Extension twinning during dynamic compression of. polycrystalline magnesium. Caleb Hustedt

Transcription

1 Extension twinning during dynamic compression of polycrystalline magnesium by Caleb Hustedt A thesis submitted to The Johns Hopkins University in conformity with the requirements for the degree of Master of Science. Baltimore, Maryland December, 2016 c Caleb Hustedt 2016 All rights reserved

2 Abstract We performed time-resolved x-ray diffraction on bulk, equal-channel angular extruded (ECAE) polycrystalline magnesium and magnesium alloy AZ31B during dynamic compressive loading at strain rates up to 2500 s 1 at the Cornell High Energy Synchrotron Source (CHESS). The crystallographic texture of the two materials was similar, but the alloy specimens had a smaller average grain size. The texture of both materials evolves during loading in a manner consistent with extension twinning on the { } twin system, but the progress of twinning with strain is slower in the alloy, probably due to its smaller grain size. After twinning the lattice strains of the twinned grains increases linearly, without evidence of yielding or plastic flow, likely because the orientation of these grains results in a small resolved shear stress for basal slip and prismatic slip. Similar experiments, with a larger detector, were performed at the Dynamic Compression Sector(DCS) at the Advanced Photon Source(APS) at Argonne National Laboratory(ANL). An orientation distribution function was calculated from the data and used as input to a reduced crystal plasticity model created by the Army Research ii

3 ABSTRACT Laboratory(ARL). The experiments and model were found to be in good agreement suggesting the simplifications to the model were justified. Primary Reader: Todd Hufnagel iii

4 Acknowledgments I am using this opportunity to express my gratitude to everyone who supported me throughout the course of this Masters degree. Special thanks are given to Paul Lamber for his continued support and assistance. I would also like to thank Vignesh Kannan and Mantong Zhao for their assistance in my research efforts. Other individuals I would like to thank include: KT Ramesh, Alexander Ananiadis, Tim Weihs, Dan Casem, Emily Huskins, Jeffery Lloyd, Richard Becker, Christopher Meredith, and Nick Sinclair. This work was sponsored in part by the Army Research Laboratory and was accomplished under Cooperative Agreement No. W911NF The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein. This work is based upon research conducted at the Cornell High Energy Syniv

5 ACKNOWLEDGMENTS chrotron Source (CHESS) which is supported by the National Science Foundation and the National Institutes of Health/National Institute of General Medical Sciences under NSF award DMR Detector development at Cornell is supported by the DOE Grant No. DE-SC , the Keck Foundation, and CHESS. P.K.L. would like to acknowledge OSD-T&E (Office of Secretary Defense-Test and Evaluation), Defense-Wide/PE D8Z National Defense Education Program (NDEP)/BA-1, Basic Research, for their support. This work was sponsored by the Center for Materials in Extreme Dynamic Environments (CMEDE), part of the Hopkins Extreme Materials Institute (HEMI). The Dynamic Compression Sector is supoorted by the DOE and NNSA under Award Number DE-NA The Advanced Photon Source is supported by the DOE under Contract No. DE-AC02-06CH v

6 Dedication This thesis is dedicated to my wife and children, Tyra, Eli and Lila Hustedt. vi

7 Contents Abstract ii Acknowledgments iv List of Tables viii List of Figures ix 1 Introduction 1 2 Background Deformation in HCP metals Effects of grain size Quasistatic testing Dynamic testing using in-situ time-resolved x-ray diffraction Experiment Cornell High Energy Synchrotron Source (CHESS) vii

8 CONTENTS 3.2 Dynamic Compression Sector (DCS) Results and discussion CHESS DCS Pole figure inversion method Computationally efficient crystal plasticity model Conclusions 33 Bibliography 35 Vita 52 viii

9 List of Tables 2.1 CRSS values of deformation mechanisms in magnesium ix

10 List of Figures 2.1 Plots of integrated intensity versus scattering vector for { }, { }, and { } planes of an AZ31B magnesium sample processed by ECAE and compressed along the extrusion direction at 1000 s 1. X-ray diffraction patterns were collected using a pixel array detector(pad) at the Dynamic Compression Sector(DCS) at the Advanced Photon Source(APS) with a single bunch, 33.5 picosecond, exposure time Example of test detector timings with a true stress-true strain curve of a rolled AZ31B magnesium alloy compressed along the rolling direction at a strain rate of 1000 s 1 overlaid with detector signal (in blue) indicating when the diffraction patterns were captured Diffraction geometries for observing { } twinning in Mg. (a) If the scattering vector k is perpendicular to the loading direction, initial scattering from the {0002{ basal planes will be replaced by scattering from the { } prism planes as the lattice reorients. (b) With k parallel to the loading direction, scattering from the prism planes decreases and scattering from the basal planes increases Experimental geometries. (a) With the incident x-ray beam k 0 perpendicular to the Kolsky bar and the scattered beam k (at angle 2θ with respect to k 0 ) in the vertical plane, k is perpendicular to the loading axis. (b) With the bar rotated about a vertical axis by ω 2θ/2 and k in the horizontal plane, k is parallel to the loading direction Pole figures along the extrusion direction for (a) magnesium alloy AZ31B and (b) pure magnesium, determined by electron backscatter diffraction (EBSD) Experimental setup for time-resolved x-ray diffraction on dynamicallydeforming specimens x

