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Study of hysteretic damping in small elastomeric structures Mossman, Michele Ann 1997

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STUDY OF HYSTERETIC D A M P I N G IN S M A L L ELASTOMERIC  STRUCTURES  by MICHELE A N N MOSSMAN B . S c . H . , A c a d i a University, 1995  A T H E S I S S U B M I T T E D LN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Physics and Astronomy) W e accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A October 1997 © Michele A n n Mossman  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  University  of  British  Columbia,  available for  copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and study. scholarly  or  her  for  of  Department  of  Phu^tXs-  The University of British Vancouver, Canada  Date  DE-6 (2/88)  &  OrSnhPA^  4  As4lfD n o  Columbia  It  gain shall not  permission.  nfl  y  that  agree  may  representatives.  financial  requirements  I agree  I further  purposes  the  be is  that  the  permission  granted  allowed  an  advanced  Library shall make  by  understood be  for  for  the that  without  it  extensive  head  of  my  copying  or  my  written  ABSTRACT This thesis investigates the damping coefficient for silicone rubber micro structures under oscillating applied stresses.  These small elastomeric structures are important to the  development o f Elastomeric M i c r o Electro Mechanical Systems, or E M E M S , which has recently become a field o f interest. Since energy is lost in a damped system, it is generally desirable to minimize the effect of the damping.  Although often overlooked, the primary mechanism o f this damping results from the hysteresis effect. This complicated phenomenon cannot be described analytically, and it is a primary objective o f this thesis to develop a computational algorithm to determine the hysteretic force based on a sequence of past displacements of the rubber.  The extent o f the damping can be determined by measuring the resonance response o f a silicone structure when an oscillating displacement is applied. It is shown that these small silicone rubber structures exhibit the unique characteristic that the damping coefficient is independent of both the amplitude and frequency of the oscillation.  This apparent independence o f this damping coefficient of small silicone structures makes the use of the elastomer in E M E M S devices look promising. The ability to predict the effect of damping on the behaviour of these structures is crucial to device design, and its independence over a wide range of operating parameters bodes well for widespread use o f elastomeric microstructures.  ii  T A B L E OF CONTENTS ABSTRACT  ii  T A B L E O FCONTENTS  iii  LIST O FT A B L E S  vi  LIST O FFIGURES  vii  ACKNOWLEDGEMENTS  x  1. I N T R O D U C T I O N  1  2. B A C K G R O U N D  5  2.1. T H E RUBBER-LIKE STATE 2.2. DEFORMABILITY AND MODULUS  2.2.1. Molecular Structure  5 5  9  2.2.2. Thermodynamics  12  2.2.3. Glass Transition  13  2.2.4. Silicones  15  2.3. DEFORMATION RESPONSE TO AN APPLIED STRESS  2.3.1. Viscoelastic Behaviour 2.3.2. Hysteretic Behaviour 2.3.3. Long term effect of an applied stress 3. B A C K G R O U N D R E G A R D I N G R E S O N A N T S Y S T E M S  16  19 23 24 27  3.1. RESONANCE BEHAVIOUR  27  3.2. RESONANCE QUALITY FACTOR  28  4. E X P E R I M E N T A L D E S I G N O F A R E S O N A N T S Y S T E M 4.1. NATURE OF EXPERIMENT 4.2. PHYSICAL FACTORS INFLUENCING DYNAMIC PROPERTIES  31 31 31  4.2.1. Size, shape and type of rubber  31  4.2.2. Amplitude 4.2.3. Temperature  32 33  4.3. FABRICATION OF SAMPLES  34  4.4. EXPERIMENTAL SET-UP  37  4.4.1. 4.4.2. 4.4.3. 4.4.4.  Determining the natural frequency Measuring the resonance peak Detecting the oscillations Creating a controlled temperature environment  5. R E S U L T S O F T H E R E S O N A N T S Y S T E M  37 37 37 40 42  5.1. DETERMINING THE NATURAL FREQUENCY  42  5.2. FORCED DAMPED HARMONIC MOTION  43  5.2.1. Size, shape and type of rubber  50  5.2.2. Amplitude  55  5.2.3. Temperature  62  6. M O D E L I N G T H E S Y S T E M 6.1. FINITE E L E M E N T ANALYSIS USING A N S Y S  64 64  iii  6.2. PRODUCING A RESONANCE PEAK  6.2.1. 6.2.2. 6.2.3. 6.2.4.  Varying Varying Varying Varying  the the the the  post diameter post height post shape type of rubber  6.3. COMPARISON WITH EXPERIMENTAL D A T A 7. H Y S T E R E S I S B A C K G R O U N D 7.1. INTRODUCTION  64  :  67 68 69 70 71 73 73  7.2. STANDARD PREISACH TREATMENT OF MAGNETIC HYSTERESIS  74  7.3. APPLICATION TO RUBBER  89  7.4. DEVELOPING THE M O D E L  95  7.5. PREDICTING THE RESIDUAL DISPLACEMENT  105  7.6. ENERGY LOSS IN AN OSCILLATING SYSTEM  106  7.7. DETERMINING THE DENSITY DISTRIBUTION  109  8. H Y S T E R E S I S D E S I G N  113  8.1. NATURE OF EXPERIMENT  113  8.2. FABRICATION OF SAMPLES  113  8.3. EXPERIMENTAL SET-UP  114  9. H Y S T E R E S I S R E S U L T S  116  9.1. VERIFYING THE SYSTEM BEHAVIOUR  9.1.1. 9.1.2. 9.1.3. 9.1.4.  Demagnetization procedure Eliminating the viscoelastic effect Residual displacement for different stretch magnitude and duration Comparison of latex and HS-II rubber bands  9.2. TESTING THE PREDICTIVE ALGORITHM  9.2.1. Power law fit of residual displacement 9.2.2. Predicting the residual displacement 9.2.3. Optimizing fit parameters  116  116 117 118 120 121  121 125 126  10. I M P L I C A T I O N S O F H Y S T E R E S I S T H E O R Y F O R O B S E R V E D O S C I L L A T I O N  129  11. C O N C L U S I O N  134  REFERENCES  138  APPENDIX A : DIMENSIONAL ANALYSIS O F A SIMPLE R E C T A N G U L A R POST  140  APPENDIX B : STRUCTURE OF SAMPLES  142  APPENDIX C : P H O T O D E T E C T O R CIRCUITS  144  APPENDIX D : C O N T R O L L E D T E M P E R A T U R E ENVIRONMENT  145  APPENDIX E : S A M P L E DIMENSIONS  148  APPENDIX F : LAB VIEW  149  APPENDIX G : A M P L I T U D E AND PHASE RESPONSE O F PIEZOSTACK  150  APPENDIX H : A N G S T R O M RESOLVER OPTICAL PROBE  155  iv  APPENDIX I: PHASE DIFFERENCE B E T W E E N F G SYNC AND OUTPUT  156  APPENDIX J : DETERMINING Q  157  APPENDIX K : PROPERTIES OF SPECIFIC SILICONE RUBBERS  158  APPENDIX L : C A L C U L A T I O N OF E N E R G Y LOSS IN AN OSCILLATING S Y S T E M  160  v  LIST O F T A B L E S T A B L E 5.1:  CHARACTERISTICS OF RESONANCE PEAKS FOR VARIOUS SAMPLES  55  T A B L E 5.2:  CHARACTERISTIC OF RESONANCE PEAKS FOR DIFFERENT OSCILLATION AMPLITUDES  61  T A B L E E . l : DIMENSIONS OF CYLINDRICAL SAMPLES  148  T A B L E E . 2 : DIMENSIONS OF RECTANGULAR SAMPLES  148  T A B L E K . l : PROPERTIES OF SILICONE RUBBERS  159  vi  LIST O F FIGURES FIGURE 2.1 : MODES OF DEFORMATION  6  FIGURE 2 . 2 : TYPICAL STRESS-STRAIN CURVE FOR RUBBER  8  FIGURE 2 . 3 : M O L E C U L A R STRUCTURE OF POLYMETHYL SILOXANE OR SILICONE RUBBER  16  FIGURE 2 . 4 : HYSTERESIS LOOPS AT LOW AND MODERATE FREQUENCIES  18  FIGURE 2 . 5  : Viscous  FLOW OF MOLECULES IN A LIQUID  FIGURE 2 . 6 : EFFECT OF TEMPERATURE ON COMPLEX MODULUS  20 22  FIGURE 2 . 7 : EFFECT OF TEMPERATURE ON LOSS FACTOR  23  FIGURE 2.8 : DEFORMATION-TIME CURVE FOR A VISCOELASTIC MATERIAL  25  FIGURE 3 . 1 : DETERMINING THE Q OF A RESONANCE PEAK  29  FIGURE 3 . 2 : MEASURING THE SHARPNESS OF THE RESONANCE PEAK  30  FIGURE 4 . 1 : CASTING PROCESS OF THIN SILICONE RUBBER FILMS  35  FIGURE 4 . 2 : CLEAVING RECTANGULAR POSTS FROM A THIN FILM OF SILICONE RUBBER  37  FIGURE 4 . 3 : SILICONE POST ATTACHED TO PIEZOSTACK  37  FIGURE 4 . 4 : TRANSMITTED LASER BEAM SET-UP  38  FIGURE 4 . 5 : REFLECTED LASER BEAM SET-UP  39  FIGURE 4 . 6 : INTERRUPTED LASER BEAM SET-UP  40  FIGURE 4 . 7 : APPARATUS FOR CONTROLLING TEMPERATURE OF NITROGEN GAS FLOW  41  FIGURE 5 . 1 : RELATIONSHIP BETWEEN POST LENGTH AND NATURAL FREQUENCY  43  FIGURE 5 . 2 : TYPICAL RESONANCE PEAK COLLECTED FOR A SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION  45  FIGURE 5 . 3 : TYPICAL PHASE DATA COLLECTED FOR A SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION  46  FIGURE 5 . 4 : TYPICAL X-COMPONENT DATA COLLECTED FOR A SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION  48  FIGURE 5 . 5 : TYPICAL Y-COMPONENT DATA COLLECTED FOR SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION  49  FIGURE 5 . 6 : COMPARISON OF RESONANCE PEAKS FOR DIFFERENT SIZES OF SILICONE RUBBER POSTS  51  FIGURE 5 . 7 : COMPARISON OF RESONANCE PEAKS FOR POSTS OF DIFFERENT HEIGHT  52  FIGURE 5 . 8 : COMPARISON OF RESONANCE PEAKS FOR SILICONE POSTS OF DIFFERENT SHAPE  53  FIGURE 5 . 9 : COMPARISON OF RESONANCE PEAKS FOR DIFFERENT TYPES OF SILICONE POSTS  54  FIGURE 5 . 1 0 :  S M A L L AMPLITUDE RESPONSE FOR HS-II SILICONE RUBBER POST WITH CIRCULAR CROSS  SECTION  57  FIGURE 5 . 1 1 : RESONANCE PEAK FOR HS-n  SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION AT STACK  INPUT AMPLITUDE OF 4 0 0 M  58  FIGURE 5 . 1 2 : RESONANCE PEAK FOR HS-II SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION AT STACK INPUT AMPLITUDE OF 4 0 M V R M S FIGURE 5 . 1 3 :  59  RESONANCE PEAK FOR HS-II SILICONE RUBBER POST WITH CIRCULAR CROSS SECTION AT STACK  INPUT AMPLITUDE OF 4 M V  R M S  60  FIGURE 5 . 1 4 : RESONANCE PEAKS FOR DIFFERENT TEMPERATURES  63  FIGURE 6.1 : DEFORMED SHAPE OF OSCILLATING SILICONE POST USING A N S Y S MODEL  66  FIGURE 6 . 2 : RESONANCE PEAKS FOR POSTS OF DIFFERENT DIAMETER AS GENERATED BY A N S Y S PROGRAM 6 8 FIGURE 6 . 3 : RESONANCE PEAKS FOR POSTS OF DIFFERENT HEIGHTS AS GENERATED BY A N S Y S PROGRAM .. 6 9 FIGURE 6 . 4 : RESONANCE PEAKS FOR POSTS OF DIFFERENT SHAPES AS GENERATED BY A N S Y S PROGRAM .... 70 FIGURE 6 . 5 : RESONANCE PEAKS FOR POSTS OF DIFFERENT TYPES OF RUBBER AS GENERATED BY A N S Y S PROGRAM  71  FIGURE 6 . 6 : COMPARING THE A N S Y S MODEL WITH EXPERIMENTAL DATA  72  FIGURE 7.1 : APPLIED MAGNETIC H E L D AND RESULTING MAGNETIZATION HISTORY OF AN IRON SAMPLE  74  FIGURE 7 . 2 : HYSTERESIS LOOP FOR AN IRON SAMPLE IN A CYCLICALLY CHANGING FIELD  75  FIGURE 7 . 3 : HYSTERESIS OPERATOR FOR SCALAR PREISACH MODEL OF FERROMAGNETIC HYSTERESIS  76  FIGURE 7 . 4 : MAPPING HYSTERESIS OPERATORS TO AB PLANE FOR PREISACH MODEL  78  vii  FIGURE 7.5 : ILLUSTRATING THE STAIRCASE INTERFACE  79  FIGURE 7.6 : REPRESENTATION OF DEMAGNETIZED STATE IN A,B-SPACE  80  FIGURE 7.7  81  : OSCILLATING, DECAYING SEQUENCE OF APPLIED MAGNETIC FIELDS  FIGURE 7.8  : INTERFACE FOLLOWING APPLICATION OF MAXIMUM H E L D  82  FIGURE 7.9  : INTERFACE FOLLOWING SUBSEQUENT APPLICATION OF MINIMUM FIELD  83  FIGURE 7.10:  STAIRCASE INTERFACE FOLLOWING ARBITRARY MAGNETIZATION HISTORY  FIGURE 7.11  : AB NOTATION FOR SEQUENCE OF EXTREMA  FIGURE 7.12  : SEQUENCE OF EXTREMA ILLUSTRATING WIPING OUT PRINCIPLE  FIGURE 7.13:  86  STAIRCASE INTERFACE PRIOR TO APPLICATION OF H E L D UI  FIGURE 7.14  : DEMAGNETIZATION SEQUENCE  FIGURE 7.15  : GENERATING T H E DEMAGNETIZED STATE  FIGURE 7.16:  84 85 86 87 88  HYSTERESIS OPERATOR FOR SCALAR PREISACH MODEL OF HYSTERESIS IN RUBBER  91  FIGURE 7.17:  MAPPING HYSTERESIS OPERATORS TO AB PLANE FOR RUBBER MODEL  93  FIGURE 7.18:  GENERATING STAIRCASE INTERFACE ASSOCIATED WITH A GIVEN DISPLACEMENT  FIGURE 7.19  : GEOMETRICAL EVALUATION OF INTEGRALS OVER AB SPACE  FIGURE 7.20  : RECTANGULAR REPRESENTATION OF STAIRCASE INTERFACE  FIGURE 7.21  : ILLUSTRATION OF THE NOTATION OF RECTANGLES  94 97 99 101  FIGURE 7.22  : EXPRESSION OF INTEGRALS AS THE DIFFERENCE BETWEEN TWO MEASURED RESIDUALS  102  FIGURE 7.23  : ARBITRARY DISPLACEMENT HISTORY  103  FIGURE 7.24  : STRESS HISTORY FOR OSCILLATING CYCLE  107  FIGURE 7.25  : A,B-SPACE  107  FIGURE 7.26  : DETERMINING THE DENSITY DISTRIBUTION  FIGURE 8.1:  RUBBER BAND MOLD  PICTURE FOR HALF CYCLE OF OSCILLATION  110 114  FIGURE 8.2 : MEASURING T H E RESIDUAL DISPLACEMENT FIGURE 9 . 1 :  115  RECOVERY OF A RUBBER BAND FOLLOWING THREE DISPLACEMENTS OF -200MM, STARTING FROM  DIFFERENT INITIAL STATES  117  FIGURE 9.2 : RESIDUAL DISPLACEMENT FOR VARIOUS EXTENSIONS  119  FIGURE9.3 : RESIDUAL DISPLACEMENT FOR VARIOUS STRETCH DURATIONS  120  FIGURE 9.4  : INTERPOLATING T H E RESIDUAL DISPLACEMENTS FOR LATEX RUBBER BAND  123  FIGURE 9.5  : INTERPOLATING T H E RESIDUAL DISPLACEMENTS FOR HS-n  124  RUBBER BAND  FIGURE 9.6 : ARBITRARY DISPLACEMENT HISTORY ILLUSTRATING WIPING OUT PROCEDURE FIGURE 9.7  126  : OPTIMIZING T H E PARAMETERS OF THE H T FOR PREDICTING RESIDUAL DISPLACEMENTS FOR  LATEX RUBBER BAND FIGURE 9.8 : OPTIMIZING PARAMETERS FOR PREDICTING RESIDUAL DISPLACEMENTS OF AN HS-n  127 RUBBER  BAND FIGURE 10.1  128 : DETERMINING T H E RESIDUAL DISPLACEMENTS FOR SMALL DISPLACEMENTS OF A LATEX RUBBER  BAND FIGURE 10.2  131 : DETERMINING THE RESIDUAL DISPLACEMENT FOR SMALL DISPLACEMENTS OF AN HS-n  RUBBER  BAND  132  FIGURE A . 1 : DIMENSIONS OF A SIMPLE RECTANGULAR POST FIGURE B . l  : STRUCTURE OF A CYLINDRICAL SILICONE POST  140 142  FIGURE B.2 : STRUCTURE OF A RECTANGULAR SILICONE POST  143  FIGURE C . l  144  : SIMPLE PHOTODETECTOR CIRCUIT  FIGURE C . 2 : UPDATED PHOTODETECTOR CIRCUIT  144  FIGURE D . 1 : R A T E OF CHANGE OF TEMPERATURE OF NITROGEN GAS FLOW WITH DEWAR RESISTOR POWER 4 5 W AND TUBE RESISTOR POWER 0 W  145  FIGURE D.2 : R A T E OF CHANGE OF TEMPERATURE OF NITROGEN GAS FLOW WITH DEWAR RESISTOR POWER 4 5 W AND TUBE RESISTOR POWER 2 0 W  146  FIGURE D.3 : R A T E OF CHANGE OF TEMPERATURE OF NITROGEN GAS WITH DEWAR RESISTOR POWER 2 0 W AND TUBE RESISTOR POWER 2 0 W FIGURE G . l : AMPLITUDE RESPONSE OF PIEZOSTACK FOR LARGE INPUT SIGNALS  147 151  FIGURE G . 2 : AMPLITUDE RESPONSE OF PIEZOSTACK FOR SMALL INPUT SIGNALS  152  FIGURE G . 3 : AMPLITUDE RESPONSE OF PIEZOSTACK FOR DIFFERENT FREQUENCIES OF INPUT SIGNALS  153  viii  FIGURE G.4 : PHASE RESPONSE OF PIEZOSTACK FOR DIFFERENT FREQUENCIES OF INPUT SIGNAL  154  FIGURE H . l : ANGSTROM RESOLVER OPTICAL PROBE TIP  155  FIGURE 1.1 : PHASE DIFFERENCE BETWEEN FUNCTION GENERATOR OUTPUT AND SYNCHRONIZED OUTPUT. ..156  ix  ACKNOWLEDGEMENTS  First and foremost, I wish to express my thanks to the past and present team members o f the Structured Surface Physics Laboratory at U B C . Here, we have a unique environment which both invites the development o f independent, original thoughts and ideas and encourages teamwork. W e attribute a great deal of the success o f the lab to D r . L o m e Whitehead, our team leader and source o f inspiration.  Special thanks to him for his  invaluable assistance throughout this project.  I would like to express my appreciation to D r . Frank Curzon for his assistance during the preparation o f this thesis. M a n y thanks also to Peter K a n for his help with lab matters, and to Alison Clark for answering endless questions in our quest to understand silicone rubber.  Thanks, all, for your patience while I monopolized the best computer over the  past few months.  I am grateful to the Natural Sciences and Engineering Research Council o f Canada for financial support over the course of this project.  Thank you to my family for nourishing my roots which are so firmly planted in the Maritimes, and for never letting me forget that home is just a phone call away. Special thanks also to Cory Proctor, who has been there from the beginning, and to Jen Thompson- here's to climbing mountains.  1.  Introduction The motivation for this work has been recent interest in Elastomeric M i c r o Electro  Mechanical Systems, or E M E M S . 1  These are a special case of the well-known Micro  Electro Mechanical Systems, called M E M S , which use electromechanical transduction in a wide variety of applications, from medical applications such as blood pressure sensors to consumer products such as scuba diving computers and fuel pressure sensors. widespread  interest  in  MEMS  was  generated  by  the  desire  to  2  The  miniaturize  electromechanical devices, coupled with the increasing availability o f nricromachining equipment and techniques arising from the microelectronics industry. However, despite this availability, the cost of such devices remains high and their design and construction are limited by the mechanical linkages or bending of cantilevered structures which restrict the relative motion of the components and are subject to fracture even with normal use. These disadvantages of M E M S are at least in part overcome by the use o f elastomers in place of these mechanical structures, leading to the interest in E M E M S .  A s a result of  their high flexibility, elastomers employed in such devices can undergo much more substantial relative motion in response to electrostatic actuation than their non-elastomeric counterparts.  Further,  there  is  a  substantial  cost  reduction  since  elastomeric  microstructures can be inexpensively replicated from micro-machined molds. The behaviour o f these elastomers under the design conditions o f E M E M S is highly important.  In the past, elastomers were used primarily in applications where the  bulk properties were of interest. F o r applications involving elastomeric microstructures, it is important to investigate the behaviour of these structures as the size decreases. It is  1  conceivable that there could be important changes in the properties arising from surface effects as the volume becomes comparable to the surface area. This could influence the deformation response o f such structures to mechanical or electrostatic forces, which has a direct influence on the operation o f an E M E M S device. T o the best o f our knowledge, this is the first attempt to quantify such issues for small elastomeric microstructures. W e have chosen to address the issue o f the primary damping mechanism o f these elastomeric structures, specifically silicone rubber structures, and the most significant factors which affect it. Generally, since damping implies loss of energy, it is favourable to minimize the loss. Knowing the dependence o f this loss on factors which we can control, such as the amplitude and frequency of oscillation and the temperature of the environment, we can not only predict the energy loss under typical circumstances, but also work to minimize its effect on the operation of the device. There are two components o f the damping mechanism in an elastomeric system, discussed in detail in Chapter 2. The most widely studied is the viscoelastic behaviour, where the elastic component of the response o f the rubber to a deforrriing force is inhibited by a viscous flow component. A s a result, the deformation of an elastomer in response to the applied force is not instantaneous.  When the elastomer is subjected to an oscillating  force, the coefficient of viscoelastic damping depends on the frequency o f the oscillations, since this is a time dependent effect, and does not depend on the magnitude o f the applied force. Using a number o f assumptions, this behaviour can be represented analytically and is therefore an attractive simple model to describe such materials. A n equally important, though often overlooked, damping mechanism arises from the hysteresis behaviour of elastomers.  When an elastomer is deformed by an externally  2  applied force, it does not return exactly to its original shape when the deforming force is removed, but rather there remains a small residual displacement of the material.  This  behaviour is similar to other hysteretic phenomena, such as ferromagnetism, in that the residual displacement depends only on the history o f applied displacements and not on their frequency or duration. If an oscillating force is applied to the system, the hysteretic behaviour has the effect o f damping the response o f the system to the driving signal. Unlike viscoelastic damping, hysteretic damping is independent  of the  oscillation  frequency, but as this is an intrinsically non-linear effect, it may depend substantially on the amplitude of oscillation. In most studies, the hysteretic damping is neglected in favour o f viscoelastic damping. However, in many circumstances hysteresis dominates and should not be ignored.  One of the objectives of this thesis is to generate a computational  algorithm to determine the hysteretic force in the rubber, based on the displacement history. A convenient method o f measuring the damping o f an elastomeric material is described in Chapters 3 and 4. The procedure involves measuring the resonance peak for the forced harmonic oscillation o f a simple structure such as a small post constructed of rubber. From the shape of each resonance curve, the relative damping of the elastomeric post can be determined. The dependence of this damping on the size, shape and type of silicone rubber post, the amplitude of the oscillation, and the temperature  of the  surrounding environment can then be studied. These relationships yield conclusions about which parameters are important in the motion of small scale elastomeric structures. A s mentioned earlier, one of the objectives of the project is to develop a model to calculate the hysteretic force in a rubber sample given the past displacements. Chapter 7  3  presents such a model, which is an extension o f the standard Preisach treatment o f hysteresis, commonly used in dealing with ferromagnetic systems.  This predictive  algorithm is tested in Chapters 8 and 9 by measuring the hysteresis effect and comparing with the predicted value generated by the model.  The most straightforward way of  measuring the hysteresis effect is to conduct an experiment at zero frequency. In this case, there is no contribution from the viscoelastic behaviour and there is no need to attempt to separate the effects.  For this static situation, a long rubber band is used as the silicone  structure in contrast to the small oscillating post used in the dynamic experiments. In Chapter 10, the results o f the hysteresis model are used to predict how the energy loss for one cycle of the oscillation rubber post should depend on amplitude and the conclusions are compared with experimental results.  Based on these results, the  prospects for successful E M E M S microstructures are shown to be good, and suggestions are made for further work.  4  2.  