On strain partitioning and micro-damage behavior of dual-phase steels

Materials Science & Engineering A 614 (2014) 180–192

Contents lists available at ScienceDirect

Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

On strain partitioning and micro-damage behavior of dual-phase steels A. Fillafer a,n, C. Krempaszky a,b, E. Werner a a

Institute of Materials Science and Mechanics of Materials, Technische Universität München, Boltzmannstraße 15, 85748 Garching, Germany Christian-Doppler-Laboratory of Material Mechanics of High Performance Alloys, Institute of Materials Science and Mechanics of Materials, Technische Universität München, Boltzmannstraße 15, 85748 Garching, Germany

b

art ic l e i nf o

a b s t r a c t

Article history: Received 9 April 2014 Received in revised form 30 June 2014 Accepted 11 July 2014 Available online 19 July 2014

This work presents a model used to study strain partitioning and micro-damage initiation of various dual-phase steel microstructures. The model is formulated via continuum micro-mechanics and is solved in a finite-element framework. A phenomenological failure criterion is employed, calibrated and used to identify micro-damage initiation during plastic deformation. What is shown is how the primary microstructural parameters responsible for microstructural stress/ strain fluctuations, namely phase strength contrast and hard phase fraction, influence phase averaged strains and phase specific strain distributions. Results of a study on micro-damage initiation show the influence of these primary microstructural parameters on strain for micro-damage initiation, the associated macro stress, and strain hardening. It is concluded that with rising microstructural mechanical heterogeneity, strain for micro-damage initiation decreases and strain hardening increases. Macroscopic stress at micro-damage initiation is shown to be primarily governed by the hard phase fraction and secondarily by the phase strength contrast. & 2014 Elsevier B.V. All rights reserved.

Keywords: Micromechanics Steel Finite-element method Plasticity Interfaces Failure

1. Introduction Dual-phase (DP) steels consist of hard martensite grains usually embedded in soft ferrite and exhibit the following mechanical characteristics (at room temperature) [1,2], which distinguish them from high-strength low-alloy (HSLA) steels: (i) High tensile strength in comparison to HSLA steels, (ii) low yield strength to tensile strength ratio, and (iii) absence of yield point elongation; Feature (i) makes them suitable for high stress (lightweight design) applications. Since the high tensile strength of DP-steels is accompanied with a low initial yield strength (feature (ii)), workpieces are most favorably manufactured by forming processes to provide high yield strength after forming. Feature (ii) is a desired characteristic in forming operations, since it results in a low average forming force. Furthermore, feature (ii)—as a measure for strain hardening—ensures a good deep drawability, since it may shift necking at a given stress state to higher strains [3,4]. Feature (iii) is attributed to transformation induced dislocations in ferrite, which result from microstructure processing (austenite

n

Corresponding author. Tel.: þ 49 89 289 15313. E-mail address: fi[email protected] (A. Fillafer).

http://dx.doi.org/10.1016/j.msea.2014.07.029 0921-5093/& 2014 Elsevier B.V. All rights reserved.

to martensite transformation) [5,6]. This property is favorable in forming applications with strict standards regarding the workpiece surface, since it prevents visible discontinuities in sheet thickness resulting from Lüders-bands. From the foregoing discussion it is obvious that plastic deformation characteristics (formability) play a significant role in DP-steel sheet applications, either during component fabrication or application. Improving formability—which is governed by microstructural damage (micro-damage) and possibly necking—at a desired strength level, is therefore one of the design objectives for DP-steel manufacturers. Superior formability of DP-steels over HSLA-steels of equal chemical composition was first reported by Rashid [7]. Fischmeister and Karlsson [8] published a comprehensive overview on aspects of plastic deformation of coarse two-phase microstructures including strain partitioning between the soft and hard phases. Davies [1] used HSLA-steel grades as basis for the processing of DP-steel microstructures and studied their mechanical properties after various heat treatments. Different types of micro-damage, which are observed prior to macroscopic fracture in DP-steels, were reported. The primary ones are decohesion of ferrite/martensite phase boundaries and fracture of martensite [5,9–14]. As a secondary mechanism, void nucleation at non-metallic inclusions is mentioned [10,14]. With increasing plastic deformation, voids so generated grow and coalesce, which finally leads to macroscopic fracture. Necking (plastic instability) is possibly involved in this sequence.

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

This, however, is a question of the material's strain hardening behavior and the respective stress state. In order to optimize a microstructure with regard to formability, it is necessary to understand how microstructural parameters influence stress/strain partitioning and by this micro-damage initiation. Experimental studies on this issue were conducted, however, mostly in an unsystematic manner: Several authors used microstructures processed from one chemical composition via different heat treatments [12,15,16], while others used microstructures processed from various chemical compositions with various heat treatments (but exhibiting similar macroscopic strength) [17]. Theoretical (numerical) studies of micro-damage behavior of DP-steels, on the other hand, are quite promising. Not only can costly test melts be avoided thereby, but microstructural parameters may be varied more easily than in the experiment. It has to be assured, however, that the significant mechanisms are captured by the modeling depth (length scales, microstructure geometry, constitutive behavior, etc.). Void nucleation is induced by microstructural stress and strain fluctuations which are primarily caused by the mechanical heterogeneity on grain level. To model stress and strain fluctuations on grain level, continuum micro-mechanics may be employed. Böhm [18] gives a thorough overview of different techniques to allow for simulation of representative, yet adequately small volume elements which can be solved with conventional computational capabilities. Among them are embedded cell, windowing and periodic microfield approaches. Either computer generated or real microstructure phase arrangements are used within these models. The latter are typically obtained from metallographic sections, serial sections or tomographic data. Both strategies have been employed to model DP-steel microstructures. Al-Abbasi and Nemes [3] studied the macroscopic stress–strain behavior of an axis-symmetric arrangement of a spherical inclusion of martensite in a cylindrical matrix of ferrite. Prahl et al. [19] employed a computer generated phase arrangement in combination with the Gurson–Tvergaard–Needleman (GTN) constitutive model [20,21] to study its macroscopic stress–strain behavior. Krempaszky et al. [22] modeled two-dimensional DP-steel microstructures via Voronoï-tessellation and subsequent phase coloring to study their macroscopic stress–strain behavior, as well as their local stress distributions associated with void nucleation. Sun et al. [23] used metallographic sections of DP-steel microstructures to model two-dimensional real microstructure arrangements, which they investigated for plastic instability, associated with ductile failure. Computer generated microstructures incorporating the GTN constitutive model, as well as cohesive zones, have been reported by Uthaisangsuk et al. [24,25]. While the above-mentioned authors used J2-plasticity to model the basic phase specific constitutive behavior, most recently several authors employed crystal plasticity to model the behavior of ferrite [26], as well as of ferrite and martensite [27]. The objective of the present work is to gain an understanding of the influence of primary microstructural parameters on

 stress and strain fluctuations on the grain level, and  micro-damage initiation. For this purpose, quasi-static, plastic deformation with restrained macroscopic plastic instability1 (necking) of various computer generated microstructures is studied. In this work, micromechanical volume elements on grain level are considered, which are subjected to periodic boundary conditions (periodic microfield approach).

2. Theoretical framework The micro-mechanical model presented in this section is based on simplifying modeling assumptions. These are explained along with the description of the model.