11 LIST OF FIGURES 3.5 Preliminary x-ray diffraction data on extension twinning from a rolled AZ31B magnesium sample compressed along the rolling direction at a strain rate of 1000 s 1. Both images are taken on an intensified CCD (ICCD) array at the Dynamic Compression Sector (DCS) at the Advanced Photon Source (APS). The red box highlights the { } basal planes undergoing a 90 lattice reorientation from perpendicular to parallel to the loading axis due to extension twinning One-dimensional diffraction patterns during dynamic compression for pure Mg with the scattering vector k (a) perpendicular and (b) parallel to the loading axis (a) Twin volume fraction V twinned as a function of strain, determined from Eqn. 4.1 from nine samples of pure Mg and ten samples of AZ31B under dynamic compression. The data points each represent a single diffraction pattern (eight per sample). Filled circles represent samples having their scattering vector perpendicular to the loading and open circles parallel. The vertical error bars correspond to the uncertainty in the integrated areas of the diffraction peaks, whereas the horizontal bars represent the strain over which the data were collected during the exposures. The solid lines are running averages for the two materials. (b) Representative stress-strain curves from one sample of each material. The end of the stress-strain curves corresponds to unloading, not fracture of the material Elastic strain in pure Mg samples as a function of true stress for (a) untwinned and (b) twinned regions with scattering vector parallel and perpendicular to the loading direction. Vertical error bars represent the stress range over which each data point was collected. Horizontal error bars are uncertainties in the fitted peak positions. Dashed lines represent linear elastic behavior of grain populations based off of single crystal elastic constants Flow of pole figure inversion method employed on in-situ x-ray diffraction experiments Flow of computationally efficient crystal plasticity modeling Twin volume fraction as a function of strain for both experimental data and simulation. The experimental data was calculated using insitu x-ray diffraction data from a rolled AZ31B magnesium sample compressed along the rolling direction at a strain rate of 1000 s 1. Simulation data are taken from a simulation run using the computationally efficient crystal plasticity model with initial textural input taken from the diffraction data of the same rolled AZ31B magnesium sample using the inverse pole figure method and sampling of the subsequent ODF xi

12 Chapter 1 Introduction The primary slip directions in hexagonal-close-packed (hcp) metals are the directions of atomic close-packing, with Burgers vectors b corresponding to the lattice vector a = These directions lie in the ( ) basal plane, although the 3 basal plane may or may not be the slip plane, depending on the particular metal in question. 9 Accommodating a general plastic deformation requires a slip system with an out-of-plane component of the Burgers vector (such as b = c + a on pyramidal planes), but the large Burgers vector and high critical resolved shear stress make these systems less favorable for slip Deformation twinning is an alternative mechanism of plastic deformation that provides a means to accommodate deformation in non-basal directions. 13 While deformation twinning under low-rate loading has been extensively studied, less is known about behavior at high strain rates that are important in many situations, ranging 1

13 CHAPTER 1. INTRODUCTION from automobile crashes to penetration of armor by ballistic projectiles. Magnesium and its alloys are becoming increasing popular in automotive, aerospace, and defense industries due to magnesium s low atomic weight. However, in magnesium, the basal slip and two non-basal slip modes, the prismatic and pyramidal, provide deformation parallel to the basal plane only. Furthermore, the secondary pyramidal c + a slip, which provides another non-basal deformation is relatively difficult to activate at room temperature due to its high critical resolved shear stress(crss). Therefore, other mechanisms, such as deformation twinning(dt), provide an additional mechanism to accommodate c-axis deformation. The most commonly observed DT mode is the { } extension twinning when the c-axis experiences tension. Extension twinning plays a key role in the deformation of magnesium owing to its low CRSS as well as crystallographic reorientation of the lattice by In this work we report the use of in situ x-ray diffraction to study deformation twinning in magnesium and magnesium alloy AZ31B under dynamic compression. Prior in situ x-ray diffraction 14 and neutron diffraction 15 work has shown that crystallographic texture evolution and deformation twinning can be tracked under quasistatic loading with data collection times on the order of minutes to hours. Here we demonstrate a similar ability for dynamic loading, using a high-intensity synchrotron source and a fast pixel-array detector to monitor deformation at strain rates of up to 2500 s 1 with data collection times of a few microseconds. The evolution of the twin volume fraction follows a similar pattern to that observed under quasi-static loading. 2

14 CHAPTER 1. INTRODUCTION At a given strain we find that samples of AZ31B have a smaller twin volume fraction than those of pure magnesium, likely due a smaller grain size in the alloy samples (1.5 µm v. 9.5 µm). We compared our observations quantitatively with predictions of the evolution of crystallographic texture from an efficient reduced crystal plasticity model for magnesium and its alloys. The model we used explicitly accounts for basal slip and extension twinning on a rate-independent basis, but treats other mechanisms (pyramidal and prismatic slip) as isotropic, rate-dependent functions. This combination yields substantial improvements in efficiency over full crystal-plasticity models while retaining key aspects of the most important deformation mechanisms. 3

15 Chapter 2 Background 2.1 Deformation in HCP metals In hexagonal close packed (HCP) materials the primary and most easily activated slip system has a Burgers vector, b = a = 1120 /3 on the basal plane as in Cd, Zn, Mg, and Be, or on a prism plane as in Ti, Zr, Hf. 16 Non-basal slip modes such as b = c+a on pyramidal planes are secondary or additional slip systems with 2, 17, 18 higher activation energies. Accommodating out of plane strain requires either non-basal slip or deformation twinning. Basal and non-basal slip systems, such as ( ) basal slip, { } prismatic slip, and { } pyramidal c + a slip, do not provide enough independent slip systems to accommodate general homogeneous plas- 19, 20 tic deformation, leading to poor formability at room temperature. Therefore, 4

