An in-situ study of stress evolution and fracture growth during compression of concrete

International Journal of Solids and Structures 168 (2019) 26–40

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An in-situ study of stress evolution and fracture growth during compression of concrete R.C. Hurley a,∗, D.C. Pagan b a b

Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA Cornell High Energy Synchrotron Source, Ithaca, NY 14853, USA

a r t i c l e

i n f o

Article history: Received 5 September 2018 Revised 20 February 2019 Available online 14 March 2019 Keywords: Concrete Fracture characterization Damage Non-destructive measurements 3D X-Ray diffraction X-Ray computed tomography

a b s t r a c t We experimentally correlate fracture nucleation with the evolution of aggregate stress tensors during the compression of concrete. The concrete sample, made with portland cement and single-crystal quartz aggregates, was compressed to failure in eight strain increments. X-ray computed tomography was used to determine the extent of fractures and their proximity to aggregates at each strain increment. Threedimensional X-ray diffraction was simultaneously used to evaluate the full stress tensor in 30 to 40 individual aggregates at each strain increment. We found that a recently-developed mean-field model provides a good prediction of average aggregate stress tensors throughout the sample but does not predict the significant stress heterogeneity observed in individual aggregates, which are likely caused by aggregate interactions and may be responsible for fracture nucleation. Stress variations in aggregates after the nucleation of particular fractures suggest that fractures divided the sample into distinct load paths that experienced either an increase or decrease in compressive stress with further strain. We discuss extensions of the experimental measurements for quantifying specific failure processes in concrete, such as the fracture of the interfacial transition zone. We also discuss the applicability of the measurements to a broad range of other composite materials, and use of the measurements for developing and calibrating theoretical and computational models for composite strength. © 2019 Published by Elsevier Ltd.

1. Introduction Composites are ubiquitous in infrastructure, automotive, and aerospace industries (Aïtcin, 20 0 0; Allison and Cole, 1993; Rawal, 2001) and concrete, arguably the oldest class of man-made composite, is the most-used construction material in the world (Aïtcin, 20 0 0). Composites contain particulate inclusions or fibers distributed in a matrix material to produce properties that deliver superior performance compared to their non-composite counterparts. However, the complex three-dimensional (3D) stress states generated by matrix-inclusion interactions can cause fracture nucleation and damage during mechanical or thermal loading (Budiansky et al., 1986; Lewandowski et al., 1989; Fu et al., 20 08; Li et al., 20 06). Predicting these failure modes is a major focus of research on composite materials and validating these predictions requires local stress measurements at specific microstructural features. We work towards satisfying this need in a sample of concrete by presenting in-situ measurements of individual aggregate stress tensors and analyzing their evolution with respect ∗

Corresponding author. E-mail address: [email protected] (R.C. Hurley).

https://doi.org/10.1016/j.ijsolstr.2019.03.015 0020-7683/© 2019 Published by Elsevier Ltd.

to an evolving fracture network. In addition, focus will be placed on how this work studying a concrete composite may be readily extended to study other specific failure processes, such as matrixinclusion debonding and matrix fracture, during mechanical and thermal loading of a broad range of other materials such as metalmatrix composites (MMCs) and epoxy-matrix composites. The partitioning of stress at the microscale between concrete’s matrix material (cement paste) and its aggregates is controlled by each phase’s properties, constitutive response, and spatial distribution. Stress partitioning constrains the overall macroscopic mechanical properties (Hill, 1963) and, because failure often initiates by aggregate-cement paste failure, is intimately linked to yield and ultimate strength. Mean-field Mori-Tanaka approximations and other analytical methods have frequently been employed to predict effective macroscopic properties of concrete and other composites by examining stress partitioning (Königsberger et al., 2014a; Mori and Tanaka, 1973; Kachanov and Sevostianov, 2013). However, these analytical approaches typically neglect the interaction of aggregates or inclusions due to their close proximity. As noted in recent work on concrete (Königsberger et al., 2014a, 2014b), experimental access to full stress tensors in aggregates would provide an opportunity to examine the accuracy of this assumption and to

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further investigate aggregate-cement paste failure. Until recently, experimental measurements of full aggregate stress tensors were not possible. Experimental techniques have been extensively used to investigate fracture processes in composites. Optical microscopy and scanning electron microscopy (SEM) have been employed to investigate fracture surfaces and matrix-inclusion interfacial failure modes in a variety of materials after failure (Hull, 1999; HerreraFranco and Valadez-Gonzalez, 2005; Mookhoek et al., 2010; Zhang et al., 2008; Quinn and Quinn, 2007). X-ray computed tomography has been used to evaluate the in-situ evolution of fractures in 3D, also called 4D imaging, in materials as diverse as bone, dentin, metals, and concrete (Withers and Preuss, 2012; Landis et al., 2007). XRCT provides the opportunity to study the 3D geometry of fracture surfaces and fracture deflection, which is an important toughening mechanism in materials with complex microstructures (Withers, 2015). While XRCT has been used to locate fractures with sub-pixel dimensions (Breunig et al., 1993; Stock, 1999), X-ray phase-contrast imaging (PCI) has more frequently been employed to examine sub-pixel cracks in composites (Buffiere et al., 1999). Several review articles discuss applications of XRCT and PCI for monitoring fracture growth in composites and other materials (Withers and Preuss, 2012; Withers, 2015; Stock, 1999, 2008). To access information beyond microstructure, complementary experimental techniques have been used simultaneously. Diffraction contrast tomography (DCT) has been used in conjunction with XRCT to study the interaction of stress corrosion cracking in an austenitic stainless steel wire with microstructure and crystallographic orientations (King et al., 2008). XRCT has also been combined with X-ray powder diffraction to study the partitioning of stress between the matrix and inclusions in fiber composites (Sinclair et al., 20 04, 20 05) and the crack-tip stresses present during fatigue loading of alloys and composites (Allison, 1979; Steuwer et al., 2010; Withers et al., 2012). This combination of X-ray measurements provides a unique opportunity to observe deviations of crack-tip stress fields from those expected by linear elastic fracture mechanics and to correlate these deviations with microstructural features (Withers, 2015). However, the method has only been used to obtain strain and stress fields in the plane of fractures, limiting its ability to unravel the complex 3D stress states that evolve ahead of and as a result of fracture processes in microstructurally complex composites. 3D X-ray diffraction (3DXRD) has recently emerged as a powerful tool in the metals community for determining the in-situ evolution of individual grains’ strain and stress tensors, positions, and crystallographic orientations (Poulsen, 2004; Oddershede et al., 2010). 3DXRD has been used to study a variety of phenomena in metals, including phase transformations and local stresses in alloys in various loading conditions (Abdolvand et al., 2018; Pagan et al., 2017; Sedmák et al., 2016; Oddershede et al., 2012; Obstalecki et al., 2014) and twinning in pure polycrystalline metals (Bieler et al., 2014). 3DXRD has also recently been adopted in the granular materials community and combined with XRCT to examine particle micromechanics: stresses, rotations, inter-particle forces, contact fabric, fracture mechanics, stress-induced twinning (Cil and Alshibli, 2014; Cil et al., 2017; Hurley et al., 2016, 2018a, 2018b). The major advantage of combining XRCT with 3DXRD, rather than powder diffraction, is access to the full elastic strain tensors that essentially serve as ’strain-gauges’ at specific locations throughout a material’s microstructure. In this work, we employed in-situ XRCT and 3DXRD to study stress and fracture evolution in concrete. XRCT provided access to microstructural details, including the locations of different material phases, voids, and fractures. 3DXRD provided the full strain tensor, stress tensor, position, and crystallographic orientation of individual crystalline aggregates. We employed these measurements dur-