2.1. Microstructure geometry For microstructure generation as pursued in this work, an approach not reliant on real microstructures is employed to ensure that microstructural parameters such as the martensite phase fraction and the geometric arrangement of martensite grains may be varied as desired. Each microstructure uses its own computer generated three-dimensional grain structure (tessellation), since two-dimensional microstructures at best capture the mechanical behavior of three-dimensional microstructures with prismatic grains (aligned parallel to each other). For the tessellation of space (here: the volume-element), different techniques are readily available and have been reported in the literature. The ordinary (non-weighted) Voronoï-tessellation [28,29] employed in this work requires only very little input, namely a definition of the volume to be tessellated and a set of “seed”-points within it. The use of this tessellation technique however implies the modeling assumption of polyhedric grain shapes (i.e. flat grain surfaces). These, according to Werner et al. [30], still approximate the shape of grains in DP-steels well. The grain generation sequence as implemented in this work (see Appendix A.1) lacks a well formulated control of grain size distribution. Grain size distributions as encountered in some DP-steel grades (e.g. very small martensite grains in relation to the ferrite grains [31]) cannot be modeled with the employed seed point patterns. This, however, is not problematic, since the phases are modeled as micro continua (see Section 2.2), and hence the influence of the density of grain boundaries on mechanical properties is non-relevant in this context. An important issue is the minimum acceptable size of the volume element (minimum number of grains per volume element) that leads to representative results. Gained from a convergence study, 150 grains per volume element are selected to assure representativity, which is why the volume elements will be termed “representative volume elements” (RVEs) in the following. Further details on the convergence study are given in Appendix C. In each tessellated volume element (“initial” grain structure) martensite and ferrite have yet to be defined. In a “coloring”process, selected grains are assigned to be martensite grains, the remaining ones compose the ferrite-phase. Two parameters control this process, namely the martensite phase fraction P M and—as a measure for the contact of the martensite grains with each other —the martensite contiguity C M A ½0; 1 [32]. Low contiguities represent microstructures in which martensite grains are quite isolated from each other, while high contiguities represent clustered grains of martensite. Two sample microstructures obtained after coloring are shown in Fig. 1. Further details on the coloring process are given in A.2.

2.2. Phase specific constitutive behavior Ferrite's and martensite's phase specific constitutive behavior are modeled assuming the following characteristics [33]:

 isotropic elasto-plastic behavior,  von Mises yield criterion (J2-plasticity) and associated flow rule (strain rate normal to yield surface),

1

This type of deformation is present in the hole-expansion test, see Section 4.1.

181

 isotropic hardening.

182

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

y z

x

Fig. 1. Microstructure A (left) and B (right) in undeformed condition. Each volume element comprises 150 grains in total. Note the periodic phase geometry (by comparing RVE faces with antiparallel normal vector). Martensite: shaded; Ferrite: M M M M M transparent; P M A ¼ 0:09, C A ¼ 0:40; P B ¼ 0:49, C B ¼ 0:43; (P A;target ¼ 0:10, C A;target ¼ M 0:40; P M B;target ¼ 0:50, C B;target ¼ 0:40).

Using this type of constitutive behavior, two elastic constants and the hardening behavior (the flowcurve) of each phase, respectively, have to be specified. These can be reliably determined from standard material tests. Here, tensile test data of various single phase ferritic and martensitic steel grades is used to derive hardness specific “master” flowcurve functions for ferrite and martensite. From these functions, particular flowcurves for the individual RVE's ferrite and martensite phase are determined by applying associated hardness parameters. Further details on the derivation of the “master” flowcurve functions and the used elastic constants are given in Appendix B. Up to this point, the generated RVEs are scale invariant. Assigning, however, a specific grain size dependent flowcurve to ferrite (see Appendix B) scales the RVE. During microstructure processing of DP-steels, typically the entire austenite-phase transforms to martensite. Transformation induced micro-strains (also referred to as geometric necessary dislocations, GND)—due to the martensite's volume increase— could be accounted for within the presented constitutive model by introducing a volume increase of martensite in an additional initial “transformation step,” as e.g. shown in Appendix D. Transformation induced micro-strains have an important impact on the microstructure's mechanical behavior at low macro strain [6,34]. However, their influence at high macro strains is negligible for the investigations of the present paper (see Appendix D). Hence, for reasons of simplicity, the transformation-step is omitted in all simulations of this work. As concluded from a qualitative comparison with published experimental results in Section 3, the presented model with the employed constitutive behavior successfully describes the primary mechanisms responsible for strain fluctuations in dual-phase microstructures.

2.3. Boundary conditions Several types of boundary conditions have been developed in order to subject volume elements to external mechanical loads [18]. The most common are uniform traction, uniform displacement and periodic boundary conditions (PBCs). The two former conditions are most simple in terms of implementation effort, however, they show poorer convergence behavior with volume element size than the latter. In other words, representativity is attained with smaller volume elements (less computational cost) when using PBCs [35,36].

PBCs ensure a congruent deformation of each periodic surface couple2 of the RVE. This is done by tying the displacement difference between congruent points on the two surfaces of such a couple to a “master node”. Forces or displacements may then be applied to these master nodes to mechanically load the RVE. In case of the threedimensional RVEs as used in this work, there are typically three master nodes. Since the periodic surface couples deform in a congruent manner, it also makes sense to employ periodic microstructures, the generation of which is described in Appendices A.1 and A.2. Further details on PBCs are given by Koutsnetsova [37]. The RVE's ability to deform non-periodically is inhibited by the use of PBCs, so necking of the RVE (macroscopic plastic instability) is restrained. While this seems unnatural from a tensile tests point of view, the hole expansion (HE) test (see Fig. 6) represents such a type of deformation. In case of the HE-test of typical DP-steels, the samples in-plane strain gradient is responsible for a shift of strain for necking beyond the fracture strain. Returning to the RVE with PBCs, the restrained necking is convenient because otherwise the RVE would experience a strain path change at the Considère-point [38], which is found at different macro strains for different microstructures. In that case, a comparison of the mechanical behavior of different RVEs would be rather difficult. 2.4. Finite-element discretization For the finite-element discretization of the RVEs, and to solve the resulting system of differential equations, the commercial finite-element solver ABAQUS is employed [39]. The exact geometric meshing3 of the RVEs has to be accomplished with tetrahedrons with linear basis-functions (denoted as C3D4), since these are the only ones which—within the framework of the finite-element software used—allow for automatic and therefore comparable meshing of the various RVEs. Automatic meshing of the RVEs with higher class finite-elements (e.g. tetrahedrons with quadratic basis-functions, C3D10, or bricks, C3D8/C3D20) could not be accomplished within the finite-element software used.

3. Strain partitioning In this section, phase specific stresses and strains are studied as a function of macro strain for the microstructures A and B of Fig. 1. Both microstructures are deformed to a log. macro strain of 81%4 (  125% eng. strain) in x-direction by applying uniaxial stress. For the following discussion, a definition of “phase averaged” quantities has to be given. With V phase as the phase volume, let e.g. the phase averaged plastic equivalent strain be evaluated as Z 1 ε ðplÞ ¼ εðplÞ dV: ð1Þ equ;phase V phase V phase equ The phase averaged plastic equivalent strain as a function of (log.) macro strain εx;macro in microstructures A and B is shown in Fig. 2 (left). Due to the different strength of both phases, the phase averaged plastic equivalent strain in ferrite is higher than in martensite, irrespective of the macro strain. The phase averaged strains furthermore display a nearly linear behavior (note the auxiliary lines I and II; also note the reference line III, indicating the phase averaged plastic equivalent strain of a mechanically 2

Here: two RVE surfaces having antiparallel normal vectors. Of interest in the present study are, among other quantities, stress fluctuations at the phase boundaries. A voxel-based meshing, despite of all its benefits, is not used here since it only approximates the shape of the phase boundaries and the local stress fluctuations. 4 This value is selected because higher strains would probably lead to significant finite-element mesh distortion. 3

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

10

0.8

¯ε e(pl) (-) qu,phase

I 8

z

x

5 4

6

3 0.2

0.4

0.4

1 2 3 4 7 6

5 8

9

10

0.2

ferrite martensite

i h e g

0.5

0.811

1.0

1.5

1.0

j

0.8

d c

f

0.2

0.6

0.0

0.8

0.6

ferrite martensite

1.0

0.8

2

1

0.0 0.0

ε¯ e(pl) (-) qu,phase

II

III

7 0.2

y

9

cumul. distr. function (-)

1.0

ferrite martensite

b

cumul. distr. function (-)