Background  2.1.  The Rubber-like State  Rubbers are a class o f materials which exhibit unique physical properties.  The  term "rubber", previously only used to describe natural rubber, has come to encompass all materials which possess mechanical properties similar to those o f natural rubber.  The  most important physical characteristic is the ability to undergo large deformations in response to relatively small stresses and recover almost completely when the deforming forces are removed. This is called "elastomeric" behaviour.  Synthetic materials having  rubber-like properties, known as "elastomers", are related physically in structure and behaviour rather than chemically. The term elastomer is now often used to describe any material exhibiting elastomeric behaviour, so it is not uncommon to see the terms elastomer and rubber used interchangeably.  2.2.  Deformability and Modulus  A stress, or deformation per unit area, applied to a material produces a strain, or unit deformation in the material. For sufficiently small deformations, in most solids the applied stress and resulting strain are directly proportional and can be related in terms o f a modulus of elasticity, where  stress=modulus  x strain  (2.1)  5  Stresses and strains take different forms, namely tensile, shearing or hydraulic, as shown in Figure 2 . 1 . deformation.  The moduli for a given material are in general different for each mode of 3  If a material is subjected to a tensile or compressive stress, the result is a change in the length o f the sample. The strain produced in the material is proportional to the applied stress by Young's modulus, E, where the stress is given by the force, F, applied per unit area, A, and the strain is represented by the fractional change in length dUL.  If the stress is in the same plane as the area to which the force is applied, the material is said to be under shear stress and the resulting strain is given by the displacement in the direction of the force as a fraction of the total length o f the sample, dx/L. In this case, the shear modulus G relates the stress and strain.  F = A  G  ^ L  (2.3)  Hydraulic compression is analogous to the pressure exerted by a fluid surrounding the object and results in a change of volume. The modulus relating the pressure P and the fractional change in volume Z\V7V for this mode of deformation is known as the bulk modulus, B.  P =B  V  (2.4)  The high degree of deformability is the most notable difference between rubber and a typical non-elastomeric material. The Young's modulus of a hard solid such as steel is about 2 x l O  n  N/m  2  and the maximum elastic extensibility is less than 1%.  A typical  stress-strain curve for rubber shown in Figure 2.2 shows that the extensibility is typically in the range of 500-1000% and Young's modulus is several orders o f magnitude lower than an ordinary solid.  4  However, the nonlinearity o f the curve shows that since Hooke's  7  law does not apply, a constant value of Young's modulus cannot be assigned except for small extensions.  The high elasticity and low modulus are responsible for the vastly  different behaviour of elastomers and non-elastomeric materials.  It should be noted that the reversible behaviour of a rubber upon release of a deforming force causing substantial deformation is fundamentally different from that of a non-elastomeric solid. Atoms composing a metal are fixed in a permanent lattice structure and therefore large forces are required to achieve deformation since they must change the interatomic distances.  Under small applied forces the deformation is reversible, but  practically negligible. When the force is sufficiently large that the deformation becomes substantial, the material reaches a yield point and there is irreversible slippage of adjacent  8  crystals in the solid. There is no such yield point in a rubber since there is no straining o f interatomic bonds because the chain-like molecules are so mobile.  Stretching causes  alignment of the chains, and at very large extensions the interatomic forces increase causing a decrease in the volume. However this decrease is not significant and as a result, the bulk modulus of rubber is not substantially lower than typical values for nonelastomeric polymers.  2.2.1.  Molecular Structure  The elastic property of a rubber-like material is a result o f its molecular structure. Materials exhibiting rubber-like behaviour are composed of a network o f small molecular units linked together into long chains. These long molecules, called polymers, have a high degree of flexibility as a result o f the vibration of individual links in the chain. A s well, all rubbers share the characteristic of weak attractive van der Waals forces between adjacent molecules in the network and as a result are relatively free to move past one another. Given this structure alone, i f the material were subject to a tensile stress it would flow like a liquid. However, while the attractive forces between the adjacent molecules in a rubber are weak, their motion is somewhat restricted by crosslinks between the molecules, in the form o f either permanent chemical bonds or mechanical entanglements of the chains. These crosslinks provide resistance to the slipping of any given molecule past adjacent molecules. With such an interconnected network of polymers, the chain elements have a high degree of freedom of motion while the bulk material resists flow. The elastic stability of a given rubber sample can be increased by introducing additional crosslinks into the  9  structure.  During this process, known as vulcanization, rubber molecules are chemically  linked, typically by sulphur, at regular intervals to adjacent molecules.  Certain physical  properties can be adjusted by vulcanization of the rubber, depending on the density and chemical nature of the crosslinks.  6  To understand how the molecular structure o f rubber gives rise to its unique physical properties, it is necessary to describe a kinetic theory of elasticity. Prior to the acceptance of the idea o f long-chain molecules, initial attempts to explain the mechanisms responsible for elastic behaviour were forced to use classical concepts of molecular structure.  Theories of elasticity were based on open structures of molecules which  allowed large  deformations  under  small applied forces,  but  these theories  unsatisfactory.  Once the concept of the long, chain-like polymer replaced that o f a rigid  structure it became possible to develop a satisfactory kinetic theory of elasticity.  were  7  Molecules in a rubber-like material are composed of a backbone chain of molecules.  The chemical nature of these molecules is relatively unimportant since the  elastic property of rubber results from the physical structure rather than the chemical composition. Smaller side groups of molecules are attached periodically to the backbone. Vibrations and rotations of various links in the resulting chain allow a great deal o f mobility for these polymers. The thermal energy o f atoms in the polymer produces lateral vibrations which are more readily thermally excited than are longitudinal vibrations. Lateral vibrations involve rotations about single bonds in the chain, which require crossing a relatively small energy barrier. Longitudinal vibrations on the other hand require strong intermolecular forces to be overcome by deforming bond lengths and angles.  In a highly elastic material, bond  10  rotations take place very quickly.  The corresponding lateral motions cause kinks in  intermediate sections o f the chain and the overall irregular configuration of the molecule is determined statistically. The resulting configuration of the molecules in the relaxed state is such that there is no overall directional dependence.  When tension is applied to the  rubber, the molecule extends and restricts lateral vibration, causing tension along its length.  A s the extension increases toward the maximum, the molecules approach their  fully extended configuration and the value o f Young's modulus increases substantially. This is consistent with the measured stress-strain curves for elastomeric materials.  8  The degree of elasticity of a rubber depends on the details o f its structure.  For  example, molecular side groups may be attached on the same side o f a double bond, in the cis- configuration or on opposite sides in the trans- configuration.  Such differences in  structure lead to different physical properties, since the cis- configuration lowers the density of packing of the molecules and reduces the tendency for the material to crystallize.  9  The side groups attached to the backbone chain can be thought of as  "bumps" along the length o f the molecule. intermediate  These groups inhibit the rotation of  sections and molecules with bulky or highly polar side groups  face  substantially larger energy barriers to rotation. While the side groups inhibit the rotation of the chain elements, a large number of double bonds within the backbone facilitates rotation about existing single bonds since there are less bonding sites to which side groups can attach and inhibit rotation. Thus, the elastic behaviour of rubber-like materials is a result o f the structure of the long chain-like molecules, and the flexibility o f this chain allows it to conform to a variety of configurations. B y the addition of specific compounds, the molecular structure  11  can be changed and consequently, the elastic properties o f a given material can be controlled.  2.2.2.  Thermodynamics  Rubber resists being stretched and relaxes when it is released.  In its relaxed,  unstretched state, the molecules of a rubber are a tangled mass o f long chains. work is done to stretch the rubber, the chains align somewhat.  When  This stretched state is a  more ordered arrangement than the unstretched state and is therefore thermodynamically less probable. Forcing the rubber into such a state is thermodynamically equivalent compressing a gas, and the result is a release of heat.  to  When the deforming force is  removed, the molecules will return to the relaxed state where entropy is maximized, and heat will be absorbed in the process. These thermo-elastic effects are typically grouped together as the Gough-Joule effect.  10  A s a result of such observations, thermodynamics  has become a convenient method of analysis of rubber-like behaviour. When rubber is stretched at constant temperature, the result is a change in entropy and internal energy. However, at all but the very lowest extensions the change in internal energy is far exceeded by the change in entropy associated with ordering the molecules by untangling them, therefore most o f the work done in stretching the rubber goes to decreasing the entropy.  A fundamental difference between elastomeric and non-  elastomeric materials is the thermodynamic tendency o f the retractive force in rubber to increase with temperature. Unlike rubber, stretching a non-elastomeric solid increases the  12  intermolecular distance, increasing the volume and internal energy but not significantly changing the entropy.  A n increase in temperature causes the tension in the solid to  decrease as a result of the thermal expansion.  The contrasting behaviour of these two  distinct types of solids arises from the difference in molecular structures.  A s the  temperature increases the particles in both structures vibrate with larger amplitude.  In  metals, where atoms are in a lattice structure, this serves to increase the intermolecular separation which relaxes the tension whereas in a rubber, the increased motion o f the chain segments allow the molecules to return to a more disordered state, causing increased tension as it tends to retract. This behaviour is observed for extensions which exceed the thermal expansion associated with the rise in temperature.  A t very low extensions, the  thermal expansion is much greater than the increase in entropy caused by the extension, and as a result the tension decreases with an increase in temperature, an effect known as thermo-elastic inversion."  2.2.3. Glass Transition  The thermal motions of molecules relative to one another will occur when the vibrational energy is greater than the potential barrier restricting the motion. probability that a chain element will overcome this barrier is proportional to Boltzmann factor e x p ( - £ ' / kT),  The the  where E is the activation energy resulting from the  chemical composition o f the molecule. The freedom of rotation is temperature dependent since as the temperature is lowered, the vibrational energy decreases and fewer chain  13  elements undergo rotation. Although van der Waals forces between molecules are small, they must be overcome to allow rearrangement of molecules when the rubber is stretched. A s the temperature decreases, the rubber enters a "leathery" state where the vibrational energy is low and the probability of overcoming the potential barrier decreases.  The  elastic response in this leathery region is sluggish because of this restricted chain element mobility. As the temperature is lowered further, the chain elements do not have sufficient energy to overcome barriers restricting bond rotation and the material no longer exhibits rubber-like elasticity since molecules are essentially frozen in position. The behaviour of the material is the same as that of a non-elastomeric solid, where the deformation is due to the straining o f interatomic bonds and requires very large forces. Thus, in this region the modulus increases rapidly. Since this "frozen" rubber is generally non-crystalline, it is said to be in a "glassy" state. The transition from the rubbery state through the leathery state to the glassy state is gradual, with a characteristic glass transition temperature Tg representing the boundary between the glassy and leathery states.  The phenomenon occurs in all rubber-like  materials and Tg varies depending on the strength of the intermolecular forces and the degree o f flexibility o f the molecules. This transition is second-order, and though there are no discontinuities which accompany a first-order transition, the rate o f change of the modulus and viscosity with temperature changes rather abruptly, temperature range of 2 ° C to 5 ° C .  12  typically over a  13  The deformation of an elastomeric material in response to an applied force is not instantaneous since the intermolecular attractions must first be overcome by the vibrational  14  energy o f the molecules. Thus, the deformation is slower at low temperatures because the vibrational energy is lower. When a cyclical deforming force is applied to the material, it responds well at low frequencies but as the frequency increases, the molecules do not have sufficient time to respond within the deformation cycle and the behaviour seems sluggish. Thus an increase in the rate of deformation has the same effect as a decrease in the temperature of a material.  14  2.2.4. Silicones  Silicone rubbers belong to the polysiloxane family, the most widely studied and commercialized class of inorganic polymers. They consisting of a backbone of alternating silicon and oxygen atoms with organic side groups, as depicted by Figure 2.3.  This  siloxane backbone is highly flexible since the S i - 0 bond is longer than the C - C bond in hydrocarbon elastomers and the motion of the oxygen atoms is not restricted by the side groups. A s well, the S i - O - S i bond angle of 143° is larger than the C - C tetrahedral angle of 110°, making single bond rotation easy even at low temperatures. Because of this flexibility, silicone rubbers exhibit highly elastic behaviour over a wide temperature range. A s a result, silicones are one of the most important classes o f elastomers. ' 15  high-performance  16  15  CH —  3  Si — 0  CH  3  CH  3  — Si — 0  CH  —  3  Figure 2.3 : Molecular structure of polymethyl siloxane or silicone rubber  2.3.  Deformation Response to an Applied Stress  As mentioned in Section 2.2.3, there are three behaviour regimes for rubbers; glassy, elastomeric and viscous, depending on the duration o f the applied stress. Under extremely high frequency cycles o f applied stress, the behaviour is considered to be glasslike, since the molecules do not have sufficient time to respond to the stress. They deform elastically in response to applied stresses at moderate frequencies, but i f the stress is held constant for a long time, they will undergo a continuous deformation similar to the viscous flow o f a liquid.  In the intermediate frequency region,  component inhibited by a viscous flow component.  the response has an elastic  The transition between the two  distinct behaviours is time-dependent, based on a time factor which is characteristic o f the material and largely independent o f the amplitude of the applied stress. The treatment o f the complicated viscoelastic response simplified by employing two assumptions.  can be significantly  Firstly, since there are two distinctly different  molecular mechanisms responsible for the viscous and elastic responses,  the total  deformation can be thought o f as the sum o f the viscous response and the elastic response. Secondly, for small deformations it can be assumed that elastomers obey Hooke's law o f  16  elasticity and Newton's law of viscosity. In other words, the elastic component can be described in terms of a single spring constant and the viscosity represented by a single constant value. These assumptions allow analytical solution of the viscoelastic behaviour of elastomers. However, there is a third component of the response to an applied stress which is usually overlooked.  This component, which shall be referred to as the "hysteretic"  component is independent  of the duration of the stress and depends rather on the  magnitude of the previous applied stresses. In a hysteretic system, the state depends on its history in a time independent manner and the irreversible behaviour can be represented by a hysteresis loop whose enclosed area is proportional to the loss associated with the process.  One common example o f a hysteretic process is a ferromagnetic material in an  applied magnetic  field.  When the applied field is removed the material has a remnant  magnetization, and the energy lost in a cyclically changing field is dissipated as heat. A similar phenomenon is encountered in rubber under applied oscillating displacements. This irreversible component of the deformation results from crosslinked molecules bumping over one another as they are stretched.  18  Since it is very complicated to describe this  effect, it is often considered negligible under the assumption of small deformations and is thus ignored. Because both visco-elastic and hysteretic effects can occur at the same time, it is easy to confuse them.  In Figure 2.4, the inner hysteresis loop is a measure of the  deformation of a piece of rubber in a cyclically changing applied field at very low frequency.  The energy loss, given by the area o f the loop, is entirely the result o f the  hysteretic effect since the viscoelastic effect becomes vanishingly small as the  frequency  17  approaches zero.  The larger outer loop corresponds to the deformation at moderate  frequency. The additional area is a result o f the viscoelastic effect, since there are energy losses due to the fact that the deformation is not instantaneous, but rather lags the applied stress. It is important to note that over most ranges o f frequency and displacement for many elastomeric materials, the viscous component is small relative to the hysteretic component.  F o r this reason, when analysing the behaviour of rubber under cyclical  applied displacements, the hysteresis effect should not be neglected.  Deformation  Figure 2.4 : Hysteresis loops at low and moderate frequencies  These two distinctly different mechanisms have the same basic effect of damping the response o f a system under an applied stress which oscillates over time. However, as mentioned earlier, these effects differ in their dependence on amplitude and frequency, and in their amenability to analytic modeling. Characterizing the damping of oscillations under  18  varying conditions is crucial to properly understand the behaviour of silicone rubber structures in dynamic applications.  2.3.1. Viscoelastic Behaviour  A number of theories have been proposed to explain flow processes. known example is Eyring's theory o f viscous flow in normal liquids.  19  One well  According to the  theory, the molecules in the liquid at equilibrium occupy states o f minimum free energy similar to lattice positions in a solid by oscillating about mean positions in the potential energy field created by interactions with neighbouring molecules. In a normal liquid, there are unoccupied free energy minima or "holes" to which molecules can jump if they acquire enough thermal energy to exceed the potential barrier E between neighbouring free energy minima.  In an unstressed material, the jumps are in random directions and there is no  preferred direction of motion, as depicted by Figure 2.5 (a).  The application of an  external stress causes each molecule to be subjected to a small force / , and the energy required to move from position a to position b in the x direction against this force is given by£,.  B.-)fdx  <-> 2  5  a  In order to jump into a hole in the direction o f the stress, the molecule needs only to acquire an energy of E-E whereas to jump backwards against the stress requires s  E+E . s  This has the effect of lowering the potential barriers in the direction of the stress,  19  increasing the probability of thermally excited jumps and causing a net flow in that direction, as shown in Figure 2.5 (b).  