the difference in elastoplastic properties between the phases influences the cyclic stress–strain behavior of DSSs. The basic idea in multiscale material modeling (see Refs. [13–15]) is that the a priori homogenized macroscale material model is replaced by the homogenized response of a numerically generated grain structure model. For polycrystalline materials, such as DSSs, the grain structure model describes the grain and phase structures of the material. The introduction of the grain structure model enables the study of the behavior of the different phases in multiphase materials and the effects of load sharing between the phases, since the phase and grain information is explicitly included via the grain structure model. Moreover, it is also possible to study effects of changes in the phase/grain structure by changing the geometry of the grain structure model. Since DSSs are crystalline materials, it is natural to use crystal plasticity for modeling the material behavior in the grains. Crystal plasticity is a continuum model of the plastic slip in the slip systems (see Ref. [16]). So the crystallographic directions of the grains are explicitly included and the plastic slip is introduced in the modeling. The aim of the present investigation is to examine the role of the difference in the elastoplastic properties and the load sharing between the phases when considering the cyclic stress–strain behavior of DSSs. Note that, in this investigation, cyclic stress–strain behavior refers to the stress amplitude as a function of number of cycles (i.e., accumulated cyclic plastic strain) and not a state of cyclic saturation as in [8]. Three different austenitic–ferritic steels (DSSs), where the difference in the elastoplastic properties between the phases varies between them, are examined both experimentally and by simulations using multiscale material modeling. 2. Materials and experimental procedures 2.1. Materials The materials used in the present work are two super duplex stainless steels UNS S32750 and UNS S32906 with the nominal chemical compositions shown in Table 1. For UNS S32750, two material conditions, as-delivered Table 1 Nominal compositions of the materials used Materials

C

Si

Mn

Cr

Ni

Mo

N

PRE

UNS S32750 UNS S32906

0.03 0.03

0.8 0.5

1.2 1.2

25 29

7 6

4 2

0.3 0.4

42 42

(2507AD) and as-aged (475 C/4 h ) (2507HT), were used. For UNS S32906 only the as-delivered material (2906AD) was investigated. Table 2 shows the mechanical properties of the materials and it can be seen that 2906AD has the highest yield stress followed by 2507HT and 2507AD. Furthermore, Table 2 presents microhardness measurements which show that in 2507AD the ferrite is slightly harder than the austenite initially. In 2507HT, the initial hardness of the ferrite has increased as a result of spinodal decomposition during aging. The increase of the nitrogen content, as in 2906AD, leads to an increased initial hardness in the austenite because of the solid solution hardening of this phase. This is in good agreement with the findings in Refs. [4,5]. However, the initial hardness of the ferrite in the 2906AD grade is lower than in the 2507AD grade and this contrasts the findings in Refs. [4,5]. The discrepancy could be explained by the fact that the addition of nitrogen has a relatively small effect on the hardness of the ferrite, since the dissolution of nitrogen in the ferrite at room temperature is very low. Furthermore, the 2906AD grade contains a lower amount of Mo compared with the other steels and this could lead to a softer ferrite phase, since Mo is a ferrite-stabilizing element. 2.2. Experimental procedures Microhardness measurements of the austenite and ferrite phases were done both for the initial undeformed materials and for monotonically loaded materials at different applied strains. The hardness measurements were performed using a Vickers microindenter with an applied load of 50–100 g, and the presented results (see Fig. 1) are the mean values of five measurements. The low-cycle fatigue tests were performed under total strain control using a computer controlled servo-hydraulic 100 kN Instron machine at room temperature. The strain was measured using an extensometer with a gauge length of 25 mm on round specimens with a diameter of 12 mm. A symmetric push–pull mode with a sinusoidal waveform and cyclic strain rates from 6 · 104 up to 1.8 · 103 s1 were applied. The tests were stopped when 20% reduction of the stress amplitude was obtained. In order to investigate the elastoplastic deformation mechanism, disks with a diameter of 3 mm were taken from both the fatigue-tested specimens and the as-received mateTable 2 Mechanical properties of the materials used Materials

Conditions

rYS (MPa)

rTS (MPa)

A (%)

