Dynamic Behavior of Materials, Volume 1: Proceedings of the 2023 Annual Conference on Experimental and Applied Mechanics (Conference Proceedings of the Society for Experimental Mechanics Series) [2024 ed.] 3031506456, 9783031506451

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Conference Proceedings of the Society for Experimental Mechanics Series

Veronica Eliasson Paul Allison Phillip Jannotti   Editors

Dynamic Behavior of Materials, Volume 1 Proceedings of the 2023 Annual Conference on Experimental and Applied Mechanics

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman Society for Experimental Mechanics, Inc., Bethel, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.

Veronica Eliasson • Paul Allison • Phillip Jannotti Editors

Dynamic Behavior of Materials, Volume 1 Proceedings of the 2023 Annual Conference on Experimental and Applied Mechanics

Editors Veronica Eliasson Colorado School of Mines Golden, CO, USA

Paul Allison Baylor University Waco, TX, USA

Phillip Jannotti U.S. Army Research Laboratory Adelphia, MD, USA

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-031-50645-1 ISBN 978-3-031-50646-8 (eBook) https://doi.org/10.1007/978-3-031-50646-8 © The Society for Experimental Mechanics, Inc 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Dynamic Behavior of Materials represents one of five volumes of technical papers presented at the 2023 SEM Annual Conference & Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics held on June 5–8, 2023. The complete proceedings also include volumes on: Additive and Advanced Manufacturing, Advancement of Optical Methods in Experimental Mechanics, Fracture and Fatigue, Inverse Problem Methodologies, Machine Learning and Data Science, Mechanics of Biological Systems and Materials, Mechanics of Composite and Multifunctional Materials, Residual Stress, Thermomechanics and Infrared Imaging and Time-Dependent Materials. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. Dynamic Behavior of Materials is one of these areas. The Dynamic Behavior of Materials track was initiated in 2005 and reflects our efforts to bring together researchers interested in the dynamic behavior of materials and structures, and to provide a forum to facilitate technical interaction and exchange. Over the years, this track has been representing the ever-growing interests in dynamic behavior to the SEM community, working towards expanding synergy with other tracks and topics, and improving diversity and inclusivity, as evidenced by the increasing number and diversity of papers and attendance. The contributed papers span numerous technical divisions within SEM, demonstrating its relevance not only in the dynamic behavior of materials community, but also in the mechanics of materials community as a whole. The track organizers thank the authors, presenters, organizers and session chairs for their participation, support and contribution to this track. The SEM support staff is also acknowledged for their devoted efforts in accommodating the large number of paper submissions this year’s Dynamic Behavior of Materials Track a success. Golden, CO, USA Waco, TX, USA Adelphi, MD, USA

Veronica Eliasson Paul Allison Phillip Jannotti

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Contents

A Data Processing Approach for Kolsky Bar Experiments on Metallic Samples . . . . . . . . . . . . . . . . . . . . . . . . Andrew R. Roginski, Cody D. Kirk, and Weinong W. Chen

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High-Rate Ductile Fracture of Al 7075 Alloy at a Range of Stress Triaxialities . . . . . . . . . . . . . . . . . . . . . . . . . Christopher S. Meredith, Daniel J. Magagnosc, and Jeffrey T. Lloyd

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High Strain Rate Tests by a 90 m Long Tension-Torsion Hopkinson Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marco Sasso, Edoardo Mancini, Gianluca Chiappini, Mattia Utzeri, and Dario Amodio

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Rate Dependence of Penetration Resistance in a Cohesive Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sophia R. Mercurio, Stephan Bless, Abdelaziz Ads, Mehdi Omidvar, and Magued Iskander

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Characterization of Shale Structure Subjected to Two Different Loading Rate Conditions . . . . . . . . . . . . . . . . Achyuth Thumbalam Guthai, Ali F. Fahem, Kyle R. Messer, and Raman P. Singh

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In-Situ Mesoscale Characterization of Dynamic Crack Initiation and Propagation Using X-Ray Phase Contrast Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrew F. T. Leong, Bryan Zuanetti, Milovan Zecevic, Kyle J. Ramos, Cindy A. Bolme, Christopher S. Meredith, John L. Barber, Marc J. Cawkwell, Brendt E. Wohlberg, Michael T. McCann, Todd C. Hufnagel, Pawel M. Kozlowski, and David S. Montgomery

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Nose Shape Effects from Projectile Impact and Deep Penetration in Dry Sand . . . . . . . . . . . . . . . . . . . . . . . . . J. Dinotte, L. Giacomo, S. Bless, M. Iskander, and M. Omidvar

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A Novel Specimen Design for Multiaxial Loading Experiments at High Strain Rates . . . . . . . . . . . . . . . . . . . . Yuan Xu, Govind Gour, Julian Reed, and Antonio Pellegrino

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Investigation and Characterization of Dynamic Energy Absorbed by Shale Materials . . . . . . . . . . . . . . . . . . . Ali F. Fahem, Achyuth Thumbalam Guthai, Kyle R. Messer, and Raman P. Singh

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Dynamic Fracture Characteristics of Cyanoacrylate Weakened Planes in Polycarbonate Material . . . . . . . . . . Kyle R. Messer, Achyuth Thumbalam Guthai, Ali F. Fahem, and Raman P. Singh

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Modal Verification and Thermal-Fluid-Structure Coupled Analysis of Centrifugal Impeller . . . . . . . . . . . . . . . Po-Wen Wang and Chang-Sheng Lin

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Multiaxial Failure Stress Locus of a Polyamide Syntactic Foam at Low and High Strain Rates . . . . . . . . . . . . Yuan Xu, Yue Chen, and Antonio Pellegrino

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Practical Considerations for High-Speed DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phillip Jannotti, Nicholas Lorenzo, and Samantha Cunningham

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Dynamic Behavior of Lungs Subjected to Underwater Explosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helio Matos, Tyler Chu, Brandon Casper, Matthew Babina, Matt Daley, and Arun Shukla

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Dynamic Behavior of Curved Aluminum Structures Subjected to Underwater Explosions . . . . . . . . . . . . . . . . 105 Matthew Leger, Helio Matos, Arun Shukla, and Carlos Javier

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The Effect of Layering Interfaces on the Mechanical Behavior of Polyurea Elastomeric Foams . . . . . . . . . . . . 111 Mark Smeets, Behrad Koohbor, and George Youssef Moderate-Velocity Response of Polyurea Elastomeric Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Paul Kauvaka, Mark Smeets, Behrad Koohbor, and George Youssef The Use of Human Surrogate for the Assessment of Ballistic Impacts on the Thorax . . . . . . . . . . . . . . . . . . . . 121 Martin Chaufer, Rémi Delille, Benjamin Bourel, Christophe Marechal, Franck Lauro, Olivier Mauzac, and Sebastien Roth

A Data Processing Approach for Kolsky Bar Experiments on Metallic Samples Andrew R. Roginski, Cody D. Kirk, and Weinong W. Chen

Abstract The split Hopkinson pressure bar (Kolsky bar) is an experimental tool that has been used to characterize materials at intermediate to high strain rates since 1949. Since the genesis of this method, several investigations into the validity of the data have led to general conclusions regarding the nuances of the method that affect the outcome of the data. The goal of this work is to review the sources of error for the Kolsky bar method and implement physically driven solutions that help to aid in the standardization of processing experimental Kolsky bar data. Correction methods for longitudinal wave dispersion in both tension and compression configurations are implemented using the analytical solutions by Pochhammer and Chree, the frequency domain method by Gorham and Follansbee and Frantz, and a special implementation of spectral analysis by James F. Doyle. Other corrections include a novel pulse detection method and strain correction due to elastic punching (compression). Keywords Kolsky bar · Data processing · Wave dispersion

Introduction For most materials, different loading/environmental conditions such as temperature, impact velocity, and pressure change their mechanical responses. Material properties such as yield strength, elongation at failure, ultimate strength, etc., are coupled with these environmental inputs. Since 1949, the Kolsky bar has helped researchers quantify the mechanical response of materials subject to different strain rates between 102 and 104 s-1 [1]. These results inform how to better design materials given a specific application. Figure 1 is the traditional setup of the Kolsky bar in the compression configuration. In general terms, a gas gun launches a striker bar into the incident bar. This action induces a stress pulse (incident wave) where the profile of the pulse is dependent on the type of pulse shaper used (or lack thereof) that traverses the incident bar until it reaches the incident bar/specimen interface. Depending on the materials, the impedance mismatch between the incident bar and the specimen will cause the incident wave to split into a reflected wave and a transmission wave. The reflected wave traverses the incident bar in the opposite direction as the incident wave and is a direct measurement of the incident bar’s strain history in time. The transmission wave describes the strain history of the transmission bar. By assuming no energy is lost to the incident and transmission bars during the loading event, it is implied that the difference between incident and reflected signals on the incident bar is equal to the transmission signal [1–10]. The equations used to process Kolsky bar experimental data for the sample are then derived by applying dynamic equilibrium of an element in a long, thin rod to 1D Hooke’s law [1, 7]. The derivations assume that bars are isotropic, homogeneous, uniform in crosssection, the neutral axis of the bar remains on axis, the stress distribution is equal across the face of the bars, and the bars experience no plasticity upon loading [7]. Assuming sample equilibrium and taking the impedance mismatch between the bar and sample materials into account, the strain rate, strain, and stress of the sample during the loading event can be described using Eqs. 1, 2, and 3, respectively, below. Here ER is the bar strain from the reflected signal, c0 is the bulk sound speed of the bar materials, L is the length of the sample, Abar is the area of the bar, Ebar is the elastic modulus of the bar, Asample is the area of the sample, and ET is the strain in the bar from the transmission signal. A. R. Roginski (✉) · C. D. Kirk School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA e-mail: [email protected] W. W. Chen School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA School of Materials Engineering, Purdue University, West Lafayette, IN, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_1

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Fig. 1 Typical Kolsky bar setup in compression configuration [1]

E_ =

Eðt Þ =

- 2c0 ER L

c0 L

σ sample ðt Þ =

t

ð1Þ

ER ðτÞdτ

ð2Þ

ðAbar E bar Þ ET ð t Þ Asample

ð3Þ

0

Processing the voltage outputs from an experiment can be particularly misleading without taking consideration of the assumptions made during the analytical derivation for the fundamental equations of Kolsky bar experiments. Although these equations provide a succinct way to calculate time history for the stress, strain, and strain rate of a sample during dynamic loading, the nuances due to the three-dimensional reality of the event lead to necessary corrections to apply to the data to describe the mechanical response best accurately. The goal of this paper is to create a method like Sure-Pulse™ from REL or the authors [11, 12] and to incorporate some of the popular correction methods from the past 70 years of Kolsky bar analysis. This work was started to supplement the ASTM Standard Group WK65916 that is developing the standard test method for dynamic compression of bulk metallic materials using a Kolsky bar. This method applies correction for data smoothing, bar indentation, several wave dispersion methods, and pulse detection on experiments for metallic samples. Inertial effects of the sample are currently deemed negligible for metallic samples.

Background Correction methods for Kolsky bar experiments have been used extensively in recent years, however, there currently is no standard for which correction methods are “correct”. Therefore, this method is to serve as a hub to compare results from experiments where a plethora of correction methods from literature can be used to correct the data and compare results. In this processing approach, the following correction methods are available: indentation correction for Kolsky bar compression experiments using the analytical method by K. Safa [13], wave dispersion methods using the Pochhammer-Chree solution [3, 4, 6, 7, 14], and a frequency method using the Love rod model [3, 15]. Data is smoothed using a cutoff frequency in the

A Data Processing Approach for Kolsky Bar Experiments on Metallic Samples

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frequency space utilizing the FFT of the data and a novel pulse detection method is performed by using the Savitzky-Golay filter [16]. For descriptions on the analytical solutions for the indentation correction method and Pochhammer-Chree wave dispersion methods, please refer to Refs. [3, 4, 6, 13, 7, 14], respectively. The frequency method application for the Love rod model introduced by James F. Doyle can be described by Ref. [15]. Further analysis in this abstract is focused on describing the pulse detection method and the Love rod frequency methods; a comparison between all correction techniques will be shown at the time of this presentation.

Analysis Pulse detection in Kolsky bar experiments has been known to be a rather iterative method that involves manual input. Techniques exist utilizing wave dispersion methods to find the beginning of pulses [8] or utilizing simulations as a basis for pulse start and duration [9]. An attempt to reduce user input has been developed by curve fitting the beginning of the incident pulse up using the Savitzky-Golay filter [16] to the oscilloscope trigger point and find the first zero crossing looking back in time. The beginning of the reflected and transmission pulses is found using the bar geometry and bulk sound speed of the bar material. The Savitzky-Golay (SG) filter uses a combination of convolution and least-squares method to create a moving average approach that conserves the shape of the signal while allowing the user to select the number of points and polynomial fit to consider for their applications. For the purposes of this paper, a frame length of five points was used and a second-order polynomial was used to fit the data. Figure 2 shows an example of modeling the incident pulse with the SG filter. The start point is found by taking the first zero point moving back in time from the oscilloscope trigger point that satisfies the first derivative and has a zero-crossing indicating an inflection point. Further testing for this sequence of mathematical logic is to be tested for optimal start point selection. Figure 3 shows the outcome of the SG filter and pulse detection selection for different types of pulse shapes. Along with the pulse detection lines (shown in red), the gray line is the original signal, and the blue line is the filtered signal using the cutoff frequency in the FFT. The Love rod frequency method described in Ref. [15] corrects for wave dispersion by calculating the phase speeds of different frequency components in frequency space and correcting the length of travel along the bar relative to the bulk sound speed. Figure 4 is an example of applying the Love rod frequency model to experimental data.

Fig. 2 Example of modeling the incident pulse with the SG filter using a second-order polynomial and a frame length of five points

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Fig. 3 Different loading profiles with pulse detection results (red line)

Fig. 4 Example of wave dispersion correction using the frequency method and Love rod solution

The Love Rod solution was utilized to illustrate the effects of differing phase velocities; this solution was selected as it possesses only a single mode – simplifying the decision-tree within the Matlab script for dispersion correction. After successful implementation of the Love Rod model, higher order models, as well as the analytical solutions to dispersion in rods (Pochhammer-Chree) were implemented.

Conclusion Although incomplete, the correction methods result in the example in Fig. 5. The pulse alignment and force equilibrium history are optimized with little user input. Further comparisons of all correction methods will be shown at the time of the conference presentation. Only the pulse detection and frequency space wave dispersion correction were discussed as they are relatively new to the Kolsky bar application in literature.

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Fig. 5 Example experiment showcasing the strain and strain rate history (top left), stress vs. strain plot (top right), force equilibrium (bottom left), and pulse alignment (bottom right)

Acknowledgments I would like to thank the ASTM Standard Group WK65916 for helping me work toward a goal of standardizing what is a complex experimental method with several nuances.

References 1. Chen, W., Song, B.: Split Hopkinson (Kolsky) Bar Design Testing and Applications, 1st edn. Springer US, New York (2011). https://doi.org/10. 1007/978-1-4419-7982-7 2. Kolsky, H.: An investigation of the mechanical properties of materials at very high rates of loading. Proceedings of the Physical Society. Section B. 62(11), 676–700 (1949). https://doi.org/10.1088/0370-1301/62/11/302 3. Davies, R. M.: A critical study of the Hopkinson pressure bar 4. S. E. Rigby, A. D. Barr, M. Clayton, A review of Pochhammer-Chree dispersion in the Hopkinson bar, Proceedings of the Institution of Civil Engineers: Engineering and Computational Mechanics, 171, 1. ICE Publishing, pp. 3–13, 2018. https://doi.org/10.1680/jencm.16.00027 5. Gong, J.C., Malvern, L.E.: Professor D A Jenkins Associate Engineer, “Dispersion Investigation in the Split Hopkinson Pressure Bar,” (1990). [Online]. http://asmedigitalcollection.asme.org/materialstechnology/article-pdf/112/3/309/5716592/309_1.pdf 6. Follansbee, P. S., Frantz, C.: “Wave Propagation in the Split Hopkinson Pressure Bar,” (1983). [Online]. http://asmedigitalcollection.asme.org/ materialstechnology/article-pdf/105/1/61/5674776/61_1.pdf 7. Gama, B., Lopatnikov, S., Gillespie, Jr J.: Hopkinson bar experimental technique: A critical review 8. Lifshitz, J.M., Leber, H.: Data Processing in the Split Hopkinson Pressure Bar Tests, vol. 15, p. 723 (1994) 9. Zhao, H., Gary, G.: On the use of SHPB techniques to determine the dynamic behavior of materials in the range of small strains. Int. J. Solids Struct. 33(23), 3363–3375 (1996). https://doi.org/10.1016/0020-7683(95)00186-7 10. Zhao, H.: A Study on Testing Techniques for Concrete-like Materials Under Compressive Impact Loading, vol. 20, p. 293 (1998) 11. Gershanik, T., Levin, I., Rittel, D.: 2BarG—A program to process split Hopkinson (Kolsky) bar test results. SoftwareX. 18, 101093 (2022). https://doi.org/10.1016/j.softx.2022.101093 12. Francis, D.K., Whittington, W.R., Lawrimore, W.B., Allison, P.G., Turnage, S.A., Bhattacharyya, J.J.: Split Hopkinson pressure bar graphical analysis tool. Exp. Mech. 57(1), 179–183 (2017). https://doi.org/10.1007/s11340-016-0191-9 13. Safa, K., Gary, G.: Displacement correction for punching at a dynamically loaded bar end. Int J Impact Eng. 37(4), 371–384 (2010). https://doi. org/10.1016/j.ijimpeng.2009.09.006 14. Bancroft, D.: The velocity of longitudinal waves in cylindrical bars. Phys. Rev. 59(7), 588–593 (1941). https://doi.org/10.1103/PhysRev.59.588 15. Doyle, J.: Mechanical engineering series wave propagation in structures. [Online]. http://www.springer.com/series/1161 16. Savitzky, A., Golay, M.: Smoothing and differentiation of data by simplified least squares procedures, (1964). [Online]. https://pubs.acs.org/ sharingguidelines

High-Rate Ductile Fracture of Al 7075 Alloy at a Range of Stress Triaxialities Christopher S. Meredith, Daniel J. Magagnosc, and Jeffrey T. Lloyd

Abstract Engineering materials are intrinsically heterogeneous owing to their processing history. For heat-treatable lightweight aluminum alloys, e.g., 7XXX alloys, the fracture behavior is governed by the size and distribution of second-phase particles. Modern empirical and micro-mechanically motivated computational failure models used for ductile fracture calculations do not include spatially heterogeneous microstructural information in any meaningful manner, despite recent significant advances in high-throughput, three-dimensional in-situ and ex-situ characterization techniques. Our hypothesis is that direct numerical simulations of particle-initiated failure, which will be informed by advanced testing and characterization, will provide the most realistic prediction of failure to date. This paper will focus on high-rate ductile fracture experiments using the mini-tension Kolsky bar with sample geometries that induce a wide range of stress triaxialities—from pure shear to plane strain tension. Samples are cut from near the surface and at the midplane of a thick plate because the size and distribution of the second-phase particles, and the texture of the matrix Al, are different at these locations. Deformation of the different geometries is recorded at 1 mfps, where the total loading time is approximately 100 μs. The experiments show there is a clear difference in the flow response between the two locations, but a clear strain to failure difference is not clear from these initial experiments. Keywords Ductile fracture · Stress triaxiality · Al 7075 · Split Hopkinson pressure (Kolsky) bar · High-speed camera

Introduction Heat-treatable aluminum alloys are widely used due to the material’s light weight and good strength and fracture resistance. The high strain rate constitutive response of these materials is reasonably well known and existing constitutive models do a good job of predicting the plastic response. However, the fracture response of these materials subjected to high-rate loading is poorly understood, with existing fracture models often unable to correctly replicate real-world observations and they offer little predictive capability. Ductile fracture occurs in regions subjected to large plastic deformation. Classically ductile fracture occurs by the nucleation and growth of voids that coalesce to form a crack, which quickly degrades the load-carrying ability of the material. But there are several different mechanisms that can govern ductile fracture—shear instability [1], intervoid necking and shearing, void sheeting, Orowan alternating slip, necking to a point, and single- and multi-plane catastrophic shear [2]. In general, multiple different fracture mechanisms are simultaneously activated as material fractures, but the relative activity will vary based on the properties of the material, stress state, strain rate, temperature, etc. For pure metals, voids nucleate at dislocation pileups [3] and grain boundaries [4]. For alloyed metals, voids typically nucleate at precipitates via particle and matrix delamination or the cracking of the particles themselves [1, 5, 6]. Thus, the size, morphology, orientation, and distributions thereof, along with the spatial distribution of precipitates, all influence the fracture behavior of ductile materials. To study ductile fracture, different specimen geometries are routinely used that generate different stress states within the specimen gage section. The specimens are often cut from sheet metal (owing to the car industry historically driving ductile research) so the stress triaxiality (ratio of the hydrostatic stress to the von Mises equivalent stress) is approximately constant over the course of the experiment [7–10]. Since these specimens are flat, digital image correlation (DIC) is used to measure the full-field strains of the specimen through the entire loading regime, including at the instance of initial failure. The typical dog bone-shaped uniaxial tension geometry is not good for fracture experiments because at the onset of necking the stress C. S. Meredith (✉) · D. J. Magagnosc · J. T. Lloyd Army Research Directorate, DEVCOM Army Research Laboratory, Aberdeen Proving Ground, MD, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_2

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triaxiality shifts to higher values (toward a plane strain tension stress state). Thus, adding a central hole to the dog bone geometry erases any necking and generates a nearly constant triaxiality up to fracture at the hole boundary [8, 11]. Additionally, flat shear specimens that utilize eccentric loading produce nearly pure shear up to failure and are readily implemented [9, 11, 12]. Finally, plane strain tension is of particular interest because that has the lowest strain to failure, a flat notched tension geometry is commonly used, even if the triaxiality tends to vary to a greater extent than the central hole or shear geometries [13], but it is easier to implement at higher strain rates than the sharp bend experiment [11, 14]. An important note is that since the experiments measure the full-field strain on the specimen surface, a phenomenological or crystal plasticity model is required so the local stress can be calculated for stress triaxiality determination. The ductile fracture of aluminum alloys has received considerable attention over the recent decades, both investigating the micromechanics of void formation, growth, and coalescence [5] and the continuum-level fracture behavior under different stress states [7, 12, 15], however the material is often in sheet metal form and exclusively at quasi-static strain rates. High strain rate ductile fracture over a range of stress triaxialities has rarely been studied for any material [16, 17], although shockinduced spall failure under uniaxial strain has been well studied [18, 19]. Additionally, ductile fracture of intrinsically heterogeneous thick plates has received considerably less study. This paper reports on the dynamic fracture behavior of Al 7075-T651 subjected to a range of stress triaxialities—from pure shear to plain strain tension—using geometries developed for sheet metals and adapted for use with a mini-tension Kolsky bar. The source material was a 30 mm thick plate where samples were cut at the midplane in the plate thickness direction and very close to either surface. A high-speed camera was used to record the deformation of the gage section and capture the initiation of failure at 1 m fps where the total loading time was approximately 100 μs for each specimen. The heterogeneous microstructure created during its processing history results in distinct fracture behaviors between the middle and surface of the source plate that cannot be ignored in applications that involve high strain rate ductile fracture.

Experimental Procedures The material investigated was a 30 mm thick plate of Al 7075-T651, where Zn, Mg, and Cu are the major alloying elements, that was obtained from a major supplier. The as-received microstructure was measured using electron backscatter diffraction (EBSD) with a step size of 400 nm, accelerating voltage of 20 kV, working distance of about 10 mm, and a 70° tilt between the sample surface and electron beam. EBSD sample preparation followed conventional methods—mechanical grinding with progressively finer grit sandpaper, then polishing with progressively finer polycrystalline diamond suspension on a soft, long nap polishing cloth, and final polishing using 0.05 μm colloidal silica solution on a soft rubber polishing cloth. Successive orientation image maps (OIMs) were collected continuously from the surface through the midplane of the plate sample. The specimens were designed specifically for mini-tension Kolsky bar testing, and were inspired by specimens used previously on sheet metals under quasi-static conditions [13, 16, 20]. Figure 1 summarizes all five specimen geometries. The cross-sectional area of the gage section of each geometry was selected such that sufficient transmitted wave signal with the Kolsky bar was achieved (from which force/stress are calculated). The length of the gage sections was kept relatively short so that fracture was readily achieved during the first loading pulse of the Kolsky bar (longer sections require greater displacement of the bars in order to cause fracture). Samples were machined using electro-discharge machining (EDM) cut from within a few mm of the midplane and surfaces of the source plate. The edges of the gage section of all specimens were cut such that the surface roughness was minimized during EDM and no subsequent sanding was performed on the edges. The thickness of the specimens was EDMed to 1 mm and the faces of the samples were sanded/polished to 0.5 mm with the final step being P4000 sandpaper to remove the rough outer surface and any oxidation that formed on the specimen faces. The surface roughness has a significant effect on the fracture behavior [11, 15, 21]. The direct tension Kolsky bar (Fig. 2) utilized a striker tube that was accelerated by opening a solenoid valve and rapidly releasing inert gas into the gun barrel. The striker tube impacts a flange at the end of the incident bar which generates an elastic tensile wave that propagates down the incident toward the sample. The bars utilized for these experiments were 6.35 mm in diameter with lengths of 1.8, 0.91, and 0.31 m for the incident, transmitted, and striker tube, respectively. The specimen slides into the slotted ends of the Kolsky bars and it is held and loaded via steel pins that pass through the bars and specimen. The pins freely slide through the holes in the bars (but with little to no slop) but are press-fit into the holes in the specimen. The incident, reflected, and transmitted wave signals are measured using strain gages and the force and stress in the sample are calculated in the usual manner [22]. Note that the usual way of calculating strain and strain rate is generally inaccurate for tension Kolsky bars because of displacement of the bars caused by non-gage section deformation and compliance in the bar-specimen gripping, and it is even more inaccurate with little physical meaning for the non-uniform fracture geometries tested here. The loading was recorded with a high-magnification lens with a field of view of ~3.2 × 2 mm2 with a Shimadzu HPV-X2 high-speed camera at 1 mfps (200 ns exposure time).

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Fig. 1 Specimen fracture geometries for dynamic (a) uniaxial tension (UT), (b) central hole (CH), (c) 1.25 mm round notched tension (NT1), (d) 3.8 mm round notched tension (NT2), and (e) shear. All the samples have a thickness of 0.020″ (0.5 mm) and are loaded axially under tension. All units are in inches in the drawings

Fig. 2 Schematic of the tension Kolsky bar setup, where the inset shows the pin loading between the bars and specimen. Note, the schematic is not to scale

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Fig. 3 Through-thickness EBSD map of the 30 mm thick plate of Al 7075-T651

Fig. 4 Engineering stress-strain data at a strain rate ~ 2000 s-1 for the UT geometry. A quasi-static response is provided, whose specimen conforms to ASTM E8. The right side is an example surface specimen just prior to failure, where the loading direction is vertical

Results Figure 3 shows the through-thickness EBSD orientation map with the rolling (RD) and normal directions (ND) indicated. The pancaked grains are typical for a rolled plate, where the grains are elongated in the RD and significantly thinner in the ND. Near the surface, there is a greater fraction of sub-50 μm diameter grains, whereas near the midplane large (>300 micrometer) grains dominate. In terms of the texture, the near-surface texture is more random but near the midplane the texture is typical for a rolled face-centered cubic material. Figures 4, 5, 6, 7 and 8 show the results for the different fracture geometries tested at both the midplane (solid) and the surface (dashed) of the source plate. In these figures, the measured force is normalized by the cross-sectional area of the samples, which does not necessarily have physical significance for all the geometries. The net displacement of the Kolsky bars is normalized by the length of each specimen gage length (and closely matches the initial gap between the bars with each sample in place but prior to loading). The strain rate listed in each graph corresponds to the calculated strain rate from the traditional equation for the Kolsky bar, which generally overestimates the strain in the gage section resulting in an overestimated rate as well. The experiments show a clear flow stress difference because of the midplane and surface. However, only the notched tension geometries appear to show a consistent difference in the strain to failure, with the surface being slightly greater. However, using the reflected wave for strain measurement likely does not have the fidelity to resolve

High-Rate Ductile Fracture of Al 7075 Alloy at a Range of Stress Triaxialities

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Fig. 5 Normalized force-displacement data at a strain rate ~ 900 s-1 for the CH geometry. The right side is an example of a surface specimen just prior to failure at the hole surface, where the loading direction is vertical

Fig. 6 Normalized force-displacement data at a strain rate ~ 1500 s-1 for the NT1 geometry. The right side is an example of a surface specimen just prior to failure which occurs in the center of the notch. The loading direction is vertical

differences that may be present at the location of initial fracture. Additionally, these figures show an example experiment for each geometry at the last image prior to the observation of initial failure of the specimen. As the specimen deforms the surface roughness increases resulting in more light being reflected back into the camera, which is shown as white-colored areas (oversaturated) in the images.

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Fig. 7 Normalized force-displacement data at a strain rate ~ 600 s-1 for the NT2 geometry. The right side is an example of a surface specimen just prior to specimen failure which occurs at the center of the notch. The loading direction is vertical

Fig. 8 Normalized force-displacement data at a strain rate ~ 700 s-1 for the Shear geometry at the midplane. The right side is an example of a surface specimen just prior to specimen failure. The loading direction is vertical

The shear responses are an outlier, where there is not a clear distinction in the behavior between locations and there is a large specimen-to-specimen variability. These samples were glued in place, in addition to the pin loading, which was necessary because the eccentric loading of this geometry caused the sample to rotate when it was being loaded. However, gluing the connection may have created these issues with the response. Additionally, this geometry appears to have more compliance outside the gage section than the other geometries which could also be a contributing factor.

High-Rate Ductile Fracture of Al 7075 Alloy at a Range of Stress Triaxialities

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Conclusions This paper described dynamic fracture experiments using a mini-tension Kolsky bar with a wide variety of specimen geometries that generate a wide stress triaxiality range in the gage section—from plane strain tension to pure shear. Specimens were cut from two locations—at the midplane and the surfaces—from a 30 mm thick plate of Al 7075-T651. Clear differences in the mechanical behavior were observed between the locations of the specimens but no clear difference in the strain to fracture could be discerned, however, using the reflected wave as the strain measurement is unlikely to have the fidelity necessary to measure any differences that might be present. In the future digital image correlation will be utilized at high framing rates such that the strain at the location of fracture can be measured in situ and compared to model predictions. Acknowledgments Thanks to Jason Garvey (ARL) for help with conducting the experiments and Saadi Habib (NIST) for many helpful discussions.

References 1. Pineau, Benzerga, Pardoen: Failure of metals I: brittle and ductile fracture. Acta Mat. 107, 424–483 (2016) 2. Noell, Carroll, Boyce: The mechanisms of ductile rupture. Acta Mat. 161, 83–98 (2018) 3. Noell, Carroll, Hattar, Clark, Boyce: Do voids nucleate at grain boundaries during ductile rupture. Acta Mat. 137, 103–114 (2017) 4. Ovid’ko: Review on the fracture processes in nanocrystalline materials. J. Mat. Sci. 42(5), 1694–1708 (2007) 5. Babout, Brechet, Maire, Fougères: On the competition between particle fracture and particle decohesion in metal matrix composites. Acta Mat. 52(15), 4517–4525 (2004) 6. Benzerga, Besson, Pineau: Anisotropic ductile fracture. Acta Mat. 52(15), 4623–4638 (2004) 7. Driemeier, Brünig, Micheli, Alves: Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mech. Mat. 42(2), 207–217 (2010) 8. Dunand, Mohr: Hybrid experimental-numerical analysis of basic ductile fracture expeiments for sheet metals. Int. J. Solids Struct. 47(9), 1130–1143 (2010) 9. Peirs, Verleysen, Degrieck: Novel technique for static and dynamic shear testing of Ti6Al4V sheet. Exp. Mech. 52(7), 729–741 (2012) 10. Wierzbicki, Bao, Lee, Bai: Calibration and evaluation of seven fracture models. Int. J. Mech. Sci. 47(4–5), 719–743 (2005) 11. Roth, Mohr: Ductile fracture experiments with locally proportional loading histories. Int. J. Plasticity. 79, 328–354 (2016) 12. Brünig, Chyra, Albrecht, Driemeier, Alves: A ductile damage criterion at various stress triaxialities. Int. J. Plasticity. 10, 1731–1755 (2008) 13. Habib, Lloyd, Meredith, Khan, Schoenfeld: Fracture of an anisotropic rare-earth-containing magnesium alloy (ZEK100) at different stress states and strain rates: experiments and modeling. Int. J. Plasticity. 122, 285–318 (2019) 14. Stoughton and Yoon: A new approach for failure criterion for sheet metals. Int. J. Plasticity. 27(3), 440–459 (2011) 15. Luo, Dunand, Mohr: Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading – part II: ductile fracture. Int. J. Plasticity. 32-33, 36–58 (2012) 16. Roth and Mohr: Effect of strain rate on ductile fracture inititation in advanced high strength steel sheets: experiments and modeling. Int. J. Plasticity. 56, 19–44 (2014) 17. Karp, Dorogoy, Rittel: A shear compression disk specimen with controlled stress Triaxiality under dynamic loading. Exp. Mech. 53, 243–253 (2013) 18. Fensin, Escobedo-Diaz, Brandl, Cerreta III, G., Germann, Valone: Effect of loading direction on grain boundary failure under shock loading. Acta Mat. 64, 113–122 (2014) 19. Becker: Direct numerical simulation of ductile spall failure. Int. J. Fracture. 208, 5–26 (2017) 20. Gorji and Mohr: Micro-tension and micro-shear experiments to characterize stress-state dependent ductile fracture. Acta Mat. 131, 65–76 (2017) 21. Wang, L., Wierzbicki: Experiments and modeling of edge fracture for an AHSS sheet. Int. J. Fracture. 187(2), 245–268 (2014) 22. Chen and Song: Split Hopkinson (Kolsky) Bar. Springer, New York (2011)

High Strain Rate Tests by a 90 m Long Tension-Torsion Hopkinson Bar Marco Sasso, Edoardo Mancini, Gianluca Chiappini, Mattia Utzeri, and Dario Amodio

Abstract This work describes the design, construction, and first experimental results of an innovative device of the Hopkinson bar type with a length of 90 m for performing high strain rate tests on metals in a combined tension-torsion state. Analogously to the classic split Hopkinson bar technique, the system configuration consists of three bars: a pre-stressed bar, an input bar, and an output bar; the measurement is also based on the classical three-wave method, where the incident, transmitted, and reflected waves are measured. The length of the bars is designed so that the tensile wave reaches the sample from the output bar side at the same time as the torsion wave comes from the input bar. A successful test has been conducted on a hollow aluminum sample; it has been possible to measure the tension-torsion stress-strain curves; in addition, the dynamic equivalent stress-equivalent strain curves have been evaluated. Keywords Hopkinson bar · Torsion test · Tensile test · High strain rate · Wave synchronization

Introduction The Hopkinson bar is the most used device for carrying out compression and tensile tests on almost any type of material, with strain rates in the order of 102–104 1/s [1]. Torsion Hopkinson bar versions have also been made. The literature is vast on these two types of SHB, and the reader is referred to the review works [2, 3]. In recent years, growing attention is paid to the execution of high strain rate tests, aimed at the development and calibration of ductile damage models for the prediction of strain at failure [4]. Besides the equivalent plastic strain and stress triaxiality, the most recent models consider the failure to be governed by other important parameters, such as the Lode angle or the third invariant of the deviatoric stress tensor, J3, which give a more complete description of the state of stress [5]. The calibration of these advanced models requires the study of the behavior of the material in the multiaxial state of stress, and the most used methods to achieve this can be divided into 2 categories: (i) adoption of samples of complex shape solicited with monoaxial testing machines [6, 7]; (ii) samples of standard geometry solicited in combined tensile-torsion stress [8] by multiaxial machines. In the context of the study of materials at high strain rate, the first method is quite widespread [9] today; conversely, performing dynamic tests in a multiaxial loading environment is far less common; the works of Cadoni et al. should be noted [10], where the dynamic solicitation is superimposed on a hydrostatic one; practically absent in the literature is the combination of tension and torsion, with the exception of the recent Oxford solution [11], where a sample is positioned immediately downstream of the loading system. As far as the authors are aware, a version of SHB has not yet been developed which allows simultaneous tensile and torsion tests up to large deformations. The device shown here instead allows to perform this type of test and measure the stresses and strains according to the classical SHB theory, and potentially reach elongations up to 30 mm and torsion angles up to 2π in a range of strain rates in the order of 100–1000 1/s.

