- Email: [email protected]
Modeling of Advanced High Strength Steels with the realistic microstructure–strength relationships
Computational Materials Science 45 (2009) 860–866
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Modeling of Advanced High Strength Steels with the realistic microstructure–strength relationships S.A. Asgari a,*, P.D. Hodgson a, C. Yang b, B.F. Rolfe b a b
Centre for Material and Fibre Innovation (CMFI), Geelong Technology Precinct (GTP), Waurn Ponds Campus, Deakin University, Geelong 3217, Victoria, Australia School of Engineering and Information Technology, Deakin University, Waurn Ponds 3217, Victoria, Australia
a r t i c l e
i n f o
Article history: Received 26 June 2008 Received in revised form 28 November 2008 Accepted 5 December 2008 Available online 18 January 2009 PACS: 46.15.-X 02.70.-C 07.05.Tp Keywords: Advanced High Strength Steels Realistic microstructure Strength
a b s t r a c t The objective of the work is to consider the first-order effects of the realistic microstructure morphology in the macroscale modeling of the multiphase Advanced High Strength Steels (AHSS). Instead of using constitutive equations at macroscale, the strength–microstructure relationship is studied in the forms of micromechanical and multiscale models that do not make considerable simplifications with regard to the microscale geometry and topology. The trade-off between the higher computational time and the higher accuracy has been offset with a stochastic approach in the construction of the microscale models. The multiphase composite effects of AHSS microstructure is considered in realistic microstructural models that are stochastically built from AHSS micrographs. Computational homogenization routines are used to couple micro and macroscale and resultant stress–strain relations are compared for models built with the simplified and idealized geometries of the microstructure. The results from this study show that using a realistic representation of the microstructure, either for DP or TRIP steel, could improve the accuracy of the predicted stress and strain distribution. The resultant globally averaged effective stress and strain fields from realistic microstructure model were able to accurately capture the onset of the plastic instability in the DP steel. It is shown that the macroscale mechanical behavior is directly affected by the level of complexities in the microscale models. Therefore, greater accuracy could be achieved if these stochastic realistic microstructures are used at the microscale models. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Advanced High Strength Steels (AHSS) are multiphase materials that are of great importance in the automotive industry for the purposes of strengthening the vehicle structure and decreasing the vehicle weight. Whereas the numerical modeling codes currently used for forming, springback and crash modeling generally represent the sheet as a homogenous material, uniform in thickness and material properties. These steels are generally characterized by their complex microstructure that has a first-order effect on the strength, ductility and formability. With the increased interest in using AHSS, there is a growing need to take into account explicitly the microscale mechanisms such as composite effects of the multiphase microstructure, deformation induced phase transformation and history dependent large deformation in forming, springback and crash simulations. Over the past few decades, numerous micromechanical models and multiscale techniques have emerged in the literature, not only for AHSS but also for a wide range of heterogeneous composite materials, which in one way or another can be used as a framework * Corresponding author. Tel.: +61 3 5227 1382; fax: +61 3 5527 1103. E-mail addresses: [email protected], [email protected] (S.A. Asgari). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.12.003
to model the microscale phenomena in AHSS. A key issue with these methods, whether analytical or numerical, is the simplification of the microstructural problem such that it can be solved in a reasonably fast and accurate way. However, with the everincreasing computational power, it might be possible to reduce the microstructural simplifications and model the AHSS with its realistic microstructure topology and configuration, and thereby, increase the accuracy for modeling some of the more difficult problems like springback. The aim of this paper, therefore, is to improve the method by introducing a stochastic numerical approach that preserves the realistic microstructural morphology of AHSS while keeping the computational costs low. 2. Realistic microstructure model In general, there are two approaches in the modeling of microstructural behavior of the heterogeneous composite materials: analytical and computational. In the analytical (and semi-analytical) methods, the properties at the microscale are embedded as parameters in the macroscale constitutive equations that are solved either numerically or in closed form. In such methods, the multiphase nature of the material is idealized by the inclusion-matrix type simplifications and a variant of the Eshelby original equivalent inclusion
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
model is used [1]. In the computational methods, the microstructure of the material is regarded in a Representative Volume Element (RVE), which can statistically represent the macroscale material point. The RVE, which is generally defined by an idealized (spherical or elliptical) geometry and topology of the microstructure, is meshed and the boundary conditions are properly imposed on it to fulfill the micro–macro couplings. Then, the solution of the boundary value problem at the RVE fine scale is homogenized and used at the coarse scale. The RVE could have a very complex microstructure consisting of grains separated by grain boundaries, voids, inclusions, cracks, different phases and moving boundary differences [2]. In this work, the RVE was represented by an actual and realistic geometrical representation of the microstructure, which was reconstructed
Table 1 Alloying elements of steel grades used in this study.