16 CHAPTER 2. BACKGROUND other mechanisms, such as deformation twinning, must be activated to allow for arbitrary plastic deformation. 21 There are two commonly observed deformation twinning mechanisms, the more prevalent { } extension twinning and { } contraction twinning. Extension twins provide an extension along the crystallographic c-axis whereas contraction twins provide a contraction along the crystallographic c-axis. Additionally, extension twins also tend to span the entire length of grains and grow to consume the grains completely whereas contraction twins tend to be more localized. Thus, extension twins are much easier to observe experimentally. 1 Experimentally measured values for the critical resolved shear stress(crss) for these slip and twin systems for magnesium have been reported by several studies. 1 3, 5 7, As can be seen in Table 2.1, the lowest values of CRSS, and therefore most easily activated mechanisms, are basal slip and extension twinning. Prismatic, pyramidal c + a, and contraction twinning have much higher CRSSs. 2.2 Effects of grain size The effect of grain size on deformation twinning has been well studied at low rates as well as a few studies at high rates It has been reported that both 25, 27, 31, 32 twinning and dislocation plasticity are grain size dependent. The grain size also influences the expected predominant deformation mode. In AZ31 it has been 5

17 CHAPTER 2. BACKGROUND Critical resolved shear stress (CRSS) of deformation mechanisms in magnesium Slip/twin plane Slip direction/shear CRSS (MPa) direction by twinning Basal slip (0 { 0 0 1) } Prismatic slip { } Pyramidal c+a { } Extension Twin { } Compression Twin Table 2.1: CRSS values of deformation mechanisms in magnesium. 1 7 shown that the stress verses strain response changes from a concave shape into a convex shape when the grain size is on the order of 3 to 4 µm. 25 This is a result of the absence of twinning deformation in specimens with smaller grain sizes. Furthermore, it has been suggested that there exists a critical grain size below which twinning is suppressed, and this critical grain size is 3 to 4 µm for ZK In pure magnesium, this critical grain size is about 2.7 µm, and for smaller grains dislocation mediated 27, 31, 32 plasticity dominates. The existence of such a critical grain size below which twinning deformation is suppressed is rationalized by suggesting that the sensitivity of twinning to grain size is greater than that for dislocation mediated plasticity. 26 This is also in agreement with the experimental observations in FCC, BCC and HCP materials. 34 Other grain size studies have shown the length of the plateau region in the stress versus strain curves was influenced by grain size and the Schmid factor, increasing with grain refinement because of the high fraction of grain boundaries. 28 At 6

18 CHAPTER 2. BACKGROUND rates of 1100 s 1 decreasing the grain size was found to result in a smaller twinning 29, 30 fraction and lower strain hardening rates in AZ31B alloy during shock loading. 2.3 Quasistatic testing The low-rate behavior of deformation mechanisms in magnesium has been 18, 25, studied extensively. This is typically done using some type of MTS system for deformation and ex-situ measurements of the microstructure done by x-ray diffraction, optical, or electron microscopy. Of particular interest here we highlight the work using neutron diffraction to study magnesium and its texture evolution under quasi-static loading Quasi-static strain rates and/or interrupt tests allow for long exposure times that yield signal to noise ratios that would not be possible on shorter timescales. This allows for processes such as Reitveld refinement and ODF estimation to be easily performed Dynamic testing using in-situ timeresolved x-ray diffraction Deformation mechanisms at high rates have not been as exhaustively studied and are not as well understood. It has been shown that there are changes in material behavior for several deformation mechanisms under high strain rate loading. 52 As 7

19 CHAPTER 2. BACKGROUND discussed previously, deformation twinning is one example of the effects of strain rate on deformation mechanisms, increasing in twin volume fraction with increasing strain 53, 54 rate. In-situ x-ray experiments at high rates have not been possible due to low signal quality and slow detectors. However, improvements in synchrotron sources, insertion devices, and x-ray detector technology in recent years have made it possible, through careful experimental design, to collect much of the same diffraction and imaging information from a sample during dynamic deformation that can be obtained quasistatically A schematic of an experimental setup utilizing these capabilities is shown in Figure 3.4. Due to the short test duration of the experiments certain test procedures are not possible at high rates, such as sample rotation. Additionally, the signal-to-noise(s- N) ratio at high rates will likely never be as good as at low rates. However, even with these and other restrictions, diffraction patterns with good enough signal-tonoise ratio to be able to perform quantitative analysis have been collected on the sub-microsecond time scale. 55 Example diffraction data are shown in Figure 2.1. These diffraction patterns can be correlated to specific points in the experiment and matched up with the stress-strain data recorded during the deformation event. This allows for a direct correlation between mircostructural properties and specific stressstrain states in the material. Signals from the x-ray detector can be simultaneously recorded on an oscilloscope along with the Kolsky bar data, allowing us to precisely 8

