Challenges in Mechanics of Time-Dependent Materials, Volume 2

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Challenges in Mechanics of Time-Dependent Materials, Volume 2 H. Jerry Qi Bonnie Antoun Richard Hall Hongbing Lu Alex Arzoumanidis Meredith Silberstein Jevan Furmanski Alireza Amirkhizi Joamin Gonzalez-Gutierrez Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Tom Proulx Society for Experimental Mechanics, Inc. Bethel, CT, USA

River Publishers H. Jerry Qi • Bonnie Antoun • Richard Hall • Hongbing Lu Alex Arzoumanidis • Meredith Silberstein • Jevan Furmanski Alireza Amirkhizi • Joamin Gonzalez-Gutierrez Editors Challenges in Mechanics of Time-Dependent Materials, Volume 2 Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-898-9 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2015 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, or reproduction 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.

Preface Challenges in Mechanics of Time-Dependent Materials, Volume 2: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics represents one of the eight volumes of technical papers presented at the SEM 2013 SEM Annual Conference & Exposition on Experimental and Applied Mechanics, organized by the Society for Experimental Mechanics and held in Greenville, SC, June 2–5, 2014. The complete proceedings also include volumes on: Dynamic Behavior of Materials; Advancement of Optical Methods in Experimental Mechanics; Mechanics of Biological Systems and Materials; MEMS and Nanotechnology; Composite, Hybrid, and Multifunctional Materials; Fracture, Fatigue, Failure and Damage Evolution; and Experimental and Applied Mechanics. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics, the Mechanics of Time-Dependent Materials being one of these areas. This track was organized to address constitutive, time (or rate)-dependent constitutive, and fracture/failure behavior of a broad range of materials systems, including prominent research in both experimental and applied mechanics. Papers concentrating on both modeling and experimental aspects of Time-Dependent Materials are included. The track organizers thank the presenters, authors, and session chairs for their participation and contribution to this track. The support and assistance from the SEM staff is also greatly appreciated. Atlanta, GA, USA H. JerryQi Livermore, CA, USA Bonnie Antoun Wright-Patterson AFB, OH, USA Richard Hall Dallas, TX, USA Hongbing Lu Evanston, IL, USA Alex Arzoumanidis Ithaca, NY, USA Meredith Silberstein Los Alamos, NM, USA Jevan Furmanski San Diego, La Jolla, CA, USA Alireza Amirkhizi Ljubljana, Slovenia Joamin Gonzalez-Gutierrez v

Contents 1 Unimorph Shape Memory Polymer Actuators Incorporating Transverse Curvature in the Substrate............................................................ 1 Jason T. Cantrell and Peter G. Ifju 2 Yield Criterion for Polymeric Matrix Under Static and Dynamic Loading........................ 11 B.T. Werner and I.M. Daniel 3 Investigating Uncertainty in SHPB Modeling and Characterization of Soft Materials ................ 21 Christopher Czech, Aaron J. Ward, Hangjie Liao, and Weinong W. Chen 4 Diffusion of Chemically Reacting Fluids through Nonlinear Elastic Solids and 1D Stabilized Solutions ............................................................ 31 Richard Hall, H. Gajendran, and A. Masud 5 Effect of Temperature on Mechanical Property Degradation of Polymeric Materials ................ 41 Tong Cui, Yuh J. Chao, John W. Van Zee, and Chih-Hui Chien 6 Small Strain Plasticity Behavior of 304L Stainless Steel in Glass-to-Metal Seal Applications .......... 49 Bonnie R. Antoun, Robert S. Chambers, John M. Emery, and Rajan Tandon 7 Observations of Rate-Dependent Fracture of Locally Weakened Interfaces in Adhesive Bonds ................................................................... 55 Youliang L. Guan, Shantanu Ranade, Ivan Vu, Donatus C. Ohanehi, Romesh C. Batra, John G. Dillard, and David A. Dillard 8 Time Dependent Response of Composite Materials to Mechanical and Electrical Fields .............. 65 K.L. Reifsnider 9 Characterizing the Temperature Dependent Spring-Back Behavior of Poly(Methyl Methacrylate) (PMMA) for Hot Embossing................................... 73 Danielle Mathiesen and Rebecca Dupaix 10 Thermomechanical Fatigue Evaluation of Haynes® 230®for Solar Receiver Applications ............ 81 Bonnie R. Antoun, Kevin J. Connelly, Steven H. Goods, and George B. Sartor 11 Viscoelastic Characterization of Fusion Processing in Bimodal Polyethylene Blends ................. 89 Aaron M. Forster, Wei-Lun Ho, Kar Tean Tan, and Don Hunston 12 Viscoelastic Properties for PMMA Bar over a Wide Range of Frequencies ........................ 95 T. Tamaogi and Y. Sogabe 13 Implementation of Fractional Constitutive Equations into the Finite Element Method............... 101 L. Gaul and A. Schmidt 14 Effect of Pressure on Damping Properties of Granular Polymeric Materials ...................... 113 M. Bek, A. Oseli, I. Saprunov, N. Holecˇek, B.S. von Bernstorff, and I. Emri vii