DP TRIP
C
Mn
Si
Al
Cu
Cr
0.035 0.120
1.199 1.392
1.127 1.768
0.037 0.031
0.10
0.868 0.020
V
Mo
Ti
Fig. 1. Stress–strain curves of single phases used in the microscale model.
Nb
861
from optical micrographs of AHSS. The procedure was similar to the digital image-based (DIB) finite element method [3], which enables the construction of accurate microstructural models by taking into account the specific effects of the microstructural morphology. Phase morphology, shape, orientation and distribution could be reflected in a realistic microstructure models compared to rather simplified geometrical representations in the idealized spherical or elliptical microstructural models. The presence of such detailed description of the microstructure has the potential to improve the accuracy; however, the improved accuracy must be offset by the time required in pre-processing, processing and post-processing of each single RVE. In this work, a specialized microstructural meshing program called OOF (object oriented finite element analysis software) created by the US National Institute of Standards and Technology (NIST) [4] was used to create the microscale models. The element types and boundary conditions in OOF are under constant improvement, with periodic boundary condition recently added to the software [4]. The main feature of OOF used in this work is the image processing capacity to create finite element meshes from AHSS micrographs. The realistic microstructure model is then imported to Abaqus for a detailed analysis of the plastic deformation and localized properties in each of the different types of RVE models studied.
Fig. 3. Generic representation of an RVE.
Fig. 2. Realistic microstructure model for TRIP steel, with three points of views meshed. Different phase properties are identified by separation of similar intensity pixels and assigning specific color and material properties to them.
862
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
3. Modeling and experimental approach A DP and a TRIP steel grade were tested in a standard MTS tensile machine with constant crosshead speed. The steel grades and their composition are listed in Table 1. The steel grades were commercially supplied, and were denoted by DP and TRIP. It was understood that these steels are generally produced by the standard cold rolling and intercritical annealing (IA) at 780 °C for 180 s. After that, the DP steel is quenched while the TRIP steel is cooled to a temperature of 400 °C and was held for 300 s followed by quenching [5]. The Cartesian ðn; g; fÞ coordinate frame of the tensile test specimen was aligned with the material symmetry directions. n was
along the rolling direction and f was the through thickness direction and perpendicular to the plate. Based on these symmetries, only one quarter of the each sample was modeled in the macroscale using Abaqus. 3.1. Finite element model and material properties The material properties of ferrite, martensite and austenite have been reported in the literature [6–8]. These material properties are generally found from the tension or compression tests of steels that approximately consist of a single phase. In our micromechanical model, the stress–strain curves of each phase from [7] were used in the microscale models. Each phase is considered as an elastoplastic material where the strain rate is decomposed into elastic and plastic components. The plastic flow stress of each phase was given by the stress–strain curves shown in Fig. 1. As such phases have different plastic properties. The models are assumed to be in a quasi-static state, where inertia effects are ignored. The von Mises yield condition is assumed for single phases given by:
f ¼ r ry : The equivalent stress,
ð1Þ
r, is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 0 r¼ rr; 2 ij ij Ferrite (Gray) 70.56% Austenite (Black) 26.60% Martensite (White) 2.84%
(a) TRIP
Ferrite (Gray) 74.80% Others (Black) 25.20%
(b) DP
Fig. 4. Simplified multiple and single inclusion RVE models for (a) TRIP and (b) DP steel.
ð2Þ
with the deviatoric stresses r0ij ¼ rij 1=3rkk . The yield stress ry is taken to be a function of the equivalent plastic strain to describe the isotropic hardening behavior of single phases, as shown in Fig. 1. The material properties of AHSS are statistically homogenous at the macroscale and characterized by a set of effective tangent moduli that do not vary from point to point in the macroscale. The effective tangent moduli relate the average stress and strain
Fig. 5. Inelastic strain (above) effective micro-stress distribution (below) in TRIP (left) and DP (right) RVEs with simplified inclusion models.
