In-situ measurement of local strain partitioning in a ...

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Y. Ososkov et al.: In-situ measurement of local strain partitioning in a commercial dual-phase steel

Yuriy Ososkova, David S. Wilkinsona, Mukesh Jainc, Todd Simpsonb a

McMaster University, Materials Science & Engineering Department, Hamilton, ON, Canada University of Western Ontario, Nanofabrication Laboratory, London, ON, Canada c McMaster University, Mechanical Engineering Department, Hamilton, ON, Canada b

In-situ measurement of local strain partitioning in a commercial dual-phase steel Dedicated to Professor Dr. Wolfgang Pompe on the occasion of his 65th birthday

This paper presents the results of an in-situ Scanning Electron Microscopy study of the local strain partitioning between ferrite- and martensite-rich regions in a commercial dual-phase steel. A Scanning Electron Microscopy tensile micro-stage, coupled with strain measurement methodologies based on gold micro-grids and digital image correlation, has been used to measure inhomogeneous strain fields at the micron scale. It has been found that when martensite is distributed non-uniformly, local strain partitioning depends significantly on the local spatial phase distribution and morphology. Strain distribution maps can be developed which provide valuable information about local strain paths for both phases. The results suggest that a rather detailed description of the two-phase microstructure of such materials is needed in order to fully understand their mechanical behaviour. Keywords: Local strain partitioning; Martensite-ferrite; In-situ SEM; Dual-phase steel

1. Introduction Dual-phase (DP) steels are effectively composite materials consisting of a soft ferrite matrix and hard martensite particles. They are produced by a process of intercritical annealing, followed by accelerated cooling, the details of which depend on the grade being produced as well as local plant 664

conditions [1]. Interest in DP steels has revived in recent years because their higher strength and work hardening characteristics can be utilized to produce automotive structures with reduced weight, thus improving performance in a cost-effective manner [2, 3]. If an external force is applied to the ferrite–martensite composite, the mechanical interactions arising from the constraints between the two different phases cause an inhomogeneous distribution of stress and strain [4 – 6]. This, in turn, dictates the macroscopic stress – strain behaviour of the material. Developing micromechanical models to predict the flow behaviour of composite materials requires knowledge of the flow properties of each constitutive phase, which can be problematic in a material such as this where the properties of martensite depend on its precise chemistry, which is in turn dependent on thermomechanical processing parameters. The properties of bulk martensite may not be representative, and therefore local measurements are needed. In addition, models for composite deformation must make some assumptions about how strain is partitioned between the two phases. However, the dependence of the stress and strain partitioning on the morphology of the microstructure is not yet clearly established [7], especially for complex morphologies such as those found in DP steels. A number of techniques have been developed to enable the measurement of local strain distributions at a microscale (for a brief review of these see Kang et al. [8]). Optical methods, including Moiré techniques [9] and laser interInt. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