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ing unconfined compression of a concrete sample made with portland cement and angular single-crystal quartz aggregates. At each of eight load steps, measurements revealed the fracture network growth, the aggregate stress tensors and their heterogeneity, and correlations between stress fluctuations and fracture nucleation. The remainder of the paper is organized as follows. Section 2 describes the experimental protocol, measurements, and data analysis. Section 3 provides results of the experiment. Section 4 provides a discussion of the results, possible future work, and extension of the measurements to a broad range of other composite materials. Section 5 provides conclusions. 2. Experimental approach and application to concrete 2.1. Experimental setup We performed the experiment described in this paper at the F2 experimental station of the Cornell High Energy Synchrotron Source (CHESS). The experimental setup at F2 consists of the second generation of the rotational and axial motion system (RAMS2) (Shade et al., 2015), a Retiga 40 0 0DC camera with 5 × Mitutoyo objective focused on an LuAG:Ce scintillator in a close sample-todetector distance geometry (8 mm, referred to as the ‘near-field’), and two Dexela 2923 NDT flat panel detectors in larger sampleto-detector distance geometry (860 mm, referred to as the ‘farfield’). A schematic of the setup is shown in Fig. 1, with an outset showing an actual Bragg diffraction peak measured on a farfield detector during the experiment presented in this paper. The RAMS2 has an air bearing rotation stage within a screw driven loading system that permits unobstructed measurement of transmitted and diffracted X-rays during a 360◦ sample rotation as mechanical loads are applied. 2.2. Sample preparation The concrete sample was prepared as follows. First, we chiseled a block of synthetic hydrothermally-grown single-crystal quartz (Sawyer Technical Materials, LLC) into fragments smaller than 1 cm. We then ball-milled the quartz fragments using a stainlesssteel ball and stainless-steel vial (SPEX SamplePrep Mixer/Mill 80 0 0D ball-mill) at room temperature without processing control agents for 30 seconds. We subjected the resulting powder to vibratory sieving for 5 minutes, retaining those captured on a number 80 (177 μm) mesh but passing a number 60 (250 μm) mesh for aggregates in our sample. We combined Type I portland cement (Quikrete Type I/II, ASTM C150 compliant) with the quartz aggregates in a 1:2 wt ratio, adding water to obtain a 1:3 water to

Fig. 1. Schematic of experimental setup at the CHESS F2 structural materials beamline. The sample is shown in the load path of the RAMS2. Transmitted X-ray intensity is recorded using a Retiga 40 0 0DC CCD camera focused on an LuAG:Ce scintillator. X-rays diffracted by single-crystal inclusions are captured by far-field Dexela 2923 NDT flat panel detectors. The outset box shows a close-up of a pixelated Bragg peak generated by an aggregate particle.

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Fig. 2. (a) The maximum load cell reading before relaxation and load cell reading during measurements. (b) Stress-strain curve for the sample with σ zz > 0 considered compressive only for this figure, calculated by dividing the load cell reading during measurements by the initial sample area, 1 mm2 .

solids weight ratio. We slowly added additional water to enhance workability until a viscous slurry could be packed into a 1 cm3 silicone mold without obvious surface voids. We vibrated the silicone molds for 30 seconds at 200 Hz with a small mechanical device in contact with the underside of the mold. The specimens cured for 24 hours in the molds while covered by saran wrap before submersion in a hydrated lime solution for 28 days. The hydrated lime solution consisted of tap water and a mixed lime solution (Type S masons hydrated lime from Carmeuse, Rockwell Operation). After 28 days, we removed the sample from the hydrated lime solution and used a diamond-tip blade to cut out an approximately 1 mm × 1 mm × 1 mm (1 mm3 ) cubic sample for testing. We kept this sample at room temperature and ambient humidity for five months prior to the experiment. 2.3. Experiment procedure We placed the 1 mm3 sample between stainless steel platens that were inserted into the RAMS loading grips. The sample was surrounded by an aluminum cylinder to prevent sample fragments from being lost. The cylinder provided no lateral support to the sample and was separated from the sample by several tens of microns. We lowered the stainless steel platen above the sample until it made contact as indicated by a nonzero reading on the load cell. With the strain held constant, we then rotated the sample first through 180◦ , then through 360◦ , while it was illuminated by a 41.991 keV X-ray box-beam 1.2mm tall and 2.5 mm wide. In the first 180◦ rotation, we recorded 1800 transmission radiographs at 0.1◦ sample rotation increments on the near-field detector. In the second 360◦ rotation we recorded 1440 diffraction patterns at 0.25◦ sample rotation increments on the far-field detectors. After these measurements, we subjected the sample to quasi-static compression by lowering the top platen in 2 μm increments until a desired load level was reached. We then held the sample strain constant while it rotated through another two rotations for transmission radiograph and diffraction pattern measurements. We repeated this procedure until the sample would no longer hold increasing loads. Immediately prior to and during the beginning of the measurements, we observed a decrease in the load cell reading, reflecting sample relaxation. However, we observed no artifacts in the XRCT images (described in Section 2.5) that would indicate significant motion within the sample’s microstructure on scales of 1 μm or greater. Furthermore, no significant additional relaxation occurred during the diffraction pattern measurements, suggesting that the 3DXRD measurements (described in Section 2.4) accurately reflected the sample’s state throughout the holding time.