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
quantities defined over the smallest sub-volume of the AHSS whose average response is representative of the response at the macroscale. This sub-volume is found from stochastic selection of the smallest possible RVEs from realistic AHSS micrographs. The minimum size of these RVEs is defined based on a trade-off between the amount of information included in the RVE and the available computational power. In this work, the minimum size is considered as 10 lm, which was represented by a 200 pixels square box in a typical micrograph shown in Fig. 2. The macroscopic stress components were then computed with a global averaging scheme over the number of RVEs, rn , such that
r ij ¼
rn 1 X r k ; r n k¼1 ij
kij is the effective stress from RVEk (RVE index number k) where r with volume X in a micrograph is given by the volume average of the microscopic stress component according to the following equation:
r kij ¼
Z X
rkij dX:
The effective strain field eij is found similarly. The relation be ij and eij is replaced with the effective tangent moduli tween r Ceff . The volume averaging of Eq. (4) has an obvious effect on the accuracy of the macroscale field variables. Prior to this work, the volume averaging was generally performed on simplified geometrical RVEs using homogenization techniques such as the self-consistent approach or the Eshelby inclusion approach [9–14]. In this work, we use a homogenization technique outlined by Kouznetsova et al. [15] to find the volume average in Eq. (4) for a RVE that can be represented by the generic shape of Fig. 3. The periodic boundary conditions on the edges of this RVE are enforced with
x34 ðn34 Þ þ x1 x4 x12 ðn12 Þ ¼ 0 for n34 ¼ n12 ; ð3Þ
ð4Þ
863
x23 ðn23 Þ þ x1 x2 x41 ðn41 Þ ¼ 0 for n23 ¼ n41 ;
ð5Þ
and the first Piola–Kirchhoff stress is then computed from:
Pkij ¼
1 X
X
fik X kj ;
ð6Þ
k
where again for the RVEk, fik and X kj are the internal forces and initial position vectors at the nodes i and j along the boundary of RVE, respectively. The resultant effective stress is then found from:
Fig. 6. Stress and strain distribution in the realistic RVEs from three points of view (POVs) of the TRIP micrograph.
864
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
1 J
T
rkij ¼ Pkij F kij ;
ð7Þ
with the Jacobian J as the determinant of F kij , the deformation gradient at the macroscale material point. 3.2. Microstructure sampling and mesh generation For microscopic analyses, etching in 2% Nital was used to reveal the difference between phases in the multiphase steel micrographs (DP and TRIP) [5]. The micrographs were used to analyze the volume fraction of each phase at the sampling points. For each steel grade, these volume fractions were used to translate the realistic microstructure to simplified single or multiple inclusion models. Three arbitrary random points of views were selected from each micrograph as shown in Fig. 2. Each point of view was meshed using the OOF software [4]. The thresholds of the desired phases were defined using pixel color intensities (Fig. 2). The calculation of volume fraction then was easily followed by measuring the area of each phase with the assumption of unit thickness. The realistic micrographs were used to build a constitutive model in Abaqus based on sets of realistic RVE models for each steel grade. In addition, two types of simplified RVE models were created for the TRIP and
DP steels using the corresponding volume fractions found from the realistic microstructure models, as shown in Fig. 4. The homogenization routine performed for the realistic microstructure models were similarly performed on these simplified RVEs to provide a comparison basis between our approach and the widely used idealized geometry approach.
4. Results and discussion The inelastic strain and effective micro-stress distribution from the simplified microstructure models are shown in Fig. 5. The simplified models are abstracted by a single inclusion of hard martensite in soft ferrite matrix in the case of DP steel and multiple inclusions of martensite and austenite in soft ferrite matrix in the case of the TRIP steel. With periodic BCs imposed on the simplified models there is no interaction between the inclusions and the effects of stress and strain partitioning are limited to the misfit between the inclusion and the matrix. It is possible to find the analytical solution for such models with approximations proposed by the Eshelby [1] or the Mori–Tanaka [16] methods. For the method proposed in this paper, several micrographs of the DP and TRIP steels were prepared before and after tensile test. With
Fig. 7. Stress and strain distribution in the realistic RVEs from three points of view (POVs) of the DP micrograph.