15 Flow of Dry Grains Inside Rotating Drums ................................................ 121 G. De Monaco, F. Greco, and P.L. Maffettone 16 Statistical Prediction of Tensile Creep Failure Time of Unidirectional CFRP...................... 131 Yasushi Miyano, Masayuki Nakada, Tsugiyuki Okuya, and Kazuya Kasahara 17 Thermal Crystallinity and Mechanical Behavior of Polyethylene Terephthalate.................... 141 Sudheer Bandla, Masoud Allahkarami, and Jay C. Hanan 18 Effect of UV Exposure on Mechanical Properties of POSS Reinforced Epoxy Nanocomposites ........ 147 Salah U. Hamim, Kunal Mishra, and Raman P. Singh 19 Overcoming Challenges in Material Characterization of Polymers at Intermediate Strain Rates ....... 153 William J. Briers III 20 Prediction of Statistical Distribution of Solder Joint Fatigue Lifetime Using Hybrid Probabilistic Approach.................................................... 165 Hyunseok Oh, Hsiu-Ping Wei, Bongtae Han, Byung C. Jung, Changwoon Han, Byeng D. Youn, and Hojeong Moon 21 Effect of Moisture and Anisotropy in Multilayer SU-8 Thin Films .............................. 171 C.J. Robin and K.N. Jonnalagadda 22 Shrinkage Coefficient: Drying Microcrack Indicator......................................... 177 Dragana Jankovic 23 Thermo-Fluid Modeling of the Friction Extrusion Process .................................... 187 H. Zhang, X. Deng, X. Li, W. Tang, A.P. Reynolds, and M.A. Sutton viii Contents

Chapter 1 Unimorph Shape Memory Polymer Actuators Incorporating Transverse Curvature in the Substrate Jason T. Cantrell and Peter G. Ifju Abstract Shape memory polymers (SMP) utilized in reconfigurable structures have the potential to be used in a variety of settings. This paper is primarily concerned with the use of Veriflex-S shape memory polymer and bi-directional carbon fiber in a unimorph actuator configuration. One of the major deficiencies of SMP unimorphs is the permanent set (unrecovered shape) after a single or multiple temperature cycle(s). The novel concept of incorporating transverse curvature in the composite substrate, similar to that of an extendable tape measurer, was proposed to improve the shape recovery. A set of experiments was designed to investigate the influence of transverse curvature, the relative widths of SMP and composite substrates, and shape memory polymer thickness on actuator recoverability after multiple thermomechanical cycles. Flat carbon fiber and shape memory polymer unimorph actuators were evaluated for performance versus actuators of increasing transverse curvature. Digital image correlation was implemented to quantify the out-of-plane deflection of the unimorph composite actuators (UCAs) during the actuation cycle. Experimental results indicate that an actuator with transverse curvature significantly reduces the residual deformation while increasing the shape memory recoverability which could be further tailored to enhance the performance of shape memory polymers in reconfigurable arrangements. Keywords Shape memory polymer • Unimorph • Transverse curvature • Digital image correlation • Composite Nomenclature CF Carbon fiber DIC Digital image correlation MAV Micro air vehicle SMP Shape memory polymer Tg Glass transition temperature u, v, w Lengthwise widthwise, and vertical displacements UCA Unimorph composite actuator VIC Visual image correlation x, y, z Lengthwise widthwise, and vertical coordinates 1.1 Introduction Shape memory polymers (SMPs) are a category of smart material with the ability to change their shape upon the application of external stimuli such as temperature electricity, magnetism, or light. Classes of smart materials include piezoelectric, shape memory alloys, and shape memory polymers. Varieties of smart materials practical for various applications include shape memory alloys in orthodontic treatments, piezoelectric actuators for control of micro air vehicles, shape memory J.T. Cantrell (*) • P.G. Ifju Mechanical and Aerospace Engineering Department, University of Florida, MAE Receiving, 134 MAE-C, Gainesville, FL 32611, USA e-mail: jasontcantrell@gmail.com; ifju@ufl.edu H.J. Qi et al. (eds.), Challenges in Mechanics of Time-Dependent Materials, Volume 2: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-06980-7_1, #The Society for Experimental Mechanics, Inc. 2015 1

