A study of microcrack formation in multiphase steel using representative volume element and damage mechanics

Computational Materials Science 50 (2011) 1225–1232

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A study of microcrack formation in multiphase steel using representative volume element and damage mechanics V. Uthaisangsuk a,⇑, S. Muenstermann b, U. Prahl b, W. Bleck b, H.-P. Schmitz c, T. Pretorius c a

Department of Mechanical Engineering, King Mongkut’s University of Technology, Thonburi, 126 Prachautit Rd., Bangmod, Tungkru, Bangkok 10140, Thailand Department of Ferrous Metallurgy, RWTH Aachen University, Intzestr. 1, D-52072 Aachen, Germany c ThyssenKrupp Steel Europe AG, Kaiser-Wilhelm-Strasse 100, D-47166 Duisburg, Germany b

a r t i c l e

i n f o

Article history: Received 5 October 2009 Received in revised form 18 May 2010 Accepted 3 August 2010 Available online 1 September 2010 Keywords: Multiphase steels Representative volume element Damage curve Microcracks

a b s t r a c t Multiphase steels have become a favoured material for car bodies due to their high strength and good formability. Concerning the modelling of mechanical properties and failure behaviour of multiphase steels, representative volume elements (RVE) have been proved to be an applicable approach for describing heterogeneous microstructures. However, many multiphase steels exhibit inhomogeneous microstructures which result from segregation processes during continuous casting. These segregations lead to a formation of martensite bands in the microstructure causing undesirable inhomogeneities of material properties. The aim of this work is to develop an FE evaluation procedure for predicting a microcrack formation provoked by banded martensitic structures. A micromechanism based damage curve was applied as a failure criterion for the softer ferritic matrix in the microstructure in order to simulate the propagation of cracks resulting from the failure of martensitic bands. The parameters of the damage curve were determined by in situ miniature bending tests and tensile tests with notched samples. The presented approach provides the basis for an assessment criterion of the component safety risk of multiphase steels with inhomogeneous microstructures. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction A variety of high strength steels like low-alloy dual phase (DP) and transformation induced plasticity (TRIP) steels have been developed for the automotive industry with respect to the increasing demand for lighter, more deformable, and high energy absorbing materials [1,2]. These steels exhibit excellent mechanical properties as they combine high strength and good ductility compared to conventional steels of similar strength. This favourable balance of properties is owing to the existence of different phases in their microstructures. Modifications of the chemical composition, thermomechanical processing or heat treatments allow various microstructure formations. Type, shape, size, fraction, and spatial distribution of the different phases are in charge of the overall behaviour [3–5]. Due to the multiphase microstructures, numerical FE simulation techniques using representative volume elements (RVE) as a submodel implemented to macroscopic global models have become a promising method for quantitatively correlating microstructure and mechanical properties of materials. However, the ⇑ Corresponding author. Address: Department of Mechanical Engineering, King Mongkut’s University of Technology, Thonburi, 126 Prachautit Rd., Bangmod, Tungkru, Bangkok 10140, Thailand. Tel.: +66 2 470 9274; fax: +66 2 470 9111. E-mail address: [email protected] (V. Uthaisangsuk). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.08.007

RVE is usually generated assuming a homogeneous distribution of microstructural constituents. Hence, the effect of segregations and similar microstructural inhomogeneities has not yet been incorporated adequately in the RVE simulations. Indeed, it has been reported that segregations significantly alter the failure behaviour of multiphase steels. It was noted that multiphase steels show a tendency to contain segregations as a result of high alloy contents and specific thermal cycles during the continuous hot dip galvanising process [6]. Assuming fast cooling conditions, there is not enough time for carbon diffusion and ferrite nucleation and thus no banded structure results. However, fast cooling only suppresses the formation of a banded microstructure, but the reason for banding, i.e. the microsegregation, cannot be removed [7]. In consequence, the bands will reappear when a specimen with an inhibited banding is reheated and cooled down slowly. The effect of cooling rate on the formation of martensite bands has also been investigated by Thompson and Howell [8]. They noticed that banding is much less pronounced in samples taken from the edge of a hot rolled sheet where the cooling rate is higher than in the centre of the sheet. It is furthermore possible to obtain banded ferrite/martensite microstructures by holding material in the austenite/ferrite two-phase region and subsequent fast cooling [9]. The effect of banded martensitic structures on mechanical properties has also been investigated by several authors. Nevertheless,