1.0

183

0.8 0.6 a 0.4

d e

c

b f

g

h

i

j

0.2

ferrite martensite

a

0.0 0.0

0.2 ≈ ε max

0.4 0.6 ≈ ε max ≈ ε max ≈ ε max

ε x,macro (-)

0.8 ε max

0.0

0.5

0.811

1.0

P90 ε

1.5

(pl)

ε equ (-)

Fig. 2. Plastic equivalent strain of ferrite and martensite in microstructures A (top) and B (bottom) as a function of macro strain. Both microstructures are assigned equal martensite as well as equal ferrite constitutive behavior, giving the same phase strength contrast (hM ¼ 525 HV and d F ¼ 12 μm used for the coefficients in Table B2). Left: Phase averaged plastic equivalent strain; Right: cumulative distribution function of plastic equivalent strain at five different macro strains (which are indicated in the according left diagram). Note that no failure criterion is applied in these simulations.

homogeneous RVE). Assuming a linear relation between plastic equivalent strain and macro strain, the phase averaged plastic equivalent strain ratio between ferrite and martensite is constant. Similar results were already obtained from experimental investigations of deformed DP-steel samples. Shen et al. [40] give results for ferrite and martensite phase averaged strains, while Su and Gurland [41] show results for martensite's phase averaged strain. The results in Fig. 2 (left) qualitatively correlate very well with the results of Shen et al., except for the initial phase averaged strain in martensite. Shen reports that this strain is zero up to a macro strain of a few percent and then rises nearly linearly with macro strain. Su and Gurland on the other hand report a similar behavior as in Fig. 2. It seems obvious that the initial averaged plastic equivalent strain in martensite is associated with the martensite yield stress, which might have been higher in the samples Shen used. The good correlation between the present microscopic strain data and the published experimental results leads to the conclusion that the model presented here is at least capable of capturing the primary deformation mechanisms in DPsteels along with the resulting field quantities in the undamaged state. Note that, although the martensite in microstructure B exhibits the shape of a “skeleton” (see Fig. 1), its phase averaged plastic equivalent strain is nevertheless significantly lower than the RVE's macro strain. A study of the phase averaged strains of several microstructures leads to the following qualitative statements:

 The angle between lines I and II correlates with the phase



strength contrast. As the phase strength contrast approaches unity (in case of a mechanically homogeneous microstructure), both lines coincide with line III if one neglects the elastic component of strain. The angle between lines II and III correlates with the martensite phase fraction. The higher the martensite phase fraction, the smaller the mentioned angle becomes. The angle between lines I and III behaves in the opposite manner.

The cumulative distribution function of plastic equivalent strain in ferrite and martensite of microstructures A and B is shown for five different macro strains in Fig. 2 (right). With increasing macro strain, the distribution functions are not only shifted to higher strains (which is clear from the foregoing discussion), but also their variance increases. One should note by comparing points number 10 with j and 5 with e, that increasing the phase fraction of martensite from 9% in microstructure A to 49% in microstructure B increases strain in both martensite and ferrite. Phase averaged von Mises stress in ferrite and martensite as a function of macro strain of microstructures A and B are depicted in Fig. 3 (left). The phase averaged von Mises stress in ferrite is always higher, and in martensite always lower, than their according flow stresses at equal strain. Bearing strain partitioning in DPmicrostructures (see Fig. 2 left) in mind, this behavior can be readily understood and even derived analytically, as was shown by Fischmeister and Karlsson [8]. From Fig. 3 (left) it is furthermore obvious that the macroscopic stress–strain behavior cannot be derived from the constituents flowcurves by a linear rule of mixture,

σ macroðεmacro Þ ¼ σ MðεðplÞ  P M þ σ FðεðplÞ Þ  ð1  P M Þ; equ Þ equ

ð2Þ

with σ M and σ F denoting the martensite and ferrite flow stress, respectively, and εðplÞ equ denoting the plastic equivalent strain. Numerous publications are dedicated to this subject, which is therefore kept very brief here. The macroscopic stress/strain response of the microstructure, however, can be derived in a good approximation by a linear mixture of the phase averaged von Mises stresses of martensite and ferrite (σ vMis;M and σ vMis;F , respectively) according to

σ macroðεmacro Þ ¼ σ vMis;Mðεmacro Þ  P M þ σ vMis;Fðεmacro Þ  ð1  P M Þ;

ð3Þ

see Fig. 3 (left). The phase averaged von Mises stress ratio as a function of macro strain is depicted for microstructures A and B in Fig. 3 (right). The behavior shown represents an increasing phase strength contrast with macroscopic deformation. It can be explained by the high strain hardening of martensite in comparison to ferrite and the accompanied

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

z

x

σM ε σF ε σ x,macro(ε

6000 5000

10

σ¯vMis,M(ε σ¯vMis,F(ε )

) )

8

(-)

7000

4000

σ¯ σ¯

3000

6 4

2000 2 1000 0.2

0.0 7000

0.4

σM ε σF ε σ x,macro(ε

6000 5000

0.6

0.8

0.0

0.2

0.4

0.6

0.8

0.4

0.6

0.8

10

σ¯vMis,M(ε σ¯vMis,F(ε )

) )

8

(-)

y

σ x,macro , σ¯vMis,ph. , σ flow,ph. (MPa)

σ x,macro , σ¯vMis,ph. , σ flow,ph. (MPa)

184

4000 3000

6 4

2000 2 1000 0.2

0. 0

0.4

0.6

0.8

0.0

0.2

ε x,macro (-)

(pl)

ε x,macro , ε equ (-)

Fig. 3. Left: Phase specific flowcurves (simulation input) of microstructures A (top) and B (bottom) and their macroscopic stress/strain response (simulation output, indicated by squares). Triangles and diamonds indicate phase averaged von Mises stress in martensite and ferrite, respectively. Right: Phase averaged von Mises stress ratio between martensite and ferrite as a function of macro strain. Note that no failure criterion is applied in these simulations and that any non-periodic deformation (e.g. necking) of the RVEs is restrained by periodic boundary conditions. 1.2

ε¯ e(pl) (-) qu,phase

0.9

ferrite martensite

0.6

both phases

0.3

0.0

0

45

90

135

180

ϕ (°) Fig. 4. Phase averaged plastic equivalent strain in ferrite/martensite/both phases of microstructure A as a function of loading angle in the xy-plane (relative to the xaxis). RVE deformed to 81% log. macro strain by applying uniaxial stress; the curves are quadratically interpolated.

(quasi) constant strain ratio between them that was already mentioned. Moreover, even though both microstructures are assigned with equal martensite and ferrite constitutive behavior, microstructure B shows a steeper increase of phase strength contrast in comparison to microstructure A. This behavior is mainly due to the higher phase averaged strains in martensite of microstructure B (see Fig. 2). At this point, one may question the results for microstructure A, since it contains non-homogeneously distributed islands of martensite: If one imagines the microstructure stacked in a periodic manner (which is done in a mechanical sense by the PBCs), one might expect scattering results if the microstructure is deformed in various directions. As shown in Fig. 4, the variation of ε ðplÞ equ is small and can be neglected in our qualitative studies. For details on the boundary conditions employed for this particular study see Appendix E.