polymers as cardiovascular stents, and a multitude of smart materials for the morphing of aircraft structures [1–5]. VeriflexS, the SMP used during the studies in this paper, uses a thermal external stimulus to allow reconfiguration and recovery. This material’s properties, and those of its higher temperature counterpart, Veriflex-E, have been studied extensively by researchers including Fulcher et al. [6–8], Nahid et al. [9], McClung et al. [10–14], Liu et al. [15], and Atli et al. [16]. The Veriflex SMPs have been utilized for notable applications including active disassembly for recycling, deployment of satellite solar panels, and deployable aircraft wings [17–19]. Veriflex can be divided into two categories of stiffness and material behavior: the high glassy modulus and low rubbery modulus [12, 13]. At temperatures below their glass transition temperature (Tg), the material is relatively stiff and has a high elastic modulus; however once the SMP is heated above Tg the modulus drops by several orders of magnitude. This transition from the glassy to the rubbery state is illustrated in Fig. 1.1. In the rubbery state, shape memory polymers can deform at levels up to 400 % and after cooling below Tg maintain this new shape indefinitely [20]. The original shape can be recovered by heating the polymer above Tg again. The glassy state is classified as the temperatures lying 10 C or more below the Tg, while the rubbery state is identified as temperatures lying 10 C or greater above the Tg [21]. The area in between the glassy and rubbery state is classified as the transition region in which the elastic modulus transitions rapidly. SMPs can change their shape from their original cast shape (flat beams in this study) to a deformed shape and return to the original shape when exposed to elevated temperatures. An illustration of an ideal shape memory thermomechanical cycle is shown in Fig. 1.2. The SMP begins in its original shape at a high modulus below Tg and then heat is applied to the sample causing the modulus to fall into the rubbery state. Once in the rubbery state the sample is bent into the desired deformed shape (a U-shaped configuration for this study) and then allowed to cool below Tg locking the current deformed shape. The sample can be stored at this configuration to await the reapplication of heat. After heating the sample will release and return to the unconstrained original form. The sample is then cooled and would ideally return to 100 % of the original shape seen before the heating cycle. However, in reality, the Veriflex SMP can achieve a final shape that is only close to the original shape. Various researchers have studied this behavior and determined that the recoverability of SMP can vary between 65 and 95 % of the original shape after repeated cycling depending upon testing conditions [12, 18, 22, 23]. Despite these hindrances SMPs are still advantageous over other shape memory materials due to the fact that they are low cost low density, and highly deformable among other benefits [24, 25]. Shape memory materials are valued for their potential use in adaptive structures in applications such as micro air vehicles (MAVs) and morphing aircraft [26]. The University of Florida has worked with adaptive structures and MAVs extensively, adopting both active adaptation Fig. 1.1 Generalized plot of the elastic modulus (E) versus temperature for Veriflex SMP. The graph is divided into the plastic (T <Tg), transition, and rubbery (T >Tg) regions Fig. 1.2 Illustration of a shape memory cycle for recovery of thermally activated SMPs 2 J.T. Cantrell and P.G. Ifju

with piezoelectric actuators and passive adaptation with flexible membrane wings [2, 3, 27–31]. Ifju et al. developed a bendable load stiffened MAV wing that is compliant in the downward direction for storing the aircraft, but uses the wing curvature to avoid buckling due to flight loads [32]. Using this knowledge of MAVs and morphing wing structures, a plan of study was devised for a multipurpose morphing actuator to determine if the same bendable composite technology used in MAV wings could assist in increasing the recoverability of the Veriflex-S SMP. In order to properly understand the overall performance of the SMP in a unimorph composite actuator (UCA) configuration, extensive digital image correlation (DIC) testing was required to determine the residual deformation present. A UCA was described as an element capable of bi-stable configuration when supplied with an external stimulus consisting of one active layer (SMP) to which the stimulus is applied and one inactive layer (carbon fiber laminate) that supports the active layer. A simple flat carbon fiber (CF) beam with SMP adhered to its surface was compared against a transversely curved CF beam with curvature similar to that of the MAV wing discussed previously. Additionally, a more detailed survey investigating the influence of other variables present was also documented via DIC. The details of the UCA analysis and the experimental procedure are explained in the subsequent sections. During the course of experimenting with flat generic unimorph actuators, research indicated that by incorporating a transverse curvature (similar to an extendable tape measure) in the composite layer one can vastly improve the shape recovery of an SMP unimorph actuator. The following paper will cover the findings from the investigation of generic SMP unimorphs. 1.2 Unimorph Composite Actuator Experimental Procedure 1.2.1 Unimorph Composite Actuator (UCA) Fabrication Each unimorph composite actuator consists of a layer of SMP bonded to a graphite/epoxy substrate. Both flat carbon fiber composite unimorphs and unimorphs incorporating transverse curvature followed the same fabrication methodology. A single layer of [ 45 ] oriented plain weave, bi-directional carbon fiber was cut and placed on a Teflon covered plate or curved tooling board. The entire assembly was covered in an additional layer of Teflon, vacuum bagged, and cured at 130 C for 4 h. After curing, the carbon fiber was cut down to the appropriate size, then a Veriflex-S shape memory polymer panel was bonded to it using Araldite 2011 two-part epoxy. After the epoxy cured the actuators were coated with a base coat of flat white spray paint then speckled for DIC using flat black spray paint. 1.2.2 Digital Image Correlation (DIC) Set-Up The primary objective of this research was to determine the deformation and shape of the composite beam samples. This was done through the use of the DIC system a non-contact, full-field shape and deformation technique developed at the University of South Carolina [33, 34]. The system uses two Point Grey Research 5-megapixel grayscale cameras to simultaneously capture images of the random speckle pattern applied to the samples. The cameras are calibrated via a high contrast dot pattern of known diameter and spacing. In these set-ups, it was a 9 9 grid of points with a separation of 10 mm. Once calibrated, the system is ready to photograph the composite beam and determine deflection as a function of time. Reference images of the beams were initially taken after the samples were painted. Subsequent images were taken before starting each testing cycle. These images were contrasted against images taken over the hour observation time to determine the deflection as the sample cooled. Images are captured via VIC Snap 2009 and processed via VIC-3D 2009 to determine deformations. Figure 1.3 shows the digital image correlation experimental set-up to measure the remaining deformation on the generic UCA specimens. 1.2.3 Environmental Chamber Set-Up The UCAs were placed in a Sun Systems Model EC12 environmental chamber and was used to regulate the temperature to the desired point above the shape memory polymer glass transition temperature. The temperature was monitored via a thermocouple inside of the chamber and confirmed via a Fluke 561 series infrared thermometer. Beam samples were placed on a Teflon plate within the chamber to allow for full expansion under elevated temperature conditions. 1 Unimorph Shape Memory Polymer Actuators Incorporating Transverse Curvature in the Substrate 3

