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Keywords:

  • cast duplex stainless steel;
  • plastic strain localization;
  • micropillar compression;
  • finite element method

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

Microstructure-based finite element modeling was performed on different microstructures of an austenitic-ferritic cast duplex stainless steel using the constitutive behavior of individual phase obtained from micropillar compression tests. A qualitative analysis of equivalent plastic strain and von Mises stress was conducted in plane stress and plane strain conditions. The simulated results reveal that the morphology and the area fraction of the second phase can affect the mechanical properties. The stress values and the equivalent plastic strain in the shear bands are higher in plane strain loading than plane stress loading condition. Stress is concentrated mostly in ferrite phase. The evolution of stress during deformation is found to be dependent on the morphologies of ferrite phase in austenite matrix.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

Modern metallic alloys having polycrystalline multi-phase microstructures can provide excellent mechanical properties.[1-3] An example is the austenitic-ferritic stainless steel which consists of a body-centered cubic ferrite and a face-centered cubic austenite phase. The combined properties of the individual phases in austenitic-ferritic stainless steel results in excellent corrosion resistance and mechanical properties, and thus, is replacing the conventional austenitic or ferritic single-phase stainless steel in a wide range of industrial applications, especially in chemical, petrochemical, and nuclear industries.[1-3]

Previous experimental results reveal that the flow behavior of austenitic-ferritic stainless steel relies on the intrinsic mechanical properties of the austenite and ferrite phases.[2, 4] It has also been reported that factors such as the volume fraction of each phase, morphological and crystallographic texture in ferrite and austenite, as well as the partitioning of the stress and strain between phases can also affect the bulk mechanical properties.[4-7] Recently, finite element models have been applied to predict the flow behavior of dual-phase steel,[8-13] duplex stainless steel[2, 14] and ferritic-pearlitic steel.[15] In these studies, the effect of uniform and banded microstructures[9, 10] and different applied boundary conditions[9, 10, 13] on the flow behavior were revealed. However, in their models, the flow curves of individual phases were obtained using crystal plasticity theory,[2] dislocation based strain hardening approaches[9, 13, 15] and GTN (Gurson–Tvergaard–Needleman) modeling.[14] Microstructure-based modeling was also used to predict the mechanical behavior of particle reinforced composites[16, 17] and sintered steels.[18] However, very few finite element models have used the flow behaviors of individual phase obtained from experiments. Sun et al.[10, 11] used high energy X-ray diffraction measurement to obtain the flow behavior of each phase to use in their finite element models. Micropillar compression is an excellent technique to obtain the stress-strain behavior of phases at small length scale.[19-25] For example, the stress strain behavior of the single crystal ferrite and martensite was studied by Stewart et al.[21] using pillar compression and was used to predict the mechanical properties of the bulk material using analytical solutions.

In this study, we aim to explore a modeling approach for predicting the effect of various microstructures on the deformation behavior of austenitic-ferritic stainless steel under uniaxial tension. The distribution and also the evolution of the strain and stress between the different phases are of our great concern. Two-dimensional microstructures, taken from optical microscopy images, were used as a framework for modeling by the finite element method (FEM). More importantly, the flow behavior of austenite and ferrite was determined from micropillar compression tests and used as an initial constitutive relationship for the individual phase in the FEM model.

2 Materials and Experimental Procedure

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

The material studied in this paper is a commercially available austenitic-ferritic cast duplex stainless steel (Z3CN20-09M). The chemical composition of this material, obtained by electron probe microscope analyzer (EPMA, JXA-8230, JEOL), is summarized in Table 1 (atomic %). The materials were machined from a 105 mm thick centrifugally cast pipe used in pressurized water reactor coolant system piping in nuclear electric power plant. The pipe was heat treated at 1105 ± 10°C for 4.5 h and then water quenched.

Table 1. Chemical composition of a Z3CN20-09M cast duplex stainless steel (atomic %)
ElementsCrNiMnSiFe
Bulk22.997.910.791.53Bal.
Austenite22.078.460.821.50Bal.
Ferrite28.214.810.601.50Bal.