20 CHAPTER 2. BACKGROUND {1011} Intensity (arb.) {1010} {0002} Scattering vector [A -1 ] Figure 2.1: Plots of integrated intensity versus scattering vector for { }, { }, and { } planes of an AZ31B magnesium sample processed by ECAE and compressed along the extrusion direction at 1000 s 1. X-ray diffraction patterns were collected using a pixel array detector(pad) at the Dynamic Compression Sector(DCS) at the Advanced Photon Source(APS) with a single bunch, 33.5 picosecond, exposure time. 8 determine when the data were being collected. An example of the timing of the detector overlaid with the stress-strain response is shown in Figure 2.2. Both the integration timings and spacings can be arbitrarily varied. 9

21 CHAPTER 2. BACKGROUND 300 True stress (MPa) True strain (%) Figure 2.2: Example of test detector timings with a true stress-true strain curve of a rolled AZ31B magnesium alloy compressed along the rolling direction at a strain rate of 1000 s 1 overlaid with detector signal (in blue) indicating when the diffraction patterns were captured. 10

22 Chapter 3 Experiment 3.1 Cornell High Energy Synchrotron Source (CHESS) The utility of diffraction for studying deformation twinning is best understood with the aid of Fig The scattering vector k is defined as the difference between the incident and diffracted x-ray beam wave vectors. The only planes that contribute to scattering in a given direction are those perpendicular to k, so by recording the diffracted intensity in different directions we obtain information about the crystallographic orientation of the grains. Here, we are concerned with { } twinning (commonly called extension twinning in Mg) which causes a reorientation of the lattice by In Mg this type of twinning is promoted by compressive 11

23 CHAPTER 3. EXPERIMENT Scattering vector {1012}<1011> Twin plane and direction Twinned orientation {1012}<1011> Twin plane and direction Twinned orientation Scattering vector Compression axis Compression axis {1010} 86.3 Initial orientation {0002} Scattering vector perpendicular to compression axis (a) {0002} 86.3 Initial orientation {1010} Scattering vector parallel to compression axis (b) Figure 3.1: Diffraction geometries for observing { 1012 } 1011 twinning in Mg. (a) If the scattering vector k is perpendicular to the loading direction, initial scattering from the {0002{ basal planes will be replaced by scattering from the { 1010 } prism planes as the lattice reorients. (b) With k parallel to the loading direction, scattering from the prism planes decreases and scattering from the basal planes increases. loading parallel to the basal plane, so if k is perpendicular to the loading direction (Fig. 3.1(a)) we would expect to see scattering from the (0002) basal planes replaced by scattering from the { 1010 } prism planes as twinning proceeds and the lattice reorients. The opposite is true if k is parallel to the loading direction (Fig. 3.1(b)). Figure 3.2 shows how these diffraction geometries can be implemented in the context of compression Kolsky bar (split-hopkinson pressure bar) dynamic testing of materials. In Fig. 3.2(a) the x-ray detector is placed to capture scattering in the vertical plane, making the scattering vector k perpendicular to the loading axis. In Fig. 3.2(b) the detector is placed to record scattering in the horizontal plane and the Kolsky bar is rotated in the horizontal plane; together, these make k parallel to the loading axis. In the Kolsky bar technique the sample is placed in initial contact with input and 12

24 CHAPTER 3. EXPERIMENT z y x Striker bar Strain Pulse Keck PAD ω Striker bar Keck PAD Incident X-rays Incident bar 2θ Specimen k ( load axis) (a) Scattered X-rays Output bar Incident bar z y 2θ x Specimen Output bar k ( load axis) (b) Figure 3.2: Experimental geometries. (a) With the incident x-ray beam k 0 perpendicular to the Kolsky bar and the scattered beam k (at angle 2θ with respect to k 0 ) in the vertical plane, k is perpendicular to the loading axis. (b) With the bar rotated about a vertical axis by ω 2θ/2 and k in the horizontal plane, k is parallel to the loading direction. output bars, which in our case were mm diameter and made of aluminum. A striker bar fired from a gas gun strikes the incident bar, creating a strain pulse that propagates down the bar and into the specimen. Both the input and output bars are equipped with strain gauges, recording the strain-time history of the reflected and transmitted pulses, respectively. From these data the stress-strain history of the specimen can be determined. 59 We installed our Kolsky bar in hutch G3 of the Cornell High Energy Synchrotron Source (CHESS). The sample was illuminated with a 1 mm 1 mm beam of 10 kev x-rays. We recorded x-ray diffraction data on the Keck pixel array detector (Keck- PAD) capable of sub-microsecond integration times 60 [Add reference to Philipp, 2016]. The detector can record eight frames before read-out (which takes much longer). For these experiments we used a 2 3 detector array with an area of mm and a pixel size of 150 µm. The two-dimensional diffraction data recorded on the detec- 13

25 CHAPTER 3. EXPERIMENT tor were azimuthally integrated to produce the one-dimensional diffraction patterns shown below. The majority of the diffraction patterns were collected for integration times of 2.56 µs with a few having integration times of 5.12 µs, which represent either one or two circulation times of the positrons around the storage ring at CHESS. Additional details of the in situ x-ray diffraction/kolsky bar experiments can be found in Reference. 61 We studied both magnesium (nominally 99.9% pure) and alloy AZ31B (Mg-3%Al- 1%Zn-0.3%Mn, expressed in weight percent) processed using equal channel angular extrusion (ECAE). After extrusion the average grain size was 9.5 µm for the pure Mg samples and 1.5 µm for AZ31B. Both materials had an initial texture in which a majority of the basal (0002) poles were oriented at 45 to both the extrusion and transverse directions (Fig. 3.3). We cut samples for dynamic compression from the extruded ingots by electric discharge machining (EDM) and mechanically polished them into right rectangular prisms ( mm) with a surface finish of 5 µm. We collected in situ x-ray diffraction patterns during dynamic uniaxial compression along the extrusion direction at strain rates of 2500 s Dynamic Compression Sector (DCS) The experimental setup at the Dynamic Compression Sector (DCS) (beamline 35) at the Advanced Photon Source (APS) at Argonne National Laboratory (ANL) 14