1.2.4 UCA Sample Holder Set-Up Once samples were removed from the environmental chamber they were folded into a U-shaped configuration as shown in Fig. 1.4, and stored in a tabletop retainer to ensure equivalent loading conditions for all actuators. This apparatus consisted of five 1/4-20 bolts in a U configuration secured to the table in order to constrain the samples from folding inwardly, and two metal blocks spaced 60 mm apart to constrain the samples in the outward direction. 1.2.5 Procedure to Measure UCA Recoverability Step-by-Step Procedure to Measure Shape Recovery of the UCA Using DIC The procedure for measuring the out-of-plane residual deformation with DIC after a temperature cycle is enumerated below. Step 1. After applying a speckle pattern to the sample take an initial (reference) image of the UCA using the DIC set-up. Step 2. Place the undeformed UCA in the environmental chamber for 1 h at 85 C. Step 3. Bend the UCA beam into a U-shaped configuration and place it within the holder to cool for 1 h in the stored configuration. Step 4. Return the sample to the environmental chamber set to 85 C and allow the beam to hold for 1 h at temperature. Step 5. Remove UCA from the oven to start recovery to original position. Step 6. Monitor via DIC while the UCA cools to room temperature. Fig. 1.3 Experimental set-up for DIC analysis of the UCAs Fig. 1.4 (a) Sample container without a sample. (b) Sample container holding a curved carbon fiber beam 4 J.T. Cantrell and P.G. Ifju

1.3 Unimorph Composite Actuator Results Out-of-plane deflection (w) was the main focus of the UCA experiments. The goal was a beam with minimal residual deformation that when stored would hold the desired shape. Initial samples consisted of 200 mm long by 38 mm wide flat (zero curvature) and 63.5 mm radius of curvature carbon fiber samples. These samples consisted of a 12.7 mm wide and 1.6 mm thickness strip of shape memory polymer adhered via Araldite 2011 epoxy to the center of the carbon fiber beam. Figure 1.5 shows a mock-up of these variables on a concave curved carbon fiber beam. Figure 1.6 shows a schematic of the flat concave, and convex actuators used for testing. Post-processing of the DIC data was required to properly determine the deflection for each UCA over time. Postprocessing was done by extracting the XYZ coordinates and UVW displacements for the centerline of each sample at the desired timestamp. Next the data are extracted to an Excel file, the deformation (W) data sorted by timestamp, and shifted to the desired coordinate system via MATLAB. Once in the desired X–Z plane, the data are rotated to eliminate rigid body motion making sure to rotate the sample in the X or lengthwise direction to maintain the correct displacement directions. After rotation the data undergo a final vertical translation to the X-axis ensuring all images can be compared in the same coordinate system. This process is illustrated in Fig. 1.7. Data for both the flat and concave UCA samples were collected in 2 min intervals for the entirety of the 30 min cool down time. The centerline shape was measured for the reference (before any temperature cycle) and at various times after the temperature cycle. To obtain the deformation the reference shape was subtracted from the shape after the temperature cycle. In order to properly control for any manufacturing defects, only the deflection from the original shape is covered in the subsequent results. Table 1.1 shows the maximum out-of-plane deflection for both the flat and concave samples while Figs. 1.8, 1.9, 1.10, and 1.11 show the centerline deformation along the longitudinal direction for both samples through Fig. 1.5 Illustration of the variables present on a UCA Fig. 1.6 Comparison of a UCA (a) without curvature (flat composite), (b) with concave transverse curvature, and (c) a UCA with convex transverse curvature 1 Unimorph Shape Memory Polymer Actuators Incorporating Transverse Curvature in the Substrate 5

30 min. The data clearly show that the concave sample has significantly less residual deformation than the flat sample over the 30 min trial. The concave sample has a maximum variation from the original sample of only 0.35 mm while the flat sample has a maximum difference of 12.7 mm. The graphs show that the concave UCA reaches a peak deflection at approximately 6 min then relaxes a distance of 60 μm by the 30 min mark. The flat UCA does not reach equilibrium in Fig. 1.7 Illustration of the process of converting the DIC data to the desired coordinate system and removing rigid body motion Table 1.1 Maximum deflections for the UCA samples at each marked time Time (min) Max deflection concave sample (recovered-reference) (mm) Max deflection flat sample (recovered-reference) (mm) Reference 0.00 0.00 2 0.28 5.89 4 0.34 9.80 6 0.35 11.4 8 0.35 12.0 10 0.35 12.1 20 0.31 12.3 30 0.30 12.7 Fig. 1.8 Lengthwise versus out-of-plane deflection for the first 10 min of the 63.5 mm concave UCA cooling cycle 6 J.T. Cantrell and P.G. Ifju