The entire loading procedure for the concrete sample is reflected in Fig. 2. Fig. 2a shows the maximum load cell reading before relaxation, and the load cell reading during the majority of measurements. Fig. 2b shows the stress-strain curve for the sample. The average macroscopic stress is calculated by dividing the load cell reading obtained during the experiment by the initial sample area, 1 mm2 . The strain is calculated in the elastic regime, load steps 0 through 3, by registration of XRCT images using open-source software (Andò et al., 2017) which employs the approach described in Lucas and Kanade (1981). In load steps 4 through 7, the sample strain is calculated from the distance measured between a fixed location on the top and bottom platens, as observed in XRCT images. Sample strain is not calculated using the distance between load platens in steps 0 through 3 because a small machining artifact on the platens (a cylindrical protrusion raised several microns) caused local crushing of the sample where it contacted the concrete, and therefore the platen displacement approach would have over-predicted bulk sample strain. Using our stress-strain measurements, we observed near-linear sample stiffness in steps 0 through 3, up to a strain of approximately 0.0017. In these load steps, the sample stiffness was approximately 15.8 GPa. Step 4 involved significant sample strain, later correlated with the nucleation of a fracture. The sample maintained a reduced stiffness as it was strained beyond a sample strain of 0.04 and finally underwent total failure between a strain of 0.065 and 0.09. 2.4. 3DXRD Analysis We processed the 1440 diffraction patterns captured on farfield detectors during each 360◦ sample rotation as follows. First, we calibrated detector orientation, tilt, and distance to the sample in HEXRD (Bernier et al., 2011) by using a X-ray powder diffraction pattern obtained from a cerium oxide (CeO2 ) powder immediately prior to the experiment. We then thresholded each diffraction pattern to eliminate background noise and used the unit cell parameters of single-crystal alpha-quartz (space group 154, a = ˚ c = 5.4071 A) ˚ in HEXRD to locate Bragg peaks pro4.9144 A, duced by the quartz crystals. We used HEXRD to associate specific diffraction peaks with individual crystal orientations (in this case, quartz aggregates) in a process known as indexing (Bernier et al., 2011). Finally, we used HEXRD to determine the centers-of-mass, orientations (with 0.05◦ resolution), and average strain tensors,  kl , (with 10−4 resolution) of grains from the measured positions of associated diffraction peaks. In this study, grain state parameters (specifically elastic strain tensors) were analyzed from grains with completeness values greater than 0.7. The completeness is the ratio of the number of found and predicted diffraction peaks for a

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Fig. 3. (a) Number of aggregate particles with at least one crystal having a completeness score of 0.7 or higher at each loading step. (b) The mean absolute value of stress component differences between crystals comprising a twinned aggregate particle.

specific aggregate. The number of predicted peaks can be determined based on the orientation of the crystal lattice of each particle and a simulation of a rotating single-crystal diffraction experiment (Bernier et al., 2011), and the number of found peaks is the number of those predicted peaks actually located in diffraction patterns. We have independently verified that we can achieve 10−4 resolution on strain tensor components for the quartz particles used in this study (Hurley et al., 2018b). As the quartz aggregates are crystalline, the stresses in the aggregates can be calculated from the elastic strain tensors using Hooke’s Law, σi j = Ci jkl kl , where Cijkl is the fourth order stiffness tensor. To begin the process, we transformed each quartz crystal’s elastic strain tensor from the sample coordinate frame to its respective crystal coordinate frame using the measured crystal orientation (see Oddershede et al., 2010; Bernier et al., 2011). The stresses were then calculated in the crystal frame and next transformed back into the sample frame. The single crystal elastic moduli of cultured α −quartz used in the calculation were taken from (Heyliger et al., 2003) (in Voigt notation and with units of GPa): C11 = 87.16, C33 = 106.00, C44 = 58.14, C66 = 40.26, C12 = 6.64, C13 = 12.09, C14 = −18.15. After calculating stresses, we matched the positions and stresses of individual crystals across the eight loading steps by pairing crystals in subsequent loading steps if their centers-of-mass changed by no more than 30 μm and their orientations changed by no more than 5◦ . After tracking, we identified a significant number of crystals (between 18 and 25) in each load step that constituted a Dauphíne twinned particle: particles with two comprising two crystals with centers-of-mass within 5 μm and orientations differing by a 60◦ rotation about the c−axis. We merged these crystal positions and stresses by unweighted averaging, as justified by twin volume measurements made in a previous study (Hurley et al., 2018b). The total number of resulting aggregate particles with at least one crystal having a completeness score of 0.7 or higher is shown for all loading steps in Fig. 3a. The drop in particle number above load step 3 is primarily due to large rotations and displacements of particles, which may create conditions in which particles no longer have a completeness score of 0.7 or higher. The mean absolute value of stress component differences between crystals comprising a twinned particle are shown in Fig. 3b. 2.5. XRCT Analysis We used the 1800 transmission radiographs to perform XRCT reconstructions with the iterative ASD-POCS algorithm in Liver-

