Investigation on the influence of loading-rate on fracture toughness of AHSS grades

Materials Science & Engineering A 726 (2018) 332–341

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

Investigation on the influence of loading-rate on fracture toughness of AHSS grades

T

⁎

Stefan Gollinga, , David Frómetab, Daniel Casellasb,c, Pär Jonsénc a

Gestamp R&D, Box 828, 97 125 Luleå, Sweden Fundació CTM Centre Tecnològic, Plaça de la Ciència 2, 08243 Manresa, Spain c Luleå University of Technology, SE 971 87 Luleå, Sweden b

A R T I C LE I N FO

A B S T R A C T

Keywords: AHSS Essential work of fracture Fracture toughness Quench partitioning Dual phase TRIP-assisted bainitic-ferritic Loading rate

The automotive industry is striving for light body-in-white structures while maintaining or improving passenger safety. The aim of this paper is to investigate the influence of the loading rate on the fracture toughness of thin steel sheet metal of three advanced high strength steels. Although steel is a heavy material it plays a significant role for lightweight solutions in car bodies. Three different advanced high strength steel (AHSS) grades, namely dual-phase (DP), quench-partitioning (Q&P) and TRIP-assisted bainitic-ferritic (TBF), are investigated in the present paper. For crash relevant components it is of importance to know the material response under high loading velocities i.e. high strain rates. A standard tensile test system is used for low loading rates, a high-speed tensile testing setup is used to obtain high loading rates. The fracture toughness of the three AHSS grades is evaluated using the methodology of the Essential Work of Fracture (EWF). The tensile specimen used in the present work is the double edge notched tensile (DENT) geometry with a pre-developed crack. High-speed imaging is applied to verify the validity of the evaluation method Essential Work of Fracture at high rates of loading. Results from this work show that knowledge of fracture toughness would improve the understanding of fracture and crack propagation mechanisms for third generation high strength steels used for automotive components.

1. Introduction In engineering applications the use of AHSS grades is favored due to there excellent strength to weight ratio. In many light weighting applications it is not possible to use materials with lower weight, like for example aluminum or magnesium, because of load bearing considerations or energy absorption capacity. The use of AHSS grades is advantageous in applications where energy absorption is of importance. Such an application is for example the low speed crash-box in automobiles. A low speed crash box is located between the bumper and the chassis of car. The main purpose of this particular crash box is the absorption of impact energy at low vehicle speeds, deformation occurs in this specially designed part and prohibits damage of the remaining chassis. At higher impact speeds the crash box absorbs the initial impact energy and transfers the not dissipated energy into the chassis. The integrity of this systems is of importance for passenger safety and advantageous for manufacturers for their insurance ratings. The influence of the rate of loading on the mechanical properties of materials has a long history in investigations. One of the earliest studies on the effect of the strain rate on the plastic flow of steel was conducted ⁎

Corresponding author. E-mail address: [email protected] (S. Golling).

https://doi.org/10.1016/j.msea.2018.04.061 Received 16 March 2018; Accepted 14 April 2018 Available online 24 April 2018 0921-5093/ © 2018 Elsevier B.V. All rights reserved.

by Zener and Hollomon [1] in 1944. In early works the split Hopkinson pressure-bar method is usually applied to achieve high loading rates. Kolsky [2] investigated different materials like rubbers, polythene and different metals at high loading rates. Twenty years later Lindholm and Yeakley [3] investigated aluminum in tension and compression at loading rates of 1000/s. Another twenty years later Kalthoff [4] studied the fracture behavior of a polymer and high-strength-steel at high rates of loading and captured the onset of crack propagation using highspeed photography and determined the fracture toughness. Investigations on the influence of the strain rate on the mechanical properties of material continued also in the next century. Bleck and Schael [5] investigated the strain rate sensitivity of eight different steel grades and could show the influence on yield and ultimate tensile strength as well as uniform and total elongation. Radwaski et al. [6] pointed out the role of the microstructure morphology and chemical composition of phase constituents on the mechanical properties of AHSS steels, with particular focus on a DP grade. Alturk et al. [7] studied the influence of the microstructure in DP and Q&P steel grades and results suggest a relationship between the phase-content of the steels grades and their response to the loading rate. Gronostajski et al. [8] investigated the