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ferometry [10], are capable of measuring very small strains in local regions of a sample. In fact, these techniques are often used to measure strain at notches or other stress concentrators. Moiré techniques are the oldest and most well known of these methods. In the direct Moiré method, small distortions in a grid of initially uniform lines are easily seen when the deformed grid is observed through the initial grid. This technique is not suitable for the study of small areas; instead one applies interferometric Moiré techniques using coherent light to measure very small strains on a sample surface. Holography and laser-speckle interferometry are similar techniques that use the special properties of coherent light to measure small distortions in a sample without an applied grid [11]. These latter methods have the additional advantage that they can measure out-of-plane deformations on a sample surface. Unfortunately, all of these methods require very stable optical setups to make accurate measurements, and the equipment is typically expensive. They are also challenging to apply in-situ because of the difficulty of moving the setup to accommodate different mechanical test configurations. Direct measurement techniques provide a solution to some of the drawbacks just mentioned. One of these relies on a grid applied to the sample surface either using lithography [12 – 15], rolling a fine screw thread across the surface [16, 17], or chemical etching [18] amongst others. Distortions in the sample due to deformation cause distortions in the grid, which can then be measured [12]. The grid method is the simplest and most direct approach for the measurement of deformation. Its use has been evolving over time, in parallel with complementary technological developments [19]. These methods of strain measurement are accurate to within a pixel or so, providing sufficient accuracy for many applications. Kang et al. [8] however note that the major difficulty in utilizing this method lies in the complexity of processing the deformed grids to extract the full-field strain distribution. An alternative to the micro-grid method involves digital image correlation (DIC) which was initially developed as a computer-vision-based, non-contact, full-field surface strain measuring method. This technique provides a measure of the displacement field by correlating a pair of digital images (before and after a deformation increment is applied) using a mathematically well defined function. The DIC method was originally proposed in the early 1980s for experimental stress analysis [20, 21]. During the last two decades, the potential of DIC for analyzing deformation behaviour has been recognized, and the method has been improved significantly through reduced computation complexity and improved accuracy [22]. DIC is a versatile technique, which offers adjustable spatial resolution, and thus it provides measuring capabilities at various length scales. Both the magnification and sensitivity of the acquisition system (optical lenses, scanning electron microscope (SEM), scanning tunneling microscope (STM), atomic force microscope, etc.) determine the resolution of the measurement [23]. Recently, DIC has been used in conjunction with SEM and STM for quantitative material characterization [8, 24 – 27]. Although in many cases the natural texture of the surface provides enough features (speckles) to perform the correlation [8, 27], artificial speckles can also be applied to enhance the accuracy of the measurements, for example by imprinting a dot map or Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

markers using an SEM electron beam [24] or a photo-lithography process [12]. Since this study is concerned with experimental evaluation of non-homogeneous local strain distribution fields between ferrite and martensite, the approach reported here goes one step further. It uses commercial DIC software to analyze the grey-scale information present in the series of images obtained on a tensile sample with the etched dualphase structure used as the pattern for cross-correlation. For comparison, a gold micro-grid with a 5 lm pitch is also deposited using e-beam lithography. The information obtained is evaluated in terms of its contribution to a better understanding of the effect of strain partitioning on the overall deformation behaviour of a commercially-produced DP600 dual-phase steel.

2. Experimental The DP600 steel was provided as 1.8 mm thick, flat coldrolled galvannealed sheet. The chemical composition was (in wt.%): 0.07C, 1.84Mn, 0.15Mo, 0.09Si, 0.03Cr, 0.01Ni and 0.01Ti. Flat dog-bone-shaped tensile samples with gage dimensions of 3.5 mm wide · 1.85 mm thick · 14 mm long (Fig. 1a) were machined from the as-received sheet with the long axis parallel to the rolling direction. Prior to testing one of the flat surfaces of the tensile sample was polished to 0.05 lm finish, and then etched using a 2 % nitric acid solution in ethanol (Nital) to reveal the microstructure. The sample was clamped at both ends and stretched under displacement control by a screw-driven mini-tensile SEM stage (Ernest F. Fullam Inc., Lantham, NY) at a constant cross-head speed of 17 lm s – 1, equivalent to an initial strain rate of about 1.2 · 10 – 3 s – 1. The mini-tensile stage

(a)

(b) Fig. 1. (a) Tensile specimen design and (b) the screw-driven mini-tensile stage used for the in-situ SEM tests.

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Basic Y. Ososkov et al.: In-situ measurement of local strain partitioning in a commercial dual-phase steel

shown in Fig. 1b has a total crosshead travel of 25 mm and a load cell with 4500 N maximum capacity. The stage was installed inside an Environmental SEM (ESEM Electroscan 2020) which enabled observations of the sample surface during the deformation. The tensile test was interrupted without unloading at approximately equal displacement increments in order to capture a secondary electron (SE) image of the surface. The imaging process took between 60 and 180 s and a slight load relaxation occurred during these pauses. Each of the recorded 8-bit greyscale digital images had a size of 1024 · 1024 pixels with a spatial resolution of 0.073 lm per pixel at a magnification of 1900 times. The recorded digital images were analyzed using the ARAMIS DIC software [28] in order to obtain a local strain distribution on the tensile sample surface for each image captured. The facet size used was 69 · 69 pixels with a step size of 41 pixels. 2.1. Gold micro-grid deposition