26 CHAPTER 3. EXPERIMENT {1010} {0002} {1010} {0002} -TD -TD -TD -TD LD (a) LD LD (b) LD Figure 3.3: Pole figures along the extrusion direction for (a) magnesium alloy AZ31B and (b) pure magnesium, determined by electron backscatter diffraction (EBSD). z y x Striker bar Strain Pulse Incident bar 2θ Incident X-rays Specimen Scattered X-rays Output bar k ( load axis) Figure 3.4: Experimental setup for time-resolved x-ray diffraction on dynamicallydeforming specimens. was extremely similar to the setup at CHESS with one key difference, a larger detector (see Figure 3.4). This larger detector, consisting of an ICCD camera array was able to capture more and full diffraction rings. Full rings allowed for not only direct observation of extension twinning (Figure 3.5) but for the estimation of an orientation distribution function (ODF) (4.2.1). The Kolsky bar technique utilized the same mm diameter aluminum bars used at CHESS. The striker was made of aluminum and tapered from mm down to mm to create a more constant strain rate. 15

27 CHAPTER 3. EXPERIMENT ε = 0.00 ε = Compression axis Compression axis Figure 3.5: Preliminary x-ray diffraction data on extension twinning from a rolled AZ31B magnesium sample compressed along the rolling direction at a strain rate of 1000 s 1. Both images are taken on an intensified CCD (ICCD) array at the Dynamic Compression Sector (DCS) at the Advanced Photon Source (APS). The red box highlights the { } basal planes undergoing a 90 lattice reorientation from perpendicular to parallel to the loading axis due to extension twinning. The experimental setup was installed in B hutch at the DCS. The size of the 22 kev x-ray beam interrogating the sample was approximately 350 µm wide 500 µm tall. The diffraction patterns were recorded on the ICCD camera array that used a scintillator follow by two beam splitters to record images on four separate cameras at the sub-microsecond time scale. Each camera was able to capture one image during the experiment and by adjusting the interframe spacing and integration times we were able to obtain images at arbitrary times during the deformation event. Each camera had a pixel size of approximately 40 µm in a array. The two dimensional diffraction patterns were binned and azimuthally integrated for input into the MTEX program (4.2.1). The diffraction patterns were taken with an integration time of 10 µs 16

28 CHAPTER 3. EXPERIMENT with 20 µs interframe spacings. We studied hot-rolled magnesium alloy AZ31B (Mg-3%Al-1%Zn-0.3%Mn, expressed in weight percent). The number average grain size is approximately 32 µm. The initial texture has the majority of basal poles oriented in the perpendicular direction with respect to the loading axis, as shown in Figure 4.4. We cut samples for dynamic compression from the rolled material by electric discharge machining (EDM) and mechanically polished them into right rectangular prisms ( mm) with a surface finish of 5 µm. We collected in situ x-ray diffraction patterns during dynamic uniaxial compression along the rolling direction at strain rates of 1000 s 1. 17

29 Chapter 4 Results and discussion 4.1 CHESS Figure 4.1 shows one-dimensional diffraction patterns obtained from pure Mg in the two geometries illustrated in Fig. 3.2 above. With k perpendicular to the scattering vector (Fig. 4.1(a)) the basal plane {0002} peak reflects a population of grains whose basal planes are parallel to the loading direction and which are therefore favorably oriented for extension twinning (Fig.3.1(a)). As deformation proceeds we observe a decrease in the {0002} peak intensity and an increase in the prism plane { } peak intensity, resulting from the reorientation of the lattice due to extension twinning. With k parallel to the loading direction (Fig. 4.1(b)) we see the converse: A decrease in the basal plane peak and an increase in the prism plane peak. This is just the same lattice reorientation due to extension twinning, seen from a different 18

30 CHAPTER 4. RESULTS AND DISCUSSION Intensity (arb.) {1010} {0002} {1011} 9 μs ε = {0002} 18 μs 10 ε = μs ε = μs ε = Intensity (arb.) {1010} 61 μs ε = μs ε = μs ε = μs ε = {1011} Scattering vector (1/Å) (a) Scattering vector (1/Å) (b) 2.6 Figure 4.1: One-dimensional diffraction patterns during dynamic compression for pure Mg with the scattering vector k (a) perpendicular and (b) parallel to the loading axis. point of view. Because the area under a diffraction peak is proportional to the volume of crystals contributing to that peak, we can quantify the extent of twinning by comparing the relative intensities of basal plane and prism plane peaks. When observing a decrease in the intensity of a peak due to deformation twinning, we can express the volume fraction of twinned material as V twinned (t) = I 0 I(t) I 0, (4.1) where I 0 is the integrated intensity of the peak in the undeformed material, and I t is the integrated intensity at time t after the onset of deformation. When expressed in this way, V twinned represents the volume fraction of those grains that were initially favorably oriented for twinning that have, in fact, twinned by time t. Note that this is not the fraction of the entire sample that has twinned, because many grains are not 19