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testing wave propagation phenomena can be seen in test specimens, for samples similar to those in the present study literature did not reveal images. Tarigopula et al. [23] investigated a DP steel using DIC and high-speed imaging during tensile tests in a split Hopkinson tension setup. Wave propagation in meta-materials is studied by Ruzzene et al. [24] and schaeffer et al. [25], both studies are conducted on materials similar to honeycomb cores. Their results show that DIC can be used to capture wave propagation in materials. The intention of the authors' is to extend the knowledge on fracture toughness of advanced high strength steel sheet metal at different loading rates. Therefore, the essential work of fracture is determined for three AHSS grades at two loading rates, ranging from quasi-static to dynamic loading. To our knowledge such a study is not available in literature. The aim of the present study is to fill this gap of knowledge and provide experimental test data and fracture toughness values for the use in industrial applications.

stress-strain relationship and effects on the microstructure of DP and TRIP steel caused by strain rate effects. Flow behavior and strain rate sensitivity are two factors of interest, the influence of the strain rate on the fracture behavior is another key point, especially if emphasizes is put on components or structures that are loaded and fracture is a possible scenario. The influence of the strain rate on fracture toughness has been investigated by Kim et al. [9] and Vendra et al. [10], both studies focus on aluminum alloys and apply different test setups for fracture toughness determination, pointing out the importance of considering loading rate on determination of fracture toughness. Knowledge on the fracture behavior, in terms of fracture toughness, is an important material property for the design of automotive components. Rahmatabadi et al. [11] evaluated experimentally the fracture toughness of ultra-fine grained aluminum in plane stress, relevant for sheet metal applications. During manufacturing of sheet metal components different operations are necessary, Efthymiadis et al. [13] contributed to the understanding of fracture toughness in forming and flanging operations while Pouranvari [12] contributed with a study on spot welds. Recently Casellas et al. [14] showed that tougher steels present higher higher resistance to edge cracking. Frómeta et al. [15] evaluated the crash resistance by means of fracture toughness and crash index. The crash index is a value describing the appearance and size of cracks in a hat profile after axial impact. A linear correlation between the crash index (CI) and the fracture toughness evaluated by means of the essential work of fracture was shown. Accordingly, fracture toughness is a relevant material property for the design of, for example, crash relevant components. It is well known that the evaluation of fracture toughness in thin sheets, in the range of 1–3 mm, is experimentally difficult and out of the conventional ASTM procedures for metals. An alternative approach is the methodology based on the evaluation of the essential work of fracture (EWF). The foundation of the EWF is that the non elastic deforming region at the crack tip can be divided into two sections, one section is the fracture process zone i.e. the region where the fracture process takes place. The second section is surrounding the first section and accommodates the plastic strains. In ductile material, the region where plastic deformation occurs is large compared to the fracture process zone. The work performed in the fracture process zone can be seen as a material constant. This material constant was first described by Broberg [16] and termed essential work. In later publications Broberg [17,18] discussed further topics and details of the essential work. For the case of plane stress, which is a common assumption for thin sheets, the fracture process zone can be identified with necking. For this case the essential work is not a true material constant as it has a dependency on the sheet thickness. The plastic deformation in a test specimen is dependent on the specimen geometry. Therefore, the plastic work in the surrounding of the fracture process zone is not a material constant. The EWF methodology is a common technique in the characterization of the fracture toughness in thin films and polymers. Some works have been addressed to metal sheets. Cotterell and Reddel [19] investigated the essential work of fracture for plane stress conditions and ductile fracture in a low alloyed, cold rolled, steel. Cotterell and Reddel [20] studied the influence of the test parameters on the measurement of the EWF in zinc sheets. Recent works by Munoz [21] and Casellas [22] show the possibility of determining fracture toughness using the essential work of fracture methodology for several AHSS grades like dual and complex phase, press hardened steels, TRIP and high manganese steels. Using a high-speed camera it is possible to capture the deformation of the specimen surface. Furthermore, it is possible to capture the crack initiation and propagation during loading. The use of digital image correlation (DIC) as post-processing tool allows the determination of the strain field on the specimen surface. For the EWF to be valid a fully yielded specimen ligament is required, visualizing the strain field allows to examine the validity of the test procedure. During dynamic