30 min as it continues to deflect until the 30 min mark. However, the data show that a majority of the deformation has already occurred after 6 min which was also true of the concave sample. The preliminary test clearly shows that the concept of applying concave transverse curvature to a unimorph substrate substantially improves shape recovery. As such in order to explore further this concept, an additional curved UCA was created but instead of the conventional concave orientation (saddle configuration) it was created with convex orientation (trough configuration). The convex sample was constructed to determine the effect on the residual deformation in an alternate orientation. An example of a convex sample was shown previously in Fig. 1.6. The convex set of samples was monitored via DIC for 30 min during the cool down like the previously tested samples. Table 1.2 shows the maximum out-ofplane deviation with respect to time for the convex sample versus the original concave sample. The data show that while the original concave sample had more initial deformation the convex sample has the larger change in residual deflection. As stated previously, the concave sample deflects only 0.35 mm whereas the convex sample deflects 1.29 mm in the same time period. The convex sample behaves similarly to the concave sample with respect to relaxation. Both samples reach maximum deflection at approximately 6 min and decrease in deflection to some extent up to the 30 min. Figures 1.12 and 1.13 Fig. 1.9 Lengthwise position versus out-of-plane deflection for times 10–30 min of the 63.5 mm concave UCA cooling cycle Fig. 1.10 Lengthwise position versus out-of-plane deflection for the first 10 min of the flat UCA cooling cycle 1 Unimorph Shape Memory Polymer Actuators Incorporating Transverse Curvature in the Substrate 7

Fig. 1.11 Lengthwise position versus out-of-plane deflection for times 10–30 min of the flat UCA cooling cycle Table 1.2 Maximum deflections for the UCA samples at each marked time Time (min) Max deflection concave sample (recovered-reference) (mm) Max deflection convex sample (recovered-reference) (mm) Reference 0.00 0.00 2 0.28 0.86 4 0.34 1.19 6 0.35 1.29 8 0.35 1.29 10 0.35 1.29 20 0.31 1.20 30 0.30 1.13 Fig. 1.12 Lengthwise position versus out-of-plane deflection for the first 10 min of the convex UCA cooling cycle 8 J.T. Cantrell and P.G. Ifju

support these findings and show the centerline deformations for the convex sample. The results indicate the original concave configuration should be used for any further testing due to the minimal deflections seen under comparable conditions. A final series of testing was done on a new concave curved sample to determine the repeatability of testing and any residual deformation as additional deflection cycles were performed on the UCA. A series of four consecutive tests were conducted and compared at the maximum out-of-plane position (Z + W) time of 6 min as well as at the end of the data collection period. Table 1.3 shows the data range was only 40 μm at 6 min and 30 μm at 60 min. Both values that are well within an acceptable range for repeatability. 1.4 Conclusion A series of tests were conducted on carbon fiber and shape memory polymer composite actuators to determine the effect of radius of curvature on the residual deformation. Digital image correlation was employed to find the out-of-plane deformation and allowed for the study of the recovery behavior of these unimorph composite actuators. In the experiments conducted a unimorph composite actuator with a 63.5 mm concave transverse curvature was able to reduce residual deformation by two orders of magnitude compared to a flat unimorph composite actuator keeping all other variables constant. A unimorph with convex transverse curvature was only able to reduce residual deformation by one order of magnitude making a concave actuator the best option for future use. Unimorph composite actuators display repeatable actuation and storage cycles as they do not increase residual deformation with increasing number of cycles. These discoveries can facilitate the expanded use of shape memory polymers on a reconfigurable folding wing micro air vehicles as well as various other applications. Future research will continue to develop the design space presented in this paper. Unimorph composite actuators with varying transverse curvature polymer thickness, substrate width, and polymer width will all be evaluated to determine the correlation between each variable and residual deformation. Fig. 1.13 Lengthwise position versus out-of-plane deflection for times 10–30 min of the convex UCA cooling cycle Table 1.3 Repeatability test data for the concave curved UCA sample Time (min) Test 1 position (mm) Test 2 position (mm) Test 3 position (mm) Test 4 position (mm) Standard deviation (mm) Coefficient of variation (%) 6 1.93 1.95 1.91 1.93 1.63E 2 0.85 60 1.81 1.80 1.78 1.81 1.41E 2 0.78 1 Unimorph Shape Memory Polymer Actuators Incorporating Transverse Curvature in the Substrate 9