Fig. 2. Illustration of DIC principle: square facet before (a) and after (b) deformation [27].

fact that the distribution of grey-scale values in a rectangular area (facet) of the un-deformed state (reference image) corresponds to the distribution of grey scale values of the same area in the deformed sample (destination image) as illustrated in Fig. 2. A starting point is defined in the reference image (Fig. 2a), which must also be defined as the starting point in the destination image (Fig. 2b). The grey values g at these two points are related by an equation of the form

The gold micro-grid deposition was performed on a polished and etched using e-beam lithography following the procedure reported in [13, 14]. Samples were bonded to a silicon carrier wafer with low melting point CrystalBond adhesive. The 4 % PMMA resist in anisole (A4) was deposited dropwise on the steel sample immediately prior to spinning at 1500 rpm for 45 seconds. The sample was removed from the carrier wafer and then baked at 160 8C for 1 h in an oven at atmospheric pressure. Electron beam lithography was performed on a LEO 1530 Field Emission (FE) SEM equipped with a Nanometer Pattern Generation System (NPGS) lithography system (JC Nabity Lithography Systems) at the University of Western Ontario nanofabrication laboratory. The sample was developed in a 3 : 1 solution of IPA:MIBK for 90 s. A 5 nm chromium adhesion layer followed by 80 nm of gold was deposited by electron beam evaporation. The sample was then sonicated in acetone to lift off the unexposed resist.

Here the values of a1 and a5 describe the translation of the facet’s centre while the other parameters describe the rotation and deformation of the facet [29]. In order to compensate for possible differences in illumination in the images, a linear radiometric transformation is adapted simultaneously while the images are being matched:

2.2. Principles of digital image correlation

g1 ðx; yÞ ¼ b1 þ b2 g2 ðxt ; yt Þ

The distribution of strain during tensile testing was obtained using a commercial whole-field optical strain mapping system, named ARAMIS [28]. ARAMIS recognises the surface structure in digital images and can allocate coordinates to every pixel in the image. The first coordinates are gathered by recording a reference image, typically taken from the sample prior to deformation. After the sample has been deformed, a second image is recorded. The software then compares the images and registers any displacement of the sample characteristics. The resulting strain field can then be derived. The advantage of the method is that images taken in-situ during testing in the SEM avoid the inaccuracies induced by contrast changes due to image distortions, variations in magnification and rotation. All of these factors are the same for the reference and subsequent images. Furthermore, the use of the SEM allows imaging of microstructural features of interest during testing of the specimen (for example, the development of microvoids), and their later correlation with the local strain distribution obtained. The ARAMIS system [28] employs the fundamental principles of digital image correlation (DIC) based on the

The parameters of the geometric and radiometric transformation are calculated in such a way that the sum of the quadratic deviation of the matched grey values is minimized, i. e. both speckle patterns can be matched to one another optimally. This can be evaluated through a correlation coefficient C, which reflects the degree of conformance between the reference and destination facet and can be calculated as follows: P g1 ðx; yÞ g2 ðxt ; yt Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð4Þ C ¼ 1 pP P g21 ðx; yÞ g22 ðxt ; yt Þ

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g1 ðx; yÞ ¼ g2 ðxt ; yt Þ

ð1Þ

The pixels of the facet in the reference image are then transformed into the destination image according to xt ¼ a1 þ a2 x þ a3 y þ a4 xy yt ¼ a5 þ a6 x þ a7 y þ a8 xy

ð2Þ

ð3Þ

A value of C = 0 corresponds to a perfect match. Thus, the correlation is optimized by searching for the minimum position of the distribution of correlation coefficients. 2.3. Strain calculation using micro-grids Strain calculations from the micro-grids were performed using a simple interactive computer program based upon an algorithm similar to that reported earlier [30, 31]. The grid consists of cells which are assumed to be perfectly Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

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Fig. 3. Undeformed triangle (top-left), deformed triangle (bottom-left) and BSE image of deformed micro-grid (right).