2. Experiment 2.1. Materials used in the investigation The present study focuses on three AHSS grades, DP, Q&P and TBF. To introduce the unacquainted reader to these steel grades a brief summary is given. Dual-phase (DP) steels are commonly found in automotive applications, as structural reinforcements for crash resistant structures. DP steels consist of a ferritic matrix containing a hard martensitic second phase in the form of islands. Usually the soft ferrite forms a continuous pattern in the microstructure, causing high ductility. During deformation of DP steels the strain concentrates into the lower-strength ferrite which is surrounding the martensite islands, this effect causes high work-hardening. Quench & Partitioning (Q&P) steels are produced by quenching and partitioning steps. In recent years Q&P steels attract high attention because of their mechanical properties. The fully austenitized steel is quenched to a temperature, termed ”quench” temperature, between the martensite start and finish temperature in order to form a controlled volume fraction of martensite. The quenched steel is then held at the quench temperature or higher during the subsequent partitioning step. The remaining austenite after quenching is considered to be stabilized. The austenite stabilization is achieved by carbon partitioning from martensite into austenite during the partitioning step. The final microstructure consists mainly of tempered martensite and retained austenite, Zhu et al. [26]. Q&P steels show a deformation induced martensitic transformation, Zou et al. [27], providing high strength and good ductility. Q&P processes can also be applied on local level, for example in sections of an automotive component, where modified mechanical properties are desired, Forouzan et al. [28]. Modifications of the heat treatment process of Q&P steel grades bridge the gap to the third steel part in the present investigation. Huang et al. [29] reports on an isothermal process introducing bainite into a Q&P process. The microstructure of the TBF (TRIP assited bainitic-ferritic) steels consist of a bainitic and/or tempered martensitic matrix containing retained austenite, which gives the TRIP assisted effect, see Bachmaier et al. [30] and Hausmann et al. [31]. TBF steel grades possess in addition to a high tensile strength a very good property combination of high deep drawability and stretch flangeability. Therefore these steel grades are suitable for the manufacturing of complicated safety related parts for the body-in-white (BIW) lightweight construction in the automotive industry, Winkelhofer et al. [32]. 2.2. Specimen preparation Two different sheet thicknesses are used in the present study for DP and Q&P steels the thickness is t = 1.38 mm , for TBF a thickness of t = 1.48 mm is available. Double edge notched tensile (DENT) specimens were extracted from the coil perpendicularly to the rolling direction. Notches in the DENT specimens are machined with a notch root 333

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continuously during the test. For low loading rate tests, the cross-head velocity is set to v = 1 mm / min ≈ 0.0167 mm / s . Force and elongation measurement are synchronized using a trigger provided from the start script of the tensile test machine.

radius of about 150μm . A schematic representation of the specimen geometries used for both loading rates are shown in Fig. 2. In total five nominal ligament lengths ranging from L = 6 − 16 mm were produced. A fatigue pre-crack was introduced at the notch tip by subjecting the specimens to cyclic loading. A crack length of approximately one millimeter is desired. The propagation of the crack is controlled manually by visual observation, hence variation in the crack length are observed. The actual ligament length was measured for each specimen. Recommendations for the relation of specimen thickness and width to ligament length are given by Cotterell and Reddel [19]. For the correct evaluation of the essential work of fracture the ligament area, i.e. ligament length times sheet thickness, must be completely yielded before crack initiation. Furthermore, the ligament must be in a plane stress state. Those requirements are fulfilled if the lower ligament length is three to five times the sheet thickness and the upper ligament length is not larger than one third of the width of the specimen or two times the radius of the plastic zone, rp , in plane stress.

3, …, 5t ≥ L ≥ min (B /3, 2rp)

2.4. Tensile testing at high loading rates Tensile testing at high cross-head speeds, i.e. high loading rates, require a different test setup. In the present work a Instron high strain rate VHS testing system is utilized. This system is an advanced servohydraulic machine using control technologies designed for a wide range of high-speed test requirements. The test frame capacity is dimensioned for loads up to 100 kN and velocities up to 25 m / s . An operating pressure of 280bar results in high acceleration velocity and load performance. The lower clamping is an in-house development and consists of a combined grip and load cell to optimize the linearity of the load signal at high frequency which occurs at high speeds. The load cell also has a built in system for compensating for the additional masses that are connected to it, in this case the specimen. The load cell clamping consists of two parallel surfaces and a bolt which applies load on the specimen head and positions the specimen in a straight aligned position relative to the upper grip. The upper grip is comparable to a standard double wedge clamping system but modified to allow for acceleration prior to clamping. The upper clamping grip is pretensioned using bolts and a wedge located between the grips. Ejector rods are used to push out the wedges which keep the grips separated. The acceleration distance between wedge and ejector rod is variable but set to 60 mm in the present work. Control of the cross-head displacement and time to reach the desired speed showed that less than 20 mm are sufficient to reach the desired speed. The optical extensometer available at the high-speed machine is designed to have a sufficient sample rate but has a deviation specification of ± 0.1 mm . Elongation of the specimens is known to be only slightly larger than this value. Hence, it is decided to use DIC to measure the elongation. Images for DIC are taken using a Phantom v1610 high-speed camera. It provides a widescreen CMOS sensor and delivers 16,000 frames-per-second (fps) at full resolution of 1280 × 800. At reduced resolutions, the camera offers frame rates of up to 647000fps . The sensor ensures high light-sensitivity which is essential in ultra-highspeed imaging. The camera is equipped with a large internal high-speed memory, pictures are transferred to a computer after the completed test by an Ethernet connection. The camera runs in a continuous recording mode and a trigger signal is used to identify pictures taken during the test. GOM Aramis software system [33] is used for digital image correlation and test evaluation. Three factors are used from this evaluation, (i) for the determination of the essential work of fracture it is necessary to guarantee that the complete ligament length is in a plastic state, (ii) the strain rate at fracture in the region of crack propagation is of interest, and (iii) a virtual extensometer is established. In the present study, the cross-head speed of the high-speed machine is set to v = 0.5 m / s = 500 mm / s , an initial study was conducted with test speeds ranging from v = 0.15 − 6 m / s . The choice of the test