20  equal flow Energy  V a  b  a  x  b  x  stressed  unstressed  (a)  (b)  Figure 2.5 : Viscous flow of molecules in a liquid  This explanation agrees with the observation that viscosity or resistance to flow decreases as temperature increases since the number of molecules which will jump to another free energy minimum will increase with the thermal excitation. Thus, the viscosity of a normal liquid depends on the temperature, the density and the size o f the free energy barrier. Although a complete molecular theory of viscosity is far more complicated, this explanation is sufficient to understand the phenomenon o f viscous flow in elastomeric materials. The viscous flow of polymers involves the jumping of chain elements rather than whole molecules into vacant sites.  In an unstressed material these chain elements,  typically five to ten monomer units, jump randomly with no preferred direction and there is no net motion. When a stress is applied, the motions of segments in the direction of the stress exceed the motions against the stress, and the net result is that the center o f mass of the molecule moves in the direction of the stress.  21  20  The high degree of flexibility of these chains allow the molecules to retain a random statistically determined configuration and the segments are free to move so as to reduce the activation energy associated with moving the center o f mass.  However, in  order to classify as flow there must be translation o f the whole molecule in the direction o f the stress.  This requires the cooperation of all the chain elements, resulting in a higher  activation energy for the whole molecule than for the individual elements. The larger the number of elements, the more cooperation is required. A s a result, the viscosity of the polymer depends on the chain length of the individual molecules. Given this viscous component, the response of a rubber to an applied stress is not instantaneous and there is an energy loss associated with the phase difference between the applied stress and resulting strain. The rotation of the chain elements and movement of the molecules past one another cause energy losses.  A s a result, part o f the energy  required to deform the rubber is lost as heat and not recovered when the stress is removed. The non-instantaneous response can be described in terms o f a complex modulus E*,  with the real component  E'  representing the in-phase elastic response and the  imaginary component E" representing the viscous response which leads the strain by 9 0 ° .  E*=E'  + iE"  (2.6)  The loss angle 8, indicating the extent to which energy input is converted into heat is determined by  21  (2.7)  The size of the loss factor depends on the temperature or equivalently the frequency o f the cyclic deformation.  A t high frequency or low temperature, the material is in the glassy  state and there is no rubber-like deformation. A s shown in Figure 2.6 and Figure 2.7, the Young's modulus in this case is high, and there is a small loss factor. frequency/temperature  In the central  range associated with the leathery state, the loss factor moves  through a maximum. A t low frequency or high temperature the material exhibits rubberlike behaviour, with low modulus and a small loss factor. refers to this as "hysteretic loss"  22  Literature in the field often  but often fails to distinguish the viscoelastic component  from the true hysteretic phenomenon discussed in the next section.  Glassy Region  Transition  Rubbery Region  *  Temperature  Figure 2.6 : Effect of temperature on complex modulus  22  Temperature  Figure 2.7 : Effect of temperature on loss factor  2.3.2. Hysteretic Behaviour  The irreversible hysteretic component of the deformation is a result o f the "bumpy" structure of rubber molecules. A s the rubber is stretched, the chain elements rotate and assume a new configuration and the bumpy molecules essentially slide past one another. In the process, some crosslinks in the form of mechanical entanglements fail and new ones are formed in the strained state, resulting in an irreversibly different network.  When the  stress is removed and the rubber is allowed to relax, the molecules regain a random configuration which is different than the original. There remains a residual displacement, also known as "set', as some of the bumpy molecules are tangled up with adjacent molecules and cannot completely relax.  The small force associated with this residual  23  strain is balanced by the spring force such that the rubber is in equilibrium. The magnitude of the residual displacement depends on the magnitude o f the applied stress since the further the rubber is stretched, more mechanical entanglements will be disturbed and more bumpy molecules will have the opportunity to become  entangled.  The residual  displacement remains after the stress has been removed, so the rubber essentially has a "memory" o f the magnitude of the past stresses. There is no such memory in a liquid since there are no crosslinks between the molecules as in a rubber. In absence o f the crosslinks, when the applied force is removed, there is no remaining small force / acting on each o f the molecules as described in Section 1.3.1. and as a result there is no preferred direction of flow resulting from the previously applied stresses.  2.3.3. Long term effect of an applied stress  When rubber is subjected to an external stress or strain well above the glass transition temperature, it undergoes relaxation processes which have long term effects on the deformation.  These effects are illustrated in the deformation-time curve for a  viscoelastic material as presented in Figure 2.8. The initial response A B is an instantaneous deformation associated with the elastic behaviour o f the rubber. If a stress or strain were applied to rubber in the absence o f crosslinks, there would be a continuous deformation as the molecules slipped past one another unimpeded.  When the molecules  are crosslinked, this slippage is inhibited, however there is still a slow change in deformation for a crosslinked rubber under an applied stress or strain.  This additional  deformation, known as creep, occurs under compression, tension and shear.  There are  24  both reversible and irreversible components  of creep deformation.  The reversible  component, or primary creep in B C , results from the slow, restricted re-orientation of chain elements whereas secondary creep, an irreversible component in C D is a result o f the viscous flow of whole molecules in the material. In this region o f the creep curve, the deformation is generally linear with log(time).  These two components result from the  viscoelastic nature of rubber.  D  a  o S3  F  B  A  G  Time  Figure 2.8 : Deformation-time curve for a viscoelastic material  25  The recovery of a rubber when the stress is removed is analogous to creep. Immediately after release in D E , the elastic portion o f the deformation recovers very rapidly, then settles to an approximately linear function on a log(time) scale in E F . The deformation curves representing creep and recovery are similar, however the creep curve has a very high upper bound corresponding to the ultimate failure of the material. Under extremely high stresses over long periods o f time, material failure can occur.  In such  cases, the deformation is associated with the breakdown of crosslinks and backbone chains to the point o f material fracture. A s well, chemical failure can result from degradation of crosslinks by oxidation.  This ageing process is accelerated by high temperatures, and  depends on the chemical stability of the material.  While the failure point for each  elastomer is different, this further deformation is not considered to be a result o f either the viscoelastic or the hysteretic effect, and at lower stresses and temperatures, this creep-tofailure phase is not reached.  A s a result o f the hysteretic effect, there is an finite  deformation even after the deforming force is removed, shown by F G . The recovery curve is therefore limited by this residual deformation, which is defined by the magnitude of the applied stress. In general, the long term effects noted in this section are not o f concern in determining the response of elastomers to an oscillatory stress of moderate frequency. In the rest o f the work described in this thesis, therefore, the three phenomena o f interest will be the elastic, viscoelastic and hysteretic forces.  26  3.  Background Regarding Resonant Systems  3.1.  Resonance Behaviour A rubber-like structure which is free to vibrate in response to a disturbance can  oscillate at one or more natural frequencies determined by its dimensions, the physical characteristics of the rubber, and the mode of deformation.  T o observe this natural  vibration, the sample need only be given an impulse to initiate the oscillations and the subsequent oscillating displacement can be observed over time. The response is damped somewhat by the external air resistance, but generally the characteristic damping forces within the rubber predominate. A s introduced in Section 2.3., there are two main components of this internal damping effect - the viscoelastic component and the hysteretic component.  Since the viscoelastic effect is linear and time dependent, it varies with the  oscillation frequency but not the magnitude of the impulse, as described in Section 2.3.1. On the other hand, as described in Section 2.3.2., the hysteresis effect may depend on the amplitude of the oscillation caused by the impulse, but will not depend substantially on the frequency of the oscillation.  Throughout the rest o f this discussion, we will focus on  systems which have a single well-defined lowest frequency mode of oscillation and we will refer to this as the "natural oscillation frequency". If rubber is subjected to a sinusoidal driving force at a frequency considerably less than the glass transition frequency, the system becomes more complicated than the case of the free oscillations since there are now two frequencies which are o f concern; the driving frequency and the natural frequency of the sample.  Under such forced vibrations, the  sample will reach a steady state oscillation at the driving  frequency.  However, the  27  amplitude o f the oscillation will be a maximum when the frequency o f the driving signal is the same as the natural frequency, a condition known as resonance.  In the absence o f  damping forces, the amplitude would increase indefinitely as the system approached resonance. However, in a real system, the motion o f the sample is limited by the damping force. A s the frequency is increased through the resonance o f the system, the amplitude o f the sample traces a smooth curve, with the peak at the resonance frequency.  3.2.  Resonance Quality Factor  T o quantify the damping effect on a particular resonance peak, the dimensionless quality factor, Q, is often reported. When it was first introduced, Q was intended to be represent the ratio o f reactance to resistance in electrical resonators.  However, its usage  has been extended as an attribute to describe the behaviour o f widely varying resonators, including mechanical vibrations, spectral lines and bouncing balls.  23  In describing a resonant system, Q is defined by the relationship,  '  Q.2X-0-  (3 1)  AU  where U is a time average o f the energy stored in the system at the resonance frequency and AU is the energy dissipated by the system during one cycle at this  frequency.  24  If the  value o f Q is greater than 4 , its value can be determined approximately from the graph o f power versus frequency as the ratio o f the resonance frequency to the full width half  28  maximum of the resonance.  In this case, the half maximum is a measure o f the power,  which corresponds to 1/V2 of the amplitude.  o ~ l ^  ( 3  A/  -  2 )  where A / i s the full width of the resonance peak at the half maximum o f the power. These parameters are depicted on the resonance peak in Figure 3.1.  1  2  Frequency  Figure 3 . 1 : Determining the Q of a resonance peak  The sharpness o f the resonance is an indication of the amount o f damping in the system and Q is a convenient method o f quantifying this sharpness.  A s Figure 3.2  illustrates, for a tall, narrow resonance peak, there is a smaller damping force and consequently a higher Q value than for a shorter, broader peak. A useful way of thinking of the Q is that the oscillation amplitude o f a natural resonator drops by a factor o f lie in about Q oscillations.  26  29  Frequency  Figure 3.2 : Measuring the sharpness of the resonance peak  30  4.  Experimental Design of a Resonant System  4.1.  Nature of Experiment The basis of this experiment is to determine the amount o f damping in an  elastomeric material under a variety o f different conditions by measuring the resonance peak o f an oscillating sample and determine the corresponding Q factor for each resonance peak. T o conduct such an experiment, it is necessary to use a structure which is easy to fabricate and convenient to measure and to model. For these reasons, the structure used was a post, which also allows convenient size scaling and variations in cross sectional shapes. The oscillations of the post were measured by detecting intensity variations in a laser beam incident on the post using a photodetector circuit. There are a number of different experimental set-ups to detect the oscillation signal, which are described in Section 4 . 5 . 1 .  4.2.  Physical Factors Influencing Dynamic Properties  4.2.1. Size, shape and type of rubber  The deformation response of a rubber sample depends not only on the type o f rubber and the magnitude and direction o f the deforming force, but also on the geometry and dimensions o f the sample. The frequency o f oscillation o f a simple rectangular post was estimated using dimensional analysis. A rigorous analytical solution is complicated and unnecessary since the system can be modeled using a finite element analysis as described in Chapter 6. However, the simple dimensional analysis shown in Appendix A  31  was sufficient initially to get an idea o f the magnitude of the frequency we will be dealing with and how it varies with the sample dimensions. The result is that the frequency o f oscillation is proportional to 1/L , where L is the length of the post. 2  4.2.2.  Amplitude  Under applied stress, the amount of deformation remaining when the stress is removed depends on the amplitude of the applied stress.  If the stress is applied at  a  frequency of effectively zero, the result is a measure of the hysteresis effect in the rubber. The oscillation can also be applied at nonzero frequency and the amount of damping in the system is determined by the Q associated with the resonance peak. In these experiments, it is not possible to separate the damping due to the viscoelastic effect from the hysteretic damping, but a significant change in the Q will indicate that the primary damping mechanism is affected by the amplitude of the oscillation.  Presumably, this would be a  hysteretic effect as it has been explained in Section 2.3 that the viscoelastic effect is substantially independent of the oscillation amplitude. If the damping is primarily due to the hysteresis effect, there will be a measurable change in the Q as the amplitude of the oscillation is reduced.  This expected amplitude  dependence follows from Section 2.3.2., where an explanation of the hysteretic damping mechanism is provided. Here, the entanglement of the bumpy molecules, induced by the applied deformation , is described as being responsible for maintaining a small force in the material after the stress is removed.  If the "bumps" were homogeneous in size and  position on the molecule, it could be argued that there is a very small amplitude o f  32  oscillation for which there is no further entanglement  and therefore  no damping.  However, the entanglements of the molecules are entirely random and it cannot be predicted if the damping will increase, decrease or remain the same as the amplitude o f oscillation is reduced.  4.2.3.  Temperature  The behaviour of the sample depends on the region o f the temperature/frequency scale under which the deforming force is applied. conducted  to investigate the three behaviour regimes - glassy, leathery and rubbery -  described in Section 2.2.3.  These experiments have measured the glass transition  temperature in terms o f a wide range of properties.  Numerous experiments have been  27  mechanical, thermal and even electrical  Using our approach of measuring the resonance frequency, we can locate the  transition to the glassy state by determining the temperature at which the peak disappears since a single glass-like structure should exhibit no resonance response in the observed frequency range. It should be noted that although we recognize that there are far more accurate ways of determining the glass transition temperature of a rubber sample, our experiment lends itself rather easily to examining the damping behaviour o f the oscillating system under the condition of varying temperature.  33  4.3.  Fabrication of Samples  Initial experiments involved making posts with circular cross section. A mold was made using the smallest available drill bits for a high-speed bench top pc board drill press, the smallest of which is a #80. The diameter o f the hole was stepped so as to control the height of the post and form an elastomeric base, as shown in Appendix C  The post  diameters ranged from 350 Lim to 500|im, with the heights determined by the aspect ratio of approximately 4. It was thought that a higher aspect ratio would make it difficult to remove the posts from the mold intact. In a further attempt to ease the removal of the posts, the mold was made from 3mm Teflon.  This material is easy to drill, but has a  tendency to relax over time, such that the accuracy o f the drilled holes is compromised. However, since this is a prototype the exact size of the posts initially was not crucial. The mold was fabricated as described, and the silicone was injected into the large diameter hole using a syringe. The uncured silicone has a sufficiently low viscosity that in a matter of minutes it flowed through the small hole and beaded on the underside. curing time of the samples was reduced by heat curing at 200°F.  The  Once cured, the bead  was cleaved with a razor blade and the excess silicone loosened from the top o f the Teflon mold. About half o f the samples remained intact, whereas the rest tore between the base and the post as a result o f the weakness at this joint. Despite this, enough posts were able to be removed intact for this to be considered a useful mold for fabricating cylindrical silicone posts on a small size scale. In order to fabricate posts with rectangular cross sections, uniform films o f silicone were cast and the films were cleaved under the stereo microscope to produce reasonably  34  uniform rectangular posts.  The thin films are cast by pressing the uncured silicone  between two pieces o f mylar, supported by thick glass plates, and separated by a spacer of the desired thickness, as shown in Figure 4.1.  spacers  x  silicone film  Figure 4.1 : Casting process of thin silicone rubber films  The best method for cleaving the films, depicted in requires removing the bottom layer of mylar, placing the film with the top layer of mylar on a microscope slide and placing a second slide along the edge of the film.  Using a second microscope slide as a  guide and working under the stereo microscope, the posts could be cleaved to approximately 300 (im in width. When the size scale is reduced to microns, the method will require re-evaluation, but for the time being it will be sufficient to cast these films and cleave posts of the desired size. Although this is not a highly accurate method since it is difficult to slice the sample without tearing it, and the finite thickness of the blade itself means that the posts cannot be made arbitrarily thin, it produces rectangular posts which are sufficiently uniform to serve our purposes.  35  razor blade -microscope slides  mylar glass support  silicone film  Figure 4.2 : Cleaving rectangular posts from a thin film of silicone rubber  4.4.  Experimental Set-up  4.4.1. Determining the natural frequency  The first experiment is to determine the natural frequency of the posts in the absence o f a driving force. released.  The base was held securely and the post brought back and  The post then oscillated at its natural frequency and the oscillations were  measured using the transmitted laser beam technique, detailed in Section  The  signal was viewed and stored on a Tektronics storage oscilloscope and from the trace, the frequency and Q were determined.  4.4.2.  Measuring the resonance peak  To determine the behaviour o f the post over a range o f frequencies, a piezostack was used to drive the post. The post was attached by its base to the piezostack so that both the amplitude and frequency of oscillation could be controlled, as shown in Figure  36  4.3.  Using this set-up, the piezostack changes length horizontally, causing lateral  deflections o f the post tip. This allows one to sweep through a range of frequencies to determine the resonance.  /  piezostack  Aluminum support silicone post  Figure 4.3 : Silicone post attached to piezostack  4.4.3.  Detecting the oscillations  The basic principle behind measuring the oscillations of the small rubber post involves detecting the intensity variations in a laser beam incident on the sample.  To  accomplish this task, there is are several possible experimental set-ups as described in this section. Transmitted laser beam technique  Initially, the oscillations o f the optically clear silicone post were measured by aiming a laser beam at the end o f the oscillating post. A s it moved back and forth, the intensity of the laser beam transmitted through the optically clear silicone changed. This change was detected by a simple photodiode circuit shown in Appendix D and displayed  37  on an oscilloscope. Although this technique, shown in Figure 4.4, was fairly crude, it showed that the oscillations of the posts could be measured by alignment with a laser beam and photodetector.  