square initially, with the grid size specified by the user. For elemental strain calculations, each square is divided into two triangles. The strain within each triangle is assumed to be homogeneous with no out-of-plane shear strains (see Fig. 3). The Lagrangian strains are computed for each triangle throughout the entire geometry, and then averaged for the two triangles corresponding to each cell. Although, the results are identical regardless the direction in which the square is divided, to simplify the computation of the strain, each square was divided in the same direction. The Lagrangian strains are then converted to true strains. Wolak and Parks [31] noted that the above development is rigorous for homogeneous strain fields. In the case of non-homogeneous strains, one way to maintain the rigor of the analysis is to restrict it to a small region. Physically, however, it is recognized that this requirement, in general, need not to be so strict, and it is sufficient that the region be small enough so as to be a local region of homogeneous strain. Often this restriction is relaxed even further to accommodate the method of measurement, and the region is referred to as a region of nearly homogeneous strain. The undeformed triangle is aligned with the coordinate system, while the deformed triangle has an arbitrary location in three-dimensional space. Let l0 equal the initial grid length, a be the new length of one leg of the original triangle, b be the new length of the second leg, and c be the new length of the hypotenuse. For a two-dimensional case, the sides of the deformed triangle are defined as follows: ~ m1 ¼ ðx2 x1 ; y2 y1 Þ ~ m2 ¼ ðx3 x1 ; y3 y1 Þ ~ m2 ~ m1 ¼ ðx3 x2 ; y3 y2 Þ m3 ¼ ~

The components of the deformation tensor can be calculated therefore from the vectors: C11 ¼

m1 j ðx2 x1 Þ2 þ ðy2 y1 Þ2 a2 j~ ¼ ¼ 2 l20 l20 l0

m2 j ðx3 x1 Þ2 þ ðy3 y1 Þ2 b2 j~ ¼ ¼ 2 ð7Þ l20 l20 l0 ~ m 2 ð x2 x1 Þ ð x3 x1 Þ þ ð y2 y 1 Þ ð y3 y1 Þ m1 ~ ¼ 2 ¼ 2l20 l0

C22 ¼ C12

¼

m2 j2 j~ m3 j2 a2 þ b2 c2 m1 j2 þ j~ j~ ¼ 2l20 2l20

The Lagrangian strains, E, are calculated from C. Since C ¼ I þ 2E, where I is the identify matrix, then: E11 ¼

C11 1 a2 1 ¼ 2 2 2l0 2

E22 ¼

C22 1 b2 1 ¼ 2 2 2l0 2

E12 ¼

C12 a2 þ b2 c2 ¼ 2 4l20

ð8Þ

where E11 and E22 are the tensile strains in the x-direction and y-direction respectively, while E12 is the shear strain in the x – y plane.

ð4Þ

and then: a ¼ j~ m1 j; b ¼ j~ m2 j; c ¼ j~ m3 j

ð5Þ

Based on the Cauchy – Green deformation tensor, C, the m2 and~ m3 can be used to calculate the Lagrangian vectors~ m1 ,~ strains [30]. The stretch ratio is defined as the ratio between the deformed length l, and the initial grid length l0. C is defined as the square of the stretch ratio: 2 l ð6Þ C l0 Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

Fig. 4. Mohr’s circle for plane strain [32].

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The true strains can now be determined by the use of Mohr’s circle for plane strain, illustrated in Fig. 4 [32]. From the distance formula, the radius and centre of the Mohr’s circles can be computed as: E11 þ E22 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðE11 E22 Þ2 þ ð2E12 Þ2 r¼ 2


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Eaverage ¼

neck, respectively) using a standard uniaxial tensile test in combination with a DIC non-contact optical system. Once flow localization occurs, the strain outside the neck remains constant, while strain accumulation inside the neck accelerates. Similar behaviour is also observed in the case of local strain measurements taken at the micron scale, as illustrated

ð9Þ

while the minimum and maximum values of the Lagrangian strain are defined as Emin ¼ Eaverage r Emax ¼ Eaverage þ r

