FABRICATION AND MECHANICS OF FIBER-REINFORCED ELASTOMERS. Larry D. Peel. A dissertation submitted to the faculty of. Brigham Young University

Transcription

1 FABRIATION AND MEHANIS OF FIBER-REINFORED ELASTOMERS by Larry D. Peel A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical Engineering Brigham Young University December 1998

2 opyright 1998 Larry D. Peel All Rights Reserved Unless Otherwise Noted

3 BRIGHAM YOUNG UNIVERSITY GRADUATE OMMITTEE APPROVAL of a dissertation submitted by Larry D. Peel This dissertation has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date David W. Jensen, hair Date Jordan J. ox Date Paul F. Eastman Date Larry L. Howell Date William G. Pitt

4 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the dissertation of Larry D. Peel in its final format and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date David W. Jensen hair, Graduate ommittee Accepted for the Department raig. Smith Graduate oordinator Accepted for the ollege Douglas M. habries Dean, ollege of Engineering and Technology

5 ABSTRAT FABRIATION AND MEHANIS OF FIBER-REINFORED ELASTOMERS Larry D. Peel Department of Mechanical Engineering Doctor of Philosophy Heightened interest in flexible composite applications such as bio-mechanical devices, flexible underwater vehicles, and compliant aircraft structures has revealed a need for improved fabrication techniques, more experimental data, and accurate analytical models for fiber-reinforced elastomeric (FRE) materials. An improved method was developed to fabricate small batches of good quality fiberreinforced elastomer prepreg. Strengths of the method include excellent fiber adhesion, medium to high fiber volume fractions, highly parallel fibers, use of traditional advanced composites fabrication methodologies, and reproducible ply thicknesses. Silicone/cotton, silicone/fiberglass, urethane/cotton, and urethane/fiberglass elastomer/fiber combinations were studied. Balanced angle-ply laminates of each fiber-reinforced elastomer system were laminated from the prepreg with off-axis angles ranging from 0 to 90 in 15 incre-

6 ments. The specimens had fiber volume fractions of 12% to 62%, using fiberglass and cotton fibers, respectively. Tensile stress-strain results of elastomers, dry and elastomer-impregnated cotton fiber, and the fiber-reinforced elastomer specimens are presented and discussed. The axial stiffness of individual cotton fibers increased 74% to 128% when impregnated with an elastomer. The stiffening trend of the silicone rubber and softening trend of the urethane rubber are reflected in the stress-strain response of their respective fiber-reinforced elastomer specimens. Nonlinear shear and transverse properties for each material combination were extracted from 45 and 90 stress-strain data. Nonlinearity of the stress-strain curves are functions of fiber angle, elastomer type and the amount of deformation. An accurate nonlinear FRE model is presented. It is based on classical lamination theory and has been modified to include geometric and material nonlinearity. Geometric nonlinearity is included in the form of nonlinear strain-displacement relations. Material nonlinearity is included in the form of nonlinear orthotropic material properties as a function of extensional strain. The nonlinear strain-displacement relations and the nonlinear material models were added to the FORTRAN code of a pre-existing composites analysis software package. Results from the nonlinear FRE model are compared with test results of balanced angle-ply specimens. orrelation between predicted and experimental results range from good to excellent. A rubber muscle which exhibits high contractive forces was also fabricated and modeled as part of the work.

7 AKNOWLEDGEMENTS I would like to express appreciation to my advisor for his efforts in my behalf and willingness to pursue this somewhat unorthodox research area. My graduate committee has been very helpful in reviewing my work, have aided in solving problems, and have provided ideas and moral support. I would like to thank Brigham Young University, the Department of Mechanical Engineering, and the Department of ivil and Environmental Engineering for the use of their facilities. I would like to thank Koichi Suzumori of the Toshiba orporation, in Japan, for his help and ideas. Last, but certainly not least, I would like to thank my wife, Makayla, for her continued encouragement and my family for their interest and support. I would also like to thank the US-Japan enter of Utah for their support. This effort was sponsored in part by the Air Force Office of Scientific Research, Air Force Material ommand, USAF, under grant number F , US-Japan enter of Utah. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. DISLAIMER The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government. Distribution State A. Approved for public release; distribution is unlimited. Fiber-Reinforced Elastomers vii 1/11/00

8 TABLE OF ONTENTS hapter 1 Introduction and General Review Synopsis Motivation and Background Scope of urrent Research Fabrication and Testing of Specimens Modeling onsiderations A Rubber Muscle Application Overview of Previous and urrent Work Modeling of Fiber-Reinforced Elastomers Experimental Work and Applications Fiber-Reinforced Elastomers and Rubber Muscles in Japan Summary...10 hapter 2 Small Batch Fabrication of Fiber-Reinforced Elastomers Synopsis Introduction ontributions to the State of the Art Intent of urrent Work onstituent Materials and haracteristics Matrices Reinforcement Rubber-to-Rubber Adhesion Vacuum-Assisted Resin Transfer Molding Process Filament Winding and Lamination Process Specimen Preparation Discussion of the Fabrication Process Fabrication and Processing Summary...30 hapter 3 The Tensile Response of Fiber-Reinforced Elastomers Synopsis...33 vii

9 3.2 Introduction ontributions to the State of the Art onstituent Materials and haracteristics Specimen haracteristics Test Specimen Dimensions Fiber Volume Fractions Experimental Procedures Test Equipment Definitions of Stress and Strain Strain alibration Experimental Stress-Strain Behavior Elastomer Behavior Reinforcement Behavior Discussion of Fiber-Reinforced Elastomer Response a otton-reinforced Silicone b Fiberglass-Reinforced Silicone c otton-reinforced Urethane d Fiberglass-Reinforced Urethane Laminate Failure Modes Influence of Fiber Angle, Matrix and Fiber Type Prediction of Initial Orthotropic Material Properties Nonlinear Orthotropic Material Properties a Extensional and Transverse Moduli b Poisson s Ratio and Shear Modulus Summary of Experimental Behavior...79 hapter 4 Nonlinear Modeling of Fiber-Reinforced Elastomers Synopsis Introduction ontributions to the State of the Art urrent ontributions Nonlinear Modeling of Fiber-Reinforced Elastomers Overview of Linear lassical Lamination Theory Material Nonlinearity a The Bi-Linear Stress-Strain Model b The Mooney-Rivlin Material Model...94 viii

10 4.4.2c The Ogden Material Model d Implementation of the Ogden Material Model Geometric Nonlinearity Implementation of Nonlinear Model a The omputer ode b Method of Solution Fiber Re-Orientation and the Rubber Muscle Summary of the Nonlinear Model hapter 5 omparison of Predicted and Experimental Data Synopsis Predicted and Experimental Stress-Strain Responses otton-reinforced Silicone Fiberglass-Reinforced Silicone otton-reinforced Urethane Fiberglass-Reinforced Urethane Discussion of Predicted Results Predictions from the Rubber Muscle Model hapter Summary hapter 6 losure and Recommendations for Future Work General omments Processing and Fabrication onclusions Future Processing and Fabrication onclusions From Experimental Work Future Experimental Work Nonlinear Modeling onclusions Future Nonlinear Model Enhancements References Appendix A PFRE3 Screen Output and Flow Appendix B PFRE3 Fortran ode Appendix B PFRE4 / Rubber Muscle Fortran ode Appendix Data Files for PFRE Linear Material Properties - MDAT2.DAT Nonlinear Material Properties - FREDAT.DAT ix

11 .3 Output Data File - FREOUT.DAT Rubber Muscle Model Output data file - FREM.dat Appendix D Stress-Strain Data from Individual Specimens D.1 otton-reinforced Silicone D.2 Fiberglass-Reinforced Silicone D.3 otton-reinforced Urethane D.4 Fiberglass-Reinforced Urethane Bibliography With Notes I Theoretical Work Relating to FRE II Fabrication Techniques III FRE Applications IV Rubbertuator Articles V General and Related References x

12 LIST OF TABLES Table Page 1.1 Fiber-reinforced elastomer specimen test matrix Prepreg and laminate thicknesses (± one standard deviation) onstituent Material Properties Thickness (± one standard deviation) for each material system Laminate and prepreg fiber volume fractions for each material system Approximate Young s modulus for dry and impregnated cotton fibers Average initial axial laminate stiffness ( ) of each FRE combination by angle Predicted and measured axial stiffness ( ) for each material system Predicted and measured initial shear stiffness ( ) for each material system Predicted and measured transverse stiffness ( ) for each material system Average Poisson s ratios (ν xy ) for each material system Ogden coefficients for each shear stiffness Ogden coefficients for each transverse stiffness Measured and predicted Poisson s ratios (ν xy ) for each material system E x o E 1 o o G 12 E 2 o xi

13 LIST OF FIGURES Figure Page Figure 1.1 Schematic and examples of the flexible microactuator [22]...8 Figure 1.2 Soft gripper schematic from Okayama University [19]...9 Figure 1.3 Schematic of the Bridgestone rubbertuator [29]...10 Figure 1.4 Figure 2.1 Figure 2.2 Figure 2.3 Robotic arm with inflated and un-inflated rubbertuators...11 Examples of flexible micro-actuators (Reprinted with permission from Toshiba orp)...16 Initial fiber-reinforced elastomer mold for the vacuum-assisted RTM process...21 Filament winding of fiber-reinforced elastomer onto a rectangular mandrel Figure 2.4 Schematic of a vacuum-bagged mandrel assembly Figure 2.5 Samples of cotton- and fiberglass-reinforced elastomer prepreg...25 Figure 2.6 Specimens with jagged edges after being cut with a utility knife Figure 2.7 Raw test results for urethane/fiberglass at 45 and silicone/cotton at Figure 3.1 Elastomer test specimen configuration Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.7 Fiber-reinforced elastomer test specimen configuration...39 Special test fixture with gripping screws...42 Specimen and equivalent spring-stiffness diagram...44 Plot of extensometer versus machine strain for a typical urethane rubber specimen, with linear fit Stress-strain curves at widely varying strain rates. Response is essentially independent of strain rate...50 Figure 3.6 Typical stress-strain response of silicone under repeated loadings Figure 3.8 Tensile stress-strain results for undamaged silicone rubber specimens...52 xii

14 Figure 3.9 Individual and average tensile results from all urethane rubber specimens. 53 Figure 3.10 Elastic moduli of pure silicone and urethane rubber as a function of tensile strain Figure 3.11 Multiple strand tensile test results of dry cotton fiber Figure 3.12 Silicone (s/c) and urethane-impregnated (u/c) cotton fiber tensile test results Figure 3.13 omparison of dry, silicone-impregnated (s/c), and urethaneimpregnated (u/c) cotton tensile test results, with best linear fits...57 Figure 3.14 Schematic of a dry (left) and impregnated (right) cotton fiber Figure 3.15 Measured tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens from 0 to Figure 3.16 Measured tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens from 45 to Figure 3.17 Average tensile test results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens from 0 to Figure 3.18 Average tensile test results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens from 45 to Figure 3.19 Average tensile test results from 0 to 45 for [± θ] 2 cotton-reinforced urethane (u/c) specimens Figure 3.20 Average tensile test results from 45 to 90 for [± θ] 2 cotton-reinforced urethane (u/c) specimens Figure 3.21 Average tensile test results from 0 to 45 for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens Figure 3.22 Average tensile test results from 45 to 90 for the [± θ] 2 fiberglass-reinforced (u/g) urethane specimens Figure 3.23 Average tensile test results from [± 30 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g) Figure 3.24 Average tensile test results from [± 45 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g) Figure 3.25 Average tensile test results from [± 60 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g) Figure 3.26 Average tensile test results from [± 75 ] 2 for silicone/cotton (s/c), silicone/fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g) xiii