31 CHAPTER 4. RESULTS AND DISCUSSION initially oriented to diffract (i.e. are not in the basal plane orientation in Fig. 4.1(a)). Figure 4.2(a) shows the extent of twinning calculated from Eqn. 4.1 as a function of strain from measurements on nine samples of pure Mg and ten samples of AZ31B, along with representative stress-strain curves from one sample of each. The evolution of the twinned volume fraction and the shape of the stress-strain curves (Fig. 4.2(b)) are similar to those previously observed in in situ neutron diffraction experiments for samples loaded in quasistatic compression 15 (although we note that the crystallographic texture of those samples differ from ours). In both materials, yielding appears to coincide approximately with the onset of twinning (perhaps slightly delayed in AZ31B). The two materials strain harden relatively slowly in the initial stages of plastic deformation, but quickly the strain hardening rate increases, especially in pure Mg. At the same time, twinning proceeds more quickly in the pure Mg. This suggests that the higher rate of strain hardening in pure Mg may be associated with twin boundaries acting as barriers to dislocation motion, and possibly also with an increase in dislocation activity necessitated by exhaustion of material available for twinning. The differences in twinning behavior between the two materials are probably due to the difference in average grain size. It has been observed that smaller grains favor dislocation slip over twinning in both pure magnesium 62 and AZ31B. 63 Several authors have argued this transition occurs because twinning shows a greater sensitivity to grain size than does dislocation slip Recently, Fan and coworkers have proposed 20

32 CHAPTER 4. RESULTS AND DISCUSSION 1.0 Twin volume fraction Pure magnesium Grain size: 9.5µm AZ31B Grain size: 1.5µm True strain (%) (a) True stress (MPa) AZ31B Pure magnesium True strain (%) (b) Figure 4.2: (a) Twin volume fraction V twinned as a function of strain, determined from Eqn. 4.1 from nine samples of pure Mg and ten samples of AZ31B under dynamic compression. The data points each represent a single diffraction pattern (eight per sample). Filled circles represent samples having their scattering vector perpendicular to the loading and open circles parallel. The vertical error bars correspond to the uncertainty in the integrated areas of the diffraction peaks, whereas the horizontal bars represent the strain over which the data were collected during the exposures. The solid lines are running averages for the two materials. (b) Representative stressstrain curves from one sample of each material. The end of the stress-strain curves corresponds to unloading, not fracture of the material. 21

33 CHAPTER 4. RESULTS AND DISCUSSION that the transition is a result of a competition between strengthening caused by the presence of twin boundaries (which act as barriers to dislocation motion) and weakening due to twin growth. 66 In their model, twin growth is suppressed when migration of twinning dislocations becomes difficult at small grain sizes. In our results, although the difference in flow stress between the two materials studied is considerable (pure magnesium σ y = 125 ± 22.3 MPa and AZ31B σ y = 178 ± 19.0 MPa), there is still substantial twinning activity despite the large difference in grain size (Fig. 4.2). However, it may also be that precipitates in the AZ31B alloy hinder twin activity, which has been observed during deformation at low strain rates. 67 From the in situ diffraction data we can also extract the component of elastic normal strain in a direction parallel to the scattering vector, ɛ = dhkl d hkl 0, (4.2) d hkl 0 where d hkl 0 is the initial (unstrained) lattice spacing of the {hkl} planes and d hkl is the corresponding lattice spacing during deformation. The elastic strain measurement is complicated by the small size of the detector (which prevents us from capturing entire diffraction rings) and the short data collection times (which makes precise determination of peak positions difficult). Nonetheless, although the strains are subject to considerable uncertainty in their absolute values, trends in elastic strain are apparent in the data, and provide additional information about deformation mechanisms. 22

34 CHAPTER 4. RESULTS AND DISCUSSION 400 Untwinned grains 400 Twinned grains True stress (MPa) Transverse direction Loading direction True stress (MPa) Transverse direction Loading direction Lattice strain x Lattice strain x10-3 Figure 4.3: Elastic strain in pure Mg samples as a function of true stress for (a) untwinned and (b) twinned regions with scattering vector parallel and perpendicular to the loading direction. Vertical error bars represent the stress range over which each data point was collected. Horizontal error bars are uncertainties in the fitted peak positions. Dashed lines represent linear elastic behavior of grain populations based off of single crystal elastic constants. Elastic strains in the loading and transverse direction of pure magnesium for grains in the original (untwinned) and twinned orientation are in Fig. 4.3(a) and (b), respectively. For the original orientation the strain in the loading direction is measured from scattering from the { } prism planes, while the { } basal planes are used for the twinned orientation. In both populations of grains the lattice strain is initially linear with the global true stress (from the Kolsky bar data) as would be expected for elastic deformation. To verify that the slope of this stress-strain curve is consistent with elastic deformation, also shown on the plot is a line corresponding to the expected slope for a simple model in which each grain is subject to a uniaxial compression stress given by the global true stress. After the initial linear region, the elastic strain in the loading direction for the untwinned orientation stops increasing after reaching a value of about , with further increases prevented by load-sharing with adjacent grains in different orienta- 23