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Chapter 2 Yield Criterion for Polymeric Matrix Under Static and Dynamic Loading B.T. Werner and I.M. Daniel Abstract A polymeric matrix (3501-6) used in composite materials was characterized under multi-axial quasi-static and dynamic loading at varying strain rates. Tests were conducted under uniaxial compression, tension, pure shear and combinations of compression and shear. Quasi-static and intermediate strain rate tests were conducted in a servo-hydraulic testing machine. High strain rate tests were conducted using a split Hopkinson Pressure Bar system built for the purpose. This SHPB system was made of glass/epoxy composite (Garolite) bars having an impedance matching the test polymer closer than metals. The typical stress–strain behavior exhibits a linear elastic region up to a yields point, a nonlinear elastoplastic region up to an initial peak or critical stress, followed by a strain softening region up to a local minimum and finally, a strain hardening region up to ultimate failure. It was observed that under multi-axial loading, yielding is governed by one characteristic property, the yield strain under uniaxial tension. Furthermore, it was found that the yield point varied linearly with the logarithm of strain rate. A general three-dimensional elasto-viscoplastic model was formulated in strain space expressed in terms of an effective strain and its yield point. A unified yield criterion was proposed to describe the onset of yielding under any state of stress and at any strain rate. Keywords Polymer-matrix • Multi-axial testing • Elastic–plastic behavior • Yield criteria • Strain rate effects 2.1 Introduction Recent and ongoing research in fiber reinforced polymer composites has dealt with material characterization, constitutive behavior and failure prediction. The process of fabrication, testing and modeling of these composites is costly and time consuming and impedes the introduction of new materials. To facilitate and accelerate the process of introducing and evaluating new composite materials, it is important to develop/establish comprehensive and effective methods and procedures of constitutive characterization and modeling of structural laminates based on the properties of the constituent materials, e.g., fibers, polymers and the basic building block of the composite structure, the single ply or lamina. Lamina characterization and modeling under multi-axial states of stress has shown that there are significant inelastic, nonlinear, viscoelastic and rate effects on the matrix dominated constitutive and failure behavior of these materials [1–3]. In the case of carbon/epoxy composites for example, the fiber itself shows little nonlinear behavior and no rate dependence in its mechanical response. This suggests that the matrix is the key element that controls the inelastic and nonlinear behavior of the composite and that its characterization and modeling are very important. Since the polymer matrix is basically isotropic, a much less costly evaluation of a composite can be achieved by characterizing the bulk matrix under multi-axial states of stress at various strain rates. The constitutive and strain rate behavior of epoxies under various loading conditions has been studied by many researchers including the authors of this paper [4–14]. Some studies describe characterization of the resin at various strain B.T. Werner Sandia National Laboratories, Livermore, CA, USA e-mail: bwerner@sandia.gov I.M. Daniel (*) Northwestern University, Evanston, IL, USA e-mail: imdaniel@northwestern.edu H.J. Qi et al. (eds.), Challenges in Mechanics of Time-Dependent Materials, Volume 2: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-06980-7_2, #The Society for Experimental Mechanics, Inc. 2015 11

rates [4–11]. Mayr et al. [12] and Behzadi and Jones [13] have focused on the yielding behavior of these polymeric materials including strain rate effects. The onset of yielding in the matrix of the composite may not be critical for quasi-static loading or even single impact, but is the limiting factor in fatigue life of a composite structure. Christensen has presented simplified and comprehensive approaches to the study of yield (failure) criteria for homogeneous materials with particular reference to polymers [14–16]. The objective of this study was to characterize the matrix resin under multi-axial loading at different strain rates and to develop a general yielding model that incorporates rate effects. It is based on experimental observations made in the process of developing a constitutive model for the polymer [17]. Emphasis was placed on development of a relatively simple yield model that would not require extensive testing for evaluation of a given polymeric matrix. 2.2 Material Characterization The polymer matrix investigated is a high stiffness, high strength epoxy (3501-6) commonly used in composites. It has a highly crosslinked structure that provides stiffness and strength but also reduces its ductility and leads to fairly brittle behaviour. The B-staged resin (supplied by Applied Poleramic, Inc.) was chipped out, weighed, and placed in the mold. A small amount of acetone was added to reduce the resin viscosity and facilitate casting of the partially cured resin into the mold. The temperature was raised at a rate of 2 C/min up to 120 C and held at that level for 1 h. During this stage the resin viscosity was low and vacuum was drawn to remove any entrapped air and to boil off any acetone that might remain in the resin. For more complex casting geometries a small amount of solvent (acetone) was used to reduce the viscosity of the partially cured epoxy and facilitate casting into the mold. The material was cast into closed molds to produce thin (3 mm) plates for tensile coupons, thick blocks for prismatic compression coupons, and thin-wall cylinders for specimens to be tested under torsion and combinations of torsion and axial tension or compression. The geometry and dimensions of the specimens used are shown in Fig. 2.1. Specimens for uniaxial tensile testing were thin dogbone coupons with a gage section of 12.7 50.8 mm machined from 3 mm thick sheets using a router with an abrasive bit. Uniaxial compressive tests were conducted on thick prismatic coupons 12.7 mm long with a cross section of 7.62 8.89 mm. Specimens with varying aspect ratios (from 0.125:1 to 2.7:1) were tested. It was found that the apparent stiffness reaches the plane strain value (C11) when extrapolated to zero aspect ratio and reaches asymptotically the Young’s modulus E of the material for larger aspect ratios (Fig. 2.2). The stiffness determined by extrapolation to zero aspect ratio, C11, was used along with the Young’s modulus to determine Poisson’s ratio ν. Tests under pure shear and combinations of shear and normal tensile or compressive stress were conducted on thin-wall cylindrical specimens. The cylinders were machined to produce a 1.27 mm wall thickness in the gage section and a fillet radius of 11.43 mm. The resulting dogboned cylinder was mounted on a steel post used to apply compression and torsion. The young’s modulus, Poisson’s ratio and shear modulus of the polymer were obtained from the uniaxial compression tests using the relations 101.6 12.70 50.80 R14.00 + + + + 7.62 12.70 8.89 50.8 f16.92 f14.38 R11.43 15.24 + + + + f22.23 a b c Fig. 2.1 Specimen geometries and dimensions. (a) Tensile specimen, (b) compressive specimen, (c) shear and combined stress specimen 12 B.T. Werner and I.M. Daniel