ð10Þ

Because the stretch ratio, l/l0, is related to the true strains through the Cauchy – Green deformation tensor the true strains can be expressed logarithmically: l e ¼ ln l0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð11Þ eminor ¼ ln 1 þ 2Emin ¼ lnð1 þ 2Emin Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 emajor ¼ ln 1 þ 2Emax ¼ lnð1 þ 2Emax Þ 2 In this strain measurement system, the possible sources of error include inaccuracies in the original grid application and errors in processing the digital image to obtain the locations of the intersection points. The advantage of this measurement method is that a single operation can give the surface strain of the entire area under observation without requiring an image of the initially undeformed region, as is absolutely necessary in the case of DIC system analysis. Limiting factors include the number of data points and the resolution of the SEM digital image. As illustrated in Fig. 5, the grid was deposited on the sample gage section as a 5 · 10 array of squares (50 squares in total) with initial dimension 250 · 250 lm. The array was arranged so that there were 5 squares across the sample width and 10 squares along the tensile axis. The SEM pictures of the micro-grid were taken in-situ for the same field of interest as for DIC analysis. In addition, the SEM imaging was performed after removing the sample from the tensile stage for other gridded areas along the tensile direction.

Fig. 5. BSE images of gold micro-grids deposited on a tensile sample: array of gridded squares (top) and a single square at higher magnification (bottom). Both pictures have been taken from a sample tested to failure. The width of each “square” after deformation has been reduced from 250 to 225 lm.

3. Results A typical engineering stress – strain curve obtained during an in-situ tensile test is shown in Fig. 6. The interruptions in the test, used to collect SEM images, are marked by small load relaxations. When the test was resumed, the previous load was quickly re-established. In this example the test was ended just before the onset of localized necking. Previous DIC experiment [33] showed that diffuse neck development is accompanied by a bifurcation of the strain distribution along the gage length of the sample. Figure 7a shows schematically a typical example of the local principal tensile strain as a function of average sample deformation plotted for two areas (located inside and outside the 668

Fig. 6. Typical engineering stress – strain curve obtained during an insitu experiment (the test has been stopped right before the on-set of necking).

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(a)

(b) Fig. 7. Local strain as a function of global true strain: (a) – standard macro tensile test [33] in two areas located outside (point 0) and inside (point 1) the localized neck, and (b) – in-situ tensile test in an area located in the uniform elongation zone.

in Fig. 7b. In this figure, the average value of local strain represents an average strain obtained by DIC over a small domain corresponding to the area of observation located away from the neck. Sample true strain corresponds to global tensile deformation calculated from the overall displacement measured with a Linear Variable Displacement Transducer (LVDT) on the tensile stage. It is seen that up to the strain corresponding to the onset of localized necking (approximately 0.18) there is a linear relationship between the average localized strain and global sample deformation. The strain remains almost constant after localization occurred. In this paper the partitioned strains reported will be plotted against the average local true strain. Figure 8 shows SEM images of a specimen area located outside of the neck prior to (first row) and after (second and third rows) tensile deformation. The tensile loading axis coincides with the vertical direction of the micrographs. Such SEM micrographs taken at different global strains are processed by the DIC system to determine the local strain. The system allows users to generate maps of local in-plane strains in the region of observation as a function of the applied global strain as well as to plot local strain profiles along arbitrary sections as a function of distance on the sample surface. The local strain maps corresponding to these SEM pictures are also presented in the first column in Fig. 8. It is seen that the DP600 microstructure exhibits a non-homogeneous distribution of ferrite and martensite. Two types of regions can be distinguished – “soft” regions corresponding to a high fraction of ferrite grains, and Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