35 CHAPTER 4. RESULTS AND DISCUSSION tions. This observation suggests that the untwinned material is able to activate some other mechanism of plastic deformation that allows these grains to accommodate the imposed plastic strain at constant stress. Most likely this is dislocation slip of some kind; one possibility is c+a slip on the pyramidal planes but even basal slip may be a possibility given its very low critical resolved shear stress. We emphasize, however, that most of the grains in the original orientation do twin (as shown by the decrease in intensity of the { } peak in Fig. 4.1(b)), so whatever plastic deformation may be occurring in these grains probably contributes only insignificantly to the global plastic strain. In the twinned orientation, in contrast, the elastic strain in the loading direction increases linearly, to at least 0.007±0.001, and never saturates (at least not to a total true strain of about 0.1, which was the limit of our experiments). This indicates that once extension twinning has occurred, a twinned region no longer deforms plastically but simply experiences elastic deformation due to the displacements imposed on it by the surrounding material. This is not entirely unexpected, because deformation mechanisms favored by this orientation (such as compression twinning and pyramidal slip) have large critical resolved shear stresses. This information is useful in evaluating the assumptions made in crystal plasticity models. For example, Becker and Lloyd have recently published a computationally efficient crystal plasticity model for hcp metals. 68 One of the assumptions in their model is that regions of the material that have twinned no longer deform plastically 24

36 CHAPTER 4. RESULTS AND DISCUSSION but continue to load elastically. The data in Fig. 4.3 and the discussion above indicate that this is a good approximation, at least for the present case of loading in uniaxial compression. 4.2 DCS In addition to being able to calculate volume fractions and elastic strains the DCS data allows for calculation of an orientation distribution function (ODF) Pole figure inversion method Analyzing the crystallographic preferred orientation in polycrystalline samples is commonly performed in materials science. 69 A complete description of the orientations of a sample can be defined by the orientation distribution function (ODF). Experimentally, the ODF can be found using methods such as electron backscatter diffraction (EBSD) in which the individual crystallographic orientations are measured directly, or neutron, synchrotron, or x-ray diffraction based techniques in which only integral information about the ODF is measured in diffraction pole figures. 53 The mathematical and computational determination of the ODF in the latter case used to be a major problem. This problem is often referred to as the pole figure inversion problem. 70 There have been a number of methods for solving the pole figure inversion prob- 25

37 CHAPTER 4. RESULTS AND DISCUSSION lem as far back as 1965 when Bunge and Roe independently developed a method for 70, 71 doing so. However, most of these methods, and their implementation into software code, do not account for high-resolution diffraction pole figures such as those captured from area detectors. Recently, a software package, MTEX has been developed for calculating solutions to the inversion pole figure problem that can handle 72, 73 high-resolution diffraction pole figures. MTEX is a free Matlab toolbox for analyzing and modeling crystallographic textures by means of EBSD or pole figure data. MTEX employs a novel pole figure inversion method with an algorithm that is especially well suited for sharp textures and pole figures measured with respect to arbitrary scattered specimen directions, e.g. by area detectors. 74 The measured diffraction intensities are modeled as a random sample of a Poisson process parameterize by the unknown ODF Then the algorithm looks for the best estimator of the unknown parameter given the observed diffraction intensities. Eventually an estimator is derived that differs from the regularized least-squares estimate by the characteristic that deviations from small diffraction intensities are more severely penalized than deviations from large diffraction intensities. 78 In this way, the method accounts for the standard deviation of measurement errors which increase with the square root of the diffraction counts. 79 The MTEX software solves the inversion pole figure problem by solving the fundamental equation of texture analysis. Mathematically, the pole density function (PDF) P corresponding to an ODF f is characterized by the fundamental equation 26

38 CHAPTER 4. RESULTS AND DISCUSSION Diffraction pattern Partial pole figures Experimental ODF Experimental pole figures (0002) (0002) (1010) (1010) Figure 4.4: Flow of pole figure inversion method employed on in-situ x-ray diffraction experiments. of texture analysis (Equations 4.3 and 4.4): 1 P (h, r) = [Rf (h, r) + Rf ( h, r)] 2 1 Rf (h, r) = 2π f (g)dg (4.3) (4.4) G(h,r) Where r = (u,v,w) corresponds to the coordinate vector with respect to the specimen coordinate system and h = (h,k,l) corresponds to the coordinate vector with respect to the crystal coordinate system. A rotation g SO(3) is the orientation of the crystal if it rotates the specimen coordinate system onto the crystal coordinate 27

39 CHAPTER 4. RESULTS AND DISCUSSION system, therefore, gh = r. The path of integration G(h, r) := {g SO(3) gh = r} is defined as the set of all rotations that map the crystallographic direction h S 2 onto the specimen direction r S 2. R is know as the totally geodesic Radon transform. Figure 4.4 shows the process by which diffraction patterns from our x-ray diffraction experiments can be binned into 72 sectors to create partial pole figures. The partial pole figures are input into the MTEX software as files defining the intensity associated with a particular polar and azimuthal angle. The MTEX software then estimates an ODF based upon a modified least squares estimator. The functional that is minimized is a modified least-squares estimator (Equation 4.5): N N i f MLS = argmin {[ ν i (f)rf(h i, r ij ) + Iij b I ij ] 2 /I ij } + λ f 2 H[SO(3)] (4.5) i=1 j=1 Where N is the number of pole figures, N i is the number of specimen directions, ν i is the unknown normalization coefficients, H are the superposed lattice planes, r the specimen directions, Iij b the background intensities, I ij the diffraction counts and λ f 2 H[SO(3)] is a regularization term. A more precise and full description of the estimator and algorithm can be found in the paper Pole Figure Inversion - The MTEX Algorithm. 72 The calculated ODF contains a complete description of all orientations in the 28