The microstructures of the samples were revealed by standard polishing followed by etching using a solution of 25 mL HCl, 25 mL HNO3 and 50 mL H2O. Optical microscopy (OLYMPUS BX51M) was used to obtain the microstructures. The area fraction of ferrite was determined from thresholded images and analyzed by image analysis software (Image J, Bethesda. MD).

Micropillar samples were polished to a 1 μm diamond finish and then to a final finish of 0.05 μm colloidal silica. Micropillars were fabricated in both ferrite and austenite phases using a focused ion beam (Nova 200 NanoLab FEG-SEM/FIB, FEI Co., OR). The ion beam current was set at 20 nA for coarse milling at the beginning, and was gradually reduced to 0.1 nA. By the standard milling approach,[20] pillars with taper angle of 2° to 5° were obtained. The “ion-lathe” technique was used to minimize the taper of the pillars.[20] The final pillars had diameters of ∼1.0 µm at the top surface, heights of ∼3 µm and a slight taper of 2°–3°.

The micropillars were compressed using a nanoindenter (MTS XP, Agilent Technologies, AZ). A four-sided pyramid diamond indenter (Micro Star Tech., Huntsville, TX) with a flat square cross-section (10 µm side) was used. The sample was mounted on aluminum stub for testing using a mounting adhesive (Crystalbond, West Chester, PA). A continuous stiffness measurement (CSM) technique was used in all tests,[26] where the continuous measurement of contact stiffness was measured instantaneously as a function of depth. The compression testing was conducted at a constant displacement rate, 10 nm/s, corresponding to an initial strain rate ∼3 × 10−3/s. In total, 4–5 pillars of each phase were compressed. The diameter at the top surface was used to calculate the stress values.

3 Microstructure-Based Modeling

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

The optical microscope images were first automatically segmented into two different phases using an image processing software (ImageJ). Then, the segmented images were imported to the Rastervect software package and converted to vectorial format suitable for input to a commercial finite element package. The vectorized lines were next imported to ABAQUS FEM software.[27]

The elastic properties of the constituent phases were obtained by nanoindentation, while the plastic properties were determined by micropillar compression. The reason for this is that in the micropillar experiment some degree of compliance exists in the elastic portion, which is difficult to subtract. Nanoindentation is more straightforward for obtaining the Young's modulus of the phases. The young's modulus of austenite and ferrite were measured as 200 ± 8 and 221 ± 5 GPa, respectively.[28] The data was obtained from more than 15 tests for each phase and for each test, the young's modulus value from the displacement range of 300–800 nm was averaged.

In this model, only two phases are considered to be present and perfect bonding is assumed between the two phases. In addition, damage in the form of cracking, void growth, etc., is not considered. In reality, the material would also deform by multiple slip that could be quantified by a more complex crystal plasticity model. In these simulations we are simply interested in the microstructural morphological and the distribution effects on local stresses and strains. The evolution of strain and stress between the ferrite and austenite phase is also one of our great concerns. More importantly, we wish to elucidate the use of the local phases' constitutive behavior as determined by micropillar compression.

In all the simulations, explicit models are adopted to investigate the strain and stress evolution between austenite and ferrite phases. The loading direction was applied to the right edge of the model and the same overall strain value, around 8.4%, was given for all the models. Both, two-dimensional generalized plane stress and plane strain loading conditions were adopted. The plane strain loading condition is similar to the loading condition in the center part of a thick specimen where the strain in the thickness direction is zero. The plane stress loading condition is more representative of what takes place at the surface of the specimen, i.e. where the stress normal to the free surface is zero. For each loading condition, the nodes of the left edge were confined to have no displacement in x direction but can move freely in the y direction.

4 Results and Discussion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

4.1 Microstructures of the Cast Duplex Stainless Steel

The microstructures of the cast pipe contain δ-ferrite in an austenite matrix are shown in Figure 1. Due to complicated solidification conditions in different parts of the pipe and also the subsequent complex phase transformation of ferrite to austenite during solidification, microstructures with different morphologies were found in the samples and two typical microstructures are shown. Figure 1a shows the typical dendrites of ferrite in austenite matrix and Figure 1b shows banded morphology of ferrite. The area fraction of ferrite is measured to be 17.5% and 19.3% for Figure 1a and b, respectively.

image

Figure 1. Microstructures showing different features: (a) complex dendrite of ferrite; (b) banded ferrite.