4. Micro-damage 4.1. Model definition and calibration After a sufficiently large plastic deformation and prior to macroscopic fracture, experimental investigations of DP-steels show

microscopic damage in their microstructure, typically in the form of voids. These voids mainly nucleate at ferrite/martensite interfaces or result from cracking of slender regions of the latter [12,17,14]. Micro-damage is defined here as a microstructural state, which barely, but not negligibly, degrades macroscopic mechanical properties. To study micro-damage initiation within the present theoretical model, a phenomenological approach—in the following termed “failure criterion’—is employed [22]. In this approach, only very few parameters need to be calibrated. Starting point for the derivation of the failure criterion is the assumption that stress peaks in the microstructure are responsible for the nucleation of voids. To identify the macro strain for this to happen, one can evaluate the Rankine-criterion, σ I  σ crit Z 0,5 locally in the microstructure. Fulfillment of this criterion at a single location in the microstructure, however, does not indicate a sufficient magnitude of micro-damage to degrade the macroscopic mechanical properties. Rather, the Rankine-criterion is to be fulfilled at several locations or in a large enough volume of the microstructure. The quantile6 of the first principal stress distribution may be used to write this in form of a criterion P q fσ I g  σ crit ¼ 0;

ð4Þ

with the parameters quantile-size q and failure stress σ crit . Since experimental observations point out two different primary void nucleation sites, namely between martensite grains and at the ferrite/martensite phase boundaries, both these sites are considered with separate failure criteria. These sites are denoted as the ferrite/martensite (FM) interfaces, and as the 5 This seems odd, given that the Rankine-criterion involving the first principal stress is typically used for brittle materials, while DP-steels are ductile. The first principal stress, however, is considered here as a local field variable and not as a macroscopic quantity. 6 The quantile of a cumulative distribution function is shown exemplarily in Fig. 2 (lower right, here: the 90%-quantile P90, or also called 90%-percentile of the cumulative distribution function of plastic equivalent strain in ferrite).

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

185

y x

z

ε (pl) equ

ε (pl) equ (-)

(-)

5 2. 0

00

00

00

00

95

00

00

0.

0.

0.

ε x,macro = 0.0038

ε x,macro = 0.8109

Fig. 5. Interface element sets of microstructure B. Finite-elements contained in the interface element sets are plotted solid, while the others (not considered in the failure criteria) are plotted semi-transparent. The field plot shows the plastic equivalent strain due to uniaxial stress in x-direction (146 235 nodes ¼ ^ 5000 elements per grain, tetrahedrons with linear basis-functions). Left: Close to the macroscopic yield point; right: at 81% log. macro strain.

Table 1 Experimental data of DP-steel grades used for failure criterion calibration. P M , C M and hM denote martensite phase fraction, contiguity and hardness, respectively, while d F and λ denote the average ferrite grain chord length and the grade's hole expansion strain. Steel grade

P M a (–)

C M a (–)

d F a (μm)

hM b,c (HV0:0002 )

λa,d (–)

DP600 DP1000

0.180 0.490

0.335 0.305

4.71 1.95

546.3 457.9

1.10 0.97

a

From [42]. From [43]. Arithmetic mean of 20 measurements. d Samples with drilled holes. b c

martensite/martensite (MM) interfaces,7 as defined by the tessellation and coloring processes. Finite-elements, which touch an MM or an FM interface (and in the latter case belong to ferrite8), respectively, are combined to element sets (see Fig. 5) for the purpose of evaluating the first principal stress distributions. The failure criteria are evaluated in these element sets throughout the RVE's deformation. The quantile size q is assumed as 90% for both interface types. This means that the critical stress must be exceeded in at least 10% of the FM or MM interface element set volumes to flag the RVE as “micro-damaged”. The failure criteria for the two interface types can be written as

although a macroscopic quantity, is approximately9 the strain for micro-damage initiation. For the calibration procedure, five RVEs of each grade are generated with the simulation input as given in Table 1, and are then subjected to a deformation resembling the strain path at the hole's outer rim (see Fig. 6). This condition is modeled by applying uniaxial stress in t-direction. With this assumption, the curvature of the hole is neglected (for a finite RVE size), as well as the contact force between the conical punch and the sheet sample. This is justified, since the edge of the hole typically separates from the surface of the conical punch with increasing penetration depth. The RVEs are deformed until the strain λ in loading direction is reached. In this configuration, the first principal stress distributions at the FM and MM interface element sets are processed and their 90% quantiles are evaluated, see Table 3. The quantiles respective mean value is used as critical stress σ crit in the failure criteria, Eqs. (5) and (6). The critical stresses are significantly higher than e.g. the macroscopic strength of the calibration steel grades. One must note, however, that the critical stresses are not macroscopic, but rather local quantities. They are in the range between the theoretical shear strength of steel, G=ð2 π Þ ffi 13 GPa (with G denoting the shear modulus) and the empiric upper limit for the tensile strength of steel, E=100 ffi 2 GPa (with E denoting the Young's modulus), which seems to be reasonable.

P 90 fσ I g  σ crit;FM Z 0 in ΩFM ; or

ð5Þ

4.2. Results

P 90 fσ I g  σ crit;MM Z 0 in ΩMM :

ð6Þ

The finite-element sets touching an interface are denoted by ΩFM and ΩMM , respectively. The critical stresses σ crit are calibrated from experimental data (see Tables 1 and 2) gained for two industrial DPsteel grades, in the following denoted as calibration steel grades. The grades' hole expansion strain λ, which denotes the engineering strain in circumferential direction at the holes outer rim (see Fig. 6) at first through-thickness macro crack appearance [44], is a particularly important input quantity for the calibration procedure. This is due to the suppression of necking of the two calibration steel grades in the hole expansion (HE) test, due to their high strain hardening. Hence, the hole expansion strain,

Before addressing results of the micro-damage study, for the purpose of validation, the stress/strain response of the calibration steel grade's RVE simulations and tensile tests should be compared, both of which are depicted in Fig. 7. It is obvious that, although the curves are similar, they are not congruent with each other (particularly in case of DP1000). The reasons for the differences observed are the data used as simulation input (particularly the phase specific flowcurves), along with the microstructural modeling assumptions. One must note, however, that no measures are taken to fit the RVE's stress/strain responses to the experimental ones, since this is not the intention of the present work. When the stress/ strain responses of DP600 and DP1000 RVE simulations are compared to each other, though, it is found that the strength of

7 Because of the rugged shape of the martensite phase, stress peaks are typically located at these interfaces. 8 This is an arbitrary specification. The elements which touch an FM interface and belong to martensite could be used instead.

9 The strain for micro-damage initiation, the definition of which is given previously, is of course somewhat lower. The difference between strain for microdamage initiation and strain for macro-crack appearance in the HE-test is neglected here.

186

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

Table 2 Nominal chemical composition (mass-%) of industrial DP-steel grades, of which experimental data is used for failure criterion calibration. Steel grade

C

Si

Mn

P

S

Al

Cr þNi þCu

DP600 DP1000

0.120 0.150

0.500 0.500

1.400 1.500

0.085 0.010

0.008 0.002

0.040 0.040

1.300

Nb

Ti

N

B

0.009 0.015

movement of conical punch

1200

t

ng) σ (xe,m acro (MPa)

1000

z

800 600

DP600 (simulation) DP600 (tensile test) DP1000 (simulation) DP1000 (tensile test)

400 200

−r

0 0.00

0.10

0.05

0.15

0.20

0.25

ε x,macro (-) μm

Fig. 6. Sketch of the hole expansion (HE) test (upper and lower blank holder not shown). The RVEs used for failure criterion calibration are deformed by applying uniaxial stress, similar to the condition at the outer rim of the sheet's hole, shown in the sketch.