M. Sasso (✉) · G. Chiappini · M. Utzeri · D. Amodio Department of Industrial Engineering and Mathematical Sciences, Università Politecnica delle Marche, Ancona, Italy e-mail: [email protected] E. Mancini Department di Industrial and Information Engineering and Economics, Università degli Studi dell’Aquila, L’Aquila, Italy © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_3

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Working Principle The fundamental principle of the device here presented is to generate an extensional and a torsion wave simultaneously, making the former travel a suitably greater distance so that their arrival on the sample is synchronized. Instead of launching a striker bar, a pre-stressed bar is statically loaded both in tension and in torsion; analogously to the direct tension SHB developed and today routinely used by the authors [12], the load is quickly released through the failure of a fragile element placed at the beginning of the pre-stressed bar. The sudden release generates a torsion and compression wave; the latter traveling faster reaches the sample but passes into the output bar which is placed in direct contact with the input bar. The compression wave reaches the free end of the output bar and is reflected as a tensile wave; at this point, it returns to the sample just as the torsion wave arrives from the input bar. In practice, a direct torsion SHB system is combined with an indirect tensile SHB system. Here follows the space-time analysis of the system. The length of the pre-stressed (LP), input (LI) and output (LO) bars must simply satisfy the requirement: (LP + LI)/Cγ = (LP + LI + 2LO)/Cε, where Cε and Cγ represent the velocity of extensional and torsion waves, respectively. The project aimed to satisfy other important requirements in addition to wave synchronization: (i) diameter of the bars and samples for practical engineering use of the order of 20 mm, (ii) possibility of achieving high elongations (up to 30 mm) and torsions (up to 2π) to achieve failure in ductile metallic materials, (iii) avoid overlapping of the transmitted waves (εT, γ T) with the respective reflected waves (εR, γ R), iv) equal time duration of the extensional and torsional waves. This led to the following final choices: bars in Ti6Al4V with a diameter of 20 mm, length LP of the pre-stressed bar of 9.8 m (of which the first 6 m is subjected to both torsion and traction), maximum preload of 100 kN and 340 Nm, length LO of the output of 21 m, input bar length LI of 57 m; the total length is then 87 m.

Application Formulas Normal and shear stress, strain, and strain rate of the specimen can be computed by measuring the waves that propagate into the bars using the strain gauge rosettes appropriately placed, which convert the stress waves into proportional analogic signals. Then, the mechanical behavior of the sample material can be evaluated by the Kolsky analysis method. Considering the scheme of Fig. 1, where the central cylinder represents the sample in contact with the extremities of the input and output bars, the displacements u1, u2 and the rotations θ1, θ 2 of the opposite faces of the sample can be obtained by the elementary theory of one-dimensional propagation of elastic waves. By the same theory, it is possible to also obtain the loads P1, P2 and torques T1, T2 exchanged between the sample and the bars at the same interfaces. If P1(t) is equal to P2(t) and T1(t) is equal to T2(t) the sample deforms uniformly and is in dynamic equilibrium, the engineering normal and shear stress, strain rate, and strain that take place in the sample can be obtained from the classical “reduced” formulae:

Ls

Ls

eI

γI

eT

D

x

eR

Ds

u2

u1 (a)

γT

D

x

γR

Ds

θ2

θ1 (b)

Fig. 1 Scheme of traveling waves and displacements for (a) Tension Split Hopkinson Bar, (b) Torsional Split Hopkinson Bar

High Strain Rate Tests by a 90 m Long Tension-Torsion Hopkinson Bar

ε_ ðt Þ = -

17

2C 0 ε ðt Þ Ls R

ð1Þ

t

2C 0 εðt Þ = Ls

εR ðt Þ . dt

ð2Þ

0

σ ðt Þ =

Ab . E b εT ð t Þ As

ð3Þ

γ_ ðt Þ =

2Ct DS . γ R ðτ Þ LS D

ð4Þ

t

2Ct DS γ ðt Þ = . LS D

½γ R ðτÞ]dτ

ð5Þ

0

τ ðt Þ =

GD3 γ T ðt Þ 8D2S t S

ð6Þ

where D is the diameter of the bar, Ds is the outer diameter of the hollow sample, LS and AS represent the initial length and cross-sectional area of the sample, respectively, while ts is its initial thickness. Excluding time from previous Eqs. (1) to (6), i.e., synchronizing the reflected and transmitted signals, the axial stress-strain and shear stress-shear strain laws of the material of the sample at high strain rate are achieved.

Design and Realization Parts CAD All the parts that make up the system have been designed internally with CAD software and made in the workshops. The most significant ones are shown here. The static load was conceived to be applied by two actuators mounted in series: an electromechanical axial jack, equipped with a thrust bearing, pulls one end of a large rod; the latter passes through a hollow shaft which receives torque from a gear motor. The rod is connected at the other end to the pre-stressed bar through a sacrificial element, that is specifically designed to sustain the desired amount of axial load and torque. Figure 2a, b shows the loading system and the sacrificial element. The static torsion block system must prevent torsion at the end of the Lpγ part, but must allow axial sliding; it was made using an external collar with an eccentric profile which abuts unilaterally on radially sliding supports. In this way, apart from a limited frictional interaction, the rod is free to rotate after the release of the static load through an indefinite angle. The drawing is shown in Fig. 2c. Given that the maximum length available among various titanium suppliers is 6 m, the system was created by joining several bars. The connection between the bars, shown in Fig. 2d, is made by M12 internal threading for the transfer of axial loads; the bars are assembled by screwing one into the other until they are fully seated; the transmission of the twisting load is further guaranteed by an external collar with a grooved profile. These collars are made of AA7075T6 with the smallest possible section to minimize the mechanical impedance variation. The supports of the bars, the static tension block, and the end arrest at the output bar extremity are substantially identical to the system routinely used by the authors [12].

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Fig. 2 (a) Static torsion block, (b) sacrificial element, (c) axial-torsional actuation, (d) bars connection

Fig. 3 Facility installed on the roof of Civil Engineering Building at Marche Polytechnic University (Lat.43.58776°, Long.13.51667°)

Installation Because of its relevant length, the system has been installed at open air on the roof of Civil Engineering building of Università Politecnica delle Marche. A protective case has been applied along the length. Larger cabinets were adopted at the beginning and the end of the bars, where the electro-mechanical actuators and the end-arrest must be placed. Furthermore, a test room was created with a small prefabricated shed, which contains the sample, the digital and PC acquisition system, and the highspeed camera. Figure 3 shows the final appearance of the entire rig.

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Instrumentation The system is substantially instrumented through strain gauge bridges like a classic split Hopkinson bar, except that the measurement stations are doubled in order to measure both extensional and torsional waves. The signals of interest are acquired in the pre-stressed bar, at approximately half of the input bar, at approximately half of the output bar, and finally close to the sample location. All 8 strain gauge rosettes are installed in full bridge configuration. The power supply and acquisition of the strain gauge signals are performed through the HBM Gen 2tB system, which allows the simultaneous sampling of all the channels of interest at 500 kHz with a resolution of 16 bit.

First Tests Void Test To evaluate the general behavior of the system and especially of the bar connections, considered the most critical part, a void test was first conducted, without a sample and without connection between input and output bars. In addition, this test was conducted to accurately measure the traveling speed of the real waves, which is essential for achieving the synchronous arrival of torsion and tension loads on the sample. The procedure consisted in applying slowly the axial load on the preloaded bar up to 25 kN, and then the torque load was slowly applied up to 160 Nm until the failure of the sacrificial element. The signals recorded by the strain gauges in the input bar and converted into axial load and torque are shown in Fig. 4. By knowing the exact position of the strain gauges, it was possible to measure the Cε and Cγ, which resulted to be 4980 and 3110 m/s, respectively. In addition, some features can be observed in the load and torque signals. First, the first incident waves appear to have a reasonably rectangular shape. Small peaks due to discontinuities among the bars are visible; they can be easily filtered out, if necessary. The incident torsional wave is characterized by a small precursor of the opposite sign. This represents a small issue which will be solved by improving the contacts in the static block systems. By integration of the strain gauge signals, the axial displacement and the rotation can be measured at the strain gauge location. It is found that the first wave’s transit determined 7 mm displacement and 80° rotation, approximately. Considering that the applied axial and torsional preloads were about 1/4 and 1/2 of the design maximum, respectively, it is argued that the maximum available stroke at the end of the input bar is about 55 mm and 320°.

Test on Aluminum Sample This paragraph shows the first test carried out on AA 7075-T651 sample. The geometry of the sample is shown in Fig. 5. The sample is hollow in order to accommodate the ends of the input and output bars, which are in contact so as to allow the transit of the incident compression wave without deforming the sample itself. The inner diameter is 12.1 mm, while the outer diameter is 14 mm; the constant section length is 12 mm.

Fig. 4 Acquired signals in the first void conducted using only the input bar

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Fig. 5 Hollow sample

Fig. 6 Acquired signals during the test on AA 7075 T651 sample

This test was actually conducted with the original 21 m long output bar. In this way, the tensile wave is observed to come with a little delay with respect to the torsion load; in future tests, the output bar will be cut to a final length of 19.8 m. Nevertheless, useful information has been recorded for determining the stress-strain curve of the material up to failure. The applied static preload was 23 kN and 190 Nm. Figure 6 shows the signals acquired from strain gauges; for the sake of clarity, the signals are time-shifted (without dispersion correction) to the sample location. It is observed that, while the incident axial strain wave had the typical rectangular shape, the torsional incident wave was characterized by a very smooth ramp and by a precursor of the opposite sign, as already shown in Fig. 4. Both these undesired effects are due to an imperfect interaction at the torsional static block. In addition, the tensile loading wave arrived at the sample approximately 600 μs after the torsional one. The temporal evolutions of stress and strain, as computed by Eq. (1) are

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Fig. 7 (a) Temporal evolution of strain (a) and stress (b); reconstructed axial and shear stress-strain curves; (c) equivalent strain-stress curve (d)

Fig. 8 Sample after the test

shown in Fig. 7a and Fig. 7b, respectively. It is noted that the material is close to yielding when the shear stress approaches 230 MPa; then the axial stress jumped in, determining a drop in the torsional load. Nevertheless, reasonable axial and shear stress-strain behaviors are achieved, as shown in Fig. 7c; the resulting equivalent stress-strain is reconstructed in Fig. 7d. The fracture occurred approximately at axial strain εx = 0.1 and shear angle γ xy = 0.20. Figure 8 shows the sample after the test.

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Conclusion The paper describes an innovative system for simultaneous tension and torsion testing at high strain rate. The system is of the pre-tensioned type, where the incident waves generation exploits the failure of a brittle sacrificial element. According to the illustrated working principle, the different speeds of longitudinal and torsional waves are compensated by forcing the former to travel for a longer path length. The design requirements led to a huge overall length of about 87 m. The precise length of the output bar, which is necessary for the simultaneous arrival of the waves at the sample location, was determined by means of a void test, which permitted to measure independently the waves’ speeds. Even if some initial issues were experienced, especially with the static block system which alters the shape of the incident torsion wave, the wave propagation and recording appear to be acceptable. One test was conducted on a hollow AA7075-T651 sample. The failure occurred at an axial strain and a shear angle of 0.1 and 0.2, respectively; the average equivalent strain rate was in the order of 100 1/s. Acknowledgements Financed by the European Union-NextGenerationEU (National Sustainable Mobility Center CN00000023, Italian Ministry of University and Research Decree n. 1033-17/06/2022, Spoke 11- Innovative Materials & Lightweighting), and National Recovery and Resilience Plan (NRRP), Mission 04 Component 2 Investment 1.5-NextGenerationEU, Call for tender n. 3277 dated 30 December 2021. The opinions expressed are those of the authors only and should not be considered representative of the European Union or the European Commission’s official position. Neither the European Union nor the European Commission can be held responsible for them.

References 1. Song, B., Chen, W.: Split Hopkinson Kolsky Bar: Design, Testing and Applications. Springer (2010) 2. Gama, B., Lopatnikov, S., Gillespie Jr., J.: Hopkinson bar experimental technique: a critical review. Appl. Mech. Rev., 223–250 (2004) 3. Yu, X., Chen, L., Fang, Q., Jiang, X., Zhou, Y.: A review of the torsional Split Hopkinson Bar. Advances in Civil Engineering. 2018, 2719741 (2018) 4. Roth, C., Mohr, D.: Effect of strain rate on ductile fracture initiation in advanced high strength steel sheets: experiments and modeling. Int. J. Plast. 56, 19–44 (2014) 5. Bai, Y., Wierzbicki, T.: A new model of metal plasticity and fracture with pressure and lode dependence. Int. J. Plast. 24(6), 1071–1096 (2008) 6. Driemeier, L., Brunig, M., Micheli, G., Alves, M.: Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mech. Mater. 42(2), 207–217 (2010) 7. Roth, C., Mohr, D.: Determining the strain to fracture for simple shear for a wide range of sheet metals. Int. J. Mech. Sci. 149, 224–240 (2018) 8. Cortese, L., Nalli, F., Rossi, M.: A nonlinear model for ductile damage accumulation under multiaxial non-proportional loading. Int. J. Plast. 85, 77–92 (2016) 9. Cortis, G., Nalli, F., Sasso, M., Cortese, L., Mancini, E.: Effects of temperature and strain rate on the ductility of an API X65 grade steel. Appl. Sci. 12, 2444 (2022) 10. Cadoni, E., Dotta, M., Forni, D., Riganti, G., Albertini, C.: First application of the 3D-MHB on dynamic compressive behavior of UHPC. Europ. Physic. J. WoC. 94, 01031 (2015) 11. Xu, Y., Farbaniec, L., Pellegrino, A., Siviour, C., Eakins, D.: The development of Split Hopkinson tension-torsion Bar for the understanding of complex stress states at high rate. Dynamic Behav. Mater. 1, 89–93 (2021) 12. Mancini, E., Sasso, M., Rossi, M., Chiappini, G., Newaz, G., Amodio, D.: Design of an Innovative System for wave generation in direct Tension-Compression Split Hopkinson Bar. J. Dynamic Behav. Mater. 1, 201–213 (2015)

Rate Dependence of Penetration Resistance in a Cohesive Soil Sophia R. Mercurio, Stephan Bless, Abdelaziz Ads, Mehdi Omidvar, and Magued Iskander

Abstract Penetration resistance of soils is described in a Poncelet framework in which it is due to the sum of an inertial drag term and a strength-related bearing stress. The problem addressed in this paper is how to determine the Poncelet bearing stress from a conventional cone-penetrometer test (CPT). For the case of a particular saturated sandy clay, it is shown that strength has an exponential dependence on strain rate, determined from unconfined compression experiments, which can be used to account for the difference in strength between a CPT carried out at 0.02 m/s and the strength resisting projectiles traveling impacting at 140–200 m/s. With that correction, the CPT tip resistance can be taken as the Poncelet strength. Keywords Clay · Poncelet · Penetration · Rate effects

Introduction Penetration of intruders into soils has been a subject of scientistic and engineering research for quite a long time. Recent reviews can be found in Ref. [1–4]. The motivation for our recent research has been to predict the depth of burial of unexploded ordnance (UXO) in former defense installations. In that context rigid projectiles penetrating clayey materials and other soft soils have been of primary interest. The experiments reported herein on a sandy clay were designed with that in mind. Penetration is analyzed in a Poncelet framework. The penetration resistance, defined as the force with which the soil resists penetration divided by the penetrator area, is computed from the measured deceleration of the projectile, according to the formula dV ρACV 2 - AR=M =M dt

ð1Þ

where V is velocity, ρ is density, A is cross-section area (which varies during nose embedment), C is the Poncelet drag, M is mass, and R is the Poncelet penetration resistance. For quasi-static penetration, R is the total penetration resistance. Penetration resistance is thus expressed as the sum of a term proportional to velocity squared, due to inertia, and a term independent, or nearly so, of velocity, due to strength. Crucial to the project is understanding how Poncelet penetration resistance varies with strain rate. The strain rate during penetration is proportional to V/D, where V is penetration velocity and D is projectile diameter at the base of the nose. There are many treatments of how Poncelet penetration resistance is related to strength [2, 4, 5], but in all cases it is proportional to shear strength. Therefore, understanding how shear strength varies with strain rate should allow one to estimate penetration resistance at high velocity (such as a UXO penetration) from penetration resistance at relatively low speeds.

S. R. Mercurio · S. Bless (✉) · M. Iskander Department of Civil and Urban Engineering, Tandon School of Engineering at New York University, Brooklyn, NY, USA A. Ads Department of Civil and Urban Engineering, Tandon School of Engineering at New York University, Brooklyn, NY, USA Department of Structural Engineering, Tanta University, Tanta, Egypt M. Omidvar Department of Civil Engineering and Environmental Engineering, Manhattan College, Riverdale, NY, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_4

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Penetration resistance at low rates can be measured in situ by means of a cone penetrometer tests (CPT). There are several commercially available platforms, a number of operators who can perform this test, and well-established protocols described in ASTM D5778-20 [6]. In order to understand the connection between shear strength, penetration resistance at high speeds, and penetration resistance at low speeds, we have separately measured compressive strength as a function of strain rate by means of unconfined compression tests.

Experiments Description of Soil Targets Wet clayey sand targets were prepared for impact testing. The soil material comprised of 70% Ottawa 50–80 Sand, a fine silica with rounded grains having a particle diameter ranging between 0.3 mm and 0.15 mm, and 30% Kaolin Clay, a white, low-plasticity fine-grained material. The water content (defined as the ratio of weight of water to that of solids) was 11%. Saturation was close to but not equal to 100%. The nominal density and moisture content were 2.05 g/cm3 and 11%, respectively. Clayey sand targets were housed in 16-gal commercial drums (355-mm diameter) and prepared to a depth of 623 mm. Static compaction of hydrated soil was performed in controlled lifts at a rate of 1 mm/min, using a large hydraulic MTS press in 100-mm-thick loose layers which corresponded to 75-mm-thick compacted layers. Specimens of the same material for compression testing were compacted in 38-mm diameter tubes at a rate of 1 mm/min. The sandy-clay material was compressed in molds. The soil was sufficiently cohesive that preparation of unconfined compression samples was possible. The prepared density was the same for both the large drums and compression-test samples. The density was uniform through all specimens to a precision of 2.05 ± 0.05 g/cm3.

High-Speed Penetration Experiments A vertically-firing air gun was used to shoot cone-cylinder projectiles into the clay targets. The main instrumentation was a PDV system that measured projectile deceleration by reflecting a laser from its rear surface. The launch apparatus and PDV was described in Grace et al. [7]. The projectiles were 34.5-g cone cylinders. Cone half angles were 30 degrees, and shank diameters were 14.2 mm. Projectiles were made from 6061 T6 aluminum. Trajectories were vertical and angles of attack were nominally zero. The measured data in each test are the velocity history, V(t), and the final penetration, DOB (depth of burial). The data were analyzed in the context of Eq. (1), within which values of C and R were varied to match the dV/dt data and also to match the observed final depth of burial.

CPT Experiments A commercial Vertek HT 10 cm2 cone was obtained and adapted to a MTS hydraulic frame. The MTS load capacity was 1000 kN. The speed of advance was 20 mm/s. The cone diameter was 35.7 mm. Hence the nominal strain rate, V/D, was 0.52 per second. Stroke limit was 150 mm, which usually necessitated stopping once per test to insert a new coupling section. The transducers on the cone allowed for decomposition of the penetration resistance into the tip stress, qt, and the lateral stress, qf. Figure 1 is a photograph of the instrumented cone tip.

Compression Testing Compression testing was carried out with a Geotac Sigma1 Automated Load Test System. Compressive strength is twice the shear strength in this geometry. Clay samples were 38 mm diameter and 76 mm high. With this machine, strain rates up to 10/s could be achieved. The top platen was also 38 mm, as is the norm for compression of soil samples. The bottom platen was 50 mm. Conventional unconfined compression tests were carried out per ASTM [8].

Rate Dependence of Penetration Resistance in a Cohesive Soil

25

Fig. 1 Instrumented cone tip

Table 1 Penetration experiments into 30% clay 11% water content soils

Test number 32 67

Impact velocity (m/s) 202 140

Depth of burial (mm) 450 275

Experimental Results Dynamic Penetration There were two experiments with the material under discussion, test #32 and #67. The parameters for these tests are given in Table 1. Listed are the impact velocities and depths of burial, which is the final penetration depth. The V(t) records from these two experiments agreed at corresponding times after impacts to within the precision of the data (Fig. 2). The integrated V(z) data are in Fig. 3. In a Poncelet fit, at high-velocity C is very tightly constrained by the experimental V(t) data, but as velocity decays, the penetration resistance becomes dominated by R. The combination of matching the V(t) or V (z) data and also the measured penetration depth tightly constrains values of both C and R. The embedment phase as discussed in [7] was also included. The data and analysis for test #67 fit are shown in Fig. 3. The value of R was very narrowly defined by the observed value of ultimate penetration. The data from test #32 have been similarly analyzed with a PDV curve and DOB fit. The result using data from both experiments is that 0.23 < C < 0.24 and 3.0 < R < 4.0 MPa. The value of R defines the Poncelet strength. The range of strain rates varies, defined as V/D, ranges from 104/s to near zero. However, as can be surmised from Fig. 2, the crucial range of strain rates for R is really in the range around 103/s, where V = 50–20 m/s.

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200

PDV 67

180

PDV 32

160

Velocity (m/s)

140 120 100 80 60 40 20 0 0.E+00

5.E-04

1.E-03

2.E-03

2.E-03

3.E-03

3.E-03

4.E-03

Time (s) Fig. 2 Velocity-time records for two penetration tests

Fig. 3 Velocity-penetration for tests #67 and #32

CPT Penetration CPT measurements of tip stress are presented in Fig. 4, which depicts a relatively uniform tip resistance throughout the specimen consistent with uniform density. There is a region in which the tip stress increases with depth, beyond which it is relatively constant. The relative constant region below 200 mm is what should be compared to the stress in the ballistic experiments. In interpreting these data, we take the value of qt as constant below 200 mm, which is approximately 5 cone diameters. In this region, the average value of penetration resistance is 1250 kPa. The interpretation is that the strength below five diameters represents a deep failure mechanism, uninfluenced by the free surface. This follows the interpretation normally applied to measurements of bearing stress, in which shallow failure and deep failure are distinguished [9].

Rate Dependence of Penetration Resistance in a Cohesive Soil

27

Tip stress (kPa)

Fig. 4 CPT tip stress for sandy clay

0

500

1000

1500

2000

0

0.1

Depth, (m)

0.2

0.3

0.4

0.5

0.6

700 600

Stress,(kPa)

500 400 300 200 100

Strain

0 0

0.05

0.1

0.15

0.2

0.25

Strain rate 10/s

Strain rate 1/S

Strain rate 0.01/S

Strain rate 0.001/S

Strain rate 0.0001A/s

Strain rate 0.1A/s

0.3

Fig. 5 Engineering stress vs strain for unconfined compression tests on sandy clay at various strain rates

Compression Tests The results from unconfined compression tests on the clay-sand mixture soil at various strain rates are shown in Fig. 5. All of the samples had a shear failure mode as shown in Fig. 6. Normally in tests like this one would plot true stress. We believe that would be misleading here for two reasons. (1) The upper platen is the same diameter as the sample, which is the convention in geotechnical testing, where the unconfined test is seen as a member of a family of confined compression tests and the sample

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Fig. 6 Typical final deformation of a sample. Sample shown is at one at 1/s

and platen are the same size. (2) In most of these tests ultimately most of the deformation took place at the bottom platen, as shown in Fig. 6. Load must be continuous, absent inertial effects. So, the load divided by the upper platen diameter is the best measure of stress in the upper portion of the sample. Hence that section of the material is at least that strong. It is apparent from the recovered samples that there has been plastic deformation in this region of the specimen, and hence we take the peak engineering stress as the unconfined compressive strength of the sample. Strain is non-uniform in a test like this, of course; hence, the displacement of the platens is not readily translated into axial strain. It has been well documented that the undrained shear strength of clay soils is strain rate dependent. In general, about 20% increase in strength can be assumed per increase in log cycle of strain rate [10–12]. Two formulations are often used to estimate the shear strength given strain rate dependencies, given reference parameters. Typically, these correlations are valid for up to 3 orders of magnitude change in strain rate [13, 14]. These rate functions, Rf, serve to scale the bearing capacity static strength as follows:

Su = Suref :Rf

ð2Þ

One relationship is semilogarithmic: Rf = 1 þ λ log

E_ E_ ref

ð3Þ

The other is exponential: Rf =

E_

E_ ref

β

ð4Þ

where Suref is strength at a reference strain, E_ and E_ ref are the strain rates for the event and reference values, and β and λ are strain rate parameters. We take the reference rate as 0.01/s. Analysis of the rate factor Rf is shown in Fig. 7. At low strain rates, the strength is likely controlled by drainage effects because loading is slow, and permeability does control the failure mechanism during undrained loading. There appears to be a change in the deformation mechanism, and it is only after a rate of 0.01/s that the standard rate hardening rules, Eqs. (3) and (4) are followed. That is probably because at lower rates the failure emphasizes a single dominant slip plane, rather than bulk failure. In order to evaluate Eq. (3), data are plotted on a semilog plot, and for Eq. (4), on a log-log plot. Confined to the data available from the tests, both Eqs. (3) and (4) provide reasonable fits. When the data in Fig. 5 are analyzed using Eqs. (3) and (4), the result depends slightly on the reference strain. Using 20% strain, as in Fig. 5, λ = 0.363, or β = 0.107.

Rate Dependence of Penetration Resistance in a Cohesive Soil

29

2.5

(A)

Su/Su,ref

2

1.5 y = 0.3631x + 0.9527 R² = 0.9854 1

0.5

0 0

0.5

1

1.5

Log (

2 ./ .

2.5

3

3.5

)

0.350

(B)

log(Su/Su,ref)

0.300 0.250

y = 0.1078x - 0.003 R² = 0.9897

0.200 0.150 0.100 0.050 0.000 0

0.5

1

1.5

Log (

2 ./ .

2.5

3

3.5

)

Fig. 7 Rate effects in a clayey sand. (a) Logarithmic (Eq. 3) factor lamda (λ = 0.363) (Top), (b) Exponential (Eq. 4) factor Beta (β = 0.107) (bottom). Note rate hardening follows a logarithmic or exponential relationship at high strain rates

Connecting CPT Stress to Dynamic Penetration Resistance By using the rate correction equation for strength, the quasi-static bearing strength obtained from CPTs was correlated to the dynamic resistance realized by the PDV measurements obtained for the dynamic impact tests by adjusting for the difference in strain rates. A good approximation can be obtained by considering that the strain rate as the projectile slows to a stop in the ballistic tests is about 1000/s, then: 1 þ λ logð1000Þ R = = 2:30, using equation ð3Þ qt 1 þ λ logð0:56Þ 1000 R = 0:56 qt

β

= 2:24, using equation ð4Þ

ð5Þ

ð6Þ

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0

Stress (kPa) 2500 5000 7500 10000

0

0

Stress (kPa) 2500 5000 7500 10000

1E+04 9E+03

0.05 8E+03 7E+03 Strain Rate (1/s)

Depth (m)

0.1

0.15

0.2

6E+03 5E+03 4E+03 3E+03

0.25 2E+03 0.3

1E+03 R( =0.108)

R(C=0.24)

Drag stress

R( =0.363)

R = 3500

Fig. 8 Variation of Poncelet terms for test 67: resistance (blue) and drag stress (orange) from deceleration measurements using equation 1 and assuming C=0.24 or as the coefficient R as a constant (red point) to the predict DOB from v(p), as well as strength predictions for resistance from CPT measurements using equations 3 (gray) and 4 (yellow)

A more nuanced treatment is to consider that the strain rate decreases during penetration. R can be estimated by solving Eq. (1) for R, and assuming C is constant and equal to 0.24. This is shown in Fig. 8, which also includes the C-term, assuming C = 0.24, and a point at the DOB and the best-fit value of R as a constant. The value of R computed this way is noisy for several reasons. One is that especially at high velocity, the drag term is almost equal to the penetration resistance, so R is computed from the small difference between two much larger quantities. Only below a depth of about 200 mm does the value of R computed this way have high confidence. There is in fact noise in the V(t) data, and it is greatly magnified when the derivative, dV/dt, is computed. The noise comes in part because of the finite windows over which frequency spectra are computed, and in part perhaps because the velocity record is not smooth because of stress waves in the projectile and of heterogeneity in the soil medium. Nevertheless, it appears the R increases during penetration instead of decreasing as would be expected if it were dominated by rate. As the projectile comes to rest, R computed this way is very close to the average value of R obtained by fitting the deceleration and DoB data. That is because in the Poncelet formulation the DoB is very sensitive to R. In other words, when Eq. (1) is used with a constant R, that best fit value of R can be predicted from CPT experiments if account is taken of the strain rate difference between the CPT and a low-speed penetration.

Rate Dependence of Penetration Resistance in a Cohesive Soil

31

Conclusions In conclusion, the power law form of rate hardening better predicts the Poncelet penetration resistance from the CPT data than the logarithmic form. In other words, the CPT resistance qt can be identified with the Poncelet resistance R with a rate correction based on Eq. 4. Acknowledgments This work was supported by the Strategic Environmental Research Development Project (SERDP) Project No: MR19-1277.