more Tomography Tools (LTT) (Sidky and Pan, 2008; Champley, 2016). Resulting 3D XRCT images were 1648 × 1648 × 821 pixels with a resolution of 1.48 μm per pixel. We first cropped these raw XRCT images to isolate the concrete sample from the aluminum cylinder and external void space. An example of a cropped XRCT image at load step 0 is shown in Fig. 4a. We then segmented cement paste/aggregate, void/fracture, and higher density constituents such as calcium silicates with simple thresholding: pixels with grey-scale values below a specified threshold were deemed voids or fractures (Fig. 4b); pixels with grey-scale values above a specified threshold were deemed high density constituents such as calcium silicates (Fig. 4c); pixels with grey-scale values between these thresholds were deemed cement paste or aggregate. We selected the threshold values heuristically by carefully examining the XRCT images. The pixels identified as voids and fractures also corresponded to some low-density zones in the interfacial transition zone (ITZ) near aggregate particles. The ITZ can be readily identified as the more diffuse, non-spherical, clusters of void space in Fig. 4b. Because cement paste and aggregate featured similar grey-scale values in XRCT images, we separated them by applying a standard deviation filter to the XRCT images. The standard deviation filter operates by calculating the standard deviation of grey-scale values in a 18 × 18 × 18 window of pixels (equivalent to 26.64μm × 26.64μm × 26.64μm) surrounding each pixel in the image and comparing it to a threshold value. We heuristically selected a threshold value that produced well-defined outlines of aggregate particles. We then used a watershed algorithm to segment the image of thresholded standard deviations into individual 52 distinct aggregate particles. Finally, we used a morphological opening algorithm with a 6 × 6 × 6 structuring element to “grow” the aggregate particles. This growing operation was necessary to restore aggregates to their approximate original size because standard deviation filtering reduced particle sizes when the window extended beyond the aggregate boundaries. An XRCT image of the 52 segmented aggregate particles at load step 0 is shown in Fig. 4d.

2.6. 3DXRD And XRCT integration We transformed 3DXRD data to the XRCT coordinate system and matched each aggregate particle identified in the XRCT image to its nearest 3DXRD center of mass. The number of matched particles per loading step is shown in Fig. 5a. The volume-averaged stresses for aggregate particles that were matched with a 3DXRD

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Fig. 4. Example of XRCT analysis at load step 0. (a) Cropped XRCT image with a horizontal slice showing two-dimensional detail. (b) The void and fractures determined by selecting pixels in the XRCT images with grey-scale values below a specified threshold. (c) High density constituents such as calcium silicates determined by selecting pixels in the XRCT images with grey-scale values above a specified threshold. (d) Aggregate particles determined using a standard deviation filter described in the text.

Fig. 5. (a) Number of XRCT particles matched to 3DXRD particles at each load step. (b) Volume-averaged aggregate stresses computed only for aggregates matched to 3DXRD particles.

particle was calculated by

1

σ¯ i j = Nagg

ple stress shown in Fig. 2b, suggest that the aggregates are bearing a portion of the macroscopic load at each step.

Nagg



Vk k=1 agg k=1

k σiagg,k Vagg , j

(1)

where Nagg is the number of matched particles shown in Fig. 5a, k Vagg is the volume of aggregate k and σi j is its stress tensor. These volume-averaged stresses are shown in Fig. 5b. The volumeaveraged stress component σ¯ zz exhibits a similar trend to the samagg,k

3. Results and analysis 3.1. Evolution of the fracture network A major feature of microstructure evolution in the concrete sample was the growth of a fracture network with applied stress.

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Fig. 6. (a) Front, top, and back views of the sample at load step 0 showing aggregates, voids, and fractures. Letters at sample corners are used to convey the orientation of the observed in each sub-figure. (b)-(e) The same sample orientations shown in (a) but for load steps 4 through 7.

Fig. 6 shows three views of the aggregate, void, and fracture networks at load steps 0 through 7. We omitted steps 1 through 3 because there was no significant change in the void and fracture network between these steps. The fracture network exhibited vertically-oriented fracture sets that coalesced during sample compaction. The vertical nature of fractures can be observed by examining the top view of the sample, in which an unobstructed view through the sample can be observed through load step 6, even as side views show extensive fracture growth in all directions.

Deviations from vertical fracture orientations appear to be most significant in regions, such as near the line B-C in the top views, where aggregates may be interacting and altering the local stress and strain fields in a complex manner. We restricted our analysis in this paper to a field of view (FOV) reflected by the initial dimensions of the sample at load step 0, shown in the Fig. 4a. We did not expand the FOV as load on the specimen increased. Therefore, as the fracture network grew and the sample expanded in the x and y dimensions, a very small

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Fig. 7. (a) The fraction of the FOV occupied by the material phases at each load step. (b) Probability distributions of the distance of void and fracture pixels to any aggregate’s surface for load steps 0, 4 through 7, as well as a probability distribution of the distance of cement paste pixels to any aggregate’s surface at load step 0.

amount of material left the FOV due to surface spallation or Poisson effects. Fig. 7a shows the fraction of the total FOV occupied by the material phases. The void and fracture network initially constituted approximately 1% of the FOV while the aggregates and calcium silicates constituted 15–20% each and the remaining cement paste constituted approximately 65%. The portion of the FOV occupied by the void and fracture network increased in each load step, beginning in load step 4, reaching approximately 15% of the FOV by load step 7. This suggests that approximately 14% of the FOV was occupied by fractures at sample failure (load step 7). The 1% initial porosity is lower than the 10% initial porosity reported for a variety of samples in Kumar and Bhattacharjee (2003). One reason for the low porosity is XRCT resolution: we cannot reliably resolve voids or fractures with characteristic dimensions less then a few pixels (one pixel is 1.48 μm) in our XRCT images, whereas in Kumar and Bhattacharjee (2003) more than half of the sample porosity is reported to be formed by pores with sizes smaller than 106 nm. Another reason for the low porosity is sampling bias: we selected a 1 mm3 cube for the current experiment that did not have obvious surface defects or porosity, removing several cubes with large voids from consideration. We note that the fracture network observed up to load step 6 appears to convey more tortuosity in lateral directions than in the vertical direction, as suggested by the unobstructed views through the sample in the top view as compared with fracture curvature and intersection in the front and back views. Investigating fracture network tortuosity in rock samples with similar properties to the concrete sample examined here may therefore reveal anisotropic permeability changes relevant to energy exploration and extract. We reserve further discussion of this matter for future work. Fig. 7b shows the probability distribution of the distance of each pixel identified as a void or fracture to the nearest aggregate surface. This figure also shows the probability distribution of the distance of each pixel identified as cement paste (including calcium silicates) at load step 0 to the nearest aggregate surface. We include this latter curve to illustrate that initial voids are composed of two parts: regions near aggregate surfaces (around 10 μm) and spherical voids that are, because their peak location is coincident with that for the cement paste, randomly distributed. A comparison between voids / fractures and cement paste curves for load steps 4 through 7 shows that voids and fractures are collectively no longer randomly distributed but are preferentially close to aggregate surfaces, because their peak trends toward lower values than that for the cement paste. This finding is consistent with the understanding of the ITZ zone as a region of material weakness in which fractures initiate through a variety of failure mechanisms (Scrivener et al., 2004; Königsberger et al., 2014b).