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
r n ¼ 3, three realizations of each micrograph were taken randomly. The stress and strain distributions from these points of view are shown in Fig. 6 for the TRIP and in Fig. 7 for the DP steels. The periodic BC effect is visible with the deformation of opposite edges on these RVEs. The stress distribution in both the DP and TRIP steels shows shear bands that are much more pronounced in the TRIP samples. Localized stress and strain regions are detected at the interface of hard and soft phases in both DP and TRIP microstructures. The stress and strain partitioning is visualized in both of the DP and TRIP microstructural models. The microstructural models of the DP steel show how the ferrite and martensite phases behave during deformation. Significant strain localization in the martensite phase is visible while ferrite has already yielded and deformed plastically. The fact that load has transferred from ferrite to martensite through their interface is due to this realistic representation of the geometrical morphology of the DP steel. The existence of three phases in the TRIP steel introduces a more complex deformation process even with the phase transformation process being neglected. The reason is that the interaction between martensite, ferrite and austenite causes a stress distribution that strongly depends on the RVE selection. In other words, if the current realization or point of view of the TRIP steel is randomly selected from a position with highly oriented phase morphology, then the resultant macroscale stress could severely drift away from the ideal solution. To avoid this problem, the random selection must be repeated to cover a balanced region of the micrograph. The micromechanical solution of the TRIP and DP realistic models result in a macroscale stress–strain curve, which is plotted versus the experimental and simplified microstructure model in Figs. 8 and 9 for the TRIP and DP steels, respectively. With the TRIP steel, both the realistic and simplified models fail to predict the correct stresses at strains below 0.01. Interestingly, the stress prediction in the realistic model is over-predicting the experimental values up to a strain of 0.15 and then similar to the simplified model, underestimates the macro-stress. The reason for this behavior could be twofold. Firstly, with the neglect of austenite to martensite phase transformation, the volume and shape changes in the austenite and martensite inclusions are not included. The phase transformation parameters, which have been the subject of numerous studies before, could strongly improve the predictions made with the realistic microstructure. Secondly, the interaction be-
Fig. 8. Macro-stress–strain relationship from experimental TRIP steel, three points of view (POVs), realistic (RM) and simplified (SM) models.
865
Fig. 9. Macro-stress–strain relationship from experimental DP steel, three points of view (POVs), realistic (RM) and simplified (SM) models.
tween the martensite, austenite and ferrite phases causes changes in the stress prediction, such that at low strains, martensite and part of the austenite stay in the elastic region, while the ferrite deforms plastically. At higher strains, austenite and martensite yield and deform plastically as well, which results in lower stress predictions. For the DP steel on the other hand, it is obvious that the hardening rate is predicted with higher accuracy using both the simplified and realistic models. Specifically at low strains, the realistic model captures the most accurate yield behavior of the DP steel. This is a direct result of the global averaging between the three point of views that collectively predict the same yield stress at strains below 0.01 (cf. Fig. 9). The hardening index (n value) from the simplified and realistic models was obtained in comparison of the resultant macroscale stress–strain curves to power law. The n values found this way are plotted in Fig. 10 for the TRIP and DP steels. As shown in this figure, the simplified and realistic models both predicted the n value of DP steel with high accuracy, while the error from the realistic model was a little lower than the simplified model. However, their difference was strongly shown in the prediction of the hardening index for the TRIP steel. The realistic model was much closer to the experimental n value than the simplified model. It is again important to note that both the simplified and realistic models neglect the phase transformation, grain boundaries and texture details and therefore fail to predict the actual behavior of the TRIP steel with the same level of accuracy obtained on modeling DP steel.
Fig. 10. Prediction of hardening index from the realistic (RM) and simplified (SM) models of DP and TRIP in comparison to experimental values.
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5. Conclusions
References
In this work, a method of using a realistic representation of the microstructure in the micromechanical modeling of Advanced High Strength Steels (AHSS) has been implemented. In most previous work, the geometrical representation of the micromechanical models is represented with simplified or idealized geometries. However, the results from this study show that using a realistic representation of microstructure, either for DP or TRIP steel, could improve the accuracy of the predicted stress and strain distribution. Sometimes the nature of strain hardening is more complex than just the composite effect of phase interactions, e.g., in TRIP steel. In these cases, the extra information could be incorporated in a realistic microstructure model rather than in a simplified model, to provide another level of improvement in accuracy. The tradeoff between the higher computational time required for the microstructural modeling and the higher accuracy has been offset with a stochastic approach in the selection of some point of views from DP and TRIP micrographs. In the current work, the number of selected RVEs is considered three with a size of 10 lm. The lower and higher bounds of selected RVEs might be found from an optimization study as an extension to the current work. The resultant globally averaged effective stress and strain fields from realistic microstructure model were able to capture the onset of the plastic instability in the DP steel, while also predicting more accurate results compared to the simplified model. Furthermore, if the phase transformation process, shape and volume change that occur at microstructure level of TRIP steel are considered, the realistic microstructural model may significantly increase the accuracy of the predicted results compared to the preliminary composite effects considered in this work for TRIP steel.
[1] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, in: Proceedings of Royal Society of London, Series A, Mathematical and Physical Sciences, vol. 241, 1957. [2] S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland Series in Applied Mathematics and Mechanics, Elsevier Science Publishers BV, Amsterdam, The Netherlands, 1993. [3] K. Terada, M. Hori, T. Kyoya, N. Kikuchi, Int. J. Solids Struct. 37 (2000) 2285– 2311. [4] National Institute of Standards and Technology (USA), Object-Oriented Finite Element Program, Version 2.4, 2007,
Acknowledgement The authors would like to acknowledge the support from Australian Research Council through the Federation Fellowship Grant for PDH (FF455846).