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the findings of these investigations are inconclusive. Caballero et al. [10] reported that banded martensitic structures cause a deterioration of material properties, as both the overall ductility and the impact toughness of the steel sheets are decreased. Furthermore, the inhomogeneous phase distribution also leads to a lowering of crack resistance [11]. In contradiction to these results, Grange [12] noted a positive effect of martensite bands on the ductility as he observed that a lower ductility regarding the reduction in area was achieved for microstructure without banding. By fractographic investigations in [13], it was shown that the improved ductility of banded steel is due to crack arresting mechanisms in ferrite bands. On the other hand, the impact energy in the ductile range of a micro-alloyed steel decreases with increasing number of bands per unit area [14]. Resulting from these inconsistent findings it can be concluded that an approach to systematically study the influence of microstructural inhomogeneity on mechanical properties needs to be developed. The aim of this work is to develop an evaluation procedure for predicting the formation of microcracks as a result of applied mechanical loading. This approach will provide fundament for an assessment criterion of the component safety risk. In these investigations, a RVE has been used to describe the multiphase microstructure within the framework of continuum mechanics. Different configurations of martensite bands in terms of size, number, and location were considered in the RVE model. A micromechanism based damage curve was applied in RVE simulations as a failure criterion for the ductile ferrite to predict the propagation of microcracks resulting from the failure of martensitic bands. The damage curve was intensively discussed by Bao and Wierzbicki [15] who performed a series of experimental tests on an aluminium alloy to provide an instruction for the dependence of fracture ductility on the stress triaxiality. They developed a damage curve which shows three distinct branches with discontinuities in the transition region. Fig. 1 shows the damage curve they provided for a 2024-T 351 aluminum alloy. In this work, parameters for the damage curve were determined by an in situ miniature bending test and tensile test of a notched sample. The results were verified by a conventional sheet stretch-forming test and a hydraulic bulge test. The influences of the banded structure induced by microsegregation on the microstructure development and microcrack initiation were investigated.

problem. Instead, the microstructural constituents have to be considered by the calculations. For this reason an RVE model was applied to represent the multiphase microstructures. Effective mechanical properties of individual phases in the RVE were defined separately. The RVE is linked to a macroscopic calculation by a weak coupling. By applying a failure criterion in the RVE calculations, the initiation of microcrack as a result of martensite segregations should be numerically predicted. The RVE was generated as a submodel of macroscopic models, for which the investigated material is simply described as a single isotropic material [16]. The deformation tensors computed from the macromodel were transferred to the submodel as boundary conditions. In the submodel, a multiphase microstructure was taken into account as different material models were defined. This RVE submodel can be arranged for the corresponding volume fraction and morphologies determined from a real microstructure. Yield behaviour of different individual microstructural phases were defined on the basis of empirical models, dislocation theory based approaches, and local chemical compositions of material [17–21]. The empirical approaches for formulating the flow curves are given in Eqs. (1)–(3) with the chemical composition in mass % for Ferrite [18,22]:

r ¼ 77 þ 5000  %Css þ 80  %Mn þ 60  %Si þ 750  %P þ 1000  e0:3 ; Bainite [19,22]:

r ¼ 0:55  ð15:4  ð16 þ 125  %C þ 15  %MnÞÞ þ 1150  e0:35 ;

In general, fracture mechanics approaches are used to describe and quantitatively predict crack initiation in engineering structures. In case of multiphase steels, pre-cracking occurs in the magnitude of these distributed phases. A macroscopic approach based on fracture mechanics is therefore not acceptable for the specified

equivalent plastic strain ε

p

1.80 f -0,46 , -1/30,4

1.60 1.40 1.20 1.00 0.80 0.60

Shear fracture

Fracture due to void formation

0.40 0.20 0.00 -0.40

-0.20

0.00

0.20 0.40 0.60 stress triaxiality h

0.80

1.00

1.20

Fig. 1. Damage curve of a 2024-T 351 aluminum alloy provided by Bao and Wierzbicki [15].