3.2. Stress evolution in aggregates Fig. 8 provides two views of the aggregate and void network at load step 0, with a majority of the 52 aggregate particles labeled numerically. Fig. 9 provides 30 plots of aggregate stress tensors, with plots labeled consistently with Fig. 8. As described in Hurley et al. (2018b), error bars for stresses, reflecting one standard deviation in possible measurement error, are approximately 10 MPa for diagonal stress tensor components and 5 MPa for offdiagonal components. With few exceptions, the plots in Fig. 9 are only shown for particles for which stress measurements are available from 3DXRD for all load steps. Load steps for which all stress tensor components are zero correspond to load steps for which a particle’s completeness score was below 0.7 and therefore insufficient to calculate an accurate value for strain and stress. The

Fig. 8. Front and back views of the aggregates, voids, and fractures and load step 0. A majority of the 52 aggregate particles are labeled numerically for comparison with Fig. 9.

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Fig. 9. Stress tensor components within individual aggregates determined using 3DXRD. Plot numbers can be matched with aggregate numbers in Fig. 8. As described in the text, in Hurley et al. (2018b), the absolute error in diagonal stress tensor components is approximately 10 MPa, while in off-diagonal stress tensor components the absolute error is approximately 5 MPa.

plots convey the significant heterogeneity of stresses within aggregates at each load step. Recent theoretical work employed a Mori-Tanaka scheme coupled with an Ehselby problem of a spherical inclusion in an infinite matrix subjected to homogeneous strains to determine aggregate stresses prior to matrix fracture (Königsberger et al., 2014a). We compared the predictions of this approach with the aggregate stresses shown in Fig. 9 by using the macroscopic sample stress tensor



i j =

0 0 0

0 0 0

0 0

zz



,

(2)

σyy = (1/3 )[(Bvol − Bdev )(zz )],

(4)

σzz = (1/3 )[(Bvol + 2Bdev )zz )],

(5)

σxy = 0,

(6)

(3)

σyz = 0,

where the terms Bvol and Bdev are given by Königsberger et al. (2014a)

Bvol =

to determine the aggregate stresses by

σxx = (1/3 )[(Bvol − Bdev )(zz )],

σxz = 0,

Bdev =

kagg

,

(7)

μagg . (α fcp + fagg )kagg + [ fcp (1 − α )]kcp

(8)

(α fcp + fagg )kagg + [ fcp (1 − α )]kcp

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Fig. 10. (a) Aggregate stress tensor components predicted using the Mori-Tanaka scheme from Königsberger et al. (2014a). (b) Volume-averaged aggregate stresses from Fig. 5b with error bars representing 25% of the standard deviation of stress across all aggregates for which 3DXRD measurements are available.

Table 1 Elastic constants of concrete sample phases from (Königsberger et al., 2014a).

3.3. Stress partitioning between phases

Property

Quartz aggregates

Cement paste

ITZ

Young’s modulus Poisson’s ratio Bulk modulus Shear modulus

Eagg = 96.0 GPa νagg = 0.08 kagg = 38.1 GPa μagg = 44.4 GPa

Ecp = 16.0 GPa νcp = 0.23 kcp = 9.88 GPa μcp = 6.5 GPa

EIT Z = 13.6 GPa νIT Z = 0.23 kIT Z = 8.4 GPa μIT Z = 5.53 GPa

The parameters α and β are given by

α=

3kcp 3kcp + 4μcp

and

β=

6(kcp + 2μcp ) , 5(3kcp + 4μcp )

(9)

where kagg , μagg , kcp , and μcp are the elastic bulk and shear moduli of the aggregates and cement paste, respectively, and fagg and fcp are the volume fractions of aggregates and cement paste, fulfilling fagg + fcp = 1. In calculations using fcp , we assumed fcp included both cement paste and calcium silicates and was equal to 1 − f agg without accounting for voids or fractures. The scatter in the data that we used to compare with the theory was large enough that further refinement of fcp did not affect our observations. The values of elastic constants for the aggregates and cement paste are shown in Table 1 and taken from Königsberger et al. (2014a). Fig. 10a shows the aggregate stress tensor predicted using the Mori-Tanaka scheme coupled with an Eshelby problem from Königsberger et al. (2014a). While this theoretical stress evolves consistently with the volume-averaged stresses from Fig. 5b, individual aggregate stresses deviate from it significantly. Fig. 10b shows the diagonal components of the volume-averaged stresses from Fig. 5b with error bars representing 25% of the standard deviation of stresses for all aggregate at each load step. We did not plot off-diagonal stresses for graphical clarity. The predictions of Königsberger et al. (2014a) fall within a standard deviation of the volume-averaged stresses at all load steps (even after the emergence of fractures) but provide no indication of the large variance in individual aggregate stresses. From this observation, we conclude that individual aggregates experienced dramatic departures from the uniform homogeneous matrix strain field assumed in Königsberger et al. (2014a) and instead experienced significant differences in matrix strains and possibly matrix strain gradients caused by neighboring aggregates and microstructural features. Therefore, the estimate of Königsberger et al. (2014a) may be improved by accounting for inclusion interactions, which is a known challenge in the study of heterogeneous materials (Kachanov and Sevostianov, 2013), but is strongly motivated by the present data.