Hv_0,1 a

c

UNS S32750 UNS S32750 UNS S32906

As-delivered, 2507AD As-aged, 2507HT

566

836

41.6

286

272

688

896

39.8

316

275

708

931

39.7

265

298

As-delivered, 2906AD

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Fig. 1. Microhardness vs. strain: (a) 2507AD, (b) 2507HT and (c) 2906AD.

rial for transmission electron microscopy (TEM) investigation. Thin foils were polished using an electrolyte consisting of 950 ml acetic acid and 50 ml perchloride acid subjected to 32 V and 22 C. In order to reduce the risk of surface absorption of carbon, the fresh thin foils were immediately inserted into the sample holder and analyzed. The dislocation structures were studied using a JEOL 2000-FX analytical transmission electron microscope (TEM/STEM) operating at 200 kV. The slip band localization was investigated using scanning electron microscopy (SEM). 3. Multiscale material modeling 3.1. Grain structure model The two-dimensional grain structure model representing the DSSs is generated using a Voronoi polygonization algorithm [17]. The same grain structure model is used for all three materials, i.e., it is assumed that the grain structure is not influenced by the heat treatment (2507HT) or the addition of nitrogen (2906AD). In

Fig. 2. Grain structure model of the austenitic–ferritic DSS. The grain structure consists of 64 grains where the dark gray grains are ferritic and the light gray grains are austenitic.

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Fig. 2, the generated grain structure model with the ferrite and austenite grains are shown. The grain structure model consists totally of 64 grains and is generated in such a way that the ferrite phase, which occupies 45% of the volume, is continuous in the grain structure. Crystallographic directions are also randomly assigned to the grains whereby the global directions of the slip systems for a grain can be determined. 3.2. Crystal plasticity model

where sy is the initial yield stress, ba is an integrity measure which will be defined later, and ja is the hardening stress which follows the evolution law j_ a ¼

Nr:slip X

ðq þ ð1  qÞdab Þhb ðAb Þ_cb

with ja ð0Þ ¼ 0

ð7Þ

b¼1

Here, q is a parameter which determines the amount of cross-hardening and dab is the Kronecker delta. The hardening function ha(Aa) is suggested by Miehe et al. (see Ref. [13]):

The crystal plasticity model is thoroughly described in Ref. [18] and therefore only the main ideas are summarized here. Note that the crystal plasticity model represents the material behavior in a material point inside one grain. First, the large deformation kinematics in the model must be defined. The classical assumption in crystal plasticity is that the plastic deformation gradient Fp only describes the plastic slip in the slip systems and that it leaves the crystal lattice undistorted. The crystal lattice distortion is described by the elastic deformation gradient Fe. In accordance with this, the deformation gradient is multiplicatively decomposed as

ha ðAa Þ ¼ h0 þ ðh1  h0 Þð1  eðnAa Þ Þ

F ¼ Fe Fp

where g determines the evolution rate and m determines the nonlinearity of the evolution. The integrity measure will represent the loss of friction in the slip system. The evolution for the plastic and damage deformation gradients are chosen as

ð1Þ

In order to introduce softening/damage into the model, the ‘‘damage deformation gradient’’ Fd is also introduced. The damage deformation gradient represents the loss of integrity compared to the undamaged material and defines b e via the effective elastic right Cauchy–Green tensor C b e ¼ FdT Ce Fd C

e

eT

e

where C ¼ F F

ð4Þ

Note that the Cauchy stress and the second Piola–Kirchhoff type stress are related by 1  e e eT  FSF r¼ ð5Þ detðFÞ Furthermore, the yield function based on the Schmid stress is introduced in each slip system a as /a ¼

jsa j  ðsy þ ja Þ ba

ð6Þ

with

ðq þ ð1  qÞdab Þ_cb

ð8Þ

b¼1

Hence, h0 is the initial hardening modulus, h1 determines the saturation of the hardening and n determines the rate at which the hardening saturates. For the evolution of the integrity measure introduced in the yield function given by Eq. (6), the following format is proposed sa b_ a ¼ _ca g m with ba ð0Þ ¼ 1 ð9Þ ba