15 Figure 3.27 Average tensile test results at [± 90 ] 2 for silicone/cotton (s/c), silicone/fiberglass (s/g), urethane/cotton (u/c), urethane/fiberglass (u/g) combinations and pure rubber...70 Figure 3.28 Average initial longitudinal [± θ] 2 laminate stiffness, E x, for each material system as a function of off-axis angle Figure 3.29 Transverse modulus, E 2, of each material system measured using a [± θ] 2 laminate, as a function of extensional strain...77 Figure 3.30 Shear modulus, G 12, of each material system measured using a [± θ] 2 laminate, as a function of extensional strain...80 Figure 4.1 oordinate systems for laminate and local (layer) axes Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Bi-modular or bi-linear stress-strain material model...94 omparison of several material models with silicone/cotton shear modulus...97 Experimental and modeled nonlinear shear stiffness for each material system with a [+45/-45] 2 layup Experimental and modeled nonlinear transverse stiffness for each material system with a [+90/-90] 2 layup Predicted Poisson s ratios as a function of angle for each material system Modeled and measured cotton/silicone stress-strain behavior from [± 0 ] 2 to [± 45 ] Modeled and measured cotton/silicone stress-strain behavior from [± 45 ] 2 to [± 90 ] Modeled and measured fiberglass/silicone stress-strain behavior from [± 0 ] 2 to [± 45 ] Modeled and measured fiberglass/silicone stress-strain behavior from [± 45 ] 2 to [± 90 ] Modeled and measured urethane/cotton stress-strain behavior from [± 0 ] 2 to [± 45 ] Modeled and measured urethane/cotton stress-strain behavior from [± 45 ] 2 to [± 90 ] Modeled and measured urethane/glass stress-strain behavior from [± 0 ] 2 to [± 45 ] xiv

16 Figure 5.8 Figure 5.9 Modeled and measured urethane/glass stress-strain behavior from [± 45 ] 2 to [± 90 ] Example of an inflated rubber muscle fabricated by Peel Figure 5.10 Modeled contractive muscle force versus pressure for each material system Figure 5.11 Modeled fiber angle change as a function of pressure for each material system Figure 5.12 Modeled contractive force as a function of pressure for different initial fiber angles Figure 5.13 Modeled fiber angle change as a function of pressure for different initial fiber angles Figure D.1 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.2 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.3 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.4 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.5 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.6 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.7 Measured and predicted tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens at Figure D.8 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at Figure D.9 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at Figure D.10 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at Figure D.11 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at xv

17 Figure D.12 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at Figure D.13 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at Figure D.14 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens at Figure D.15 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.16 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.17 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.18 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.19 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.20 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.21 Measured and predicted tensile results for [± θ] 2 cotton-reinforced urethane (u/c) specimens at Figure D.22 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at Figure D.23 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at Figure D.24 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at Figure D.25 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at Figure D.26 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at Figure D.27 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at Figure D.28 Measured and predicted tensile results for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens at xvi

18 HAPTER 1 INTRODUTION AND GENERAL REVIEW 1.1 SYNOPSIS The research presented in this dissertation is intended to provide fabrication methodologies, experimental knowledge, and analytical tools for those who desire to use the unique characteristics of fiber-reinforced elastomers (FRE). ontributions include an improved fabrication methodology, experimental stress-strain results from four elastomer/fiber combinations, and an accurate nonlinear model of fiber-reinforced elastomer composites. A rubber muscle actuator was created and modeled. The rubber muscle model includes the effects of fiber re-orientation. The rubber muscle actuator exhibits high contractive forces when inflated at relatively low pressures. 1.2 MOTIVATION AND BAKGROUND The need for new materials, experimental data and analytical tools is never satisfied. Emerging applications, or the desire for specific characteristics in an application, often create this need. Fiber-reinforced elastomers are emerging materials that show great potential in tailoring specific characteristics such as stiffness, deformation, stress-strain nonlinearity, and Poisson s ratio. Applications that could benefit from these capabilities include adaptive and inflatable structures and bio-mechanical devices. Recently, attention [1] has been given to compliant or flexible structures because of their ability to mimic nature. In nature many structures, such as trees, bird wings, ligaments and turtle shells are 1

19 very strong, yet have some flex. Since fiber-reinforced elastomeric composites have the ability to imitate these structures and their characteristics, they are also well suited for flexible and compliant structures. Many elastomers increase in stiffness when stretched. This stiffening capability varies with the type of elastomer and filler, and can be enhanced by the proper use of fiber reinforcement and structural configuration. For example, flywheels are being considered as energy storage devices in electric vehicles. If the diameter of a stiffness-tailored FRE flywheel were to increase rather than its speed, energy storage could increase without increasing angular velocity, decreasing the complexity of energy recovery. As the diameter of the flywheel reached a pre-determined size, the stiffness of the flywheel structure would increase significantly, limiting the radial expansion. Tailoring the stiffening capability of some elastomers, would satisfy a complaint that inflated structures are generally too flexible, the stiffening-vs.-strain" tailoring could be applied to make these structures less compliant when fully inflated. The ability to tailor or change the stiffness of a fiber-reinforced elastomeric laminate along one axis relative to another is orders of magnitude greater than for typical composites. For example, on aircraft or missiles it would be advantageous to have a flexible control surface that is extremely compliant about one axis, but stiff about the other axes. Aircraft designers are suggesting compliant wings [1] for highly maneuverable aircraft. Using conventional aerospace composites fabrication techniques married with elastomer processing knowledge and with accurate analytical capabilities; rubber muscle actuators, flexible wings, tailored prosthetics, and better rubber fingers can be fabricated. 2

20 1.3 SOPE OF URRENT RESEARH The research presented in this dissertation is intended to be broad and exploratory in nature, yet have enough depth for the interested researcher. Since the analysis of fiberreinforced elastomeric specimens is of limited worth without the ability to fabricate and obtain good quality fiber-reinforced elastomer specimens, considerable effort was placed on the fabrication of high quality and high fiber volume fraction test articles. As the research evolved, it became obvious that each area of research, fabrication, testing, and modeling could take years of work. To properly balance the work, the scope of the current research was restricted and divided into areas that describe the emphasis of the research Fabrication and Testing of Specimens To validate an improved theoretical model, angle-ply specimens needed to be fabricated, however, many processing and fabrication issues arose. To satisfy these issues, a successful fabrication method was developed that combined aerospace composites fabrication techniques with elastomer processing knowledge. The fabrication method not only allows the creation of good quality test specimens, but can also be used to fabricate many types of fiber-reinforced elastomer applications. Pure rubber and fiber-reinforced elastomer specimens were fabricated. otton and fiberglass were used as fiber reinforcements. These reinforcements were combined with urethane and silicone rubber matrices. The effect of fiber volume fraction was observed by the use of the four material combinations, which had varying amounts of fiber reinforcement. The variation of other fabrication parameters, such as laminate thickness, number of plies, specimen size, etc. were minimized except as noted below in the test matrix of Table

21 Four specimens at each angle and material combination were considered adequate to obtain valid results. All fiber-reinforced elastomer test specimens have four plies with a [± θ] 2 lay-up with angles and material combinations shown in Table 1.1. Additional test data was obtained from raw elastomer specimens and from plain and rubber-impregnated cotton fibers. TABLE 1.1 Fiber-reinforced elastomer specimen test matrix Elastomer type otton fiber reinforcement Fiberglass reinforcement urethane rubber 4 angle-ply specimens at angles of 0, 15, 30, 45, 60, 75, and 90 degrees. silicone rubber 4 angle-ply specimens at angles of 0, 15, 30, 45, 60, 75, and 90 degrees. 4 angle-ply specimens at angles of 0, 15, 30, 45, 60, 75, and 90 degrees. 4 angle-ply specimens at angles of 0, 15, 30, 45, 60, 75, and 90 degrees Modeling onsiderations lassical lamination theory was modified to include geometric nonlinearity and an effective nonlinear material model. The improved nonlinear model was implemented in an existing composites analysis computer program. Fiber re-orientation is a function of geometry and boundary conditions, and is implemented with a rubber muscle model A Rubber Muscle Application A rubber muscle actuator was fabricated by embedding fibers in the elastomer matrix. It uses a rubber that has good fiber-to-rubber adhesion and allows high elongation. The actuator was filament-wound on an appropriate mandrel. The rubber muscle demonstrates fiber rotation (re-orientation) is a function of geometry and boundary conditions. The rubber muscle could be used as a contractive-type actuator. The displacements and 4

22 high forces developed by this actuator are quantified using fiber re-orientation and the nonlinear model. 1.4 OVERVIEW OF PREVIOUS AND URRENT WORK The literature review in this chapter has been separated into theoretical and experimental areas. Most of the experimental work, however, has a theoretical or applied basis. Because of an opportunity to view FRE-related work in Japan, some Japanese fiber-reinforced elastomeric applications are also discussed. In addition to the references listed after hapter 6, a general bibliography is included after Appendix D, that contains other references which might be of interest to readers. The references in the bibliography are arranged according to subject area. Some references have notes that discuss the article and its relevance to fiber-reinforced elastomer research. hapters 2, 3, and 4 contain discussions of past and current research specifically relating to the chapter topic Modeling of Fiber-Reinforced Elastomers The modeling of elastomers (rubber) and directional reinforcement as fiber-reinforced elastomers or elastomer composites has traditionally been restricted to belting and tire research. Lee at Penn State [2] and others have spent considerable effort characterizing the laminates in tires. Because of the cords in tires and high concentrations of fillers in the rubber, however, there is little elongation of the reinforced rubber; hence, linear strain-displacement relations and linear material properties are assumed and are adequate in most cases. 5

23 Most basic FRE theoretical work has involved the inclusion of material nonlinearity in classical lamination theory. lark [3] at the University of Michigan used a bi-linear stressstrain model of the elastomer in his application of composite theory to reinforced elastomers but does not include viscoelastic, hyperelastic or large deformation effects. This model does not always predict stiffness properly and again is directed primarily towards cord-rubber applications. Woo [4] at the University of Pittsburgh has conducted extensive characterizations of human and animal ligaments and has developed viscoelastic strain models that describe the response of ligaments very well. Woo s ligament models could be incorporated into a model for artificial ligaments created using fiber-reinforced elastomers. hou at the University of Delaware and Luo [5-7] at the University of Nevada- Reno have conducted the most extensive work on the finite deformation and the nonlinear elastic behavior of flexible composites. Their work deals primarily with wavy fibers in an elastomeric matrix, using a polynomial material model. Although they assume that the fiber-reinforced elastomer exhibits geometrically linear behavior, nonlinearity is introduced through the wavy fibers Experimental Work and Applications Philpot [8] published an interesting article on filament winding with an elastomeric resin. Several different elastomer resins and curing methods are discussed, but with little detail. Epstein [9], Ibarra [10], Krey [11], and Shonaike [12] discuss processing and fabrication methods. All discussed methods (except Philpot s method) used hand fiber placement methods which produced specimens with very small fiber volume fractions. There are a number of novel applications involving fibers and elastomer. In the 1960 s the Soviets used a rubber-impregnated fabric to create an inflatable airlock on the Voskhod 6

24 spacecraft, although the rubber-fabric simply unfolded rather than stretched. The Japanese are also investigating fiber-reinforced elastomers for applications such as actuators, discussed later. Researchers [13,14] have used elastomers as the matrix for composite flywheels. Sharpless and Brown [15] developed curved flexible tubes to hold tents up, and patented the idea. Their tubes follow essentially the same idea as rubber fingers by Vasiliev [16], and by the researchers at Okayama University [17,18] and Okayama Science University [19]. Potential for other new applications are wide-ranging and include inflatable toys, flexible aircraft structures, flexible space structures, numerous bio-mechanical applications and marine-related vehicles. The presented research will help interested researchers to develop these and other exciting applications Fiber-Reinforced Elastomers and Rubber Muscles in Japan Fiber-reinforced elastomeric technology in Japan includes rubber actuators at Tokyo Institute of Technology, Saga University, Kyushu Institute of Technology, and Okayama University. Fiber-reinforced rubber fingers" and related flexible micro-actuators can be found at Toshiba, University of Tokyo, Okayama University and the Okayama Science University. Suzumori is conducting some very well known work with FRE articles at Toshiba. His flexible micro-actuators (FMA) can be considered as rubber fingers" [20-26]. Figure 1.1 illustrates how they are made. They consist of a rubber cylinder that is divided lengthwise into three chambers. Fiber is wound circumferentially around the three chambers, and more rubber is applied. The tip of the finger" can be made to rotate by varying the air pressure in the three chambers. Some of Suzumori s FMA s are used in an endoscope 7

25 and the Toshiba Science Museum has operated some of his FMAs for seven years in one of their robots to pick up and move objects. Figure 1.1 Schematic and examples of the flexible microactuator [22]. At Okayama University Gofuku and Tanaka [17,18, 27] have developed a different type of actuator or grasping finger" than Suzumori. It also consists of a rubber tube wrapped with circumferential fibers, but a fiber is laid axially along one side of the single tube, and more rubber is applied over it. The finger will bend in the direction of the fiber when inflated. Air is used to inflate and provide the necessary internal pressure. They are also looking at electro-rheological fluids to provide the necessary inflation energy. Gofuku and Tanaka also applied a tactile sensor, made of two copper coils separated by a layer of conductive rubber, to the end of the rubber fingers. 8