(1)

The shortest nominal ligament chosen is Lnom = 6 mm , with propagated fatigue crack the recommendation for the lower boundary given in Eq. (1) is fulfilled. In order to determine the essential work of fracture with good accuracy five different ligament lengths are used. To accommodate sufficient spacing between those ligaments the specimen is designed with a width of B = 45 mm , see Fig. 1. The largest nominal ligament length is Lnom = 16 mm , taking the propagated fatigue crack into account the actual ligament length is L = 15 mm or less and therefore within the recommendation. The actual, effective, ligament length is measured under a light optical microscope. The presented results in Section 5 use the effective ligament length. The specimen geometries presented in Fig. 1 are governed by (i) the requirements of the essential work of fracture test procedure and (ii) the clamping system of the tensile test machines. The specimen for low loading rate, Fig. 1a, has a length of H = 200 mm . Mechanical clamping in the test machine allows a compact specimen design. For high loading rate tests a different specimen geometry is necessary due to a different machine layout. The specimen length is increased to H = 300 mm and the width beneath and above the ligament section is decreased. The hole in the lower part of the specimen, on the left hand side in Fig. 1b, is used for clamping. A bolt compresses the grip, the load is transferred by friction and mechanical locking of the bolt. The upper and lower grip of the machine have a width of w = 25 mm , hence a wider specimen in this region does not contribute to the load transfer. Furthermore, in dynamic tests it is of advantage to have low weight in connection to the load cell, which is located in the lower grip, to minimize dynamic effects. 2.3. Tensile testing at low loading rate A standard tensile test machine is used for low loading rates. Elongation of the test specimen is measured using a video extensometer with a gauge length of L0 = 50 mm . Force and elongation are recorded

(a) Specimen for low loading rate.

(b) Specimen for high loading rate.

Fig. 1. Schematic representation of the DENT specimen geometry used for the determination of the EWF. 334

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DENT specimens at the root of the notch is discussed by Noda et al. [34] in detail. Using digital image correlation, it is possible to compute the local strain rate at a measurement point. The size of the region where this value applies depends on the chosen facet and step size. In the present study, images for DIC evaluation are available for high speed testing. In the previous section properties of the image recording and DIC evaluation are given. Using the calculated strain values and the time increment between images it is possible to determine the strain rate on the specimen surface. The strain rates of the presented tests calculated with the respective gauge length or the extensometers are ε˙low = 0.33·10−3/ s for low speed and ε˙high = 25/ s high speed. For straight tensile specimens the low speed corresponds to quasi static testing. Seen as single number the value for high speed testing is low, but as will be shown in the result section the actual present values of the strain rate exceed this single number. A summary of the used strain rates and the controlling parameters, crosshead speed and extensometer length, are given in Table 1.

Table 1 Summary of strain rates, cross head speed and extensometer length for the test setups.

Low High

ε˙ [1/ s]

v mm/ s

L0 [mm]

0.334·10−4 25.0

0.0167

50

500.0

20

speed is governed by the obtained strain rate and the number of images possible to capture during the test. Limiting factor is the sample rate of the high-speed camera. Increasing the number of frames per second leads to a decrease in resolution, the resolution cannot be decreased below 640 x 290 px due to the virtual extensometer length of L0 = 20 mm . The choice of the virtual extensometer length is governed by the image size at the necessary camera speed and the size of the plastic zone in the ligament area. For EWF testing the length of the extensometer has only a minor impact as the deformation of the specimen is confined to a small region. Therefore, it is possible to use different extensometer length without producing a significant error. The frame rate at a resolution of 640 x 290 px is 72 k which corresponds to a time increment of 13.3μs between images. The settings for the DIC evaluation in ARAMIS are set to facet size of 12 px and a step size of 2px . Strain values are calculated within a 3x 3 matrix representing three adjacent facets.