1 F_ p  Fp ¼

ð2Þ

The effective elastic Cauchy–Green tensor determines the second Piola–Kirchhoff type stress tensor Se via the elastic part of the Helmholz free energy function: def b e be Se ¼ o W ð C Þ. Here, a neo-Hooke type of elastic free energy Ce ob is chosen (see Ref. [19]), b e Þ  2 lnðdetð C b e Þ ¼ G ðtrð C b e ÞÞ  3Þ b eðC W 2 k b e Þ2  2 lnðdetð C b e ÞÞ  1Þ ð3Þ þ ðdetð C 4 where G and k are the Lame´ constants. The plastic slip ca in the undistorted slip system (sa, ma) is driven by the resolved shear (Schmid) stress that is obtained as sa ¼ ðCe  Se Þ : ðsa  ma Þ:

A_ a ¼

Nr:slip X

Nr:slip X

ðsb  mb Þ_cb signðsb Þ

with Fp ð0Þ ¼ d

ð10Þ

ðsb  mb Þb_ b signðsb Þ

with Fd ð0Þ ¼ d

ð11Þ

b¼1 1 F_ d  Fd ¼

Nr:slip X b¼1

where d is the Kronecker delta. A viscoplastic format of Perzyna type is adopted for the evolution of the plastic slip:  n  1 h/a i def 1 c_ a ð/a Þ ¼ þ1 1 with hxi ¼ ðx þ jxjÞ t rya þ ja 2 ð12Þ where the relaxation time t* is chosen sufficiently small that a rate-independent response is obtained and n is the creep exponent. Hence, viscosity is only introduced for numerical regularization. Finally, the pertinent material parameter values and the slip systems must be chosen in order to model the different steels. The material parameter values in Table 3 are used for simulating the behavior of the 2507AD steel. The values have been chosen in order to mimic qualitatively the experimentally found cyclic stress– strain behavior of this steel. Furthermore, the hardening parameters for the austenite and ferrite phases were chosen such that the hardening is largest in the austenite. This choice is motivated by the microhardness measurements (see Fig. 1) which clearly show this behavior. For the softening part, the ferrite phase was thought to have a larger softening than the

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Table 3 Material parameter values for 2507AD used in the simulations

Austenite Ferrite

G (GPa)

k (GPa)

sy (MPa)

q

n

h0 (MPa)

h1 (MPa)

g

m

n

t* (s)

71 71

106 106

210 250

0.1 1.1

75 50

4000 100

0 0

0.0001 0.001

1.0 1.0

1.0 1.0

Small Small

austenite. This is in accordance with the findings in Ref. [20]. In order to include the effect of the heat treatment for the 2507HT steel, the yield stress in the ferrite phase is increased to sy = 300 MPa compared with 250 MPa for the 2507AD steel (see Table 3). Note that the yield stress of the austenite in the 2507HT steel is assumed to be the same as that in the 2507AD steel. For the 2906AD steel, the effect of the addition of nitrogen is modeled by increasing the yield stress of the austenite to sy = 310 MPa and decreasing the yield stress of the ferrite to sy = 220 MPa. The choices for the numerical values of the yield stresses are motivated by the microhardness measurements given in Table 2. If 2507AD is considered to be the unaltered steel, then we can conclude that in the 2507HT steel it is only the initial hardness in the ferrite phase that is changed by the heat treatment. However, for the 2906AD steel, the initial hardness of the ferrite decreases while that of the austenite increases, which motivates the choices of the changes in the yield stresses of both the austenite and the ferrite. The slip systems are given by the atomic packing scheme. In DSSs the austenite has a face centered cubic crystal structure characterized by the {1 1 1} slip planes and the h1 1 0i slip directions which gives a total of 12 slip systems. The ferrite in DSSs have a body centered cubic crystal structure from which the {1 1 0} slip planes and the h 1 1 1i slip directions are chosen, leading to a total of 12 slip systems. 3.3. Computational homogenization

Fig. 3. The pertinent experimental set up of the LCF tests with a prescribed strain. The underlying phase/grain structure is identified and a grain structure model is generated and discretized using CST elements.