26 The Okayama Science University has a similar type of rubber grasper or finger called a soft gripper as seen in Figure 1.2. Rather than use a separate axial fiber to cause the gripper to bend when inflated, it uses the same fiber for both axial and circumferential directions. This was done by first wrapping the fiber circumferentially, then pulling the fiber a short direction axially, wrapping circumferentially, tucking the end of the fiber under the beginning of the loop, and continuing axially, much like crocheting. The researchers also developed a crude high elongation strain gage out of elastomer and a conductive paint to use with the soft gripper. The special strain gage allowed the measurement of deflection and forces [19]. Figure 1.2 Soft gripper schematic from Okayama University [19]. Another visible area in Japan where researchers are using FRE techniques, is with a rubber pneumatic actuator called the rubbertuator, formerly made by Bridgestone. This actuator, shown in Figures 1.3 and 1.4, basically consists of an inner rubber tube, surrounded by a outer braided fiber layer. The actuator is capped at the ends with metal fittings which allowed air to enter and leave, and provided attach points. Although the fibers 9

27 are not embedded in the rubber or elastomer, the behavior of the actuator is the same as if the fibers were embedded. The rubbertuator rubber muscles are a form of the McKibben pneumatic rubber actuator [28]. The rubbertuators were fabricated several years ago by Bridgestone [29]. Bridgestone stopped making the rubber actuators due to financial losses on the venture. The rubber actuators or muscles are still used at Tokyo Institute of Technology, Saga University, Kyushu Institute of Technology, and Okayama University [30]. Figure 1.3 Schematic of the Bridgestone rubbertuator [29]. 1.5 SUMMARY Information gained from the review of the various processes and types of FRE applications, and from contact with Japanese researchers was very helpful in developing a good quality FRE processing and fabrication method. The fabrication method is discussed in detail in hapter 2. Experimental stress-strain results from the fabricated specimens are discussed in detail in hapter 3. Because of the volume of test data, only average test results at each angle and material type are shown in hapter 3. A nonlinear model, including geometric and material nonlinearity is presented in hapter 4. Stress-strain predic- 10

28 Figure 1.4 Robotic arm with inflated and un-inflated rubbertuators. tions from the nonlinear model are compared with experimental results in hapter 5, and are discussed. general conclusions and recommendations for future work are presented in hapter 6. The menus of the nonlinear model, as implemented in a computer program PFRE3 are shown in Appendix A. Appendix B gives relevant FORTRAN code from PFRE3, and Appendix shows several input and output data files from PFRE3. Individual specimen stress-strain results, in a less refined form, can be found in Appendix D. 11

29 12

30 HAPTER 2 SMALL BATH FABRIATION OF FIBER-REINFORED ELASTOMERS 2.1 SYNOPSIS Heightened interest in flexible (elastomeric) composite applications such as biomechanical devices, flexible underwater vehicles, and inflatable space structures highlight the need of improved fabrication techniques for fiber-reinforced elastomeric materials (FRE). Previous methods have generally been limited to fiber volume fractions of less than 2%, or used calendering manufacturing methods that are not generally suitable for non-tire fiber-reinforced elastomeric composites applications. Other researchers have noted problems with fiber-elastomer adhesion. The current work demonstrates a method for making small batches of good quality fiber-reinforced elastomer pre-preg. Strengths of the method include good fiber adhesion, fiber volume fractions of 12% to 62%, highly parallel fibers, use of traditional advanced composites fabrication methodologies, and reproducible ply thicknesses. The method combines standard techniques of filament winding, wet lay-up techniques, and autoclave curing with pertinent knowledge of elastomers to produce fiber-reinforced elastomer prepreg. Fiber-elastomer adhesion was enhanced by the proper choice of fiber/elastomer combinations, autoclave pressure, and the application of a primer. Fiber parallelism and straightness were accomplished by use of a filament winder. Fiber-reinforced elastomer prepreg and laminated specimens were fabricated using fiberglass and cotton fibers, respectively. Manufacturing quality was ver- 13

31 ified by increased fiber volume fractions, reproducible prepreg thicknesses, and consistent experimental results from fabricated specimens. 2.2 INTRODUTION Non-tire fiber-reinforced elastomers show promise for use in a broad range of applications, including safer flywheels, flexible underwater vehicles, variable camber wings, rubber muscle actuators, inflatable aerospace structures, flexible robotic skeletons that mimic the human body, and numerous bio-mechanical applications. Advantages that fiber-reinforced elastomers have over conventional stiff materials such as metals and advanced composites include increased damping and the ability to tailor physical characteristics such as elongation, nonlinearity and stiffness over a much broader range. This enhanced tailoring ability is able to provide increased capability to adaptive structures. Fabrication techniques, however, have not been adequate to reach the full potential range of applications and physical characteristics. Typical cord-rubber composites, (e.g., tire and belting) are fabricated by a process called calendering. In this process raw gum rubber is masticated in a huge vat, additives are mixed in, and the resulting thick, viscous slab is flattened and compressed by passing it through a series of rollers. As the slab becomes thinner, fibers are fed in and embedded in the rubbery sheet. The fiber-reinforced sheets can then be cut, stacked (laid up at desired angles) and calendered again to form the reinforced part of a tire. Belting is fabricated in a similar manner, except that the fibers are uni-directional. The flexible composite applications listed above could use two-part liquid elastomers that are cured through chemical reactions and heat. Such elastomer matrices do not lend themselves to the calendering/masticating process. In addition the size and cost of mixing and calendering equip- 14

32 ment make it prohibitive for most firms and universities to consider such processes. The pressure applied to the rubber/fiber combination by the rollers, however, aids in rubberfiber adhesion and should be included in a fabrication method for fiber-reinforced elastomer composites. The cords (fiber groups) in tire composites are twisted to improve fatigue resistance and change transverse or three-dimensional properties. Such twisting, however, reduces the effective strength and stiffness of the composite; hence, the current method does not employ twisted cords. Since fiber reinforced elastomers/flexible composites are extensions of typical advanced composites, the use of standard composites manufacturing methods, where possible, is desirable. 2.3 ONTRIBUTIONS TO THE STATE OF THE ART Suzumori, et al. [1-4], have fabricated several types of flexible micro-actuators at Toshiba, that use fiber-reinforced elastomers. Some examples are shown in Figure 2.1. To produce the actuating tube, fibers were wound circumferentially around a three-chambered rubber cylinder. Silicone rubber and polymer fibers were used, with a primer on the fiber and adjoining non-rubber surfaces to improve rubber adhesion. Similarly fabricated single-chambered rubber fingers were fabricated at Okayama University [5] and at Okayama Science University (more information about the FRE-related work in Japan has been reported in Reference 6). Krey and Shonaike [7,8] created low fiber-volume fraction specimens where the fibers were laid in a zig-zag pattern around nails or rubber pegs in a mold and liquid elastomer was poured over the fibers to form angle-ply specimens. When tested, the fibers tended to tear through the matrix. Kuo, et al. [9,10], fastened fibers to a frame and immersed the fibers in a tray of liquid silicone rubber. They also arranged 15

33 Figure 2.1 Examples of flexible micro-actuators (Reprinted with permission from Toshiba orp). 16

34 fibers in sinusoidal patterns and poured elastomer over them. Fiber-volume fractions were on the order of one to two percent. At low fiber volume fractions, the combination may not act as a continuum, as needed to analyze the FRE material using a modified form of classical laminated plate theory. Philpot, et al. [11], discuss filament winding of fibers impregnated with various types of elastomers. Their primary purpose was to explore the feasibility of using and curing urethane rubber in situ. Krey and Kuo fabricated their specimens such that there were no or few cut fibers along the specimen longitudinal edges. This introduced additional nonlinearity due to fiber rotation relative to the longitudinal axis of the specimen. 2.4 INTENT OF URRENT WORK Relatively little has been published on the fabrication of fiber-reinforced elastomers specimens and applications. The intent of this work is to present, in a concise manner, sufficient information for one to make small batches of high-quality fiber-reinforced elastomer prepreg and use that prepreg to make specimens and applications. The improved method demonstrates excellent fiber adhesion, simplicity, the ability to vary fiber volume fractions, use of typical advanced composite fabrication methodologies, highly parallel fibers, and reproducible ply thicknesses. A filament winder can be used to lay down fibers in a highly parallel manner using circumferential windings. Once a prepreg is made and an inter-ply adhesive is selected, hand lay-up, vacuum bagging, and autoclave curing are the obvious choices to produce high quality specimens. The present work uses these techniques, coupled with understanding of fiber sizings and fiber-rubber adhesion to produce good quality specimens with fibervolume fractions varying from 12 to 62%. Initial limited-success efforts to fabricate FRE 17

35 specimens using a vacuum-assisted resin transfer molding (RTM) process will be mentioned since the information may be useful to other researchers. 2.5 ONSTITUENT MATERIALS AND HARATERISTIS Materials used in this study included cotton and fiberglass reinforcement, urethane rubber, silicone rubber, and a primer for the silicone rubber. The rubber materials were selected for their nonlinear stress-strain characteristics and for their low pre-cured viscosities, which would have aided the vacuum-assisted RTM method. The advantages and disadvantages of each fiber and rubber (elastomer) are discussed briefly Matrices Urethane Rubber: iba RP two-part urethane rubber was chosen for its low pre-cured viscosity, high elongation (330%), and nonlinear-softening (stress-vs.-strain) characteristics. This rubber has a usable pot life of approximately forty minutes, and curing can be accelerated by the addition of heat. Typically this rubber is used in mediumtemperature ( ) mold-making applications. The low viscosity aided mixing and wet-out of fibers. The RP 6410 rubber is a light yellow color when cured. Silicone Rubber: A two-part Dow-orning Silastic S room-temperature vulcanizing (RTV) mold-making rubber was chosen as a contrasting elastomer matrix. This silicone RTV was chosen because of its extremely high elongation (700%), low uncured viscosity, and nonlinear stiffening (stress-vs.-strain) characteristics. The rubber has a usable pot life of approximately one hour and, like the urethane, curing can be accelerated by the addition of heat. This and similar silicone rubbers are typically used in high-temperature (175 ) composite molds. The cured rubber is green in color. 18

36 2.5.2 Reinforcement otton: otton was chosen for its good adhesion characteristics and availability. Researchers in Okayama, Japan indicated success in using cotton fiber as reinforcement for some of their grasping fingers. otton has long been used as a belting reinforcement. The advantages of cotton fibers include widespread availability and good adhesion to the rubber matrix because of the hairs or fibrils on the cotton strands. Some disadvantages include lower strengths and stiffnesses than typical composite fibers and a potential difficulty in reproducing results because of variation in twine properties. For this research large rolls of cotton twine (Wellington construction twine) were obtained, and only fiber from the same roll was used with a particular rubber matrix. Although the cotton fibers are lower in stiffness and strength than fiberglass or graphite, their elastic modulus is still several orders of magnitude higher than either silicone or urethane rubber. Testing showed that cotton fiber strength and stiffness from roll to roll were consistent. Fiberglass: PP&G 1062 fiberglass was chosen because of its widespread use in industry, its high strength and stiffness relative to the cotton fibers. The silane sizing on the fiberglass is intended for typical epoxy resins, and showed good adhesion to the urethane. Very poor adhesion, however, was initially noted to the silicone rubber. The application of an appropriate primer alleviated this problem. Primer: Discussions with representatives from PP&G and Dow orning led to the use of a primer on the fiberglass, which enabled excellent adhesion to the silicone rubber. The original sizing was stripped from the fiberglass by running the fibers through a bath of commercial grade acetone and winding the fibers on a spindle, using a filament winding machine. This process was repeated to ensure a clean fiber. Dow-orning 1200 primer 19