4. Evaluation of fracture toughness Cotterell and Reddel [19] and Marchal and Delannay [20] evaluated the essential wok of fracture for a set of experiments and suggested a methodology which is adapted in the present work. The total work of fracture(Wf ) during ductile fracturing can be separated into two components. The first part is the essential work of fracture (we ) which is dissipated in the fracture process zone, the second part is termed nonessential plastic work (wp ) and is spent in the region outside the fracture process zone as a consequence of plastic deformation. If the material front of the crack tip is completely yielded and the plastic zone confined into the ligament, then the plastic work carried out for total fracture is proportional to the plastic volume fracture initiation and the work performed in the fracture process zone correlates with the fractured area. The total work of fracture Wf can than be written as,

3. Determination of strain rate Strain is a measure of deformation a material undergoes during loading, strain rate is the change of strain with respect to time. The strain rate measures the change of distance during time between two adjacent points. The influence of the strain rate on the plastic flow and fracture of materials has been a subject of research under long time, see for example Zener and Hollomon [1] for a pioneering study. For simple deformation, like in a straight tensile test specimen, the strain and therefore the strain rate can be described by a single number. In this case, the definition of strain as measure of the change in distance between two points is obvious. An extensometer measures the change of distance between two points in one direction. This allows to express the strain rate as

ε˙ (t ) =

dε d ⎛ L (t ) − L0 ⎞ v (t ) = = dt dt ⎝ L0 L0 ⎠ ⎜

Wf = we Lt + wp βL2t

(3)

where L is the ligament length, t the specimen thickness and β a shape factor depending on the shape of the plastic zone. The essential work of fracture and the non-essential plastic work scale differently with the sample size. Therefore, if a series of geometrically similar specimens are tested then the two works can be separated. For thin sheets the double edge notched tensile (DENT) specimen is usually used because the transverse stress between the notches is tensile which avoids buckling. Eq. (3) can be normalized by the cross-section area which allows the determination of the essential work of fracture.

⎟

(2)

Here v (t ) is the cross-head speed, L0 the initial gauge length and L (t ) the length of the specimen at time t. For straight samples this assumption is valid during uniform elongation, upon necking validity is lost. In loading cases where the material deforms in various directions, like in notched specimens, the strain and therefore the strain rate at a point within the material cannot be expressed by a single number. For such cases, the rate of deformation is expressed by a tensor that expresses how the relative velocity of the material changes when a point moves by small distance in a given direction. This strain rate tensor can be defined as the time derivative of the strain tensor. The strain rate is not only depending on the rate of loading but also on the shape of the test specimen. Hence, the strain rate varies depending of the local position on a notched specimen. For classification of test speeds, it is convenient to describe the displacement rate of the cross-head of the test machine. Applying the cross-head speed to a measurement length allows calculating a strain rate but this value might be misleading. The material experiences at the local region of the notch and at the tip of a crack different, much higher, strain rates than would be measured using a extensometer. Specimens used in the present study have a notch and a pre-developed crack, causing stress and strain concentration at the crack tip and hence showing high strain rates at the vicinity of the crack tip. The strain rate concentration for

Wf / Lt = wf = we + wp βL

(4)

From experimental results load F, displacement s and ligament length L of the DENT specimens are known. The total work of fracture can be defined as an integral with boundaries at zero displacement, s0 , and displacement at fracture sf .

Wf =

∫s

sf

0

Fds

(5)

If the specific total work of fracture wf is plotted against the ligament length L, a straight line with positive intercept is obtained. The positive value at intercept is the essential work of fracture. A schematic representation of the evaluation of the EWF is shown in Fig. 2a. However, there are some restrictions that must be met in order to use Eq. (4), these were already pointed out in Section 2.2. The EWF methodology also permits to easily separate the energetic contributions from crack initiation and crack propagation. The specific work for fracture initiation, wfi , can be obtained by calculating the work of fracture up to the onset of crack propagation and dividing by the cross-section area Lt . From wfi values the specific work of fracture at initiation of propagation, wei , is obtained. Mai and Cotterell [35] showed 335

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(a)

(b)

Fig. 2. A schematic representation of the determination of the essential work of fracture, Fig. 2a and determination of wei in Fig. 2b.

that wfi is constant and independent of the ligament length. Thus, only mean values of wfi are necessary to calculate wei .