The stress–strain state in the grain structure model is motivated from the fact that the performed experiments are uniaxial LCF tests (see Fig. 3), so a state of macroscale uniaxial stress is said to prevail in the grain structure, i.e., P yy ¼ P zz ¼ 0

ð17Þ

Furthermore, the deformation is prescribed as a uniaxial linear displacement on the boundary C of the grain structure model, i.e., u ¼ ðF  IÞX;

8 X on C with

In computational homogenization the volume average integrals relating the macroscale first Piola–Kirchhoff stress P and deformation gradient F to the corresponding microscale first Piola–Kirchhoff stress P and deformation gradient F are given by Z 1 P¼ P dV ð13Þ V X Z 1 F¼ F dV ð14Þ V X

F xx ¼ 1 þ exx

In practice, these integrals are computed using the finite element method. The generated grain structure model is discretized using CST elements (see Fig. 3) and the equilibrium equation

Fig. 4a–c shows the experimental and simulated cyclic stress–strain behavior with an applied strain amplitude of 1%. The simulated responses are qualitatively similar to the experimental responses although the initial softening of the 2906AD steel is underestimated in the simulations. All three steels show cyclic hardening in the first three cycles followed by cyclic softening. The experimental results show that the steel with the highest maximum stress, i.e., 2906AD, also shows the highest softening rate. Comparing 2507AD and 2507HT, we note that 2507AD, which has the smallest maximum stress, shows a slightly higher softening rate than 2507HT.

divðPÞ ¼ 0

ð15Þ

is solved in the domain of the grain structure model (X). Note that the macroscale first Piola–Kirchhoff and Cauchy stress tensors are related via ¼ r

1 PFT detðFÞ

ð16Þ

and

F ij ¼ 0; i 6¼ j i; j ¼ x; y; z

ð18Þ

where I is the identity tensor of the same dimension as F. The numerical algorithm that is used to fulfill Eq. (17) is described in detail in Ref. [21]. 4. Results 4.1. Cyclic stress–strain behavior

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Fig. 4. Experimental and simulated stress–strain response curves for the 2507AD, 2507HT and 2906AD DSS with De/2 = 1% up to 30 cycles: (a) 2507AD, (b) 2507HT and (c) 2906AD.

Because multiscale material modeling is used, it is possible to study the cyclic stress–strain behavior of the pure austenite and ferrite phases by volume averaging over the pure phases. In Fig. 4a–c, the simulated cyclic stress–strain behavior is also shown for the pure phases of the three different steels. For 2507AD as well as for 2507HT the austenite is the softest phase initially but becomes the hardest phase during the cyclic loading. However, the initial difference in the maximum stress in the first cycle is larger for 2507HT than for 2507AD, which leads to a delay of the transition point until loading cycle 18. In 2906AD the austenite is the harder of the phases, both initially and during the subsequent cyclic deformation. 4.2. Slip characteristics The SEM investigations show that the appearance of the dislocation slip bands is quite different in the three steels (Figs. 5a, 6a and 7a). For 2507AD, slip bands appear mainly in the ferrite phase, but they are also observed in small parts of the austenite phase (Fig. 5a). For 2507HT, the slip bands are observed in both the ferrite and austenite phases (Fig. 6a). For 2906AD, the slip bands are almost

exclusively found in the ferrite phase (Fig. 7a). By assuming that a large amount of plastic slip in the model corresponds to the observation of slip bands, the total accumulated plastic slip vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uNr:slip Z T 2 uX t ~c ¼ ð19Þ c_ b dt b¼1

0

is introduced, where c_ b is obtained from Eq. (12). This measure is then used to color the grain structure model such that the simulations and SEM results can be compared qualitatively. Figs. 5b, 6b and 7b show the total accumulated slips in the grain structures for the three different materials. The simulated results show that the amount of the total accumulated plastic slip is somewhat larger in the austenite phase of the 2507HT steel than in the 2507AD steel. In the 2906AD steel, the accumulated plastic slip is almost exclusively situated in the ferrite phase. 4.3. Dislocation structures Fig. 8a shows a TEM bright-field image of the 2507AD steel. As can be seen, the austenite phase in 2507AD has a

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Fig. 5. Fatigue damage in 2507AD steel: (a) SEM picture shows the slip bands formed during cyclic loading at De/2 = 1% and (b) simulation of the total accumulated plastic slip after cycle 30. The darkness corresponds to the amount of total accumulated plastic slip.

Fig. 6. Fatigue damage in 2507HT steel: (a) SEM picture shows the slip bands formed during cyclic loading at De/2 = 1% and (b) simulation of the total accumulated plastic slip after cycle 30. The darkness corresponds to the amount of total accumulated plastic slip.