laser beam technique  In an effort to obtain more sensitive measurements, the reflected laser beam was detected instead of the transmitted beam.  The apparatus set up, shown in Figure 4.5,  ensures that the reflected beam is incident on the photodetector at a long distance, so that a small variation in the narrow transmitted beam corresponds to a much larger fluctuation in the reflected beam. When measuring smaller posts, it was necessary to focus the laser beam in order to ensure that the beam was incident on the post and the reflected beam was incident on the photodetector.  T w o lenses with focal lengths o f approximately 0.1m  accomplished this task.  38  photodetector  oscilloscope photodetector circuit Figure 4.5 : Reflected laser beam set-up  laser beam technique  In order to measure posts of small diameter accurately, a narrow laser beam is required. For this purpose, we obtained a laser diode coupled to a 9u.m optical fibre. This allows more flexibility in terms of experimental set-up since the light can be aimed directly at the sample while the source of the light can be kept some distance away. Using this socalled "pigtailed" laser, the experimental set-up was altered so that instead of measuring the reflected beam, the updated photodetector circuit, shown in Appendix D , once again detects the incident beam. This set-up, shown in Figure 4.6, is similar to the transmitted laser beam technique, however it does not require that the post is optically clear as it is the interruption o f the beam which shall be detected by the photodetector circuit. A s the post oscillates, it interrupts the beam. If the post is constructed from an optically clear silicone, some o f the incident light will be transmitted through the post as it interrupts the beam and  39  there will be a measurable change in the signal intensity when this occurs since the transmission through the silicone is not 100%. However, the signal to noise ratio is much improved if an opaque rubber is used as the sample instead of an optically clear sample since there will be no transmission through the sample when it is interrupting the beam, and the R M S voltage of the photodiode signal will be proportional to the amplitude of the oscillation. For these reasons, this became our preferred diagnostic technique. laser  silicone  photodetector circuit  oscilloscope  |  1  computer Figure 4.6 : Interrupted laser beam set-up  4.4.4.  Creating a controlled temperature environment  To investigate the effect of temperature on the oscillating rubber posts, it was necessary to build a temperature control apparatus.  For this purpose, a length of thin  walled stainless steel tubing was secured in a hole in the Styrofoam lid o f a liquid nitrogen dewar.  A 2 0 W resistor near the bottom o f the dewar was used to boil off the liquid  40  nitrogen at a controlled rate while a second resistor inside the tube was used to heat up the exiting nitrogen gas. B y adjusting the power dissipated in each o f these resistors, both the flow rate and the temperature of the nitrogen gas could be controlled and the temperature measured by placing a thermocouple within the column of flow.  This basic set-up is  shown in Figure 4.7. T o achieve constant temperature gas flow, the system requires about ten minutes to stabilize. Once the temperature has reached the desired level, it fluctuates by only a couple of degrees.  Measurements of the temperature behaviour over time are  provided in Appendix E .  Figure 4.7 : Apparatus for controlling temperature of nitrogen gas flow  41  5.  Results of the Resonant System  5.1.  Determining the natural frequency  The natural frequencies of three rectangular posts, labeled R - l , R-2 and R-3 were measured using the transmitted laser beam method.  The cross sectional dimensions o f  these posts, detailed in Appendix F , range from widths o f 300u.m to 2000|lm and the length o f each post was trimmed from about 2mm down to about 1mm. A t shorter lengths, it was not possible to align the laser accurately with the post tip in order to obtain a measurable signal. T o test the predicted relationship in Section 3.2.1., the reciprocal of the square root of the natural frequency, or more simply the square root of the period, is plotted in Figure 5.1 as a function o f the length.  Even with this crude measurement  method, in all three cases the resulting plots were linear, indicating that in fact the frequency does vary as 1/L , in agreement with the predicted relationship. 2  42  0.35 0.3 0.25  • •  0.2 +  •  t  • •  A  I•  A  •  E 0.15 ex  I  A  1  R-1 lR-2  AR-3  0.1 -0.05 --  •  •• A  A  0 10  15  20  Post length (mm)  Figure 5.1 : Relationship between post length and natural frequency  5.2.  Forced  Damped  Harmonic  Motion  Using the piezostack to drive the post and the reflected laser beam set-up to measure the response, data was taken to determine the frequency response of a number of posts.  The data collection procedure was improved by interfacing the appropriate  instruments using the LabV I E W software, as detailed in Appendix G , and the result is an easily measured resonance peak. When a sinusoidal voltage is applied to the piezostack, it changes length in response to the voltage. Since the base o f the silicone post is attached  43  to the stack, as illustrated in Figure 4.3, the post responds to the oscillating displacement and there is lateral deflections of the post tip. A s the tip oscillates, it interrupts the laser beam and results in a measurable signal on the photodetector and the amplitude o f this photodetector signal is proportional to the tip deflection, and we refer to this amplitude as R. The phase o f the oscillation, 6, represents the difference in the phase of the deflection of the tip compared to the displacement of the base. A typical resonance peak for a sample with circular cross section is shown in Figure 5.2 and associated phase plot is shown in Figure 5.3.  These plots have been  corrected for the amplitude and phase response of the piezostack, detailed in Appendix H , by expressing the resonance peak as a ratio,  R tw ra  o f the deflection o f the rubber, R, to the  motion of the stack, R at a given frequency. s  ratio  Similarly, the phase response, 6, is expressed as a difference between the phase of the tip deflection, Q, , and the displacement of the stack, 6 . including a correction for the phase ip  S  difference between the function generator output and sync, 6FG, as detailed in Appendix J.  d = e,, -d -e p  s  FG  (5.2)  Note that the fluctuations in the curves near 650 H z appear to result from electronic noise in the measurement equipment and do not represent any physical behaviour of the silicone post.  44  04  500  600  700  800  1000  900  1100  Frequency (Hz)  Figure 5.2 : Typical resonance peak collected for a silicone rubber post with circular cross section  W e l l below resonance there is a non-zero amplitude and the motion of the rubber is approximately in phase with the motion of the stack.  A s the frequency o f oscillation  passes through resonance, the phase passes through 9 0 ° , as expected for a resonant system. Beyond resonance, we would expect the amplitude to drop to zero, since the tip of the post cannot keep up with the motion of the base and as a result there would be very small amplitude. However, the signal above the resonance peak does not drop to near zero in any o f the resonance curves we have measured.  This is may be related to  additional modes of deformation at the higher frequencies since as the frequency is  45  increased, there are other less significant peak responses.  These additional modes most  likely result from non-uniformity of the post structure.  500  600  700  800  900  1000  1100  Frequency (Hz)  Figure 5.3 : Typical phase data collected for a silicone rubber post with circular cross section  In addition to the amplitude and phase, the behaviour of the post can be expressed in terms of the x and y components o f the oscillation. If we consider that the amplitude, R, and phase, 6, represent the motion of the post in polar coordinates, then by simple trigonometry we can express the same information in terms of the Cartesian coordinates, x and y. x = Rcosd  (5.3)  46  y = Rsind  (5.4)  The Cartesian plots corresponding to the sample in Figure 5.2 and Figure 5.3 are shown below in Figure 5.4 and Figure 5.5. In these plots, we would expect the x-component o f the oscillation to taper to zero on either side o f resonance and the y-component to pass through zero at resonance. Xratto,  In Figure 5.4, the x-component is represented by the ratio,  of the x-component of the motion of the rubber to the motion of the stack, x. . (  x ratio  X  ~~  (5-5)  Similarly for the y-component, the motion o f the rubber is corrected for the motion o f the stack, y , by expressing it as a ratio, y ratios  y  . = X  y ratio  (5.6) v  y,  '  47  1.5  500  600  700  800  900  1000  1100  Frequency (Hz)  Figure 5.4 : Typical x-component data collected for a silicone rubber post with circular cross section  48  Frequency (Hz)  Figure 5.5 : Typical y-component data collected for silicone rubber post with circular cross section  The Q of the oscillation is estimated by fitting the data with an appropriate best-fit line. This is most accurately accomplished by modeling the dimensions of each particular post using A N S Y S and adjusting the Young's modulus and damping ratio until the peaks match, as explained in detail in Chapter 6. While informative, this is a time-consuming procedure.  Rather, the peak was approximately fit using the analogy of an L R C  resonance , given by Equation (5.7), 28  A  /  =  (  aco  V  b  (5.7)  \  coj  2  +c  49  where A is the amplitude o f the response, co is the angular frequency o f the oscillation, and a, b, c, and / are constants.  Knowing the resonance frequency and the corresponding  maximum amplitude from the experimental resonance curve, b and I can be calculated. The remaining two constants, a and c are adjusted until a good visual fit to the experimental data is attained. This allows us to fit each resonance curve and determine the Q factor quickly and easily, as detailed in Appendix L . Throughout this results section, the fit is based on the x component o f the data. It could equivalently be fit using the amplitude o f the response, with the adjustment o f a constant offset added to Equation (5.7). F o r this study, we are comparing the effect o f different experimental parameters on the Q value, therefore the relative Q values in a series o f resonance peaks are at least as important, i f not more important, than the absolute Q values themselves.  F o r this reason, a consistent method o f fitting the  experimental data is required. In Figure 5.4, the fit to the experimental data is shown, and performing the calculation i n Appendix L , the Q is determined to be 8.0.  5.2.1. Size, shape and type of rubber  The resonance peaks o f various sizes, shapes and types o f rubber posts were measured and compared. The parameters o f interest when comparing the peaks are the Q value, indicating the magnitude o f the damping coefficient, and the resonance frequency. The amplitude o f the peak is a measure o f the maximum deflection o f the post, however it is measured in terms o f the intensity o f the interrupted laser beam relative to the uninterrupted beam.  Since the degree to which the post interrupts the beam is highly  dependent on the position o f the sample with respect to the pigtailed fibre, and therefore  50  can vary for different samples, the difference in the amplitude o f the peaks does not provide much useful information as to the actual behaviour of the system. In Figure 5.4 the resonance peaks are shown for two samples o f the same height and type of rubber but different cross sectional diameters.  A s indicated in Appendix F ,  sample C - l has a diameter of 440 urn and C-2 has a diameter of 330 urn. The resonance frequency for the post of smaller diameter is less than that o f the larger diameter post, as predicted by the dimensional analysis in Appendix A .  2.5  •  •  •  04  1.5 +  •  w  • • •  • C-l • C-2  ••••••  Jl0.5  o -I—*400  600  800  -t1000  1200  1400  Frequency (Hz)  Figure 5.6 : Comparison of resonance peaks for different sizes of silicone rubber posts  51  Similarly, in Figure 5.7 the resonance peaks for samples C-3 and C-4, having different lengths, were measured. The resonance frequency is lower for the longer post C 4, further supporting the prediction in Section 3.2.1. that natural frequency of oscillation goes as 1/L . 2  2.5  -i  ^  ,  •  C-3  • C-4  0 -I 0  1  1  1  1  1  1  200  400  600  800  1000  1200  Frequency (Hz)  Figure 5.7 : Comparison of resonance peaks for posts of different height  The resonance peaks were also compared in Figure 5.8 for two samples of similar size but different cross sectional shape. For both the rectangular cross section R-4 and the circular cross section C - l , there was one primary resonance peak, and it does not appear as though either shape was particularly susceptible to additional degenerate modes o f oscillation. However, because the cylindrical samples are easier to fabricate with uniform  52  dimensions, as described in Section 4.2, they were used for further experiments. Note that although it is difficult to make any interpretation o f the comparison o f the position o f the peaks because of the difference in cross sectional shapes, the resonance frequencies are similar.  3.5 + 3 2.5 •  Pi  •  •  •  •  2  c-i  • R-l ••••  1.5 1 •  0.5 0 100  300  500  700 .  900  1100  1300  1500  Frequency (Hz)  Figure 5.8 : Comparison of resonance peaks for silicone posts of different shape  Finally, posts o f equivalent dimensions were fabricated from different types o f rubber. Sample C - l used the softer, opaque HS-II rubber whereas sample C-5 used the stirrer, clear R T V 6 1 5 .  The important differences between these two types of rubber, as  well as some details about the manufacturer, are outlined in Appendix K . In Figure 5.9, the higher resonance frequency for the R T V 6 1 5 agrees with the fact that the Y o u n g ' s  53  modulus is larger since it is a stiffer rubber. A s well, the R T V 6 1 5 peak is slightly higher and somewhat noisier than the HS-II rubber presumably because it is clear and therefore does not fully interrupt the laser beam during the oscillations, as does the opaque HS-II.  • C-l • C-5  04  100  300  500  700  900  1100  1300  1500  Frequency (Hz)  Figure 5.9 : Comparison of resonance peaks for different types of silicone posts  In all cases, the resonance peak was shifted but the Q values, shown in Table 5.1 remained about the same for each of the resonance peaks.  F r o m this observation, it  appears that the damping is not significantly affected by changes in the size, shape and type of rubber within the ranges we have used.  Resonance Sample  Q  Frequency (Hz)  C-l  9.0  970  C-2  8.5  650  C-3  8.8  840  C-4  8.1  235  C-5  11.1  1225  R-l  9.2  625  Table 5.1 : Characteristics of resonance peaks for various samples  5.2.2.  Amplitude  We would like to see how the Q of the oscillation changes with the amplitude of the stack, in order to determine how the primary damping mechanism is affected by the oscillation amplitude. If the damping is primarily due to the hysteresis effect, then there could be a noticeable change in the damping as a function o f the amplitude.  A s the  amplitude decreases, one might expect a trend in the data indicating that the damping o f these small structures is a result of the hysteresis effect. First, the linearity o f the piezostack in terms o f the amplitude of the driving signal was verified.  This was accomplished by varying the amplitude o f the sinusoidal voltage  applied across the stack through an amplifier. A s a result o f this experimental set-up, the  55  input voltage amplitudes to which we refer in this section are the amplitude o f the input to the stack amplifier, which is directly proportional to the voltage applied across the stack. It was found that the motion o f the stack was linear with the driving signal up to an input amplitude of 400mV, as shown in Appendix H . Once confident of the linear operating region o f the stack, the response of the post was measured as the amplitude o f the driving force was decreased. A s the data will span a few orders o f magnitude in amplitude, the best representation to see a trend is to plot the ratio o f the output to the input versus the input amplitude. If the damping remains the same, the results should trace a horizontal line. However, it is difficult to measure the small amplitude response accurately for input amplitudes o f less than lmVrms. A s shown in Figure 5.10, the ratio o f the response for amplitudes o f about lmVnns is fairly constant, indicating that there is no significant change in the damping as the amplitude is decreased. Note that the values in Figure 5.10 have been normalized to the ratio at lOOmY,™-  56  1.0  10.0  100.0  1000.0  Input to stack amplifier ( m V ^ )  Figure 5.10 : Small amplitude response for HS-II silicone rubber post with circular cross section  The next obvious step is to trace out the resonance curves for different amplitudes along the curve in Figure 5.10. These resonance curves are shown in Figure 5.11 through Figure 5.13 for input signals to the stack amplifier o f 400 mVrms, 40 mVrms, and 4 mVrms, respectively.  For those curves corresponding to larger amplitudes, the main resonance  peak is clearly defined at about 860Hz, preceded by a secondary peak at about 640Hz. This secondary peak is presumed to be an artifact o f the electronics, however it is possible that it is another mode of deformation which happens to be nearly degenerate with the  57  main peak, as a result o f some non-uniformity in the cylindrical structure, since geometrically there is no preferred mode of oscillation in the azimuthal plane.  1.5  500  600  700  800  900  1000  1100  Frequency (Hz)  Figure 5.11 : Resonance peak for HS-II silicone rubber post with circular cross section at stack input amplitude of 4 0 0 m V ms  58  0.08  -0.04  J  — —  Frequency (Hz)  Figure 5.12: Resonance peak for HS-II silicone rubber post with circular cross section at stack input amplitude of 4 0 m V r m s  59  0.006  • •  -0.004 --  •  -0.006 -•  -0.008 -I  •  •  Frequency (Hz)  Figure 5.13 : Resonance peak for HS-II silicone rubber post with circular cross section at stack input amplitude of 4 m V r m s  The Q values and resonance frequencies for the three resonance plots are compared in Table 5.2. There might be a slight change in the damping o f the oscillating post at small amplitude, indicated by the slight increase in the ratio in Figure 5.10 and the slight increase in the resonance frequency. However, there is no significant change and it is difficult to determine the Q factor o f the oscillations confidently at these small amplitudes since as the amplitude decreases the signal to noise ratio decreases, resulting in noisy resonance peaks.  Although we are unable to determine the Q factor reliably for these  small oscillations, it is remarkable that by using such a crude detection method we can discern a peak at all since the lmVrms stack amplifier input corresponds to a maximum  60  stack displacement of about l n m , as shown in Appendix H . If the amplitude o f the tip o f the post is considered to be approximately Q times the driving amplitude, then the maximum displacement of the tip is lOnm for a Q of about 10. A t this level of motion, we can resolve the resonance peak but there is too much noise to determine a best line fit with any certainty.  Input Amplitude  Resonance Q  (mV^)  Frequency (Hz)  400  8.0  865  40  8.0  873  4  8.5  875  Table 5.2 : Characteristic of resonance peaks for different oscillation amplitudes  Thus, the damping coefficient of the silicone post does not appear to be amplitude dependent since the Q o f the oscillations did not change significantly as the amplitude o f the driving signal was decreased.  Although we anticipated a possible change in the  damping coefficient at small amplitudes, this amplitude independence is not a surprising result since the complexity of the molecular structure o f the rubber makes it impossible to predict its influence on the damping.  61  5.2.3. Temperature  The behaviour of the silicone rubber post was investigated over a range o f different temperatures.  Although the glass transition temperature for some silicones can be  extremely low, for typical silicone rubbers is in the range of -50°C to -60°C.  This  transition temperature can be measured for HS-II silicone using the oscillating post and the temperature control apparatus described in Section 4.3.4. Recall from Section 2.2.3. that the transition between the rubbery state and the glassy state is not instantaneous and rather occurs over a reasonably wide temperature range of 2°C to 5°C. This transition can be observed by comparing the resonance curves in Figure 5.14. The amplitude decreases with the temperature, which indicates that in fact the sample is becoming stiffer at lower temperatures. A s well, the resonance frequency increases which is consistent with the fact that Young's modulus is higher for stiffer materials . This transition occurs around -43°C, however the experimental results indicate that it takes place over a range o f 1°C, which is narrower than would be expected. However, pinpointing this range is complicated by the fact that the resonance peak might disappear in the noise before the material actually completes the transition to glass-like behaviour.  