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Modeling of Advanced High Strength Steels with the realistic microstructure–strength relationships S.A. Asgari a,*, P.D. Hodgson a, C. Yang b, B.F. Rolfe b a b
Centre for Material and Fibre Innovation (CMFI), Geelong Technology Precinct (GTP), Waurn Ponds Campus, Deakin University, Geelong 3217, Victoria, Australia School of Engineering and Information Technology, Deakin University, Waurn Ponds 3217, Victoria, Australia
a r t i c l e
i n f o
Article history: Received 26 June 2008 Received in revised form 28 November 2008 Accepted 5 December 2008 Available online 18 January 2009 PACS: 46.15.-X 02.70.-C 07.05.Tp Keywords: Advanced High Strength Steels Realistic microstructure Strength
a b s t r a c t The objective of the work is to consider the first-order effects of the realistic microstructure morphology in the macroscale modeling of the multiphase Advanced High Strength Steels (AHSS). Instead of using constitutive equations at macroscale, the strength–microstructure relationship is studied in the forms of micromechanical and multiscale models that do not make considerable simplifications with regard to the microscale geometry and topology. The trade-off between the higher computational time and the higher accuracy has been offset with a stochastic approach in the construction of the microscale models. The multiphase composite effects of AHSS microstructure is considered in realistic microstructural models that are stochastically built from AHSS micrographs. Computational homogenization routines are used to couple micro and macroscale and resultant stress–strain relations are compared for models built with the simplified and idealized geometries of the microstructure. The results from this study show that using a realistic representation of the microstructure, either for DP or TRIP steel, could improve the accuracy of the predicted stress and strain distribution. The resultant globally averaged effective stress and strain fields from realistic microstructure model were able to accurately capture the onset of the plastic instability in the DP steel. It is shown that the macroscale mechanical behavior is directly affected by the level of complexities in the microscale models. Therefore, greater accuracy could be achieved if these stochastic realistic microstructures are used at the microscale models. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Advanced High Strength Steels (AHSS) are multiphase materials that are of great importance in the automotive industry for the purposes of strengthening the vehicle structure and decreasing the vehicle weight. Whereas the numerical modeling codes currently used for forming, springback and crash modeling generally represent the sheet as a homogenous material, uniform in thickness and material properties. These steels are generally characterized by their complex microstructure that has a first-order effect on the strength, ductility and formability. With the increased interest in using AHSS, there is a growing need to take into account explicitly the microscale mechanisms such as composite effects of the multiphase microstructure, deformation induced phase transformation and history dependent large deformation in forming, springback and crash simulations. Over the past few decades, numerous micromechanical models and multiscale techniques have emerged in the literature, not only for AHSS but also for a wide range of heterogeneous composite materials, which in one way or another can be used as a framework * Corresponding author. Tel.: +61 3 5227 1382; fax: +61 3 5527 1103. E-mail addresses: [email protected], [email protected] (S.A. Asgari). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.12.003
to model the microscale phenomena in AHSS. A key issue with these methods, whether analytical or numerical, is the simplification of the microstructural problem such that it can be solved in a reasonably fast and accurate way. However, with the everincreasing computational power, it might be possible to reduce the microstructural simplifications and model the AHSS with its realistic microstructure topology and configuration, and thereby, increase the accuracy for modeling some of the more difficult problems like springback. The aim of this paper, therefore, is to improve the method by introducing a stochastic numerical approach that preserves the realistic microstructural morphology of AHSS while keeping the computational costs low. 2. Realistic microstructure model In general, there are two approaches in the modeling of microstructural behavior of the heterogeneous composite materials: analytical and computational. In the analytical (and semi-analytical) methods, the properties at the microscale are embedded as parameters in the macroscale constitutive equations that are solved either numerically or in closed form. In such methods, the multiphase nature of the material is idealized by the inclusion-matrix type simplifications and a variant of the Eshelby original equivalent inclusion
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
model is used [1]. In the computational methods, the microstructure of the material is regarded in a Representative Volume Element (RVE), which can statistically represent the macroscale material point. The RVE, which is generally defined by an idealized (spherical or elliptical) geometry and topology of the microstructure, is meshed and the boundary conditions are properly imposed on it to fulfill the micro–macro couplings. Then, the solution of the boundary value problem at the RVE fine scale is homogenized and used at the coarse scale. The RVE could have a very complex microstructure consisting of grains separated by grain boundaries, voids, inclusions, cracks, different phases and moving boundary differences [2]. In this work, the RVE was represented by an actual and realistic geometrical representation of the microstructure, which was reconstructed
Table 1 Alloying elements of steel grades used in this study.