Fig. 7. Fatigue damage in 2906AD steel: (a) SEM picture shows the slip bands formed during cyclic loading at De/2 = 1% and (b) simulation of the total accumulated plastic slip after cycle 30. The darkness corresponds to the amount of total accumulated plastic slip.

planar array dislocation structure with relatively low dislocation density. The ferrite phase in the same steel shows dense dislocations with a near parallel dislocation wall structure, as shown in Fig. 8b. The austenite phase in the

2507HT steel showed micro twins and planar array dislocation structures (see Fig. 9a). However, the dislocation density is much higher than that of the austenite phase in the 2507AD steel. The ferrite phase in the 2507HT steel

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Fig. 8. TEM photographs of the 2507AD steel after cycle 1807 showing: (a) planar array dislocations in the austenite g = (1 1 1)c and (b) dislocation channels in the ferrite g = (0 0 2)a.

Fig. 9. TEM photographs of the 2507HT steel after cycle 1058 showing: (a) micro twins in the austenite and (b) linear dislocations in the ferrite g ¼ ð 1 0 1Þa.

Fig. 10. TEM photographs of the 2906AD steel after cycle 2748 showing: (a) planar array dislocations in the austenite g = (1 1 1)c or ð1 1 1Þc and (b) cell structures in the ferrite.

showed dense dislocation walls with dense dislocations between them (see Fig. 9b). It is noteworthy that the morphology of the walls appears very different from the one observed for the 2507AD steel. However, these two were from different crystallographic orientations and the images

were obtained from different beams. The austenite phase in 2906AD also shows planar array dislocation structures with high dislocation density (see Fig. 10a). The distribution of dislocations in this sample is not homogenous as in the 2507AD steel. In the ferrite phase, however, small

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subcell structures were observed with the size of 1 lm. These are considerably smaller than the original size (see Fig. 10b). A careful analysis of the electron diffraction, with small splitting in diffracted spots, revealed that it was ferrite grain with very small angle grain boundaries. 5. Discussion 5.1. Cyclic stress–strain behavior Although all three steels consist of austenite and ferrite phases, their cyclic stress–strain behaviors are different. The dissimilarity is caused by the fact that the elastoplastic properties of the pure austenite and ferrite phases are different between the three steels. These changes will alter the behavior both directly via an influence on the hardening/softening behavior of the DSSs and indirectly via changing the load sharing between the phases. From the microhardness measurements (see Fig. 1), it is noted that the hardening of the austenite is more rapid than that of the ferrite. Hence, it is possible that the initially softer austenite phase can become the harder of the two phases, depending on the applied strain and the initial difference in the hardness between the phases. The heat treatment performed on the 2507HT steel increases the initial yield stress (hardness) of the ferrite phase such that a larger difference between the initial yield stresses of the ferrite and austenite phase is obtained compared with the 2507AD steel. Hence, the transition point where the austenite becomes the harder of the two phases is delayed (see Fig. 4a and b), and the austenite will initially experience most of the plastic deformations. This leads to an evolution of the hardening/softening behavior of the austenite. After the transition point, the ferrite will experience most of the plastic deformations, and hence the hardening/softening behavior in the ferrite will evolve. Since much of the evolution of the hardening/softening behavior takes place in the ferrite, this will dominate the evolution of the 2507HT steel. However, there will also be an influence of the initial evolution of the hardening/softening behavior in the austenite. In the 2507AD steel, the difference of the initial hardness and therefore also the initial yield stress between the two phases is small. Since the hardening rate of the austenite is larger than that of the ferrite, it becomes the harder of the two phases almost immediately, which leads to the ferrite experiencing most of the plastic deformations (see Fig. 4a). Hence, the hardening/softening behavior of the ferrite evolves and this evolution dominates the behavior of the 2507AD steel. The influence of the austenite in 2507AD is smaller than in the 2507HT steel. The nitrogen content in the 2906AD steel is higher than in the 2507AD steel. From the measurements of the initial hardness given in Table 2, it can be seen that the initial hardness of the ferrite decreases and that of the austenite increases, so that austenite is the harder of the two phases initially. Since the austenite also has a higher hardening