To further explore stress partitioning between microstructural phases, we examined the diagonal components of the areaweighted aggregate stress tensor given by Nagg  (k ) agg Aagg σi j , (k ) A k=1 agg k=1

1

σ¯ i j = Nagg

(k ) where Aagg is the cross-sectional area of the kth aggregate in a particular horizontal slice through the sample. We performed this averaging only using grains for which stresses were available from 3DXRD. The resulting area-weighted stresses provide an approximation of the portion of total stress carried by aggregates versus cement paste. In a homogeneous linear-elastic solid subjected to the same macroscopic stresses as the sample we examined, the area-weighted stresses would be equal in any equal-area portion of any horizontal slice through the specimen. Figs. 11a-c shows the resulting stresses at six heights in the sample for all load steps. Fig. 11d shows the fraction of the crosssectional area at each height occupied by aggregate particles. Horizontal sample stresses,  xx and  yy , were equal to zero for all load steps because the sample was subjected to unconfined compression. However, we observed the area-averaged aggregate stresses, σ¯ xx and σ¯ yy , to be nonzero for all load steps. We also observed these average stresses to increase with load, suggesting that aggregates provided tensile support for the surrounding cement paste, which remained in compression during loading. The sample may have also experienced some lateral confinement as a result of friction between the loading platens and the top and bottom surfaces of the sample, as observed in similar experiments in the past (van Vliet and van Mier, 1996). The variability of average stresses with height, without a clear correlation with changes in the fraction of cross-sectional area occupied by aggregates (see Fig. 11d), highlights the complex interaction of aggregates with their surrounding cement paste and with one another. The vertical sample stress,  zz , shown in Fig. 2b, is shown again at the top of Fig. 11c. At load step 0, the area-averaged aggregate stresses, σ¯ zz , are nearly zero, exhibiting slightly positive (tensile) stresses at the top and bottom of the sample and slightly negative (compressive) stresses in the center of the sample. As sample stress became more compressive, σ¯ zz decreased considerably. At some vertical locations within the sample, σ¯ zz was significantly less than the sample stress,  zz , suggesting that the aggregate particles were bearing more of the compressive stress than the surrounding cement paste at that location which is consistent with the fact that the aggregates are generally stiffer than the cement paste. At other vertical locations, notably z/h = 0.4, σ¯ zz remained

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Fig. 11. (a)-(c) Area-averaged aggregate stresses in six horizontal slices of the sample, as described in the text. The lines at the top of (c) correspond to the average macroscopic vertical stress, σ zz , measured from the load cell and shown in Fig. 2b. (d) The fraction of each horizontal slice occupied by the aggregates over which averaging was performed.

less compressive than  zz , suggesting that the cement paste was bearing more of the compressive stress than the surrounding cement paste. As with horizontal area-averaged aggregate stresses, we did not observe a clear correlation between these stresses and changes in the fraction of cross-sectional area occupied by aggregates.

models and motivate the development of new models for capturing stress partitioning during fracture. We hypothesize that the partitioning of samples into distinct load carrying paths may be determined, in larger samples, by a complex combination of aggregate to sample size ratios, aggregate volume density, and particular details of aggregate-aggregate stress-field interactions.

3.4. Stress fluctuations and fracture growth

3.5. ITZ Stresses

We examined stress fluctuations and their relationship to the nucleation of fractures in the sample by isolating aggregates whose stress tensor component in the direction of loading, σ zz , increased or decreased in a particular load step by more than 10 MPa when compared to the prior load step. 10 MPa was chosen because it is the nominal absolute error in diagonal stress tensor components for individual aggregates, as discussed in Section 3.2 and Hurley et al. (2018b). Fig. 12 shows aggregates that exhibited an increase in σ zz at load steps 4, 5, 6, and 7, considering only aggregates for which 3DXRD measurements are available in the respective load step and prior load step. Such aggregates experienced a release of compressive load. Several aggregates exhibited this decrease in compressive load at each of load steps 4 through 7. A notable location of compressive load release was near the origin (x = y = z = 0) in load steps 4 and 5. Aggregates in this location were also observed to be nearly isolated from the rest of the sample by the major fractures nucleated during these load steps. This suggests that the fractures partitioned the sample into distinct load paths experiencing either increases or decreases in compressive stresses in the loading direction. This hypothesis is supported by examination of Fig. 13. Fig. 13 shows aggregates that exhibited a decrease in σ zz of at least 10 MPa at load steps 4, 5, 6, and 7. Such aggregates experienced an increase in compressive load. In steps 4 and 5, these aggregates were located on the +x side of the major fractures that nucleated during these steps, while almost all aggregates experiencing a release of compressive load are located on the −x side of the major fractures. In load steps 6 and 7, the fracture network continues to grow more tortuous and it is difficult to see clear trends in the location of aggregates experiencing compressive load increases or decreases relative to the nucleated fractures. Future work with larger samples may provide additional support for the hypothesis that fractures partition concrete into distinct load carrying paths. Such a finding may validate existing

If stress fields within aggregates were available, it would be possible to accurately calculate the stress tensor at each point within the ITZ of each aggregate, σiITj Z (x ). These stresses are important for understanding aggregate-cement paste failure criteria (Königsberger et al., 2014a, 2014b). However, the full intra-crystal stress field variation across aggregates are not available from the 3DXRD measurements described in this paper; only average stress tensors within each aggregate are known. Possible extensions of our measurements to obtain stress fields within aggregates are discussed in Section 4. Because stress fields within aggregates are not available, we illustrate calculation of σiITj Z (x ) for completeness by assuming stresses within individual aggregates were approximately uniform. We first used XRCT images to compute the unit outward normal vector, nj (x), at each point on the surface of each aggregate by calculating the gradient of the map of distances of each aggregate’s interior pixels to its surface pixels. We employed stress equilibrium at the aggregate-ITZ interface to obtain three equations agg relating σiITj Z to σi j :

σiITj Z n j (x ) = σiagg n j ( x ). j

(10)

We used strain compatibility at the aggregate-ITZ interface (Königsberger et al., 2014a) to obtain an additional three equations agg relating σiITj Z to σi j :

ti(1) (x )iITj Z t (j 1) (x ) = ti(1) (x )iagg t (j 1) (x ) j

(11)

ti(2) (x )iITj Z t (j 2) (x ) = ti(2) (x )iagg t (j 2) (x ) j

(12)

ti(1) (x )iITj Z t (j 2) (x ) = ti(1) (x )iagg t (j 2) (x ) j

(13)

where ti(1 ) (x ) and ti(2 ) (x ) are two orthonormal vectors forming a basis for the plane tangent to the aggregate surface at point x, and

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Fig. 12. (a) Front, top, and back views of the sample at load step 4 showing voids, fractures, and aggregates with an increase in σ zz (less compression) equal to or greater than 10 MPa. (b)-(d) Same as (a) but for steps 5, 6, and 7.

are each thus perpendicular to ni (x), and strain tensors  ITZ and  agg are assumed to be related to stress tensors by Hooke’s law

σi j = 3 K

1 3

   1 kk δi j + 2G i j − kk δi j , 3

(14)

equivalently written as

1 E

i j = (σi j − ν (σkk δi j − σi j )).