References 1. Iskander, M., Bless, S., Omidvar, M.: Rapid Penetration into Granular Media. Elsevier (2015) 2. Omidvar, M., Bless, S., Iskander, M.: Recent insights into penetration of sand and similar granular materials, 137–163 (2019). https://doi.org/ 10.1007/978-3-030-23002-9_5 3. Glößner, C., Moser, S., Külls, R., et al.: Instrumented projectile penetration testing of granular materials. Exp. Mech. 57, 261–272 (2017). https://doi.org/10.1007/s11340-016-0228-0 4. Bragov, Balandin, V.V., Igumnov, L., et al.: Impact and penetration of cylindrical bodies into dry and water-saturated sand. Int J Impact Eng. 122, 197–208 (2018). https://doi.org/10.1016/j.ijimpeng.2018.08.012 5. Forrestal, M.J., Luk, V.K.: Penetration into soil targets. Int J Impact Eng. 12, 427–444 (1992) 6. ASTM D5778-07: Standard Test Method for Electronic Friction Cone and Piezocone Penetration Testing of Soils. ASTM Int (2020) 7. Grace, D., Mercurio, S., Omidvar, M., et al.: A vertical ballistics range with photon doppler velocimeter instrumentation for projectile penetration testing in soils. In: Dynamic Behavior of Materials, Volume 1: Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics, pp. 93–100. Springer (2022) 8. ASTM D2166-06: Standard test methods for unconfined compressive strength of cohesive soils. Annu B ASTM Stand. 4, 163–167 (2006) 9. Vesic, A.B.: Bearing Capacity of Deep Foundations in Sand. Highw Res Rec (1963) 10. Einav, I., Randolph, M.F.: Combining upper bound and strain path methods for evaluating penetration resistance. Int. J. Numer. Methods Eng. 63, 1991–2016 (2005). https://doi.org/10.1002/nme.1350 11. Kim, Y.H., Hossain, M.S., Wang, D.: Effect of strain rate and strain softening on embedment depth of a torpedo anchor in clay. Ocean Eng. 108, 704–715 (2015). https://doi.org/10.1016/j.oceaneng.2015.07.067 12. Teh, C.I., Houlsby, G.T.: An analytical study of the cone penetration test in clay. Geotechnique. 41, 17–34 (1991) 13. O’Beirne, C., O’Loughlin, C.D., Gaudin, C.: Assessing the penetration resistance acting on a dynamically installed anchor in normally consolidated and overconsolidated clay. Can. Geotech. J. 54, 1–17 (2017). https://doi.org/10.1139/cgj-2016-0111 14. Suescun-Florez, E., Omidvarm, M., Iskander, M., Bless, S.: Review of high strain rate testing of granular soils. Geotech. Test. J. 38, 511–536 (2015). https://doi.org/10.1520/GTJ20140267

Characterization of Shale Structure Subjected to Two Different Loading Rate Conditions Achyuth Thumbalam Guthai, Ali F. Fahem, Kyle R. Messer, and Raman P. Singh

Abstract Hydraulic fracturing is an important technique to produce oil and gas through the stimulation of unconventional hydrocarbon reservoirs, typically shale deposits. Quantifying elastic properties of bi-modulus shale under static and dynamic loading in different orientations is consequential. The in-situ bulk modulus of Wolfcamp shale is tested at two different loading rates. First, a universal testing machine was used for a quasi-static uniaxial compression test of a Brazilian disk, and then SHPB was utilized for a dynamic compressive load. Far-field loading data and local full-field deformation are used to calculate the physical properties through the linear elastic model. Static and dynamic mechanical properties of shales, such as elastic bulk modulus, compressive and tensile strength, were analyzed in two bedding orientations. The full experimental results are shown in detail, and in general, the elastic properties of shale under parallel and normal bedding conditions are different, as well as the static and dynamic in-situ elastic modulus are different. Keywords Shale · SHPB · Quasi-static · Dynamic · Bi-modulus · In-situ bulk elastic modulus · Brazilian disk

Introduction Understanding mechanical response and quantifying properties of rock are significant in numerous disciplines such as geomechanics, geophysics, hydraulic fracturing, mining, civil engineering, geology, petroleum exploration, and reservoir engineering. In all these fields, different rock properties are of interest for process optimization. For instance, elasticity and mechanical properties of the rock are important in geophysics and geomechanics, characterization of kerogen and organic matter in shale gas exploration and finally microfractures and brittleness in hydraulic fracturing [1]. Different techniques have been employed to measure these properties of interest [2]. Shale is a clastic sedimentary rock consisting of many layers of constituents generally oriented parallel to horizontal plane deposited over a large time scale. Consequently, they exhibit anisotropy in stiffness parameters and strength due to the one-dimensional nature of sediment compaction, particle reorientation under stress rotation, depositional environment, and mineral fabric [3, 4]. This special case of anisotropy is generally termed transversely isotropic response where each layer’s response is isotropic. Hence, elastic properties and strength of shale are generally different within the bedding plane and normal.

A. Thumbalam Guthai Department of Mechanical and Aerospace Engineering, Stillwater, OK, USA e-mail: [email protected] A. F. Fahem Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA Department of Mechanical Engineering, College of Engineering University of Al-Qadisiyah, Al-Diwaniyah, Iraq K. R. Messer Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA R. P. Singh (✉) Department of Mechanical and Aerospace Engineering, Stillwater, OK, USA Department of Mechanical and Aerospace Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Stillwater, OK, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_5

33

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Bond rupture is a rate process and fractures in all materials are rate-sensitive. Thus, the rate effect on the mechanical response of materials is very important for practical applications in mining, geotechnical engineering, and geology [5]. Most practical applications like tunneling and fracking involve dynamic fracturing of rocks, hence, understanding dynamic strength and elastic properties of rocks is crucial. Dynamic bulk modulus for shale has been estimated using ultrasonic and sonic log velocity measurements [6, 7]. They reported an increase in dynamic bulk modulus with confining pressure and is linearly related to volume fraction of organic matter and volume of clay. Studies exploring rate-dependent response in different directions are missing. In this work, quasistatic and dynamic elastic bulk modulus, tensile and compressive strengths within and perpendicular to the bedding at different loading rates are studied using two different methods described in detail in the next section followed by the methodology implemented to determine these parameters. Finally, results and conclusions are presented.

Experimental Approach Material and Specimen Preparation The material investigated in this work was obtained from Delaware Basin that is a part of Permian Basin in New Mexico, USA and procured from Kocurek Industries [8]. Wolfcamp shale was provided as cylindrical rods cored parallel to the bedding with D = 25.4 mm diameter and L = 50.8 mm in length. Brazilian disk specimens were cut to final dimensions of d ≈ 25.4 mm in diameter and the thickness is t ≈ 5 mm, i.e., t/D = 0.2 according to the ASTMD3967 [9].

Quasi-Static Experimental Setup The mechanical and elastic properties of Wolfcamp shale were studied under Quasi-static condition in two loading orientations (90° and 0°), perpendicular and parallel to the bedding planes. These specimens were subjected to diametral compression in a screw-driven mechanical test frame (Instron Model 5567, Norwood, MA, USA) under displacement control with a cross-head speed of 50 μm/min resulting in a strain rate of 3.28 × 10-5 s-1. The load P, was monitored using a 30 kN load cell. Load vs displacement data was recorded at a time step of 200 μs [10]. The cross-head displacement v was also recorded but not used for calculation of material properties. During loading the specimen was held in a custom-built frame, as shown in Fig. 1. The lower jaw was held fixed at the bottom platen of the mechanical test frame while the upper jaw moves with the upper platen. The jaws were machined from high-strength steel in accordance with the standardized procedure suggested by ISRM (1978) [11] in which the ratio of the disk’s radius to that of the jaw is 0.67 to provide a curved contact to minimize compressive damage at the point of loading.

Dynamic Experimental Setup A standard Split Hopkinson Pressure Bar (SHPB) system [12–14] was employed to subject Wolfcamp shale to longitudinal elastic compressive stress waves which works on the principle of 1D-elastic wave propagation theory. A typical SHPB consists of a striker, incident, and a transmitted bar. The bars were made of 25.4 mm diameter high-strength aluminum 7075T651 and supported on a horizontal plane. The Brazilian disk specimen was sandwiched between the bars (incident and transmitted) as shown in Fig. 2 and the assembly was used to load the specimen at the strain rate of around 1500 s-1. Linear strain gauges were used to measure the strain waves of incident and transmitted bars (quarter Wheatstone bridge). In equilibrium conditions, and no inertia effect, dynamic forces on the incident end (P1(t)) and the transmitted end (P2(t)) of the specimen are calculated from Eq. 1: P1 ðt Þ = Ab Eb ½εi ðt Þ þ εr ðt Þ]; and P2 ðt Þ = Ab E b εt ðt Þ

ð1Þ

The average effective load, Pef f on the specimen is given by Eq. 2: Pef f ðt Þ = ½P1 ðt Þ þ P2 ðt Þ]=2

ð2Þ

where Ab is the cross-section area of the bar; Eb is the elastic modulus of the bar; εi(t),εr(t), and εt(t) are the incident, reflected, and transmitted strain waves respectively.

Characterization of Shale Structure Subjected to Two Different Loading Rate Conditions

35

Fig. 1 Quasi-static experimental setup showing the camera located perpendicular to the specimen, light, jaws fixture with the circular disk specimen mounted in the test frame to measure load and full-field strain

Methodology Load determined from quasi-static and dynamic loading configurations as described in Sects. 2.2 and 2.3 combined with specimen dimensions were used as input to the Muskhelishvili Eqs. 3 and 4 to calculate full-field stress, tensile and compressive strengths within and perpendicular to the bedding planes [17]. In a Brazilian disk test, a biaxial stress state is evolved and σ zz and ezz are approximately zero. Digital image correlation (DIC) method was used to measure full-field deformation and strain in both quasi-static and dynamic experiments, Figs. 1 and 2. The details of the specimen preparation, testing procedure, and camera setup for DIC measurement can be found in [18–22]. Stresses and strains from the disk center were used with Eqs. 5, 6, and 7 to evaluate in-situ elastic bulk modulus of shale at two different loading rates and configurations. Longitudinal direction refers to bedding parallel and transverse to bedding normal in this study.

σ xx =

2P πt

σ yy =

2P πt

ðr - yÞx2 x2 þ ð r - yÞ 2

ð r - yÞ3 x2 þ ð r - yÞ 2

þ 2

þ 2

ðr þ yÞx2 x2 þ ðr þ yÞ2

2

-

1 2r

ð3Þ

2

-

1 2r

ð4Þ

ð r þ yÞ 3 x2 þ ðr þ yÞ2

σ=K * Δ

ð5Þ

σxx þ σyy þ σzz 3

ð6Þ

Δ = exx þ eyy þ ezz

ð7Þ

where x and y are the Cartesian coordinates, P is the radial compressive load, t, r are the thickness and radius of the disk, Δ is the volumetric strain, K is the bulk or compressibility modulus, and ¯p is the hydrostatic stress.

36

A. Thumbalam Guthai et al.

Fig. 2 A schematic of the standard split Hopkinson pressure bar (SHPB) and specimen orientation setup [15, 16]

Results and Discussion Tensile and compressive strengths of Wolfcamp shale in the longitudinal and transverse directions at two different loading rates can be observed in Table 1. As expected, compressive strengths in the longitudinal as well as transverse directions are ðcÞ ðt Þ ð cÞ ðt Þ ðcÞ ðt Þ higher than tensile strength at both quasi-static and dynamic loading rates, σT = 3 × σT , σL = 3 × σL and σT,dyn = 3 × σT,dyn , ð cÞ

ðt Þ

σL,dyn = 3 × σL,dyn . This is mainly because rocks are stronger in compression due to the presence of flaws, micro and macro cracks resulting in weaker response in tension. Also, for this shale, strengths are similar at quasi-static loading rate in both material directions, indicative of an isotropic response in strength. In a microscale level, the response of constituents are similar in both material directions and out of scope for this study. However, at dynamic loading rate, material displays slightly higher strength in 90° loading configuration, indicating a transition to anisotropic strength response. This may be due to

Characterization of Shale Structure Subjected to Two Different Loading Rate Conditions

37

Table 1 Quasi-static and dynamic tensile and compressive strength of Wolfcamp shale in parallel and normal bedding directions Material tested

Wolfcamp

Q-s tensile strength (t) (t) σ T (MPa) σ L (MPa) 13.27 12.8

Q-s compressive strength (c) (c) σ L (MPa) σ T (MPa) 39.83 38.5

Dynamic tensile strength (t) (t) σ L (MPa) σ T (MPa) 48.97 53.93

Dynamic compressive strength (c) (c) σ L (MPa) σ T (MPa) 161.80 146.91

Fig. 3 Diametral compression load versus cross-head displacement in quasi-static setup and effective load vs time in dynamic setup of Wolfcamp shale at 90° and 0° orientations. (a) Typical test results from quasi-static testing setup at 90° and 0° orientation. (b) Typical test results from dynamic testing setup at 90° and 0° orientation

different responses of cracks and flaws at higher loading rates. Tensile and compressive strengths of the material within and ð cÞ ðcÞ ðt Þ ðt Þ perpendicular to the bedding increases to varying degrees with loading rate, σT,dyn = 3:81 × σT,q - s , σT,dyn = 3:86 × σT,q - s , ðt Þ

ðt Þ

ð cÞ

ð cÞ

σL,dyn = 3:82 × σL,q - s and σL,dyn = 4:06 × σL,q - s . This can be explained by different thermally activated fracture mechanisms, differences in stress concentration and redistribution mechanisms during static and dynamic failure of rocks and failure stress decreasing with increasing time. Also, at both q-s and dynamic loading rates tensile strengths are higher in transverse direction and compressive strengths are higher in longitudinal direction. In quasi-static case, though strengths are similar, 90° exhibits slightly higher stiffness. However, in dynamic case, 0° loading exhibits higher stiffness and can be observed in Fig. 3. Bulk modulus of Wolfcamp shale in the longitudinal and transverse directions at two different loading rates can be observed in Fig. 4. Wolfcamp shale exhibits transversely isotropic response in bulk-modulus elastic values. The data from the ∘ Þ boxes (dashed lines) are considered to identify the average bulk modulus of Wolfcamp shale, where K ðq90 - s = 18:32 ~ GPa, ð90∘ Þ

∘

ð0∘ Þ

K ðq0-Þs = 15:54 ~ GPa, K dyn = 26:84 ~ GPa, and finally, K dyn = 29:95 ~ GPa. The differences in bulk modulus values at different orientations at q-s rate can be attributed to bedding orientation with loading and micro, macro crack orientations and porosity at microstructure level and out of the scope of this study. Similarly, values of dynamic bulk modulus are dissimilar at different orientations. In both loading conditions, dynamic bulk modulus is significantly higher than the corresponding quasi∘ ð90∘ Þ Þ static values indicating rate-sensitivity and its influence on Wolfcamp shale elastic properties K dyn = 1:46 × K ðq90 - s and ð0∘ Þ

∘

K dyn = 1:92 × K ðq0-Þs .

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A. Thumbalam Guthai et al.

Fig. 4 Quasi-static and dynamic bulk modulus of Wolfcamp shale at 90° and 0° orientations. (a) Typical bulk modulus results from quasi-static testing setup at 90° and 0° orientation. (b) Typical bulk modulus results from dynamic testing setup at 90° and 0° orientation

Conclusion In this work, bulk modulus, tensile, and compressive strength of Wolfcamp shale were investigated experimentally at two different loading rates, quasi-static and dynamic and bedding directions, longitudinal and transverse. Quasi-static tests were conducted on a Brazilian disk using a universal testing machine and dynamic tests were performed by using a SHPB apparatus to generate a compressive stress wave and applied to the Brazilian disk. Full-field stress and strain measurements were performed and data from the center of the disk were used for analysis. The main conclusions are: (1) At two different loading rates and conditions, compressive strengths of Wolfcamp shale are higher than tensile strength by a magnitude of 3. (2) The compressive and tensile strengths of Wolfcamp shale are stress-rate sensitive to varying degrees. The strength increased by a factor of 3.81 in compression and 3.86 in tension in transverse direction, 4.06 in compression and 3.82 in tension in longitudinal direction, as a result of increasing the strain rate from 3.28 × 10-5 s-1 to 1500 s-1. (3) Dynamic bulk modulus is significantly higher than the corresponding quasi-static values indicating the influence of rate∘ ð90∘ Þ ð0∘ Þ Þ ð0∘ Þ sensitivity on Wolfcamp shale elastic properties K dyn = 1:46 × K qð90 - s and K dyn = 1:92 × K q - s . Acknowledgments This material is based upon work supported by the Department of Energy under Award Number DE-FE0031777. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

References 1. Javadpour, F., Moravvej Farshi, M., Amrein, M.: Atomic-force microscopy: a new tool for gas-shale characterization. J. Can. Pet. Technol. 51(04), 236–243 (2012) 2. Messer, K.R., Fahem, A.F., Guthai, A.T., Singh, R.P.: The experimental methods and elastic properties of shale bedding planes materials stateof-the-art review. Al-Qadisiyah J. Eng. Sci. 15(2) (2022) 3. Gautam, R., Wong, R.C.: Transversely isotropic stiffness parameters and their measurement in Colorado shale. Can. Geotech. J. 43(12), 1290–1305 (2006)

Characterization of Shale Structure Subjected to Two Different Loading Rate Conditions

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4. Wong, R.C., Schmitt, D.R., Collis, D., Gautam, R.: Inherent transversely isotropic elastic parameters of over-consolidated shale measured by ultrasonic waves and their comparison with static and acoustic in situ log measurements. J. Geophys. Eng. 5(1), 103–117 (2008) 5. Bazant, Z.P., Shang-Ping, B., Ravindra, G.: Fracture of rock: effect of loading rate. Eng. Fract. Mech. 45(3), 393–398 (1993) 6. Lo, T.-W., Coyner, K.B., Toksoz, M.N.: Experimental determination of elastic anisotropy of Berea sandstone, Chicopee shale, and Chelmsford granite. Geophysics. 51(1), 164–171 (1986) 7. Omovie, S.J., Castagna, J.P.: Relationships between dynamic elastic moduli in shale reservoirs. Energies. 13(22) (2020) 8. Kocurekindustries.com, “Samples of Wolfcamp Shale” 9. ASTM: D 3967: Standard Test Method for Splitting Tensile Strength of Intact Rock Core Specimens, pp. 1–3. ASTM International, Philadelphia, PA., no. Reapproved (1992) 10. Thumbalam Guthai, A., Fahem, A.F., Singh, R.P.: Use of full-field strain measurements to determine mechanical properties of shale under repeated cyclic loading. In: Beese, A., Berke, R.B., Pataky, G., Hutchens, S. (eds.) Fracture, Fatigue, Failure and Damage Evolution, vol. 3, pp. 67–72. Springer International Publishing, Cham (2023) 11. Komurlu, E., Kesimal, A.: Evaluation of indirect tensile strength of rocks using different types of jaws. Rock Mech. Rock. Eng. 48(4), 1723–1730 (2015) 12. Chen, W., Bo, S.: Split Hopkinson(Kolsky) Bar Design, Testing and Application. Springer (2011) 13. Messer, K.R., Guthai, A.T., Fahem, A.F., Singh, R.P.: Mixed-mode fracture interactions along centrally cracked weakened planes. In: Beese, A., Berke, R.B., Pataky, G., Hutchens, S. (eds.) Fracture, Fatigue, Failure and Damage Evolution, vol. 3, pp. 61–66. Springer International Publishing, Cham (2023) 14. Xia, K., Yao, W.: Dynamic rock tests using split hopkinson (kolsky) bar system – a review. J. Rock Mech. Geotech. Eng. 7(1), 27–59 (2015) 15. Fahem, A., Singh, R.: “Dynamic Damage Evolution in Shale in the Presence of Pre-existing Microcracks,” Dynamic Behavior of Materials. In: Conference Proceedings of the Society for Experimental Mechanics Series, vol. 1, pp. 39–45. Springer, Cham (2022) 16. Fahem, A.F., Guthai, A.T., Singh, R.P.: Experimental investigation of the nonlocal dynamic damage mechanism in shale. In: Mates, S., Eliasson, V., Allison, P. (eds.) Dynamic Behavior of Materials, Volume 1, pp. 57–61. Springer International Publishing, Cham (2023) 17. Muskhelishvili, N.I., et al.: Some Basic Problems of the Mathematical Theory of Elasticity, vol. 15. Noordhoff Groningen (1953) 18. Fahem, A., Tg, A., Singh, R.P.: “A Novel Method to Evaluate Elastic Properties of Heterogeneous , Orthotropic and Bi-Modulus Materials with Applications to Shale,” in preparation, pp. 1–32 (2020) 19. Fahem, A., Kidane, A., Sutton, M.A.: A novel method to determine the mixed mode (I/III) dynamic fracture initiation toughness of materials. Inernational Fracture Mechincs. 224(May), 47–65 (2020) 20. Fahem, A., Kidane, A.: A General Approach to Evaluate the Dynamic Fracture Toughness of Materials, vol. 1, pp. 185–194. Dynamic Behavior of Materials, Conference Proceedings of the Society for Experimental and Applied Mechanics (2017) 21. Fahem, A., Kidane, A., Sutton, M.A.: Mode-I dynamic fracture initiation toughness using torsion load. Eng. Fract. Mech. 213(May), 53–71 (2019) 22. Sutton, M.A., Orteu, J.J., Schreier, H.W.: Image Correlation for Shape, Motion and Deformation Measurements- Basic Concepts, Theory and Applications, p. 341. Image Rochester NY (2009)

In-Situ Mesoscale Characterization of Dynamic Crack Initiation and Propagation Using X-Ray Phase Contrast Imaging Andrew F. T. Leong, Bryan Zuanetti, Milovan Zecevic, Kyle J. Ramos, Cindy A. Bolme, Christopher S. Meredith, John L. Barber, Marc J. Cawkwell, Brendt E. Wohlberg, Michael T. McCann, Todd C. Hufnagel, Pawel M. Kozlowski, and David S. Montgomery

Abstract Predicting and controlling the failure of brittle materials against impacts have important applications in defense, mining, and medicine. To that end, the key is understanding at the mesoscale the events of crack initiation, propagation, branching, multiple crack interactions, and coalescence. Therefore, we developed an x-ray phase contrast imaging-based technique to directly visualize and quantify the cracking process. We chose to test our technique on single-crystal quartz because it has well-defined material and mechanical properties which computational models can use to accurately simulate the cracking process. Also, quartz serves as an ideal model material to developing experimental techniques/analysis and highfidelity damage models for energetic materials and heterogenous geomaterials. To achieve the micron and nanosecond resolution required to resolve and track cracks in real-time, we use the high brilliance, spatially coherent synchrotron source at the Dynamic Compression Sector (Advanced Photon Source, Argonne National Laboratory) and the 8-frame LANL/DCS detector system coupled to a 150-μm thick single crystal LYSO scintillator. Quartz samples are uniaxially compressed at 103– 104 s-1 strain rates with a custom-built Kolsky bar and stress-strain histories are measured using PDV probes. To characterize the evolving crack morphology, a physics-based inverse model is developed that converts the phase contrast-enhanced image intensity of the cracks into crack volume orientation distributions inside the sample. Using this model, we study how sample surface finish affects the dynamic behavior of cracks. Keywords Synchrotron · Ultrafast · Phase Contrast · Crack · Quartz

Introduction Mesoscale structural defects including interfaces, dislocations, pores, cracks, and interstitials connect atomic, molecular, and nanoscale origins of macroscopic behavior. In the last century, there has been growing consensus that mesoscale defects have a large influence on the macroscopic behavior of materials under extreme conditions [1]. By building an understanding connecting the atomic to the nanoscale level, it may soon become possible to construct innovative architectures of mesoscale A. F. T. Leong (✉) · P. M. Kozlowski · D. S. Montgomery Physics Division, Thermonuclear Plasma Physics (P-4), Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] B. Zuanetti · C. A. Bolme Dynamic Experiments Division, Shock and Detonation Physics Group (M-9), Los Alamos National Laboratory, Los Alamos, NM, USA M. Zecevic Theoretical Division, Fluid Dynamics and Solid Mechanics (T-3), Los Alamos National Laboratory, Los Alamos, NM, USA K. J. Ramos Weapon Stockpile Modernization Division, High Explosives Science and Technology (Q-5), Los Alamos National Laboratory, Los Alamos, NM, USA C. S. Meredith Soldier Protection Sciences Branch, DEVCOM Army Research Laboratory, Aberdeen Proving Ground, Adelphi, MD, USA J. L. Barber · M. J. Cawkwell Theoretical Division, Physics and Chemistry of Materials (T-1), Los Alamos National Laboratory, Los Alamos, NM, USA B. E. Wohlberg · M. T. McCann Theoretical Division, Applied Mathematics and Plasma Physics (T-5), Los Alamos National Laboratory, Los Alamos, NM, USA T. C. Hufnagel Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, MD, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_6

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Fig. 1 Schematic of experimental setup showing x-rays illuminating a crack recorded downfield onto the detector that is parallel to the xy-plane. * *0 The laboratory coordinates system is given by r = ðx, y, zÞ and that of the crack is given by r = ðx0 , y0 , z0 Þ

defects that can enhance material performances and functionality needed to, for example, generate clean energy [2, 3], explore space [4, 5], and understand planetary formation and evolution [6, 7]. In this work, we employed propagation-based x-ray phase contrast imaging (XPCI) to directly visualize and quantify crack growth evolution (damage) of single crystal quartz (SXQ) under high strain rate uniaxial compression. Recently, XPCI has combined with the highly brilliant, coherent synchrotron x-rays and high-speed cameras to achieve the spatial resolution and image contrast necessary to visualize cracks and track their behavior in real-time [8, 9]. The popularity of propagation-based XPCI over other XPCI imaging modalities stems from its simple setup that consists of a spatially coherent x-ray source, sample, and detector system arranged in-line (Fig. 1). The process by which images are formed begins when spatially coherent x-rays are generated and directed toward the sample. The x-rays absorb and scatter in different amounts depending on the composition and density of the sample. Upon exiting the sample, the modulated x-rays are recorded by the detector positioned some distance away from the sample. Cracks are filled with air surrounded by quartz. Consequently, differential absorption of x-rays between the crack and its surrounding makes the crack appear as a 3D projection of its shape on the 2D detector plane. Whereas x-rays scattered from the cracks and their surroundings constructively and destructively interfere with each other. For a single crack, this produces bright and dark fringes around the image of the crack boundary. This effectively produces an edge enhancement that sharpens the appearance of the cracks. These two mechanisms combined provide sufficient visibility of cracks to overcome the high amount of image noise and artifacts associated with the photonstarved conditions created by high-speed imaging. Interpreting and quantifying damage from single XPCI images of SXQ is complicated by having many cracks projected onto a 2D plane. It is difficult to delineate individual cracks and study their behavior. Instead, we developed a physics-based XPCI statistical crack model to quantify the total damage accumulated and help constrain and validate micromechanical damage-based.

XPCI Crack Model To derive our physics-based XPCI crack model relating the total volume of cracks (i.e., damage) to its XPCI image, we first model a crack as a rectangular cuboid embedded in an absorbing medium of uniform thickness T along z * * V r ¼ 1 everywhere , G r : *

x0 y0 z0 rect rect , a b c

*

G r = V r - rect

ð1Þ

where

rectðt Þ =

0 1

*

1 2: 1 if jt j ≤ 2 if jt j >

ð2Þ

The laboratory coordinate system is defined by r = (x, y, z) where the x-ray beam propagates along z and the detector plane *0 *0 is parallel to (x, y). The crack has dimensions (a, b, c) = (width, length, thickness) along r = (x′, y′, z′), respectively. r * represents the directions normal to the surface of the crack, and is related to r via the three-dimensional rotation matrix:

In-Situ Mesoscale Characterization of Dynamic Crack Initiation and Propagation Using X-Ray Phase Contrast Imaging

x′ y′ z′ 1

cos β cos γ cos β sin γ = - sin β 0

sin α sin β cos γ - cos α sin γ sin α sin β sin γ þ cos α cos γ cos β sin γ 0

cos α sin β cos γ þ sin α sin γ - sin α cos γ þ cos α sin β sin γ cos α sin β 0

43

0 0 0 1

x y z 1

ð3Þ

where (α, β, γ) are rotation angles around positive axes (x, y, z), respectively. The wavefield immediately after the crack (z = 0) is given by: z=0 *

ψ ðx, y, 0Þ = exp - ðμ þ ikδÞ

ð4Þ

G r dz z= -T

where the complex refractive index is n = 1 - δ + iμ, where δ is the refractive index decrement and μ is the attenuation coefficient and the XPCI image recorded at z = L is given by *

*

I r ⊥ , z = L = F - 1 PF ψ r ⊥ , z = 0

2

ð5Þ

,

where P is the free-space propagator [10]. We will simplify Eq. (5) by making and justifying the following three approximations. First approximation: During dynamic loading of SXQ, cracks typically nucleate and grow rapidly in two directions relative to the third direction. Consequently, they resemble thin rectangular plates with length (b) and thickness (c) 100’s of microns and width (a) only a few microns. As a consequence, in the power spectrum of Eq. (1), most of the signal along the y′ and z′ is concentrated near the zero-frequency term such that they can be approximated to follow Dirac Delta distributions like functions centered at zero-frequency. Second approximation: Since the crack morphology resembles thin rectangular plates, its XPCI image would be highly *0 * dependent on its pose. To demonstrate this, consider a 258 × 0.3 × 20 μm3 crack initially oriented along r = r . That is, the x′ z′-plane is parallel to the xz plane and β = 0. Then, if we rotate the crack around y (i.e., increase β) and calculate the area under the power spectrum of its simulated XPCI image (PSArea), we find that, after adding Gaussian noise with a standard deviation measured from recorded XPCI images of SXQ, the crack is only visible over a small angular range (~2∘) (Fig. 2(a)). This angular range would increase with the crack thickness. However, early in the dynamic failure of SXQ, most cracks have a thickness at most of only a couple microns. Consequently, XPCI images are only sensitive to cracks whose x′y′-plane is close to parallel with the xy-plane. Third approximation: The peak x-ray energy that we used in our synchrotron experiment was ~24 keV. At this energy, x-rays weakly scatter from SXQ and can therefore approximate x-rays to be mostly forward scattering. Consequently, the projection approximation is assumed [10]. After applying all three approximations to Eq. (5), we arrive at *

I r = expð- μT Þ j expðikLÞ + abðexpð½0:5μ + ikδ]cÞ - 1Þℑ - 1 Psincð0:5kx aÞj2

ð6Þ

where the wavevector k = 2πλ is inversely proportional to the x-ray wavelength, λ. We generalize Eq. (1) to describe a spatial distribution of cracks, considering only those with x′y′-crack planes parallel to * the xy-plane, GN( r ): *

180=Δγ ηðγ j Þ

GN r =

*

V r - rect j=0

i=1

x 0j y 0j z 0j rect rect aji bji cji

*

*

⊗ δ r - r ji

ð7Þ

*

where r ji is the position of the ith crack that is oriented γ j = jΔγ from the positive x-axis. Δγ is the fixed angular increment in γ and j = 0, 1, 2, . . ., 180/Δγ, N =

180=Δγ j=0

η γ j and ⊗ is the convolution operator. If the cracks are assumed to not overlap in

projection, then the area under the power spectrum of its XPCI image is given by [11]:

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Fig. 2 (a) PSArea vs β angle of a 258 × 0.3 × 20 μm3 crack with respect to the x-ray direction. Inset shows XPCI images of a simulated crack at β = 0°, 2° and 4° from the x-ray direction. PSArea calculated using Eq. (8) and is shown to be proportional to the crack (b) length (width fixed at 0.1 μm and thickness fixed at 1 mm), (c) width (length fixed at 1 mm and thickness fixed at 1 mm) and (d) thickness (length fixed at 1 mm and width fixed at 0.1 μm). For (d), PSArea oscillates with crack thickness. Often, however, multiple cracks with a distribution of thicknesses are analyzed at a time in an XPCI image. Their crack thicknesses collectively is proportional to PSArea. This is represented by the orange curve in (d) that is calculated by smoothing the blue curve in (d) using robust linear regression over a 0.1 mm window. Hence, damage is proportional to PSArea

PSArea ðγ Þ = expð - 2μT Þ

ηðγ Þ i=1

ℑ expðikLÞ + abðexpð½0:5μ + ikδ]cÞ - 1Þℑ - 1 Psincð0:5kx aÞ

2 2

dk r

ð8Þ

kr > 0

where kr = kx sin γ + ky cos γ. Equation (8) shows that cracks oriented at an angle γ only contribute to the power spectrum at angle γ. It also shows that the variation in PSArea with crack length, width and thickness are plotted in Fig. 2 (b-d) shows that damage is proportional to PSArea. Consequently, we can study the directions cracks (damage) preferentially grow. But to *0 * reiterate, our crack model is only sensitive to cracks oriented along r = r and therefore the total damage of the sample is not known.

Experimental Method and Image Processing Dynamic XPCI experiments were performed in the Dynamic Compression Sector (Sector 35) at the Advanced Photon Source, Argonne National Laboratory. SXQ samples were cut into 1 × 1.1 × 1.2 mm3 cubes along the crystallographic planes 1 10 , (558), and (111) by Boston Piezo-Optics, Inc. The surfaces were polished using fine lapping to reach different degrees of surface roughness. These were prepared to investigate the effects of surface roughness on SXQ fracture behavior. SXQ samples were uniaxially compressed with our LANL-built desktop-size Kolsky bar [12] (for details, see Zuanetti et al. (2024) Conference proceedings of SEM). Photon Doppler velocimetry (PDV) measured the particle velocity of the incident and transmitted bars before processing and converting the signal into stress-strain history curves of the SXQ samples. XPCI images of the sample failure were captured with single pulse x-rays arriving every 153 ns and recorded by an 8-frame LANL/ DCS detector system coupled to a 150-μm thick single crystal LYSO scintillator [13]. The sample-to-scintillator propagation

In-Situ Mesoscale Characterization of Dynamic Crack Initiation and Propagation Using X-Ray Phase Contrast Imaging

45

Fig. 3 XPCI images showing dynamic compression of a 1.0 × 1.1 × 1.2 mm3 SXQ 13 with a 7 μm surface finish on the 110 and (110) surface planes. Damage is quantified in each direction for each image. Radial limits = [0 3e-6]

distance was fixed at 320 mm and the pixel size was measured to be 2.58 μm. We performed post-image alignment correction with a 51 μm pitch copper grid (Ted Pella, Inc.) to align each camera to the first camera. Each camera PSF was measured using a JIMA RT RC-05B resolution target plate (NDT Supply, Inc.). The PSF between cameras was slightly different but insignificant to affect quantitative comparisons of the images. However, we plan to account for the PSF in future works to improve quantitative accuracy.

Results and Discussion Two SXQ samples with different surface roughness were dynamically compressed and imaged. Figure 3 presents a sequence of XPCI images recorded depicting an SXQ sample dynamically compressed on its 1 10 plane with a surface finish of 7 μm (SXQ 13). Similarly, Fig. 4 shows XPCI images of a dynamically compressed SXQ sample with a surface finish of 3 μm (SXQ 10) on the 1 10 plane. The time stamp on each image represents the time elapsed since the stress wave arrived at the incident bar-interface. Below each image is a polar plot where the plotted line represents the damage accumulated in the sample as a function of the azimuthal angle (γ). That is, the polar plot indicates the crystallographic planes on which cracks are propagating. The stress-strain and strain-rate-strain curves are plotted in Fig. 5 with vertical lines indicating the recording time for each frame in Figs. 3 and 4. SXQ 13 reached a peak stress of ~4.3 GPa just before t6 = 4.695 μs. However, cracks began to nucleate from the incident bar-sample interface along the 111 plane before that at t3 = 2.391 μs. The peak stress was marked by cracks growing along the (110) plane and thereafter in multiple directions as it underwent catastrophic failure and granular flow. SXQ 10 reached a peak stress of ~3.6 GPa between t3 = 3.685 μm and t4 = 4.451 μm. Similarly to SXQ 13, cracks was already nucleating from the incident bar-sample interface along the 111 plane starting at t3 = 3.685 μs. Unlike SXQ 13, however, failure of SXQ 10 seems to have been initiated from cracking at the top right corner of the sample ~ t4 = 4.451 μs. This may have arised from a pre-crack introduced during sample polishing or the sample was not sufficiently cut square to prevent a stress concentration at that top right-hand corner of the sample. Regardless, subsequent cracking at the top righthand corner of the sample may have led to shear cracking at approximately along the 094 plane.