62  1.5 1.4 1.3 1.2  X  04  1 + 0.9  *xx  AA* ¥ •  0.8 +  •  X  •  1.1  •  •  •  • A A*t  .A*  A  X *A A  ^  *  *  x  ^  A A A  4  -31.7  A-42.7  x  4  0.7  degrees Celsius  A  A  X -45.5 X -46.5  A A  A  X  ^  0.6 0.5  4-  0  500  1000  1500  2000  2500  Frequency (Hz)  Figure 5.14 : Resonance peaks for different temperatures  Our primary interest in this study is the effect of the temperature of the post on the damping of the oscillations. For the well-defined resonance peaks corresponding to the higher temperatures of -31.7° and -42.7°, the Q values are approximately the same. However, it is clear to see that the signal to noise ratio substantially deteriorates the glass transition temperature is approached, making it impossible to determine the Q of the oscillations confidently near the transition.  63  6.  Modeling the System  6.1.  Finite Element Analysis using AN SYS  The oscillating post can be modeled most accurately using a finite element analysis program.  In this instance, we are using the commercially available software package  A N S Y S which allows us to model the resonance behaviour o f the oscillating silicone posts. The basic procedure involves geometrically constructing the sample, applying the appropriate material properties, imposing boundary conditions and applying loads or forces specific to the particular analysis. The model is then divided into small elements, or meshed, and the equations of interest are solved for each element. This greatly simplifies the solution, however it requires hundreds or thousands of calculations, depending on the number o f elements.  The result is clearly most accurate for an extremely small mesh,  however the mesh size is dictated mainly by time constraints since the time required to perform the calculations increases quickly with the number of elements.  Finite element  analysis techniques can be applied to a variety of static and dynamic phenomena, including mechanical stress and strain analysis, fluid flow, and electromagnetic fields.  6.2.  Producing a Resonance Peak  In this application, we geometrically construct  a post  o f the  appropriate  dimensions and specify material properties which are characteristic o f the silicone rubber. A harmonic displacement is applied to the base of the post to simulate the operation of the  64  piezostack. The model is then solved by determining the displacement o f each element in response to different frequencies of oscillation of the base. T o determine the resonance response, the real and imaginary components of displacement for one element near the top of the post are examined. This allows us to plot the amplitude and phase of the oscillation as the driving frequency moves through resonance.  The deformed shape o f a post  subjected to an oscillating displacement of its base is shown below in Figure 6.1. In this figure, the dashed line represents the undeformed post, and the solid line represents the deflection of the post at one point in the oscillation cycle when the base is displaced in the -x direction and the frequency of oscillation is below resonance. Note that for illustrative purposes, the scale of the deflection has been increased by a factor of 100.  65  Figure 6.1 : Deformed shape of oscillating silicone post using A N S Y S model  The results of the model for different sizes, shapes and types of silicone samples can be compared to the experimental results in Section 5.2 and the parameters of these models are based on the post dimensions provided in Appendix E . Note, however, that it is acceptable that the curves do not match perfectly since the peaks are generated by A N S Y S using only the approximate Young's modulus, damping ratio, and sample  66  dimensions.  The height, width, and position of the resonance peak are controlled by  adjusting the Young's modulus o f the material and the damping ratio. Thus, by matching the peak to experimental data, we can determine the modulus and damping ratio of a real post, as demonstrated shortly. This is, however, a computationally expensive procedure which is the primary reason for using Equation (5.7) to determine the Q value instead o f an A N S Y S model.  The results must be scaled since the model determines the actual  deflection of the post whereas the experimental peaks indicate the amount of deflection in terms of the intensity of the interrupted laser beam relative to the direct beam.  6.2.1. Varying the post diameter  The resonance peaks in Figure 6.2 are determined using the A N S Y S model by varying the post diameter. In agreement with the experimental results in Figure 5.6., the resonance frequency is greater for the large diameter post. A s in Section 5.2., recall that the variable  R tw ra  represents the ratio o f the deflection o f the post tip to the displacement  of the base at a given frequency.  67  o  2.5 |  • C-1 • C-2  0  200  400  600  800  1000  1200  1400  1600  Frequency (Hz)  Figure 6.2 : Resonance peaks for posts of different diameter as generated by A N S Y S program  6.2.2. Varying the post height  If the post height is varied, the peaks in Figure 6.3 show that the longer posts result in a lower resonance frequency, verifying the relative position of the peaks in Figure 5.7.  68  4.5  Frequency (Hz)  Figure 6.3 : Resonance peaks for posts of different heights as generated by A N S Y S program  6.2.3. Varying the post shape  The results o f the A N S Y S model in Figure 6.4 confirms that the shape o f the post does not greatly affect the frequency response o f the post, as determined experimentally in Figure 5.8.  69  4.5 4  3.5 3 +  2.5  04  • C-1  •  2  • R-4  • •  •  1.5 1  0.5 0  :—I 400  H 200  0  1 600  h 800  1000  1200  1400  1600  Frequency (Hz)  Figure 6.4 : Resonance peaks for posts of different shapes as generated by A N S Y S program  6.2.4. Varying the type of rubber  The type o f rubber is varied by changing the Young's modulus to the approximate value for the different silicones. In modeling the HS-II rubber, the Young's modulus was taken to be 0 . 8 x l 0 N / m , whereas 6  2  the R T V 6 1 5 rubber has a higher modulus o f  1 . 8 x l 0 N / m . In agreement with the experimental results in Figure 5.9, the model of the 6  2  rubber with the higher modulus is shown in Figure 6.5 to have higher  resonance  frequency.  70  4.5 4 + 3.5 3 2.5 +  04  • C-l • C-5  2 1.5 + 1 0.5 0 + 0  1000  500  1500  2000  2500  Frequency (Hz)  Figure 6.5 : Resonance peaks for posts of different types of rubber as generated by A N S Y S program  6.3.  Comparison  with Experimental  Data  A resonance peak generated using the A N S Y S experimental data in Figure 6.6.  model is compared to  the  Since we have only approximate knowledge of the  Young's modulus and the damping ratio, these parameters are adjusted until the resulting peak matches the experimental data.  For this particular data set, the model specifies a  Young's modulus o f 0.77x10 N / m , which is a reasonable value o f HS-II silicone rubber, 6  2  as well as a damping ratio of 9%.  71  • data • ansys  500  600  800  700  900  1000  1100  Frequency (Hz)  Figure 6.6 : Comparing the A N S Y S model with experimental data  Using finite element techniques, we can model the behaviour of the silicone rubber post when it is subjected to an oscillating displacement. This model provides an accurate method for determining the resonance behaviour of a post of given shape, dimensions and material properties, in agreement with our predictions based on simple dimensional analysis and experimental observations.  Thus, we feel confident that we understand the  details of the post behaviour.  72  7.  7.1.  Hysteresis Background  Introduction  When rubber is subjected to oscillating stress cycles, the energy loss resulting from the viscoelastic behaviour is generally small compared with the loss associated with the hysteresis effect, as described in Section 2.3. However, the hysteresis effect is generally neglected, mainly because the viscoelastic energy losses can be treated analytically. B y ignoring the contribution of the hysteresis effect, the analysis does not take into account a large portion o f the loss. In contrast to the elegant viscoelastic solution, the hysteresis effect depends not on the oscillation frequency but rather on the history of oscillation extrema. This is a much more complicated problem to solve and since there is no straightforward analytical description o f the mechanism, there is a need for an efficient computational algorithm based on easily obtained experimental data. The system can be described by a mathematical model analogous to the scalar Preisach treatment of magnetic hysteresis, a phenomenological treatment not intended to explain the physical causes of the hysteresis. Rather, the model provides a method for predicting the hysteretic force in terms o f the past extrema values o f displacement of the rubber.  It is necessary to introduce a new adaptation and application o f the standard  Preisach m o d e l  29  to overcome a number o f drawbacks associated with its application to  the phenomenon of hysteresis in rubber.  73  7.2.  Standard Preisach Treatment of Magnetic Hysteresis The magnetic behaviour of a piece of iron in a changing magnetic field is perhaps  the best known example of a hysteretic phenomenon. Under the influence of the magnetic field, the magnetic dipoles within an initially demagnetized sample align with the applied field, and there is a net magnetization within the material. When the field is removed, the dipoles do not all return to their original orientation, leaving a remnant magnetization which is largely independent of the duration o f the applied field. The application of an external field in the opposite direction results in a different magnetization state which depends on the remnant state before this new field was applied. It is found experimentally that the magnetization of the sample depends only on the extrema of the magnetization history, which allows a substantial economy in recording past history for the purposes o f predicting residual magnetization.  30  The magnetization o f a ferromagnetic material under  an arbitrary sequence of applied fields is depicted in Figure 7.1, where B represents the magnetic flux in the material, H represents the applied field intensity, and t is the time at which the field was applied.  Figure 7.1 : Applied magnetic field and resulting magnetization history of an iron sample  74  In a cyclically changing field, there is an energy loss associated with each cycle, and the behaviour traces out a hysteresis loop such as the one shown in Figure 7.2, where the energy loss is given by the area enclosed within the loop.  31  In order to model this behaviour, it is necessary to represent the hysteresis curve in the simplest mathematical form.  The approach taken by Mayergoyz uses the Preisach  model , where one first considers a hysteresis loop that switches between the up and 32  down branches at thresholds a and /?. A s developed later in this section, the switching between "up" and "down" branches of the hysteresis loop can be thought o f in terms of the up and down spins o f the magnetic dipoles in the material, depending on the parallel and antiparallel spin orientations with respect to the direction o f the applied field, H. The  75  appropriate branch is chosen by comparing the input parameter u(t) to the threshold values and the function y p produces an output of + 1 or -1 for a given input. This function will a  be referred to as a "hysteresis operator". The hysteresis operator used successfully in the case of ferromagnetism is illustrated in Figure 7.3. Conceptually, one can think o f this hysteresis operator as relating to a particular dipole, but the model is intended to be much more general, as w i l l be seen later. Yap  -<-  +1 a  u(t)  -1  Figure 7.3 : Hysteresis operator for scalar Preisach model of ferromagnetic hysteresis  The hysteresis operator y p is a function which operates on a previous time history a  of the state of the system, represented by u{t), and has only two output states, + 1 and - 1 . A s an aid to understanding this curve, consider a simple time history in which the applied field begins at  increases to +°<>, and then returns to -«>. In the initial state at  u(t) is  less than /3. When jap operates on the function u(t) in this case the result is - 1 , and the output is considered to be on the negative branch o f the hysteresis curve. A s the field is increased, the result o f the operation remains - 1 until u(t)=a. A t this point, the result o f the operation switches to the positive state or + 1 and remains in this state as u(t) is  76  increased to +«>.  If the applied field is then decreased, the output of the hysteresis  operator remains at +1, or follows the positive branch o f the curve until w(r)=/3. A t this point, the output switches back to the value of -1. Thus, for each value of u(t) between a and /3 there are two possible output results of the operation o f y p on the function u(t). a  This value is determined by the applied field is currently being increased or decreased. The  hysteretic  behaviour  of  a  ferromagnetic  system  can  be  described  mathematically using a set of these operators having a distribution of values of a and j3. In such a model, applied magnetic fields cause transitions between up and down spin states of the magnetic dipoles in the material, and the remnant magnetization depends on the sequence of applied fields. A s described earlier in this section, the "up" and "down" states refer to the parallel or antiparallel spin orientation of the magnetic dipoles, with respect to the direction of the applied field. A sequence of applied fields give rise to a distinct subset of operators in the + state, and a complementary subset in the - state. The ranges a  and /3 for these subsets are determined  by the extrema values in the  magnetization history. It is helpful to consider mapping the set of hysteresis operators onto a plane representing the possible combinations of thresholds a,/5 so that each operator can be thought of as located at a certain point on this plane, as depicted by the plot in Figure 7.4. Consider also that there is a density distribution or weighting function  fJ.(oc,($) across  plane so that operators located in a higher density region make a large contribution to the final state whereas those in a lower density region have less impact. For illustration of the  the  density distribution, the contours in Figure 7.4 represent arbitrary regions o f equal density  }i(a,P). Note that by definition, co/J, so fi(a,P)=0 for values of (3>a.  Figure 7.4 : Mapping hysteresis operators to a(5 plane for Preisach model  Based on this model, the mathematical treatment describing the hysteresis response is summarized by the Preisach-Krasnoselskii relation in Equation (7.1), which determines 33  the final magnetization f{t) of the ferromagnetic material. In this equation, y ,$u(f) is the a  result of the hysteresis operation on the applied field u(t) at time t. Note, however, that this relation requires instantaneous knowledge o f y ,pn(t), which for each individual a  element is dependent on the past extrema history as w i l l be extensively discussed shortly.  (7.1)  78  This model makes no assumptions about the physical nature o f the  ferromagnetic  hysteresis. Referring to Figure 7.5, at any instant o f time, the set of points plane can be divided into two distinct subsets.  (p,cc)  on the half  For a given point, if the past fields have  caused a transition to a positive branch, the result o f y pu(t) is + 1 and the point a  (P,oc)  is  considered to be in the "up" state. Similarly, if the transition is to a negative branch, the result o f Yapu(f) is - 1 and the point is considered to be in the "down" state. These two subsets are labeled as S for those in the up state and S" for those in the down state and in +  general they are separated by an interface which has the shape o f a staircase, as will be shown in the next section. Each additional field causes some of these states to switch.  a  Figure 7.5 : Illustrating the staircase interface  Prior to the application o f external fields u(t), the material is considered to be in a demagnetized state. The boundary between the S , S" regions in the demagnetized state +  lies along the line  oc=-p,  as shown in Figure 7.6 and explained in more detail later in this  79  section.  T o see how this interface emerges, we must first consider how the staircase  interface evolves from a sequence of inputs.  s-  a  /o^p  s  +  P  Figure 7.6 : Representation of demagnetized state in a,P-space  Consider the following oscillating, decaying sequence of fields u(t), shown in Figure 7.7 where the subscript 0 refers to the most recent extremum.  80  u(t)  Figure 7.7 : Oscillating, decaying sequence of applied magnetic fields  A s the field increases, the hysteresis operators y p a  for which the applied field is  greater than the upper threshold, or u(t)>a, switch to the "up" state.  Once the local  maximum value w is reached, a portion of the down states have been switched to up states 5  and there is an new interface between the S  +  and S" sets, shown by the bold line in  Figure 7.8.  81  a  Figure 7.8 : Interface following application of maximum field  Continuing with the sequence in Figure 7.7 the field then decreases, causing a transition of hysteresis operators to the "down" state for those operators whose lower limit (5 is greater than the applied field, as illustrated in Figure 7.9. While some of the operators which switch to the down state were those which were switched up by the field w , there remain some operators in the up state which are not changed by decreasing the 5  field to « - In this manner, the past fields influence the final result. 4  82  a  A s time progresses, the oscillating field causes transitions o f the appropriate hysteresis operators between the "up" and "down" states. The resulting interface separating the up and down states is a staircase, shown in Figure 7.10, with vertices du numbered to correspond to the extrema. A local extremum is characterized by — = 0 , dt du with —— < 0 indicating a maximum and creating a horizontal link in the staircase, and dt 2  du —— > 0 indicating a minimum and creating a vertical link. 2  83  Figure 7.10 : Staircase interface following arbitrary magnetization history  Using this geometric interpretation, Equation (7.1) can be more simply written as Equation (7.2).  34  (7.2)  This equation describes the magnetization of the material as the difference between the integrals of the density function ji{a,P) over the S up states and the S" down states, as +  determined by the past field extrema. The vertices of the staircase in Figure 7.10 can also be labeled in terms o f the (a,p) coordinates, with CCJ corresponding to the j' maximum and h  j3j corresponding to the f  h  minimum. Figure 7.11 shows the same sequence of extrema  using the (a,fi) notation.  84  fc  Pi  P  Figure 7 . 1 1 : ocf3 notation for sequence of extrema  Each local extremum in a sequence has the ability to "wipe out" the staircase vertices corresponding to previous smaller extrema. magnetization history shown in Figure 7.7.  Consider a modified version o f the  Here, the oscillating field is no longer  decaying in magnitude, and the most recent maximum field u\ is larger than the previous maximum u . 3  85  Figure 7.12 : Sequence of extrema illustrating wiping out principle  Before the application of u\, the interface forms the staircase shown in Figure 7.13.  a  \ 1  p  u  4  Figure 7.13 : Staircase interface prior to application of field Ui  86  However, once the field reaches the new maximum value u  u  all the states which  switched because of the minimum u and the maximum w have returned to their original 2  3  state and the states appear to be affected only by the new maximum u\. This erases the effects associated with the smaller extrema, eliminating them from the sequence of significant extrema.  Such a "wiping out" technique can be used to erase the memory of  past history in the sample.  35  When the memory o f past applied fields has been erased, the net magnetization within the sample is said to be demagnetized.  This state can be achieved by applying an  oscillating magnetic field, with an amplitude initially much larger than those previously, applied and gradually decaying to zero, as depicted in Figure 7.14.  Applied field time  Figure 7.14 : Demagnetization sequence  When the material is in a demagnetized state, the S ,S" regions occupy equal areas o f cc,(5 +  space and the integrals of  fi{oc,p) over these regions are equal, indicating that the net  87  magnetization is zero.  This demagnetization procedure is represented schematically in  Figure 7.15.  (e)  (d)  (f)  Figure 7.15 : Generating the demagnetized state  In Figure 7.15(a), the sample has memory o f some arbitrary past history. A local maximum w„ is applied in (b), followed by a local minimum u„.  u  magnitude than u  n  slightly smaller in  shown in (c), followed again by a slightly smaller local maximum u .  n 2  as in (d). A s the magnitude of the oscillations gradually fade to zero, the demagnetized  88  interface in (e) emerges. If the difference between the successive oscillatory amplitudes is infinitesimal, the interface approaches the line a=-fi  and the demagnetized state is  represented i n cu,/?-space, as shown in (f).  7.3.  Application  to  Rubber  Using this scalar Preisach model, the magnetized state of a ferromagnetic sample can be determined following an arbitrary sequence o f applied fields.  