ð2Þ

Retained austenite [22]:

r ¼ 650 þ 1200  e0:4 ;

ð3Þ

In these equations, %Css is the carbon content in solid solution and e is the equivalent plastic strain. These functions are related to an estimation of yield behaviour depending on the chemical compositions and can be applied for each given type of microstructure. The Eq. (4) is a modelling of yield behaviour as a function of microstructural constituents and alloying composition for the ferritic and martensitic phase [19,20].

pffiffiffi

r ¼ r 0 þ Dr þ a  M  l  b  2. Method

ð1Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  expðM  k  eÞ kL

ð4Þ

a is a constant, M is the Taylor factor (M = 3), l is the shear modulus (l = 80,000 MPa), and b is the Burger’s vector (b ¼ 2:5  1010 m). L is the dislocation mean free path, k is the recovery rate, and Dr is the additional strengthening with regard to the precipitation and carbon in solution. These three parameters for ferrite and martensite are different and were discussed in [19,20,22]. r0 is the Peierls stress which is a function of the alloying elements in solid solution as presented in the equation:

r0 ¼ 77 þ 80  ð%MnÞ þ 750  ð%PÞ þ 60  ð%SiÞ þ 80  ð%CuÞ þ 45  ð%NiÞ þ 60  ð%CrÞ þ 11  ð%MoÞ þ 5000  N:

ð5Þ

It was shown that the modelling of the flow curve by means of these functions is significantly depending on the local concentration of interstitial dissolved alloying elements. Therefore, the carbon partitioning was firstly estimated according to the mass balance and phase fraction distribution. For modelling of ductile failure processes according to the mechanism of void nucleation, growth and coalescence, several empirical failure criteria and micromechanical damage models can be used. A reliable criterion is the damage curve that was for instance used successfully in [23,24]. Damage curves describe the critical local stress–strain response leading to the crack initiation,

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as the sustainable equivalent plastic strain is defined as a function of the stress triaxiality h [25,26], which is calculated as

h¼

rm 13  ðr1 þ r2 þ r3 Þ ¼ : r r

ð6Þ

r is the effective stress that is proportional to the root-mean square of the principal shear stress. rm is the hydrostatic stress, which is responsible for a volume increase of cavities in a porous matrix. The hydrostatic stress is an invariant of the stress tensor and does not contribute to the plastic state. The effective plastic strain epl is proportional to the root-mean square of the principal plastic strains e1, e2, and e3. Arndt et al. [27] modified the damage parameter introduced by Rice and Tracey [28] by defining the failure strain in an easier form:

eplc ¼ A  eBh þ ei :

ð7Þ

eplc is the critical plastic strain at which failure initiates. A and B are the material dependent constants representing the position and the exponential shape of the damage curve. The function is represented in a diagram as a limit curve, so-called damage or failure curve, describing the arbitrary state of ductile crack initiation or the first coalescence of neighbouring voids for a given stress–strain field. Below this curve the material is saved and no ductile crack initiation will occur. To determine the damage curve, specimens are deformed up to ductile crack initiation. Subsequently, the site of the crack initiation is identified by metallographic investigations. The effective stress and strain values in this critical area at the moment of the crack initiation are calculated with numerical simulations. By repeating this extensive procedure using samples with different notch geometries, sufficient interpolation points along the damage curve are eventually obtained. An applicable function according to the method of mean square error can be then derived. 3. Results 3.1. Microstructure based description of the strain hardening behaviour A TRIP steel of grade HT 700T with a sheet thickness of 1 mm was investigated in this work. The chemical composition is given in Table 1. Firstly, metallographic analyses were carried out to