37 was diluted with hexane reagent to approximately 1% (by weight) active ingredient. Fiberglass was pulled through the primer bath, wound on a spindle and allowed to dry. Because of the moisture-sensitive nature of the primer, the treated fibers were used within 24 hours of primer application (if humidity is high, the treated fiber should be used as soon as it is dry). Both the Dow-orning and PP&G representatives emphasized that less primer is better; since too much primer can actually hinder adhesion Rubber-to-Rubber Adhesion Some types of rubber do not adhere well to themselves or other materials such as plastics or metal. To test the selected rubbers, specimens of cured silicone and urethane rubber were placed in separate containers. Liquid urethane and silicone rubber were poured over the respective samples and allowed to cure. The bond lines between the old and new silicones, and old and new urethanes were examined. For both rubbers, the bond lines were virtually imperceptible. Simple pull tests also indicated good rubber-to-rubber adhesion. These observations demonstrated that the respective liquid rubbers could be used as an adhesive between layers of fiber/rubber prepreg. 2.6 VAUUM-ASSISTED RESIN TRANSFER MOLDING PROESS Vacuum-assisted resin transfer molding (VA-RTM) involves aiding resin flow through a mold by pulling a vacuum at an outlet point as the resin is inserted. For this process a three-part mold was machined from plexiglass. The mold employed top and bottom plates which encapsulated a center section with dog-bone shaped openings (see photograph of mold in Figure 2.2). All plates were aligned, clamped together, and sealant applied at possible leaking points. The mold was attached by hoses to a vacuum pump. At the mold inlet, hoses were attached to a container of liquid elastomer. Air bubbles had already been 20

38 Figure 2.2 Initial fiber-reinforced elastomer mold for the vacuum-assisted RTM process. removed from the elastomer, using vacuum. At this point clamps were removed from exit hoses to enable the vacuum to draw the elastomer into the mold. The vacuum enhanced the flow of the elastomer through an inserted fiber preform, and out a tube at the other end of the mold. The VA-RTM process had the potential to make very high quality, reproducible specimens, but was set aside due to challenging complications. The flowing elastomer caused the fiber preforms to move, and bunch up against the mold outlet. Increasing the preform density decreased fiber movement but impeded flow of the highly viscous elastomers. Additionally, it was virtually impossible, using the present configuration, to eliminate all voids and bubbles from the specimens. Since this process is com- 21

39 monly used to make high-quality traditional composite components, these problems are clearly not insurmountable, but the following method is simpler and better suited to small batch fabrication of fiber-reinforced elastomer. 2.7 FILAMENT WINDING AND LAMINATION PROESS A more reliable method of making specimens was developed, which can also be used to fabricate fiber-reinforced elastomer applications. The fabrication process involves winding elastomer-impregnated fibers onto a rectangular mandrel, curing the assembly in an autoclave under high pressure, and laminating the resulting uni-directional prepreg in a manner similar to traditional advanced composites fabrication techniques. Fabrication steps were similar for all elastomers and fibers employed in this study, with the addition of a primer on the fiberglass when used with silicone rubber. To begin, a rectangular aluminum mandrel with dimensions 35.6 by 17.8 by 12.7 cm (17 by 7 by 5 in) was coated with a wax-like release agent (unlike most epoxies, the rubbers for this work do not adhere well to aluminum, but this procedure protects the relatively fragile pre-preg from tearing). Then, in preparation for winding, a thin layer of elastomer is applied to the mandrel. As shown in Figure 2.3, two tows of reinforcing fibers are wrapped circumferentially around the mandrel. The lead or advancement of the fiber placement head is such that consecutive tows were placed side-by-side. The cotton fiber stayed essentially round, but the fiberglass tows spread or flattened considerably, so the lead or movement of the fiber placement head relative to mandrel rotation was increased until no tow overlap was observed. Tension of the cotton tows as they were wound around the mandrel was accomplished by passing the fibers through a series of guide rings. The friction generated about 9 N (2 lbs.) of tensile force on each tow. The 22

40 fiberglass tows laid down better with 22 N (5 lbs.) tensile force. Additional tension was obtained by increasing the pressure of a plate on the end of the creel on which the fiberglass was stored. Figure 2.3 Filament winding of fiber-reinforced elastomer onto a rectangular mandrel. The elastomer resins were too viscous to use in a regular filament-winding bath, so the resin was applied to the fibers using a plastic scraper, until all were covered and all crevices were filled. A sheet of teflon-coated porous peel-ply was tightly wrapped around the mandrel and fibers. The teflon-coated cloth does not adhere to the rubber and aids in separation of subsequent layers. Fibers were again wound circumferentially around the mandrel to form a second layer. This process was repeated until four or five layers of fiberreinforced elastomeric prepreg were completed with each layer only one tow thick. A 23

41 final sheet of non-porous peel-ply was applied prior to removal of the mandrel from the filament winder. Bleeder cloth was wrapped around the mandrel and four flat caul plates, matching the dimensions of the mandrel, were placed on the sides. The assembly was vacuum bagged, and a full vacuum was drawn to remove air bubbles, and to provide pressure that would hold the caul plates in place. A representative assembly is shown in Figure 2.4. VAUUM BAGGING BLEEDER LOTH PEEL-PLY RIGID AUL PLATES FRE FILAMENT WINDINGS RETANGULAR MANDREL (MULTIPLE LAYERS OF FIBER-REINFORED ELASTOMER SEPARATED BY PEEL-PLY) Figure 2.4 Schematic of a vacuum-bagged mandrel assembly. The filament-wound material was cured in an autoclave at 276 MPa (40 psi) and 71 (160 F). Pressure and temperature were allowed to ramp up from ambient during the first 24

42 fifteen minutes, held constant for thirty minutes and then returned to ambient levels. Although the silicone and urethane rubbers cure at slightly different rates, the combination of pressure and temperature during the cure cycle was sufficient to cure both elastomers to a point that the uni-directional prepreg could be cut off the mandrel. The added pressure of the autoclave was very beneficial in forcing out trapped air and increasing adhesion between fibers and elastomer. Additionally, the caul plate, under outside pressure, flattens the laminae and ensures more uniform layer thickness by forcing excess resin to the corners of the mandrel. Examples of resulting unidirectional prepreg are shown in Figure 2.5. Figure 2.5 Samples of cotton- and fiberglass-reinforced elastomer prepreg. To form a fiber-reinforced elastomer laminate the uni-directional sheets were laid up in the orientation desired, with additional liquid elastomer used as the adhesive between 25

43 layers. The laminates were vacuum-bagged and cured in the autoclave using the same cure cycle as used for the prepreg. In this study all specimens had an angle-ply or (+θ/-θ) 2 lay-up, where θ is the ply orientation angle and each laminate consisted of four layers. A review of lamina and total laminate thicknesses is illustrative of the importance of processing parameters, such as autoclave pressure. In Table 2.1 prepreg, nominal (four times the prepreg thickness) and measured laminate thickness for each type of specimens, with standard deviations, are presented. Autoclave cure pressure for each material system is also presented, as is a percent difference between nominal and measured laminate thicknesses. TABLE 2.1 Prepreg and laminate thicknesses (± one standard deviation) Material Autoclave ure Pressure [MPa (psi)] Measured Prepreg Thickness a [mm. (in)] Nominal Laminate Thickness b [mm. (in)] Measured Laminate Thickness c [mm. (in)] Thickness Difference (%) otton / Urethane 345 (50) 1.72 ± ( ± ) 6.88 (0.271) 6.02 ± (0.237 ± ) otton / Silicone 276 (40) 1.63 ± ( ± ) 6.53 (0.257) 7.24 ± (0.285 ± Fiberglass / Urethane 276 (40) ± ( ± ) 3.43 (0.135) 3.99 ± (0.157 ± ) Fiberglass / Silicone 276 (40) ± ( ± ) 3.15 (0.124) 4.01 ± (0.158 ± ) a Average of five thickness measurements of each prepreg, except the cotton/urethane prepreg, which consists of an average of three thickness measurements. b Based on four times measured prepreg thickness. c Twenty-eight thickness measurements of each laminated FRE material system were taken. The actual laminate thickness for the cotton/urethane rubber system, 6.02 mm (0.237 in), is less than the cotton/silicone rubber laminate thickness, and is also less than 6.88 mm 26

44 (0.237 in), the nominal laminate thickness. This difference was partly due to an increase of the autoclave pressure, 345 MPa (50 psi), during the cure cycle of the cotton/urethane laminate. The twenty five percent higher pressure relative to the other laminates, squeezed out excess resin and compressed the somewhat cured (but still soft) prepreg. As the laminate cured, it remained in the thinner state. For the other laminates, differences between nominal laminate thickness (four times the prepreg thickness) and the measured laminate thickness were due to liquid rubber being used as an adhesive. The slightly greater prepreg thickness of the urethane rubber prepregs relative to the silicone rubber prepreg may be due to the higher initial modulus of the urethane rubber matrix. Other factors such as the cured stage of the elastomer when it is vacuum bagged and autoclaved, and vacuum pump pressure, could also affect the final thickness of the laminate. Standard deviations in thickness of the laminates for each material system were typically less than 10%. Such thickness variations are comparable to standard composites cured using vacuum bagging. 2.8 SPEIMEN PREPARATION Dog-bone shaped specimens were cut from the cured FRE laminates. Experimentation showed that cutting of the elastomer composites was easier with a sharp utility knife than with a machine, such as a band saw. The soft elastomer tends to deform, producing a jagged edge when cut with a band saw. utting the specimens with a knife, however, entails applying considerable pressure to the laminate. The pressure causes the oriented layers to deform in different directions and after a specimen is cut and released, the layers contract differentially to form a non-uniform edge. Specimen edges with this problem are shown in Figure 2.6. This problem was solved by using a water-jet process to cut all spec- 27

45 imens into a dog-bone shape. The resulting specimens have a very smooth edge in the dog-boned or test region. Figure 2.6 Specimens with jagged edges after being cut with a utility knife. After water-jet cutting, all specimens were post-cured in an oven at 60 (140 F) for six hours and allowed to cool to room temperature. Representative samples of the test results, including urethane/fiberglass at 45 and silicone/cotton at 60, are shown in Figure 2.7. The complete test results are quite extensive and are presented in hapter DISUSSION OF THE FABRIATION PROESS Raw test results from laminated specimens fabricated with the same materials and with the same off-axis angles were consistent, indicating good quality fabrication. The cotton-reinforced silicone and urethane elastomer specimens were fabricated with fiber volume fractions of 52% and 62%, respectively. The fiberglass-reinforced silicone and urethane specimens were fabricated with 12% and 18% fiber volume fractions, respectively. Fiber volume fractions of the cotton-reinforced specimens were obtained by count- 28

46 Urethane/Glass 45 Silicone/otton Stress (kpa) Stress (psi) Strain (mm/mm) Figure 2.7 Raw test results for urethane/fiberglass at 45 and silicone/cotton at 60. ing the number of cotton fiber ends shown, and comparing cotton cross-section area with total cross-section area. Fiber volume fractions of the fiberglass-reinforced specimens were found using the immersion method. are was taken to avoid air bubbles, which could change volumetric measurements. The lower fiberglass/elastomer fiber volume fractions were due to a decision to not overlap the fiberglass tows, rather than limitations in the fabrication process. Higher fiberglass fiber volume fractions can be obtained by increasing tow tension, which 29

47 decreases tow spreading, and allowing overlap of adjacent tows by decreasing the filament winder head advancement relative to mandrel rotation. Because some angle-ply specimens failed by scissoring (shear) along lamina bondlines, prepreg layers should be roughened before lamination. Bond-line strength could be further increased by: 1) reducing the autoclave cure time of the filament wound prepreg for the urethane composites; and, 2) increasing cure cycle times for the silicone prepreg. Inadequate mixing or incomplete curing of the silicone rubber may prevent total polymerization of the rubber constituents. The oils or constituents left can actually hinder rubber adhesion; hence, new-to-old silicone rubber adhesion is best when the old rubber is fully cured. New-to-old urethane rubber adhesion, on the other hand, is best when the old rubber is not fully cured. A final lesson learned is that the water-jet process is the preferred procedure for cutting the whole dog-bone specimen from a laminate. Although thickness variations of the prepregs and laminates were acceptable, and comparable to typical advanced composite laminates, further improvement is still possible. Thickness variations could be further reduced by using thicker caul plates, using higher autoclave pressures, and metering the elastomer resin onto the mandrel FABRIATION AND PROESSING SUMMARY A non-calendering method for fabricating good quality, medium to high fiber volume fraction, fiber-reinforced elastomer (FRE) specimens has been demonstrated. Fiber-reinforced elastomer specimens with fiber volume fractions of 12% to 62% have been fabricated. The manufacturing quality has been verified by prepreg uniformity and tensile tests on fabricated specimens with representative test results shown. The fabrication method uses a combination of filament winding, standard lamination techniques, autoclave curing, 30