40 CHAPTER 4. RESULTS AND DISCUSSION Experimental ODF Sampled orientations Simulation ODF Simulation pole figures (0002) (1010) Figure 4.5: Flow of computationally efficient crystal plasticity modeling. sample. As a result, full pole figures can be created from the ODF giving a much greater understanding of the evolving microstructure than the diffraction pattern alone Computationally efficient crystal plasticity model Comparison of experimental data with models can lead to better understanding of experimental data and predictions of untested materials behavior leading to creating materials by design. 80 We compared our experimental results with predictions based on a computationally-efficient crystal plasticity model created by R. Becker and 29

41 CHAPTER 4. RESULTS AND DISCUSSION 1.0 Twin volume fraction Experiment Simulation True strain Figure 4.6: Twin volume fraction as a function of strain for both experimental data and simulation. The experimental data was calculated using in-situ x-ray diffraction data from a rolled AZ31B magnesium sample compressed along the rolling direction at a strain rate of 1000 s 1. Simulation data are taken from a simulation run using the computationally efficient crystal plasticity model with initial textural input taken from the diffraction data of the same rolled AZ31B magnesium sample using the inverse pole figure method and sampling of the subsequent ODF. J.T. Lloyd from the Army Research Laboratory (ARL). The input to the model is a sample of orientations taken from the ODF created from the x-ray diffraction data. Figure 4.5 shows a schematic of how the ODF can be sampled to provide an arbitrary number of crystal orientations as an initial texture for input into the model. The model can then perform its calculations, stopping at arbitrary amounts of strain to match the strains at which the x-ray data were acquired, and pass back new crystal orientations from which a new ODF can be calculated. These two ODFs, experimental and computational, can then be qualitatively and quantitatively compared in MTEX. This computationally efficient crystal model does not take a full detailed approach 30

42 CHAPTER 4. RESULTS AND DISCUSSION to deformation with full crystal kinematics, as does one such as Zhang and Joshi s model shown in Equation 4.6: 2 N tw N s L p = (1 f β ) β α=1 N tw γ α (s (α) m (α) )+ β=1 N tw γ β (s (β) m (β) )+ β=1 f β N s α=1 γ ( α) (s ( α) m ( α) ) (4.6) Where the first term accounts for slip in the parent crystal, the second term twinning in the parent crystal, and the third term slip in the twinned region. The computationally efficient model makes simplifications to a full detailed approach as shown in Equation 4.7: L p = N s α=1 γ α (s (α) m (α) ) + N tw β=1 γ β (s (β) m (β) ) (4.7) Where now the model explicitly accounts for basal slip and extension twinning on a rate-independent basis, but treats other mechanisms (pyramidal and prismatic slip) as isotropic, rate-dependent functions and does not have slip occuring on the crystallographically reoriented lattice. This combination yields substantial improvements in efficiency over full crystal-plasticity models while retaining key aspects of the most important deformation mechanisms. 2 Even though the rate effect of twinning is of prime importance to our investigation and it is treated as rate-independent in our choice of model, the parameters of the model can be modified to approximate rate effects. This can be done by either coupling strength to pressure via the shear modulus 31

43 CHAPTER 4. RESULTS AND DISCUSSION or fitting strength values to the regime we are interested in. This type of approach of using a rate-independent model to fit high strain rate experiments has been used in the literature Figure 4.6 shows a quantitative comparison of the twin volume fraction as a function of strain for both the experimental x-ray data and simulation results. The agreement in twin volume fraction as a function of strain appears to be quite good and suggests that the model is capable of capturing the key aspects of the microstructural evolution of the sample. In addition to allowing us to gain a better understanding of the twin volume fraction during deformation the model can also shed insight into the relative activity of slip systems, allowing for a more full understanding of mircostructural behavior. 32

44 Chapter 5 Conclusions We have demonstrated the ability to track the evolution of microstructure in magnesium in situ during dynamic deformation using x-ray diffraction. In both pure magnesium and the AZ31B alloy we observe that deformation twinning begins at small global strains (< 0.01) and continues, approaching saturation at the largest strains considered here ( 0.1). Twinning develops somewhat more slowly in the alloy than in the pure metal, possibly due to the alloying content but more likely due to a smaller grain size in the alloy. We are also able to track the elastic strains in different populations of grains and in different directions with respect to the loading axis. We observe that once a region of material has twinned, it experiences only elastic deformation, without additional plasticity. We have also demonstrated the ability to calculate orientation distribution functions from data collected at DCS. This data can act as an initial microstructural input 33

45 CHAPTER 5. CONCLUSIONS for the reduced crystal plasticity model written at ARL. The experimental data and modeling results are in good agreement demonstrating that the assumptions made in the model can still reasonable reflect the actual microstructural evolution of the sample. 34

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63 Vita Caleb J. Hustedt was born in Plano, Texas in He received the B. S. in Physics from Brigham Young University in 2011, and enrolled in the Materials Science and engineering graduate program in His research focuses on magnesium and its alloys under dynamic compression using in situ x-ray diffraction as well as three dimension serial sectioning of materials using a scanning electron microscope with integrated femtosecond laser. 52