Table 3 90% quantile of first principal stress distribution in interface element sets of the calibration steel grade RVEs at (macro) hole expansion strain λ (deformed by applying uniaxial stress). Arithmetic mean of 5 simulations each. The bottom line gives the total mean values used as critical stresses in the failure criteria (5) and (6). Steel grade

P 90 fσ I;FM g (MPa)

P 90 fσ I;MM g (MPa)

DP600 DP1000

7229 5591

7709 7014

Mean value

σ crit;FM ¼ 6410

σ crit;MM ¼ 7361

DP1000 is approximately 40% higher than the strength of DP600, as one would expect. Hence, despite its simplicity, our model is capable of predicting strength properties reasonably well The relation between the failure criteria (5) and (6) and macro strain for microstructure B of Fig. 1 is shown in Fig. 8. The MM failure criterion is fulfilled first at a log. macro strain of approximately 48%. This strain to failure, denoted as εðengÞ , is shown for various x;macro;fail microstructures in the field plot of Fig. 9. The parameter field is M

spanned by the martensite phase fraction P and the phase strength contrast σ 0:025;M =Rm;F .10 According to Byun and Kim [45], these parameters represent the primary influence on stress and strain fluctuations in DP-steel microstructures. The four wireframe graph inserts indicate the volumes in which the failurecriterion is fulfilled (shaded black). Microstructure A of Fig. 1 is located at the left center of the parameter field, whereas microstructure B is located in the center of the field plot. ðengÞ Fig. 10 shows the engineering macro stress at failure, σ x;macro;fail ,

and as a measure for strain hardening, the yield stress ratio,

σ

ðengÞ =Rp0:2 . x;macro;fail

10 The quantity σ 0:025;M in this rather odd definition of phase strength contrast is used because the more intuitive Rm;M is very high for the employed martensite flowcurves. σ 0:025;M is used instead because the martensitic single phase grades used for experimental input fracture at ffi 2:5% log. strain.

Fig. 7. Comparison of stress/strain behavior from tensile test and RVE-simulation (uniaxial stress applied to the RVEs) of the steel grades DP600 and DP1000 used for calibration of the failure criteria. A comparison between tensile test and simulation is feasible up to the point of uniform elongation in the experiment.

P90 σ I,intfc − σ crit,intfc (MPa)

10

8000 6000 4000 2000 0.0 −2000 −4000 −6000 −8000 0.0

ferrite/martensite intfc. martensite/martensite intfc.

0.182

0.336

0.470

0.588

0.693

0.788

ε x,macro (-) Fig. 8. Relation between failure criteria in interface element sets of microstructure B and macro strain (RVE deformed by applying uniaxial stress). The MM-failure criterion, P 90 fσ I;MM g σ crit;MM Z 0, is fulfilled when entering the shaded region, at which point micro-damage initiation is anticipated.

These result quantities are shown for “sectioning” 1 and 2 of Fig. 9 in normalized form in Fig. 11. The normalizations with respect to microstructure B are defined as follows:

 normalized engineering strain to failure, ε~ fail;B ¼

ðengÞ εx;macro;fail ; ðengÞ εx;macro;fail;B

ð7Þ

σ ðengÞ x;macro;fail ; ðengÞ σ x;macro;fail;B

ð8Þ

 normalized engineering stress at failure, σ~ fail;B ¼

 and normalized yield stress ratio, n~ fail;B ¼

σ ðengÞ x;macro;fail Rp0:2



Rp0:2;B ðengÞ σ x;macro;fail;B

:

ð9Þ

4.3. Discussion Typical commercially available DP-steels exhibit martensite phase fractions between about 20 and 50%, see Table 1. Despite

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

(eng)

ε x,macro,fail (-)

Sectioning 2

C

187

D 0.4

4.5

4.0

0.5

0.6

3.5 B

0.7

A

y

0.8

E

3.0 0.9

(-)

1.1

2.5

Rm,F

x

σ 0.025,M

z

1.0

Sectioning 1 0.2

0.4

PM (-)

0.6

0.8

first crit. interface.: MM intfc. FM intfc. non-crit. up to 125 % eng. strain

Fig. 9. Macro engineering strain to failure (i.e. for micro-damage initiation) for microstructures with various phase fractions and phase strength contrasts, deformed by applying uniaxial stress in x-direction. The symbols signify at which interface-type (martensite/martensite, “MM” or ferrite/martensite, “FM”) the failure criteria are fulfilled first. The field plot is linearly interpolated. Four wire-frame graph inserts (phase boundaries and RVE edges shown) indicate the interface volumes of the respective deformed microstructure in which the failure-criterion is fulfilled, i.e. the first principal stress is higher than the critical stress. Ferrite's and martensite's constitutive behavior is given M via d F ¼ 12 μm and 394 r hM r 700 HV for the coefficients of Table B2. C M target ¼ 0:40, C ffi 0:4.

their significantly different tensile strength (DP600: Rm ffi600 MPa, DP1000: Rm ffi 1000 MPa) and martensite phase fractions, it is remarkable that their hole expansion strains λ are similar. This behavior correlates well with the results shown in Fig. 9. In the following, we assume the macro strain to failure, εðengÞ , to be x;macro;fail equivalent to the hole expansion strain λ. According to Fig. 9, a microstructure with a high martensite phase fraction must have a lower phase strength contrast than one with a low martensite phase fraction in order to exhibit the same hole expansion strain. This is indeed true for the steels DP600 and DP1000, as confirmed by their martensite hardness and ferrite grain size (Table 1), and their approximately equal carbon-content (Table 2). In order to simplify the interpretation of Fig. 9, a scalar value, which correlates with macro strain to failure is introduced. Guided by Fig. 11, which suggests a parabolic relation between ε~ fail;B and martensite phase fraction (left) and a linear relation between the former and phase strength contrast (right), the mechanical heterogeneity of a dual-phase microstructure can be defined as

Φ¼4

σ 0:025;M Rm;F

 ðP M  ðP M Þ2 Þ;

ð10Þ

or alternatively (which is easier to determine experimentally) as

Φh ¼ 4

hM  ðP M  ðP M Þ2 Þ; hF

ð11Þ

with hF denoting the ferrite hardness. Hence, microstructures with high mechanical heterogeneity Φ (high phase strength contrast and a martensite phase fraction of about 50%) exhibit low values of macro strain to failure and vice versa. It should be noted that failure is predicted by our model only if the failure criteria at the ferrite/martensite and/or martensite/ martensite interfaces are fulfilled. In some DP-steel grades, and particularly in nearly single phase “soft” martensitic grades, experimental investigations show micro-damage in the form of voids at sufficiently large non-metallic inclusions [10,14,42]. It is

reasonable to assume that the number of voids, which nucleate by this particular mechanism that is not considered in the present model, correlates with the number of non-metallic inclusions (provided they are of sufficient size). Therefore, as P M -1, the results of the present study are valid only under the assumption that non-metallic inclusions are of negligible size and/or number. Fig. 10 shows that for a given phase strength contrast, macro stress at failure increases with rising martensite phase fraction. This result correlates with findings of Davies [1] and Kim and Thomas [5]. An overall strength increase can also be achieved by increasing the phase strength contrast, even though this strategy is less effective because simultaneously strain to failure is reduced significantly. Fig. 11 illustrates this behavior clearly by the marked dependence of σ~ fail;B on martensite phase fraction (left), and a much less pronounced dependence on the phase strength contrast (right). Strain hardening or, equivalently, the yield stress ratio (Fig. 10, right and n~ fail;B in Fig. 11) shows a nearly opposite behavior to the strain to failure. Microstructures with a high mechanical heterogeneity (large Φ) show higher strain hardening than microstructures with a low mechanical heterogeneity. This is plausible, considering that the higher the mechanical heterogeneity, the higher is the phase averaged plastic equivalent strain in ferrite at a given macro strain, see Section 3 and Fig. 2. This ensures high macroscopic strain hardening, even if ferrite exhibits low strain hardening on the micro-level. Of particular interest is the nonmonotonous behavior of n~ fail;B in Fig. 11, left. For martensite phase fractions below 30% there is a marked dependence of n~ fail;B on the phase fraction, while it is nearly independent of the phase fraction above 50% martensite content. Most likely responsible for this behavior is the topology of the martensite phase, which is either composed of islands of martensite grains or represents an interconnected martensite-skeleton. The latter is typical above roughly 35% martensite content (see Fig. 1, right). A martensite-skeleton is not primarily deformed by stresses at the ferrite/martensite

188

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

(eng)

σ x,macro,fail Rp0.2 (-)

(eng)

σ x,macro,fail (MPa)

C

C

D

D

4.5

4.0

3000

2000

3.5

σ 0.025,M Rm,F

(-)