48 and a knowledge of elastomer cure parameters to produce good quality parts. The challenge of fiber-to-elastomer adhesion was overcome by a careful choice of fibers and resins, selection of autoclave cure cycle parameters, and application of a primer on the fiberglass to aid adhesion to the silicone rubber. Fiber parallelism and straightness were accomplished by using circumferential windings on a filament winder. The present approach allows any researcher with a working knowledge of advanced composites fabrication skills and common composites fabrication equipment to fabricate FRE specimens and applications. Processing parameters such as autoclave pressure, vacuum pressure, cure stage of the elastomer matrix, and elastomer stiffness also affect adhesion, prepreg thickness, and laminate thickness. Fabricated prepreg and laminate thicknesses are consistent, with variations similar to those observed in typical advanced composite material manufacturing processes. Fiber volume fractions can be adjusted by changing filament winder parameters. Nonlinear material properties from the tests are being used to validate a modified nonlinear laminated plate model discussed in hapter 4. The complete test results and comparison with the enhanced theory are being reported in hapter 3 and in papers [12-13]. 31

49 32

50 HAPTER 3 THE TENSILE RESPONSE OF FIBER-REINFORED ELASTOMERS 3.1 SYNOPSIS The mechanical behavior and basic response mechanisms of fiber-reinforced elastomers (flexible composites) can be significantly different from those of typical advanced stiff composites. This chapter presents experimental results of elastomer (rubber) matrices, dry and impregnated fibers, and four sets of fiber-reinforced elastomeric composite, summarizes the corresponding initial and nonlinear orthotropic constitutive properties, and sheds light on fundamental response mechanisms. Silicone and urethane rubber were combined with fiberglass and cotton reinforcing fibers. Balanced angle-ply laminates of each material system were fabricated with off-axis angles ranging from 0 to 90 in 15 increments. Dog-boned test specimens, 76 mm (3 in) long, were fabricated with fiber volume fractions ranging from 12% to 62% using a previously documented non-calendering fabrication method [1, 2]. The average extensional stiffness of individual twisted cotton fibers increased 74% to 128% when impregnated with an elastomer. Fiber-reinforced elastomer laminate stiffness and nonlinearity can vary significantly with fiber angle. The nonlinear stiffening or softening trends of the silicone and urethane rubbers are reflected in their respective fiber-reinforced elastomers. Longitudinal stiffness at low off-axis angles is a function of the reinforcement stiffness. At high off-axis angles, longitudinal stiffness and strength are functions of fiber type, fiber volume fraction and elastomer. 33

51 3.2 INTRODUTION Recent developments of non-tire fiber-reinforced elastomeric (FRE) composites can enable a broad range of exciting applications, such as safer flywheels, underwater flexible vehicles, flexible aerodynamic surfaces on aircraft, rubber muscle actuators, flexible robotic skeletons that mimic the human body, and numerous bio-mechanical applications. The primary advantage of fiber-reinforced elastomers is the ability to tailor physical characteristics such as stiffness, deformation and nonlinearity over a much broader range than is possible with traditional stiff composites, plastics, and metals. These characteristics are determined by elastomer selection, fiber selection, fiber orientation, and fibervolume fraction. Material properties for each elastomer/fiber combination, however, are generally nonlinear and must be measured experimentally. Due to large differences in stiffness between matrix and fiber, deformation of a FRE laminate can change dramatically with fiber orientation. Typical stiff composites have fiber reinforcements with an axial Young s modulus on the order of 70 to 400 GPa (10 to 60 million psi). Matrix moduli are on the order of 3.5 GPa (0.5 million psi). The difference in matrix and fiber stiffness is approximately one to two orders of magnitude. The current elastomers have elastic moduli as low as 0.5 MPa (70 psi), or about five orders of magnitude lower than fiberglass. Such large ranges in stiffness suggest the possibility of behavior not seen with conventional stiff composites. Poisson s ratios, which affect transverse deflection, can also vary considerably with fiber angle. Because of these unusual behaviors, successful use of FRE materials depends on accurate nonlinear material properties, and the exploitation of response mechanisms of such laminated materials. Tailored flexible composites provide a new category of materials that are especially suited for 34

52 smart structures. Knowledge of the response characteristics of FRE composites will aid in the development of new or improved smart structures. 3.3 ONTRIBUTIONS TO THE STATE OF THE ART Published test results for fiber-reinforced elastomers (flexible composites) are fairly limited, except for tires and belting. Several challenges contribute to the shortage of valid test data. These include fully engaging the fibers, measuring large strains and gripping the specimens without crushing while preventing slippage. Krey and Friedrich [3] fabricated specimens with the fibers placed in a zig-zag pattern around nails or pegs of cured rubber. They varied the number of fiber bundles (fiber volume fraction) and rubber types. Fiber volume fractions were on the order of one or two percent. As the specimens stretched, the fibers straightened (reoriented). Their test results showed a severe nonlinear stiffening effect that was mostly due to fiber re-orientation rather than material or geometric (strain-displacement) nonlinearity. Increasing the fiber volume fraction of the specimens produced stiffer stress-strain curves, with maximum strains between twenty and thirty percent. Due to the specimens extremely low fiber volume fractions, fibers tended to tear through the elastomer matrix. Kuo et al. [4], Luo, and hou [5] give perhaps the most complete test data on FRE materials. They fabricated both straight fiber and wavy-fiber specimens. Again, Kuo s specimens had very low (one to two percent) fiber volume fractions. Test data from the straight fiber specimens is limited. Their wavy-fiber specimens were laminated at various off-axis angles. Because of straightening of the fibers, similar to Krey, extreme stiffening of the stress-strain curve was also noted. Fiber ends were not embedded in the elas- 35

53 tomers; tensile tests were conducted by clamping and transferring load directly to the fibers. Average extension of the specimens was approximately twenty percent. Additionally, lark [6] presents experimental results from several cord-rubber material systems. Because of the stiffer rubbers used, strains were less than ten percent. ompression-tension tests of uniaxial specimens produced stress-strain curves that were essentially bi-linear in nature. ontributions of the current work include tensile results of constituent elastomers and fibers along with the examination and comparison of the tensile response of four distinct fiber-reinforced elastomer composites. The experimental data includes initial and nonlinear orthotropic material properties that are used to explore the effects of contrasting elastomers and fibers on the response mechanisms of FRE composites. The FRE composites contain moderate to high fiber volume fractions and show specimen extensions greater than two hundred percent. More information on the fabrication of specimens can be found in hapter 2 and References 1, 2. The experimental results and resulting nonlinear material properties are being used to validate a modified nonlinear lamination theory discussed in hapter 4. The flexible composite materials are intended to be similar to those that might be used in emerging FRE applications. Stress-strain nonlinearity due to fiber-reorientation was separated as much as possible from geometric (large strain-displacement relations) and material nonlinearity. Materials and test specimen configuration were chosen such that fiber-reorientation was not significant. 36

54 3.4 ONSTITUENT MATERIALS AND HARATERISTIS Materials used in this study include cotton and fiberglass reinforcement, urethane rubber, silicone rubber, and a primer for the silicone rubber. iba Geigy RP two-part urethane rubber was chosen because of its low pre-cured viscosity, high elongation (330%), and nonlinear-softening (stress vs. strain) characteristics. A two-part Dow-orning Silastic S (RTV) mold-making rubber was chosen because of its extremely high elongation (700%), low uncured viscosity, and nonlinear stiffening (stress vs. strain) characteristics. Tests demonstrated the respective liquid rubbers could be used as adhesives between layers of fiber/rubber prepreg. otton was chosen because of its adhesion characteristics and ease of use. Although the cotton fibers are lower in stiffness and strength than fiberglass or graphite, the cotton strength and stiffness are still several orders of magnitude higher than those of the rubber matrices. PP&G 1062 fiberglass, used widely in industry, was chosen for its high strength and stiffness relative to the cotton fibers. The silane sizing on the fiberglass is intended for typical epoxy resins and showed good adhesion with the urethane. Use of Dow orning 1200 TM primer on the fiberglass enabled excellent adhesion with the silicone rubber. Material properties for each of the elastomers and fibers are given in Table 3.1. Fiberglass properties were obtained from Reference 7. The rubber Poisson s ratios are based on the incompressibility condition, and the cotton Poisson s ratio was obtained from Reference 8. All other properties were obtained through testing for the current work. The elastomer properties are based on test results at initial strains since the elastomer properties will vary significantly with strain. Two sets of properties are given for the cotton twine (fiber). The impregnated axial stiffness of the cotton fiber is higher than non-impregnated 37

55 (dry) cotton fiber, explanations for the difference are discussed later. The shear moduli were calculated using Young s moduli and Poisson s ratios with the isotropic relation: G = E/2(1+ν) (3.1) TABLE 3.1 onstituent material properties Material Young s Modulus [MPa (psi)] Poisson s Ratio Shear Modulus [MPa (psi)] Failure Strain (approx.) Silicone rubber (133) (44.3) ~ 700% Urethane rubber 1.65 (239) (79.6) ~ 330% Fiberglass a (10.5e+6) (4.3e+06) ~ 2% otton (dry) 337 (48,900) 0.33 b 127 (18,400) ~ 9% otton (impregnated) 526 (76,300) 0.33 b 198 (28,700) ~ 9% a - properties obtained from Reference [7], b - Poisson s ratio obtained from Reference [8] 3.5 SPEIMEN HARATERISTIS Pure elastomer specimens with the geometry shown in Figure 3.1, were poured and cured using a special mold. The specimen tabs gradually taper into a narrow test section area. The tapering prevents premature failure of the elastomer specimens while insuring an reasonable test region. The actual straight test section is approximately 4.45 cm (1.75 in) long, similar to the FRE specimen test section length. Five specimens of each elastomer type were fabricated Test Specimen Dimensions The FRE test specimens were 7.6 by 2.5 cm (3 by 1 inch) rectangles cut out of the appropriate FRE laminates. All specimens were cut into a dog-bone shape with a highpressure water-jet. The dimensions of the fiber-reinforced elastomer specimens are shown in Figure

56 11.4 cm (4.5 in) 2.54 cm (1.0 in) 2.22 cm (0.875 in) cm (0.25 in) Figure 3.1 Elastomer test specimen configuration cm (3.0 in) 1.26 cm (0.5 in) 2.54 cm (1.0 in) 1.91 cm (0.75 in) Figure 3.2 Fiber-reinforced elastomer test specimen configuration. Specimens fabricated out of the four elastomer/fiber combinations were laminated with angle-ply, (+θ/-θ) 2, lay-ups from 0 to 90 in 15 increments. Four specimens at each off-axis angle were fabricated for each material system. Average thicknesses with standard deviations and specimen count for each laminated material system are presented in Table

57 TABLE 3.2 Thickness (± one standard deviation) for each material system. Material System Measured Laminate Thickness [mm (in)] Number of Test Specimens otton/urethane 6.02 ± (0.237 ± ) 28 otton/silicone 7.24 ± (0.285 ± Fiberglass/Urethane 3.99 ± (0.157 ± ) 28 Fiberglass/Silicone 4.01 ± (0.158 ± ) Fiber Volume Fractions Fiber volume fractions of the cotton-reinforced elastomer specimens and prepreg were obtained by measuring average fiber diameter, counting the number of cotton fiber ends shown, and comparing cotton cross-section area with total FRE cross-section area. Fiber volume fractions of the fiberglass-reinforced elastomer prepreg and laminated specimens were obtained using the immersion method. This method involved weighing the specimens and measuring volume by immersing the specimens in a graduated cylinder of water. are was taken to avoid air bubbles which would distort volumetric measurements. Using known fiber and matrix densities, fiber volume fractions were calculated. Prepreg and laminate fiber volume fractions (V f ) for each material combination are given in Table 3.3. The differences between prepreg and laminate fractions are due to the extra rubber resin used as adhesive during lamination. Notice that the cotton/urethane laminate had the same fiber volume fraction as its prepreg. This material combination was cured at twenty percent higher pressure. The higher pressure forced excess rubber out from the layers, hence prepreg and laminate fiber volume fractions are the same. 40