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4.2 Constitutive Relationships of Austenite and Ferrite by Micro-Compression Tests

Examples of the fabricated pillars of ferrite and austenite phases are shown in Figure 2a and b, respectively. Figure 2c is the representative stress-strain curves of ferrite and austenite phases obtained from micropillar compression. The results show that ferrite has higher strength than austenite. The higher stress values in ferrite have also been reported in a neutron diffraction study of a similar duplex stainless steel.[7] It can be noticed here that the stress values of both austenite and ferrite are slightly higher than the bulk material.[29] We believe that this relatively high stress is due to the relatively small volume of the pillar used in this study. The high values of stress due to the so-called size effect have been reported widely in micropillar compression tests.[19, 20, 24] The smaller strain values reported here might be due the activation of the single slip system with multiple slip bands leading to the large flat region in the stress-strain curves.[30, 31] As mentioned above, the goal of this paper is to show the microstructural influences on local stresses and strains, and that stress-strain data for the microstructures can be obtained from micropillar compression experiments for input as constitutive behavior in FEM models.

image

Figure 2. Fabricated micropillars of (a) ferrite and (b) austenite; (c) stress-strain curves of austenite and ferrite phases obtained by micropillar compression tests.

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4.3 Simulation Results of Different Microstructures

Figure 3a and b show the vectorized images of Figure 1a and b, respectively. Finite element discretization is generated on the vectorized image, as shown in Figure 3c. A study on the effect of element size on the convergence of results was carried out. Element sizes of 1, 1.5, and 3.5 with a total number of 83801, 50285, and 34105 elements, respectively, were chosen. The simulated results for the three groups were found to be similar for the models with different element sizes. The mesh size of 1.5 was adopted in all the simulations.

image

Figure 3. (a, b) Vectorized images, and (c) finite elements of two phases for microstructure (a); (d) detailed finite elements in the area as indicated in (c).

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Figure 4 shows the simulated flow curves of different microstructures as shown in Figure 1 in plane stress and plane strain loading conditions. We notice that stress values are a little higher compared to our previous experimental results conducted on the bulk material.[29] This might have been caused due to slightly high values of stress for both ferrite and austenite obtained from micropillar compression tests. In bulk material, deformation could be compensated by the multiple slip bands and the migration and rotation of grain boundaries. The inclusions in the material can also result in early failure during deformation which is not considered in the models. The simulated curves also show that the stress is higher in plane strain than in plane stress loading condition in both banded and dendritic microstructures. The flow curves also show that the stress in banded microstructure is a little higher than in the dendrite microstructure. This might be attributed to the higher area fraction of ferrite in banded microstructure (19.3%) than in dendritic microstructure (17.5%).

image

Figure 4. (a) Simulated engineering stress-strain curves for the dendrite and banded microstructures under tensile loading in the x direction; (b) the detailed information of the region as indicated in (a).