and indicated by a lower and upper bound. Different colors indicate the different loading rates. In Table 2 the results from the evaluation are summarized. A general observation from Fig. 6 is that there is an influence of the loading rate on the essential work of fracture. With increased loading rate the wei and we values increase, showing a rate sensitivity of the fracture toughness. The non-essential plastic work βwp shows different behavior for the three steel grades. In Fig. 6a the result of the DP steel is summarized. The value of we increases significant with the loading rate. The scatter of the measured data points is relative low which leads to a narrow confidence interval and good fit using R2 1 as a measure of quality of linear fitting. The combined value of the shape factor and the non-essential plastic work βwp shows similar gradients for the low and the high loading rate suggesting that the plastic work is not sensitive to the loading rate in the DP grade. In Fig. 6b the result of the TRIP-assisted Bainitic-Ferritic steel is presented. Again, the values for we increase with increasing loading rate. Compared to the DP steel the difference in the values is slightly higher. For low loading rates the scatter of the data is comparable to the results from DP testing. The high loading rates show a scatter of data points, this is caused by specimens where load and/or elongation is higher although the ligament length is comparable to other specimens in the same ligament group. Naturally the confidence interval is wider and the R2 -value is smaller for high loading rate tests. The gradient βwp at higher loading rate is lower in comparison to low rate testing. Taking the scatter of data points into account similar gradients are possible and it is concluded that the loading rate does not influence the non-essential plastic work significant. The third steel investigated is the Q&P grade which is presented in Fig. 6c. The influence of the loading rate on we is similar to the other steel grades while βwp shows a clear change in the gradient. The scatter for the low rate tests is small allowing a high R2 value which is an indicator for a good linear fit of the data, similar for the high loading rate where a good R2 value is obtained. The prediction interval shows a narrow span comparable between the loading rates. The determination of the wei value is conducted using DIC images where the onset of crack initiation is visually observed, due to the number of pictures taken it is possible to observe this point with good reliability. Also, the DIC evaluation loses the image recognition if the random speckle pattern is strongly disturbed, indicating that a crack has initiated or is about to initiate. A sensitivity study is conducted to evaluate how wei is influenced if the data point taken for the calculation is changed. It is found that the influence is small and within the error

5. Results and discussion This sections is intended to summarize and discuss the test results obtained on the three different steel grades, tested at different loading rates. In the first section the results from tensile testing are presented. In the following section the results from the evaluation using the essential work of fracture approach are discussed. Furthermore, a discussion on DIC observations on the high loading rate samples is given. 5.1. Tensile testing of DENT specimens The tensile tests of the DENT specimens are conducted on two different machines using different technological approaches. For low loading rates a screw driven machine is used, while high speed tests are conducted on a hydraulic machine. In addition the load-cell and the extensometer are different comparing low with high speed testing. These differences are caused by the large span of test speeds and reliability of measurement equipment at respective cross-head speed. The experimental setup for the low speed measurements are presented in Section 2.3 and for high speed testing in Section 2.4. The tensile test results, load versus elongation, for all loading rates and materials are presented in the following figures, see Fig. 3 to 5. DP and TBF grades show similar load levels and elongations for all ligament length. The Q&P steel shows higher load levels but less elongation prior to fracture compared to the other grades. A general observation is that a clear peak load at low strain rates is visible for all steel grades. For tests conducted at high strain rates DP and TBF show clear peak loads, the Q&P grade shows clear peak loads for most specimens. 5.2. Evaluation of the essential work of fracture In the previous section the test results from tensile testing of DENT specimens are presented for the three investigated steel grades. Furthermore, the data from load and displacement measurement is subsequently used to calculate the total work which is then used to determine the essential work of fracture and the non-essential plastic work. In Section 4 the procedure is introduced, this section is intended to show the results of this evaluation. In Fig. 6 the result for all steel grades are summarized, every dot indicate the result of the evaluation of one load-displacement curve i.e. the total work versus the measured ligament length. Using linear regression a best fit to experimental results is obtained providing the essential work of fracture at zero ligament length and non-essential work together with the shape factor from the gradient of the linear fit. From linear regression analysis also a confidence interval is calculated

1 The coefficient of determination, R2 , is a number between zero and one and provides a measure of how well a regression line fits a set of data. An indication for high fitting quality is a value close to one, while a value close to zero indicates inadequate fit of the regression line to the data.

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Fig. 3. Result of DPDENT specimens tested at different loading rates.

Fig. 4. Result of TBFDENT specimens tested at different loading rates.

Fig. 5. Result of Q&PDENT specimens tested at different loading rates.