“hard” regions containing agglomerates of very fine ferrite and martensite particles. The cumulative strain map is obtained by comparing the micrograph of the deformed specimen with the micrograph of the undeformed specimen at each arrest point. Even after a global tensile strain of 3.5 % a significant strain partitioning emerges between soft and hard regions in the microstructure. Figure 9 better illustrates this by plotting the local major and minor strains as a function of position along a line (shown on the top micrograph in Fig. 8). These curves display the evolution of the local strain profiles at different levels of global (engineering) strain. It is clear that strain partitioning can be observed from the onset of deformation but that it becomes much more pronounced at higher global strain. The DIC software also enables one to monitor the development of local strain for any area of interest on the sample surface by performing a point analysis. This shows the evolution of local strain at a single point as a function of increasing global strain during testing. An example is presented in the plot of Fig. 10. In this case the average values of local strain are plotted for a soft and a hard region, as marked in Fig. 8. The strain can also be calculated from metallographic measurements of the projected length of the martensite islands along the tensile direction (labeled as the Y-projection strain). This data is also included on the plot, for comparison. It correlates well with the DIC data although the scatter is significantly greater. We observe that the soft region exhibits a rapid increase in strain as deformation proceeds whereas the strain in the hard region increases gradually. At a tensile strain of around 5 %, the ratio between local strains in the soft and hard regions equals 5, decreasing to about 4 once the average local strain reaches 17 %. While it is not a purpose of this paper to analyze the mechanistic basis for the inhomogeneous deformation, they are clearly correlated with the clusters of martensite particles and individual large martensite particles. Information generated by DIC analysis can be also used to analyse deformation strain paths for each microstructure constituent. Thus, in Fig. 11 we compare the deformation strain path plotted in major and minor surface strain space for a soft region (Ferrite grain 1 in Fig. 8) and a hard region (M Agglomerate 2 in Fig. 8). It is seen that the soft region deforms initially along a tensile strain path with a strain ratio (minor/major strain) of about – 0.3, while the hard region is initially subjected to plane strain deformation. However, both regions deform along the uniaxial strain path (strain ratio of – 0.5) at higher global deformation. Figures 12 and 13 show gold micro-grids deposited on the sample surface before and after plastic deformation. The grid is seen as an array of crosses on the etched sample surface. It is seen that the grid is of high quality and it is easily recognizable under SEM observation even using secondary electron imaging (Fig. 12). The displacements of the grid points have been used as the basis for local strain calculations. It is seen in Fig. 13 that strain partitioning between soft and hard regions of the microstructure can easily be visualized. Squares number 1 and 2 correspond to the soft region, whereas squares number 3 and 4 are in the hard region of the ferrite-martensite agglomerate. Figure 13 demonstrates the advantage of using backscattered electron mode (BSE) for micro-grid imaging. The BSE image shows much better contrast, making it easier to define grid points and so to improve the accuracy of local strain measurements. 669

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Figure 14 shows the variation of local strain for soft and hard regions as a function of the average local strain. The plot consists of data obtained using both DIC and microgird techniques. The DIC values in Fig. 14 have been calculated from in-situ SEM measurement for the sample area shown in Fig. 8 (points without error bars). The local strain values from the micro-grid (points with error bars) have

been obtained from measurements performed on a sample after a tensile test for 65 different areas located within the sample gage length and, therefore, subjected to different amounts of plastic deformation. Because some of the micro-grids are within the neck this data extends to much higher strains than the DIC-only data plotted in Fig. 10. This data shows that local strain partitioning between ferrite


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Fig. 8. Local strain maps and corresponded SE images: first row – undeformed state; second row – after 0.035 of deformation; third row – after 0.207 of deformation. Areas representing different features of steel microstructure (“soft areas” – ferrite grains 1 and 2, and “hard areas” – ferrite–martensite (M) agglomerates 1 and 2) are marked on the maps.

Fig. 9. Local strain profiles plotted along the cross-section of a sample taken along the line shown in the top micrograph in Fig. 8. The strain profile has been measured at two times, corresponding to global strains of 0.035 and 0.207.

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Fig. 10. Variation of partitioned strain with average local true strain. Data represent the average values for soft and hard regions shown in Fig. 8. The solid curve represents iso-strain condition.


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Fig. 11. Deformation strain paths for soft and hard regions in Fig. 8 obtained by DIC.

Fig. 12. Imprinted micro-grids: undeformed state (left) and after applied deformation (right). The microstructure etched with 2 % Nital solution.


Fig. 14. Combined plot showing variation of local partitioned strain for soft and hard regions with average true local strain. The results obtained from micro-grid (points with error bars) and DIC (points without error bars).