identify the emerged martensite bands induced by segregation. Fig. 2 shows the observed microstructures in different regions over the strip width. In general, in the middle area of the sheet plate segregation induced martensite band formation with higher magnitude than in the border area was found. In industrial applications, multiphase steels are often loaded several times. Firstly, a forming process is carried out to produce arbitrary geometries from sheet steels. Later, during the service lifetime of component, it might be subjected to further severe loading conditions, for example in case of crash. Therefore, the investigated steel sheet was pre-stretched up to about 20% strain. Flat bar specimens were manufactured and elongated under uniaxial tensile deformation until reaching a plastic strain of 20%. In the further investigations, phase fractions and carbon distribution were quantitatively determined by means of light optical metallography and X-ray diffraction method. These data were required for the geometric model generation of the RVE as well as for the modelling of the flow curves of all individual phases, ferrite, bainite, martensite, and retained austenite. Table 2 outlines the results of the phase fraction and the carbon partitioning. The carbon contents in ferrite and retained austenite were calculated from the measured lattice constants and for the other phases they were estimated using the mass balance. On the basis of these metallographic results, two different RVEs were developed. The significant difference between both RVE models is the configuration of the martensite. The first RVE model was assumed to include segregation induced martensitic banded structures, whereas the second RVE model was defined with randomly distributed martensitic microstructures. In this manner, the extreme cases of the microstructural morphology could be investigated. The RVE model being free from segregation bands is depicted in Fig. 3, while in the RVE model including segregation martensite with banded structure was generated manually. The modelled flow curves of the individual phases according to the mentioned material models are shown in Fig. 4. In fact, the martensite has a very high hardness value; however it can be slightly plastically deformed. Therefore, the flow curve for the martensite was modified by increasing the strain hardening in order to better represent its local stress–strain response. The validation of the modelled flow curves was performed by means of uniaxial tensile test of specimens without pre-straining. Concerning these experiments, macroscopic flow curves of the investigated steel Table 2 Microstructural composition and carbon content distribution of the investigated steel.

Table 1 Chemical composition of the investigated steel, mass contents in %.

Retained austenite

Martensite

C

Mn

Si + Al

P

Cr

Ni

Cu

Fraction

%C

Fraction

%C

Fraction

%C

Fraction

%C

0.23

1.61

1.50

0.01

0.03

0.02

0.02

71

0.02

10

0.03

17

1.3

2

0.24

position A strip centre

Ferrite

position B between strip centre and strip edge

Bainite

position C strip edge

Fig. 2. Microstructures of the investigated TRIP steel in different regions over the strip width.

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71% α

+

10% αB

17% γR

+

+

2% α

=

RVE

Fig. 3. Configuration of the RVE model without segregation.

were determined and used for a macroscopic simulation of the tensile test. Afterwards, the RVE was coupled to the central part of the sample. Therefore, the calculated deformation fields from the tensile test simulation were used as boundary condition for the submodel RVE calculation without martensite band. In the RVE simulation, the isotropic flow curves of individual phases were applied under J2 von Mises plasticity. By evaluating the load–elongation behaviour of the RVE, effective stress–strain response can be determined. With appropriate RVE and models of the individual flow curves, the resulted stress–strain response has to be consistent with the macroscopic flow curve of the TRIP steel obtained by experiments. As shown in Fig. 4, the experimentally determined flow curve can be well described by the overall calculated curve of

3600 3200

True stress [MPa]

2800 2400 2000 1600 1200 800 400

martensite - %C = 0.24

retained austenite - %C = 1.3%

ferrite - %C = 0.02

bainite - %C = 0.03

experiment with extrapolation

RVE

0 0

0.2

0.4

0.6 True strain [-]

0.8

1

1.2

Fig. 4. Modelled flow curves for the individual phases of the investigated TRIP steel and their validation by a uniaxial tensile test.

the RVE. This observation leads to the conclusion that the modelled individual flow curves are valid to describe the strain hardening of the selected TRIP steel in RVE simulations. 3.2. Determination of failure criteria To determine the damage mechanics based failure criteria, sheet samples had to be deformed up to the onset of crack initiation. Subsequently, the evaluation of the loading characteristics regarding the equivalent strain and stress triaxiality for plotting the damage curve was carried by FE simulations. The determination of the failure criteria was conducted by means of miniature three point bending tests and tensile tests of flat bar sheet samples with sharp notch. The samples for these experiments were prepared from the sheet strip which was firstly pre-stretched about a uniaxial strain of 20%. These samples were taken from the sheet plates longitudinal to the rolling direction. In the miniature bending tests, it is possible to monitor the sample surface during the experimental procedure, since the bending tests were performed in a scanning electron microscope (SEM). Fig. 5a illustrates the used bending test installation and Fig. 5b shows a schematic representation of the test procedure and the sample geometry. A small groove with a radius of 0.25 mm in the centre of the sample was prepared to facilitate the observation of the crack initiation. During the bending tests, the applied force was increased incrementally. However, the load was held stepwise after a specified deformation in order to record the surface images of the samples in the SEM. In this manner, the mechanical loading causing the crack formation could be identified precisely. The diagram series in Fig. 6 outlines representatively the determination of the critical load leading to the crack initiation. Each of the six diagrams contains the load–elongation curve obtained in the bending experiments and a SEM image of the notch root of the sample. These