46

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Fig. 4 XPCI images showing dynamic compression of a 1.0 × 1.1 × 1.2 mm3 SXQ 10 with a 3 μm surface finish on the 110 and (110) surface planes. Damage is quantified in each direction for each image. Radial limits = [0 3e-6]

6

40 t2

t3 t0

5

t4 t1

t5 t2

t6 t3

t7 t4

t5

t6

t7

35

Stress [GPa]

4

25 Stress (SXQ 13)

3

Stress (SXQ 10) SR (SXQ 13) SR (SXQ 10)

2

20 15 10

1

0

5

0

0.02

0.04

0.06

0.08

Strain

0.1

Fig. 5 Stress vs. Strain and Strain-rate (SR) vs. Strain plots corresponding to Figs. 3 and 4

0.12

0.14

0

Strain-rate [×103 s-1]

30

In-Situ Mesoscale Characterization of Dynamic Crack Initiation and Propagation Using X-Ray Phase Contrast Imaging

47

Conclusions and Future Works We studied the dynamic failure of SXQ at strain rates of the order of 104 s-1 using a custom-built Kolsky bar. Specifically, we focused on using XPCI to understand how surface roughness could potentially affect its macroscopic properties on the mesoscale by tracking and quantifying microcracks growth during loading. We found that the SXQ sample with a 7 μm surface roughness on the loading surface (3 μm surface finish on the remaining 4 planes) had a higher peak stress than that with a 3 μm surface roughness on all surfaces. This is converse to what is expected since a higher surface roughness should introduce larger stress concentrations and therefore fail at a lower loading stress. However, the XPCI images indicate that there may have been a large pre-crack in the sample. This highlights the sensitivity of brittle material dynamic behavior to its initial geometry. In the future, better care will be taken to prepare and pre-characterize samples to ensure only the macroscopic property of interest of the sample is varied and studied. Nevertheless, we demonstrated that the measured damage from our crack model agreed at least qualitatively to the volume (damage) and direction of cracks observed in the XPCI images. We plan to quantitatively validate this model on simulated and experimental XPCI images of cracks with known sizes. Also, further investigation will be performed to determine if the crack size distribution can be recovered from XPCI images given that the image intensity is also dependent on the crack size. This will help in modeling crack kinematics for predicting the behavior of more complex geomaterial materials such as sandstone and granite. Acknowledgments AFTL acknowledges funding from Laboratory Directed Research & Development (LDRD 20200744PRD1). The authors acknowledge the funding of the CHE Grand Challenge project through Weapons Systems Safety Analysis at Los Alamos National Laboratory. This publication is based upon work performed at the Dynamic Compression Sector, which is operated by Washington State University under the U.S. Department of Energy (DOE)/National Nuclear Security Administration award no. DE-NA0003957. This research used resources of the Advanced Photon Source, a DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract no. DE-AC02-06CH11357.

References 1. Hemminger, J., Crabtree, G., Sarrao, J.: From Quanta to the Continuum: Opportunities for Mesoscale Science, pp. ii–77. A Report from the Basic Energy Sciences Advisory Committee (2012). https://doi.org/10.2172/1183982 2. Murphy, T.J., et al.: Progress in the development of the MARBLE platform for studying thermonuclear burn in the presence of heterogeneous mix on OMEGA and the National Ignition Facility. J. Phys. Conf. Ser. 717(1), 012072 (2016). https://doi.org/10.1088/1742-6596/717/1/012072 3. Olson, R.E., et al.: A polar direct drive liquid deuterium-tritium wetted foam target concept for inertial confinement fusion. Phys. Plasmas. 28(12), 122704 (2021). https://doi.org/10.1063/5.0062590 4. Coakley, J., et al.: Femtosecond quantification of void evolution during rapid material failure. Sci. Adv. 6(51), 4434–4450 (2020). https://doi. org/10.1126/sciadv.abb4434 5. Sui, H., Yu, L., Liu, W., Liu, Y., Cheng, Y., Duan, H.: Theoretical models of void nucleation and growth for ductile metals under dynamic loading: A review. In: Matter and Radiation at Extremes, vol. 7, p. 018201. China Academy of Engineering PhysicsCAEP (2022). https://doi. org/10.1063/5.0064557 6. Hawreliak, J., Erskine, D., Schropp, A., Galtier, E.C., Heimann, P.: Using Phase Contrast Imaging to Measure the Properties of Shock Compressed Aerogel, vol. 1793, p. 090006. AIP Conference Proceedings (2017). https://doi.org/10.1063/1.4971625 7. Iskander, M., Bless, S., Omidvar, M.: Rapid Penetration into Granular Media: Visualizing the Fundamental Physics of Rapid Earth Penetration. Elsevier Inc. (2015) 8. Leong, A.F.T., Robinson A.K., Fezzaa, K., Sun, T., Sinclair, N., Casem, D.T., Lambert, P.K., Hustedt, C.J., Daphalapurkar, N.P., Ramesh, K.T., Hugnagel, T.C.: Quantitative in situ studies of dynamic fracture in brittle solids using dynamic x-ray phase contrast imaging. Exp. Mech. 58, 1423–1437 (2018). https://doi.org/10.1007/s11340-018-0414-3 9. Feng, Z.D., Zhou, Y.H., Tan, R., Hou, H.M., Sun, T., Fezzaa, K., Huang, J.Y., Luo, S.N.: Dynamic damage and fracture of a conductive glass under high-rate compression: A synchrotron based study. J. Non-Cryst. Solids. 494, 40–49 (2018). https://doi.org/10.1016/j.jnoncrysol.2018.04. 030 10. Paganin, D.: Coherent X-Ray Optics. Oxford University Press (2006) 11. Leong, A.F.T., Asare, E., Rex, R., Xiao, X.H., Ramesh, K.T., Hufnagel, T.C.: Determination of size distributions of non-spherical pores or particles from single x-ray phase contrast images. Opt. Express. 27(12), 17322 (2019). https://doi.org/10.1364/OE.27.017322 12. Zuanetti, B., Wang, T., Prakash, V.: A compact fiber optics-based heterodyne combined normal and transverse displacement interferometer. Rev. Sci. Instrum. 88(3) (2017). https://doi.org/10.1063/1.4978340 13. Jensen, B.J., et al.: Impact system for ultrafast synchrotron experiments. Rev. Sci. Instrum. 84(1), 013904 (2013). https://doi.org/10.1063/1. 4774389

Nose Shape Effects from Projectile Impact and Deep Penetration in Dry Sand J. Dinotte, L. Giacomo, S. Bless, M. Iskander, and M. Omidvar

Abstract The goal of this study was to demonstrate the role of nose shape on the embedment phase of long rod projectiles penetrating densely packed sand targets. Conical nose projectiles having various apex angles were launched into sand targets at an impact velocity of approximately 200 m/s. A vertical firing range was designed and calibrated for launching projectiles with a diameter of 14.3 mm. Aluminum rod projectiles with conical nose apex angles ranging from 30 to 180° were tested. Soil targets were prepared by means of dry pluviation. Velocity-time histories were resolved using a photon Doppler velocimeter (PDV). High-fidelity velocity-time histories were obtained through measurement of the frequency shift in the laser light wave reflecting from the back of the projectile. Highly reflective retroreflective tape was applied to the back of the projectiles to enhance the intensity of the reflected light. Optical probes were used to collect the reflected light. The results of the experiments revealed that both the magnitude of the peak deceleration and the depth corresponding to peak deceleration were a function of the nose shape. A decrease in the apex angle of a conical projectile led to a reduction in the peak deceleration, with the resulting peak stress occurring later in penetration for projectiles with smaller apex angles. Upon impact on the soil target surface, a near conical-shaped mass of compacted comminuted sand formed on the nose of the projectiles tested. These observations can be used to improve predictions of projectile depth of burial in phenomenological modeling of penetration and depth of burial. Keywords PDV · Sand · Penetration · Nose shape

Introduction Projectile penetration into granular soils has a wide range of scientific and engineering applications spanning from large-scale planetary impact studies to defense and military applications [1, 2]. Numerous experiments of projectile penetration in soil targets have been reported in the literature. Data from these studies have aided the development of numerical and analytical models for prediction of the depth of burial of projectiles in soil targets. The experiments reported in this study were motivated by the need to develop robust models to accurately predict the terminal penetration depth of unexploded ordnance (UxO) at Formerly Used Defense Sites (FUDS). Penetration of a projectile into a soil target can be analytically described using Newton’s second law of motion. The sum of the forces acting on a projectile during soil penetration can be written as a second-order polynomial composed of an inertial resistance term, a viscous resistance term, and a velocity-independent frictional bearing resistance term [3–5]. The resulting equation of equilibrium is expressed as: m

dv = mg - αv2 - βv - γ dt

ð1Þ

where α controls the inertial resistance, β controls the viscous resistance, γ is the bearing resistance with dimension of force, and g is the gravitational acceleration. In modeling high-velocity penetration into soil targets, the gravitational term and the viscous term in Eq. (1) are often neglected, as their contributions are considered to be negligible [2]. Additionally, it is commonly assumed that the bearing resistance term remains constant with penetration depth. The latter assumption is J. Dinotte · L. Giacomo · M. Omidvar (✉) Department of Civil and Environmental Engineering, Manhattan College, Riverdale, NY, USA e-mail: [email protected] S. Bless · M. Iskander Civil and Urban Engineering Department, Tandon School of Engineering, New York University, Brooklyn, NY, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_7

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inaccurate for granular soils. The quasi-static resistance to penetration relies on geostatic stresses, which increases with depth. Moreover, natural soil is typically stratified, resulting in depth-varying strength. Nevertheless, the simplifying assumption of a constant bearing resistance with depth allows for integration of Eq. (1) to obtain the well-known Poncelet equation describing penetration depth at time t following impact, P, as a function of penetration velocity, v, and can be expressed as follows: P=

R v20 þ Cρ m ln 2 R 2CρA v þ Cρ

ð2Þ

where v0 is the impact velocity of the projectile, m is the mass of the projectile, A is the projected cross-sectional area of the projectile, ρ is the density of the soil, C is the inertial drag coefficient, and R is the bearing resistance term with dimension of stress. The use of the Poncelet model requires the calibration of the C and R coefficients with high-fidelity experimental velocity-penetration data. Curve-fitting to such data for projectiles results in C and R as a function of projectile geometry and soil target properties. The cross-sectional area term, A, in Eq. (2) varies during the nose embedment phase into the soil target. In this study, several tests were carried out on densely packed soil targets using conical nose rod projectiles with various cone angles to determine C and R as a function of the nose apex angle.

Experimental Setup A state-of-the-art vertical ballistic range was developed to launch projectiles vertically into soil targets. The ballistic range was comprised of an electro-pneumatic gas launcher, a Photonic Doppler Velocimetry system, and a dry sand pluviator, as shown in Fig. 1. A labeled diagram of the ballistic range can be seen in Fig. 1. A high-speed camera was used to obtain footage of impact and embedment into the soil target surface.

Launcher

High Speed Camera

Oscilloscope

Soil Target Barrel PDV Setup Fig. 1 Ballistic range used to launch projectiles into sand targets

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Regulator Solenoid Valve

Chamber

Safety Clamp

Breech Mounting Frame

Barrel

Fig. 2 Components of the launcher

A single-stage electro-pneumatic gas launcher capable of firing projectiles at velocities up to 1000 m/s was used to conduct the ballistic experiments. The main components of the launcher are shown in Fig. 2. The launcher was operated by filling the pressure chamber with high-purity helium to a calibrated target pressure. The isentropic expansion of the pressurized helium gas into the barrel accelerates the projectile to a muzzle velocity, v., as follows: v=η

V0 2 P0 V 0 1V 0 þ Alb m γ-1

γ-1

þ 2glb

ð3Þ

where m is the mass, V0 is the reservoir volume, P0 is the reservoir pressure, γ is the specific heat ratio of heat capacity at constant pressure to heat capacity at constant volume (1.66 for helium), A is the cross-sectional area, lb is the length of the barrel, and g is the acceleration due to gravity [6]. For the experiments described in this study, a pressure of 70 bar was used to launch projectiles having a mass of 34.5 g at a muzzle velocity of approximately 200 m/s. The launcher was used to propel 14.3 mm diameter aluminum projectiles with conical nose apex angles of 30°, 60°, 90°, and 120°. A blunt-nose projectile was also tested. Note that increasing the nose sharpness also increased the projectile length-to-diameter ratio, l/D, from 5.34 in the blunt projectile to 6.82 in the 30° cone. The ballistic experiments were carried out in densely packed dry soil targets comprised of Ottawa sand passing the number 50 sieve and retained on the number 80 sieve. Dry sand pluviation was used to prepare all sand samples. The pluviator used to prepare the dense sand is shown in Fig. 3. The pluviator comprised a cylindrical hopper suspended from an aluminum frame by a pulley system. The hopper could be raised and lowered manually to achieve a desired pluviation height. The hopper was raised to a height of roughly 0.6 m to achieve densely packed soil targets with an average dry bulk density of 1.82 g/cm3 and uniform density distribution with depth, as shown in Fig. 3. Densities as low as 1.57 g/cm3 could be achieved, although not used herein. The pluviation height of 0.6 m was empirically determined to allow the falling sand to reach terminal velocity during pluviation, thereby ensuring uniform and repeatable density with depth within the soil target. The cylindrical sample chambers used for these tests had a diameter of 310 mm and a depth of 620 mm. To obtain the desired density, aluminum

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Hopper

Diffuser

Pulley Soil Target Barrel

(a)

(b)

Fig. 3 (a) Pluviator; (b) soil density

shutters with precise porosity were designed to control the rate of soil pluviation from the hopper. A shutter porosity of 1% was used for the dense sand samples used in the experiments reported in this study. Additionally, two wire screen diffusers were used to ensure the sand was adequately dispersed after release from the hopper. The two wire screen diffusers were oriented with a 45° offset from one another to achieve uniform dispersion of sand into the soil target. The velocity-time history of the projectile at impact and during penetration was obtained using photon Doppler velocimetry (PDV). PDV takes advantage of the Doppler shift that occurs when light is reflected from a moving object. Combining the original and the Doppler-shifted light wave produces a new waveform, known as the beat signal. The frequency of this beat signal can be used to calculate the instantaneous velocity of the moving projectile [7]. The PDV setup used in this study utilized a coherent laser that was aimed at the back of the projectile during penetration, referred to as the target laser, as well as a second reference laser light used to produce the beat signal. The target laser was operated at a wavelength, λ0, of 1550 nm, which corresponds to a frequency, f0, of 1.93548 × 105 GHz. An optical probe was used to aim the target laser at the back of the projectile during penetration. As the light is reflected from the projectile during its flight within the vision range of the laser probe, the laser light is Doppler-shifted, increasing its wavelength to a Doppler-shifted frequency, fD. The Doppler-shifted laser light reflected from the projectile was collected by the same optical probe. The light was then directed by an optical circulator into a coupler. The coupler combined the dopplershifted laser light and the reference laser light, creating a beat signal with a frequency, fB [8, 9]. The beat signal was then sent to an optical-to-electric converter, converting it to a digital signal, and sent to a fast oscilloscope (Tektronix MSO64 6-series). The velocity-time history of the projectile can then be calculated using the beat signal frequency according to the following equation: vðt Þ =

1 λ f ðt Þ 2 0 B

ð4Þ

The velocity-time data obtained following the above-described procedure were integrated and differentiated to produce penetration-time data and acceleration-time histories, respectively.

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Fig. 4 Velocity-time and penetration time histories for two identical tests

In recognition of the potential hazards inherent in performing ballistic experiments, appropriate safety measures were implemented during the operation of the launcher and PDV, as well as the preparation of soil samples. These measures encompass the adherence to a comprehensive standard operating procedure (SOP) and the adoption of personal protective equipment (PPE) when preparing samples and conducting ballistic experiments. In addition to these safety measures, the launcher used in this study was operated from a separate room adjacent to the laboratory.

Repeatability The PDV setup described in previous sections produced highly repeatable measurements of velocity-time data. Preliminary tests to establish repeatability were carried out in dry soil targets with a density of 1.82 g/cm3 using the dry sand pluviation technique described previously. A conical projectile with an apex angle of 60° was launched at a velocity of 200 m/s into the sand target. Both the target and reference laser were operated at a laser intensity of 10 mW. The reference laser was upshifted by 0.5 GHz from the target laser frequency. The oscilloscope was operated using a bandwidth of 1 GHz in both experiments. The velocity-time and penetration-time histories of the two experiments described above can be seen in Fig. 4. A comparison of the velocity-time and penetration-time data shows that the two experiments produced nearly identical results. The PDV signal was lost at a velocity of approximately 4 m/s in both tests. The postmortem measurements of the depth of burial (DoB) for the two projectiles were 143 mm and 140 mm, respectively. Postmortem measurements of tilt also yielded similar results; the projectile exhumed from test PDV 47 was tilted approximately 14° from the vertical, while PDV 49 was found tilted approximately 12°.

Results A set of ballistic impact and penetration tests were conducted on sand targets using the experimental setup outlined in the preceding sections. The purpose of these experiments was to examine the effect of nose shape on the penetration resistance of sandy targets. The parameters used in the experiments as well as the DoB measured from postmortem analysis of the soil targets are summarized in Table 1.

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Table 1 Soil target and projectile parameters used in the ballistic experiments Test ID PDV50 PDV49 PDV53 PDV54 PDV51

Soil density [g/cc] 1.82

Projectile nose geometry 30° cone 60° cone 90° cone 120° cone Blunt

Impact velocity, v0 [m/s] 201.4 202.7 203.1 203.5 199.6

Depth of burial [m] 0.175 0.140 0.140 0.144 0.145

The velocity-time, penetration-time, and acceleration-time data for the ballistic tests conducted in this study are shown in Fig. 5. Analysis of the velocity-time data shows that the shape of the projectile nose has limited influence on the velocity-time response in dense sand, except when the projectile nose apex angle is 30°. The velocity-time data were comparable for projectiles with a nose apex angle of 60° or higher. However, the projectile with a nose apex angle of 30° exhibited less velocity decay during the initial penetration. Despite an initial deviation in the velocity-time response, the velocity-time data for the 30° cone projectile converged with the other projectiles at a velocity of approximately 20 m/s. This suggests that the shape of the projectile nose does not significantly impact low-velocity penetration. Further analysis of the acceleration-time data reveals that the sharpness of the projectile nose has a significant impact on the magnitude of peak deceleration. Increasing the nose sharpness decreases the magnitude of peak deceleration and increases the time to reach peak deceleration. The blunt projectile reached its peak deceleration of 1.35 × 106 m/s2, that is, nearly instantaneously following impact on the soil target surface, whereas the 30° cone projectile took 0.16 ms to reach peak deceleration of 3.54 × 105 m/s2, which is a decrease in peak deceleration of approximately 74%. Analysis of the acceleration-penetration data in Fig. 5 indicates an increase in the depth at which peak deceleration occurs as the projectile nose apex angle decreases. The depths of peak deceleration were recorded for various projectile nose apex angles. A comparison of the depth at which peak deceleration occurred for a given projectile and the length of that projectile nose is shown in Fig. 6. It can be seen that a strong correlation exists between the depth of peak deceleration and the nose length. The linear fit in Fig. 6 suggests that the peak deceleration occurs at 1.15 the nose length for the projectiles tested. The results of curve-fitting Eq. (2) can be seen in Fig. 7, while the resulting best fit C and R values are shown in Table 2. The results indicate that the best-fit Poncelet predictions are insufficient in accurately predicting the velocity-penetration response for projectiles with decreasing nose apex angle. Both C and R are influenced by nose sharpness. Specifically, the best-fit C value is positively correlated with nose sharpness, while the best-fit R value is inversely correlated with nose sharpness. While the inertial drag coefficient may be dependent on nose sharpness, it is important to note that the bearing resistance term in the Poncelet equation is primarily affected by soil strength and, consequently, the penetration depth for coarse-grained soils. The observed pattern of the best-fit R values obtained by fitting the Poncelet equation to the data collected in this study does not support this understanding, as all projectiles except for the 30° cone projectile exhibit comparable terminal penetration depths. A possible explanation for the poor modeling of early penetration, as well as the patterns found in the best fit C and R values, may be the use of the projected cross-sectional area of the projectile. As the projectile nose penetrates the soil, the cross-sectional area of the projectile in contact with the soil changes until the maximum cross-sectional area is achieved. To predict the velocity-penetration response of conical projectiles penetrating dense sand more accurately, Eq. (1) was implemented in an incremental form with the cross-sectional area of the projectile defined as a function of the nose height. The resulting fits are shown in Fig. 8, with C and R reported in Table 2. It can be seen in Fig. 8 that the implementation of the incremental Poncelet fits substantially improves the predictions for the early velocity-penetration data. Additionally, the new best-fit R values show that the inverse correlation found when bestfitting the conventional poncelet equation disappears. The new best-fit R values indicate that nose sharpness has no effect on the best-fit R value. The new best-fit R values also corroborate the understanding that the bearing resistance term is dependent on penetration depth. A comparison of the best-fit C values indicates that nose sharpness has little to no effect on the drag coefficient for projectiles penetrating dense sand unless they are adequately sharp. This result can be attributed to the formation of a false nose for sufficiently blunt projectiles. Postmortem inspection of the blunt projectile, as well as the 120° and 90° conical

Nose Shape Effects from Projectile Impact and Deep Penetration in Dry Sand

(a)

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(b)

(c) Fig. 5 Results of ballistic tests: (a) velocity-time history; (b) penetration-time history; (c) acceleration-time history

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Fig. 6 Penetration depth corresponding to peak deceleration as a function of the conical nose height

projectiles, revealed a false nose that was affixed to the front face of the penetrators. Several studies have reported the formation of an attached cone on the projectile made of crushed soil during penetration into sand [2, 10, 11]. High pressures and high strain rates ahead of the projectile lead to particle comminution [12]. The false nose formed around the blunt and 120° projectiles was roughly conical with an apex angle of approximately 60°. This phenomenon was not observed in the tests conducted with the 60° and 30° cone projectiles, suggesting that the conical nose forms at approximately a threshold nose angle of 60°. Sharper projectiles do not lead to the formation of a false cone, and blunter projectiles form a near 60° false cone nose, which then renders the nose shape insignificant in determining the depth of burial.

Conclusion In this study, a vertically mounted ballistic launcher was employed to fire projectiles of varying nose sharpness into dense sand targets. The dense sand targets were prepared by dry pluviation, which resulted in an average soil density of 1.82 g/cm3. Photonic Doppler velocimetry was utilized to record the instantaneous velocity of the projectile during penetration. Analysis of the data revealed that the depth of burial of long rod projectiles penetrating densely packed dry sand is not influenced by the nose sharpness except when the nose of the projectile is sufficiently sharp. The examination of the acceleration data revealed that the highest deceleration experienced by the projectile is inversely related to the sharpness of the nose. Additionally, the point of peak deceleration can be predicted to occur at a penetration depth of approximately 1.15 times the projectile nose length. A penetration model based on an incremental implementation of Eq. (1) was successfully used to predict the experiments reported herein. A variable cross-section area was able to capture the embedment phase accurately. The resulting R values suggested a depth-dependence. Furthermore, it was found that the resulting C values were not reliant on the nose sharpness except if the nose is sufficiently sharp. This result was attributed to the formation of a false nose during penetration for sufficiently blunt projectiles.

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(a)

(b)

(c)

(d)

(e) Fig. 7 Predictions using Eq. (2) for (a) 30° conical projectile, (b) 60° conical projectile, (c) 90° conical projectile, (d) 120° conical projectile, (e) blunt projectile

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(a)

(b)

(c)

(d)

(e) Fig. 8 Best-fit incremental Poncelet predictions for (a) 30° conical projectile, (b) 60° conical projectile, (c) 90° conical projectile, (d) 120° conical projectile, (e) blunt projectile

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Table 2 Best fit C and R coefficients Shot ID PDV50 PDV49 PDV53 PDV54 PDV51

Conventional Poncelet parameters C R [MPa] 0.9 6.9 1.6 3.3 1.8 2.3 1.8 2.3 2.0 1.4

Incremental Poncelet parameters C R [MPa] 1.4 3.0 2.1 1.7 2.1 1.5 2.0 1.6 2.0 1.5

Acknowledgments The authors gratefully acknowledge the support of the Strategic Environmental Research and Development Program (SERDP) of the United States of America, Grant No: MR19-1277.

References 1. Ruiz-Suárez, J.C.: Penetration of projectiles into granular targets. Rep. Prog. Phys. 76(6), 066601 (2013) 2. Omidvar, M., Iskander, M., Bless, S.: Response of granular media to rapid penetration. Int. J. Impact Eng. 66, 60–82 (2014) 3. Omidvar, M., Malioche, J.D., Bless, S., Iskander, M.: Phenomenology of rapid projectile penetration into granular soils. Int. J. Impact Eng. 85, 146–160 (2015) 4. Iskander, M., Bless, S., Omidvar, M.: Rapid Penetration in Granular Media. Elsevier (2015) 5. Allen, W.A., Mayfield, E.B., Morrison, H.L.: Dynamics of a projectile penetrating sand. J. Appl. Phys. 28(3), 370–376 (1957) 6. Cave, A., Roslyakov, S., Iskander, M., Bless, S.: Design and performance of a laboratory pneumatic gun for soil ballistic applications. Exp. Tech. 40, 541–553 (2016) 7. Peden, R., Omidvar, M., Bless, S., Iskander, M.: Photonic Doppler velocimetry for study of rapid penetration into sand. Geotech. Test. J. 37(1), 20130037 (2014) 8. Dolan, D.H.: Extreme measurements with Photonic Doppler Velocimetry (PDV). Rev. Sci. Instrum. 91(5) (2020) 9. Strand, O.T., Goosman, D.R., Martinez, C., Whitworth, T.L., Kuhlow, W.W.: Compact system for high-speed velocimetry using heterodyne techniques. Rev. Sci. Instrum. 77(8) (2006) 10. Tanaka, K.: Phenomenological Studies of the Response of Granular and Geological Media to High-Speed Projectiles. Report no. AOARD104115. Asian Office of Aerospace Research and Development (2011) 11. Sharma, A., Penumadu, D., Glößner, C.: Projectile Penetration in Granular Material. AIP Conference Proceedings (2020) Vol. 2272. No. 1 12. Omidvar, M., Bless, S., Iskander, M.: Stress-strain behavior of sand at high strain rates. Int. J. Impact Eng. 49, 192–213 (2012)

A Novel Specimen Design for Multiaxial Loading Experiments at High Strain Rates Yuan Xu, Govind Gour, Julian Reed, and Antonio Pellegrino

Abstract A specimen geometry that has four flat dog bones circumferentially arranged around the axis of the sample is proposed for combined tensile-torsional loading experiments. Finite-element modelling was implemented to optimise the design and achieve appropriate deformation and failure in both tensile and torsional loading conditions. The capability of the proposed specimen configuration is demonstrated via an experimental campaign on commercially pure titanium at various strain rates. The quasi-static tests were conducted using a universal screw-driven testing machine, whereas the high-rate experiments were carried out on an in-house designed combined tension-torsion Hopkinson bar system. A wide range of stress states were obtained using the ligament specimen, covering uniaxial tension, shear, and different combinations of tension and shear. Three distinct failure modes of the ligament specimens subjected to monotonic tension, monotonic torsion, and combined tension-torsion loading at high strain rates are presented and discussed. The quasi-static and high-rate ultimate stress loci will be constructed using direct experimental measurements to assess the strain rate sensitivity of the material. Keywords Combined tension-torsion · Specimen design · Hopkinson bar · Rate dependence · Failure mode

Introduction Materials in many engineering applications are commonly subjected to complex loading that results in a triaxial stress state. Combined tension-torsion loading experiments at a strain rate ranging from 10-3 to 103 s-1 have been developed to simulate any arbitrary stress state in a laboratory environment [1–4]; however, the geometry of the specimen appropriate for both tensile and torsional loading remains challenging. Existing attempts in the literature mainly are solid cylinder [1] and the thinwalled tube [2, 3], while no international standard exists yet. This drives us to investigate the optimal geometry of a versatile specimen valid for both tensile and torsional experiments, which then can be applied to combined tension-torsion loading scenarios. As the standard tension-testing specimen suggested by ASTM and ISO, cylindrical dog bone [5, 6] is not appreciated in conventional torsional experiments due to the complex shear stress and strain across the section. Although analytical solutions and further modifications have been established to measure the engineering shear stress and strain of a solid cylinder [7–9], it is poorly understood from experiments if the strict assumptions can be satisfied. On the contrary, the thin-walled tubular specimen has been widely used in torsion experiments as uniform distribution of the shear stress can be approximated at the engineering level. However, thin-walled tubular specimens tend to buckle before undergoing large shear deformation. It might introduce unseen issues when used in uniaxial tension testing, such as stress concentration resulting from over-constraints near the gauge-fillet transition of the specimen. A novel specimen geometry that combines the advantage of the dog-bone specimen for tension testing and that of the thinwalled tube for torsion testing is proposed for the combined tension-torsion loading experiments. This geometry is designed to have four small dog-bone plates circumferentially arranged at the gauge section. The four dog bones connect two gripping shoulders, giving the name of the geometry the ‘ligament specimen’. Commercially pure titanium (CP Ti) is taken as an example material to investigate the capability of the ligament specimen in combined multiaxial loading at high strain rates. Quasi-static experiments were conducted to give a reference to the high-rate behaviour of the material. The progressive failure process of CP Ti is discussed when it is subjected to high-rate monotonic tension, monotonic torsion, and combined tensiontorsion loading. Y. Xu (✉) · G. Gour · J. Reed · A. Pellegrino Department of Engineering Science, University of Oxford, Oxford, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_8

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Fig. 1 Specimen geometry and dimensions: (a) ligament specimens; (b) cylindrical dog-bone specimens; and (c) tubular specimens

Methodology Three distinct geometries were employed in this study. The ligament specimen (Fig. 1a) has four flat dog bones circumferentially arranged at the gauge section, each with a gauge length of 1 mm, width of 1.5 mm, and thickness of 0.75 mm. The fillet radius has a varying dimension from 0.5 to 2.0 mm. The dog-bone specimen (Fig. 1b) has the standard dimensions, with the gauge length of 8 mm and diameter of 3 mm. The thin-walled tubular specimen (Fig. 1c) has a gauge length of 2 mm, thickness of 0.25 mm, fillet radius of 2 mm, and inner diameter of 14.5 mm. Quasi-static experiments were conducted in laboratory conditions using a Zwick/Roell Z250 screw-driven universal testing machine (Fig. 2). The loads were applied under displacement control at a speed on the order of 10-3 mm/s in tension and 0.02 degree/s in torsion. Four iDs1 UEye Cameras were synchronised to video-record the macroscopic deformation mechanism and to capture the failure initiation on the circumference of the specimens. The applied force and torque were recorded via the

1

iDS Image Development Systems GmbH.

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Fig. 2 Experimental setup of quasi-static tests

respective resistive load cells,2 while the history of the axial and shear strain within the gauge section of the specimen was measured from the video footage by means of digital image correlation (DIC) analysis. High-rate tensile, torsional, and combined loading experiments were carried out using a combined Tension-Torsion Hopkinson Bar (TTHB) system [10] that was designed and developed at the Impact Engineering Laboratory, University of Oxford. The TTHB system is schematically shown in Fig. 3a. The working principle, technical specification, as well as data interpretation of a TTHB experiment, can be found in our previous work [10–12]. The experimental setup and the data acquisition system employed in this study are presented in Fig. 3b. A set of strain gauges were attached to both the incident and transmitted bars to measure the longitudinal and shear stress waves propagating along the bars and through the specimen. Additionally, two synchronised Photron SA-5 high-speed cameras would be triggered by the incident wave signal and video record the high-speed deformation of the specimen. The high-rate variation of strain within the gauge section of the specimen was retrieved from the footage via digital image correlation (DIC) techniques. The history of the tensile and torsional loads was calculated using one-dimensional wave theory based on D’Alembert’s solution [13].

Analysis The deformation and fracturing of the CP Ti ligament specimen subjected to high-rate tension, torsion, and combined tensiontorsion loading are presented in Figs. 4, 5 and 6, respectively. Four instants, including the original state, the ultimate point, the fracture initiation or fracture through, and the post fracture are selected to depict the progressive failure process. The full field of the longitudinal strain Exx and/or shear strain Exy obtained from DIC analysis are also shown on the ligament. In monotonic tension (Fig. 4), uniform distribution of the longitudinal strain Exx can be identified within the gauge section of the ligament at the ultimate point, at a value of around 3.8%. Subsequently, the ligament went through necking till the point of fracture initiation. Strain localisation took place around the middle of the ligament. The instant of post fracture indicates the monotonic tensile failure as an inclined snap through the ligament at the middle. It is worth noting that the fracture surface showed high roughness. In monotonic torsion (Fig. 5), the ligament deformed with nearly homogeneous distribution of shear strain Exy at the ultimate point, at a value of around 59%. The edge of the gauge section however barely showed deformation. Afterwards, the increasing torsion started to fracture the ligament near the fillet, while the specimen can still sustain the load, until the instant

2

Zwick force transducer 100 kN, Zwick orque transducer 1000 Nm.

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Fig. 3 Experimental setup of high-rate tests

Fig. 4 Progressive failure process in monotonic tension

Fig. 5 Progressive failure process in monotonic torsion

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Fig. 6 Progressive failure process in combined tension-torsion

of fracture through when the whole ligament snapped under shear. Post fracture indicates the monotonic torsional failure as a straight snap through the ligament near the fillet. Again, rough fracture surface can be identified. In combined tension-torsion loading (Fig. 6), the area of the uniform distribution of strains Exx and Exy narrowed down to a diagonal band across the gauge section of the ligament. Significant strain localisation can be observed since the instant of the ultimate point, and adiabatic shearing predominated the deformation of the ligament till the sharp fracture of the ligament. Post failure showed a clean diagonal snap through the gauge section.