Fig. 15. Distribution of deformation between soft and hard regions: just before necking (left), and after the failure in the area adjacent to fracture surface (right). Almost no deformation and no voids/cracks are observed within agglomerates. The ferrite grains have accommodated most of the deformation (right picture).

4. Discussion

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Fig. 13. Visualization of strain partitioning using micro-grids (0.55 average local true strain – BSE imaging). Squares 1 and 2 – soft regions, squares 3 and 4 – hard regions.

and martensite continues over a wide range of deformation but that the strain ratio between phases decreases to about two. The martensite-rich regions deform slowly and appear to saturate once a strain of about 0.3 is achieved, while the ferrite strain increases continuously. Figure 15 shows the microstructure after deformation in two different areas of the same sample: the first taken in the area of uniform elongation just before the onset of localized necking, whereas the second was acquired after the sample failure in an area adjacent to the fracture surface. The deformation of the ferrite regions is evident on both pictures. It is seen that even in the vicinity of the fracture surface where the deformation reaches over 0.7, the ferrite regions have accommodated most of the deformation. Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

Measuring large strain deformation in multi-phase materials over small domains in the range from about 10 lm to 1 mm is a complex problem of significant technological interest. It has been reported by Gonzales and Knauss [34] that the DIC method is ideal for this purpose so long as the deformation is not so large that convergence of the correlation algorithm is no longer guaranteed. This starts to become a problem once the deformation involves local strains much larger than 10 %. On the other hand, strains of the order of 50 – 100 % are typical in the necking region in the materials of interest here [35]. Accordingly, an incremental application of the DIC method that is capable of analyzing large deformations in many small steps has been developed [34, 36]. In addition, Tong [36] has reported that the incremental strain mapping approach is sensitive in monitoring in-situ deformation processes. Thus the method was capable of uncovering the existence of non-stationary deformation bands in a binary aluminum alloy even though the bands were not visible by direct observation. A similar strategy has been employed recently by Kang et al. [8] to measure large deformation within the necked area of an aluminum sheet tensile sample, in this case using shear bands themselves as the defining feature. They showed that the best results were obtained using an incremental DIC approach in which two successive images, separated by a 671

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modest amount of global deformation (3 – 5 %), were used to map local strains, thereby minimizing the effect of emerging new features on the sample surface due to deformation. However, the results of our tests have shown that as deformation increases significantly, it becomes more difficult to calculate a strain distribution for the entire field of observation, even using the incremental DIC approach. As deformation proceeds, the resolved images do not completely cover the representative domain, due to tensile stretching of the sample. Similar problems also exist for the micro-grid methodology; however for micro-grid analysis this can be addressed by moving the observation area. Another advantage of the grid measurement method is that one does not need an initial picture in the undeformed condition in order to calculate strain for any area of interest. This allows measuring strain over a wide range of plastic deformation using just one sample. Assuming that the resolution of the digital image (here 1024 · 1024 pixels) and the grey value depth (8 bit in the present work) are constant, the spatial resolution of the strain map depends on the magnification used to acquire the images. In the present work, this was chosen as a compromise between maximizing resolution and the field of observation. It was not always possible to obtain the strain distribution in a given phase at the magnification used in the present study, as the size of a facet is a limiting factor. The martensite islands for example were often smaller than 5 lm, the size of grid. Therefore, the local strains reported here are referred to as representing soft (mainly ferrite) and hard (mainly ferrite–martensite agglomerate) regions. An additional difficulty arises from the stochastic nature of necking with respect to the position of the grids. In most samples the actual fracture origin was not in one of the small regions of the sample under observation during insitu testing. As a result, it was not possible to compare data obtained from the DIC calculations with the values from the micro-grid analysis at high strains typically encountered inside a neck. Figure 16 shows the result of a comparison of the local strain computed using the DIC system and from micro-grids. The main limitation of the micro-grid technique is the accuracy of the local displacement measure-

Fig. 16. Comparison between values of local true strain obtained from micro-grid and DIC measurements (R2 = 0.9912).