1000 3.0 3.5

2.5

B E

A

B E

A

3.0

2.5

0.2

0.4

0.6

0.4

0.2

0.8

PM (-)

0.6

0.8

P (-) M

ε˜ fail,B (-), σ˜ fail,B (-), n˜ fail,B (-)

Fig. 10. Macro engineering stress at failure (left) and yield stress ratio (right) for the same microstructures as in the field plot of Fig. 9. The field plots are linearly interpolated. The symbols have the same meaning as in Fig. 9.

sectioning 1

2.5 2.0

sectioning 2 2.5

ε˜ fail,B σ˜ fail,B n˜ fail,B

E 2.0

A

1.5

1.5 B

B

1.0

1.0

0.5

0.5

0.0 0.2

0.4

0.6

0.8

0.0

2.0

2.5

3.0 σ 0.025,M Rm,F

PM (-)

3.5

4.0

4.5

(-)

Fig. 11. Sectioning 1 and 2 of Fig. 9, showing the behavior of the normalized engineering strain to failure (~ε fail;B ), normalized engineering stress at failure (σ~ fail;B ) and the normalized yield stress ratio (n~ fail;B ) as a function of phase fraction and phase strength contrast, respectively (curves are quadratically interpolated; symbols have the same meaning as in Fig. 9).

interface, as in case of isolated martensite islands (see Fig. 1, left), but rather by being directly subjected to the external load, since the grains are interconnected. Byun and Kim [45] discuss a difference in strain hardening behavior of DP-steels above and below a martensite phase fraction of P M ffi 0:3. Although in their study the martensite phase fraction varies along with the strength contrast (this would mean an inclined sectioning through Fig. 10, right), their finding may just as well be explained by the existence of a martensite-skeleton at martensite phase fractions above 30%. The wire-frame graphs of Fig. 9 indicate a deficiency of the failure criterion as implemented here: By comparing the left with the right wire-frames, one will note that the shaded (microdamaged) areas clearly are of different size. This is explained by Eqs. (5) and (6), since the failure criteria are evaluated in the interface volumes ΩFM and ΩMM , which correlate with martensite phase fraction. Hence, if the interface volume is high, then so is the micro-damaged volume when the respective failure criterion is fulfilled.

5. Conclusions Although the theoretical framework of the present contribution is based on simplifying modeling assumptions, reasonable results in accordance with experimental findings indicate that these assumptions do not violate the primary mechanisms that govern microstructural stress and strain fluctuations in DP-steels. The impact of secondary effects, such as different dislocation densities at phase boundaries during the deformation of coarse and fine grained microstructures, non-continuous (crystal-plastic) deformation of grains of both phases, or micro-damage at non-metallic inclusions are neglected within the presented model. Keeping these assumptions in mind, the following conclusions can be drawn:

 Although it is hard to imagine how plastic deformation is taking place within dual-phase microstructures, it may be grasped, if one considers phase averaged strains and the superimposed strain distributions in parallel, see Fig. 2. In this

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192







approach it is most important to consider the primary influences on the deformation mechanisms, namely the phase strength contrast and the hard phase fraction. In order to obtain high strength dual-phase steel microstructures (here: macro stress at micro-damage initiation, see Fig. 10, left), it is favorable to employ a high martensite phase fraction. This, however, is compromised by poor formability of the martensite phase (governed by e.g. void nucleation at nonmetallic inclusions). A high overall strength may also be obtained by increasing the phase strength contrast; however, not so effectively, since along with this the strain for micro-damage initiation (see Fig. 9) decreases significantly. Macroscopic strength may in theory even be increased while keeping martensite phase fraction and phase strength contrast constant, if the strength of both phases is raised (e.g. increasing martensite hardness and decreasing ferrite grain size). Along with this approach, however, strain for micro-damage initiation decreases. Whether the first or the last of the suggested three modifications is more favorable cannot be concluded here, since the employed model delivers qualitative results. In order to obtain a high hole expansion strain (here: strain for micro-damage initiation), microstructures with low mechanical heterogeneity (low phase strength contrast, very high or very low martensite phase fraction) are favorable, as long as they do not exhibit necking in the HE-test. Microstructures with a high mechanical heterogeneity on the other hand suffer from high average and peak strain in both ferrite and martensite, which leads to micro-damage at rather low macro strains. In order to obtain high strain hardening (e.g. in cases where necking is to be avoided in a given forming application, viz. a good deep drawability is desired) a high mechanical heterogeneity (high phase strength contrast and martensite phase fraction of around 50%) is favorable. By this measure, ferrite is subjected to significantly higher average strain than the macro strain, letting the microstructure significantly strain harden. Note, however, that the strain for micro-damage initiation displays the opposite behavior.

Acknowledgments The authors express their appreciation to their former colleague B. Regener, who contributed to this work by sharing his code scripts for seed-point generation and tessellation, as well as his (and T. Taxer's) “XPBC”-script. We also thank R. Wesenjak, who implemented the pre-processing script, a convenient framework for the “Scarlet”-Code, and gave access to his computer for additional power during particularly busy times. Special thanks goes to Y. Granbom of SSAB and A. Pichler of voestalpine, for the constructive input and for providing the necessary industrial steel samples. C. Murphy's proofreading is highly appreciated. We are particularly grateful for the support from the Research Fund for Coal and Steel through grant RFSR-CT-2008-00027.

Appendix A. Microstructure geometry generation A.1. Grain geometry

189

minimum distance di to their spatially next seed-point neighbor. The minimum distance di is element of a truncated Gaussiandistribution with parameters  0:8  0:2 ðA:1Þ μd ¼ p3 ffiffiffiffiffiσ d ¼ p3 ffiffiffiffiffi; ns ns and 1 ffiffiffiffiffi: dmin ¼ 0jdmax ¼ p 3 ns

ðA:2Þ

The seed-points are then copied in all directions of periodicity of the volume element, so that the resulting group of seed-points (containing nps ¼ 27  ns seed-points) leads to a periodic grain structure in the volume element. The ordinary Voronoï-tessellation [28,29] associates each point in the volume element space to uniquely one point out of the periodic seed-point set and thereby defines the grain geometries. Let the periodic seed-point set be defined as P ps ¼ fp1 ; …; pnps g and let xi denote the position vector of the individual seed-point pi. Further, let the indices i and j be A I n ¼ f1; …; nps g. Then, the space belonging to grain i is defined as V i ðpi Þ ¼ fxjJ x  xi J r J x xj J 8 j A I n \figg:

ðA:3Þ

A.2. Coloring process (Grain to Phase Assignment) The coloring process assigns selected grains of the initial grain structure to martensite (the remaining grains compose the ferrite). Two coloring parameters are employed to control this process, namely the martensite phase fraction P M and the martensite contiguity CM ¼

2 SMM V FM 2 SMM V þ SV

;

ðA:4Þ

and SFM denote the volume specific martensite/ where SMM V V martensite grain boundary and ferrite/martensite phase boundary area, respectively. In the present case the number of grains is finite, so the contiguity cannot be varied freely between 0 and 1 for a given martensite phase fraction (and vice versa). The employed coloring algorithm in its numerical frame uses “soft” criteria for acceptance of a coloring pattern, jP M  P M target jr P tol PM target

and

jC M C M target j rC tol ;

ðA:5Þ

CM target

and denote the desired coloring parameters, where P tol and C tol denote the tolerance limits for accepting a coloring pattern, and P M and C M denote the coloring parameters the actual pattern exhibits. Note that a periodic initial grain structure may become nonperiodic by poor coloring. Periodicity after coloring may be ensured by treating the periodic grain fragments that intersect the RVE's surface as a single grain in the coloring process. Appendix B. Derivation of “master” flowcurve functions Flowcurves of ferrite and martensite in this work are based on tensile tests of single phase industrial steel grades, see Table B1. Each grade's tensile test stress–strain response (up to uniform elongation) is fitted with the generalized Voce-function [46] ðplÞ

 εequ θ0 =σ 1 σ F ¼ σ y þ ðσ 1 þ θ1  εðplÞ Þ; equ Þ  ð1  e 3

The volume element (here:  R , ½0; 1 ½0; 1 ½0; 1) which is to be divided into grains, is filled with seed-points of quantity ns (theoretically: 2 r ns o 1, here, however, 150). The seed-points are placed randomly in the volume element, but must have a

ðB:1Þ

denote the to give sets of parameters σ y , σ1, θ1 and θ0. σ F and ε flow stress and the plastic equivalent strain, respectively. The parameter sets of the martensite grades are used to model the martensite “master” flowcurve parameters σ y , σ1, θ1 as a linear ðplÞ equ

190

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

Table B1 Nominal chemical composition (mass-%), hardness and average grain chord length of single phase industrial steel grades, used for derivation of “master” flowcurve functions. ULC, ELC and ULC-P denote ultra-low carbon ferritic steel grades, while designations starting with “M” denote martensitic grades.