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Figure 5 and 6 show the evolution of equivalent plastic strain and von Mises stress in the microstructures at various strain values. The results are shown only for plane stress loading condition. For both microstructures, detailed examination of the deformation process during simulation indicates that plastic strain localization occurs in the austenite phase first, close to the inter-phase boundaries, as shown in Figure 5a and Figure 6a. The strain localization in the inter-phase boundaries might be caused due to incompatible deformation between the phases.[10, 13] The deformation bands in austenite seem to distribute along the long shaped ferrite grains. Although the bonding between austenite and ferrite is assumed in this study, for ductile materials, the predicted sites for strain localization can potentially be viewed as the sites for void nucleation. Decohesion is also likely to take place in the inter-phase boundaries. The decohesion phenomenon has already been observed experimentally in our previous study in this material.[29] The simulation results also indicate that yielding occurs mainly in the austenite phase during the early stages of the deformation process while ferrite exhibits much less deformation. The strain localization starts at the left edge for dendritic microstructure, whereas it starts at multiple positions for banded microstructure. After strain localization, the strain spreads out along the shear bands observed in Figure 5 and 6, which will lead to final fracture. One might notice that the shear bands are likely to propagate along the long shaped ferrite grains as indicated in Figure 5b and 6b. It is interesting to note that the stress starts to concentrate in the ferrite arms that are parallel to the loading direction in the initial stage and then spreads out to the other arms eventually as shown in Figure 5e–h. This inhomogeneous stress distribution in microstructure was also observed in a study of microstructured-based modeling of Ag3Sn phase in solder which shows that needle-like Ag3Sn phase along the loading axis carries more load.[32] The distribution of the von Mises stress in ferrite in dendritic microstructure is not observed obviously in banded microstructure due to fewer branches in the banded ferrite. But one could expect that the banded ferrite phase would carry more load when parallel loading is applied during the tension process as has already been revealed by previous study.[32] The way of stress distribution in different arms in ferrite means the distribution of the ferrite (or the loading direction) may affect the stress concentration in ferrite and finally, affect the deformation behavior.

image

Figure 5. Evolution of equivalent plastic strain and von Mises stress in the dendrite microstructure. The average strain values are (a) 1.0%, (b) 1.3%, (c) 1.7%, (d) 2.6%, (e) 0.4%, (f) 0.9%, (g) 1.7%, and (h) 8.4%.

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image

Figure 6. Evolution of equivalent plastic strain and von Mises stress in the banded microstructure. The average strain values are (a) 1.0%, (b) 1.3%, (c) 1.7%, (d) 2.6%, (e) 0.4%, (f) 0.9%, (g) 1.7%, and (h) 8.4%.

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For comparison, the distribution of von Misses stress and equivalent plastic strain for plane stress and plane strain loading conditions at an average strain value of 8.4% is shown in Figure 7 and 8. Careful examinations reveal that the stress values of ferrite phase are slightly lower in the case of plane stress loading condition (Figure 7a and b) than the plane strain loading condition (Figure 7c and d). It is interesting to note that the overall stress is higher in plane strain as shown in Figure 4. This might be attributed to the relatively higher stress of ferrite in plane strain than in plane stress loading. Figure 8 clearly shows strain localization along few slip bands, which can result in final fracture. It is also evident that the extent of strain localization is more or the strain value in the strain bands is higher in plane strain loading (Figure 8c and d) than in plane stress loading (Figure 8a and 8b), due to the enhancement in triaxiality of stress in plane strain, i.e. there is more constraint in the material.

image

Figure 7. Distribution of von Mises stress between phases under: (a, b) plane stress loading, and (c, d) plane strain loading conditions. The strain level is around 8.4%.

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image

Figure 8. Distribution of plastic strain under: (a, b) plane stress loading, and (c, d) plane strain loading conditions. The strain level is around 8.4%.

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5 Conclusions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

In the present study, we adopted the finite element modeling approach by using the flow curves for individual phase obtained from pillar compression testing and applied this model to an austenitic-ferritic stainless steel. Based on this study, the following conclusions are obtained:

  1. The microstructural features, such as the morphology and the distribution, can affect the plastic strain localization. Deformation bands in austenite are more likely to propagate along the long shaped ferrite grains.
  2. The predicted flow curves show higher stress values in plane strain loading than in plane stress loading condition.
  3. Stress tends to concentrate in the ferrite arms that are parallel to the loading direction than those that are perpendicular to the loading direction.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References

The financial support from National Science and Technology Major Project of the Ministry of Science and Technology of China, under Project Nos. 2011ZX04014-052 and 2012ZX04012011, is greatly appreciated. En-Yu Guo also thanks the Chinese Scholarship Council for financial support during his stay at Arizona State University (USA). Helpful discussions with Carl Mayer at Arizona State University on pillar compression are also appreciated.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Materials and Experimental Procedure
  5. 3 Microstructure-Based Modeling
  6. 4 Results and Discussion
  7. 5 Conclusions
  8. Acknowledgments
  9. References