DP TRIP
C
Mn
Si
Al
Cu
Cr
0.035 0.120
1.199 1.392
1.127 1.768
0.037 0.031
0.10
0.868 0.020
V
Mo
Ti
Fig. 1. Stress–strain curves of single phases used in the microscale model.
Nb
861
from optical micrographs of AHSS. The procedure was similar to the digital image-based (DIB) finite element method [3], which enables the construction of accurate microstructural models by taking into account the specific effects of the microstructural morphology. Phase morphology, shape, orientation and distribution could be reflected in a realistic microstructure models compared to rather simplified geometrical representations in the idealized spherical or elliptical microstructural models. The presence of such detailed description of the microstructure has the potential to improve the accuracy; however, the improved accuracy must be offset by the time required in pre-processing, processing and post-processing of each single RVE. In this work, a specialized microstructural meshing program called OOF (object oriented finite element analysis software) created by the US National Institute of Standards and Technology (NIST) [4] was used to create the microscale models. The element types and boundary conditions in OOF are under constant improvement, with periodic boundary condition recently added to the software [4]. The main feature of OOF used in this work is the image processing capacity to create finite element meshes from AHSS micrographs. The realistic microstructure model is then imported to Abaqus for a detailed analysis of the plastic deformation and localized properties in each of the different types of RVE models studied.
Fig. 3. Generic representation of an RVE.
Fig. 2. Realistic microstructure model for TRIP steel, with three points of views meshed. Different phase properties are identified by separation of similar intensity pixels and assigning specific color and material properties to them.
862
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
3. Modeling and experimental approach A DP and a TRIP steel grade were tested in a standard MTS tensile machine with constant crosshead speed. The steel grades and their composition are listed in Table 1. The steel grades were commercially supplied, and were denoted by DP and TRIP. It was understood that these steels are generally produced by the standard cold rolling and intercritical annealing (IA) at 780 °C for 180 s. After that, the DP steel is quenched while the TRIP steel is cooled to a temperature of 400 °C and was held for 300 s followed by quenching [5]. The Cartesian ðn; g; fÞ coordinate frame of the tensile test specimen was aligned with the material symmetry directions. n was
along the rolling direction and f was the through thickness direction and perpendicular to the plate. Based on these symmetries, only one quarter of the each sample was modeled in the macroscale using Abaqus. 3.1. Finite element model and material properties The material properties of ferrite, martensite and austenite have been reported in the literature [6–8]. These material properties are generally found from the tension or compression tests of steels that approximately consist of a single phase. In our micromechanical model, the stress–strain curves of each phase from [7] were used in the microscale models. Each phase is considered as an elastoplastic material where the strain rate is decomposed into elastic and plastic components. The plastic flow stress of each phase was given by the stress–strain curves shown in Fig. 1. As such phases have different plastic properties. The models are assumed to be in a quasi-static state, where inertia effects are ignored. The von Mises yield condition is assumed for single phases given by:
f ¼ r ry : The equivalent stress,
ð1Þ
r, is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 0 r¼ rr; 2 ij ij Ferrite (Gray) 70.56% Austenite (Black) 26.60% Martensite (White) 2.84%
(a) TRIP
Ferrite (Gray) 74.80% Others (Black) 25.20%
(b) DP
Fig. 4. Simplified multiple and single inclusion RVE models for (a) TRIP and (b) DP steel.
ð2Þ
with the deviatoric stresses r0ij ¼ rij 1=3rkk . The yield stress ry is taken to be a function of the equivalent plastic strain to describe the isotropic hardening behavior of single phases, as shown in Fig. 1. The material properties of AHSS are statistically homogenous at the macroscale and characterized by a set of effective tangent moduli that do not vary from point to point in the macroscale. The effective tangent moduli relate the average stress and strain
Fig. 5. Inelastic strain (above) effective micro-stress distribution (below) in TRIP (left) and DP (right) RVEs with simplified inclusion models.