5.3. Observations from the high-speed imaging and digital image correlation

margin. This is mainly caused by the number of data points available during loading, up to the peak load the number of data points is dense but with crack initiation the elongation of the specimen increases at a higher rate and only fewer images, i.e. data points, are available until final fracture. The determination of the last data point, i.e. where a full crack has developed, is also obtained by visual observation of the DIC images. During the last stages of deformation the spacing between images increases and hence the distance between elongation values, see Fig. 9 for distance between data points at equal time increments. The determination of the last valid data point has a significant impact on the value of the total work and therefore on the essential work of fracture.

The test setup used for high rate loading is equipped with a highspeed video camera. Images taken during the test are post-processed using digital image correlation (DIC). The DIC output is used to generate a virtual extensometer which is used for load-displacement figures. Initial tests showed a more reliable and accurate measurement using a virtual extensometer compared to the standard optical extensometer system available. Additional to extensometer measurements it is possible to evaluate the strain field on the specimen surface. The strain values are directly calculated from the deformation gradient which in its turn is obtained from image correlation. The strain rate the 337

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Fig. 6. Result of EWF evaluation for different steel grades. Table 2 Summary of the evaluation of DENT specimen test results. Essential fracture initiation energy wei , essential work of fracture we , R-squared R2 and non-essential plastic work with the shape parameter βwp .

DP TBF QP

Low High Low High Low High

wei

we

R2

βwp

[kJ / m2]

[kJ / m2]

[-]

[kJ / m2]

114 ± 8 202 ± 26 111 ± 10 218 ± 31 168 ± 20 288 ± 29

136 ± 20 263 ± 23 146 ± 13 284 ± 55 196 ± 12 351 ± 15

0.91 0.80 0.98 0.74 0.96 0.81

20.7 18.5 28.4 22.8 20.8 9.0

material undergoes can be calculated by dividing the strain by the time increment between images. The time increment between images is controlled by the user and a fixed parameter in the test setup. In Fig. 7 a cropped image of the DIC evaluation is presented. The image size is adjusted as no additional information is found further away from the ligament length. Chosen ligament length in the figure is for all steel grades ten millimeters. The position of the ligament is indicated by the black sections in the image, which corresponds to the opened pre-developed crack. In all images the scale is set to two percent equivalent von Mises strain. In accordance with the requirements of EWF testing the ligament is fully yielded prior to fracture. Hence, the evaluation of the essential work of fracture at high loading rates is valid as the requirement of a fully yielded ligament prior to crack initiation is fulfilled. The images presented in Fig. 7a to 7c are taken at or close to peak load. The position of the image on the corresponding load-displacement curve is indicated by a solid mark in Fig. 8.

Fig. 8. Load-displacement curves with indication of points were images for yielded ligament length (solid mark) in Fig. 7, and crack propagation (circle) in Fig. 9, are taken.

A decrease in load bearing capacity in DENT specimens is usually attributed to crack initiation and propagation. The assumption that crack initiation coincides with the peak load is evaluated for the high rate samples. Due to the number of images taken during the test it is possible to determine the point of crack initiation with reasonable accuracy by visual observation. An example for crack initiation and crack growth during testing is given in Fig. 9. However, the images presented are taken from a TBF test with a ligament length of ten millimeters. In Fig. 8 the points matching the images are indicated by circles. A general observation is that the peak load and the crack initiation correlate relatively well taking the subjective observation of the first visible crack

Fig. 7. Fully yielded cross section at approximately peak load and prior to crack initiation for the three different steel grades. The position of the images presented in relation to the corresponding load-displacement curves is indicated in Fig. 8. A ligament length of ten millimeters is chosen for the display. The maximum value of the legend is set to εvM = 2% equivalent von Mises strain. 338

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Fig. 9. Crack propagation in a TBF sample with ten millimeter ligament length tested at high loading rate. The position of the images on the load-displacement curve are indicated in Fig. 8, t-values indicate the time during the test when the image was captured.

ligament an arc shaped pattern originates. This pattern can be explained by wave propagation in the specimen. The pattern is visible in earlier stages than depicted in Fig. 10 but with crack initiation the pattern intensifies. The actual strain rate acting in the material at a notch tip is significant different from the value calculated using an extensometer. In Fig. 11 two possible approaches for quantifying the strain rate of the ligament are shown. Furthermore, Fig. 11a shows the strain rate along a section perpendicular to the ligament at centered position between the crack tips i.e. at half of the width of the specimen as shown in Fig. 1a. The different colored lines indicate different time steps prior to fracture. For this particular figure, the same sample is chosen as in Fig. 9. The last images before the crack has fully propagated through the specimen show high strain rates in the ligament length. However, the band width in which the strain rate reaches high values is narrow, in the presented figure about 1.6 mm which is close to the blank thickness. This shows that the fracture process zone shows a relationship to the blank thickness. If one single time step of the section is plotted isolated the pattern seen in Fig. 10 is visible as peaks and valleys.