C11 ¼ E 1 ν ð Þ 1þν ð Þ 1 2ν ð Þ G¼ E 2 1þv ð Þ ð2:1Þ Experiments were conducted at various strain rates ranging from 10 5 to 1,500 s 1. Lower rate experiments at less than 1 s 1 were conducted in a servo-hydraulic testing machine. Higher rate testing was conducted in a Split Hopkinson (Kolsky) Pressure Bar (SHPB) system. Glass/epoxy composite (G-10) bars were used to minimize the impedance mismatch and reduce the noise-to-signal ratio. A wave propagation analysis similar to that described by Daniel et al. [18, 19] was conducted to establish the limits of validity of dynamic testing in the SHPB system. The stresses at the ends of the specimen are not equal initially, but approach each other as the wave pulse is reflected back and forth within the specimen. The ratio of stresses at the two specimen ends was calculated and plotted versus number of wave transits through the specimen. The analysis shows that, for the selected configuration, the difference between stresses at the two ends is less than 10 % after four wave transits (one wave transit equals the time needed for the wave to propagate over one specimen length). This occurs at a time corresponding to less than 10 % of the pulse duration. Once an acceptable stress equilibrium state is reached, the stress strain curve for the material (epoxy) is obtained from the classical Hopkinson bar equations for stress, strain rate and strain in the specimen: σs ¼ EA0 2As εi þεr þεt ð Þ¼ EA0 As εt _εs ¼ C0 Ls εi εr εt ð Þ¼ 2C0 Ls εr εs ¼ C0 Ls ð t 0 εi εr εt ð Þdt ¼ 2C0 Ls ð t 0 εrdt ð2:2Þ where εi, εr, εt ¼incident, reflected and transmitted strain pulses in input and output bars A0, As ¼cross sectional areas of pressure bar and specimen E ¼pressure bar modulus Ls ¼specimen length C0 ¼wavespeed in pressure bar Stress–strain curves were obtained over a wide range of strain rates. Figure 2.3 shows stress–strain curves under tension and pure shear loading at two strain rates. Figure 2.4 shows stress–strain curves under three states of biaxial compression and Fig. 2.2 Effect of aspect ratio on compressive response 2 Yield Criterion for Polymeric Matrix Under Static and Dynamic Loading 13

shear and Fig. 2.5 shows similar curves under biaxial compression and shear at two strain rates. The compressive stress–strain curves shown in Fig. 2.6 have the same overall shape and share many key features. Each curve has a linear elastic region up to the yield point. This is followed by a plastic region up to a critical point of material instability, the “critical stress.” Next, the resin reaches a minimum at the plateau point (local minimum) of the stress–strain curve and then it begins to harden. All curves share the same initial modulus. They only differ in the nonlinear behavior that shows pronounced rate dependence. The key features on the curves expand radially from the origin with increasing strain rate. This is highlighted in Fig. 2.6 where it can be seen that the yield point, the critical point, and the plateau point all fall along these radial lines. These characteristic properties vary linearly with the logarithm of strain rate as shown in Fig. 2.7. The resulting rate dependence can be expressed as P _εð Þ¼P _ε0 ð Þ mlog10 _ε _ε0 þ 1 ð2:3Þ where P _εð Þ is the rate dependent property, P _ε0 ð Þ is that property at the reference strain rate, and mis the slope of the linear logarithmic relation. A similar relationship has been observed for the matrix dominated behavior of composites and for polymeric foams [3, 19]. In the present case, a reference strain rate of 10 4 s 1 was used and mwas found to be 0.096. Fig. 2.3 Tension and shear stress–strain curves at two strain rates Fig. 2.4 Stress-stain curves under biaxial compression (a) and shear (b) at three stress ratios 1–3 14 B.T. Werner and I.M. Daniel

Fig. 2.5 Stress–strain curves under biaxial compression and shear at two strain rates Fig. 2.6 Compressive stress–strain curves at various rates showing radial alignment of characteristic features 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.00001 0.001 0.1 Strain Rate, ε (S−1) e0 = 10 −4 s −1 e0 e log + 1 P(e) = P(e0) σy σcrit 10 1000 m = 0.096 m P(ε)/P(ε0) . . . . . . . . Fig. 2.7 Variation of normalized critical and yield stresses with strain rate

Using the above relation, the stress and strain at any strain rate can be transformed into equivalent values at the reference strain rate by the following transformation relations ε _ε0 ð Þ¼ ε _εð Þ K σ _ε0 ð Þ¼ σ _εð Þ K ð 2:4Þ where K=mlog _ε _ε0 þ 1 The stress–strain curves of Fig. 2.5 can thus be transformed into one master curve at the reference strain rate as shown in Fig. 2.8. 2.3 Yield Criterion The yield strains under uniaxial quasi-static tension, compression and shear are identified in the corresponding stress–strain curves of Figs. 2.3 and 2.6. The states of strain for these cases are represented by the Mohr circles of Fig. 2.9. Under uniaxial tension, the maximum normal strain is the yield strain of approximately εt y ¼0.006 and the maximum compressive strain is νεt y ¼ 0.35 0.0066 ¼ 0.002. Under uniaxial compression (from Fig. 2.6) the maximum compressive strain at yield is εc y ¼ 0.0165 and the corresponding transverse tensile strain is νε c y ¼0.006 ¼ε t y. Similarly, from the shear stress–strain curve of Fig. 2.3, it is seen that the shear strain at yield is γy/2 0.006 ¼ε t y. Thus, yield strains and stresses under uniaxial tension, compression and shear are related. These experimental observations lead to a hypothesis that yielding behavior of a quasi-brittle polymer could be described in terms of only one parameter, the tensile yield strain or the tensile yield stress σy t ¼ε y t E¼F ð2:5Þ This represents a further simplification to the two property yielding theory presented by Christensen [14, 15]. The pattern of yielding governed by the uniaxial tensile yield strain (stress) holds true even under biaxial loading conditions. Results from compression/shear biaxial stress–strain curves shown in Fig. 2.4 are shown in Fig. 2.10. The yield criterion Fig. 2.8 Master compressive stress–strain curve at reference strain rate 16 B.T. Werner and I.M. Daniel