58 TABLE 3.3 Laminate and prepreg fiber volume fractions for each material system. Material System Prepreg V f (%) Laminate V f (%) otton/urethane otton/silicone Fiberglass/Urethane Fiberglass/Silicone EXPERIMENTAL PROEDURES Initial tests with elastomer and FRE specimens showed that grips used with standard advanced composites test equipment did not work well with highly extensible (soft) specimens. When such specimens were gripped in the fixtures and pulled, the rubber contracted in the transverse direction, as it elongated in the axial direction, and pulled out of the grips. If the grips were tightened to prevent slippage of the specimens, crushing or permanent deformation of the specimens occurred, causing premature failure. To minimize this problem, a fixture, shown in Figure 3.3, was fabricated that enclosed the tabbed ends of the specimen, and had a series of pointed screws that penetrated the specimen ends to provide additional gripping force. Rods on the ends of the fixture attach easily to universal testing machines. The fixture worked quite well, except with uniaxial fiberglass specimens. With these axially stiff specimens, the extremely compliant matrix could not transmit the total force from the test fixture, by shear, to the fibers. The specimen deformed and eventually failed by matrix tearing. The solution to this problem for specimens with axial fibers is to positively engage the fibers by clamping, or embed the fibers at the specimen tabbed ends into epoxy or another stiff matrix. 41

59 Figure 3.3 Special test fixture with gripping screws Test Equipment An Instron testing machine with a 4400 N (1000 lb) load cell was used to test all specimens. Axial force, head displacement, and extensometer strain were routed through Labview into an Excel spreadsheet. An extensometer with a range of 1.27 cm (0.5 in) was used to directly measure strain for stiffer specimens (that experienced little deformation), and to calibrate strain obtained from the machine head displacement for soft specimens (that experienced high deformations). Strain calibration is discussed in more detail later. The extensometer was attached by clips to the test section (narrow part) of the specimen. The specimen was elongated to failure if total deflection was less than 1.27 cm (0.5 in); otherwise, the specimen was extended to approximately 25% of its maximum deflection (or 30% of max load) and returned to a zero displacement condition. Then the extensom- 42

60 eter was removed and the specimen was loaded to failure. To measure failure loads and deflections, and to determine if the extensometer clips introduced premature failure, two of the four specimens at each angle and material system were tested without the extensometer. No difference was noted between extensometer and non-extensometer specimen stress-strain characteristics or failure modes. Because elastomers can show viscoelastic responses, all tensile tests were conducted at 2.54 cm/min (1 in/min), unless otherwise noted. This rate is low enough to be considered quasi-static Definitions of Stress and Strain The large deformations and significant reductions in cross-sectional area require selection of the appropriate definitions for stress and strain. The Lagrangian description considers properties relative to initial positions, while the Eulerian description considers properties relative to the current position. Since rubber is highly deformable, each description has its advantages. Standard rubber models [9], such as the Mooney-Rivlin, Ogden, and Peng use the auchy (engineering) stress, σ i, which is defined as: σ i = F i A o (3.2) where F i is the applied force in the i direction and A o is the original cross-sectional area. The previously mentioned rubber models use extension ratio (stretch) instead of strain. Extension ratio, a i can be defined as: a i = 1 + ε i (3.3) and ε i L = L o i (3.4) 43

61 where ε i is engineering extensional strain in the i direction, L is the change in length, and L o is the original gage length of the specimen. To maintain consistency with classical lamination theory, and to be able to use one of the above rubber material models, results are presented using the Lagrangian description (engineering stress and strain). Extensometer clipped to specimen A g A tab L g L tot K tabs K g F L tot Figure 3.4 Specimen and equivalent spring-stiffness diagram Strain alibration With silicone strains up to 700%, and gage lengths from 3.6 cm to 4.6 cm (1.4 in to 1.8 in), specimen deformation was frequently greater than 1.3 cm (0.5 in). To obtain accurate strains over the whole range of deflections, a method was devised to calibrate the strain obtained from the testing machine head movement using the extensometer strain. A sim- 44

62 ple illustration, shown in Figure 3.4, will aid in the explanation of the two strains. Total axial deflection of the specimen is defined by: L tot = L tabs + L g (3.5) where L tabs is the deflection of the tabbed regions and L g is the deflection of the gage length. The machine strain, which is the total strain, is defined by: ε m = L tot L tot (3.6) The extensometer strain is defined by: ε g = L g L g (3.7) These strains may not be equal to each other. Assuming that the tabbed ends are from the same material as the rest of the specimen (same Young s modulus), a relation between the two strains can be derived based on the specimen length, gage length, cross-sectional area of the ends, A tabs, and the gage length cross-section area, A g. The specimen can be modeled as two springs in series; one for the end or tabs and one for the test section. In the linear range, the total load F applied to the specimen is equal to the respective spring stiffness coefficients multiplied by the corresponding changes in length, i.e., F = K tabs L tabs = K g L g = K eq L tot (3.8) For two springs in series, the equivalent stiffness, K eq, can be defined as: K eq = K tabs K g + K tabs K g (3.9) Using the relation for change in length of a rod under an axial load: 45

63 L = FL or equivalently L = F K (3.10) AE where L is the length of the rod, A is its cross-sectional area, E is the Young s modulus, and K is the spring stiffness, we can define the tab and gage spring stiffnesses as: K tabs A tabs E A = and K g E (3.11) L g = tabs L g Remembering from equation 3.8 that: K eq L tot = K g L g (3.12) we can substitute the relations of equation 3.9 and 3.11 into equation 3.12 and find an equation that relates the change in length of the gage section to total change in length of the specimen: L g = A tabs L g ( L tot L g )A g + L g A tabs Ltot (3.13) By using the definitions for machine and extensometer strain, and rearranging this equation, we will get the extensometer (gage length) strain as a function of the total (machine strain): ε g = A tab L tot ( L tot L g )A g + L g A tab εm (3.14) Notice that the force applied and the stiffness of the material has dropped out, with terms relating only to geometry left. In equation 3.12, the equivalent stiffness, K eq, can be obtained for any geometry, and a relation similar to equation 3.14 could be determined for any geometry, hence equation 3.14 can be generalized and put in the following form: ε g = g ε m + ε o (3.15) 46

64 The coefficient g is a geometric correction coefficient relating the machine strain to the extensometer strain and ε o is a constant that is included to compensate for the lag or offset between the two strain-measuring instruments Extensometer Strain (m/m) Urethane Rubber Experiment Linear Fit (y=1.2208x-0.005) Machine Strain (m/m) Figure 3.5 Plot of extensometer versus machine strain for a typical urethane rubber specimen, with linear fit. If extensometer strain is plotted against machine strain, with extensometer strain on the vertical axis, and a best-fit linear analysis applied, the resulting linear equation will give both the geometric correction coefficient, and the lag or offset between the two strains. Figure 3.5 shows such a plot, obtained from a representative urethane rubber specimen. Note that the plotted data is very straight. This is typical of all of the elastomer 47

65 and FRE tests conducted. The exceptions are at initial strains, and strains near failure (where permanent changes are occurring). These exceptions are to be expected. At very small strains the accuracy of data gathering equipment is lower due to static electricity and machine resolution limitations. At strains close to failure, or when permanent deformation has occurred, there is no guarantee that local strains are proportional to global strains because local hardening or tearing of the specimen may occur. The preceding strain correlation method was used to correct machine strains from all test results, including the fiber-reinforced elastomeric composite test results. orrelation for all test data, including the fiber-reinforced elastomer specimens, was excellent except for some pure silicone rubber specimens. orrections to machine strains of the silicone specimens worked very well for initial pulls or extensions of the silicone rubber specimens. Specimens were typically not extended to failure during initial tests. Because of the polymeric chain (bond) breakage, test results from subsequent pulls to failure correlated better if the machine strains were used directly. The urethane and FRE specimens did not show this abnormality. 3.7 EXPERIMENTAL STRESS-STRAIN BEHAVIOR Tensile tests for three categories of materials were conducted: 1) pure elastomers, 2) dry and impregnated fibers, and 3) fiber-reinforced elastomers (flexible composites). The primary intent of this work was to obtain high quality test data for fiber-reinforced elastomer specimens. The elastomer and fiber data also aided understanding of FRE response mechanisms, and in some cases revealed new insights. 48

66 3.7.1 Elastomer Behavior Rubbers (elastomers) are a special subset of polymers. Polymers get their name from repeated groups of mers or molecules. For most polymers or plastics these chains of mers lay together in a fairly regular pattern (semi-crystalline), or are totally random and strongly cross-linked (amorphous). Elastomers get their name because the elastic chains are curled or coiled up, and allow considerable untangling or stretching before the chain links break. As the tangled chains stretch, they straighten, becoming more regular and usually stiffer. The silicone rubber in the current work acts in such a way, in that it stiffens as it is stretched. The urethane rubber used in the current work appears to include additional mechanisms which prevent it from stiffening at higher elongations. For most elastomers at higher elongations, chains can reposition or some chain links will break their bonds and reform with other chains. The elastomers will still deform elastically after this, but their stress-strain curves will be altered. The silicone rubber tests, as shown in Figure 3.6, demonstrate that something is causing the stress-strain behavior of the specimens to change when subjected to repeated higher and higher loads. Such behavior is not readily apparent with the urethane rubber. For the current work, five silicone rubber specimens were extended (pulled) to the maximum distance allowed by the extensometer, 13 mm (0.5 in), and returned to zero strain. The extensometer was taken off and the specimens were extended further. To learn more about the silicone rubber stress-strain response, some specimens were pulled (cycled) multiple times before extension to failure. Figure 3.6 shows the classic stressstrain response of multiple pulls [10, 11] of a single silicone rubber specimen. Figure 3.7 shows stress-strain results from silicone rubber specimens tested at vastly different strain 49

67 Specimen, Pull #, Rate 3500 sr2 (1st pull, 1in/min) sr2 (2nd pull, 1in/min) sr2 (3rd pull, 1.5in/min) Stress (kpa) sr2 (4th pull, 3in/min) sr2 (5th pull, 9in/min) No bond damage Stress (psi) Residual loading effects Strain (m/m) Figure 3.6 Typical stress-strain response of silicone under repeated loadings mm/min (9 in/min) Stress (kpa) mm/min (1 in/min) Stress (psi) mm/min (25 in/min) Strain (m/m) Figure 3.7 Stress-strain curves at widely varying strain rates. Response is essentially independent of strain rate. 50

68 rates. All initial tests were conducted at a rate of 2.5 cm/min (1.0 in/min). This was considered slow enough to be quasi-static. oncerns about the viscoelastic nature of the silicone rubber led to further pulls at higher strain rates of the same specimens. As shown in the figure, there was no appreciable difference between results for specimens tested at 25 mm/min (1.0 in/min) and results for specimens tested at 635 mm/min (25 in/min). The test results appear to be bounded by an upper curve for a specimen that has been lightly loaded, and a lower curve for a specimen that has repeatedly been heavily extended and has some residual bond breakage. The area between the two curves can not be considered a sort of hysteresis loss because the lower curve is due to residual elastomer chain damage. When fibers are embedded in a rubber, and the rubber is loaded, most repeated deformations are less than 150% while pure rubber deformation is approximately 700%. Because the FRE deformations are relatively small, or less than 150%, an average of the upper curves was taken to obtain nonlinear material properties for the Silastic S rubber. The upper bounds and averaged results are shown in Figure 3.8 for the five silicone rubber specimens. An initial silicone rubber Young s modulus, shown in Table 3.1, was calculated from the averaged results. A series of five specimens fabricated from urethane rubber were also tested in tension. Each specimen was extended partially, released, and then extended to failure. Some specimens were extended multiple times before failure. Two of the specimens were made using a previous batch of urethane rubber. The two stress-strain curves from these specimens were grouped together, but were distinct from the three specimens fabricated using the same batch of urethane as the urethane FRE specimens. Only results from the latter 51

69 Stress (kpa) Silicone Rubber Experiment Average Stress (psi) Strain (m/m) 0 Figure 3.8 Tensile stress-strain results for undamaged silicone rubber specimens. three specimens are shown and discussed. All results from all pulls of the three specimens and averaged results are shown in Figure 3.9. Several differences between the urethane and the silicone rubber tests are readily apparent: 1) there is no apparent chain breakage between first and subsequent pulls; 2) the urethane rubber starts out stiff, and softens as it is strained while the silicone rubber softens, then stiffens; and 3) the urethane rubber was susceptible to surface flaws. As the specimens were loaded, very small flaws on the surface of the specimens became visible, and eventually caused premature failure. This is evident in Figure 3.9, where failure 52