0.002 0.020 0.002

0.200

0.120 0.180 0.500

M900 M1200 M1400 M1700

0.050 0.110 0.170 0.280

0.200 0.200 0.200 0.200

2.000 1.700 1.400 0.700

S

0.002 0.002 0.002

Table B2 Coefficients of “master” flowcurve functions (utilizing the generalized-Voce function (B.1)) and elastic constants of martensite and ferrite. d F : average chord length of ferrite grains (μm); hM : Vickers hardness of martensite (HV). Coefficient

Unit

Ferrite

Martensite

σy

MPa

521:8 77:0 þ qffiffiffiffiffiffi a dF

 224:7 þ 2:267  hM b

σ1 θ1

MPa MPa

145c

 170:3 þ 0:983  hM b  8570:5 þ 30:583  hM b

1051:7 126:4 þ qffiffiffiffiffiffi c dF

Nb

Ti

0.035 0.040 0.035

0.070 0.010 0.010 0.010

Al

0.040 0.040 0.040

–

24.0c

589.9b

E ν

GPa –

202d 0.3d

206d 0.3d

a Obtained from [47], since the ULC-P grade in Table B1 differs significantly in Si and P contents from the other ferritic steel grades. b Determined by using tensile test data of single phase martensitic grades of Table B1. c Determined by using tensile test data of single phase ferritic grades of Table B1. d From [48].

function of hardness, while the ratio θ0 =σ 1 is found to be approximately constant. Similarly, the parameter sets of the ferrite grades are used to modelq ferrite “master” flowcurve parameter θ1 ffiffiffiffiffi as linear function of 1= d F , while σ1 and θ0 are found to be approximately constant and σ y is obtained from literature. Table B2 summarizes the results. Note that the master flowcurve functions extrapolate the experimental stress–strain data input beyond uniform elongation according to Eq. (B.1).

Appendix C. Convergence study Volume elements with various number of grains and mesh densities are deformed by applying uniaxial stress. The resulting macro- and field quantities at 23% engineering strain are shown in Fig. C1. Since there is no significant difference between the results of RVEs comprising 53 ¼ 125 and 73 ¼ 343 grains, and in order to reduce computational costs it is sufficient to use RVEs with 150 grains to achieve convergent results. The finite-element mesh density on the other hand has to be selected more cautiously, since numerical convergence is markedly affected by the mechanical contrast between the two phases [36]. The mesh density for the RVEs found to be sufficient in this work is  5000 elements per grain.

h (HV0:01 ; HV0:005 )

d (μm)

0.001

93 110 180

20.0 19.8 8.1

0.065 0.065 0.015 0.015

408 505 544 625

0.025 0.025

1.0 0.8 0.6 0.4

33 73

0.2

elements grain

103 0.0

θ0 σ1

B

P99 σ I,M / P99 σ I,M,max (-)

ULC ELC ULC-P

P

(-)

Mn

(eng) x,macro,0.23,max

Si

(eng)

C

σ x,macro,0.23 /σ

Steel grade

√9

√125

√ngrains (-)

√343

√9

125

343

√ngrains (-)

Fig. C1. Results of a convergence study employing volume elements with different number of grains and finite-element mesh densities (using tetrahedron elements with linear basis functions, denoted as C3D4). The volume elements are deformed to 23% macro engineering strain by applying uniaxial stress. Shown are the arithmetic mean and standard deviation of 6 simulations each. Left: Macro engineering stress in deformation direction; right: 99%-quantile of first principal stress distribution in martensite; P M CM d F ¼ 6:3 μm, target ¼ 0:35, target ¼ 0:22, hM ¼ 546:3 HV (resembling a DP600 steel grade).

Appendix D. Influence of transformation induced microstrains on failure criteria To quantify the influence of transformation induced microstrains on results studied in the foregoing sections, microstructure B of Fig. 1 is subjected to the following procedure: In addition to the microstructure's properties as given in the foregoing sections, martensite is modeled to exhibit a 2.6% isotropic volume increase [6] alongside the thermal contraction (with ath ¼1.2  10  5 1/K for martensite as well as for ferrite), during an initial cooling from 400 1C to 20 1C. The Young's moduli of ferrite and martensite are assumed to be linearly temperature dependent and to amount 170 GPa at 400 1C. The ferrite flowcurve is assumed to be linearly temperature dependent, exhibiting 50% of its original flowstress at 400 1C, while the martensite flowcurve is assumed to be temperature invariant. The cumulative distribution of plastic equivalent strain in ferrite and martensite resulting from the initial transformationstep is depicted in Fig. D1, denoted with ⋆. The distributions denoted with ♣ result from deforming the microstructure to 125% engineering strain (by applying uniaxial stress) after the initial transformation-step. The distributions denoted with ♠ show the results of the same microstructure, without performing the initial transformation-step (i.e. disregarding transformation induced micro-strains). The relations between the failure criteria and macro strain of microstructure B are shown in Fig. D2. The curves denoted with ♣

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

For a macroscopically isotropic volume element the transverse strains

1.0

εy and εz will each be  12εx , if one neglects elastic components.

cumul. distr. function (-)

♠ 0.8 0.6 0.4

♣

0.2

ferrite martensite

0.0

0.5

0.811 (pl) ε equ

1.0

1.5

(-)

(MPa) P90 σ I,intfc − σ crit,intfc

Fig. D1. Cumulative distribution function of plastic equivalent strain in ferrite and martensite of microstructure B. ♠ and ♣: microstructure deformed to 125% macro engineering strain by applying uniaxial stress; ♠: simulation disregarding transformation induced micro-strains (equal as in Fig. 2); ♣: simulation taking into account initial transformation induced micro-strains; ⋆: initial transformation induced micro-strain distributions (due to 2.6% volume increase of martensite).

0

− 2000

♠

♣

−8000 0.0

To load the RVE in the same mode but in an arbitrary direction of the xy-plane, the deformation gradient has to be rotated about the z-axis, giving 0 2 ε 1 c e x þs2 eεy csðeεy  eεx Þ 0 B C ðE:3Þ F ¼ @ csðeεy eεx Þ s2 eεx þc2 eεy 0 A; φ

0

eεz

with c ¼ cos φ and s ¼ sin φ. The according master node displacements are 0 2 ε 1 0 1 csðeεy  eεx Þ c e x þ s2 eεy  1 B C B C u Mx ¼ @ csðeεy  eεx Þ A; u My ¼ @ s2 eεx þ c2 eεy 1 A: ðE:4Þ 0 0 Since only component εx is specified and εy adjusts according to the RVE's mechanical behavior, the master node displacements for a tilted loading direction are implemented as three additional equations into the numerical framework.

[1] [2] [3] [4]

♣

ferrite/martensite interface martensite/martensite intfc. 0.182

0.336

0.470

0.588

ε x,macro (-) Fig. D2. Relation between failure criteria in interface element sets of microstructure B and macro strain (RVE deformed by applying uniaxial stress). ♠: Simulation disregarding transformation induced micro-strains (equal as in Fig. 8); ♣: simulation taking into account initial transformation induced micro-strains (due to 2.6% volume expansion of martensite).