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rate than the ferrite, it will always be the harder of the two phases (see Fig. 4c). Therefore, the ferrite will experience almost all of the plastic deformations and the hardening/softening behavior will be almost exclusively determined by the evolution of the hardening/softening of the ferrite. There will also be a direct effect of the increase in the nitrogen content. According to Ref. [4], the nitrogen dissolves almost exclusively in the austenite phase and it promotes planar slip in the austenite, thus inhibiting cross-slip. This will lead to the fact that the cyclic deformation is less localized in persistent slip bands. Therefore, the reversibility of slip is increased, leading to a higher softening rate in the austenite. It should be noted that the hardening/softening parameters are not changed between the three steels in the simulations. The only difference between the three steels is that the initial yield stresses for the pure phases differs. 5.2. Slip characteristics and dislocation structures The SEM investigation of the slip bands relates to the discussion of the hardening/softening behavior above if the formation of slip bands and extensive plastic deformation are correlated. In the 2507AD steel, the slip bands concentrate in the ferrite phase although some small parts of the austenite phase also show slip bands. In the heat-treated steel 2507HT, the slip bands again occur in the ferrite, although compared with the 2507AD steel larger parts of the austenite phase also show slip bands. The 2906AD steel shows slip bands almost exclusively in the ferrite phase. It is thus evident that it is the softest of the phases, i.e., the phase with the highest amount of plastic deformations, which shows the slip bands. Furthermore, it is also noted that the total accumulated plastic slip (see Eq. (19)) could give a clue as to which phase will develop most slip bands, since these results and the SEM investigation are similar to each other. Further, by studying the dislocation structures in the individual austenite and ferrite phases it is concluded that the austenite phase will always form a planar dislocation structure. However, the dislocation density is strongly dependent on the amount of plastic deformation in the austenite phase (see Figs. 8a, 9a and 10a). The austenite in the 2507AD steel undergoes a smaller amount of plastic deformation (according to the discussion above) as compared with the 2507HT steel. In addition, the dislocation density is lower in this steel than in the 2507HT steel. The large plastic deformation of the austenite in the 2507HT steel may also be explained by the formation of micro twins, which is seen in Fig. 10a (see also Ref. [5]). Although the austenite phase in 2906AD undergoes small plastic deformations, the high nitrogen content in the austenite phase can promote the formation of planar dislocation structures. Therefore, the dislocation density is quite high in the 2906AD steel, and this can be seen in Fig. 10a, where some planar dislocation structures with high dislocation density in two slip directions are shown.

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Kruml and Polak [22] have shown that the dislocation structures in the ferrite are strongly dependent on the cyclic loading and change from no clear partial dislocation to dislocation channels and subcell structure with increasing cyclic loading. The results from the present investigation show similar dislocation structures in the ferrite phases (see Figs. 8b, 9b and 10b). Here the loading is kept the same whereas the elastoplastic properties of the materials are changed. In addition, the chemical composition and material condition between the three materials are different and this may affect the formation of dislocation structures. However, it seems that the main effect in promoting the different dislocation structures is the load sharing between the phases. This will affect the amount of plastic deformation experienced by the ferrite. The dislocation structures will evolve from dislocation walls in 2507HT to dislocation channels in 2507AD and cell structures in 2906AD. This is due to the increasing plastic deformation in the ferrite phase between the three materials. 6. Conclusions Experimental investigations and simulations using multiscale material modeling have been carried out in order to study the cyclic stress–strain behavior of the austenite and ferrite phases and the load sharing between the phases in three DSSs. The following main conclusions are drawn. The hardening rate of the austenite is higher than that of the ferrite. This can lead to a transition point where the austenite becomes the harder of the two phases if the austenite is the softer of the two phases initially. The location of the transition point depends on the applied strain and the elastoplastic properties of the two phases. The load sharing between the phases in the three materials causes different hardening/softening behavior of the DSSs. Hence, the softer of the two phases will experience most of the plastic deformations and therefore most of the slip bands will form in this phase and the hardening/ softening behavior will be dominated by this phase. The dislocation structures obtained in the austenite and ferrite phases in the three materials depend strongly on the elastoplastic properties and the load sharing between the phases. The austenite phase will always show a planar dislocation structure but with different dislocation density. However, the ferrite will show dislocation channels, walls and cells depending on the amount of plastic deformation it experiences.

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