sample site motor

3 point accessory

Source: Juelich research centre

(a) Fig. 5. Used bending test installation (a), bending test procedure and sample geometry (b).

(b)

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After the experimental testing, macroscopic simulation of this three-point bending test was carried out. Subsequently, RVE simulations using both the RVE model with and without segregation induced martensite band were performed in order to determine the critical stress and strain values. These simulations were conducted under consideration of the inhomogeneous strain distributions resulted between different phases after the 20% pre-straining. Therefore, the strain distribution in the RVE after uniaxial pre-stretching was firstly analysed. According to these results, the pre-deformations of each individual phase were incorporated by modifying the yield stresses of the flow curves for each individual phase in

70

70

60

60

50

50

40

40

Load [N]

Load [N]

SEM images correspond to different deformation states indicated as points 1–5 on the presented load–elongation curves. It was considerably observed that the crack formation occurred approximately during the deformation state of point 4. Multiple parallel bending experiments were carried out, in which a good agreement among each other was found. Furthermore, it was verified whether there is a significant influence of the area where samples were taken, since the segregation profile varied over the strip width of the sheet plate. Nevertheless, the results showed that the sampling location has an inferior effect on the crack initiation in the bending test.

30 20

1

30 20

10

10

Begin

0

0 0

500

1000

1500

2000

2500

3000

0

500

(a) before deformation

1000

1500

2000

2500

3000

2500

3000

(b) end of elastic zone 70

70

3 2

60

50

50

40

40

Load [N]

Load [N]

60

30

30

20

20

10

10

0

0 0

500

1000

1500

2000

2500

3000

0

(c) after ca. 0.7 mm punch stroke

500

1000

1500

(d) after ca. 1.2 mm punch stroke 70

70 60

60

5

4 50

50

40

40

Load [N]

Load [N]

2000

30

30

20

20

10

10

0

0 0

500

1000

1500

2000

2500

3000

(e) crack initiation after ca. 1.7 mm punch stroke

0

500

1000

1500

2000

2500

(f) crack propagation

Fig. 6. In situ bending test for determination of the deformation state leading to a crack initiation.

3000

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V. Uthaisangsuk et al. / Computational Materials Science 50 (2011) 1225–1232 Von Mises stress [MPa]

Von Mises stress [MPa]

a martensitic island

martensite band

(b) martensite band

(a) randomly distributed martensite

Fig. 7. Deformed RVE without (a) and with martensitic banding (b) in three-point bending test.

the RVE. Fig. 7 depicts the deformed RVE without (a) and with martensite band (b) after bending condition. The martensite band was defined slightly underneath the notch root. The upper region of the

Equivalent plastic strain εpl

1.2 1 0.8 0.6 0.4 0.2 0 0.0

εpl = 0.098*h-2.714 - 0.42

0.4

0.8

1.2 1.6 Stress triaxiality h

2.0

2.4

Fig. 8. Damage curve determined using miniature bending test and tensile test of notched sample.

RVE close to the groove exhibits localisation of stress due to the notch geometry. Obviously, higher local stress gradients within the martensitic islands and the areas around them were observed in case of the RVE with banded structure. Since the investigated TRIP steel exhibits ductile failure behaviour at the test temperature, the damage curve can be adequately used as a failure criterion. For the determination of the damage curve stress and strain values from the performed miniature bending tests were taken. Nevertheless, other additional experiments had to be considered in order to obtain more interpolation points for plotting the failure limit curve. Thus, flat bar tensile samples with sharp notch were manufactured and investigated. As an in situ identification of deformation leading to crack initiation was not possible due to unavailable clamping for the SEM installation, the crack initiation was identified by means of the direct current potential method [31]. The principle of this method is based on a measurement of electric resistance of the sample. The electric resistance changes according to the Ohm’s law when the conductive cross section of the sample decreases, for instance, because of a damage occurrence. This method is frequently applied in fracture mechanics testing. In this manner, the crack initiation in

martensite band (1) homogeneous distribution without martensite band

(2) inhomogeneous distribution with a martensite band underneath the sheet sample surface

(3) inhomogeneous distribution with two parallel martensite bands

Fig. 9. RVE configurations with and without martensite bands.