[5] [6] [7] [8] [9] [10] [11] [12] [13]

result from deforming the microstructure after the initial transformation-step, while the curves denoted with ♠ result from deforming the same microstructure, without performing the initial transformation-step.

[14] [15] [16] [17] [18]

Appendix E. Master node displacement for tilted loading direction Let a volume element be deformed by applying uniaxial stress in x-direction. This is accomplished by either placing a force in xdirection on the x-master node,11 or by displacing it by 0 εx 1 e 1 B C u Mx ¼ @ 0 A ðE:1Þ 0 (with εx as the macroscopic log. strain in x-direction and assuming an RVE length of 1), and not constraining the two other master nodes. Assuming that no shear deformations will result from this loading mode, the according deformation gradient can be written as 0 εx 1 0 0 e B C ðE:2Þ F ¼ @ 0 eεy 0 A: 0 0 eεz

[19] [20] [21] [22]

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

11

0

References

− 4000 − 6000

191

Displacement of this node equals the displacement difference of the two RVE-surfaces which have the x and  x axes, respectively, as their normal vector.

[36]

R.G. Davies, Metall. Trans. A 9 (1978) 41–52. M. Sarwar, R. Priestner, J. Mater. Sci. 31 (1996) 2091–2095. F.M. Al-Abbasi, J.A. Nemes, Int. J. Mech. Sci. 45 (2003) 1449–1465. A. Col, T. Balan, in: Numiform: Materials Processing and Design: Modeling, Simulation and Applications, Parts I and II, vol. 908, AIP Conference Proceedings, 2007, pp. 147–152. N.J. Kim, G. Thomas, Metall. Trans. A 12 (1981) 483–489. U. Liedl, S. Traint, E.A. Werner, Comput. Mater. Sci. 25 (2002) 122–128. M.S. Rashid, GM 980X-A Unique High Strength Sheet Steel with Superior Formability, Society of Automotive Engineers Inc., Warrendale, PA, 1976. H. Fischmeister, B. Karlsson, Zeitschrift für Metallkunde 68 (1977) 311–327. A.F. Szewczyk, J. Gurland, Metall. Trans. A 13A (1982) 1821–1826. P. Uggowitzer, H.P. Stüwe, Mater. Sci. Eng. 55 (1982) 181–189. X.J. He, N. Terao, A. Berghezan, Metal Sci. 18 (1984) 367–373. D.L. Steinbrunner, D.K. Matlock, G. Krauss, Metall. Trans. A 19A (1988) 579–589. P. Tsipouridis, E.A. Werner, C. Krempaszky, E. Tragl, Steel Res. Int. 77 (2006) 654–667. J. Kadkhodapour, A. Butz, S.Z. Rad, Acta Mater. 59 (2011) 2575–2588. M. Westphal, J.R. McDermid, J.D. Boyd, J.D. Embury, Can. Metall. Q. 49 (2010) 91–97. M. Azuma, S. Goutianos, N. Hansen, G. Winther, X. Huang, Mater. Sci. Technol. 28 (2012) 1092–1100. G. Avramovic-Cingara, Y. Ososkov, M.K. Jain, D.S. Wilkinson, Mater. Sci. Eng. A 516 (2009) 7–16. H.J. Böhm, A Short Introduction to Basic Aspects of Continuum Micromechanics, ILSB Report 206, Technical Report, Vienna University of Technology, 2009. U. Prahl, S. Papaefthymiou, V. Uthaisangsuk, W. Bleck, J. Sietsma, S. Van der Zwaag, Comput. Mater. Sci. 39 (2007) 17–22. A.L. Gurson, J. Eng. Mater. Technol. 99 (1977) 2–15. A. Needleman, N. Tvergaard, Acta Metall. 32 (1984) 157–169. C. Krempaszky, J. Ocenasek, V. Espinoza, E.A. Werner, T. Hebesberger, A. Pichler, in: Materials Science & Technology: Formability of Advanced Automotive Alloys, Association for Iron & Steel and TMS, 2007, pp. 431–444. X. Sun, K.S. Choi, W.N. Liu, M.A. Khaleel, Int. J. Plast. 25 (2009) 1888–1909. V. Uthaisangsuk, U. Prahl, W. Bleck, Comput. Mater. Sci. 43 (2007) 27–35. V. Uthaisangsuk, U. Prahl, W. Bleck, Eng. Fract. Mech. 78 (2011) 469–486. J. Kadkhodapour, A. Butz, S. Ziaei-Rad, S. Schmauder, Int. J. Plast. 27 (2011) 1103–1125. J.H. Kim, D. Kim, F. Barlat, M.G. Lee, Mater. Sci. Eng. A 539 (2012) 259–270. G. Voronoï, J. für die reine und angew. Math. 134 (1908) 198–287. A. Okabe, B. Boots, K. Sugihara, S.N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, John Wiley & Sons, Ltd, Chichester, 2000. E.A. Werner, T. Siegmund, H. Weinhandl, F.D. Fischer, Appl. Mech. Rev. 47 (1994) 231–240. R.G. Davies, Metall. Trans. A 9 (1978) 671–679. H. Weinhandl, E.A. Werner, Neue Hütte 37 (1992) 141–145. J. Rösler, H. Harders, M. Bäker, Mechanisches Verhalten der Werkstoffe, Vieweg þTeubner, Wiesbaden, 2008. P. Tsipouridis, L. Koll, C. Krempaszky, E.A. Werner, Int. J. Mater. Res. 102 (2011) 674–686. K. Terada, M. Hori, T. Kyoya, N. Kikuchi, Int. J. Solids Struct. 37 (2000) 2285–2311. T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Int. J. Solids Struct. 40 (2003) 3647–3679.

192

A. Fillafer et al. / Materials Science & Engineering A 614 (2014) 180–192

[37] V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials (Ph.D. thesis), Technische Universiteit Eindhoven, 2002. [38] M. Considère, Mémoire sur l'emploi du fer et de l'acier dans les constructions, Vue Ch. Dunod, Paris, 1885. [39] ABAQUS, Manuals, Version 6.10EF, Dassault Systèmes Simulia Corp., 2010. [40] H.P. Shen, T.C. Lei, J.Z. Liu, Mater. Sci. Technol. 2 (1986) 28–33. [41] Y.L. Su, J. Gurland, Mater. Sci. Eng. 95 (1987) 151–165. [42] L. Ryde, D. Lindell, J.G. Ferreno, Y. Granbom-Vilander, A. Fillafer, E. A. Werner, R. Wesenjak, A. Pichler, Micro-Scale Damage Tolerance of AHS Steels as Function of Microstructure and Stress/Strain State – Final Report, Technical Report RFSR-CT-2008-00027, Research Programme of the Research Fund for Steel RTD, 2011. [43] L. Ryde, J.G. Ferreno, Y. Granbom-Vilander, P. Tsipouridis, A. Nitschke, M. Engman, Micro-Scale Damage Tolerance of AHS Steels as Function of

[44] [45] [46] [47] [48]

Microstructure and Stress/Strain State – Mid-Term Report, Technical Report RFSR-CT-2008-00027, Research Programme of the Research Fund for Steel RTD, 2009. A. Karelova, C. Krempaszky, E. Werner, P. Tsipouridis, E. Tragl, A. Pichler, in: Materials Science & Technology: Materials and Systems, vol. 2, pp. 17–27. T.S. Byun, I.S. Kim, J. Mater. Sci. 28 (1993) 2923–2932. E. Voce, Metallurgia 51 (1955) 219–226. T. Gladman, F.B. Pickering, Yield, Flow and Fracture of Polycrystals, Applied Science Publishers, London, pp. 141–198. U. Liedl, Anfangsverformung- und Alterungsverhalten von Dual-Phasen Stahl (Ph.D. thesis), Technische Universität München, 2003.