Conclusion A ligament specimen is proposed to understand the multiaxial response of materials in highly dynamic loading environments. Three different failure modes of the ligament specimens subjected to monotonic tension, monotonic torsion, and combined tension-torsion loading at high strain rates are presented and discussed. Post-mortem examination will be implemented to assess the evolution of the microstructure before and after the high-rate combined tension-shear loading. Further work will also include assessing the ligament specimen design via numerical simulations, regarding the stress state of each ligament, stress concentration at the fillet radius, the evolution of stress triaxiality during plastic deformation, and eventually the validity of the ligament to represent the material behaviours. Acknowledgements The authors would like to thank Rolls-Royce plc and the EPSRC for the support under the Prosperity Partnership Grant \Cornerstone: Mechanical Engineering Science to Enable Aero Propulsion Futures, Grant Ref: EP/R004951/1.

References 1. Chen, H., Li, F., Zhou, S., Li, J., Zhao, C., Wan, Q.: Experimental study on pure titanium subjected to different combined tension and torsion deformation processes. Mater. Sci. Eng. A. 680, 278–290 (2017) 2. Faleskog, J., Barsoum, I.: Tension-torsion fracture experiments – part I: experiments and a procedure to evaluate the equivalent plastic strain. Int. J. Solids Struct. 50(25–26), 4241–4257 (2013) 3. Papasidero, J., Doquet, V., Mohr, D.: Determination of the effect of stress state on the onset of ductile fracture through tension-torsion experiments. Exp. Mech. 54, 137–151 (2014) 4. Xu, Y., Farbaniec, L., Siviour, C., Eakins, D., Pellegrino, A.: The development of split Hopkinson tension-torsion bar for the understanding of complex stress states at high rate. In: Lamberson, L., Mates, S., Eliasson, V. (eds.) Dynamic Behavior of Materials, vol. 1, pp. 89–93. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham (2021) 5. Standard Methods of Tension Testing of Metallic Materials. E 8, ASTM

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6. John, M. (Tim) Holt: Alpha Consultants and Engineering, Uniaxial Tension Testing, vol. 8, pp. 297–337. ASM Handbook, ASM International (1985) 7. Nadai, A.: Theory of Flow and Fracture of Solids, 3rd edn. McGraw-Hill, New York (1950) 8. Wu, H.C., Xu, Z., Wang, P.T.: The shear stress-strain curve determination from torsion test in the large strain range. J. Test. Eval. 20, 396–402 (1992) 9. Lu, W.Y., Jin, H., Foulk, J., et al.: Solid cylinder torsion for large shear deformation and failure of engineering materials. Exp. Mech. 61, 307–320 (2021) 10. Xu, Y., Zhou, J., Farbaniec, L., Pellegrino, A.: Optimal design, development and experimental analysis of a tension–torsion Hopkinson Bar for the understanding of complex impact loading scenarios. Exp. Mech. (2023). https://doi.org/10.1007/s11340-023-00942-1 11. Xu, Y., Aceves Lopez, M., Zhou, J., Farbaniec, L., Patsias, S., Macdougall, D., et al.: Experimental analysis of the multiaxial failure stress locus of commercially pure titanium at low and high rates of strain. Int. J. Impact Eng. 170, 104341 (2022) 12. Zhou, J., Xu, Y., Lopez, M.A., Farbaniec, L., Patsias, S., Macdougall, D., et al.: The mechanical response of commercially pure copper under multiaxial loading at low and high strain rates. Int. J. Mech. Sci. 224, 107340 (2022) 13. Kolsky, H.: Stress waves in solids. J. Sound Vib. 1, 88–110 (1964)

Investigation and Characterization of Dynamic Energy Absorbed by Shale Materials Ali F. Fahem, Achyuth Thumbalam Guthai, Kyle R. Messer, and Raman P. Singh

Abstract This work investigates the dynamic energy absorbed by a shale material, which is generally classified as transversely isotropic, as a function of bedding orientations using experimental and analytical analysis. Experimentally, circular disks or Brazilian specimens of shale are tested under a high-impact diametral compression load in the split Hopkinson pressure bar, SHPB. Analytically, fundamentals of uniaxial stress and elastic wave propagation theory are utilized to derive the dynamic energy absorbed by the material. Thus, the experimental output data is used as input to the analytical damage model. The model was applied to two different orientations of shale related to the bedding directions. The experimental setup and the analytical analysis are presented. In general, results display a significant difference in the total absorbed energy and peak values that are required for exciting and increasing the cumulative micro and macro cracks in shale. The average strain and local energy results are presented and discussed. Keywords Dynamic energy absorbed · Stress wave · SHPB · Shale · Transversely-isotropic

Introduction Shale is a class of energy or organic material since it is saturated by oil and gas. These hydrocarbons are currently and at least for the next 30 years, the world’s primary sources of energy. The formation of shale is mainly a result of the organic material deposited and transformed over millions of years. Thus, the mechanical properties are different and depend on the location and depth. These properties are of interest for a researcher as they are required to enhance the most state-of-the-art technology [1–4]. Damage mechanisms of shale at different length scales are one of the highly important physical phenomena that are being explored to understand, since it is related to many industries like carbon dioxide (CO2) stored underground, between the ground layers and cracks, and oil and gas extraction from the shale porosity and cavities [5]. Shales, in general, are treated as a transversely isotropic material since they are comprised of many layers of sediments that are made from a mixture of rocks and minerals and each layer’s response isotropic. Most of the damage models developed depend on the elastic properties of the

A. F. Fahem (✉) Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA Department of Mechanical Engineering, College of Engineering University of Al-Qadisiyah, Al-Diwaniyah, Iraq e-mail: [email protected] A. T. Guthai Department of Mechanical and Aerospace Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA K. R. Messer Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA R. P. Singh Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA Department of Mechanical and Aerospace Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_9

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Fig. 1 Shale specimen orientation and preparation

shale, and in general, these can work for a specific shale and can be considered a general model. In this work, the dynamic damage of shale subjected to compressive stress wave is investigated in two material directions, longitudinal and transverse. This work has four sections: Sect. 2 describes the details regarding the experimental setup and material used in this study. Section 3 explains the fundamental of strain energy applied to develop a new model for damage related to the impact velocity and duration. Also, the methodology includes relationship between the experimental data and the damage parameter. Finally, results and discussions in Sect. 4 with conclusions and recommendations in Sect. 5 are presented.

Experimental Setup A standard split Hopkinson pressure bar (SHPB) was used to test Oklahoma shale specimens under diametral dynamic compressive load [6]. Circular disk specimens of shale were tested in two different orientations, longitudinal 0° and transverse 90° direction which are related to the principal material direction and at room temperature, Fig. 1. This shale was collected from Oklahoma area, USA, and called Yost shale. The dimensions of shale that were tested with SHPB were 15 mm in diameter and 5 mm in thickness, i.e., t/D = 0.33 according to the ASTM-D3967 (Re-approved 2001) [7]. The outline of SHPB is presented for completeness. SHPB consists of two bars called input and output bars made from aluminum T-7075. These bars are supported on a horizontal plane using an I-Beam (10 × 5 × 0.5 in) as shown in Fig. 2. The specimen is held between the bars and the incident p wave is generated by impacting the striker bar using air pressure. This wave travels in the incident bar at a speed of C0 = E∕ ρ , E is the elastic modulus and ρ is the density of the bar. As the incident wave reaches the boundary of the input- specimen, part of the wave is reflected, and the rest is transmitted to the output bar through the specimen. The loading waves were recorded and converted from voltage to strain waves by the 2310 amplifier formula. The total strain energy absorbed by the specimen was calculated based on the linear-elastic, fundamentals of energy and equilibrium consideration assumption as shown briefly in the next section.

Methodology As the incident energy wave impacts the specimen, part of this energy is absorbed by the specimen and causes deformation and damage [9], called elastic strain energy, δQ. Thus, based on the fundamentals of the conservation of energy, Eq. 1, the elastic strain energy carried by the incident and the transmitted bar are Qi, Qr, and Qt, Eqs. 2, 3, and 4.

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Fig. 2 A schematic of the standard split Hopkinson pressure bar (SHPB) and specimen orientation setup [8]

ρ ε_ = σ ji ε_ij

ð1Þ

Qi = 1=2Ao C o E bo t o ε2i

ð2Þ

Qr = 1=2Ao Co E bo t o ε2r

ð3Þ

Qt = 1=2Ao Co Ebo t o ε2t

ð4Þ

where ρ is the density, ε_ is the strain rate, and σ ji are nine components of stresses [10]. Also, Ao, Co, and Ebo are the cross section, elastic wave speed, and the elastic modulus of the bar, respectively. to is the period of the loading wave and it depends on the striker length, to = 2Lstriker/Co . εi, εr, and εt are the incident, reflected, and transmitted strain waves. Based on the dynamic equilibrium condition of SHPB, (εi + εr = εt), the energy that deforms the specimen, i.e., elastic strain energy absorbed by the specimen and calculated using Eq. 5. δQ = - A0 C 0 E bo t 0 ðεr εt Þ

ð5Þ

The reflected wave is negative which makes the final energy positive. The strain waves are measured experimentally and used with Eq. 5 and discussed in the next section.

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Fig. 3 Strain and absorbed energy waves of two experimental setups at 0° and 90° orientation

Results and Discussion Typical transmitted and reflected waves as well as the absorbed energy waves of two different experimental setups can be observed in Fig. 3. In the case of 90° orientation, transmitted and reflected strain waves as shown in Fig. 3a are similar, however, their amplitudes are different. Thus, the maximum reflected strain is εr ≈ 2000 με, but the maximum transmitted strain is εt ≈ 6500 με which is approximately 3× the reflected strain. The maximum absorbed strain energy in the specimen reaches Q90 = 350 KJ. On the other hand, for 0° orientation, the maximum amplitude of the transmitted to the reflected strain wave is almost 2×. Similarly, the absorbed strain energy for the 0° test is Q0 = 650 KJ which is considerably higher than the other direction as illustrated in Fig. 3b, d. In general, results indicate that the cohesive forces within the shale layers are higher than the adhesive forces between the layers at the macroscale level. In other words, Yost shale can absorb more energy when the loading wave is parallel to the bedding layers, hence, higher energy is required to damage under compressive impact load in this direction. However, when the load is applied perpendicular to the layers, lower magnitude strain energy and almost half is absorbed by the material and can damage significantly easily. From a physical standpoint, there are micro reflected waves between the layers, which cannot be captured in a macroscale test. Thus, when the waves travel from one layer to another, due to the micro reflected waves, residual stresses are induced locally, and plastic deformation of shale is reached faster with lower values of impact load.

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Conclusion In this work, it is recommended to use compressive impact loading wave perpendicular to the shale bedding layers to generate more damage and enhance micro and macro cracks. A transversely isotropic shale material was investigated experimentally to identify the amount of energy absorbed by the material. SHPB apparatus was utilized to generate a compressive stress wave and applied to the Brazilian disk. The energy absorbed by the material was calculated for two different bedding directions related to shale, longitudinal and transverse. The conservation of energy principle was applied to estimate the maximum energy. The main conclusions are: (1) The transmitted/reflected wave is εt/εr ≈ 2 in the case of 0° configuration and εt/εr ≈ 3 in the case of 90°. (2) The amount of energy absorbed by shale in 0° is 2× the energy absorbed by 90°. As shown above, stress wave applied perpendicular to the shale bedding layers is more effective and could cause more damage and micro-crack propagation than a wave that passes parallel to the layers. Acknowledgments This material is based upon work supported by the Department of Energy under Award Number DE-FE0031777. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

References 1. Thumbalam Guthai, A., Fahem, A.F., Singh, R.P.: Use of full-field strain measurements to de- termine mechanical properties of shale under repeated cyclic loading. In: Beese, A., Berke, R.B., Pataky, G., Hutchens, S. (eds.) Fracture, Fatigue, Failure and Damage Evolution, vol. 3, pp. 67–72. Springer International Publishing, Cham (2023) 2. Messer, K.R., Guthai, A.T., Fahem, A.F., Singh, R.P.: Mixed-mode fracture interactions along centrally cracked weakened planes. In: Beese, A., Berke, R.B., Pataky, G., Hutchens, S. (eds.) Fracture, Fatigue, Failure and Damage Evolution, vol. 3, pp. 61–66. Springer International Publishing, Cham (2023) 3. Fahem, A.F., Guthai, A.T., Singh, R.P.: Experimental investigation of the nonlocal dynamic damage mechanism in shale. In: Mates, S., Eliasson, V., Allison, P. (eds.) Dynamic Behavior of Materials, vol. 1, pp. 57–61. Springer International Publishing, Cham (2023) 4. Messer, K.R., Fahem, A.F., Guthai, A.T., Singh, R.P.: The experimental methods and elastic properties of shale bedding planes materials stateof-the-art review. Al-Qadisiyah J. Eng. Sci. 15(2), 127–131 (2022) 5. Feng, S., Zhou, Y., Li, Q.M.: Damage behavior and energy absorption characteristics of foamed concrete under dynamic load. Constr. Build. Mater. 357(March) (2022) 6. Fahem, A.A.F., Guthai, A.T., Messer, K.R., Raman, P.: Full-field strain measurement integrated with two dimension regression analysis to evaluate the bi-modulus elastic properties of orthotropic materials. Exp. Mech., 1–20 (2022) 7. ASTM: D 3967: Standard Test Method for Splitting Tensile Strength of Intact Rock Core Specimens, pp. 1–3. ASTM International, Philadelphia, PA., no. Reapproved (1992) 8. Fahem, A., Singh, R.: Dynamic Damage Evolution in Shale in the Presence of Pre-existing Microcracks. In: Dynamic Behavior of Materials. Conference Proceedings of the Society for Experimental Mechanics Series, vol. 1, pp. 39–45. Springer, Cham (2022) 9. Song, B., Chen, W.: Energy for specimen deformation in a split Hopkinson pressure bar experiment. Exp. Mech. 46(3), 407–410 (2006) 10. Graff, K.: Wave Motion in Elastic Solids. Courier Corporation (1975)

Dynamic Fracture Characteristics of Cyanoacrylate Weakened Planes in Polycarbonate Material Kyle R. Messer, Achyuth Thumbalam Guthai, Ali F. Fahem, and Raman P. Singh

Abstract Weakened plane fracture has been an interesting topic for understanding unconventional oil extraction. Thus, the dynamic fracture characteristics of weakened planes are studied experimentally to investigate the failure mechanism when exposed to stress waves at varying angles. A split-Hopkinson pressure bar (SHPB) and digital image correlation (DIC) are used to test a range of cyanoacrylate-weakened plane specimens that are created between the two-polycarbonate sheet. The weakened plane specimens are made with different inclined angles. Theoretically, a crack surface displacement method using various angles, (β), and crack tip opening displacements are shown to identify the dynamic stress intensity factors of openingmode KI(t), and in-plane shear mode KII(t). The SHPB is dynamically loaded to fracture the specimen while full-field strains are recorded using digital image correlation, and the data is imputed to the crack surface displacement formulas. A fracture envelope containing θ = 15°, 30°, 45°, and 60° angles is created, and the result of mode I, mode II, and mixed-mode (I/II) are discussed, and the major mode of each inclined angle is identified. Keywords Dynamic · Weakened plane · DIC · SHPB · Fracture

Introduction In the prospect of an energy-efficient future, it is a common thought that, as the world develops, oil and natural gas are going to be left in the past. However, while oil and natural gas may change for the auto industry, the use of oil and natural gas is abundant in the manufacturing sectors; [1]. For example, petroleum-based polymers are still in heavy production. Some wellknown petroleum-based polymers are nylon, epoxy, polypropylene (PP), polytetrafluoroethylene (PTFE), polyethylene (PE), and polyester (PS) [2]. However, petroleum demands have not been able to match its production, and the petroleum industry has begun to broaden its horizons to help meet the demand. One of the most feasible methods is unconditional extraction methods through shale regions which is the second largest mean source of oil and gas [3]. However, unconditional extraction methods have downfalls and need to be understood. There have been many studies to characterize shale unitized in petroleum extraction such as [4–7]. However, fracture characteristics on shale-weakened planes have not been studied dynamically in detail. Shale petroleum extraction can fall by 60\%–90\% in the first year of production [8]. The cause is derived from the K. R. Messer (✉) Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA e-mail: [email protected] A. T. Guthai Department of Mechanical and Aerospace Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA A. F. Fahem Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA Department of Mechanical Engineering, College of Engineering, University of Al-Qadisiyah, Al-Diwaniyah, Iraq R. P. Singh Department of Materials Science and Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA Department of Mechanical and Aerospace Engineering, College of Engineering, Architecture, and Technology, Oklahoma State University, Tulsa, OK, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_10

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porosity and capillary forces of shale, preventing recovery during oil extraction by 3–7% [9]. Understanding dynamic fracture characteristics will allow for more efficient fracture of the weakened plane of shale to prevent the effects of porosity and capillary forces from extraction. Subsequently, it is challenging to measure shale directly due to anisotropy, and a model material is needed to use an experimental technique to study shale confidently. This research aims to characterize dynamic weakened plane fracture characteristics as a function of angle on a model material, a polycarbonate bulk, and a cyanoacrylate weakened plane by use of SHPB and DIC. The benefits of understanding weakened plane fracture can increase efficiency and productivity of shale oil extraction. This research serves as a starting point for understanding the fracture mechanism and new research approaches on shale material.

Specimen Preparation A sliding compound miter saw (Ryobi® TSS103), a three-way precision angle vise, and milling machine (Accupath AC-3KV), a three-flute mill bit (Speed Tiger® Iaue 1/2″ 3), water-cooling spray system (OriGlam 130103020Q) were utilized to cut large sheets of Tuffak® polycarbonate (McMaster-Carr®) at 15°, 30°, 45°, and 60° angle to create two sides of the bulk material. The sides are then glued back together using Loctite glue (Loctite® Super Glue Liquid Brush 5 gr 852882 en-US) with TaegaSeal PTFE tape placed in the middle of the weakened plane to create a starter crack. To create the white base of the speckle patten, a spray paint (Rust-Oleum® Painter’s Touch® 2X Ultra Cover336098 Flat White) was sprayed across the bulk material. A fine tip sharpie (Sharpie® 37101PP) was used to hand speckle the specimen. The specimen dimensions were selected at a height 25.4 mm, the same height of diameter of the bar, and the length was variable to allow for additional length to account for the extra length caused by the angle of the weakened plane angles. The was a minimum length of 25.4 mm to allow for fully developed stress wave by the time of interaction with the weakened plane. A rectangular specimen allows for a uniform load transfer between the SHPB and the specimen (Fig. 1).

Methods This study uses classical SHPB and DIC methods in tandem to measure the dynamic weakened plane fracture as a function of angle and crack tip opening displacements. Two sets of tests of 15°, 30°, 45°, and 60° weakened plane angles were conducted in a total of eight test. A HPV-X2 Shimadzu high-speed camera was utilized for digital image correlation processing to measure the local displacements near the crack tip on the polycarbonate-cyanoacrylate specimen. The experimental setup included a SHPB which contains an aluminum striker, incident, and transmitted bar, a stop block, and relic Hex- 700 W light source to provide adequate lighting. The experimental procedure begins with the striker bar striking the incident creating an elastic pressure wave that transmits into the specimen, then into the transmitted bar. The validity of the SHPB experimental apparatus results depends on dynamic equilibrium and steady-state conditions. The incident bar sends a stress wave through the polycarbonate bulk material and when the wave encounters the cracked weakened plane, it will fracture.

DIC Analysis The HPV-X2 Shimadzu high-speed camera is utilized throughout the duration of the experiment by tracking the speckled pattern deformation caused by the stress wave passing. A crack surface displacement method is utilized to calculate the dynamic stress intensity factor using DIC as shown in Eqs. (1) and (2) which are deviated from [10, 11] with a modification of the geometric sine and cosine as a function of weakened plane angle. Two data points at the crack boundary are selected to extract the crack opening displacement and used with Eqs. (1) and (2) to provide fracture toughness values of the cyanoacrylate weakened plane. Fig. 1 A centrally cracked polycarbonate 45° weakened plane and handspeckled pattern specimen

Dynamic Fracture Characteristics of Cyanoacrylate Weakened Planes in Polycarbonate Material

K I ðt Þ =

ðU 1 - U 0 Þ  cosðβÞ

K d,II ðt Þ =

k stress þ1 μ

xc 2π

ðV 1 - V 0 Þ  sinðβÞ

K Mixed ðt Þ =

k stress þ1 μ

xc 2π

K I ðt Þ2 þ K II ðt Þ2

75

ð1Þ

ð2Þ

ð3Þ

where β is the angle of weakened planes, kStress is plane stress conditions, μ is the shear modulus, xc is the distance away from the crack tip, U is the horizontal displacement, and V is the vertical displacement. The mixed mode stress intensity factor is shown in Eq. (3), is described in [10], and is a function of the mode I and mode II fracture toughness values.

Results and Discussion Figures 2, 3, 4, and 5 are actual specimens post fracture showing weakened planes and associated angles. It can be seen from the figures that the fracture path followed through the weakened planes when the stress wave propagated through the polycarbonate bulk.

Fig. 2 A 60° angled specimen post fracture

Fig. 3 A 45° angled specimen post fracture

Fig. 4 A 30° angled specimen post fracture

Fig. 5 A 15° angled specimen post fracture

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Nearfield vertical CTOD (mm)

1.5

1 Point 1 Point 2

0.5

0

-0.5

-1

-1.5 50

70

90

110

130

150

170

130

150

170

Time (μs) Fig. 6 Vertical CTOD of two points showing separation through progression of time

Nearfield horizontal CTOD (mm)

1.5

1.2 Point 1 Point 2

0.9

0.6

0.3

0

-0.3 50

70

90

110

Time (μs) Fig. 7 Horizontal CTOD of two points showing separation through progression of time

An example stress wave of a 60° angle weakened plane is shown in Figs. 6 and 7 produced by DIC data. The figures are based on crack tip opening displacements measured using two points across the weakened plane boundary. When the stress wave progresses across the bulk and weakened plane, the two points will separate indicating fracture. It can be seen that Fig. 7 has a positive slope compared to Fig. 6. The slope is caused by rigid body motion across the bar before impact. Table 1 shows the weakened plane angles, the averaged mode I, mode II, and the mixed mode fracture toughness values from the weakened planes and the standard deviation of the experiments. From these values a fracture envelope was created and shown in Fig. 8. The trend is shown that for a low angle, 15° and 30°, a low mode II fracture toughness. For a 45° angle,

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Table 1 DIC results of centrally cracked 15°, 30°, 45°, and 60° angled specimens for mode I, mode II, and mixed-mode fracture toughness displaying similar failure characteristics Angle ° 15 30 45 60

KI(t) MPa√m 1.10 ± 0.02 1.07 ± 0.11 1.25 ± 0.08 1.19 ± 0.39

KII(t) MPa√m 0.12 ± 0.02 0.34 ± 0.10 0.76 ± 0.14 2.25 ± 0.16

15° Critical fracture toughness Threshold fracture line No fracture zone 30° Critical fracture toughness Threshold Fracture Line No fracture zone 45° Critical fracture toughness Threshold fracture line No fracture zone 60° Critical fracture toughness Threshold fracture line No fracture zone

2

Dynamic stress intensity factor KI(t) (MPa m)

KMixed(t) MPa√m 1.75 ± 0.03 1.47 ± 0.13 1.19 ± 0.07 1.77 ± 0.05

1.75 1.5 1.25 1 0.75 0.5 0.25 0 0

0.25 0.5 0.75

1

1.25 1.5 1.75

2

2.25 2.5 2.75

KII(t) (MPa m) Dynamic stress intensity factor Fig. 8 Dynamic fracture toughness envelope of a 15°, 30°, 45°, and 60° weakened plane

the fracture values are approximately equal, and as the angle reaches 60°, mode II is higher than mode I. For the 60°, the amount of loading is much higher than the inverse 30°. This is due to the 60° angle being able to rely on and transmit loading to the bulk material resulting in a higher fracture value.

Conclusion Weakened planes have a direct effect on materials’ fracture toughness. The results show that there is a low mode II fracture for lower angles, 15° and 30°. When the angle of the weakened plane is at 45°, the results are about equal in the mode I and mode II directions. Angles such as 60° show that the mode II fracture is prominent, and mode I is about half of the mode II fracture. However, there is not an inverse relationship between the change of angles for fracture toughness due to the bulk material allowing higher angles to transfer stress values. The ability for the 60° angle to withstand fracture by utilization of the bulk polycarbonate material is the reason for the high standard deviation shown in Table 1 when compared to the other angles. Acknowledgments This material is based upon work supported by the Department of Energy under Award Number DEFE0031777. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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References 1. Frequently Asked Questions (Faqs) – U.S. Energy Information Administration (EIA). Frequently Asked Questions (FAQs) – U.S. Energy Information Administration (EIA), 1 June 2021., https://www.eia.gov/tools/faqs/faq.php?id=34&t=6 2. Nagalakshmaiah, M., et al.: Biocomposites: present trends and challenges for the future, pp. 197–215. Green Composites for Automotive Applications (2019) V. Kramarov, P. N. Parrikar, and M. Mokhtari, Evaluation of fracture toughness of sandstone and shale using digital image correlation, Rock Mechanics and Rock Engineering, vol. 53, no. 9, pp. 4231–4250, 2020 3. Fahem, A., Singh, R.: Dynamic damage evolution in shale in the presence of pre-existing microcracks. SEM Annual Conference. 1(1), 1–10 (2021) 4. Messer, K.R., et al.: Mixed-mode fracture interactions along centrally cracked weakened planes. In: Fracture, Fatigue, Failure and Damage Evolution, vol. 3, p. 2023. Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing, Cham 5. Messer, K.R., et al.: The experimental methods and elastic properties of shale bedding planes materials state-of-the-art review. Al-Qadisiyah J. Eng. Sci. 15, 2 (2022) 6. Guthai, T., Achyuth, Fahem, A.F., Singh, R.P.: Use of full-field strain measurements to determine mechanical properties of shale under repeated cyclic loading. In: Fracture, Fatigue, Failure and Damage Evolution, Volume 3: Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing, Cham (2023) C. A. Ross, J. Tedesco, et al., Split-hopkinson pressure-bar tests on concrete and mortar in tension and compression, Materials Journal, vol. 86, no. 5, pp. 475–481, 1989 7. Fahem, A.F., Guthai, A.T., Singh, R.P.: Experimental investigation of the nonlocal dynamic damage mechanism in shale. In: Dynamic Behavior of Materials, Volume 1: Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing, Cham (2022) 8. Jia, C., Zheng, M., Zhang, Y.: Some key issues on the unconventional petroleum systems. Pet. Res. 1(2), 113–122 (2016) 9. Technically Recoverable Shale Oil and Shale Gas Resources: An Assessment of 137 Shale Formations in 41 Countries Outside the United States. EIA, U.S. Energy Information Administration, https://www.eia.gov/analysis/studies/worldshalegas/ 10. Ma, F., et al.: A CTOD-based mixed-mode fracture criterion, pp. 86–110. ASTM Special Technical Publication 1359 (1999) 11. Sun, C.-T., Jin, Z.: Fracture Mechanics. Elsevier Science, Ukraine (2011)

Modal Verification and Thermal-Fluid-Structure Coupled Analysis of Centrifugal Impeller Po-Wen Wang and Chang-Sheng Lin

Abstract The topic of this article is to investigate mechanical behaviors and dynamic characteristics under working conditions. The verification of finite element model (FEM) performed by experimental modal analysis (EMA) is discussed. The finite volume model (FVM) is established to analyze the aerodynamic and thermal effects under working conditions by computational fluid dynamics (CFD). With the consideration of loading conditions coupled with the results of CFD, the deformation, stress distribution, and modal parameters of the impellers are estimated by finite element analysis (FEA). To verify the reliability of FEM of the fluid-thermal-structure coupled analysis, modal verification is employed to ensure the consistency between the FEM and the actual structure. The modal assurance criterion (MAC) is applied as an indicator to quantify the consistency of mode shapes. In coupled stress analysis. The results show that the centrifugal load is the main factor of stress concentration, and the corresponding location of maximum stress is on the center of the impeller. In coupled modal analysis, the centrifugal force is also sensitive to the change of dynamic characteristics, however, only obviously influences the natural frequency of structures. Keywords Centrifugal compressor impeller · Experimental modal analysis · Finite element analysis · Modal verification · Thermal-fluid-structure coupled analysis

Introduction Centrifugal compressor is widely used in industry because of their characteristics of large-scale, large air intake, and ease of maintenance. The impeller of the centrifugal compressor is under coupled loading conditions during operation. The dynamic characteristics are difficult to measure in enclosed spaces under working conditions. Therefore, computational-aided engineering (CAE) is often used for simulation. The rotating mechanical usually work under high temperature, centrifugal and aerodynamic load, where centrifugal and aerodynamic load are the main factors of stress concentration [1]. The variation of temperature also obviously influences dynamic characteristics, especially in natural frequencies [2]. The method of coupled analysis is usually used for analyzing the mechanical behaviors after aero-mechanical optimization [3, 4]. EMA can reflect the conditions without consideration in the FEM [5]. Therefore, it can be used to improve the reliability of the FEM. Also, the non-contact methods enable the elimination of the impact of sensors on the behavior of the test system [6]. Through the comparison between modal shapes, the MAC is widely used, which can quantify the consistency from 0 to 1 [7]. MAC is used to verify the agreement between mode shape vectors, but can’t give information about the accuracy of natural frequencies. Therefore, the comparison of natural frequencies and mode shapes of each mode are available in modal verification [8]. In this paper, the aerodynamic force, thermal load, and centrifugal force are considered as loading conditions. We estimate the strength and dynamic characteristics under working conditions by using ANSYS. To confirm the consistency between the actual structures and CAE models, experimental modal analysis is applied to verify the reliability of finite element models. MAC is used to confirm the consistency between mode shapes containing different directions of vibration in space.

P.-W. Wang (✉) · C.-S. Lin Department of Vehicle Engineering, National Pingtung University of Science and Technology, Pingtung, Taiwan © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_11

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Numerical Method for Coupled Analysis In fluid-thermal-structure coupled analysis, the fluid and thermal effect must be coupled separately. Therefore, the aerodynamic load and centrifugal load are considered in the fluid-structure coupled analysis, the equation neglect the damping effect can be written as ½M ]

½ 0]

M fs

MP

f€ug € P

þ

½K ] þ K R

K fs

f ug

½0]

KP

fP g

=

fF g f 0g

ð1Þ

where [M] and [K] are the mass and stiffness matrixes, respectively. [M fs] and [K fs] are the fluid-structure interaction mass and stiffness matrix. [MP] and [KP] are the equivalent fluid mass and stiffness matrixes, [KP] is the centrifugal stress matrix, f€ug, {P}, and {F} are the displacement, pressure, and the loading vectors, respectively. The deformation of the impeller due to the fluid-structure coupled effect can be determined from the above equation. Also, the thermal effect has to be considered in the analysis. The equation of thermal-structure coupled analysis that neglects the damping effect can be written as ½M ] ½0]

½0] ½0]

f€ug T€

þ

½K ] ½K ut ]

f ug

½ 0]

fT g

½K t ]

=

f 0g

ð2Þ

fQ g

where [Kut] is the thermal expansion coefficient matrix, [Kt] is the thermal conductivity matrix, and {Q} is the thermal flow vector. The thermal deformation can be determined from the above equation. Considering the complex loading condition, we used modal analysis to evaluate the eigenvalue problem. Combining the aerodynamic, the thermal, and the centrifugal stresses by superposition principle, the equation of the eigenvalue problem can be written as ½K þ S] - ωi 2 ½M ] fϕgi = f0g

ð3Þ

det ½K ] - ωi 2 ½M ] = 0

ð4Þ

where ([K + S] - ωi2[M]) is the characteristic matrix of the discrete system, and [S] is the stress stiffness matrix from fluidstructure and thermal-structure coupled analysis. When the characteristic determinant is equal to zero, the natural frequencies and mode shapes can be determined as eigenvalue ωi and eigenvector {ϕ}i.

Computational Model The semi-open type impeller of a centrifugal compressor is analyzed in this paper. The height and diameter of the structure are 84 mm and 280 mm. The whole structure has 20 blades made of SUS630 stainless steel with H1050 heat treatment. The temperature-dependent mechanical properties are shown in Table 1. In computational fluid dynamics, the fluid domain model is established by ANSYS TurboGrid. We extended the inlet and outlet region to investigate the aerodynamic effect completely. The value of Y+ is set as 5 for calculating the thickness of the element. For accuracy, the grid systems are Table 1 The temperature-dependent mechanical properties of SUS630 Temperature °F (°C) Mechanical properties Density (kg/m3) Young’s modulus (GPa) Poisson’s ratio (-) Yield strength (MPa) Ultimate strength (MPa) Coefficient of thermal expansion (10–6 m/m°C) Thermal conductivity (W/m/K)

72 (22) 7800 201 0.29 1158 1200 11.3 –

100 (38) – 200.2 – 1120.2 1158.5 11.3 –

200 (93) – 196.6 – 1082.3 1117.0 11.3 –

300 (149) – 193.6 – 1044.5 1075.5 11.7 17.9

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Fig. 1 (a) Real structure, (b) grid system in structure, and (c) fluid domain of centrifugal impeller

Fig. 2 Boundary conditions in fluid domain and structure

established by structure mesh. In structure, the linear hexahedral element SOLID185 is used for efficiency in structure analysis. The grid systems are shown in Fig. 1, which consists of 257,920 elements and 292,815 nodes in the fluid domain, while the number of elements and nodes in the structure are 471,076 and 543,452, respectively. The boundary types of the fluid domain are chosen in ANSYS CFX based on measurement data. The inlet mass flow rate, temperature, outlet pressure, and rotating velocity are 7.11 kg/sec, 303 K, 613 kPa, and 20,285 rpm, respectively. In the stress analysis, the boundary condition is considered by cylindrical coordinates to estimate the deformation induced by centrifugal force, the components of tangential direction on the center surface of the impeller are zero; the components of axial direction on the top and bottom surface are zero as shown in Fig. 2.