While it does not  explain the mechanism of the hysteresis, the model is a convenient method for analysing complicated hysteretic systems. hysteresis effect  The same basic method can be used to describe the  in rubber where instead o f applied fields,  we consider applied  displacements. Instead of a magnetic force, the molecules within the sample are subject to a spring force as they are stretched, as well as a hysteretic force as the various side groups and entanglements bump past one another.  The spring force is smooth and continuous  whereas the hysteretic force only acts when two bumpy molecules interact and does not depend on the duration of the stretch. Rather, it depends on the extreme displacements which determine how many bumps with which each side group interacts as the molecules are stretched. Using an adapted version o f the standard Preisach treatment o f hysteresis, the hysteretic force in the rubber following a sequence of displacements can be predicted. There are a number of disadvantages associated with the ferromagnetic model in terms o f its application to rubber. The following approach to overcoming these difficulties is, we believe, new. The first problem to overcome is that the hysteretic force in rubber  89  cannot be described by the two-state ferromagnetic hysteresis operator. The total force FT exerted by rubber under controlled displacements is the sum of the spring force Fs and the hysteretic force FH • F (x) T  = F (x) + F (x) s  H  (7.3)  The hysteretic force resulting from the accumulation of past displacements can be expressed as a sum,  ;=o  where each /,-(jc(f)) term is an individual hysteretic element associated with the past displacements x(t). In the ferromagnetic case, the hysteretic operator can be thought of as describing the up and down spin states of the molecules. When a magnetic force acts to switch the state to up, it remains in that state as the force is increased. F o r the analogous case in rubber, there are no up or down states. Rather, consider that the molecules can be in either a "stuck" or "unstuck" position. In a stuck position, there is some mechanical entanglement o f chain elements caused by stretching the molecules. Further stretching can cause the chain elements to untangle and the molecules return to the unstuck position. The hysteretic force only acts while the molecule is stuck. Therefore, in contrast to the ferromagnetic hysteresis operator, the force is zero both below the lower limit j8 and above the upper limit a, as shown in Figure 7.16. This new definition is made possible by separating the spring force component responsible for maintaining a non-zero force for  90  displacements greater than the upper threshold and less than the lower threshold from the hysteretic force component which results from the bumpy molecules sliding past one another.  F  H  +1  a -1  X  -  Figure 7.16 : Hysteresis operator for scalar Preisach model of hysteresis in rubber  A second drawback o f the ferromagnetic model is that the origin of the applied force and the origin of the hysteresis are considered to be the same mechanism. result, the system is overconstrained.  This problem is addressed  ferromagnetism described by Mayergoyz, Friedman and Sailing . 36  As a  in a model of They propose a  nonlinear Preisach model consisting of a fully reversible component and a partially reversible component.  While this model has the ability to describe the reversible  properties of hysteresis nonlinearities, it makes no distinction between the mechanisms of the applied force and the hysteretic force. In the ferromagnetic application it is plausible to use such a description as it is difficult to describe the physical distinction between the two.  However, the overconstraint in this model makes it inappropriate to describe the  analogous phenomenon in rubber where the mechanism o f the applied spring force is clearly different from the hysteretic force of the bumpy molecules.  91  Lastly, the ferromagnetic model is computationally inefficient since it requires twodimensional input data to predict results, since the density o f the y p operators are a  expressed as a function of both a and j5. W e have attempted to overcome this problem by simplifying this density function. Since our hysteresis operators depend on the relative displacement o f the sample rather than its absolute position, the significance o f each operator depends not on the values a and /?, but primarily on the difference between them, which is equivalent to saying that the output of the hysteresis operation in Figure 7.16 is substantially independent of its location with respect to x. This reduces the complexity of the model since (i(a,P) can be represented as ii{a~P), which is a function in onedimensional space rather than two-dimensional space. Since we are considering relatively small displacements or, more appropriately, vibrations o f low hysteresis rubber, it is reasonable to hypothesize that F  H  typical cycles.  «  F  s  over  Furthermore, we can assume that viscoelastic effects can be avoided by  careful measurement techniques at sufficiently low frequency, as will be described more fully in Section 9.1.2. In the case of the ferromagnetic hysteresis operator where the upper threshold a is greater than the lower threshold j8, the one-to-one correspondence to points in a,P space is valid over the half plane a > (3. However, in the case of rubber, where the force is zero for x <fi and x > a, we need only consider the mapping on the unshaded quarter plane shown in Figure 7.17. In the three shaded quadrants, the density distribution fi(oc,P) is zero, thus we need not consider the contribution of integrals over this region.  92  Figure 7.17 : Mapping hysteresis operators to afi plane for rubber model  Similar to ferromagnetic materials, there is a "demagnetized" state o f the rubber which we shall refer to as the "de-strained" state. In this state, the only crosslinks and mechanical entanglements are those which arise from the molecular structure rather than from an applied displacement. This state is achieved by applying a series o f oscillatory displacements, initially much larger than those which were previously applied and gradually decaying to zero. This is an analogous procedure to the demagnetization of a piece of iron by an oscillatory, decaying magnetic field. Recall from Figure 7.6 the "demagnetized" or analogous "de-strained" state, where all past history has been erased. When the rubber is displaced a distance d, the boundary between the S and S" states shifts such that it intersects the +  a—fi=x+d,  a=fi line at the point where  and the intermediate values o f the displacement correspond to points along the  a=p line. A s a result, the applied displacements can be measured in  ct,fl  space using the  93  a=fi line as a "ruler". A s an illustration, the staircase interface between S and S" states +  generated by two separate extensions +d and -d are shown in Figure 7.18.  (a) de-strained state  (b) + extension (d)  (d) de-strained state  (e) - extension (-d)  (c) release extension  (f) release extension  Figure 7.18 : Generating staircase interface associated with a given displacement  Figure 7.18(a) refers to the de-strained state where all past history has been erased and the equilibrium position is considered to be x. The rubber is then displaced a distance d along the a=P line to the point x ,, where x =x+d as shown in (b). During the extension, a  F  s  »  F  H  a  and the displacement is essentially all due to the spring force. When the  94  rubber is released without overshoot, it reaches an equilibrium at x  new  where F  T  depicted in (c)  is zero. A small spring force remains to balance the hysteretic force associated  with displacement d, and the corresponding residual displacement d is noted. O n the afl r  plane the shaded triangles in (c) and (f) represent the operators whose state has been changed as a result of the extension to x . a  A triangle which lies in the S region of the +  plane was generated by a stretch in the +x direction. A comparison of the two sequences in Figure 7.18 shows that the residual displacement is -d if the rubber is displaced from r  the de-strained state a distance -d, and the value differs only in sign.  Note that these  diagrams are somewhat misleading since as a result of the scale, it is not possible to see the residual displacement which is considerably smaller than the applied displacement. In the next section, the relationship between the extension and the residual displacement will be exaggerated for illustrative purposes. However, in Figure 7.18, since d « d , it can be r  ignored in this sequence o f diagrams.  Using the geometrical interpretation provided in  Equation (7.2) over the adapted region a,j5 space, we wish to develop an expression for the hysteretic force of the rubber in terms of the displacement extrema.  7.4.  Developing the Model  The goal is to avoid evaluating the integrals in Equation (7.2) by developing an explicit formula based on measurements of the residual displacement and the known spring constant  to  determine  the  hysteretic force.  Here we  discuss a procedure  for  experimentally obtaining the value of the integrals using the geometrical representation on the cc,P - plane.  95  Beginning with the de-strained state in Figure 7.19 (a), the rubber is displaced to a large negative value, returned, and released without overshoot, resulting in state (b). A further extension d beyond xa in (c) causes more operators to change state. When the rubber is released without overshoot, it settles at a new equilibrium in (d), where the hysteretic force associated with the additional displacement d is balanced by a residual spring force. This force causes a residual displacement, dr, relative to point xa, where dr is much smaller than the original extension d. For illustrative purposes, the magnitude of the residual displacement dr with respect to the applied displacement d has been greatly exaggerated. The points  (P,oc) enclosed by the  shaded rectangle in (d) correspond to the  hysteresis operators which have switched to the positive state as a result of the additional extension d.  96  Figure 7.19 : Geometrical evaluation of integrals over oc(3 space  The residual spring force can be written in terms of Equation (7.2).  Since the  shaded region is both added to S and subtracted from S \ the residual spring force is equal +  to twice the integral of the density function over the shaded rectangle.  Since only the  spring constant is required to relate the residual hysteretic displacement to the residual hysteretic force, either could be used to determine the final state as a result of a given sequence of past applied displacements.  Throughout the rest of the thesis, we will  consider the results in terms o f the residual displacement o f the rubber.  The integral is a  measure of the hysteretic force associated with a displacement to + d, and is determined by  97  measuring the residual displacement which will from now on be referred to as the "residual", denoted by R(d).  Since the integral term in the equation expresses  the  hysteretic force, it is necessary to divide by the spring constant, k , to determine the s  associated residual.  (7.5)  The final state requires knowledge o f the residual associated with individual displacements in a sequence.  Since this is an intrinsically non-linear problem, the  measurement o f the residual R(x) following a displacement x in a sequence is generally different than the residual R(x) resulting from an extension x applied to the demagnetized state.  However, as will be shown, experimental knowledge of the one-dimensional  function R(x) captures all the required information for predicting the hysteretic force arising from an arbitrary past history.  This function will typically be different under  varying experimental circumstances and will be determined by measuring the residual R(x) associated with a number of x values. To derive an expression for the residual in more general terms, consider two successive extrema in a displacement sequence, xi+\ and x„ where  is the more recent  displacement and the difference between these two extrema is d. d =x  M  - x,  The residual R(d) associated with the application o f the new extrema x, can be written in terms of the x measurements,  98  R(d) = R(x  1  -x )  (7.6)  0  and it follows from the two sequences in Figure 7.18 that R(d)=-R(-d) for d<0. Once a sequence of arbitrary displacements has been applied, the area o f the quarter-plane can be divided into rectangular regions as shown Figure 7.20 and the total integral is the sum o f the integrals over the rectangles. The advantage of this approach is that the contribution of each individual rectangle can be expressed in terms of measured residuals R(x).  The integrals over the triangular shaded regions are equal and opposite  and therefore make no contribution to the sum. The point on the a,/^plane which refers to the destrained state prior to any displacement is defined as the origin, O.  In the  rectangular representation of Figure 7.20, the quadrant over which the density distribution is non-zero is defined by the position x along the a=j5 line. 0  Thus only the rectangles  within this quadrant contribute to the total integral.  Figure 7.20 : Rectangular representation of staircase interface  99  It is clear in the rectangular representation in Figure 7.20 that the rectangle representing the most recent displacement will touch the oc=f3 line whereas  those  representing the past displacements will not touch the line. The integrals over these other rectangles cannot be measured directly since the density distribution is unknown. However, the integral over the rectangle which does touch the cc=P line is easily determined by measuring the residual for this displacement. Since the density distribution is uniform along the a=(5 line, the position o f the rectangle is unimportant. Therefore, the approach taken to determine the contribution of the other integrals is to express them in terms o f known residual values.  Again, it should be emphasized that although the  integrals represent contributions to the hysteretic force, these integrals can be expressed in terms of the residuals by dividing by the spring constant.  Therefore, in the following  development of the model, when we refer to the residuals, we are in fact referring to the integrals over the various rectangles on the a,/3-plane. W e make the distinction shown in Figure 7.21 between the two types of residuals on this plane by using the notation R(x) for those which are easily measurable and I(x) for those which must be determined indirectly. Note that in both cases, the value x refers to the difference between the most recent maximum and the most recent minimum.  100  Figure 7.21 : Illustration of the notation of rectangles  The shaded rectangle in Figure 7.22 (a) corresponds to changes in state resulting from the application o f the maximum displacement denoted by x , followed by minimum 2  displacement x and a most recent displacement of x . x  is given by R(x]-xo). R(xi-x ). 2  0  A s in Equation (7.6), this residual  Similarly, the residual corresponding to the shaded region in (b) is  These two residuals are easily obtained by measuring the residual displacement  for a displacement of the same magnitude from the demagnetized state. geometrically in Figure 7.22, the unknown residual in (c), labeled as  A s shown  I{X -XQ) 2  is the  difference between the two measured residuals, and the value is positive for maximum displacements and negative for minimum displacements.  B y similarly expressing the  remaining rectangles, we can write to total residual as the difference between  the  measured residuals.  101  R = R(x -x ) T  l  +  0  ^I(x,-x ) 0  i=\  R(x,-x ) 0  (a)  (b)  (c)  Figure 7.22 : Expression of integrals as the difference between two measured residuals  I(x -x ) 2  0  = R(x -x )-R(x -x ) l  2  1  0  (7.8)  T o express the hysteretic force in terms of known residuals, consider the displacement history shown in Figure 7.23.  102  I(Xj-XO)  Figure 7.23 : Arbitrary displacement history  In general terms, i f the /' extremum is a maximum, its contribution to the sum is:  /(*,. -x ) 0  Since  JCJ.]<JC,-  = +(R(x  M  - * , . ) - - * , . ) )  (7  following a maximum displacement, this expression becomes:  (i  T  whereas i f the  x  ~ o) x  = (M R  X  ~ t) x  +K(Xi  -  )  extremum is a minimum, the corresponding contribution is:  (7.  I(x -x ) i  = -(/?(*,. - x )-  0  R(x, -  M  ))  (7.11)  Since x,+i>x, following a minimum displacement, we can write:  /(*,. -x )  = R(x  0  i+i  - x , . ) + R(x, - x  M  )  (7.12)  Thus, the expression for the residual I(Xi-xo) is the same regardless of whether the i extremum is a minimum or a maximum.  Summing over all i values for  (n+1)  displacements, we have:  R =R(x -x ) T  n+l  + 2^R{x -x _ )  n  i  i  (7-13)  l  i=0  Since the sequence begins from the de-strained state and the first displacement is x , the n  value x  n+]  corresponds to the point on the a=~P  line where the displacements begin.  Thus, x , =-x  (7.14)  and  R = -R(2x„) T  + 2 ^ R{  Xj  /=o  -  )  (7-15)  From this residual, we can directly determine the hysteretic force F#.  (7.16)  (  R(2x )-2j R(x  F =k H  H  s  i  l  )  Using this remarkably simple relationship, the hysteretic force on the rubber is calculated based on the past extrema of displacement.  This expression allows us to  calculate the energy loss associated with an oscillation history and thereby determine the energy lost as result o f the hysteresis, as detailed in Section 7.6. It is remarkable that the effect of such a complicated phenomenon as the hysteresis mechanism on a sample of rubber can be so easily calculated using this expression.  7.5.  Predicting  the Residual  Displacement  It is proposed that the residual displacements could obey a power law relationship in terms of the applied displacements, as given in Equation (7.17). There is no specific physical rationale for a power law relationship, but it is postulated that this may be a convenient and reasonable fit to the experimental data.  R(x)  =  c^x ' ndn  2  (7.17)  105  where x dm is a dimensionless number representing the extension x divided by l m . There is n  no way of predicting what this power law relationship might be, but it will be dependent on the type of rubber and the dimensions of the sample. It can be easily determined by fitting a curve through a set of experimental data, as shown in Section 9.2.1.  7.6.  Energy  Loss in an Oscillating  System  The primary effect of the hysteresis of rubber is to damp the motion o f the sample when it is subjected to an oscillating deforming force. This results in an energy loss which is dependent in some way on the amplitude of the applied force.  We can identify this  dependence using the known hysteretic force in the rubber to calculate the energy loss under one cycle o f an oscillating stress. T o determine the energy loss, the hysteretic force can be calculated using Equation (7.16) as well as the power law relationship in Equation (7.17) which estimates the residual displacement associated with each extension.  Using  this result, we can identify the dependence of the loss on the amplitude o f the deforming force and compare this to the spring energy as a means o f determining the effect o f amplitude on the damping of the system. To accomplish this task, we shall consider one half cycle, from -a to +a, of the stress history in Figure 7.24 below.  106  Figure 7.24 : Stress history for oscillating cycle  In terms of the a,j8-space picture, at the beginning of the cycle the only relevant past extremum is -a. a  /  +a  -a  Figure 7.25 : a,/3-space picture for half cycle of oscillation  The hysteretic force, based on Equations (7.5) and (7.16) for one past displacement, is  107  F =^-{R'(2x )-2R'(x x  H  -*„))  x  (7.18)  where  R'(x) = -R(\x\) \x\  (7.19)  /J(|x|) = c , | ^  (7.20)  and  With one past displacement of x = -a and the most.recent displacement of x = x, the {  0  hysteretic force can be written as,  k  F  H  = -^-(- R(2a) + 2R(a + x))  (7.21)  The energy lost as a result of this hysteretic force, determined as the integral of this force over the range of -a to +a, leads to  loss = Ca  (7.22)  C2+1  where C is a constant having the value  C = 2  C 2 + 1  ^  C  l +c  (7.23)  2  108  Thus, the energy loss depends on the exponent of the power law in Equation It is well known that the spring energy varies as a.  It is interesting to note that i f  C2, the exponent of the power law in Equation (7.17), is close to one, then the energy loss due to hysteretic damping also goes as a . 2  This implies that in this case the Q of the  oscillations would be independent of the amplitude. Although it is speculation whether in fact the experimental results for the residual displacements resulting from different extensions will even behave a power law relationship, it is interesting to note that the amplitude dependence o f the damping mechanism will depend on the detailed mechanism of the hysteresis, and that in one special case the result will be an amplitude independent Q.  7.7.  Determining  the Density  Distribution  Using Equation (7.16) to generate the hysteretic force associated with a known sequence of past displacements, it is not possible to provide a general analytic solution of the density distribution  li(oc,f5)  over the a,/J-plane.  However, we can determine a  function which could reasonably represent the density distribution in the special case considered in Section 7.6., where the residual displacement varies linearly with the applied displacement. Consider the integral, 7, shown Figure 7.26 (a), associated with a displacement xu which corresponds to a distance d=Xd 1^2 , in the / direction.  This integral can be  determined from the measured residual using the spring constant as expressed in Equation  109  (7.5).  