into account. Image Fig. 9a is taken prior to crack initiation i.e. before a first visible crack could be detected, due to loading of the specimen the predeveloped fatigue notch is opened and clearly visible. Crack growth starts from the pre-developed fatigue crack tip towards the center of the specimen. Two possible directions for crack initiation starting from the pre-developed crack are possible. Either the crack starts, with reference to the images, in direction left or right. The two possible crack paths and the onset of crack initiation in two directions is visible in Fig. 9f at the lower notch to the right, a crack initiated but the dominant crack occurred to the left. In Fig. 9d crack growth from both notch tips are visible, with the dominant crack path on the left specimen side. In images following Fig. 9i the crack tips meet and final fracture is reached. Besides the strain field also the strain rate field on the specimen surface can be determined. The strain rate is calculated using the equivalent von Mises strain and the stage time increment. In Fig. 10 the strain on the specimen surface is visualized. The scale is adjusted to represent values between zero and two hundred. From the notch and 339

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Fig. 10. Visualization of the strain rate pattern taken from DIC results of a TBF sample with ten millimeter ligament length tested at high loading rate. The color for the comparison scale indicates in red 200/ s for the upper boundary and in blue the lower boundary which is equal to zero. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

biased results the residuals are checked for eventual patterns. However, in the evaluation no residual patterns are observed and hence it is assumed that the results are not biased. Furthermore, in Fig. 6 the prediction interval is plotted alongside the linear function, the interval contains at least 95% of future predictions. Also, in the determination of the point of crack initiation and also of final fracture is subjective judgment and therefore a possible source of error. For crack initiation the error is comparable small as many data points with little difference in load and elongation are available, final fracture is more difficult to asses as the data points are scarce. Furthermore, the test data is evaluated independently by the authors and good agreement between the obtained results was found. Also, samples where no clear decision could be made are removed from evaluation.

A second approach to put a number on the strain rate is to average the strain rate value over an area. In Fig. 11b a rectangular area between the notches is selected, calculated strain rate values of this area are averaged to a single number for every time step. Naturally the values are lower compared to the first approach. For all three steel grades a clear trend is visible, with the strain rate rapidly increasing towards the end of the test when the crack starts to propagate. The authors want to emphasize that both approaches shown in Fig. 11 are not intended to quantify the strain rate i.e. no statement about the value of the fracture toughness is given at a specific strain rate. Therefore, the authors refer to low and high rate loading for test conditions which cause different strain rates at the specimens notch.

5.4. Uncertainties 6. Conclusion The test data is evaluated using the trapezoidal rule to obtain the the total work of fracture. However, to reduce the error by noise of the measurement the number of data points used for the calculation of the area are reduced and a curve fit is used to smooth the data. The obtained total work of fracture is evaluated using a linear regression analysis. A linear function is sufficient as the essential work of fracture is determined on geometrically similar specimens and hence non-linearity would indicate an error in the experiment. The goodness-of-fit of the linear model is evaluated using the R2 value. The R2 gives an indication for the goodness-of-fit but must be handled with care. To avoid

From the results summarized in the present paper, the following conclusions can be drawn. It has been observed that the essential work of fracture increases with the loading rate. At high loading rate all the studied steel grades have shown a significant increase on we , with respect to the low rate values. This leads to a higher crack propagation resistance at high loading rates. The difference of we for DP and TBF is about the same at low and high loading rate while comparing to Q&P a larger difference is found. Crack initiation energy wei increases for DP and Q&P grades with about seventy percent by increasing the loading

Fig. 11. Evaluation of strain rate along a centered section and averaged across an area. 340

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rate, the TBF grade shows a doubling. The Q&P grade shows the highest resistance against crack initiation for both loading rates. Using highspeed imaging and digital image correlation an evaluation of the validity of the EWF approach at high loading rates is possible. It has been shown that the methodology of EWF is applicable at high loading rates as all requirements can be fulfilled. Using DIC and high speed imaging the crack propagation in DENT specimens can be tracked and correlated to load and displacement values. The determination of the strain rate in notched specimens is possible using DIC, but has to be used with care if correlated to material properties.

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Acknowledgments

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The test at high strain rates would not have been possible without the assistance by Mr. Jan Granström, his support during testing and evaluation is highly appreciated. The suppliers of the AHSS blanks are acknowledged for their support with material.

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