Modal Testing and Verification In this study, modal testing is used to verify the reliability of finite element model of the centrifugal impeller. To determine the measurement bandwidth of frequency range and position, The modal parameters are evaluated in modal analysis under freefree boundary condition. The harmonic response analysis is employed to simulate the excitation and response in modal testing; however, the local coordinate systems that are perpendicular to blades are considered during the solution. In modal testing, the angle between the direction of excitation and response must be parallel. However, the blades of the impeller conclude different angles which are hard to excite the mode shape completely. To discuss the angle between the blades, we calculate the dot product of the unit vector which is from the local coordinate system. The equation is shown as

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Fig. 3 (a) Local coordinates systems and (b) angle between the blades

Fig. 4 Experimental set-up with non-contact sensor

θaf = cos - 1 a ∙ f

ð5Þ

where θaf is the angle between the excitation and response; f and a represent the unit vector of excitation and response, respectively. The solution of the above equation is shown in Fig. 3. The dot product is zero when excitation is perpendicular to the response. To excite all the modes of the impeller, we consider the excitation on the disk. Considering the thin blade of the impeller, which is sensitive to mass effect. The non-contact Doppler vibrometer is applied in testing, which types are Polytec OFV-5000 and OFV534. The impact hammer PCB 086C01 is used to excite the impeller. During the measurement, the Brüel & Kjær analyzer is employed for data acquisition. The sampling rate and measurement time are set in the analyzer, which is 16,384 Hz and 2 s for the signal processing. The effective frequency of modal testing is 6400 Hz based on the Nyquist theorem and anti-aliasing filter (AAF) in the analyzer. To verify the modal parameters under free-free boundary condition, the cushion is applied as a boundary in modal testing (Fig. 4). Consider the modal parameters in the frequency range to be verified. The 80 measurement points are employed, where 60 and 20 points are on blades and disk, respectively. The single input single output (SISO) method is used to calculate the frequency response function (FRF). During the modal testing, the coherence function is employed to check the correlation between excitation and response, which equation is shown as follows: 2

Gaf ðf Þ γ ðf Þ = Gaa ðf ÞGff ðf Þ 2

ð6Þ

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where Gff( f ) and Gaa( f ) are the auto-power spectrum function of excitation and response, respectively. Gaf( f ) is the crosspower spectrum function of excitation and response. The value of coherence function will be close to 1 if the correlation is high between two signals. In contrast, if the function is close to zero, which represents the signal is no correlation between the two signals. The FRF is also been calculated after the correlation is greater than 0.9, the equation is shown as follows: H af ðf Þ =

Gaf ðf Þ Gff ðf Þ

ð7Þ

Through the modal testing, the natural frequencies, mode shapes, and modal damping ratios are evaluated from frequency response functions by curve fitting in ME’scope. To quantify the comparison between parameters, the error is employed to calculate the difference in natural frequencies; and the modal assurance criterion (MAC) is applied to quantify the consistency between mode shapes, which the equation is shown as follows: MAC ϕiA , ϕjX =

ϕiA T ϕjX *

2

ϕiA T ϕiA * ϕjX T ϕjX *

ð8Þ

where superscript T and * represent the transpose matrix operator and the conjugate complex operator, respectively, while the ϕiA and ϕjX are the mode shapes from finite element analysis and experimental modal analysis, respectively. The agreement between mode shapes is good when MAC reaches 1; on the contrary, the mode shapes are orthogonal when MAC is 0. In this paper, the consistency is confirmed when MAC is greater than 0.7.

Result and Discussion The coherence function, phase, and frequency response function are measured by the processes that we discussed in the above section. Figure 5a shows that coherence and phase change significantly correspond to the characteristics in the frequency response function. The phase change is less than 180° because of the close mode in 2600 Hz to 2800 Hz bandwidth, which influences the result of modal parameters. The MAC between EMA and FEA is shown in Fig. 5b. In natural frequency verification, we filter the modes in which MAC is less than 0.6. The finite element model will be modified if the MAC and error percentage is unacceptable. The corresponding modal parameter is shown in Table 2. The result of CFD is shown as a colormap in Fig. 6, which indicates the maximum velocity is 266 m/sec at the tip of the blade. The flow velocity is as low as close to the outlet, which is a process that the pressure is converted dynamically into statically. The isentropic efficiency is 73.9%, which error percentage between measurement and analytical solution is -8.3%. Converting the velocity into pressure acts on the surface of the impeller as an aerodynamic load, where the maximum pressure

Fig. 5 (a) Coherence function, phase, and frequency response function; (b) bar chart of MAC

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Table 2 Comparison of modal parameters between FEA and EMA FEA mode No.7

FEA natural freq. (Hz) 2609.30

No.7 No.8 No.26

2609.30 2609.40 2688.10

No.27 No.28 No.30 No.31 No.32 No.33 No.34 No.35 No.36 No.37

3184.90 3184.90 3801.40 4132.20 4132.20 4254.60 4468.40 4468.50 4846.80 4846.90

EMA mode No.1 No.2 No.4 No.2

EMA natural freq. (Hz) 2584.99 2605.91 2654.56 2605.91

No.6 No.8 No.9 No.10 No.11 No.12 No.13 No.14 No.15 No.16 No.17 No.18 No.19

2686.41 2705.82 2713.35 3216.77 3228.00 3799.14 4099.03 4117.97 4237.07 4415.43 4432.22 4766.95 4785.13

MAC (-) 0.60 0.75 0.70 0.75 0.77 0.76 0.71 0.79 0.77 0.87 0.82 0.65 0.87 0.85 0.76 0.82 0.76 0.82

Freq. Error (%) 0.93 0.13 -1.70 0.13 0.13 0.06 -0.66 -0.93 -0.99 -1.34 0.06 0.81 0.35 0.41 1.20 0.82 1.68 1.29

Fig. 6 (a) Velocity streamline and (b) distribution on different section

Fig. 7 The distribution of (a) aerodynamic and (b) thermal loads employed to the impeller

is 556.7 kPa at the end of the passage; the maximum thermal load is 78.59 °C shown in Fig. 7, which is also located at the bottom of the blade. The maximum deformation is 0.19 mm located at the bottom of the blade under coupled loading conditions; the maximum von-mises stress is 456.19 MPa in the center of the impeller, and the local stress distribution of the blade, where the stress is concentrated at the root of the blade is shown in Fig. 8. To investigate the weighting of each loading condition, the

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Fig. 8 The (a) deformation distribution, (b) von-mises stress distribution, and (c) local stress distribution of the impeller Table 3 The modal parameters under different conditions

No. of mode 1 21 22 24 26 27 29 31 33

Mode type (No.) Local-1 (1–20) Global-1 (21) Global-2 (22–23) Global-3 (24–25) Global-4 (26) Global-5 (27–28) Global-6 (29–30) Global-7 (31–32) Global-8 (33–34)

ω 2462.28 2746.38 3064.40 3377.53 3395.36 3703.65 4005.03 4328.58 4694.71

ωp 2463.09 2747.03 3064.44 3377.32 3395.55 3703.10 4004.08 4327.21 4692.96

ωT 2462.02 2747.09 3066.57 3376.75 3398.69 3697.73 3992.40 4308.28 4666.39

ωr 2475.78 2750.19 3085.66 3413.99 3415.15 3755.55 4072.68 4411.76 4792.23

ωtotal 2476.38 2751.53 3087.86 3413.02 3418.65 3749.18 4059.34 4390.55 4762.89

Diff (%) 0.57 0.19 0.77 1.05 0.69 1.23 1.36 1.43 1.45

MAC (-) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

aerodynamic load, thermal load, and centrifugal load are separately considered in the coupled analysis. The result shows that centrifugal load is the main factor of stress concentration, while the thermal and aerodynamic load influenced the toot and middle of the blade, respectively. The cylindrical coordinate system is considered to substitute the cartesian coordinate system to calculate the directional stresses at the bottom of the impeller. The results show that tangential stress is the main factor of stress concentration. The pre-stressed modal analysis is employed to estimate the modal parameters under the different loading conditions to investigate the change of dynamic characteristics. The MAC and error percentage are used for comparison between modal parameters as Table 3, where ω is the natural frequency; ωp, ωT, and ωr are the natural frequency under operational conditions with aerodynamic, thermal, and centrifugal loadings, respectively, and ωtotal is the natural frequency under working condition. The centrifugal load leads the frequency difference of 1.43%. The MAC corresponding to different loading conditions is 1, approximately.

Conclusion In this paper, the coupled loading condition is considered in the centrifugal compressor impeller, which includes aerodynamic load, thermal load, and centrifugal load. The finite element analysis (FEA) is employed to analyze the mechanical behaviors and dynamic characteristics under coupled effect. Otherwise, the experimental modal analysis (EMA) is employed to validate the finite element model (FEM) by comparison of modal parameters. The result under the coupling effect shows that the maximum von-mises stress is less than the yield stress of SUS630, which represents that plastic deformation will not occur during operation; meanwhile, the centrifugal load is the main factor influencing the mechanical behaviors and dynamic characteristics, but the mood shape remains the same in any coupled condition. From the result of modal verification, the natural frequency of EMA well corresponds to the FEA in bandwidth which wants to verify. However, the MAC is less than the desired value in 2500 Hz to 2800 Hz bandwidth, where are the close modes without a complete phase change. Given the comparison of frequency error and MAC, we can conclude that the finite model is valid.

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Acknowledgments This research was supported by China Steel Corporation (CSC) under the grant RE110728. The authors would like to thank all members in Modal Identification & Failure Diagnosis Lab, Department of Vehicle Engineering, National Pingtung University of Science and Technology (NPUST), Taiwan.

References 1. Xie, Y., Lu, K., Liu, L., Xie, G.: Fluid-thermal-structural coupled analysis of a radial inflow micro gas turbine using computational fluid dynamics and computational solid mechanics. Math. Probl. Eng. 640560, 2014 (2014) 2. Zhang, Q., Zhao, W., Liu, T., Xie, Y.: Modal analysis of a turbine blade based on fluid-thermal-structural coupling. Appl. Mech. Mater. 641–642, 300–303 (2014) 3. Li, C., Wang, J., Guo, Z., Song, L., Li, J.: Aero-mechanical multidisciplinary optimization of a high speed centrifugal impeller. Aerosp. Sci. Technol. 95(105452), 105452 (2019) 4. Kang, H.S., Kim, Y.J.: Optimal design of impeller for centrifugal compressor under the influence of one-way fluid-structure interaction. J. Mech. Sci. Technol. 30, 3953–3959 (2016) 5. Lin, C.S., Chiang, H.T., Hsu, C.H., Lin, M.H., Liu, J.K., Bai, C.J.: Modal verification and strength analysis of bladed rotors of turbine in rated working conditions. Appl. Sci. 11(6306) (2021) 6. Chen, Y., Griffith, D.T.: Experimental and numerical full-field displacement and strain characterization of wind turbine blade using a 3D scanning laser Doppler Vibrometer. Opt. Laser Technol. 158(129969), 108869 (2023) 7. Teter, A., Gawryluk, J.: Experimental modal analysis of a rotor with active composite blades. Compos. Struct. 153, 451–467 (2016) 8. Allemang, R.J., Brown, D.L.: A correlation coefficient for modal vector analysis. Engineering. (1982) 9. Pastor, M., Binda, M., Hararik, T.: Modal assurance criterion. Proc. Eng. 48, 543–548 (2012)

Multiaxial Failure Stress Locus of a Polyamide Syntactic Foam at Low and High Strain Rates Yuan Xu, Yue Chen, and Antonio Pellegrino

Abstract The mechanical response of a polyamide syntactic foam under combined tension-torsion loading is measured experimentally at quasi-static (10-3 s-1) and high rates of strain (103 s-1). The dynamic experiments were conducted on a newly developed tension-torsion Hopkinson bar (TTHB) equipped with high-speed photography equipment. Quasi-static experiments were carried out using a universal screw-driven machine. The multiaxial high-rate experiments demonstrate the ability to achieve synchronization of longitudinal and shear stress waves upon loading of the specimen. The capacity to achieve force and torque equilibrium under combined loading on materials characterized by relatively low-stress wave propagation velocities is also demonstrated. Approximately constant strain rate conditions were attained. The failure envelope of the polyamide foam studied was analyzed over a wide range of stress states including pure torsion, shear-dominated combined tension-shear, tension-dominated combined tension-shear, and plain tension. Additional low rate and dynamic experiments in tension, compression, and torsion were conducted at higher than ambient and sub-ambient temperature conditions using bespoke temperature conditioning equipment, to assess the temperature dependence of the material. The multiaxial failure stress locus was constructed in the normal versus shear stress space as well as in the principal stress space from experiments conducted at low and high rates of strain. The failure stress locus of a polymer syntactic foam and its rate and temperature dependence are presented for the first time. The newly developed TTHB apparatus allows for the direct measurement of the failure stress locus of lightweight materials and therefore for the evaluation of existing failure criteria. Keywords Split Hopkinson tension-torsion bar · Multiaxial failure · Failure stress envelope · Rate dependence · Syntactic foam

Introduction Polymeric syntactic foams (PSF) are lightweight materials made up of a polymer matrix that is strengthened by hollow thinwalled glass microspheres [1]. These materials are commonly used in the aerospace and submarine industries due to their low density, low moisture absorption, high specific strength, and stiffness, which make them suitable for use in environments with varying temperature conditions. However, because the polymer matrix is temperature-sensitive, changes in temperature can have a significant impact on the physical and mechanical properties of PSFs. Additionally, the strain rate has a considerable effect on their dynamic response. Polymer syntactic foams consist of three phases: polymer, micro-balloons, and air. Their mechanical behavior at the macroscopic level is thus determined by the way these components interact during the deformation process [2]. The rate of deformation has a significant effect on the mechanical behavior of these materials due to the strain rate dependency of the polymer matrix and the dissipative nature of deformation and failure of the microspheres. Measuring the dynamic properties of polymer-based materials can be challenging because of their unique mechanical and acoustic properties. Typically, their mechanical response at high strain rates is determined using the split Hopkinson bar apparatus [3, 4]. A considerable number of studies have examined the behavior of polymer matrix syntactic foams (PSFs) at room and higher than ambient temperatures under different conditions [5, 6]. By utilizing suitable pulse shaping techniques, Song and Chen [6, 7] conducted experiments to assess the compressive behavior of an epoxy syntactic foam at both high and Y. Xu · Y. Chen · A. Pellegrino (✉) Department of Engineering Science, University of Oxford, Oxford, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_12

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intermediate strain rates. The dynamic compressive behavior of novel cenosphere polyurethane syntactic foams was studied by Fan and colleagues [8], with a focus on the impact of cenosphere size and internal lateral constraint. Several studies have examined how physical properties, including thermal and electrical conductivity, as well as coefficient of thermal expansion, vary with changes in temperature [9–12]. The quasi-static compressive response of Rohacell foams with varying densities was analyzed by Arezoo et al. [13] to determine the effect of temperature on the mechanical response. Additionally, the researchers assessed the effect of strain rate using split Hopkinson pressure bar (SHPB) experiments conducted at room temperature. In a more recent investigation, Zhang and colleagues [14] conducted a study to explore how the compressive response of an epoxy syntactic foam is affected by strain rate, temperature, and confinement. However, in real operating conditions, impact loading is not applied in a uniaxial manner. To the best of the authors’ knowledge, there is a lack of comprehensive research available in the public domain that examines the combined influence of stress state and strain rate under various loading conditions. The strain rate dependence of a syntactic polyurethane foam in tension, compression, and torsion loading was presented in [15]. None of the previous studies available have investigated the mechanical response of syntactic polyurethane foams under combined tension-torsion loading at low and high strain rates. In this study, the mechanical characteristics and failure envelope of a polyamide foam are analyzed over a wide range of stress states including pure torsion, shear-dominated combined tension-shear, tension-dominated combined tension-shear, and plain tension using the tension-torsion Hopkinson bar apparatus presented in [16, 17]. The multiaxial failure stress locus is presented in the normal versus shear stress space as well as in the principal stress space. The failure stress locus of a polymer syntactic foam and its rate and temperature dependence are presented for the first time.

Experimental Setup In this study, the authors obtained experimental data using a novel testing device, a tension-torsion split Hopkinson bar (TTHB) capable of generating both tensile and torsional stress waves during a single loading event. The TTHB is intended to measure material properties that are representative of actual impact scenarios. To produce both longitudinal and shear waves, an energy storage and release mechanism was used, which involved the rapid release of a customized clamp assembly. The time gap between the arrival of longitudinal and shear waves is largely dependent on the position of the clamp relative to the specimen, owing to the varying wave speeds. In the present TTHB configuration, the clamp is located in close proximity to the sample to minimize the time discrepancy between the instances at which the longitudinal and shear waves reach the specimen. Axial and torsional elastic energies were generated using two loading units consisting of a hydraulic actuator and an electrically controlled harmonic drive respectively. A schematic of the apparatus is provided in Fig. 1. Details on the apparatus and its design can be found in [18].

Fig. 1 Schematic representation of the TTHB setup employed during the experiments

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The syntactic foam was manufactured in the shape of thin-walled tubular samples. The samples were then bonded to lightweight aluminum endcaps to allow for the transmission of axial load via threaded connection and torsional load by means of octagonal couplings.

Results and Discussion The experimental results show an evident dependence on strain rate and temperature of the mechanical response of the investigated reinforced polymer foam. The failure envelope of the material expands with increasing strain rate and decreasing temperature showing some degree of coupling between the two effects. The failure stress loci appear to be significantly affected by the stress state, delineating an observable compression-tension asymmetry and an appreciable sensitivity to the ratio between the applied direct and shear stresses. The measured results allow for the assessment of existing failure criteria and motivate the development of novel constitutive models. Acknowledgments The authors would like to thank Rolls-Royce plc and the EPSRC for the support under the Prosperity Partnership Grant \Cornerstone: Mechanical Engineering Science to Enable Aero Propulsion Futures, Grant Ref: EP/R004951/1. The authors are grateful to Mr. S. Carter, Mr. J. Fullerton, Mr. P. Tantrum, and Mr. D. Robinson for their assistance in the manufacturing of the apparatus and specimens, Mrs. K. Bamford for her immense help with procurement.

References 1. Gupta, N., Zeltmann, S.E., Shunmugasamy, V.C., Pinisetty, D.: Applications of polymer matrix syntactic foams. JOM. 66(2), 245–254 (2014) 2. Gladysz, G.M., Perry, B., McEachen, G., Lula, J.: Three-phase syntactic foams: structure-property relationships. J. Mater. Sci. 41(13), 4085–4092 (2006) 3. Kolsky, H.: An investigation of the mechanical properties of materials at very high rates of loading. Proc. Phys. Soc. Sect. B. 62(11), 676 (1949) 4. Albertini, C., Cadoni, E., Solomos, G.: Advances in the Hopkinson bar testing of irradiated/non-irradiated nuclear materials and large specimens. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 372(2015), 20130197 (2014) 5. Gupta, N., Shunmugasamy, V.C.: High strain rate compressive response of syntactic foams: trends in mechanical properties and failure mechanisms. Mater. Sci. Eng. A. 528(25–26), 7596–7605 (2011) 6. Song, B., Chen, W., Frew, D.J.: Dynamic compressive response and failure behavior of an epoxy syntactic foam. J. Compos. Mater. 38(11), 915–936 (2004) 7. Song, B., Chen, W.W., Lu, W.Y.: Mechanical characterization at intermediate strain rates for rate effects on an epoxy syntactic foam. Int. J. Mech. Sci. 49(12), 1336–1343 (2007) 8. Fan, Z., Miao, Y., Wang, Z., Zhang, B., Ma, H.: Effect of the cenospheres size and internally lateral constraints on dynamic compressive behavior of fly ash cenospheres polyurethane syntactic foams. Compos. Part B Eng. 171, 329–338 (2019) 9. Shabde, V., Hoo, S., Gladysz, K.: Experimental determination of the thermal conductivity of three-phase syntactic foams. J. Mater. Sci. 41(13), 4061–4073 (2006) 10. Porfiri, M., Nguyen, N., Gupta, Q.: Thermal conductivity of multiphase particulate composite materials. J. Mater. Sci. 44(6), 1540–1550 (2009) 11. Zeltmann, S.E., Chen, B., Gupta, N.: Thermal expansion and dynamic mechanical analysis of epoxy matrix–borosilicate glass hollow particle syntactic foams. J. Cell. Plast. 54(3), 463–481 (2018) 12. Shunmugasamy, V.C., Pinisetty, D., Gupta, N.: Thermal expansion behavior of hollow glass particle/vinyl ester composites. J. Mater. Sci. 47, 5596–5604 (2012) 13. Arezoo, S., Tagarielli, V.L., Siviour, C.R., Petrinic, N.: Compressive deformation of Rohacell foams: effects of strain rate and temperature. Int. J. Impact Eng. 51, 50–57 (2013) 14. Zhang, L., Townsend, D., Petrinic, N., Pellegrino, A.: The dependency of compressive response of epoxy syntactic foam on the strain rate and temperature under rigid confinement. Compos. Struct. 280, 114853 (2022) 15. Pellegrino, A., Tagarielli, V.L., Gerlach, R., Petrinic, N.: The mechanical response of a syntactic polyurethane foam at low and high rates of strain. Int. J. Impact Eng. 75, 214–221 (2015) 16. Xu, Y., Lopez, M.A., Zhou, J., Farbaniec, L., Patsias, S., Macdougall, D., et al.: Experimental analysis of the multiaxial failure stress locus of commercially pure titanium at low and high rates of strain. Int. J. Impact Eng. 170, 104341 (2022) 17. Zhou, J., Xu, Y., Lopez, M.A., Farbaniec, L., Patsias, S., Macdougall, D., et al.: The mechanical response of commercially pure copper under multiaxial loading at low and high strain rates. Int. J. Mech. Sci. 224, 107340 (2022) 18. Xu, Y., Farbaniec, L., Siviour, C., Eakins, D., Pellegrino, A.: The development of split Hopkinson tension-torsion bar for the understanding of complex stress states at high rate. In: Dynamic Behavior of Materials, Volume 1: Proceedings of the 2020 Annual Conference on Experimental and Applied Mechanics, pp. 89–93. Springer International Publishing (2021) 19. Xu, Y., Zhou, J., Farbaniec, L., Pellegrino, A.: Optimal design, development and experimental analysis of a tension–torsion Hopkinson bar for the understanding of complex impact loading scenarios. Exp. Mech. (2023). https://doi.org/10.1007/s11340-023-00942-1

Practical Considerations for High-Speed DIC Phillip Jannotti, Nicholas Lorenzo, and Samantha Cunningham

Abstract Digital image correlation (DIC) is an imaging technique that enables full-field measurements of a material motion and deformation (displacement, velocity, strain, and strain rate). Advances in high-speed (HS) and ultra-high-speed (UHS) camera technology have driven widespread adoption of HS and UHS DIC especially when sampling rates in the kHz to MHz are required. A common challenge in the application of DIC to high-rate material loading are issues with the applied speckle pattern failing during high-rate loading. A primary requirement of DIC is that the applied speckle pattern follows the underlying surface. When materials of interest are subjected to high-velocity impact, the applied pattern can debond or otherwise fail to follow the motion and deformation of the material surface. This work will evaluate common speckle pattern preparations for both distributed loading and localized deformation. The study will outline unique challenges associated with characterization high-rate material testing and describe a laser-based methodology for probing time-resolved pattern integrity. Photonic Doppler velocimetry will be used to study the effectiveness of the various patterns in tracking the surface deformation including woven composite panels and thin metal plates. The technique offers a simple, objective means of quantifying pattern integrity at selected measurement points and directly comparing different pattern techniques. Keywords High-speed DIC · Pattern failure · Best practices · Dynamic testing

Introduction Digital image correlation (DIC) is an imaging technique that enables full-field measurements of displacement, velocity, strain, and strain rate. The technique relies on tracking motion and deformation of a semi-random speckle pattern applied to the surface of materials or structures of interest. The popularity and application of this technique is continuously growing in academic and industry research because DIC provides valuable insights into time-resolved material and structural behavior. Advances in high-speed (HS) and ultra-high-speed (UHS) camera technology have driven widespread adoption of HS and UHS DIC especially when sampling rates in the kHz to MHz are required. At present, DIC has been used with high-rate mechanical evaluation using Kolsky bar [1] and Taylor impact [2, 3] as well as more extreme loading due to impact [4, 5] and blast [6]. Owing to the fact that DIC is a surface measurement technique, a primary assumption of the testing is that the applied speckle pattern precisely follows the underlying material surface. This is particularly relevant for dynamic material testing where materials of interest may be subjected to extreme dynamic loading, e.g., shock loading or large strain deformation. Under these loading conditions, the applied pattern can debond or otherwise fail to follow the motion and deformation of the material surface. Understanding the limitations of various pattern application techniques and pattern failure is of utmost interest for properly characterizing the relevant dynamic behavior of materials. The focus of the current study was establishing best practices when conducting HS and UHS DIC testing, specifically understanding pattern failure during dynamic mechanical testing. Various pattern preparation techniques were examined for specimens subjected to impact loading such as spray paint and temporary tattoos. Woven composite panels were chosen to evaluate large strain deformation and deformation gradients, while metal plates were used to investigate shock loading and fragmentation. The in situ material response will be recorded using high-speed imaging and photonic Doppler velocimetry (PDV). PDV is a laser-based method of measurement surface velocity at a “point” based on Doppler shifts in the light reflected from the moving surface. It was considered to determine pattern failure through DIC analysis and by placing tracers at prescribed points on the region-of-interest (ROI); however, DIC post-processing and analysis parameters can affect the results P. Jannotti (✉) · N. Lorenzo · S. Cunningham U.S. Army Research Laboratory, Aberdeen Proving Ground, MD, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_13

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Fig. 1 Schematic representation of the experimental test setup

and complicate the comparison. Thus, it was decided to use PDV to provide direct and objective indication of pattern failure. PDV can measure multiple velocities simultaneously within the measurement “spot” at each sampling time point. This study is not simply a DIC study or direct comparison of DIC and PDV data but an understanding of the limitations of different patterning techniques when applied to HS and UHS DIC testing and how to examine their robustness during testing.

Method Pattern deformation will be evaluated in two different series of experiments: (i) woven composite panels of polyethylene (PE) and (ii) thin Al metal plates. A speckle pattern was digitally generated and scaled based on the field-of-view (or image scale) to provide approximately 5 pixel size features. The PE panels were 13 mm thick and patterned using temporary tattoos. The tests compared professionally printed and DIY printable tattoos. Temporary tattoos were chosen for this testing as they are well-suited to relatively large ROIs of 150 by 150 mm, adhere well for distributed deformation, and enable a repeatable means of pattern application. The Al panels were 6.4 mm thick and patterned using spray paint. These tests similarly used digitally generated patterns with feature sizes of at least 5 pixels. Three types of spray paint were used for both the base coating and circular features: acrylic, vinyl, and peelable rubber. Following base coat application, the patterns were applied using a 3 mm thick 3D-printed mask held against the Al plate and several thin coats of each respective paint type. All panels of each material type were procured from the same material/fabrication lot. The dynamic material testing involved launching a 12.7 mm Cu sphere at the flat panel specimens at a nominal velocity of 500 m/s for the PE panels and 1000 m/s for the Al plates. PDV was used to measure both the impact velocity and rear surface specimen motion, shown schematically in Fig. 1. One PDV probe was positioned ahead of the test specimen and aligned to the shot line at an angle of 30° to measure the impact velocity of the Cu sphere. Another PDV probe was positioned directly behind the specimen and aligned to the impact axis. The PDV probes consisted of a GRIN lens attached to a fiber pigtail (AC Photonics 1CL15A070LSD01). PDV data was collected using a high-speed digital oscilloscope (Keysight DSOS604A Infiniium S-Series). An ultra-high-speed optical camera (Shimadzu HPV-X2) was used to qualitatively assess speckle pattern integrity during the sample deformation. A common trigger was provided to the oscilloscope and camera using a p piezopolymer film (PVDF) adhered to the impact face of the material of interest. Illumination was provided by a moonlight strobe (Photogenics 2500DR). More details of the experimental method can be found elsewhere [7].

Results Composite Panels and Temporary Tattoos Figure 2 shows selected high-speed images of the dynamic response of the composite panel and associated temporary tattoo deformation and/or failure. The images were taken at approximately 50 microseconds after the onset of loading. This illustrates the representative behavior of several different temporary tattoos and clearly demonstrated the feasibility of

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Fig. 2 Selected high-speed images illustrating speckle pattern deformation at comparable times after the onset of loading: (a) professional temporary tattoo and (b, c) DIY temporary tattoos. Note the apparent failure of the DIY tattoo substrate compared to the professional tattoo

Fig. 3 Velocity spectrogram captured on the rear surface of the loaded specimens using PDV: (a) professional tattoo and representative trace from DIY tattoo specimen. Noise in the data and observation of multiple velocity paths indicate pattern failure as PDV measures the velocity of all moving surfaces within the measurement spot

using professional temporary tattoos. As observed in Fig. 2a, professional tattoos remained properly adhered throughout the experiment which would enable proper extraction of the surface deformation using DIC. However, the DIY temporary tattoos, shown in Fig. 2b, c, experienced catastrophic premature failure and spallation. PDV provided a quantitative means of evaluating the pattern motion and deformation (see Fig. 3). Figure 3a shows the pattern response from a professional tattoo. The PDV spectrogram exhibits a single strong velocity trace indicative of the deformation observed on the rear surface of the specimen. Figure 3b is a representative PDV spectrogram for the DIY tattoo behavior and exhibits a “cloud” of velocity distributions within 10 s of microseconds after initial loading. The red arrow in Fig. 3b denotes the initial pattern separation which leads the measured peak velocity to dramatically overshoot the “true” specimen rear surface motion (see Fig. 3a). Rather than ~350 m/s, the PDV spectrogram illustrated a velocity jump to 550 m/s as the tattoo substrate was ejected ahead of the specimen surface. Given the catastrophic nature of the pattern failure for the DIY tattoos, DIC analysis would not be feasible and would only allow incipient specimen motion to be measured.

Metal Plates and Spray Paint Figure 4 shows selected high-speed images of bulging and plugging of the Al plate during high-velocity loading. The images were taken approximately 30 microseconds after the onset of loading. The severe high-rate loading caused spallation and

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Fig. 4 Selected high-speed images illustrating speckle pattern deformation at comparable times after the onset of loading: (a) acrylic paint, (b) vinyl coating, and (c) rubber peal coat. Note the apparent failure of the acrylic paint and vinyl coating compared to the rubber coating

Fig. 5 Velocity spectrograms captured from the rear surface motion of the loaded specimens using PDV: (a) acrylic paint, (b) vinyl coating, and (c) rubber peel coat. Noise in the data and observation of multiple velocity paths indicate pattern failure as PDV measures the velocity of all moving surfaces within the measurement spot. Enlarged views of the velocity data from (a–c) and given in (d–f), respectively

fragmentation of the brittle paints like acrylic and vinyl, as seen in Fig. 4a, b. On the other hand, the rubber spray paint (see Fig. 4c) was able to more ductily conform and track the rear surface motion. Representative PDV spectrograms are given for the three different spray paint types. Figure 5a–c shows the overall motion of the pattern surface from initial bulging to plugging and fragmentation. The red-dotted boxes indicate regions that were enlarged for closer examination and the red arrow denoted observation of pattern failure. From the PDV data, the acrylic and vinyl pattern survived only 10 microseconds of deformation which enables measurement of only initial plate bulging, while the rubber coating remained well-adhered for nearly 30 microseconds until plugging and fragmentation.

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Conclusions A methodology was presented for time-resolved characterization of DIC speckle pattern deformation and failure. The general principles of the techniques are easily applied to a range of different test cases, including brittle and ductile material behavior, loading states, and strain rates. The concurrent use of ultra-high-speed imaging allowed for cleared interpretation of the PDV spectrograms. The technique provided objective, quantitative evaluation of pattern integrity and would be useful for the initial screening of pattern techniques prior to running dynamic material studies using high-speed and ultra-high-speed DIC. This is intended to alleviate wasted time performing DIC experiments and trying to interpret analyzed data as means of determining pattern suitability which requires user input for analysis and post-processing parameters. Acknowledgments The authors sincerely thank Eric Pierce, Gregory Beatty, and Eugene Norton for their assistance in conducting this work.

References 1. Moy, P., Walter, T.: Ultra-high speed imaging for DIC measurements in Kolsky bar experiments. In: Advancement of Optical Methods & Digital Image Correlation in Experimental Mechanics, Volume 3: Proceedings of the 2018 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing (2019) 2. Jannotti, P.: Time-resolved characterization of impact testing. In: Challenges in Mechanics of Time-Dependent Materials & Mechanics of Biological Systems and Materials, Volume 2: Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing, Cham (2022) 3. Jannotti, P., Lorenzo, N., Meredith, C.: Time-resolved characterization of Taylor impact testing. In: Challenges in Mechanics of Time Dependent Materials, Volume 2: Proceedings of the 2020 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing (2021) 4. Jannotti, P., Lorenzo, N., Walter, T., Schuster, B., Lloyd, J.: Role of anisotropy in the ballistic response of rolled magnesium. Mech. Mater. 160, 103953 (2021) 5. Jannotti, P., Schuster, B., Doney, R., Walter, T., Andrews, D.: Instrumented penetration of metal alloys during high-velocity impacts. In: Dynamic Behavior of Materials, Volume 1: Proceedings of the 2016 Annual Conference on Experimental and Applied Mechanics. Springer International Publishing (2017) 6. Spranghers, K., Kakogiannis, D., Ndambi, J., Lecompte, D., Sol, H.: Deformation measurements of blast loaded plates using digital image correlation and high-speed photography. In: EPJ Web of Conferences. Vol. 6. EDP Sciences (2010) 7. Jannotti, P., Doney, R., Schuster, B.: Time-resolved measurement of deformation of metal plates due to high-velocity impacts. Proc. Eng. 204, 276–283 (2017)

Dynamic Behavior of Lungs Subjected to Underwater Explosions Helio Matos, Tyler Chu, Brandon Casper, Matthew Babina, Matt Daley, and Arun Shukla

Abstract An experimental investigation was performed on an artificial human lung to evaluate its response and behavior after being subjected to an underwater explosive blast. The experiments were performed using a 63 mg TNT equivalent explosive charge placed 0.5 m from the front of the lung. The specimen used was a to-scale lung model representative of a 50th-percentile male. The experiments were performed on an 8200 l water tank. The artificial lung was instrumented with internal pressure sensors to record changes in the cavity pressure. Additionally, the underwater tank was instrumented with underwater blast transducers to measure the pressure from the explosive charge. Results show a significantly delayed response to the underwater blast due to the lung’s inertia. The lung initially contracts after the underwater shock, followed by an expansion showing a 50% change in relative volume within 7 ms. Results and observations qualitatively relate to the types of injuries observed during preexisting case studies. Keywords Underwater explosive · Primary blast injury · Human lungs · 3D printed organs · Surrogate organs

Introduction Underwater blasts propagate further and injure more readily than equivalent air blasts. Hence the response to underwater shock needs to be well understood independently from air blasts [1]. Injuries from direct interaction with underwater blast are referred to as primary blast injuries (PBI). The organs most vulnerable to PBI are the gas-filled organs, namely the ear, the lungs, and the gastrointestinal tract [2]. This work focuses its analysis and discussions on the lungs due to their importance and vulnerability to underwater blasts. Blast injuries have interested the medical community since the early 1920s. Underwater blast injuries specifically became of interest after World War II, when these types of injuries became more prominent. There are detailed recorded instances that describe the consequences of being exposed to an underwater blast, which include: abdominal and chest pain, transient paralysis, nausea or vomiting, difficulty breathing, and coughing blood, among other symptoms [3]. Since the 1950s, many analyses have been done to understand underwater blast injuries based on unfortunate case studies [3–8] so better treatment, prevention methods, and guidelines can be developed. Experiments that quantitatively measure the blast interaction with the human body do not exist for clear ethical reasons. For this reason, much of what is known about underwater PBI is based on previously mentioned case studies. Direct measurements are needed for numerical or analytical work to avoid unnecessary assumptions and over-simplifications. It is unclear if such assumptions or simplifications are reasonable without direct verification. Fortunately, to date, an extensive range of experimental platforms are available to study blast injury [9]. In addition, new technologies such as additive manufacturing allow for the manufacturing of surrogates that have nearly identical mechanical properties to human organs, which are created to the same shape and scale as the actual human organ, as this study did. These advancements allow for ethical experimentation to be performed on accurate surrogate to human organs. This work seeks to understand the human lung’s response after being subjected to an underwater blast. To accomplish this goal, underwater blast experiments were performed on an artificial to-scale printed adult lung instrumented with internal pressure sensors.