(15)

The elastic constants, K, G, E, and ν are shown in Table 1 and taken from (Königsberger et al., 2014a). For simplicity, we did not account for the known anisotropy of single-crystal quartz in reagg agg lating i j to σi j , although this anisotropy was taken into account in determining these quantities during 3DXRD. Furthermore, the elastic constants are ideal values and may differ for our samples due to differences in preparation techniques like water content. More accurate elastic constants may be determined in future studies by comparing XRCT data with finite-element simulations, or by nanoindentation (Velez et al., 2001). The elastic constants themselves may even be spatially varying, as shown in Landis et al. (2016) for the ITZ.

Fig. 14 shows the minimum (compressive) and maximum (tensile) normal stress and the maximum shear stress (resolved in the plane of the aggregate surface) at any location in the ITZ for several of the aggregate particles. The use of stress fields as a function of x within aggregate particles would alter these trends and provide a possible method for evaluating the local ITZ fracture or aggregate-cement paste failure criterion. In its current form, however, the trends in Fig. 14 provide similar information and reflect similar trends to those in Fig. 9, because only a uniform stress tensor was available within each aggregate (the full intra-crystal stress tensor variation as a function of x was not available from 3DXRD measurements). For instance, aggregate 1, which is in close proximity to the fracture nucleating in step 4, experienced a decrease in absolute value of ITZ shear stress and compressive stress between steps 3 and 4. Aggregate 7, which is also in close proximity to the fracture nucleating in step 4, experienced an increase in absolute values of tensile, shear, and compressive ITZ stresses. It is difficult to draw further conclusions from these curves because they are not computed using spatially varying stress tensors. However, the possibility of using other techniques to obtain spatially varying stress tensors within aggregates would permit a further investigation of local aggregate-ITZ failure criterion and is discussed briefly in Section 4.

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Fig. 13. (a)-(d) The same as Fig 13 but showing aggregates with a decrease in σ zz (more compression) equal to or greater than 10 MPa.

4. Discussion This paper has demonstrated the use of combined in-situ XRCT and 3DXRD measurements to study fracture network growth, aggregate stresses (and their heterogeneity), and correlations between aggregate stress fluctuations and fracture nucleation. We note that this combined in-situ XRCT and 3DXRD measurement approach can readily be applied to a broad range of composite materials to study failure mechanisms such as matrix-inclusion debonding, matrix fracture, and inclusion fracture. For example, these failure mechanisms have recently been studied using XRCT and/or finite-element simulations in metal matrix composites (MMCs) containing an aluminum matrix and silicon carbide (SiC) or Al2 O3 particles (Buffiere et al., 1999; Beckmann et al., 2007), which find application in variety of industries. With sufficiently large SiC or Al2 O3 inclusions, such as the 150 μm particles featured in Buffiere et al. (1999), both XRCT and 3DXRD can readily be applied during mechanical or thermal loading to simultaneously reveal matrix microstructure changes and inclusion stress tensor evolution. More generally, any MMC or matrix-based composite sample featuring crystalline inclusions may be studied with combined in-situ XRCT and 3DXRD if it satisfies the following criteria:

• X-ray attenuation coefficients of the matrix and inclusions, and the sample size, must be suitably selected to permit X-ray transmission through the sample; • X-ray diffraction peaks of the matrix and inclusions in 2θ space must be distinct to permit identification of distinct Bragg peaks during 3DXRD measurements; • Regions of uniform crystal orientation within inclusions must be sized such that less than ≈ 50 0 0 are illuminated simultaneously to ensure that individual Bragg peaks can be resolved (Oddershede et al., 2010); These attributes are common to many MMCs used in a variety of industries (e.g., Withers and Preuss, 2012; Buffiere et al., 1999), suggesting that combined XRCT and 3DXRD may provide valuable insight into failure processes in these materials. In the remainder of this section, we discuss how future extensions of these measurements may be used to further quantify strength and failure mechanisms, such as ITZ fracture, in concrete, as well as to aid in developing and calibrating theoretical and numerical models. In Section 3.2, we showed that the meanfield Mori-Tanaka scheme for calculating aggregate stresses provided an accurate estimate average aggregate stresses measured in our experiments. However, the model provided no insight into the

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Fig. 14. Extremal values of various stresses in each aggregate’s ITZ. Plot numbers can be matched with aggregate numbers in Fig. 8.

significant stress heterogeneity in the aggregates. Because extreme, rather than average, values of stress are central to fracture and strength, and because stress heterogeneity is caused by the interaction of neighboring aggregates, we suggest that local corrections to the theory of Königsberger et al. (2014a) be made that account for aggregate interactions (Kachanov and Sevostianov, 2013). Future measurements may aid in determining the accuracy of such theories in concrete and other composites. Aggregate-cement paste and ITZ failure remains incompletely understood (Königsberger et al., 2014a, 2014b). Furthermore, inclusion interactions and inclusion-matrix failure in composites remains difficult to analyze analytically for complex microstructures (Kachanov and Sevostianov, 2013). Combining the experimental method we employed in this paper with computational techniques, as well as extending the experimental method, can help overcome these challenges. One possible approach is to employ numerical simulations (e.g., with meshfree or finite-element methods) to model the microstructures imaged with XRCT, as suggested in Garboczi and Kushch (2015). The simulation’s treatment of ITZ strength or stiffness properties, as well as matrix-ITZ boundary conditions, can be varied to reproduce the average stresses measured within aggregates. The aggregate stress gradients and ITZ stresses can then be calculated and used to understand ITZ stresses at incipient failure. Extensions of the experimental method to further investigate inclusion-matrix failure may include studies of Bragg peak distortion in 3DXRD measurements, or the use of scanning 3DXRD microscopy (Hayashi et al., 2017), to provide intragrain strain, and thus stress, fields. Finally, experiments in which multi-axial loading or smaller strain increments are performed at incipient fracture may help to isolate states in which fractures have nucleated but have not yet fully propagated through the concrete’s microstructure. Mesoscale modeling methods have recently been proposed for predicting failure processes in brittle materials (e.g., Homel and Herbold, 2017; Grassl and Jirásek, 2010). A variety of continuum