70 Final Pulls Stress (kpa) Urethane Rubber Experiment Average Stress (psi) 200 Initial Pulls Strain (m/m) Figure 3.9 Individual and average tensile results from all urethane rubber specimens. strains range from 125% to 175%. Susceptibility to surface flaws isn t as great a concern with urethane composites because the interspersed fibers tend to prevent flaw growth during elongation. Young s modulus as a function of strain for the silicone and urethane rubbers is presented in Figure The Young s modulus for silicone rubber is initially high, decreases and increases again as strain increases. The softening characteristic of urethane, however, is more dominant, as the urethane stiffness drops off and then becomes quite constant at about 100% strain. 53

71 Urethane Stiffness Silicone Stiffness 250 Elastic Modulus (kpa) Elastic Modulus (psi) Strain (m/m) 0 Figure 3.10 Elastic moduli of pure silicone and urethane rubber as a function of tensile strain. Data from the urethane supplier suggest that elongations as high as 330% are possible [12], while the current tests show no more than approximately 200% maximum strain. It is highly likely that by improving the specimen surface finish, using special elastomeric test fixtures, and with additional experience, strains of 330% can be obtained. For the current work, however, the urethane rubber test results demonstrate expected trends and yield the necessary properties Reinforcement Behavior The PPG fiberglass used in the current study has been well characterized with accurate linear material properties that are readily available. The cotton fiber used in the current 54

72 study was obtained from a retail source and needed to be tested for material properties and variability. The cotton fiber (twine), is very large in diameter, approximately 1.5 mm (0.060 in), relative to a single fiberglass strand (tow). The large diameter of the fibers, and their relative softness enabled firm clamping by the grips of the Instron testing machine. No slippage was noted, and failure of the cotton did not usually occur in the grip area. Several groups of fibers from all sets of cotton rolls were tested. Two and four fibers were tested at a time. Average diameter of the dry and impregnated fibers were the same. Stressstrain data from the dry fiber tests are shown in Figure There was no significant difference between fibers from different rolls, the two-fiber results, or four-fiber results Stress (MPa) Stress (psi) Multiple Fiber Results 2 strands 4 strands Average Strain (m/m) Figure 3.11 Multiple strand tensile test results of dry cotton fiber. 55

73 Stress (MPa) Individual Fiber s/c 1 s/c 2 s/c 3 S/ Average u/c 1 u/c 2 u/c 3 U/ Average Stress (psi) Strain (m/m) Figure 3.12 Silicone (s/c) and urethane-impregnated (u/c) cotton fiber tensile test results. Several pairs of individual silicone- and urethane-impregnated cotton fibers were tested using the same test setup. Stress-strain results for the impregnated fibers are shown in Figure The silicone rubber impregnated fibers were not as stiff as the urethane rubber impregnated fibers, which suggests that the impregnated fiber stiffnesses are functions of the stiffness of the matrix (rubber) material as well as the cotton. A linear curve fit of the initial averaged stress-strain data was used to obtain approximate Young s moduli. These results are given in Table 3.4, and are shown graphically in Figure 3.13, the solid lines represent the portions of curves used in calculation of initial moduli. The stiffest curve is for averaged urethane/cotton test results. The middle curve is for the averaged 56

74 silicone/cotton results. The right curve represents the dry cotton stress-strain results. The stiffness of the dry fiber is the lowest of the group. This is very unexpected, a simple ruleof-mixture calculation combining a soft matrix and a stiff fiber, for any fiber volume fraction, will predict the axial (extensional) stiffness of the impregnated fiber to always be lower or the same, depending on fiber volume fraction Stress (MPa) Stress (psi) Impregnated Fiber Results s/c (Average) u/c (Average) Dry otton Fit of Initial Young s Moduli Strain (m/m) 0 Figure 3.13 omparison of dry, silicone-impregnated (s/c), and urethane-impregnated (u/c) cotton tensile test results, with best linear fits. TABLE 3.4 Approximate Young s modulus for dry and impregnated cotton fibers. Fiber Type Young s Modulus [MPa (ksi)] % Increase of Stiffness Dry otton 324 (47) 0% otton with Silicone 565 (82) 74% otton with Urethane 738 (107) 128% 57

75 radial contraction shearing due to tension Figure 3.14 Schematic of a dry (left) and impregnated (right) cotton fiber. A brief discussion of this phenomenon may be helpful. A fiberglass tow is not a single strand, but many fibers that are bundled together and treated like a fiber. The fibers are parallel to each other, are not twisted, and are held loosely by a binder that also aids adhesion with most resins. The cotton tow or fiber is likewise a collection of individual strands. Instead of being straight, however, these strands are twisted to keep them together, and are relatively large in size compared to the diameter of the tow. A simple illustration of a dry twisted cotton fiber is shown on the left side of Figure An illustration of a twisted cotton fiber which has elastomer embedded in it is shown on the right in the same figure. When the dry cotton fiber is loaded in tension, the individual strands tend to straighten out. As this happens the strands slide against each other, decrease in diameter and move towards the center of the bundle. The overall effect is to reduce the effective diameter of the fiber or cotton tow. When the same fiber has an elastomer 58

76 embedded in it, even one with a low modulus of elasticity, its response changes. The individual strands are not allowed to slide freely, but are restrained by the elastomer. The elastomer in the center of the group of strands resists the radially inward movement of the strands. The two mechanisms combine to increase the axial stiffness of the cotton fiber, but also increase nonlinearity, as evidenced by the fiber test results in Figure Discussion of Fiber-Reinforced Elastomer Response Experimental stress-strain behavior of the four fiber-reinforced elastomer combinations are of primary importance for this work. Due to the extensive amount of data obtained, however, only average test results for each angle are presented and discussed. Individual specimen stress-strain results, in a less refined form, can be found in Appendix D. Significant characteristics such as nonlinearity, stiffness and failure modes are discussed, as are the contributions of matrix and reinforcing fiber. Because of the narrow specimen test-section width and because fibers were cut at the left and right edges of the test region little fiber-reorientation was observed except at strains approaching failure a otton-reinforced Silicone Silicone rubber and cotton fibers represent the most compliant matrix and fiber components of the four combinations investigated. Average stress-strain results from the 0 to 45 balanced angle-ply specimens are shown in Figure Specimens with fibers at 0 are the stiffest and are the left-most curve. Stiffness decreased, and nonlinearity increased as the off-axis angle increased. At low off-axis angles, the full strength of the cotton fiber was not realized due to slippage at higher strains, but realistic laminate stiffnesses were obtained. At 30 some nonlinearity (stiffening) of the laminate is evident. The stiffening effect of the silicone rubber, is most visible at 30 through 60 but, as shown in Figure 59

77 12 s/c 0 avg s/c 15 avg s/c 30 avg 1400 Stress (MPa) s/c 45 avg Stress (psi) Strain (m/m) Figure 3.15 Measured tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens from 0 to s/c 45 avg s/c 60 avg 600 Stress (kpa) s/c 75 avg s/c 90 avg silicone rubber Stress (psi) Strain (m/m) Figure 3.16 Measured tensile results for [± θ] 2 cotton-reinforced silicone (s/c) specimens from 45 to

78 3.16, is evident at 75 as well. Average stress-strain results from 45 to 90 specimens are shown in Figure 3.16 (specimen results at 45 are provided to facilitate comparison between the two related graphs). Near or at failure stresses, for 30 to 60 specimens, some softening was observed. This is also the range where shearing action between adjacent layers of the angle-ply laminates is the greatest. Observations during testing suggest the softening is due to shear failure that has started to occur, rather than material nonlinearity. Up to approximately 50% strain, stress-strain curves for 60 to 90 specimens are very similar. This has ramifications for elastic tailoring, since at 60, a laminate has considerably more shear and transverse stiffness b Fiberglass-Reinforced Silicone The fiberglass-reinforced silicone composite specimens combine a stiff fiber and a compliant matrix. The average test results from the 0 to 45 off-axis angles are shown in Figure The average results from the 45 to 90 off-axis angles are shown in Figure The effects of the fiberglass are immediately obvious in the failure strains at low off-axis angles. The corresponding cotton-reinforced silicone failure strains are significantly higher. Laminate stiffening is evident, starting at 30, and continuing through 60, except for very small strains. As shown in Figure 3.18, stress-strain curves at 60 and 75 at higher elongations become almost linear. Like the cotton reinforced specimens, there is considerable more strength of the 90 specimens than the pure silicone rubber c otton-reinforced Urethane Average stress-strain results from urethane/cotton specimens 0 to 45 off-axis angles are shown in Figure As the angle is increased, specimen stiffness and strength decrease. At 30 and 45 the stress-strain curves are virtually linear, indicating no fiber 61

79 s/g 0 avg s/g 15 avg s/g 30 avg s/g 45 avg Stress (MPa) Stress (psi) Strain (m/m) Figure 3.17 Average tensile test results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens from 0 to s/g 45 avg s/g 60 avg s/g 75 avg s/g 90 avg silicone rubber Stress (kpa) Stress (psi) Strain (m/m) Figure 3.18 Average tensile test results for [± θ] 2 fiberglass-reinforced silicone (s/g) specimens from 45 to

80 reorientation. Figure 3.20 shows test results for 45 through 90 specimens. The softening effect of the urethane rubber is readily apparent for 60 and greater angle-ply specimens. Similar to the silicone/cotton results, at strains less than 25%, there is little difference between the 75, the 90, and to a lesser extent, the 60 stress-strain curves d Fiberglass-Reinforced Urethane Average stress-strain results for urethane/fiberglass specimens at 0 to 45 off-axis angles are shown in Figure The stress-strain curves are fairly linear through 45. Results for 45 to 90 specimens are shown in Figure The curves for 60 through 90 show characteristic urethane softening. Due to manufacturing error the 30 specimens were actually 37, and the 60 specimens were Stress (MPa) u/c 0 avg u/c 15 avg u/c 30 avg u/c 45 avg Stress (psi) Strain (m/m) Figure 3.19 Average tensile test results from 0 to 45 for [± θ] 2 cotton-reinforced urethane (u/c) specimens. 63

81 Stress (kpa) u/c 45 avg u/c 60 avg u/c 75 avg u/c 90 avg urethane rubber Stress (psi) Strain (mm/mm) Figure 3.20 Average tensile test results from 45 to 90 for [± θ] 2 cotton-reinforced urethane (u/c) specimens u/g 0 avg u/g 15 avg u/g 37 avg 2000 Stress (MPa) u/g 45 avg Stress (psi) Strain (m/m) 0 Figure 3.21 Average tensile test results from 0 to 45 for [± θ] 2 fiberglass-reinforced urethane (u/g) specimens. 64

82 u/g 45 avg u/g 53 avg u/g 75 avg u/g 90 avg urethane rubber Stress (kpa) Stress (psi) Strain (m/m) 0 Figure 3.22 Average tensile test results from 45 to 90 for the [± θ] 2 fiberglassreinforced (u/g) urethane specimens Laminate Failure Modes Failure modes for all four material systems were very similar. At low off-axis angles, in particular 0 and 15, the specimens elongated, then slipped in the grips, hence artificially low failure stresses were noted. In addition, the stiffness of the fiberglass-reinforced specimens at 0 appears suspect. Stiffness of the 0 fiberglass-reinforced specimens will be discussed in more detail in a following section on initial properties. All other fiberglass and cotton reinforced elastomer stiffness values are considered valid. The expected lowangle failure mode was brittle fiber breakage, but instead was typically due to specimen slippage. At off-axis angles from 30 to 60, the primary failure mode was shear failure of the matrix, due to the angle-ply nature of the specimens. At the higher off-axis angles 65

83 (75 to 90 ), failure is due to tensile failure of the rubber matrix, with translation (parallel fibers moving farther apart) of the fibers. Failure stresses at the higher off-axis angles are functions of fiber volume fraction and fiber characteristics. Obvious improvements in strengths at low off-axis angles can be expected. For FRE applications such strengths are functions of geometry, boundary conditions, and loading parameters Influence of Fiber Angle, Matrix and Fiber Type omparisons to this point have been between test results within the same material combination, except for general comments. Average test results at each angle, for the various FRE combinations are now plotted and compared. Average results at 30 are shown in Figure 3.23, average results at 45 are shown in Figure 3.24, and average results at 60 are shown in Figure Average results at 75 are shown in Figure 3.26, and average results at 90 and pure elastomer results are shown in Figure Average results at 0 and 15 aren t shown since they are quite linear for all material systems. The FRE test results at moderate and higher off-axis angles showed nonlinear trends similar to their respective elastomers. The fiber-reinforced silicone composites showed considerable stiffening at all but the highest (90 ) and lowest angles (0, 15 ). onversely, the fiber-reinforced urethane composites were more linear in nature at 30 and 45, and showed softening at angles greater than 30. At equivalent fiber angles, the silicone composites allowed more strain before failure and were consistently lower in stiffness than their urethane counterparts, similar to the elastomer test results. Part of the lower stiffness of the silicone composites, however, is due to their lower fiber-volume fractions relative to the urethane composites. 66