H. Matos (✉) · T. Chu · A. Shukla University of Rhode Island, Kingston, RI, USA e-mail: [email protected] B. Casper · M. Babina · M. Daley Naval Submarine Medical Research Laboratory, Groton, CT, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_14

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Fig. 1 Printed human lung

Experimental Procedures Lung Models The artificial lung used in this experimental study was made and instrumented by Creare LLC (Hanover, NH). The artificial lungs used in this study are shown in Fig. 1. This lung was made to scale representative of a 50th percentile male. The lung was created using multi-material additive manufacturing with material systems that mimic human tissues (silicon rubber Ecoflex, FlexFoam, and Dragonskin systems by Smooth-On, Inc.). The lung system was printed on a 1-inch diameter stainless steel rod with two eye loops to be supported throughout the experimental procedures. The lung was instrumented with hydrophone pressure sensors (Reason TC4013 by Teledyne Technologies) on each cavity so that the pressure variations within each cavity could be measured during blast loading.

Experimental Setup The experimental facility consists of a 2.1 m semi-spherical pressure vessel, as illustrated in Fig. 2. The lung specimen is suspended at the center of the tank, and then the tank is filled with 8200 l of water. During the experiments, lungs were submerged at one-meter depth, which resulted in an average hydrostatic pressure of approximately 10 kPa (1.5 psi). For this reason, the lungs were pressurized to an internal cavity pressure of 10 kPa, measured with a high-accuracy low-pressure gauge (McMaster-Carr 4026K16). This pressurization ensures that the initial volume is approximately equal to a 50th percentile’s male functional residual capacity (FRC) during the underwater explosion (UNDEX) event. The explosive used during these experiments was an RP-87 detonator (from Teledyne Technologies, Thousand Oaks, CA). The explosive charge had a 0.49 ± 0.01 m standoff from the lungs. Additionally, the charge was placed at the midspan of the tank, aligning with the lower sternum’s location. During the UNDEX event, three PCB W138A06 blast transducer sensors (PCB Piezotronics, Inc., Depew, NY) captured localized pressure histories at 2 MHz, along with the internal lung instrumentation mentioned in the previous section (through an Astro-med Dash® 8HF-HS from Astro-Med Inc., West Warwick, RI). The blast transducers are located at the tank’s midspan, approximately 0.3, 0.5, and 0.8 m from the charge and 0.3, 0.1, and 0.7 m from the lungs, respectively.

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Fig. 2 Experimental setup schematic

Experimental Considerations The artificial lung systems are heavily instrumented, and custom-made to full scale, which makes this system cost-prohibiting. Hence, all cases and trials were performed in the same systems. For this reason, the standoff distance was strategically chosen for a relatively “low” UNDEX shock (based on the weight of the RP-87 charge). The lung systems were also inspected before and after each experimental trial to ensure the lung systems remained in good condition.

Results and Discussions Pressure Loading The incident shockwave generated from the RP-87 UNDEX charge was consistent throughout all experiments. The incident shock profile is shown in Fig. 3, where the incident pressure is plotted as a function of time for different sensor locations. In Fig. 3 and subsequent images, zero time indicates the time the charge ignited. In addition, in Fig. 3 and subsequent plots, the average value from three different trials is plotted as a solid line, and the standard deviation from these trials is shown as a shaded region for the same respective color. The shockwave generated from the small charge propagates into a spherical pressure front, in which its magnitude decays exponentially as the wavefront expands outwards. The pressure profile shows an instantaneous rise as the shock front passes through the pressure sensor, followed by the exponential decay, which is typical UNDEX behavior. The pressure reaches a peak of approximately 2.2 MPa (325 psi) at the lung system standoff distance. In addition, since the UNDEX was ignited in an enclosed tank, and the charge was placed relatively far from the lungs, hence, close to the tank’s walls, there are notable reflection pulses from the tank’s wall that trails the initial incident pulses after 0.7 ms. After the initial shockwave and pressure pulses, an UNDEX cavitation bubble is formed due to the extremely high pressures at the source location, leading to a phase change in the water and off-gassing of the explosive burn-off [10, 11]. This bubble expands outward until its internal pressure falls below the equilibrium pressure and collapses on itself. Meanwhile, during the bubble’s expansion and collapse cycle, the fluid boundary generated by the bubble leads to fluid particle motion in the water. This fluid particle motion is initially outwards relative to the charge’s location during bubble expansion, then inwards once the bubble begins to collapse. When the bubble fully collapses, a dynamic pressure pulse is generated due to the rapid change in fluid particle’s momentum. This dynamic pressure pulse leads to subsequent cavitation bubbles, in a reversible and repeatable manner.

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Fig. 3 UNDEX incident shock and bubble pulse

The charge is far enough from the lung system for any bubble interaction with the lungs to occur. However, the pressure waves generated from this phenomenon do reach the lung system. Though the pressure magnitude from the bubble pulse is significantly lower than the initial shock pressures, it is longer in duration. Hence the bubble impulse is comparable to the initial shock impulse [11]. It is worth noting that the standard deviation plotted in Figure 3 through the shaded is primarily due to variations in the timing of the bubble pulse (±0.1 milliseconds between different trials) rather than variations in magnitude. The average pulse magnitude is consistent with the magnitude of the different trials.

Lung’s Internal Pressure and Volume UNDEX Response As mentioned, the lung system was manufactured to represent a 50th percentile male and to have air cavities that represent the functional residual capacity (FRC). For the average human male, the FRC is around 2.5 L [12]. Based on the mold volume ratios used to manufacture the lungs, we can estimate that the left lung has 1.13 L and the right lung has 1.37 L of air when kept at environmental pressure (10 kPa of cavity pressure when submerged 1 m below water). When this internal pressure is neglected, the lungs contract due to the weight of the surrounding water, leading to a lower initial volume, as illustrated in Fig. 4.

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Fig. 4 Front view of the lungs with (a) no pressure and (b) 10 kPa of pressure, as well as the side view with (c) no pressure and (d) 10 kPa of pressure

Fig. 5 (a) Pressure and (b) relative volume variation within the lung system after being subjected to an UNDEX shock

The experimental cases shown in this work include the lung system with the 10 kPa internal pressure that supports the water weight and without the internal pressure. The air cavity overpressure (pressure above its ambient pressure) from each case is shown in Fig. 5a as a function of time. In addition, using the ideal gas law and assuming an adiabatic process (PVγ = Constant; where γ= ratio of specific heats, assuming 1.4 for air), the change in air volume ratio within each lung is calculated using Eq. (1) and is shown in Fig. 5b. During the volume ratio calculation, the value of one indicates the original system volume.

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Air Volume Ratio =

V P = o P Vo

1 γ

ð1Þ

Within the lungs, transmitted shockwaves in the order of 5 kPa are seen as the shockwave, and reflected pressures, interact with the lungs during the first two milliseconds. After the initial shock interaction, there is a large delay in response due to the lungs’ inertia. The response from the shock is initially an increase in overall pressure that peaks around 40 kPa at 5 ms, followed by a pressure decay. The pressure decays to about -20 kPa when the pressure wave from the first bubble pulse interacts with the lung. The pressure history indicates that the bubble pulse provides additional momentum for the lungs to reach -30 kPa at 12 ms, after which further contraction and expansion cycles occur until around 40 ms, as illustrated in Fig. 5a. After 40 ms, changes in overpressure and volume are negligible relative to what is shown in Fig. 5, hence, the focus on analyzing the response is for the first 40 ms. The right and left lung cavities follow a similar pressure history; interestingly, the internal lung pressure did not significantly alter the pressure response. This indicates that the change in internal pressure did not substantially change the lungs’ compliance. The inertial effects and the rate at which they occur were also very similar, with nearly the same compliance and the same mass. Some of the minor differences in pressure behavior include: the right lung is slightly more sensitive to the change in momentum during the first bubble collapse; the lungs without the internal pressure showed a slightly faster response; and the upper range of pressure for all trials was 60 kPa for the lungs with 10 kPa of internal pressure and 40 kPa without internal pressure. The relative change in volume shown in Fig. 5b is inversely proportional to the change in internal pressure, as calculated from Eq. (1). The initial -20% relative compression is related to the 40 kPa initial rise in pressure. The subsequent drop in pressure to -30 kPa at 12 ms is associated with the lung expanding 30% of its original volume. This shows that the subsequent pressure drop, though smaller in absolute magnitude than the initial pressure rise, is related to a more significant relative change in relative volume. Also, when looking at 5–12 ms, the relative volume change is around 50% on average for such a short duration. It is important to note that this is not enough time for inhaling or exhaling to occur, hence, this compression and expansion is happening with the same amount of mass within the lungs. Moreover, since the pressure histories for both the pressurized and unpressurized systems are similar, the relative volume change is also similar. However, it is important to note that the unpressurized system has a lower initial volume, as illustrated in Fig. 4, which means that its absolute volume change is also proportionally lower, and that the unpressurized lung system undergoes less severe compression and expansion cycles (in terms of displacements or strains). This shows that lowering the air cavity volume through exhaling could be considered a mitigation measure to proportionally reduce stresses from the subsequent expansion and contraction cycles.

Conclusion This work aimed to understand the human lungs’ response after being subjected to an underwater blast. To accomplish this, underwater blast experiments were performed on artificial lung systems with and without a supporting ribcage. The key findings are summarized as follows: • Transmitted shockwaves in the order of 5 kPa are seen inside the lung cavities as the shockwave and reflected pressures interact with the lungs during the first two milliseconds. • The lungs show a large delay in response due to its inertial effects, leading to delayed expansion and contraction cycles after the shockwave has passed. • The bubble pulse provides additional momentum to the lungs that impact its expansion and contraction cycles. • The right and left lung cavities follow a similar pressure history as they respond to the UNDEX shockwave. • The internal lung pressure did not significantly alter the pressure response within the lung cavity. However, since the lower pressure is related to a lower initial volume, the subsequent expansion and contraction cycles have lower magnitudes. • After the initial pressure increase within the lung cavities, the drop in pressure leads to a greater relative change in cavity volume. • The relative change in lung volume between the expansion and contraction cycles is 50% on average. Acknowledgments The authors would kindly like to acknowledge the financial support provided by ONR.

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References 1. Lance, R.M., Capehart, B., Kadro, O., Bass, C.R.: Human injury criteria for underwater blasts. Plos One. 10(11) (2015). https://doi.org/10.1371/ journal.pone.0143485 2. Argyros, G.J.: Management of primary blast injury. Toxicology. 121(1), 105–115 (1997). https://doi.org/10.1016/s0300-483x(97)03659-7 3. Huller, T., Bazini, Y.: Blast injuries of the chest and abdomen. Arch. Surg. 100(1), 24 (1970). https://doi.org/10.1001/archsurg.1970. 01340190026008 4. Wolf, N.M.: Underwater blast injury – a review of the literature. (1970). https://doi.org/10.21236/ad0722666 5. Coppel, D.L.: Blast injuries of the lungs. Br. J. Surg. 63(10), 735–737 (1976). https://doi.org/10.1002/bjs.1800631003 6. Phillips, Y.Y.: Primary blast injuries. Ann. Emerg. Med. 15(12), 1446–1450 (1986). https://doi.org/10.1016/s0196-0644(86)80940-4 7. Mackenzie, I.M., Tunnicliffe, B.: Blast injuries to the lung: Epidemiology and management. Philos. Trans. R. Soc. B Biol. Sci. 366(1562), 295–299 (2011). https://doi.org/10.1098/rstb.2010.0252 8. Grant, M., Ladner, J., Marenco, C., Roberge, E.: Transcavitary penetrating trauma—comparing the imaging evaluation of gunshot and blast injuries of the chest, abdomen, and pelvis. Curr. Trauma Rep. 6(2), 83–95 (2020). https://doi.org/10.1007/s40719-020-00192-9 9. Nguyen, T.-T., Pearce, A.P., Carpanen, D., Sory, D., Grigoriadis, G., Newell, N., Clasper, J., Bull, A., Proud, W.G., Masouros, S.D.: Experimental platforms to study blast injury. J. R. Army Med. Corps. 165(1), 33–37 (2018). https://doi.org/10.1136/jramc-2018-000966 10. Matos, H., Javier, C., LeBlanc, J., Shukla, A.: Underwater Nearfield Blast Performance of hydrothermally degraded carbon–epoxy composite structures. Multiscale Multidiscip. Model. Exp. Des. 1(1), 33–47 (2018). https://doi.org/10.1007/s41939-017-0004-6 11. Javier, C., Galuska, M., Papa, M., LeBlanc, J., Matos, H., Shukla, A.: Underwater explosive bubble interaction with an adjacent submerged structure. J. Fluids Struct. 100, 103189 (2021). https://doi.org/10.1016/j.jfluidstructs.2020.103189 12. Siriwardena, M., Fan, E.: Descent into heart and lung failure. Mech. Circ. Respir. Support, 3–36 (2018). https://doi.org/10.1016/b978-0-12810491-0.00001-1

Dynamic Behavior of Curved Aluminum Structures Subjected to Underwater Explosions Matthew Leger, Helio Matos, Arun Shukla, and Carlos Javier

Abstract This research arises from the concern of damage to submersible marine structures such as underwater vehicles and pipelines and the need to understand dynamic failure. This work focuses on analyzing the dynamic bubble-to-structure interaction of curved metallic plates subjected to air-backed underwater explosive loading. This work aims to expand the current understanding of gas bubble formations during nearfield underwater explosion events with experimental analysis. The experiments were performed in an underwater explosive facility using high-speed cameras to measure full-field displacement and velocities during deformation through a digital image correlation technique. In addition, during the experiments, pressure transducers were used to record the pressure pulses emanated from the explosive charge. Experiments were performed for two plate curvatures (112 and 305 mm) and three standoff distances from the plate’s surfaces (55, 73, and 110 mm) for each curvature. The experiments show that deformations are higher if the explosive standoff is smaller or the structural radius of curvature is higher. In addition, the increase in structural deformation also leads to distorted bubble shapes that increase their repulsion from the structural specimen and decrease the bubble collapse period. Nearfield bubble dynamics are highly sensitive to the proximity of nearby structures and the compliance of such structures. Though they have higher deformations in general, complicated structures may be able to mitigate bubble attachment or jetting for specific explosive charges Keywords Underwater explosion · Fluid structure interaction · Cavitation · Bubble dynamics

Introduction An experimental investigation was conducted to evaluate the dynamic response of air-backed curved aluminum plates subjected to nearfield blast from an underwater explosive (UNDEX) loading. This research arises from the concern of damage to submersible marine structures such as underwater vehicles and pipelines and the need to understand dynamic failure. In undesirable circumstances, marine structures may be subjected to dynamic pressure loads near an explosive or adjacent to systems that collapse dynamically, such as implodable volumes. Moreover, curved marine structures are prevalent in the marine community. Therefore, this work is essential to improve the current knowledge based on this event, which can be used to enhance underwater structural systems design. The shockwave and subsequent cavitation bubbles of an UNDEX in a free-field environment have been extensively studied since the 1940s. At the detonation time, the explosive creates intense pressure. These high pressures cause the fluid to undergo gaseous phase changes in combination with the gaseous by-products of the explosive, resulting in a gas cavitation bubble [1], referred to as UNDEX bubble. The energy released by different phases of an UNDEX has also been experimentally calculated in past work [2]. Furthermore, the energy associated with an UNDEX bubble is reversible [2]. Reversible energy means the bubble reverts many cycles after the initial detonation in a free-field environment. Hence, it undergoes expansion and collapse cycles until energy within the system is lost. In addition, when the UNDEX bubble is near a rigid surface, the differential pressure caused by resistance to water flow close to the surface may result in the bubble collapsing onto the surface and producing a high-speed water jet [3, 4].

M. Leger · H. Matos (✉) · A. Shukla Dynamic Photomechanics Laboratory, Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI, USA e-mail: [email protected] C. Javier Naval Undersea Warfare Center (Division Newport), Newport, RI, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_15

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Fig. 1 URI’s experimental setup schematic

To the authors’ knowledge, experimental work does not exist where curved metallic plates were subjected to air-backed UNDEX loading that focuses on the bubble-to-structure interaction. This research aims to expand the current understanding of gas bubble formations during nearfield UNDEX events and fill the current knowledge gap with experimental data and analysis. The present work contributes to the UNDEX topic on air-backed structures by experimentally studying the interaction between structures with curvatures and UNDEX bubbles. To do this, three different UNDEX standoff distances were experimentally evaluated on two different plate curvatures.

Experimental Methods The Underwater Explosive facility utilized for the experiments is illustrated in Fig. 1. Two cameras were used to view the specimen deformation, and a third high-speed camera was also used to record the expansion and collapse cycle of the UNDEX bubble. The specimen target was fully clamped with the speckled side facing the high-speed camera pair. Furthermore, the experimental pressure data was measured and recorded by pressure transducers positioned around the explosive at predetermined intervals. The underwater explosion was created using an RP-87 detonator from Teledyne RISI. The tourmaline sensors element was placed in line with the explosive and positioned at the target midspan of the specimen.

Results The initial pressure pulse and the first bubble collapse from the UNDEX charge are the two major contributors to damage to a nearby structure during an UNDEX event. Most of the subsequent discussions and analyses will focus on the behavior of the structure and the bubble within this period. To aid in these discussions, the average pressure from the initial UNDEX shock and the bubble collapse are all summarized and presented in Table 1 for all experimental cases. Sensors 1, 2, and 3 represent 55, 73, and 110 mm distances from the explosive. The sequence of events of an UNDEX bubble’s first period is shown in Fig. 2 (a) for the 112 mm curvature and (b) for the 305 mm plates. Figure 2 illustrates the impact of the structural proximity on the bubble shape. The images show the bubble characteristics at key moments during the UNDEX event, including the maximum displacement of the plate after the initial explosive shockwave, the maximum bubble size, and the first bubble collapse. During the bubble growth phase, when near

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Table 1 Summary of peak pressures and bubble collapse period Explosive standoff (mm) 55

73

110

Sensor 1 2 3 1 2 3 1 2 3

112 mm curvature Shock pressure (MPa) 18.70 13.69 7.54 21.22 13.87 8.53 22.40 13.15 8.70

Bubble pressure (MPa) 1.44 0.80 0.57 0.69 0.86 0.54 3.42 2.19 1.15

Bubble period (ms) 11.5

11.7

12.0

305 mm curvature Shock pressure (MPa) 21.23 13.02 8.01 21.51 12.93 8.27 24.55 15.30 9.47

Bubble pressure (MPa) 1.17 0.59 0.96 1.67 1.12 1.28 5.55 2.73 1.94

Bubble period (ms) 10.7

11.0

11.5

Fig. 2 Bubble’s behavior near the (a) 112 and (b) 305 mm curvature plates

another structure, the bubble’s natural spherical shape is lost due to the limited fluid between the bubble and the plate. The change in spherical shape will be referred to as blunting of the bubble. The blunting is affected by standoff distance and plate geometry. When comparing the two curvatures, the noticeable differences from the images are the level of blunting the surface inflicts on the bubble and the characteristics of the bubble at bubble collapse. The structural interaction from the initial shock pressure as well as subsequent bubble pressure and momentum is illustrated in Fig. 3. In Fig. 3, the center point displacement of each plate and each standoff distance is shown. The values illustrated in Fig. 3 are for one representative trial for each case. The center point displacement is the maximum displacement of the plate during shock and bubble loading, and it is used herein as a representative value to describe the overall deformation

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Fig. 3 Center point displacement of all standoff distances for the (a) 112 mm curvature and (b) 305 mm plates

behavior of each plate. The displacement values for the structure were obtained using the 3D DIC analysis on the air-backed side of the plate. Furthermore, positive displacement in Fig. 3 means displacement away from the explosive change. The shockwave from the explosive causes the plate to deform away from the explosive. With closer standoff distances, the deformation increases as the standoff distance decreases due to the larger pressure shown in Table 1. Furthermore, the structures are seen to retract near their original position after reaching their maximum displacements between 2 and 4 ms. Meanwhile, the UNDEX bubble reaches its maximum size of around 5 ms, as illustrated by Fig. 2, when it transitions from expansion to contraction. During the bubble’s contraction phase, it pulls the plate inwards toward it until it collapses and emits the first bubble’s pressure pulse. After the bubble collapses, the pressure pulses emitted from the bubble lead to a subsequent deformation away from the bubble. The plateau region observed in the displacements after the initial deformation is due to the bubble’s contraction phase [5]. For the 112 mm curvature plate illustrated in Figure 3a, the closer standoff distances of 55 and 73 mm show permanent deformations after the first bubble collapse. In contrast, for the 110 mm standoff, the plate is seen to deform in sync with the subsequent bubble expansion and collapse cycles. All cases show permanent deformation after the first bubble collapse cycle for the 305 mm curvature illustrated in Fig. 3b. Furthermore, the larger structure is also more compliant. Hence, the larger displacements for the 305 mm plate compared to the 110 mm displacements for the same standoff distance.

Conclusion This study investigated an UNDEX near an air-backed curved flexible structure to determine the relationship between plate curvature and explosive standoff distance in plate loading and UNDEX bubble characteristics. The experiments were performed for two different structural curvatures and three standoff distances. The key findings within the confines of this study include the following observations: • The more compliant the surface (larger radius), the less sporadic the bubble behaves between detonation and the second bubble period. • The bubble tends to migrate toward stiffer (smaller radius) structures and away from more compliant (larger radius) structures due to the changes in structural deformations. Acknowledgments The authors would like to acknowledge the financial support provided by the Office of Naval Research and the Naval Undersea Warfare Center (Division Newport) for supporting the project.

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Availability The full analysis of this work has been submitted for Springer’s Experimental Mechanics journal and is pending 2023 publication. More information can be found by searching the journal titled “Experimental Evaluation of Curved Aluminum Structures Subjected to Underwater Explosions”.

References 1. Cole, R.H., (Robert H.: Underwater Explosion. Princeton (1948) 2. Arons, A.B., Yennie, D.R.: Energy partition in underwater explosion phenomena. Rev. Modern Phys. 20(3) (1948). https://doi.org/10.1103/ RevModPhys.20.519 3. Javier, C., Galuska, M., Papa, M., LeBlanc, J., Matos, H., Shukla, A.: Underwater explosive bubble interaction with an adjacent submerged structure. J. Fluids Struct. 100, 103189 (2021). https://doi.org/10.1016/j.jfluidstructs.2020.103189 4. Reid, W.: The Response of Surface Ships to Underwater Explosions. Department of Defence (1996) http://oai.dtic.mil/oai/oai?verb=getRecord& metadataPrefix=html&identifier=ADA326738 5. Matos, H., Javier, C., LeBlanc, J., Shukla, A.: Underwater nearfield blast performance of hydrothermally degraded carbon–epoxy composite structures. Multiscale Multidiscip. Model. Exp. Des. 1(1), 33–47 (2018). https://doi.org/10.1007/s41939-017-0004-6

The Effect of Layering Interfaces on the Mechanical Behavior of Polyurea Elastomeric Foams Mark Smeets, Behrad Koohbor, and George Youssef

Abstract The performance of density-graded elastomeric foams has been a cynosure of the pursuit of superior impact mitigation materials and structures. Elastomeric foams exhibit a remarkable mechanical response, including resilience, toughness, and recoverability. However, recent research has only focused on the performance of uniform-density foam paddings in response to various strain rates. Concurrently, the body of research on the potential of density gradation has been burgeoning, suggesting an untapped potential to achieve higher levels of protection than those offered by their ungraded counterpart. This research aims to elucidate the layering interfaces effect on the performance of density-graded elastomeric foams in response to quasi-static and impact loading. The approach is to manufacture foam laminates consisting of bi- or tri-layered polyurea elastomeric foams using two different layering techniques. In one set of samples, the foam was natively adhered by casting subsequent layers with different densities by adjusting the mixing and pouring ratios. In the second set of samples, separately cast polyurea sheets were adhered using ultrathin polyurea adhesive to mimic the configuration of the first set. All foam samples were submitted to quasi-static loading up to densification and impact loading at 7 J. The static and dynamic stress-strain curves were accompanied by full-field digital image correlation analysis, revealing the contributions of the density gradation and layering interfaces to the overall deformation. While the primary outcomes include insights into the mechanistic processes responsible for the mechanical behavior, the natively bonded density-graded polyurea foams provide an exciting platform to explore additional mechanics of elastomeric foams. Keywords Density-graded · Foams · Quasi-static · Impact

Introduction The severity of impacts hinges on the duration and amplitude of applied force to match the dynamic response of the protected structures, i.e., allowing ample time for the natural response to take place, shielding the structure from imminent failure. An effective and practical approach to reducing the impact severity is padding the structure with cellular solids to increase the absorbed energy while broadening the impact duration through different deformation mechanisms, e.g., cell walls collapse [1]. Elastomeric foams have excellent impact mitigation properties, given their ability to undergo large recoverable deformations at a broad range of strain rates [2, 3]. Polyurea foams, made from a hyper-viscoelastic elastomeric polymer, have recently emerged as promising candidates for dynamic loading scenarios, inheriting the superior mechanical properties of the base material [4–7]. When foamed, polyurea possesses high recoverability even after exhibiting significant deformations, up to densification, under different loading conditions, including quasi-static and impact loadings [8–11]. The densitydependence of the mechanical properties in cellular solids opens avenues for optimizing their impact efficacy through density gradation [12]. That is to say, combining multiple foam layers with different densities, irrespective of the bonding approach, potentially improves the overall mechanical performance. The strategic layering of different densities can improve the performance of polyurea foams without increasing the overall weight. However, the effect of laying interfaces due to the construction of density-graded structures remains an open research question, requiring further investigations, especially when laying elastomeric foams; hence, the motivation of the research led to this presentation. It is imperative to note that this

M. Smeets · G. Youssef (✉) Experimental Mechanics Laboratory, Mechanical Engineering Department, San Diego State University, San Diego, CA, USA e-mail: [email protected] B. Koohbor Department of Mechanical Engineering, Rowan University, Glassboro, NJ, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_16

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research builds on the previously reported quasi-static and dynamic behavior of density-graded foams [13, 14], emphasizing elastomeric polyurea foams. In this research, two sets of density-graded polyurea foams were fabricated and tested. The first sample set was naturally adhered through sequential casting based on the strong adhesive property of polyurea and the conformability of the foam slurry [7, 15]. Since this sample configuration was adhesives-free, it is referred to as “seamless” thereinafter. The second sample set was assembled using an ultra-thin layer of bulk polyurea after sheets were separately fabricated with different densities. This sample configuration is referred to as “adhered”. Irrespective of the bonding approach, aforesaid sample sets comprised bi- and tri-layered, density-graded structures that were tested in the positive gradation orientation, i.e., the layers’ density increasing from top (loaded side) down. In addition, mono-density polyurea foam samples and a benchmark foam were also tested for comparison.

Materials and Methods In this experimental research, seven different polyurea foam configurations were tested by varying the number of graded layers and interface type. All foam sheets were produced in-house from a mixture of modified methylene diisocyanate (Isonate 143L MDI, Dow Chemical) with oligomeric diamine (Versalink P1000, Evonik) using a 1:4 weight ratio, respectively. The two constituents were violently mixed in deionized water, at which point the frothed foam slurry floated atop the remaining water. The drained foam slurry was rapidly cast into a Teflon-coated aluminum mold. The pour weight and mold cavity volume dictated the final thickness and density of the polyurea foam sheets. After a curing period of 24 h, a new foam pour was added to the previously cured sheet to construct the seamless foam sets. The seamless foam was removed once the final layer was cured for 24 h. For the adhered set, individual sheets were de-molded after the 24 h curing period and later bonded together using thin layer polyurea prepared by slowly mixing the same ratios of diamine and isocyanate for ~1 min. The adhered foam sheets were kept under an even pressure of 22 kPa for 36 h to ensure effective bonding. Foam samples of ca. 18 × 18 mm (height varied depending on the final configuration, e.g., bilayer vs. trilayer) were cut off the sheets using a table saw. Each sample subset consisted of five specimens. Table 1 summarizes the different sample configurations with the corresponding abbreviation, the density of each layer, and the effective density for each configuration. The quasi-static testing was done using a universal load frame (Instron 5843) with a 1 kN capacity, recording the force and displacement. The applied compression and decompression cycles were force-controlled at a rate of 250 N/min and a peak load of 500 N. Each sample was tested only once to avoid fatigue or time-dependent effects. The dynamic testing was achieved using a modified CEAST 6990 Dart Tester drop-weight impact tower, which recorded the force at an acquisition rate of 15 kHz. A 950 g mass was dropped from approximately 1000 mm onto the foam sample. Similar to the quasi-static tests, each configuration was tested for five samples. After obtaining the loading portion of the mechanical response for quasi-static and dynamic tests, the average stress-strain curves were calculated and used for performance analysis using two metrics: specific energy absorbed and efficiency. The specific energy absorbed (SEA) is the mass-normalized strain energy density and corresponds to the energy dissipation capacity. SEA was calculated as the area under the stress-strain curve divided by the effective density of the corresponding sample.

Table 1 List of all sample configuration and benchmark foams investigated under quasi-static and dynamic loading, including abbreviations, density of each layer (ρf1/ρf2/ρf3 in kg/m3), effective density accounting for layer heights (ρf in kg/m3). Also listed the specific energy absorption (SEA in J/kg) and peak efficiency (ηmax) as a function of the loading scenario Samples/gravimetric data Sample configuration Seamless Mono-density Mono-density Seamless Bilayer Adhered Bilayer Seamless Trilayer Adhered Trilayer D3O (benchmark foam)

Quasi-static Abb. SM M SB1 SB2 AB ST AT D3O

ρf1 /ρf2 /ρf3

ρf

294/292/255/-/332/287/332/268/309/253/273/248/234 309/253/231 397/-/-

294 255 311 301 281 252 265 397

SEA 934 1095 912 908 881 1014 935 665

ηmax 0.25 0.28 0.24 0.27 0.25 0.21 0.25 0.22

Dynamic SEA 3603 3915 3585 3574 3357 3674 2893 4011

ηmax 0.23 0.25 0.23 0.26 0.23 0.20 0.24 0.23

The Effect of Layering Interfaces on the Mechanical Behavior of Polyurea Elastomeric Foams

SEAðεÞ =

1 ρeff

ε

σ ðεÞdε

113

ð1Þ

0

The energy absorption efficiency, η(ε), compares the energy absorption of a real foam to the ideal energy absorber counterpart with a flat plateau stress extending across all strains. The efficiency as a function of strain is calculated from Eq. 2. ηðεÞ =

ε

1 σ max ðεÞ

σ ðεÞdε

ð2Þ

0

Accompanying the dynamic tests, digital image correlation (DIC) was performed by recording the impact event using a high-speed camera (Photron FASTCAM SAI 1.1) at a rate of 30 kfps. The DIC analysis was done using commercial software (Instra 4D, Dantec Dynamics), where the in-plane strain components were calculated within each layer and surrounding the interfaces.