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For the application of the damage curves on the sheet metal forming tests, RVE configurations as described in the following were investigated, Fig. 9: (1) homogeneous microstructure, (2) microstructure including a martensite band that is underneath the sample surface, (3) microstructure including two martensite bands that are in the middle of the RVE. The martensite bands have a thickness of 18 lm and a distance between two martensite bands is 10 lm. These dimensions are in agreement with the metallographic observation. To verify the damage curve, both bulge tests and stretch-forming tests with different Nakazima sample geometries were experimentally carried out and simulated subsequently with a macroscopic model. In Fig. 10 the developments of pressure and time measured in experiment and calculated from the macroscopic simulation are shown. Furthermore, the critical loads, by which the damage curve was reached in the RVE simulations with different configurations, were also indicated on the curves. Obviously, the failure criterion was achieved for the case of the RVE with segregation band earlier than for the RVE without segregation band. The formation of microcrack was predicted primarily for a deformation state shortly before the occurrence of the macroscopic fracture. The macroscopic failure is the moment when the experimental curves fell down. It was found that the microstructure with two martensite 28 formation of macrocrack

24 20

Pressure [MPa]

the notched tensile samples could be identified. By comparing the experiment and FE calculation of the tensile test, critical stress– strain conditions were quantitatively evaluated. For these investigations, the presented two RVE models were used to be able to incorporate both limit cases, with and without martensite banding. As ductile crack initiation in TRIP steels develops primarily in the ferritic matrix, critical stress and strain values were evaluated for the ferritic phase FE elements in the immediate vicinity of the notch root of the RVE. After the local plastic strain and stress triaxiality were interpreted from the RVE simulations, the damage curve for the steel HT 700T could be determined. Unfortunately, the tensile test provided stress and strain values at the similar level as the ones from the bending test. Regardless, the damage curve was plotted using these data, as presented in Fig. 8. By the interpretation the stress–strain responses were averaged over the volume under the crack tip in order to reduce the influences of mesh and element size. Since the RVE represents a small section of the investigated sample, even a mean value calculation over the entire RVE will certainly lead to a few deviated values. The resulted damage curve exhibits that the influence of the martensite banding configuration on the position of the damage curve can be neglected, since the local stress and strain values determined from different RVEs for the miniature bending test and the tensile test of notched sample have the same magnitude. However, the loading situation in the ferrite strongly depends on the distribution of the martensitic phase, and this is why a pronounced effect of banded martensitic structures on the mechanical properties of TRIP steels can be still expected. The damage curve was applied for further RVE simulations in order to predict the effect of banded martensitic structures on the ductile crack initiation during sheet metal forming with pre-stretched samples. However, it must be noticed that this damage curve was approximated by means of a mathematical fitting of only a few measuring points. Other determination with more different interpolation points could make the description of the crack initiation with damage curve obviously more reliable.

16 formation of microcrack

12 8

4. Application on bulge test and sheet metal forming test

4

The verification and application of the introduced failure criteria was done, while the formation of the microcracks in a hydraulic bulge test and a Nakazima stretch-forming test was predicted. The investigated specimens were not manufactured from the predeformed sheet metal strip, but from a whole sheet plate of HT 700T with a dimension of 1300  200 mm2 pre-stretched in a large scale tensile machine. This steel plate could be clamped only transversely in the testing machine due to its size. Thus, the testing direction is not consistent with the testing direction of the small sheet sample strips. In addition, only a maximum deformation of 10% could be realised by the pre-straining with the large scale machine, otherwise macroscopic crack formation already took place in the sheet plate. The local plastic deformation developed during pre-stretching and the resulted strain hardening had to be considered for the further investigations. As previously the strain distributions of different phases in the RVE were evaluated, but this time for a pre-straining of 10% instead of 20%. Again, the flow curves of the individual phase were modified with regard to these pre-strains. The implementation of a kinematic strain hardening model in the macroscopic simulation would be necessary to obtain a precise calculation of the deformation behaviour. However, the kinematic model was not incorporated for the application in this work due to a large number of parameters in such a model.