was expressed in terms of the normal and shear strain components for a general biaxial state of stress noted on the Mohr circle of Fig. 2.10 as εy t ¼ εx þεy 2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifi εx εy 2 2 þ γ2 xy 4 s or 2ε y t ¼ 1 ν ð Þεy þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifffifi 1þν ð Þ 2ε2 yþγ2 xy q ð2:6Þ -1.2 -0.8 -0.4 0 0.4 0.8 1.2 -2 -1.5 -1 -0.5 0 0.5 1 Shear Strain, g/2 (%) Normal Strain, e (%) Compression Tension Shear Compression Shear Tension Fig. 2.9 Mohr circles for strain transformation under uniaxial tension, compression and shear Fig. 2.10 Mohr circle based on normal and shear yield strains recorded in a biaxial test 2 Yield Criterion for Polymeric Matrix Under Static and Dynamic Loading 17

or, in terms of stresses εy t E¼F¼σ1 νσ2 ¼ 1 ν ð Þσy þ 1þν ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifi σ2 y þ4τ2 xy q or ν σy F 2 þ 1 ν ð Þ σy F þ 1þν ð Þ 2 τxy F 2 ¼1 ð2:7Þ In the case of a general biaxial state of stress, the yield criterion would be 1 F2 ν σ2 x þσ 2 y 1þν 2 σ xσy h iþ 1 ν ð Þ σx þσy F þ 1þν ð Þ2 τxy F 2 ¼1 ð2:8Þ provided the maximum principal stress is tensile. This criterion accounts for strain rate effects without any modification since the characteristic properties of the material such as yield stresses and strains vary linearly with the logarithm of strain rate as depicted in Fig. 2.7 and described by Eqs. (2.3) and (2.4). This is true if normal and shear strain rates in a biaxial test are not significantly different from each other. In that case, the criteria of Eqs. (2.7) and (2.8) become independent of strain rate. Figure 2.5, for example, shows stress–strain curves from biaxial compression/shear tests at two different strain rates. In these tests, the normal strain rates (0.75 10 4 and 0.75 10 2 s 1) were selected to be 75 % of the corresponding shear strain rates (10 4 and 10 2 s 1). The normalized ratios of stresses and uniaxial tension yield stress, F, are not changed with strain rate. The small difference in shear strain rate from the normal strain rate has a very small effect on the yield criterion of Eq. (2.8). 2.4 Summary and Conclusions A quasi-brittle polymer material used as matrix in composites was characterized over a wide range of strain rates. It was shown that characteristic properties, such as the yield point, vary linearly with the logarithm of strain rate. Furthermore, it was observed that the onset of yielding under any state of stress is governed by the tensile yield stress or yield strain. This led to the proposal of a simple one property yield criterion which may be independent, or at least not sensitive to strain rate. Acknowledgments This work was supported by the Office of Naval Research (ONR). We are grateful to Dr. Y. D. S. Rajapakse of ONR for his encouragement and cooperation. References 1. Cho J-M, Fenner JS, Werner BT, Daniel IM (2010) A constitutive model for fiber reinforced polymer composites. J Comput Mater 44(26):3133–3150 2. Daniel IM, Cho J-M, Werner BT, Fenner JS (2011) Characterization and constitutive modeling of composite materials under static and dynamic loading. AIAA J 49(8):1658–1682 3. Daniel IM, Werner BT, Fenner JS (2011) Strain-rate-dependent failure criteria for composites. Compos Sci Technol 71(3):357–364 4. Arruda EM, Boyce MC, Jayachandran R (1995) Effects of strain-rate, temperature and thermomechanical coupling on the finite strain deformation of glassy-polymers. Mech Mater 19(2–3):193–212 5. Liang YM, Liechti KM (1996) On the large deformation and localization behavior of an epoxy resin under multiaxial stress states. Int J Solids Struct 33(10):1479–1500 6. Chen W, Zhou B (1998) Constitutive behavior of Epon 828/T-403 at various strain rates. Mech Time-Dependent Mater 2(2):103–111 7. Buckley CP, Harding J, Hou JP, Ruiz C, Trojanowski A (2001) Deformation of thermosetting resins at impact rates of strain. Part I: experimental study. J Mech Phys Solids 49(7):1517–1538 8. Gilat A, Goldberg RK, Roberts GD (2003) High strain rate response of epoxy in tensile and shear loading. J Phys IV 110:123–127 9. Gilat A, Goldberg RK, Roberts GD (2007) Strain rate sensitivity of epoxy resin in tensile and shear loading. J Aerosp Eng 20(2):75–89 10. Littell JD, Ruggeri CR, Goldberg RK et al (2008) Measurement of epoxy resin tension, compression, and shear stress–strain curves over a wide range of strain rates using small test specimens. J Aerosp Eng 21(3):162–173 18 B.T. Werner and I.M. Daniel

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