Results and Discussion Figure 1 shows the average stress-strain curves for the seven polyurea and benchmark foam configurations under quasi-static (Fig. 1a) and impact (Fig. 1b) loading scenarios. Figure 1 also plots the corresponding efficiency (bottom panels) as a function of strain, which was calculated using Eq. 2. Irrespective of the strain rate and foam configuration, the mechanical behavior in Fig. 1 embodies a typical foam response, divided into the linear elastic, plateau, and densification regions. In the case of quasi-

Fig. 1 Average stress-strain curves (top) and efficiency (bottom) for all investigated foam configurations under (a) quasi-static and (b) impact loadings. The maximum efficiency is marked by hollow circles

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static compression, Fig. 1a shows a consistent stress-strain behavior between all considered configurations, exhibiting limited linear elasticity, extended rising plateau, and steep densification initiated at ~70% strain. Nonetheless, slight differences were noted between the sample subsets. For example, the mechanical response of SB1 and SB2 reported ~5% difference in the densification strains despite their similar densities and configuration, suggesting intra-batch variability. In general, bilayer samples outperformed the trilayer counterparts, while the mono-density controls achieved the highest performance metrics among all polyurea foam configurations. These results agree with previous research, where a lower-density polyurea foam exhibited higher specific energy absorptivity [8]. While the performance metrics of polyurea foams eclipsed by those of the benchmark foam, it is imperative to reiterate the density difference, as compiled in Table 1. Results in the performance metrics between seamless and adhered samples indicated no evident difference at these strain rates. This provided evidence that the fabrication of seamless interface polyurea foams can be a viable substitute for the ultrathin adhesive layer, which is substantiated by micrographic analysis using the scanning electron microscope that will be discussed in the presentation. The full-field analysis (e.g., DIC) indicates the adverse effect of layering interfaces, as discussed in the presentation. Figure 1b shows the average stress-strain responses and the corresponding efficiency of foam samples submitted to impact loading using a drop tower. For polyurea foams, the dynamic mechanical response rapidly transitions into the plateau region oscillation in the stress due to localized buckling (i.e., undulation) superimposed up to ~50% strain. Also notable is the delayed densification of polyurea foams compared to the previously discussed quasi-static results. Irrespective of gradation configuration, the maximum strain exceeded 80% due to continued compression-induced shearing in the densification region failing. The energy absorption efficiency as a function of strain in Fig. 1b (bottom panel) shows the differences in performance among the various configurations. Under both quasi-static and dynamic loading, the peak efficiency corresponds to 0.5–0.6 strain (e.g., the onset of densification). In general, some of the polyurea foam configurations (e.g., mono density and seamless bilayer) achieved the highest efficiency values.

Conclusion This research explored the mechanical efficacy of density-graded polyurea foams consisting of two and three layers assembled in two configurations: seamless or adhered interfaces. The seamless interfaces were a byproduct of a sequential mold cast process, where foam layers with different densities were naturally assembled during manufacturing. On the other hand, the adhered interfaces resulted from assembling foam sheets with different densities using an ultrathin layer of bulk polyurea. Micrographic analysis using the scanning electron microscope delineates the structural difference between seamless and adhered interfaces. The mechanical performance was assessed based on two metrics (specific energy absorption, SAE, and efficiency) for quasi-static and impact loading scenarios. Regardless of gradation, lower effective densities yielded higher values of SEA for the strain rates investigated herein and higher peak efficiency in quasi-static compression. The interfaces showed an adverse effect on the efficacy due to localized shear forces, further substantiated by the DIC analysis. That is, results from the quasi-static and dynamic loading, accompanied by full-field digital image correlation provided evidence of minimal differences in the performance between the naturally bonded seamless and adhered layering interfaces. In closing, a seamless interface is preferable over adhered due to the relative ease of fabrication and a better control over the sharp strain gradients developed in the interface vicinity in adhered structures. Acknowledgments The authors acknowledge the support by the National Science Foundation under Grant No. 2035663 (G.Y.) and Grant No. 2035660 (B.K.). The authors are also grateful for internal funding from San Diego State University and Rowan University. Funding from the Department of Defense (G.Y.: W911NF1410039 and W911NF1810477) is also acknowledged.

References 1. Bowen, I.G., Fletcher, E.R., Richmond, D.R.: Estimate of Man’s Tolerance to the Direct Effects of Air Blast. Lovelace Foundation for Medical Education and Research, Albuquerque (1968) 2. Gupta, V., Youssef, G.: Orientation-dependent impact behavior of polymer/EVA bilayer specimens at long wavelengths. Exp. Mech. 54, 1133–1137 (2014) 3. Mills, N.: Polymer Foams Handbook: Engineering And Biomechanics Applications and Design Guide. Elsevier (2007) 4. Youssef, G.H.: Dynamic Properties of Polyurea. University of California, Los Angeles (2011) 5. Do, S., Rosenow, B., Reed, N., Mohammed, A., Manlulu, K., Youssef, G.: Fabrication, characterization, and testing of novel polyurea foam. PU Magazine Int. 16, 104–107 (2019)

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6. Reed, N., Huynh, N.U., Rosenow, B., Manlulu, K., Youssef, G.: Synthesis and characterization of elastomeric polyurea foam. J. Appl. Polym. Sci. 137, 48839 (2020) 7. Youssef, G., Reed, N.: Scalable manufacturing method of property-tailorable polyurea foam. Google Patents. (2021) 8. Ramirez, B.J., Gupta, V.: Energy absorption and low velocity impact response of open-cell polyurea foams. J. Dyn. Behav. Mater. 5, 132–142 (2019) 9. Youssef, G., Reed, N., Huynh, N.U., Rosenow, B., Manlulu, K.: Experimentally-validated predictions of impact response of polyurea foams using viscoelasticity based on bulk properties. Mech. Mater. 148, 103432 (2020) 10. Koohbor, B., Youssef, G., Uddin, K., Kokash, Y.: Dynamic Behavior and Impact Tolerance of Elastomeric Foams Subjected to Multiple Impact Conditions. J. Dyn. Behav. Mater., 1–12 (2022) 11. Youssef, G., Kokash, Y., Uddin, K.Z., Koohbor, B.: Density-dependent impact resilience and auxeticity of elastomeric polyurea foams. Adv. Eng. Mater. 25, 2200578 (2022) 12. Uddin, K.Z., Youssef, G., Trkov, M., Seyyedhosseinzadeh, H., Koohbor, B.: Gradient optimization of multi-layered density-graded foam laminates for footwear material design. J. Biomech. 109, 109950 (2020) 13. Rahman, O., Uddin, K.Z., Muthulingam, J., Youssef, G., Shen, C., Koohbor, B.: Density-graded cellular solids: mechanics, fabrication, and applications. Adv. Eng. Mater. 24, 2100646 (2022) 14. Tan, X., Rodrigue, D.: Density graded polymer composite foams. Cell. Polym. 42, 02624893221143507 (2022) 15. Jain, A., Youssef, G., Gupta, V.: Dynamic tensile strength of polyurea-bonded steel/E-glass composite joints. J. Adhes. Sci. Technol. 27, 403–412 (2013)

Moderate-Velocity Response of Polyurea Elastomeric Foams Paul Kauvaka, Mark Smeets, Behrad Koohbor, and George Youssef

Abstract The suitability of cellular solids for a specific energy absorption application, whether packaging or sports gear padding, depends on their dynamic mechanical behaviors under impact loadings. The latter is imperative not only to simulate real-life loading conditions but also to interrogate the realistic response of the material, contributions of the geometry, and determination of prominent deformation mechanisms. This research aims to extend the application domain of polyurea elastomeric foams through a mechanistic understanding of their response to loading scenarios at moderate impact velocities. Recent research focused on either leveraging quasi-static stress-strain response to forecast the impact efficacy of these foams or submitting the foam pads to low-velocity impacts. Hence, the approach here is to develop a small-scale shock tube to release a projectile into polyurea foam plugs ~31 cm apart. The shock tube was mounted vertically to (1) reduce the logistical impact of the setup and (2) leverage gravity-assisted increase in impact velocity. The impact velocity was controlled by adjusting the pressure in the driver (high pressure) section of the tube. The impact-induced deformation was captured using a high-speed camera. The velocity-time profiles were used to calculate the stress, while the high-speed images were analyzed using digital image correlation (DIC) to report the evolution of strains and inertia stresses. Samples were also examined postdeformation using optical microscopy to assess the induced structural damage. The outcomes of this research extend the property map of polyurea elastomeric foams, gearing them closer to transition into realistic sports protective gear applications. Keywords Shock response · Full field analysis · Polyurea foam

Introduction The severity of mechanical loading based on its amplitude and rate plays a significant role in selecting suitable material candidates and the efficacy of the designed structures to withstand these loading scenarios. In impact mitigation applications, cellular solids, irrespective of their base materials, cell geometry, and cellular structure, are ubiquitous, given their high specific attributes, e.g., stiffness-to-weight and energy-absorption-to-weight ratios [1]. Stochastic (random) and ordered (lattice) cellular solids, specifically those made from polymers, are common in sports applications since they provide high energy absorption at low weight penalty while being easily shaped in complex geometries to conform with the complexity of the human body form. Polymeric foams found broad utility in other impact mitigation applications such as packaging, shoding, and crashworthiness in the automotive industry [2]. Hence, this research strives to explore the shock response of polyurea elastomeric foams to assess their efficacy in moderate and high-velocity impact applications. Polyurea foams have gained increased scientific and technological attention since the recent disclosure of a novel manufacturing process relying on the self-foaming property of frothed emulsions in water [3]. The base material, i.e., polyurea, is a thermoset elastomer that has been vigorously investigated over the past two decades under a broad range of strain rates, showing superior mechanical performance in civilian and military applications [4]. This mechanical performance was attributed to the micro-segmental structure of polyurea consisting of hard domains of isocyanate suspended within a soft matrix of diamine, which is further enhanced by interchain hydrogen bonding [5]. Hence, the interest in foaming polyurea to retain the desirable mechanical behavior while further reducing the weight. In recent reports, several impact case studies have been reported, ranging from helmet liners for American football [3] to protective pads for lateral falls of the elderly [6] and P. Kauvaka · M. Smeets · G. Youssef (✉) Experimental Mechanics Laboratory, Mechanical Engineering Department, San Diego State University, San Diego, CA, USA e-mail: [email protected] B. Koohbor Department of Mechanical Engineering, Rowan University, Glassboro, NJ, USA © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_17

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118 Table 1 List of foam configurations used in this investigation along with the corresponding abbreviations, the individual sheet densities (ρ1/ρ2/ρ3 in kg/m3), and the effective density (ρ in kg/m3)

P. Kauvaka et al. Configuration Seamless Mono-density Mono-density Seamless Bilayer Adhered Bilayer Seamless Trilayer Adhered Trilayer Benchmark foam

Abb. SM M SB1 SB2 AB ST AT BF

ρ1/ρ2/ρ3 294/292 255 332/287 332/268 309/253 273/248/234 309/253/231 397

ρ 294 255 311 301 281 252 265 397

density-graded orthotics [7]. However, the mechanical responses of polyurea foams, single density or density-graded plates, to moderate and high-velocity impacts are currently absent in the open literature. Therefore, the objective of this research is to submit mono-density and density-graded polyurea foams to moderate and high-velocity impacts using a small-scale gas gun that was fitted with high-speed photography to facilitate full-field strain measurements and analysis, i.e., digital image correlation (DIC).

Materials and Methods Polyurea foams with different densities and gradation configurations were prepared using a process developed in-house [8], by violently mixing a Versalink P1000® (oligomeric diamine, Evonik) with an isocyanate [modified Methylene Diphenyl Diisocyanate (MDI), DOW Industrial] at the ratio of 4:1, respectively, and deionized water. The frothed foam slurry was quickly transferred into 31 × 31 cm, Teflon®-coated aluminum mold, where the poured amount and the size of the mold cavity dictated the final sheet density. The mold was covered with a polyethylene plate. Sheets were cured in the closed mold for 24 h, followed by an additional 24 h in ambient conditions to dehydrate the foam before testing. In this research, six polyurea foam configurations were prepared, including monolayer (i.e., single sheet with uniform density), uni-density bilayer (two sheets with same density assembled), bilayer (two sheets with two different densities assembled), and trilayer (three sheets with different densities). Table 1 lists the different configurations, the density of each layer, and the effective density for the density-graded plates. In one density-graded configuration, the sheets were prepared by sequentially casting foam plates on the top of each other, relying on the self-adhesive property of the uncured foam slurry. This configuration is referred to as “seamless” since the interface is naturally created upon casting a new sheet. In the other density-graded configuration, the individually prepared sheets were adhered together using polyurea adhesive (mixing the diamine and isocyanate at the same ratio as stated above without using any water); hence, this configuration is denoted as “adhered” (Fig. 1). All samples were tested in a small-scale gas gun built-in-house that was installed vertically to facilitate experimental logistics. Irrespective of the foam configuration, the specimen was positioned on a rigid platform and centered with the muzzle of the gas gun. The aluminum projectile with flattop (~203 g) was launched onto the sample once a single Mylar® film suddenly ruptured as the pressure build up in the pressure chamber. Concurrently, high-speed images were acquired at a rate of 100 kfps. The projectile speed was estimated to be 27 m/s based on photographic analysis. The high-speed photographs were analyzed using commercial digital image correlation software (Istra4D, Dantech) with a grid size of 31 px and gird spacing of 23 px. The in-plane strain components, including engineering axial, lateral, and shear strains, were extracted by defining virtual gauges within each of the layers (i.e., based on the number of density-graded layers) and the interface areas.

Results and Discussion Figure 2 shows the axial strain as a function of impact event time for several investigated foam configurations, including uni-density single and bi-layer and seamless and adhered bi- and tri-layer samples. Figure 2a is a plot of the axial strain in the uni-density samples, comparing the difference in the mechanical response between single and bi-layer plates to a projectile at a speed of 27 m/s. The axial strain close to the projectile impact site appears to dominate the response, given the close resemblance with the monolayer deformation, as exemplified in Fig. 2a. The strain within the top layer of the bilayer uni-density samples manifested earlier than the deformation in the bottom layer that is distal from the impact site, which is generally in good agreement with previous results based on testing of rigid polyurethane foams [9]. In the presentation, a

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Fig. 1 Schematic of the experimental setup, showing the small-scale gas gun, the safety cage, and assorted samples

discussion is dedicated to the contributions of the interface to the dichotomy in the mechanical response of mono and bilayer elastomeric foams with unified density. On the other hand, Fig. 2b, c demonstrate the effect of interface on the mechanical deformation of graded bi and tri-layer polyurea sheets. In the latter, the deformation in the seamless, graded foam samples appeared to be greater than the adhered counterparts, signifying the influence of the interface quality (e.g., stiffness) on the overall compliance of the tested foam samples. In general, this experimental research shed new light on the mechanical behavior of elastomeric polyurea foams (graded vs. conventional), enabling new directions for improving their impact efficacy.

Conclusion The research leading to this report focused on extending the property map of elastomeric polyurea foams in response to moderate and high-velocity impacts. Specifically, graded and conventional polyurea foam samples were extracted from made in-house sheets and submitted to impact by releasing a flattop projectile using a small-scale gas gun. Two sets of interface types were realized, including seamless (i.e., naturally bonded sheets) and adhered (i.e., bonded using ultrathin polyurea adhesive) interfaces to assess the effect of interface on the overall deformation of these foam structures. High-speed imaging was used to digitally capture the impact event, allowing post-loading full-field strain analysis using digital image correlation. The DIC analysis reported the mean axial strain within each layer while delineating the influence of the interface stiffness on the strain transduction between the layers of the graded foam structures. The results show that at such moderate-velocity impact, the interface type (i.e., seamless vs. adhered) plays a notable role in the overall mechanical deformation, prompting a novel pathway to improve the impact efficacy of elastomeric polyurea foams in developing superior impact-mitigating mechanisms. Acknowledgments This research was supported by the National Science Foundation under grant no. 2035663 (G.Y.) and grant no. 2035660 (B.K.). The authors are also grateful for internal funding from San Diego State University and Rowan University. The authors are grateful to Mr. Jacob Gould for his assistance with the initial construction of the gas-gun.

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Fig. 2 Temporal deformations of (a) uni-density, (b) bilayer-graded, and (c) trilayer-graded polyurea foam samples, showing the effect of the interface types on the overall response

References 1. Gibson, L.J., Ashby, M.F., Harley, B.A.: Cellular Materials in Nature and Medicine. Cambridge University Press (2010) 2. Newacheck, S., Youssef, G.: Synthesis and characterization of polarized novel 0–3 Terfenol-D/PVDF-TrFE composites. Compos. Part B Eng. 172, 97–102 (2019) 3. Reed, N., Huynh, N.U., Rosenow, B., Manlulu, K., Youssef, G.: Synthesis and characterization of elastomeric polyurea foam. J. Appl. Polym. Sci. 137, 48839 (2020) 4. Gupta, V., Youssef, G.: Orientation-dependent impact behavior of polymer/EVA bilayer specimens at long wavelengths. Experimental Mechanics. 54, 1133–1137 (2014) 5. Do, S., Canilao, J., Stepp, S., Youssef, G.: Thermomechanical investigations of polyurea microspheres. Polym. Bull., 1–15 (2021) 6. Youssef, G., Kokash, Y., Uddin, K.Z., Koohbor, B.: Density-dependent impact resilience and auxeticity of elastomeric polyurea foams. Adv. Eng. Mater. 25, 2200578 (2023) 7. Uddin, K.Z., Youssef, G., Trkov, M., Seyyedhosseinzadeh, H., Koohbor, B.: Gradient optimization of multi-layered density-graded foam laminates for footwear material design. J. Biomech. 109, 109950 (2020) 8. Rahman, O., Uddin, K.Z., Muthulingam, J., Youssef, G., Shen, C., Koohbor, B.: Density-graded cellular solids: mechanics, fabrication, and applications. Adv. Eng. Mater. 24, 2100646 (2022) 9. Koohbor, B., Ravindran, S., Kidane, A.: In situ deformation characterization of density-graded foams in quasi-static and impact loading conditions. Int. J. Impact Eng. 150, 103820 (2021)

The Use of Human Surrogate for the Assessment of Ballistic Impacts on the Thorax Martin Chaufer, Rémi Delille, Benjamin Bourel, Christophe Marechal, Franck Lauro, Olivier Mauzac, and Sebastien Roth

Abstract This study described the creation of a physical human thorax surrogate dedicated to blunt ballistic impacts called SurHUByx. The geometry of this new surrogate is based on an existing numerical model, named HUByx, which consists of a biofidelic 50th percentile human torso finite element model. In order to build the physical dummy, and to choose appropriate materials for anatomical structures, able to reproduce correctly the human behavior, a reverse engineering procedure was applied. Material properties of the numerical model (especially the bone structures, as well as internal organs) were simplified in order to match properties of manufacturable materials: Trabecular and cortical bones, as well as costal cartilage were merged, and then modeled with a homogeneous material, whereas internal organs were made of synthetic gel, which has already proven its biofidelity. These simplifications lead to the creation of a new numerical “simplified” biomechanical model (named SurHUByx FEM). It was then used to replicate experimental reference cases conducted on Post Mortem Human Subjects. Results were consistent with the experimental biomechanical corridors. The model being validated, the construction of the physical dummy began: Internal organs (heart, lungs, liver, and spleen) were molded in 3D printed molds with SEBS gel. Muscle and Mediastinum were molded with 3D printed molds using various concentrations of SEBS gel. Skin was made with the vinyl skin of the Hybrid III dummy. In the same way as for the numerical validation, the new physical dummy was submitted to the same impact cases. The whole procedure allowed creating a biofidelic dummy with manufacturable materials, which can be used for ballistic accident reconstruction, as it was already performed in the literature in the crashworthiness framework. Keywords Human torso · Blunt trauma · Ballistic impact · BABT

Introduction In recent years, optimization has become a key focus in the development of new technologies, particularly in the field of safety and protection. In the context of ballistics, the optimization of protective devices, such as body armor, can help to improve the performance of police officers and soldiers while also reducing their weight. Similarly, the development of new less-lethal kinetic energy (LLKE) weapons requires a thorough understanding of the human thorax behavior under blunt impact to ensure both non-lethality and enough stopping efficiency. In both cases, it is necessary to study Behind Armor Blunt Trauma (BABT). However, conducting experiments on humans or cadavers is governed by strict ethical rules, making it complex to conduct. Therefore, researchers have developed surrogates that can be used to mimic the human thorax behavior behind armor or a LLKE weapon. Two types of surrogates exist: numerical and physical ones. Recently, the use of numerical surrogates has gained wide interest, leading to the creation of several biofidelic finite element models dedicated to blunt ballistic impacts such as HUByx (Hermaphrodite Universal Body YX) [1–3], SHTIM (Surrogate Human Thorax for Impact Model) [4] or WALT (Waterloo Thorax Model) [5] which are consistent with biomechanical corridors and/or various field impact cases. However, the difficulty of such development lies in the characterization and numerical modeling of body armor. Therefore, physical surrogates are sometimes preferred to compare armor performance responses. Until now, only clay has been standardized by M. Chaufer (✉) · S. Roth Interdisciplinary Laboratory Carnot of Bourgogne, site UTBM, UMR 6303, CNRS / Univ. Bourgogne Franche-Comté (UBFC), Belfort, France e-mail: [email protected] R. Delille · B. Bourel · C. Marechal · F. Lauro Laboratory LAMIH UMR 8201 CNRS, Univ. Polytechnique Hauts-de-France, Valenciennes, France O. Mauzac French Ministry of the Interior, CREL/SAILMI, Paris, France © The Society for Experimental Mechanics, Inc 2024 V. Eliasson et al. (eds.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50646-8_18

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the NIJ standard [6]. Other materials, such as 10% or 20% ballistic gelatin, Permagel, ballistic soap, Roma Plastilina No. 1 clay, or styrene-ethylene-butylene-styrene (SEBS) gel-based have also been used, but these surrogates consist of a cubic block [7]. To take into account human anthropometry, anthropomorphic human surrogates have also been developed, including Ausman [8], SSO (Skin-Skeleton-Organs) [9], MHS (Modular Human Surrogate) [10], HSTM (Human Surrogate Torso Model) [11], and BTTR (Blunt Trauma Torso Rig) [12]. To the author’s knowledge, there are only a few physical surrogates in the literature that are consistent with ballistic biomechanical corridors, such as BTTR [12]. However, since these surrogates do not include internal organs, only global data can be recorded. Roberts et al. attempted to create a more detailed human torso model by developing a numerical finite element model (HTFEM) and using it to construct their physical surrogate (HSTM) [13, 14]. However, the similarities between these two models were limited, as the initial numerical model was not considered biofidelic, and thus the physical model was not considered biofidelic either. To overcome this, the authors proposed a reverse engineering method, using a biofidelic numerical model as the basis for developing a biofidelic physical surrogate of the human thorax. Following the proposed reverse engineering method, the structure of the HUByx finite element model was simplified to make it more feasible for manufacturing [15]. The material properties used in the initial FE model were extracted from human tissues and were replaced with materials that have similar properties and are readily available in the industry. The simplified model, known as the SurHUByx FEM (Surrogate HUByx), combined the trabecular and cortical parts of bones and cartilage of HUByx FEM into a single entity, modeled the spine as a single part, and only included essential organs such as the lungs, heart, liver, and spleen. Finally, the behavior of the SurHUByx FEM was evaluated using Bir et al.’s experiments [16], which are widely used as a reference in the ballistic field. The results showed that the SurHUByx FEM exhibited consistent behavior in terms of force and deflection over time within the established corridors, and its anthropometry was consistent with the 50th percentile. This confirmed that SurHUByx FEM can be used as a basis for building a physical surrogate. Therefore, this study proposes to go through the creation of the SurHUByx physical model, which will be the physical twin of the SurHUByx FEM. After it is created, this surrogate will be compared to the Bir et al. corridors to assess its biofidelity. And to validate it for protection assessments.

Methods The HUByx FEM [1, 2] served as a reference and starting point. It was then streamlined to create a new FE model called SurHUByx FEM. This new model was designed to have a manufacturable structure and uses material laws based on readily available materials in the industry. These two factors were crucial for the creation of the physical twin: SurHUByx. To achieve this, the properties of bones, cartilage, and trabecular were homogenized. After computing the homogenized properties, several manufacturable materials were tested to find the ones with the closest mechanical properties. They were then implemented in the SurHUByx FEM. The model was compared with literature to estimate its anthropometry, and wellknown tests were replicated to validate its response against experimental biomechanical corridors. Results obtained with SurHUByx FEM showed good agreement with biomechanical corridors, allowing to go further and begin the development of the SurHUByx. Once fabricated, such as SurHUByx FEM, the SurHUByx dummy was compared to biomechanical corridors using Bir et al. experiments [16]. Figure 1 proposes an overview of the reverse engineering method used to develop the SurHUByx.

Preparation of the Surrogate In order to create the physical surrogate, a computer-aided design (CAD) model of the SurHUByx was created. The SurHUByx geometry was based on the mesh of the SurHUByx FEM, and surfaces were imported into CATIA V5 software. The junctions between parts were created using a mortise and tenon system with glue for the ribs/cartilage and cartilage/ sternum junctions. In the SurHUByx FEM, the mesh was continuous at these junctions, so these parts were inseparable and the change of material was straight. In the physical surrogate, the ribs/spine junction was designed using a system of holes and two axes to maintain the ribs in position, allowing for sufficient movement to mimic breathing movements. In the SurHUByx FEM, the mediastinum was modeled with sph particles to transfer efforts to the internal organs. In the physical surrogate, the mediastinum was made of solid material with shaped holes to allow space for the organs and transfer efforts to them. The mediastinum was also made in two parts (front and rear) to insert the organs inside. In the SurHUByx FEM, the muscle was

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Fig. 1 Reverse engineering procedure: from finite element model to its physical twin

Fig. 2 SurHUByx FEM (a), SurHUByx CAD model (b), SurHUByx (c), SurHUByx without skin (d) and SurHUByx without muscle (e)

not in direct contact with the ribs, but in the physical surrogate the intercostal muscles were embedded in the muscle and the mediastinum. Skin, which was meshed continuously with the muscle in the SurHUByx FEM, was tight and adjusted all around the muscle of the physical surrogate. A visual comparison of the SurHUByx FEM and the SurHUByx CAD model is shown in Fig. 2a, b. Concerning materials, as SurHUByx FEM material properties were directly issued from manufacturable materials, SurHUByx was build using these materials. Numerical properties of the SurHUByx FEM and the corresponding manufacturable materials are gathered in Table 1. Thus a polyurethane resin was used for bones, a polymeric resin for costal cartilage, a gel based on Styrene-Ethylene-Butylene-Styrene (SEBS) with various concentrations for organs, mediastinum and muscle and vinyl material for skin. To build parts of the SurHUByx, several manufacturing processes were used. First, a casting method was used to create the bones (ribs, sternum, and spine) of the SurHUBy. Due to bones suppleness, soft molds were required to help the demolding process (Fig. 3). First, the parts were 3D printed in polylactic acid (PLA) using the fused deposition modeling (FDM) method (Fig. 3a). These printed parts were then used as the positive to create molds in silicon (Fig. 3b, c). After the curing time, the parts in PLA were gently removed by cutting the molds on one side (Fig. 3d). Then, the material was poured into the molds to form the final bones (Fig. 3e). After demolding, the sprue was cut, then the area was sanded until a clean and continuous surface was obtained (Fig. 3f). Secondly, the cartilage was created using a molding process. As the cartilage material was soft, rigid molds were used. Molds of the cartilage were manufactured using an additive manufacturing process. Each mold was made of two parts, and a silicon spray was applied on the mold surfaces to facilitate demolding. Material was poured into the mold and strategically placed vents helped to extract air bubbles.

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Table 1 Material parameters used in SurHUByx FEM Tissues Manufacturable material

Cartilage Elastomeric resin Elastic

Mediastinum Gel based on SEBS Elastic

Internal organs Gel based on SEBS Ogden

Muscle Gel based on SEBS Elastic

Density (kg/m3) Young modulus (MPa) Fracture plastic strain (–) Yield stress (MPa) Poisson ratio (–) Mu 1 Mu 2

Bones Polyurethane resin Elastic plastic Johnson Cook 1220 2225 0.03 16.5 0.33 – –

1000 16 – – 0.35 – –

1000 0.05 – – 0.45 – –

880 – – – 0.495 (Time dependent)

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Skin Vinyl Ogden 1500 – – – 0.499 0.318 0.401 1.492 3.316

Fig. 3 Overview of the casting process

Third, the gel based on SEBS used for the mediastinum, internal organs, and muscle were molded using 3D printed molds made of high-temperature polyamide carbon fiber reinforced filament. This material had to be heated up to 145 °C to be in a liquid form. Each internal organ was created using a 2-part mold. Due to the large size of the mediastinum and muscle, the molds were printed in several pieces and assembled using a silicon sealant. A silicon spray was applied inside all the molds to facilitate demolding, then the molds were heated before pouring the SEBS into. Then, molds were cooled down slowly to minimize shrinkage, once molds reached room temperature, parts were demolded. Fourth, the skin was made up of vinyl pads which were welded together as needed using an iron. A zipper was sewn in the back of the surrogate allowing to easily mount and unmount the skin. This system also ensures the skin to be well adjusted to the muscle. Finally, all the parts were assembled. Figure 2 shows similarities between SurHUByx FEM (a), SurHUByx CAD model (b) and SurHUByx physical surrogate (c and d). Figure 2e shows a view of the SurHUByx without the muscle layer, allowing to see the organs embedded into the rib cage and mediastinum.

Experimental Setup To validate the consistency of the physical surrogate with the biomechanical behavior, the authors proposed to recreate the well-known experimental tests conducted by Bir et al. [16]. These tests involved thirteen post-mortem human subjects impacted over the sternum by various rigid projectiles at different velocities. Three impact cases were conducted: case A with a projectile of 140 g launched at 20 m/s, case B (140 g–40 m/s) and case C (30 g–60 m/s). These tests allowed Bir et al. to draw

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Fig. 4 Point of impact (a), experimental setup (b), projectiles (c)

biomechanical corridors. To recreate these impacts similar conditions were used: a pneumatic launcher was used to launch projectiles at the desired speed. The SurHUByx was placed on an inclined surface so that the impact would occur directly anterior to the 8th thoracic vertebrae. Projectiles were launched over the mid sternum, skin was removed to precisely adjust the impact location (Fig. 4a). The distance from the launcher to the surrogate was 50 cm (Fig. 4b). To control the speed of projectiles inside the pneumatic launcher, guide rings were necessary. The projectile for case A and B was 140 g (sabot projectile and rings included) and 100 mm long and 36.5 mm in diameter. The projectile used for case C was 30 g (tracking rod included), 28.5 mm long (without the tracking rod), and 36.5 mm in diameter. The projectiles were made of Rubber Baton L5A7 (Pains Wessex Schermuly (UK)) (Fig. 4c). To follow the projectile and to record data, images were recorded using a lateral camera recording at 22000 fps. A tracking method using marquers on projectiles was used to catch displacements. Force was computed using the mass of projectiles and displacement to compute acceleration. Finally, a comparison between cadavers, SurHUByx, and SurHUByx FEM responses was conducted regarding force-time, deflection-time curves, and Vcmax values over the three impact conditions.

Results and Discussions The tracking method used allowed the projectiles to be precisely monitored. Once the contact between the projectile and the skin ended, the tracking of projectile was stopped. Force/time, deflexion/time curves were plotted for each impact cases. Deflexion time curve of case A and C for the SurHUByx showed a compression and a relaxation phase, which was not observed with SurHUByx FEM. In addition to deflection and force measurements, the typical parameter for thoracic impacts, VCmax (Maximal Viscous Criterion), was also calculated for the three impact conditions. Results obtained from tracking were consistent with corridors for all the 3 impact cases. SurHUByx was in the upper part of the displacement/time corridors for case A and C and in the middle of the corridor for case B (Fig. 5). SurHUByx was in the lower part of the force-time corridors for case A and B and in the upper part for case C (Fig. 6). Regarding VCmax values, for case A SurHUByx showed a higher VCmax than the corridors, for case B and C VCmax values were on the upper part of the corridors. Figure 7 (left) presents SurHUByx FEM, SurHUByx, and the experimental range VCmax values. Moreover, the sternal fracture observed on cadavers by Bir et al. was also observed on SurHUByx for case B only (Fig. 7 (middle and right)). The consistency of the surrogate response regarding corridors and sternal fracture allowed to validate SurHUByx behavior in terms of global response. The rebound observed in case A and C may be due to the material used to build muscle and mediastinum. Material with a higher level of energy absorption would delay the rebound. Indeed, this event did not occur with SurHUByxFEM in which muscle and mediastinum materials were represented as linear elastic materials. Discrepancies between SurHUByx and

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Fig. 7 VCmax comparisons between cadaveric experiments, SurHUByx and SurHUByx FEM (left), fracture pattern over the sternum for case B: with muscle (middle), sternum only (right)

SurHUByx FEM highlighted the necessity to fully characterize the SEBS gel-based materials used in mediastinum and muscle. Nevertheless, as corridors used in this study were built using old adults (from 56 to 88 years old), authors wonder if such rebound is consistent with active human thorax behavior. To take into account living effect, some researchers used living

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animal to conduct such experiments. Both methods have their advantages, with cadavers providing the best morphologic similarities and live animals offering the best pathophysiologic similarities [16, 17]. While some studies have compared the results of cadaver and live animal experiments in ballistic impacts [18], to the best of the authors’ knowledge, no studies have compared the response of cadavers, live animals, and living humans in ballistic situations. No conclusion can be drawn concerning which subject (living animals or cadavers) provided the most consistent result regarding living humans. It also may be interesting to replicate impact cases described in [18] in order to compare SurHUByx behavior to living animal and cadaver responses regarding the same impact condition. Further work will consist to add sensors inside the surrogate organs and replicate field impact cases in order to build the probability function of injury for each organ. Impact cases and methodology described in [3, 5, 8, 19] may be used. In addition, materials used to represent muscle and mediastinum in the SurHUByx have to be fully characterized and implemented in the SurHUByx FEM in order to reduce discrepancies between the physical and the numerical model.

Conclusion A physical surrogate based on the existing biofidelic finite element model SurHUByx FEM, which is a simplified version of the HUByx FE model, was constructed. The process involved exporting and reconstructing all surfaces of the SurHUByx FEM to create a computer-aided design (CAD) model. This was necessary to construct the geometries at the junctions of materials, and the type of junction used was chosen to be as close as possible to the numerical model. Once the CAD model was built, the creation of molds began. The molded parts were then assembled to create the physical twin of the SurHUByx FEM. Once created, the SurHUByx was confronted with Bir et al. experiments. The recreation of experiments conducted on post-mortem human subjects allowed the comparison of displacement/time and force/time curves with biomechanical corridors. Results showed good agreements with experimental data and corridors, and the observed sternal fracture in case B was also observed on SurHUByx. This procedure allowed the creation of a biofidelic 50th percentile human torso surrogate. In the future, replication of field case impacts on the SurHUByx coupled with adding sensors inside the surrogate may allow the building of injury criteria, which would help in protection assessment. Acknowledgments This study was conducted as part of a PhD thesis research project supported by the French “Direction Générale de l’Armement” (DGA). The authors also acknowledge the financial support of the French Ministry of Interior.

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