However, since the aim of this section is to determine the density distribution  mathematically, we w i l l refer only to the integrals and not to the measured residuals.  Recall from Section 7.3. that the density function,  n(oc,P)  depends not on the  absolute values of a and j3 but rather on the difference between them, a-fi.  Accordingly,  we can define / as the distance to any point on the a,(5 - plane along a line perpendicular to  a=P, defined in terms of the (a,P) coordinates,  j_a-P  (7-18)  4i  110  and density function is defined in terms of this length / as fi(l). Recalling from Equation (7.5), the integral, 7, in Figure 7.25 (a) is expressed as the integral o f the density function over the shaded rectangle.  T o determine this integral, the shaded region is divided into  two regions, as shown in Figure 7.26 (b), so that J=J\+J , where J represents the region 2  2  for which l<d and J\ represents the remainder of the integral where l>d.  7  ( -19) 7  7, = d] n(l)dl  d f  J =]ln(l)dl  (7.20)  2  Rather than solve the integrals explicitly, we will chose a function which satisfies the requirements o f the density function, namely that the integral must converge as / approaches ° o and when multiplied by / it must converge as / approaches 0. A convenient function which satisfies the criteria is the exponential function,  H{l) = e-  kl  (7.21)  Using this form for the density distribution, the value of the integral, J is  111  (7.22)  and it is clear that given a density distribution of an exponential form, for small values of d, the value o f the integral varies linearly with d. This result is interesting as we can in fact specify a density distribution which is consistent with the suggestion in Section 7.6. that the damping due to the hysteretic effect is independent o f amplitude if the residual displacement is linearly proportional to the applied displacement.  112  8.  Hysteresis Design  8.1.  Nature of Experiment  In order to investigate the hysteresis effect in rubber, we would like to obtain experimental data with which to compare the predictions generated by Equation (7-16). It is reasonably straightforward to separate the hysteresis effect from the viscoelastic effects by conducting experiments at essentially zero frequency.  Since the effect is quite small  compared to the applied displacements, a large structure, rather than the small posts used for the oscillation experiments, is required for the effect to be easily measurable.  A long  rubber band was chosen as the structure and the hysteresis effect was quantified by measuring  the  residual displacement  associated  with  a  much  larger longitudinal  displacement applied to the midpoint of the band.  8.2.  Fabrication of Samples  In order to conduct preliminary experiments, a length of commercial latex tubing was used as the rubber band sample. Once the basic principle had been demonstrated, it was necessary to construct rubber bands using different kinds of silicone rubber.  A  number of different methods were tried and the final modified method produced a rubber band of reasonably uniform square cross section. T o construct the mold shown in Figure 8.1, a strip of mylar approximately 0.1 m wide and 1 m in length was treated with a release coating. T w o strips of acrylic about 6mm x 20mm x 1000mm were fastened to the mylar using silicone sealant to create a 6mm wide trough. The uncured silicone was squeezed  113  into the bottom of the trough in a uniform layer. The method is successful for producing uniform bands o f silicones of moderate viscosity. If the silicone is too viscous, it will not distribute itself along the trough in a uniform layer. It is necessary to ensure that the seals between the mylar and acrylic strips are secure so that the silicone does not "ooze" into any cracks in the mold as it cures.  silicone  Figure 8 . 1 : Rubber band mold  8.3.  Experimental  Set-up  A rubber band of unstretched length L is stretched by a ratio S and held fixed at its endpoints. T o measure a residual displacement associated with an applied displacement, the equilibrium position x o f the midpoint is noted, a controlled displacement d is applied a  along the line joining the endpoints and gradually released so the midpoint moves without overshoot to a new equilibrium position x , shown in Figure 8.2. In general, the rubber r  displays a residual displacement d -x x r  r  a  which is a function of the displacement history.  114  The residual associated with an applied displacement x, as introduced in Section 7.4, is denoted by R(x).  midpoint  —k  k  1 1 1 k  1 1 1 1 H  midpoint  k  d i i  midpoint  k  IX i i  k  r  H H  Figure 8.2 : Measuring the residual displacement  For this particular experiment, we have used two different rubber bands, the first a 3 m length of 3mm diameter latex tubing and the second a 0.44m length of 5 m m square cross section. Both bands were stretched by a factor of 2.  The procedure for measuring  the integral R(x) requires that the midpoint be displaced to  In our case, we will  displace the rubber to -0.2m which approximates the fully-stretched state. The midpoint is then brought back in the +x direction until the total force is zero, extended a further distance x past this point and the residual displacement measured.  115  9.  Hysteresis Results  9.1.  Verifying  the system  behaviour  Before the residuals R(x) were measured, it was necessary to conduct a number of experiments to determine that the system was behaving as expected.  In particular, the  procedure required to "de-strain" the rubber, the elimination of viscoelastic effects, and the appropriate stretch magnitudes and duration are investigated.  9.1.1. Demagnetization procedure  It is convenient to consider all displacement sequences as starting from the destrained state. T o generate this state, the midpoint must be displaced in one direction past any previously applied extrema.  The midpoint should then be smoothly moved to an  opposite and slightly smaller displacement past the equilibrium position and continually displaced in an oscillating fashion, slightly decreasing the amplitude for each oscillation until the amplitude decays to zero. While effective, this procedure is time-consuming and tedious. In the interest of efficiency, it is preferable to avoid this step whenever possible. When measuring the residual displacement, the large initial displacement to -0.2m will wipe out any effects of the past motion so it is not necessary to demagnetize the rubber before each measurement. Experimental results for the latex band are shown in Figure 9.1 where the position of the midpoint was measured at ten-second intervals after an extension to -0.2m, with trial 1 beginning from the de-strained state and the following trials  116  beginning from the endpoint of the previous trials. The midpoint position does in fact settle to the same equilibrium regardless of the initial state.  Trial 1 Trial 2 Trial 3  Figure 9.1 : Recovery of a rubber band following three displacements of -200mm, starting from different initial states  9.1.2. Eliminating the viscoelastic effect  T o measure the displacement o f the midpoint due to the hysteresis  effect  accurately, a sufficiently long time period must be allowed before the measurement in order for the displacement due to the viscoelastic effect to become negligibly small. This  117  is the same as requiring that the experiment be conducted at effectively zero frequency. A number o f trials were conducted, with the position of the midpoint measured at 10 second intervals. For all of these trials, such as the one shown in Figure 9.1 within time period of two minutes following the release o f the applied stretch the viscoelastic effects had died out and the position of the midpoint was reasonably stable. The small fluctuations in the data past the two minute point are attributed to localized thermal fluctuations along the rubber band.  9.1.3.  Residual displacement f o r different stretch magnitude a n d duration  In order to investigate the behaviour of the system, the position o f the midpoint was measured following extensions o f different magnitude and duration.  These two  factors should exhibit the difference between the time-dependent viscoelastic effect and the amplitude-dependent hysteresis effect. The plot in Figure 9.2 shows the position of the midpoint following extensions of the indicated magnitudes for a duration o f 5 seconds.  These results show that the  viscoelastic effect dies out over the same time period regardless of the magnitude of the stretch. The residual displacement on the other hand is highly dependent on the magnitude of the stretch, as expected.  118  30 29 +X 28  X  27 o T3  C  o o 0*  26  • +10cm  -I*  I +20cm A +30cm  X  25  x +40cm x +50cm  24 23 +,  • +60cm  AVA  22  »•••»•••••«  21 20 0  Time ( x l O s) 2  Figure 9.2 : Residual displacement for various extensions  When the position o f the midpoint is tracked after being held at 100mm extension for different durations o f time as indicated in Figure 9.3, the results are not surprising. The time required for the viscoelastic effect to die out increases with the duration o f the stretch, but the residual displacement remains the same. These results indicated that the system is in fact behaving as expected and provided guidelines for the stretch duration and waiting periods required to measure the residuals as accurately as possible in order to test the model of the hysteresis effect.  3.5  2.5 +  o CL, -a ^  x  •  A  x X X  1.5 +  X  x  X  X  a  o :  i  o  1 + 0.5 +  0.5  1.5  2.5  Time (xlO s) Figure 9.3 : Residual displacement for various stretch durations  9.1.4. Comparison of latex and HS-II rubber bands  There is a fair amount o f scatter in the data for the latex rubber band, as shown in Figure 9.1 through Figure 9.3.  Once it was possible to fabricate rubber bands from other  types of rubber, the same experiments were performed using a rubber band made from HS-II rubber, since from previous experience it appeared that the residual displacement for this type of rubber was fairly large. To compare the hysteresis effect of the two different types o f rubber, a set of residual displacements were measured for each rubber following each displacement in a  120  arbitrary sequence.  F r o m the results, it is clear that the hysteresis effect is about three  times larger for the HS-II rubber than for the latex rubber. Using the HS-II band should reduce the scatter in the predicted residual data.  9.2.  Testing the predictive algorithm  The predictive algorithm established in Equation (7.16) is used to determine the residual displacement following a series of arbitrary displacements o f the midpoint of a rubber band.  T o test the algorithm, the residual displacements are measured and  compared with the predicted values, as detailed in the following sections.  9.2.1. Power law fit of residual displacement  To measure the integrals R(x) over a range o f x values, the midpoint was stretched to -0.2m for five seconds and brought back to the point where the total force is zero. After a two minute wait, the rubber reached the initial equilibrium as the viscoelastic effects became negligibly small and ten position measurements were taken. The midpoint was then stretched to +x for five seconds and after another two minute wait, ten more measurements were taken to determine the final equilibrium. The measurements were averaged to determine the equilibrium position and the residual displacement was determined as the difference between the initial and final equilibria. This experiment was conducted for both the latex rubber and the HS-II rubber band.  The resulting data sets were fit using the same convenient power-law relationship as in Equation (7.17).  Again, as explained in Section 7.6., there is no obvious physical  principle which suggests that the power-law relationship must be obeyed, and the behaviour o f the rubber is questionable near zero extension as it is difficult to Omeasure the residual effect  confidently following extremely small displacements.  However,  although there is no strict physical reason for obeying a power law relationship, given the experimental data it appears to be a reasonable guess. Once the power-law relationship has been determined for a given set o f data, the integrals R(x) can be evaluated by interpolation for any displacement. For the case of the latex rubber, shown in Figure 9.4, the relationship was found to be R(x)=(0.165 x 10' )m x dm' . 3  725  n  Similarly for the case o f  the HS-II rubber, shown in Figure 9.5, the relationship between the two variables was determined as R(x)=(0.440 x 10' )m x dm' . 3  650  n  (Recall from Section 7.5. that x dm is the n  dimensionless number representing the extension x divided by lm.)  122  Figure 9.4 : Interpolating the residual displacements for latex rubber band  123  3  Extension of midpoint (mm)  Figure 9.5 : Interpolating the residual displacements for HS-II rubber band  In both cases, the data was fit by visual inspection of the data and the power relationship.  A rigorous procedure could be used to determine the parameters a and b  from the best fit relationship, however this was not deemed necessary since the power law relationship is only a representation of the data and is not intended to determine any specific physical properties of the system.  124  9.2.2. Predicting the residual displacement  A n algorithm is required to determine the significant local extrema o f the displacement history, based on the wiping out principle described i n Section 7.2.  Each  additional displacement requires that the history be updated since the new extremum can wipe out the effect o f previous extrema.  B y definition, the extrema must alternate  between maxima and minima, and the first and last extrema can be maxima or minima and need not be the same.  Consider a history o f n + l extrema with x„ being the first  displacement and x being the most recent. Each time a new extremum x 0  new  list, it must be compared to the previous extrema. If x  new  is added to the  is a maximum and is larger than  the previous maximum JCI, then the new displacement wipes out the previous maximum x\ and subsequent minimum x . Similarly, i f x 0  new  is a minimum and more negative than the  previous minimum x\, then again the new displacement wipes out the previous minimum JCI and subsequent maximum x and the new displacement is compared with the next previous 0  like extremum. The wiping-out procedure continues until the criterion o f an alternating sequence which decreases in magnitude is met.  The notation is updated such that x  new  becomes x , the most recent extrema and continues to the first displacement x , 0  n  where  n+l is the number of remaining significant extrema. The wiping out procedure is depicted in Figure 9.6 where x\ is larger than the previous maximum and therefore the preceding maximum and minimum are deleted from the list. A computer program implementing this algorithm was used to generate a list o f significant extrema  from  any arbitrary  displacement history. The resulting hysteretic force following each additional extremum was predicted using the formula in Equation (7.16) and compared to experimental results.  125  x(t) X3  xo  t  Figure 9.6 : Arbitrary displacement history illustrating wiping out procedure  9.2.3. Optimizing fit parameters  The data sets in Figure 9.4 and Figure 9.5 were fit visually using the power law in Equation (7.17). However, as a result o f the scatter in the data, there exists a range o f appropriate values for the parameters a and b. The parameter choice was optimized by mmimizing the error between a set of experimental data and the corresponding predicted values. The measure of error used in this case was the x value, or the sum o f the squares 2  of the differences between the measured and predicted values.  T o determine the  experimental data, a sequence of thirty displacements within the range o f -400mm to +400mm was randomly generated.  Beginning with the demagnetized state, the first  displacement was applied and the residual displacement was measured. Each successive  126  displacement on the list is applied and the residual displacement is recorded until the final extremum is reached. The corresponding prediction was made following each extremum and compared to the measured values. Figure 9.7 and  Figure 9.8 show the predicted  values versus the measured values for latex and HS-II rubbers respectively, based on the a and b values provided in Figure 9.4 and Figure 9.5. A s expected, the points are scattered about the y=x line, meaning that the values calculated from the expression in Equation (7.16) are a reasonable prediction of the measured residuals.  Measured residual displacement (mm)  Figure 9.7 : Optimizing the parameters of the fit for predicting residual displacements for latex rubber band  127  1.5 -  /  •  4  1 • /  0.5 - - • <• • /  3  T3  C/J  •  i  •  •  n  <D  I  S-l  data y=x  1  J  />0.5 -  T3  /  O  •  /  /  •  •  -1.5-  Measured residual (mm)  Figure 9.8 : Optimizing parameters for predicting residual displacements of an HS-II rubber band  128  10.  Implications of Hysteresis Theory for Observed Oscillation  The two types o f experiments we have conducted, the oscillating post and the displacement of the rubber band, measure the effect of the hysteretic mechanism by which energy is lost when the rubber is subjected to a deforming stress. In the case of the rubber band, the residual displacement resulting from an extension o f the rubber is measured. Since this measurement is made at essentially zero frequency, only the hysteretic effect is responsible for the  residual displacement  since the  viscoelastic effect  makes  no  contribution. This allowed us to generate a formula for calculating the hysteretic force in the rubber as a function of the magnitude of the past displacements.  In the case of the  oscillating rubber band, the hysteretic effect tends to damp the motion o f the post. Since this experiment relies on the resonance behaviour of the post, by nature it cannot be conducted at zero frequency, and therefore the hysteretic damping cannot be easily distinguished from the viscoelastic damping. B y measuring the resonance peak, we can determine the amount of damping in the system under different oscillation amplitudes. W e have identified the dependence o f the loss on the amplitude o f the deforming force using the known hysteretic force in the rubber to calculate the energy loss under one cycle of an oscillating stress. Based on the calculation in Section 7.6., i f the hysteretic damping also goes as a , or in other words the exponent, c , o f the power law dependence 2  of R{x) is approximately 1, then the Q is independent of the amplitude, in agreement with the experimental results in Chapter 5. Recall the plots o f the residual displacement versus the extension in Figures 9.5 and 9.6. In these cases, we have reasonably fit the data using  129  power law relationships where the exponent c is less than 1. 2  However, the fractional  exponents are required to fit the residuals associated with the larger extensions. W e have already stated in Section 7.3. that the approximations we have used require small displacements only.  With this in mind, let us look only at the regions of small  displacements, shown in Figure 10.1 and Figure 10.2. In these regions, given the scatter in the data, we can achieve a reasonable fit using an exponent of 1, or a straight line. This suggests that for small extensions the residual displacement varies linearly with the extensions, although there is no reason why this relationship should be true and therefore no reason why we should force this exponent to be one.  Note that the best fit line  representing the data in Figure 10.1 would have a non-zero intercept.  This is likely the  result of the scatter in the data for the latex rubber band experiment since it is expected that there is no residual displacement for an applied displacement o f zero. For this reason, the relationship between the residual displacement and applied displacement is considered to be a line passing through the origin.  130  4  Extension of midpoint (mm)  Figure 10.1 : Determining the residual displacements for small displacements of a latex rubber band  131  3  2.5  +  Extension (mm)  Figure 10.2 : Determining the residual displacement for small displacements of an HS-II rubber band  Using a linear relationship between R(x) and x, we can say that the hysteretic damping varies with a , as does the spring energy. 2  Thus, in the region of small  displacements o f the rubber band, the effect o f the hysteretic damping becomes neither more nor less significant as the amplitude of the oscillation varies. It is interesting to note that this is the same result as the oscillating post experiments, where the Q remained approximately constant as the amplitude o f the oscillation was decreased.  A s well, i f we  consider that the relationship between the residual and the applied displacement is linear,  we can recall the result from Section 7.7. which indicates that the density distribution is such a case can be a decreasing exponential function based on the applied extension. It is worthwhile considering that this as yet another interesting characteristic of silicone elastomers.  Over a wide range of size, amplitude, and frequency, the damping  coefficient due to hysteresis, an intrinsically non-linear effect, is substantially constant.  133  11.  Conclusion  The main objective o f this thesis is to investigate the mechanisms responsible for damping the motion of elastomeric materials, primarily the hysteresis effect.  