models also employ random heterogeneity or flaws (e.g., Tonge and Ramesh, 2016). One important application of the method presented in this paper is the calibration of mesoscale models that can use XRCT data to generate simulated microstructures (Homel and Herbold, 2017) and 3DXRD data to validate or calibrate matrixinclusion interfacial mechanics models. Another application is investigating the relative importance of microstructural details by determining their influence on local stresses and damage processes. This latter application may aid in the development and calibration of continuum models. 5. Conclusion In this paper, we have shown the following: 1. aggregate stresses in a sample of concrete were highly heterogeneous and deviated significantly from predictions of a meanfield Mori-Tanaka approximation coupled with an Eshelby problem, but agreed with the predictions well on an average basis; 2. aggregates carried a varying fraction of applied loading in a complex manner that did not correlate with the area of sample cross-section occupied by the aggregates; 3. some stress fluctuations appeared to correlate with fracture nucleation and suggested that fractures partitioned the sample into distinct load paths experiencing either increases or decreases in compressive stresses in the macroscopic loading direction; 4. the method of combining in-situ XRCT and 3DXRD can simultaneously quantify fracture network growth, local stress evolution, and aggregate-cement paste interactions in concrete, as well as a broad range of other composites; 5. the method provides opportunities to validate theories of strength and failure, inclusion interactions, and mesoscale and continuum models for composites containing crystalline inclusions.

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Acknowledgments This work is based upon research conducted at the Cornell High Energy Synchrotron Source (CHESS) which is supported by the National Science Foundation (NSF) and the National Institutes of Health/National Institute of General Medical Sciences under NSF award DMR-133208. R.C.H. thanks financial support from the Johns Hopkins University’s Whiting School of Engineering and the Hopkins Extreme Materials Institute. R.C.H. thanks Dr. Eric Herbold and Dr. Michael Homel of Lawrence Livermore National Laboratory for helpful discussions. All authors thank Dr. Chongpu Zhai of Johns Hopkins University for aiding with the experiment and Dr. Sriramya Nair and Dr. Kelly Nygren of CHESS for aiding with the initial data analysis. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijsolstr.2019.03.015. References Abdolvand, H., Wright, J., Wilkinson, A.J., 2018. Strong grain neighbour effects in polycrystals. Nat. Commun. 9 (1), 171. Aïtcin, P.C., 20 0 0. Cements of yesterday and today: concrete of tomorrow. Cem. Concr. Res. 30 (9), 1349–1359. Allison, J.E., 1979. Measurement of crack-tip stress distributions by x-ray diffraction. In: Fracture Mechanics: Proceedings of the Eleventh National Symposium on Fracture Mechanics: Part I. ASTM International. Allison, J.E., Cole, G.S., 1993. Metal-matrix composites in the automotive industry: opportunities and challenges. JoM 45 (1), 19–24. Andò, E., Cailletaud, R., Roubin, E., Stamati, O., contributors, S., 2017. SPAM: the software for the practical analysis of materials. https://ttk.gricad-pages. univ- grenoble- alpes.fr/spam/. Beckmann, F., Grupp, R., Haibel, A., Huppmann, M., Nöthe, M., Pyzalla, A., Reimers, W., Schreyer, A., Zettler, R., 2007. In-situ synchrotron x-ray microtomography studies of microstructure and damage evolution in engineering materials. Adv. Eng. Mater. 9 (11), 939–950. Bernier, J., Barton, N., Lienert, U., Miller, M., 2011. Far-field high-energy diffraction microscopy: a tool for intergranular orientation and strain analysis. J. Strain Anal. Eng. Des. 46 (7), 527–547. Bieler, T.R., Wang, L., Beaudoin, A.J., Kenesei, P., Lienert, U., 2014. In situ characterization of twin nucleation in pure ti using 3d-xrd. Metallur. Mater. Trans. A 45 (1), 109–122. Breunig, T., Stock, S., Guvenilir, A., Elliott, J., Anderson, P., Davis, G., 1993. Damage in aligned-fibre sic/al quantified using a laboratory x-ray tomographic microscope. Composites 24 (3), 209–213. Budiansky, B., Hutchinson, J.W., Evans, A.G., 1986. Matrix fracture in fiber-reinforced ceramics. J. Mech. Phys. Solids 34 (2), 167–189. Buffiere, J.-Y., Maire, E., Cloetens, P., Lormand, G., Fougeres, R., 1999. Characterization of internal damage in a mmcp using x-ray synchrotron phase contrast microtomography. Acta Mater. 47 (5), 1613–1625. Champley, K., 2016. Livermore Tomography Tools (LTT) Technical Manual. Technical Report. LLNL Technical Report under development, (Dec 11, 2015). Cil, M., Alshibli, K., 2014. 3D evolution of sand fracture under 1d compression. Géotechnique 64 (5), 351. Cil, M.B., Alshibli, K.A., Kenesei, P., 2017. 3D experimental measurement of lattice strain and fracture behavior of sand particles using synchrotron x-ray diffraction and tomography. J. Geotech. Geoenviron. Eng. 143 (9), 04017048. Fu, S.-Y., Feng, X.-Q., Lauke, B., Mai, Y.-W., 2008. Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulate–polymer composites. Compos. Part B 39 (6), 933–961. Garboczi, E., Kushch, V., 2015. Computing elastic moduli on 3-d x-ray computed tomography image stacks. J. Mech. Phys. Solids. 76, 84–97. Grassl, P., Jirásek, M., 2010. Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension. Int. J. Solids Struct. 47 (7–8), 957–968. Hayashi, Y., Setoyama, D., Seno, Y., 2017. Scanning three-dimensional x-ray diffraction microscopy with a high-energy microbeam at spring-8. In: Materials Science Forum, 905. Trans Tech Publ, pp. 157–164. Herrera-Franco, P., Valadez-Gonzalez, A., 2005. A study of the mechanical properties of short natural-fiber reinforced composites. Compos. B 36 (8), 597–608. Heyliger, P., Ledbetter, H., Kim, S., 2003. Elastic constants of natural quartz. J. Acoust. Soc. Am. 114 (2), 644–650. Hill, R., 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11 (5), 357–372. Homel, M.A., Herbold, E.B., 2017. Field-gradient partitioning for fracture and frictional contact in the material point method. Int. J. Numer. Methods Eng. 109 (7), 1013–1044.

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