On the Response of Rubbers at High Strain Rates ... High Strain-Rate Stress-Strain Diagram for Latex

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Transcript of On the Response of Rubbers at High Strain Rates ... High Strain-Rate Stress-Strain Diagram for Latex

  • SANDIA REPORT SAND2010-1480 Unlimited Release February 2010

    On the Response of Rubbers at High Strain Rates Johnathan Greenberg Niemczura Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. Approved for public release; further dissemination unlimited.

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    Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: 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, nor any of their contractors, subcontractors, or their employees, make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represent 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, any agency thereof, or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof, or any of their contractors. Printed in the United States of America. This report has been reproduced directly from the best available copy. Available to DOE and DOE contractors from U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 37831 Telephone: (865) 576-8401 Facsimile: (865) 576-5728 E-Mail: reports@adonis.osti.gov Online ordering: http://www.osti.gov/bridge Available to the public from U.S. Department of Commerce National Technical Information Service 5285 Port Royal Rd. Springfield, VA 22161 Telephone: (800) 553-6847 Facsimile: (703) 605-6900 E-Mail: orders@ntis.fedworld.gov Online order: http://www.ntis.gov/help/ordermethods.asp?loc=7-4-0#online

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    SAND2010-1480 Unlimited Release

    February 2010

    On the Response of Rubbers at High Strain Rates

    Johnathan Greenberg Niemczura University of Texas-Austin Campus Executive Fellow

    Sandia National Laboratories

    Computational Structural Mechanics and Applications P.O. Box 5800

    Albuquerque, New Mexico 87185-0372

    Abstract

    In this report, we examine the propagation of tensile waves of finite deformation in rubbers through experiments and analysis. Attention is focused on the propagation of one-dimensional dispersive and shock waves in strips of latex and nitrile rubber. Tensile wave propagation experiments were conducted at high strain-rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in the rubber strips. Analysis of the response through the theory of finite waves and quantitative matching between the experimental observations and analytical predictions was used to determine an appropriate instantaneous elastic response for the rubbers. This analysis also yields the tensile shock adiabat for rubber. Dispersive waves as well as shock waves are also observed in free-retraction experiments; these are used to quantify hysteretic effects in rubber.

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    ACKNOWLEDGMENTS I would like to thank Sandia National Laboratories for supporting me through the Sandia National Laboratory Campus Executive Fellowship.

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    CONTENTS

    1. INTRODUCTION ............................................................................................................... 15

    2. RUBBER ELASTICITY ..................................................................................................... 19 2.1 Rubber Structure ......................................................................................................... 19 2.2. Quasi-Static Tensile Testing ....................................................................................... 22 2.4. Relaxation ................................................................................................................... 32

    3. THEORY OF ONE-DIMENSIONAL WAVE PROPAGATION....................................... 36 3.1 Equations of Motion ................................................................................................... 36 3.2 General Solutions for a Semi-infinite Strip ................................................................ 37

    3.2.1 Fan Solution .................................................................................................... 37 3.2.2 Constant Solution............................................................................................ 38

    3.3 Solution by Method of Characteristics ....................................................................... 38 3.4 Approximate Material Model for Rubber ................................................................... 40 3.5 Shock Jump Conditions and Driving Force ................................................................ 45 3.6 Shocks in a Cubic Material ......................................................................................... 46 3.7. Kinetic Relation .......................................................................................................... 48 3.8. Analysis of Free-Retraction Experiment in Rubber.................................................... 49

    3.8.1. Governing Equations and General Solutions.................................................. 49 3.8.2. Unloading Waves in Cubic Material Model ................................................... 51

    4. DISPERSIVE WAVES........................................................................................................ 52 4.1. Experimental Scheme for Generation of Impact-Induced Tensile Waves in Rubber. 53 4.2. Comparison of Measured and Calculated Particle Trajectories.................................. 56 4.3. Power-Law Model ...................................................................................................... 58 4.4. Dispersive Waves in the Power-Law Material ........................................................... 59 4.5. Tensile Waves in Finite Length Specimens – Reflections at a Fixed Boundary ........ 62 4.6. Impact on Prestrained Rubber Specimens Generating Dispersive Waves.................. 66 4.7. Discussion................................................................................................................... 70

    5. SHOCK WAVES................................................................................................................. 74 5.1. Impact on Rubber Specimens Generating Tensile Shocks ......................................... 74 5.2. Reflection of Shocks ................................................................................................... 84 5.3. Interpretation of Shocks in Phase Transforming Materials ........................................ 86

    6. HYSTERESIS...................................................................................................................... 90 6.1. Free-Retraction Experiments in Latex and Nitrile Rubber Specimens....................... 90 6.2. Power-Law Model for Free-Retraction....................................................................... 91 6.3. Free Retraction in Nitrile Rubber ............................................................................... 94 6.4. Free Retraction by Pure Shock in Latex Rubber ........................................................ 98 6.5. Free Retraction by Dispersive and Shock Waves in Latex Rubber ............................ 99 6.6. Free Retraction by Dispersive Waves in Latex Rubber............................................ 105 6.7. Dynamic Loading-Unloading Response in Nitrile Rubber....................................... 107

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    7. KOLSKY EXPERIMENT................................................................................................. 110 7.1. Experimental Setup................................................................................................... 110 7.2. Experimental Results ................................................................................................ 111 7.3. Analysis of Wave Propagation in Hysteretic Materials............................................ 123

    8. CONCLUSIONS................................................................................................................ 133

    9. REFERENCES .................................................................................................................. 137

    APPENDIX A: Riemann Numerical simulation......................................................................... 139

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    FIGURES Figure 2.1. Monomer Units for Polyethylene, Polyisoprene (natural latex rubber), and

    Butadiene-acrylonitrile (nitrile rubber)....................................................................................19 Figure 2.2. Structure of Polyisoprene of (a) cis-configuration and (b) trans-configura