During this  investigation, various factors were changed to observe the effect on the damping o f an oscillating post of silicone rubber. It was determined that the size and shape o f the post do not appear to affect the damping coefficient of the post substantially. The temperature of the surroundings had a large effect on the behaviour of the rubber since as the glass transition was approached, the resonance peak disappeared.  However, near the glass transition temperature, the  signal to noise ratio is substantially lower than at room temperature, making it difficult to determine the Q of the oscillation near the transition with any certainty.  Thus we can  draw no conclusions about the effect o f temperatures near the glass transition on the damping of the rubber. It was originally predicted that the amplitude of the oscillation could have an effect on the hysteretic damping o f the post.  This prediction arises from the explanation in  Section 4.2.2. that the hysteretic response o f the rubber depends on the detailed density distribution o f hysteretic elements, and in general may yield an amplitude dependent loss factor. T o study this, we measured the resonance curves and determined the Q for each as the amplitude of oscillation of the rubber post was decreased. It was found that the Q did not change significantly as the amplitude o f the oscillation was changed over two orders of magnitude.  This leads us to believe that  although there may be some nonlinearities in the behaviour of the rubber under small  134  amplitude oscillations, they are not significant and the behaviour can be considered reasonably linear. Another objective o f the thesis is to develop a method of determining the hysteretic force in a rubber sample following a series of displacements. Using the standard Preisach model, we developed an equation for the hysteretic force based on the past displacements applied to the rubber. The prediction was tested by measuring the residual displacement o f the midpoint o f a silicone rubber band following various applied displacements and comparing them to the calculated value. The prediction was found to be in reasonable agreement with the observed residual displacements. This led us to study how the energy loss in the rubber during one cycle of an oscillating applied stress should depend on amplitude.  It was determined that if the  residual displacement is directly proportional to the applied displacement, then the energy lost to hysteretic damping is proportional to the square of the amplitude. Since the spring energy is also proportional to the square of the amplitude, we expect that in this case the Q o f the oscillation will be independent o f the amplitude. While there is no reason why the residual displacement should necessarily vary linearly with the applied displacement of the rubber, it is interesting to note that a linear relationship is consistent with our experimental observations with stretched rubber bands..  This is consistent with the results of Section  5.2.2, where the Q remains approximately the same regardless o f the amplitude of the driving signal. To our knowledge, this is the first time the hysteretic force in rubber has been expressed based on the past displacement extrema.  Similar expressions have been  generated for ferromagnetic hysteresis, but the application o f the standard Preisach model  135  to the hysteretic behaviour o f rubber is, we believe, an original contribution to the field. Using this knowledge of the hysteretic force, the energy loss associated with hysteretic damping can be determined for an elastomeric structure when subjected to an oscillating stress.  The understanding of the damping mechanism is crucial for the design and  application of elastomeric micro structures in Elastomeric M i c r o Electro Mechanical Systems ( E M E M S ) devices. This study is among the first to investigate the behaviour of small elastomeric structures.  Although the properties o f the bulk material are well-documented, interest in  the behaviour of elastomeric microstructures was sparked by the promise of the E M E M S field, and the interest continues to grow as the devices are applied to a wide range of new areas. While the behaviour of elastomeric structures on the size scale of micrometers is beyond the scope of this study, it is interesting to note that the loss factor for small oscillating elastomeric structures is largely independent frequency of the oscillation.  o f both the amplitude and  This is a unique behaviour, as most resonant systems are  dependent on either one factor or the other, and is yet another interesting characteristic of silicone elastomers. Based on our observations, the damping coefficient resulting from the hysteresis effect is substantially constant over a wide range o f sizes, amplitudes and temperatures. The apparent independence of the damping coefficient o f silicone rubbers makes their use in E M E M S even more attractive from the point o f view of device design.  In  many applications, damping o f oscillations is an undesirable characteristic since it indicates a loss o f energy.  A s a result, the predictability of the damping coefficient over a wide  range of operating circumstances for E M E M S structures is an important advantage.  136  The results of this thesis open up opportunity for further study of the behaviour o f small elastomeric structures. these  structures  microstructures.  on  a  The next important step is to examine the behaviour of  much  smaller  size  scale,  specifically  into  the  realm o f  Other studies have shown that as the size of the structure decreases,  surface effects become increasingly important. In particular, it has recently been shown that for 25|im thick elastomeric membranes, substantially to the dynamics.  37  the  surface  tension can  contribute  In other experiments, where silicone structures  are  brought into contact with other surfaces for the purpose o f frustrated total internal reflection, the interfacial surface energy is a crucial factor.  38  Based on the work in this  thesis, the damping coefficient o f small silicone structures does not appear to be influenced by a wide range o f sizes, amplitudes, and frequencies. Thus, questions regarding surface effects of silicone microstructures can be addressed without the influence o f these experimental considerations.  137  References 1  2  3  4  5  6  7  8  9  1 0  11  1 2  13  1 4  15  1 6  1 7  18  1 9  2 0  2 1  2 2  2 3  2 4  2 5  2 6  2 7  2 8  2 9  3 0  3 1  3 2  L . A . Whitehead et al., U S Patent #5 642 015, June 24, 1997 J. Bryzek et al., "Micromachines on the March", I E E E Spectrum, p.20, M a y 1994 D . Rosenthal, Resistance and Deformation in Solid Media, Pergamon Press Inc., N e w York, p. 11,1974 P. Meares, Polymers, D . V a n Nostrand Company Ltd., London, p. 161, 1965 ibid, p. 161 Use of Rubber in Engineering, P . W . A l l e n , P . B . Lindley, A . R . Payne, eds., Proceedings of a Conference held at Imperial College of Science and Technology, London, p.86, 1966 L . R . G . Treloar, The Physics of Rubber Elasticity, Oxford University Press, p.44, 1958 Meares, p. 162 Treloar, p.4 Meares, p. 160 ibid, p. 164 R . P . B r o w n , The Physical Testing of Rubbers, Applied Science Publishers Ltd., Essex, England, p.247, 1979 Meares, p.255 P . K . Freakley, A . R . Payne, Theory and Practice o f Engineering with Rubber, Applied Science Publishers Ltd., p. 17, 1978 Polymer Science and Engineering, National Academy Press, Washington, D C , p. 101, 1994 Freakley and Payne, p. 17 Treloar, p.5 A . R . Payne, Engineering Design with Rubber, MacLauren and Sons, L t d . , London, p.13, 1960 Treloar, p.296 Meares, p.244 ibid, p.247 Freakley and Payne, p. 10 E.I.Green, "The Story of Q " , The American Scientist, p.584, 1955 W . P . Crummet, A.B.Western, University Physics, W m . C . Brown Publishers, p. 913, 1994 Green, p.587 ibid,p.586 Use of Rubber in Engineering, p.92 Crummet and Western, p.914 I.D.Mayergoyz, "Mathematical Models of Hysteresis", I E E E Transactions on Magnetics, V o l 22, N o 5, 1986 H.E.Burke, Handbook of Magnetic Phenomena, V a n Nostrand Reinhold Company Inc., p.60, 1986 Crummet and Western, p. 8 82 I.D.Mayergoyz, p.603  138  33 34 35  36 37  38  39  ibid ibid I.D.Mayergoyz, G.Friedman, and C.Sailing, "Comparison of the Classical and Generalized Preisach Hysteresis Models with Experiments", I E E E Transactions on Magnetics, V o l 25, N o 5, p 3925, 1989 ibid A.J.Clark, " A Variable Spacing Diffraction Grating Created with Elastomeric Surface Waves", M S c Thesis, The University of British Columbia, 1997 R.Coope, Structured Surface Physics Laboratory, The University of British Columbia, personal communications, 1997 Freakley and Payne, p.649  139  Appendix A : Dimensional Analysis of a Simple Rectangular Post  To estimate the expected frequency of oscillation of the simple silicone rubber post shown in Figure A . 1, a dimensional analysis was used. Note that this is not intended to be a detailed analysis, rather a quick estimate of the natural frequency.  Figure A . 1 : Dimensions of a simple rectangular post  The angular frequency, ft), is estimated as that o f a mass, m on a spring with a spring constant, k. (A.l) co =  The mass of the post is given by the volume and the density, p, of the rubber,  m = abLp  (A.2)  140  and the spring constant, k, is expressed in terms of the Young's modulus, E, and an unknown constant, C. k = Ca'b L E j  k  (A.3)  In order of the units of the spring constant to be N / m , the exponents must sum to one. i+j+k=l  (A.4)  F r o m studies of deflecting beams, it is reasonable that the spring constant is proportional to L " , that is, k=-3. Also, for deflection in the b direction, it is reasonable that the spring 3  constant is proportional to a, hence i=l. This implies that j=3. For the case o f a square post, in which a=b, applying these exponents to Equation (A.3) and substituting (A.2) and (A.3) into ( A . l ) , we obtain:  (A.5)  Thus the natural frequency of oscillation of a simple square post increases as the post is widened and decreases as it is lengthened.  141  Appendix B : Structure of Samples  The basic structure of the silicone rubber posts used in the experiments in Chapter 5 are shown in Figure B . l and Figure B . 2 . Note that in the case o f the rectangular post structure, there is no elastomeric base. This is a result of the preparation method, since as the posts were cleaved from a bulk silicone film, it was neither possible nor necessary to construct a base for the samples. Rather than attaching the base to the piezostack, as was the case with the cylindrical posts, the post itself was attached to the piezostack, and the length of the post determined by the mobility of the structure once in position.  Figure B . l : Structure of a cylindrical silicone post  142  w  H  Figure B.2 : Structure of a rectangular silicone post  143  Appendix C : Photodetector Circuits  Initially, a simple photodetector circuit was constructed using a photodiode, as shown in Figure C. 1. 100 k Q  Figure C . l : Simple photodetector circuit  Measures were taken to eliminate noise in the circuit since for the shorter posts the signal was less than lOmV, in danger of being washed out by the inevitable 60Hz noise. These measures included building a battery pack o f 9 V batteries to power the amplifier and placing a capacitor across the feedback resistor in the photodetector circuit shown in Figure C.2. O.OluF  nc 100Q  0.1 UF  1  r^Wv-  +9V  o  V out  0.1 uF  -9V  Figure C.2 : Updated photodetector circuit  144  Appendix D : Controlled Temperature Environment  The temperature control apparatus described in Section 4.4.4. was tested to determine the stability at various temperatures.  T o achieve low temperature gas flow  requires an equilibration time of about ten minutes, as indicated by the plot in Figure D . l .  40 -, 20  .  A  C3 /  Time (minutes)  Figure D . l : Rate of change of temperature of nitrogen gas flow with dewar resistor power 4 5 W and tube resistor power 0 W  Similarly i f the temperature of the gas is then raised by increasing the power in the tube resistor, the equilibration requires about ten minutes and the resulting gas flow is stable at the desired temperature, as shown in Figure D.2.  145  Figure D.2 : Rate of change of temperature of nitrogen gas flow with dewar resistor power 4 5 W and tube resistor power 20W  Note that the temperature remains steady if the thermocouple remains in the same place. However, i f the thermocouple is disturbed, the temperature measured by the device changes by up to 5 degrees, as indicated by the kink in the graph in Figure D.3.  146  40 4-  -60 -I  1  1  1  1  1  1  0  5  10  15  20  25  30  Time (minutes)  Figure D.3 : Rate of change of temperature of nitrogen gas with dewar resistor power 2 0 W and tube resistor power 2 0 W  Thus, although the temperature of the flow is not uniform within the column, it is reasonably steady at each position. A s long as both the thermocouple and the sample are placed in the center of the flow, the apparatus is considered capable of providing a controlled temperature environment.  147  Appendix E : Sample Dimensions The following tables give the dimensions of the various post samples. The structures of the posts are detailed in the diagrams in Appendix B .  Sample  Diameter of post D ([im) 440  Height of transition H (nm) 500  Height of base H (Hm) 250  Diameter of base D Qua) 2250  Type of rubber  C-l  Height of post Hp dim) 1375  C-2  1375  330  500  315  2250  HS-II  C-3  1000  440  375  125  1750  HS-II  C-4  2125  440  375  125  1750  HS-II  C-5  1375  440  500  315  2250  615  p  t  b  b  HS-II  Table E . l : Dimensions of cylindrical samples  Sample  Width W(Lim)  Thickness  R-l  Height H(um) adjustable  1000  1000  615  R-2  adjustable  300  300  615  R-3  adjustable  2000  1000  615  R-4  1375  560  440  HS-II  Type of rubber  T(nm)  Table E.2 : Dimensions of rectangular samples  148  Appendix F : LabVIEW  T o improve the efficiency and accuracy of the data collection, the procedure was automated using Laboratory Virtual Instrument Engineering Workbench, or L a b V I E W , a graphical programming environment from National Instruments for data analysis and instrument control. For our application, this software program provides an easy way to control both the function generator and the multimeter with the computer, since the frequency o f sampling is sufficiently low that the instrument control need not be done in machine language.  The basic algorithm sets the frequency and amplitude of the driving  signal to the stack, waits for a specified time period and measures the response amplitude as an R M S voltage with the multimeter. The data is collected over a specified frequency range which a specified step size so that the  measurements are consistent  and  reproducible.  149  Appendix G : Amplitude and Phase Response of Piezostack  To determine the response of the piezostack, the Angstrom Resolver optical probe, described in Appendix H , was positioned above the stack to measure the motion in response to a given input signal. Initially the response was measured while varying the amplitude of the input signal to the piezostack, and the results are provided in Figure G. 1. For an input o f about 0 . 5 % ^ to the stack amplifier, the sinusoidal probe signal deteriorates to a distorted periodic signal, indicating that the stack no longer responds properly to the input signal.  This upper bound for the linear stack response is about  400mVrms, as shown by the elbow of the graph in Figure G . L  150  0 -I  1  1  1  1  1  1  0  0.5  1  1.5  2  2.5  3  Amplitude of input signal to stack amplifier ( V  m s  )  Figure G . 1 : Amplitude response of piezostack for large input signals  The lower bound o f the linearity o f the stack was checked using the lock-in amplifier. Figure G . 2 shows that the motion o f the stack decreases linearly with the input amplitude to about O.lmVrms. While it may remain linear for even smaller amplitudes, the motion of the stack cannot be reliably measured using this experimental set-up.  151  1  0.1  0.01 +  1 ID  0.001  0.0001 4• •  0.00001 0.01  4-  0.1  1  10  100  1000  Input to stack amplifier ( m V ^ )  Figure G.2 : Amplitude response of piezostack for small input signals  The frequency response o f the stack was also measured.  It was found that the  amplitude of the motion increases over the low frequency range, but remains fairly constant at higher frequencies.  152  0  200  400  600  800  1000  1200  1400  1600  Frequency (Hz)  Figure G.3 : Amplitude response of piezostack for different frequencies of input signals  The phase difference between the input signal to the stack amplifier and the motion of the stack decreases over a significant range as the frequency o f the input signal is increased to 1500 H z , as shown in Figure G.4.  153  200  400  600  800  1000  1200  1400  1600  Frequency (Hz)  Figure G . 4 : Phase response of piezostack for different frequencies of input signal  The measured frequency response o f the stack in Figure G . 3 and Figure G . 4 are used to correct the measurements o f the oscillations of the silicone posts so that the results presented in Chapter 5 represent the motion of the posts relative to the stack.  154  Appendix H : Angstrom Resolver Optical Probe  The Angstrom Resolver optical probe from Opto Acoustic Sensors, Inc. was used to measure the distance o f the probe from a surface by measuring the intensity o f the light reflected from the surface. The probe is composed of a bundle o f seven optical fibres, as shown in Figure H . l . One source fibre carries a near-infrared L E D signal which is incident on the surface.  The remaining fibres surround the source fibre and detect the  reflected light, the intensity of which depends on the distance o f the probe from the surface. B y measuring the amplitude of the probe signal, the profile of a static sample or the deflection of an oscillating sample can be determined.  Figure H . 1 : Angstrom Resolver optical probe tip  155  Appendix I: Phase Difference Between FG Sync and Output  The synchronized output of the function generator was used as the reference signal for the lock-in amplifier which measured the photodiode signal.  The phase  difference, shown in Figure 1.1, is taken into account when the phase o f the post oscillation is presented in Chapter 5. Note that the fluctuations in the phase angle near 450Hz are a result of electronic noise in the measurement equipment and do not represent any physical behaviour of the function generator.  40 35 30 25 j2 20 + OH  15 10 5 0 200  400  +  +  600  800  1000  1200  1400  1600  Frequency (Hz)  Figure 1.1 : Phase difference between function generator output and synchronized output  156  Appendix J : Determining Q  The Q for the resonance peak is determined in Equation (3.2) and depicted in Figure 3.1 as  (J.l)  fres  Q=  A/  where  A / = /  2  (J.2)  - / ,  and / / a n d / are the frequencies corresponding to the half power points detailed in Figure 2  3.1. These frequencies are determined using the L R C fit in Equation  aco  h  -  — C  \  A  co -b = Q h  (J.3)  h J  157  Appendix K : Properties of Specific Silicone Rubbers  The silicone rubbers used in this thesis, namely R T V 615 and HS-II are two-part, liquid compounds,  intended  for both electrical and mechanical uses in industrial  applications. These R T V , or room temperature vulcanizing, silicones cure, upon mixing, into strong, durable rubbers. They can withstand a wide temperature range, and possess an inherent release property which makes them ideal for detailed molding. R T V 615, obtained from G E Silicones, is initially a clear, colourless liquid.  It cures to a tough,  transparent rubber which is ideal for applications requiring optical clarity. R T V HS-II from D o w Corning is a high strength moldmaking rubber. Since it is much more viscous than the R T V 615, the catalyst is coloured pink to ensure uniform blending.  Its  exceptional release characteristics and detail reproduction make it ideal for moldmaking. The physical properties of interest in this thesis are provided below in Table K . 1. Note that the value of the Young's modulus is only approximate as it is not provided by the manufacturer.  Instead, the specification sheets provide a measure of the hardness of the  rubber, as is conventional in engineering practices.  The hardness refers to the elastic  resistance to indentation by a rigid body, and serves as an indirect measure o f the elastic modulus. There are a number of standard methods o f measuring hardness, all of which are based on the procedure of indenting the rubber by pressing a ball against the surface with a specified force for a specified duration.  39  One of these measurements o f hardness, known  as the Shore Durometer (model A ) is used by the manufacturers o f these particular silicones.  158  Type of Rubber  Approximate Young's Modulus (kg/m )  R T V 615  Durometer Hardness, Shore A 44  1.8 x 10  6  R T V HS-II  16  0.8 x 10  6  2  Density (kg/m ) 1020  Manufacturer  1210  D o w Corning  3  G E Silicones  Table K . l : Properties of Silicone Rubbers  159  Appendix L : Calculation of Energy Loss in an Oscillating System  The amplitude dependence o f the energy loss associated with one cycle o f oscillation o f a silicone rubber post is calculated in Section 7.6. The hysteretic force resulting from one half cycle o f oscillation is expressed in Equation (7.21). Thus the work against this force is determined by integrating over dx from -a to +a.  a  a  W = -jJR(2a)dx  + 2JR(a + x)dx  Recall from Equation (7.17) that the residuals can be expressed  ( L  -  ! )  as a power law  relationship such that  R{2a) = c (2af-  (L.2)  x  and R(a + x) = c (a + x)  (-)  C2  L  l  The solution to these simple integrals lead to the result that the loss depends on a , Cl+x  3  as  expressed in Equation (7.22).  160  


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