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ments at low global deformation. However, the results showed that the local strain values generated from microgrid for global deformations over 2 % are close to those obtained using DIC. The distribution of inhomogeneities defines a macroscopic size scale below which the assumption of homogeneous material properties is not justified [34]. Homogeneous or mean-field properties can only be assigned to regions that are several times larger than the scale of the inhomogeneity. Consequently, the effect of morphology and spatial distribution on local strain partitioning cannot be ignored and must be taken into consideration in developing continuum based finite element models of fracture. A first attempt to describe the spatial arrangement of the phases might use a pair-distribution function assuming only two classes of material behaviour. However, a more complete description could be obtained with the help of two-point probability functions [37]. The technique presented in this paper will help to understand the deformation behaviour of dual-phase steels, and will aid the development of adequate FE models, which still require the measurement of extensive experimental data.

5. Conclusions DIC and micro-grid techniques, in tandem with SEM in-situ tensile testing, have proven to be effective techniques in determining the distribution of strain between ferrite- and martensite-rich regions in dual-phase steel. We have shown experimentally that the etched steel microstructure can be successfully used as a pattern for cross-correlation using commercial DIC software. The technique enables one to obtain both major and minor strains and thus the strain path followed by each phase during macroscopically tensile deformation. In general, since one does not know a priori the location where necking will occur it is hard to capture local strains in the necking region. One approach to resolve this would involve hour-glass shaped or shallow notched samples with a narrow centre sufficient to locate the neck in a well defined region. When the DIC methodology is combined with post-deformation strain analysis using microgrids one can also obtain large volume statistics over a large range of strain, which makes it possible to evaluate the strain partitioning between phases in multi-phase materials with a high level of confidence, especially at very large deformation, such as in the localized necking area where eventually DIC methods break down. The main drawback of the micro-grid approach is that the resolution is limited by the grid spacing whereas in DIC it is limited by the scale of the microstructural features of interest. DIC therefore enables one to assess local strain distribution on a finer scale. For this reason the combination of the two methods produces the most accurate analysis of strain partitioning behaviour. In terms of the specifics of strain partitioning in DP600, we have found that this occurs starting at low overall deformation and that it is significantly affected by spatial distribution of the ferrite and martensite phases. It is not possible to determine the mechanical response of the martensite phase in isolation. Rather we have analyzed the response of “hard areas” having a high local area fraction of martensite mixed with fine ferrite grains. The strain in the ferriterich (soft) areas increases continuously as deformation proInt. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8

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ceeds, whereas the strain in the hard areas increases much more slowly, initially in plane strain. It only starts to deform rapidly once necking commences. The hard areas force the material to deform inhomogeneously by restricting plastic flow to the ferrite-rich regions, as supported by the strain path analysis. Based on these results, it is clear that a full understanding of the deformation behaviour of dual phase steels requires a rather complete description of the spatial arrangement of the constituent phases. Moreover, modified thermomechanical processes which change the distribution without necessarily altering the macroscopic yield strength or the overall martensite volume fraction could have a profound effect of the large strain behaviour and ductility of these materials. The methodology developed provides a means of quantifying such effects. Indeed the data presented here can be used to develop the constitutive laws for martensite and ferrite deformation needed as input to a microstructurally based finite element model for the deformation of DP600. The authors gratefully acknowledge technical assistance of Dr. Steve Koprich and Mr. Klaus Schultz (McMaster University) with SEM operation. The financial support for this research was provided by the Auto21 Network of Centers of Excellence program in Canada. The material for this investigation was kindly supplied by Stelco, Hamilton, ON. One of the authors (DSW) is especially grateful for many years of friendship and scientific discussion with Prof. Wolfgang Pompe, who taught him a great deal about humility, integrity and modeling the flow in inhomogeneous solids.

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DOI 10.3139/146.101526 Int. J. Mat. Res. (formerly Z. Metallkd.) 98 (2007) 8; page 664 – 673 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282 Correspondence address Professor Dr. David S. Wilkinson McMaster University Department of Materials Science & Engineering 1280 Main Street West, Hamilton, ON L8S 4L7 Tel.: +905 525 9140, Ext. Fax: +905 528 9295 E-mail: [email protected]

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