S.A. Asgari et al. / Computational Materials Science 45 (2009) 860–866
quantities defined over the smallest sub-volume of the AHSS whose average response is representative of the response at the macroscale. This sub-volume is found from stochastic selection of the smallest possible RVEs from realistic AHSS micrographs. The minimum size of these RVEs is defined based on a trade-off between the amount of information included in the RVE and the available computational power. In this work, the minimum size is considered as 10 lm, which was represented by a 200 pixels square box in a typical micrograph shown in Fig. 2. The macroscopic stress components were then computed with a global averaging scheme over the number of RVEs, rn , such that
r ij ¼
rn 1 X r k ; r n k¼1 ij
kij is the effective stress from RVEk (RVE index number k) where r with volume X in a micrograph is given by the volume average of the microscopic stress component according to the following equation:
r kij ¼
Z X
rkij dX:
The effective strain field eij is found similarly. The relation be ij and eij is replaced with the effective tangent moduli tween r Ceff . The volume averaging of Eq. (4) has an obvious effect on the accuracy of the macroscale field variables. Prior to this work, the volume averaging was generally performed on simplified geometrical RVEs using homogenization techniques such as the self-consistent approach or the Eshelby inclusion approach [9–14]. In this work, we use a homogenization technique outlined by Kouznetsova et al. [15] to find the volume average in Eq. (4) for a RVE that can be represented by the generic shape of Fig. 3. The periodic boundary conditions on the edges of this RVE are enforced with
x34 ðn34 Þ þ x1 x4 x12 ðn12 Þ ¼ 0 for n34 ¼ n12 ; ð3Þ
ð4Þ
863
x23 ðn23 Þ þ x1 x2 x41 ðn41 Þ ¼ 0 for n23 ¼ n41 ;
ð5Þ
and the first Piola–Kirchhoff stress is then computed from:
Pkij ¼
1 X
X
fik X kj ;
ð6Þ
k
where again for the RVEk, fik and X kj are the internal forces and initial position vectors at the nodes i and j along the boundary of RVE, respectively. The resultant effective stress is then found from:
Fig. 6. Stress and strain distribution in the realistic RVEs from three points of view (POVs) of the TRIP micrograph.
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1 J
T
rkij ¼ Pkij F kij ;
ð7Þ
with the Jacobian J as the determinant of F kij , the deformation gradient at the macroscale material point. 3.2. Microstructure sampling and mesh generation For microscopic analyses, etching in 2% Nital was used to reveal the difference between phases in the multiphase steel micrographs (DP and TRIP) [5]. The micrographs were used to analyze the volume fraction of each phase at the sampling points. For each steel grade, these volume fractions were used to translate the realistic microstructure to simplified single or multiple inclusion models. Three arbitrary random points of views were selected from each micrograph as shown in Fig. 2. Each point of view was meshed using the OOF software [4]. The thresholds of the desired phases were defined using pixel color intensities (Fig. 2). The calculation of volume fraction then was easily followed by measuring the area of each phase with the assumption of unit thickness. The realistic micrographs were used to build a constitutive model in Abaqus based on sets of realistic RVE models for each steel grade. In addition, two types of simplified RVE models were created for the TRIP and
DP steels using the corresponding volume fractions found from the realistic microstructure models, as shown in Fig. 4. The homogenization routine performed for the realistic microstructure models were similarly performed on these simplified RVEs to provide a comparison basis between our approach and the widely used idealized geometry approach.
4. Results and discussion The inelastic strain and effective micro-stress distribution from the simplified microstructure models are shown in Fig. 5. The simplified models are abstracted by a single inclusion of hard martensite in soft ferrite matrix in the case of DP steel and multiple inclusions of martensite and austenite in soft ferrite matrix in the case of the TRIP steel. With periodic BCs imposed on the simplified models there is no interaction between the inclusions and the effects of stress and strain partitioning are limited to the misfit between the inclusion and the matrix. It is possible to find the analytical solution for such models with approximations proposed by the Eshelby [1] or the Mori–Tanaka [16] methods. For the method proposed in this paper, several micrographs of the DP and TRIP steels were prepared before and after tensile test. With
Fig. 7. Stress and strain distribution in the realistic RVEs from three points of view (POVs) of the DP micrograph.
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r n ¼ 3, three realizations of each micrograph were taken randomly. The stress and strain distributions from these points of view are shown in Fig. 6 for the TRIP and in Fig. 7 for the DP steels. The periodic BC effect is visible with the deformation of opposite edges on these RVEs. The stress distribution in both the DP and TRIP steels shows shear bands that are much more pronounced in the TRIP samples. Localized stress and strain regions are detected at the interface of hard and soft phases in both DP and TRIP microstructures. The stress and strain partitioning is visualized in both of the DP and TRIP microstructural models. The microstructural models of the DP steel show how the ferrite and martensite phases behave during deformation. Significant strain localization in the martensite phase is visible while ferrite has already yielded and deformed plastically. The fact that load has transferred from ferrite to martensite through their interface is due to this realistic representation of the geometrical morphology of the DP steel. The existence of three phases in the TRIP steel introduces a more complex deformation process even with the phase transformation process being neglected. The reason is that the interaction between martensite, ferrite and austenite causes a stress distribution that strongly depends on the RVE selection. In other words, if the current realization or point of view of the TRIP steel is randomly selected from a position with highly oriented phase morphology, then the resultant macroscale stress could severely drift away from the ideal solution. To avoid this problem, the random selection must be repeated to cover a balanced region of the micrograph. The micromechanical solution of the TRIP and DP realistic models result in a macroscale stress–strain curve, which is plotted versus the experimental and simplified microstructure model in Figs. 8 and 9 for the TRIP and DP steels, respectively. With the TRIP steel, both the realistic and simplified models fail to predict the correct stresses at strains below 0.01. Interestingly, the stress prediction in the realistic model is over-predicting the experimental values up to a strain of 0.15 and then similar to the simplified model, underestimates the macro-stress. The reason for this behavior could be twofold. Firstly, with the neglect of austenite to martensite phase transformation, the volume and shape changes in the austenite and martensite inclusions are not included. The phase transformation parameters, which have been the subject of numerous studies before, could strongly improve the predictions made with the realistic microstructure. Secondly, the interaction be-
Fig. 8. Macro-stress–strain relationship from experimental TRIP steel, three points of view (POVs), realistic (RM) and simplified (SM) models.