0

homogeneous (case 1) one martensite band (case 2) two martensite bands (case 3)

0

10

20

30

40

50

Time [s] Fig. 10. Application of the damage curve to the bulge test.

80 experiment 70

simulation crack initiation - case 1

60 Force [kN]

crack initiation - case 2 50

crack initiation - case 3

40 30 20 10 FLC50 - b = 115mm

0 0

5

10 15 Displacement [mm]

20

25

Fig. 11. Application of the damage curve to the stretch-forming test (sample width 115 mm).

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80 experiment

70

simulation

Force [kN]

60

crack initiation - case 1 crack initiation - case 2

50

crack initiation - case 3 40 30 20 10 FLC50 - b = 30mm

0 0

5

10 15 Displacement [mm]

20

25

Fig. 12. Application of the damage curve to the stretch-forming test (sample width 30 mm).

bands has a stronger effect on a deterioration of the mechanical loading capacity than the microstructure with one martensite band, as the microcrack formed earlier in the RVE containing two martensite bands. In other words, the microstructure with two martensite bands has a lower crack initiation resistance. The influence of the martensite band on the development of the ductile microcrack in microstructure can be quantitatively described by these means. Force and displacement curves determined in experiments and calculated by macroscopic simulations of the stretch-forming test for the sample widths of 115 and 30 mm are depicted in Figs. 11 and 12, respectively. The predictions of the microcrack formation for different RVE configurations were included in the diagrams. As expected, all investigated cases showed the same tendency that the initiation of the microcrack takes place in the microstructures with martensite band earlier than in the homogeneously distributed microstructure. In the examined cases it can be established that a decrease of load carrying capacity more than 20% can arise due to the formation of microcracks, when a microstructure containing heavy segregated bands is considered instead of a microstructure without banded martensitic structures. In the simulations, high stress concentration within the martensite banding was observed for all sample dimensions due to the high strength and low ductility of the martensite. The localisation is especially high in the middle of the bands due to the symmetry conditions of the RVE model. In case of the RVE including two martensite bands the matrix material in the crossing section between both martensite plates show significantly increased stress values. This region is the critical site where crack formation and propagation will appear. 5. Conclusion In this investigation, the damage curve was used as a failure criterion to predict the ductile crack initiation due to multiaxial deformation in a multiphase steel with segregation induced martensite banding. For this purpose, the failure criterion was not applied for a macroscopic calculation, but a representative volume element (RVE) was defined to carry out microstructure simulations in combination with a damage mechanics approach. The following results were obtained:  The mechanical deformation state in term of load and displacement required for the initiation of a microcrack could be determined for different forming tests. This method allows a quantitative evaluation of the microstructure influence on the failure behaviour.

 Novel techniques for experimental determination of the failure criterion were used. In particular, the miniature three-point bending test was successfully established in SEM. It is a promising approach for prospective investigations.  By means of the developed analysis procedure, a quantitative description of the influence of real microstructural morphologies on the formation of the microcrack is possible. The RVE technique can be well used for the characterisation of the mechanical properties of multiphase microstructures.  Nevertheless, homogenisation methods are necessary for the microstructure simulation to effectively connect the results from different scales. With the introduced evaluation technique the description of the effect of microcrack on the macroscopic component failure behaviour is still not possible.  For future investigations, a new modified simulation method is required that is able to couple the macro- and microscale and to describe backwards the influence of the microscopic crack initiation on the macroscopic component behaviour. In this work the calculations were carried out up to the crack initiation and the influence of crack growth was not taken into account. To obtain a more practical evaluation criterion for the risk of component safety relating to the segregation induced microcrack, an approach has to be further developed, which not only can predict the onset of the stable crack growth in the microstructure, but also describe the effect of this microcrack on the macroscopic behaviour.

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