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Fig. 9. Macro-stress–strain relationship from experimental DP steel, three points of view (POVs), realistic (RM) and simplified (SM) models.
tween the martensite, austenite and ferrite phases causes changes in the stress prediction, such that at low strains, martensite and part of the austenite stay in the elastic region, while the ferrite deforms plastically. At higher strains, austenite and martensite yield and deform plastically as well, which results in lower stress predictions. For the DP steel on the other hand, it is obvious that the hardening rate is predicted with higher accuracy using both the simplified and realistic models. Specifically at low strains, the realistic model captures the most accurate yield behavior of the DP steel. This is a direct result of the global averaging between the three point of views that collectively predict the same yield stress at strains below 0.01 (cf. Fig. 9). The hardening index (n value) from the simplified and realistic models was obtained in comparison of the resultant macroscale stress–strain curves to power law. The n values found this way are plotted in Fig. 10 for the TRIP and DP steels. As shown in this figure, the simplified and realistic models both predicted the n value of DP steel with high accuracy, while the error from the realistic model was a little lower than the simplified model. However, their difference was strongly shown in the prediction of the hardening index for the TRIP steel. The realistic model was much closer to the experimental n value than the simplified model. It is again important to note that both the simplified and realistic models neglect the phase transformation, grain boundaries and texture details and therefore fail to predict the actual behavior of the TRIP steel with the same level of accuracy obtained on modeling DP steel.
Fig. 10. Prediction of hardening index from the realistic (RM) and simplified (SM) models of DP and TRIP in comparison to experimental values.
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5. Conclusions
References
In this work, a method of using a realistic representation of the microstructure in the micromechanical modeling of Advanced High Strength Steels (AHSS) has been implemented. In most previous work, the geometrical representation of the micromechanical models is represented with simplified or idealized geometries. However, the results from this study show that using a realistic representation of microstructure, either for DP or TRIP steel, could improve the accuracy of the predicted stress and strain distribution. Sometimes the nature of strain hardening is more complex than just the composite effect of phase interactions, e.g., in TRIP steel. In these cases, the extra information could be incorporated in a realistic microstructure model rather than in a simplified model, to provide another level of improvement in accuracy. The tradeoff between the higher computational time required for the microstructural modeling and the higher accuracy has been offset with a stochastic approach in the selection of some point of views from DP and TRIP micrographs. In the current work, the number of selected RVEs is considered three with a size of 10 lm. The lower and higher bounds of selected RVEs might be found from an optimization study as an extension to the current work. The resultant globally averaged effective stress and strain fields from realistic microstructure model were able to capture the onset of the plastic instability in the DP steel, while also predicting more accurate results compared to the simplified model. Furthermore, if the phase transformation process, shape and volume change that occur at microstructure level of TRIP steel are considered, the realistic microstructural model may significantly increase the accuracy of the predicted results compared to the preliminary composite effects considered in this work for TRIP steel.
[1] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, in: Proceedings of Royal Society of London, Series A, Mathematical and Physical Sciences, vol. 241, 1957. [2] S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland Series in Applied Mathematics and Mechanics, Elsevier Science Publishers BV, Amsterdam, The Netherlands, 1993. [3] K. Terada, M. Hori, T. Kyoya, N. Kikuchi, Int. J. Solids Struct. 37 (2000) 2285– 2311. [4] National Institute of Standards and Technology (USA), Object-Oriented Finite Element Program, Version 2.4, 2007,
Acknowledgement The authors would like to acknowledge the support from Australian Research Council through the Federation Fellowship Grant for PDH (FF455846).