84 Stress (kpa) s/c 30 avg s/g 30 avg u/c 30 avg u/g 37 avg Stress (psi) Strain (m/m) Figure 3.23 Average tensile test results from [± 30 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g) Stress (kpa) Stress (psi) s/c 45 avg s/g 45 avg u/c 45 avg u/g 45 avg Strain (m/m) Figure 3.24 Average tensile test results from [± 45 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g). 67

85 Stress (kpa) s/c 60 avg s/g 60 avg u/c 60 avg u/g 53 avg Stress (psi) Strain (m/m) Figure 3.25 Average tensile test results from [± 60 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g) Stress (kpa) s/c 75 avg s/g 75 avg u/c 75 avg u/g 75 avg Stress (psi) Strain (m/m) Figure 3.26 Average tensile test results from [± 75 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), and urethane/fiberglass (u/g). 68

86 Average initial laminate longitudinal ( E x o ) stiffnesses for each material system, and for each set of specimens (each angle), are given in Table 3.5. The average initial laminate longitudinal stiffnesses are also plotted as a function of angle and are shown in Figure 3.28 as a log-linear graph. The trends shown are consistent with stiff composites. Other than general observations, however, too much emphasis should not be placed on individual initial values. Since all of the FRE material systems exhibit material nonlinearity, laminate stiffness can change significantly with small amounts of additional strain. TABLE 3.5 Average initial axial laminate stiffness ( angle. E x o ) of each FRE combination by Off-axis angle Silicone/cotton [MPa (psi)] Silicone/fiberglass a [MPa (psi)] Urethane/cotton [MPa (psi)] Urethane/fiberglass a [MPa (psi)] (39,200) 1830 (265,000) a 341 (49,400) 3400 (493,000) a (15,300) 263 (38,200) 209 (30,300) 636 (92,200) (2400) 19.4 (2810) 33.1 (4801) 33.3 (4830) b (460) 7.65 (1110) 9.86 (1430) 21.4 (3110) (222) 3.09 (448) 6.21 (901) 5.11 (741) c (374) 2.70 (391) 5.32 (772) 2.96 (430) (546) 1.94 (282) 6.23 (904) 2.25 (327) a - Stiffness is considered low, may be inaccurate, b - Actual angle is 37, c - Actual angle is 53. omparison of test results by reinforcing fiber show interesting trends. Since fiberglass is over two orders of magnitude greater in axial stiffness than cotton fiber, it is not surprising that at lower off-axis angles, the fiberglass-reinforced elastomers showed higher stiffness. At off-axis angles of 75 or greater (as seen in Table 3.5, Figures 3.27 and 3.28) the higher fiber volume fractions of the cotton-reinforced elastomer composites begin to dominate. The cotton fibrils may also contribute to increased transverse stiffness 69

87 Stress (kpa) s/c 90 avg s/g 90 avg u/c 90 avg u/g 90 avg Urethane Rubber Silicone Rubber Strain (m/m) Stress (psi) Figure 3.27 Average tensile test results at [± 90 ] 2 for silicone/cotton (s/c), silicone/ fiberglass (s/g), urethane/cotton (u/c), urethane/fiberglass (u/g) combinations and pure rubber. Initial Longitudinal Stiffness, Ex (MPa Silicone/cotton Silicone/fiberglass Urethane/cotton Urethane/fiberglass Off-Axis Angle, θ Figure 3.28 Average initial longitudinal [± θ] 2 laminate stiffness, E x, for each material system as a function of off-axis angle. 70

88 at 90. Axial stiffness for all material systems varies greatly with off-axis angles less than 45. The same trend is noted with conventional stiff composites, but the rate at which the stiffness changes is much greater with fiber-reinforced elastomer composites, with the highest changes noted in the fiberglass-reinforced elastomers Prediction of Initial Orthotropic Material Properties The initial axial lamina (fiber direction) stiffness, E 1 o, for each material system was predicted using three separate models: 1) the rule of mixtures [13]; 2) Hyer s concentric cylinders model [13]; and 3) a method set forth for flexible composites by hou [14]. The rule of mixtures model, when the fiber is considered isotropic, can be presented as: E 1 o = E f V f + E m ( 1 V f ) (3.16) The values E f and E m are the fiber and matrix Young s moduli, respectively, and V f is the fiber volume fraction. Hyer uses the theory of elasticity and a concentric cylinders model to obtain another form. The expression for E 1 o can be written in the form: E 1 o = E f ( 1 + γ)v f + E m ( 1 + δ) ( 1 V f ) (3.17) where γ and δ are functions of the extensional moduli, the fiber volume fraction, and Poisson s ratios. They are given by: γ = 2ν f E m ( 1 ν f 2( ν f ) 2 )V f ( ν f ν m ) E f ( 1 + ν m )( 1 V f ( 1 2ν m )) + E m ( 1 ν f 2( ν f ) 2 )( 1 V f ) (3.18) and δ = 2ν f E f ν m V f ( ν m ν f ) E f ( 1 + ν m )( 1 V f ( 1 2ν m )) + E m ( 1 ν f 2( ν f ) 2 )( 1 V f ) (3.19) 71

89 where ν m and ν f are the matrix and fiber Poisson s ratios, respectively. hou [14], gives a very simple approximation: E 1 o = E f V f (3.20) This approximation is possible because the rubber stiffnesses are so small that they contribute little to the lamina extensional stiffness. Initial elastomer Young s moduli, average impregnated cotton Young s modulus, and published fiberglass properties were used in the calculation of E 1, along with measured fiber volume fractions. Good correlation between predicted and test results for the cotton-reinforced elastomer material systems was obtained, but the predictions of E 1 o for the fiberglass-reinforced elastomer material systems were four to five times higher than test results. TABLE 3.6 Predicted and measured axial stiffness ( E 1 o ) for each material system. Silicone/ cotton [MPa (psi)] Silicone/ fiberglass [MPa (psi)] Urethane/ cotton [MPa (psi)] Urethane/ fiberglass [MPa (psi)] Test Results 270 (39,200) 1830 a (265,000) 341 (49,400) 3400 a (493,000) Rule of Mixtures 273 (39,600) 8761 (1,270,600) 329 (47,700) (1,879,700) oncentric ylinders 273 (39,600) 8760 (1,270,500) 329 (47,700) (1,879,300) hou s Approximation 272 (39,500) 8760 (1,270,500) 328 (47,600) (1,879,500) a - Stiffness is considered low, may be inaccurate. The comparison of measured axial stiffnesses with predictions using the several methods is shown in Table 3.6. These comparisons, inspection of the failed 0 fiberglass test specimens, and evaluation of each stress-strain curve led to the conclusion that the test fixture did not fully engage or load the fiberglass-reinforced specimens, as desired. Re-testing of the 0 fiberglass-reinforced specimens is possible, but the accepted accuracy of any 72

90 of the above mentioned models are considered sufficient. More of a major concern is determining what caused such low test results. It is thought, that with these axially stiff specimens, the extremely compliant matrix could not transmit the total force from the test fixture, by shear, to the fibers. The solution to this problem for specimens with axial fibers is to positively engage the fibers by embedding the fibers at the specimen tabbed ends into epoxy or another stiff matrix. Prediction of initial shear stiffness ( o G 12 ) for each material system used the rule-ofmixtures model [13], a modified rule-of-mixtures model [13], and a method set forth for flexible composites by hou [14]. The rule-of-mixtures model for o G 12 is expressed as: o G 12 = V f 1 V f G f G m (3.21) where G f and G m are the fiber and matrix shear moduli, respectively. The modified ruleof-mixtures, with a partitioning factor η, can be expressed as: o G 12 = V ---- f η( 1 V f ) [ V f + η( 1 V f )] G f G m (3.22) For this model, a partitioning factor of η=0.6, as suggested by Hyer [13], was used. hou [14] based the shear stiffness only on matrix properties. He expresses the lamina shear modulus as: o G 12 G m = ( 1 + V f ) ( 1 V f ) (3.23) Experimental results and predictions for lamina shear modulus are summarized in Table 3.7. A standardized method [15] using tensile results of 45 o specimens was used to 73

91 obtain an initial shear stiffness for each material combination. Initial elastomer Young s moduli, laminate fiber volume fractions, average impregnated cotton Young s modulus, published fiberglass properties and published Poisson s ratios were used in the calculation of o G 12 (urethane and silicone rubber are considered incompressible, hence their Poisson s ratios are 0.5). Reasonable correlation between predicted and test results for the cottonreinforced elastomer material systems was obtained using the modified form of the ruleof-mixtures, based on measured laminate fiber volume fractions. hou s model works almost as well. All of the constitutive models work better when the difference in stiffness between fiber and matrix is three orders of magnitude or less. As the difference in stiffness increases, very small variations in angle, or very small amounts of misaligned fibers can make a huge difference in stiffness. TABLE 3.7 Predicted and measured initial shear stiffness ( system. o G 12 ) for each material Silicone/ cotton [MPa (psi)] Silicone/ fiberglass [MPa (psi)] Urethane/cotton [MPa (psi)] Urethane/ fiberglass [MPa (psi)] Test Results (95) 1.95 (283) 2.10 (304) 5.34 (774) Rule of Mixtures (91.7) (50.4) 1.45 (211) (97.0) Mod. Rule of Mixtures (123) (56.4) 2.05 (298) (114) hou s Approximation (139) (56.4) 2.37 (344) (114) The Vanyin model [16] is a more refined constitutive model that is used by some researchers in the tire rubber industry. Predictions of initial shear and transverse stiffnesses by the Vanyin model, however, were virtually identical to the modified rule of mixtures model, hence they are not presented. 74

92 Predictions of initial transverse stiffness ( ) for each material system were made using the rule-of-mixtures model [13], modified rule-of-mixtures [13], and hou s model [14]. The rule-of-mixtures model for can be expressed as: E 2 o E 2 o o E 2 = V ---- f 1 V f E f E m (3.24) where E f and E m are the fiber and matrix moduli respectively. The modified rule-of-mixtures, with a partitioning factor η, can be expressed as: o E 2 = V ---- f η( 1 V f ) E f E m V f + η( 1 V f ) (3.25) For this model, a partitioning factor of η=0.6 was used, as suggested by Hyer. hou also bases the lamina transverse stiffness completely on matrix properties. He expresses the lamina transverse modulus as: o ( V E f ) E 2 = m ( V f )( 1 V f ) (3.26) Experimental results and predictions using Equations 23 through 25 are presented in Table 3.8. Measured fiber volume fractions for the laminated specimens were used. Reasonable correlation between predicted and test stiffnesses for the fiber-reinforced urethane specimens was obtained using the modified form of the rule-of-mixtures and hou s model, using measured laminate fiber volume fractions. Stiffness predictions for the fiberreinforced silicone specimens, however, were mixed and somewhat lower than measured. 75

93 TABLE 3.8 Predicted and measured initial transverse stiffness ( material system. E 2 o ) for each Silicone/ cotton [MPa (psi)] Silicone/ fiberglass [MPa (psi)] Urethane/cotton [MPa (psi)] Urethane/ fiberglass [MPa (psi)] Test Results 3.76 (546) 1.94 (282) 6.23 (904) 2.25 (327) Rule of Mixtures 1.90 (275) 1.04 (151) 4.36 (632) 2.01 (291) Mod. Rule of Mixtures 2.55 (370) 1.17 (169) 6.15 (892) 2.36 (343) hou s Approximation 2.53 (367) 1.14 (165) 6.05 (877) 2.27 (329) Nonlinear Orthotropic Material Properties One of the basic objectives of the current work was to obtain orthotropic material properties for each material system. Since the fiber-reinforced elastomer composites exhibit material nonlinearity, the material properties such as E 1, E 2, G 12, and ν 12 become functions of strain. The methods used to obtain these nonlinear material properties for each material combination are discussed in the following sections a Extensional and Transverse Moduli Inspection of the test results from every material system at 0 show linear stress-strain curves. This indicates that a single constant, E 1, can be used for the extensional stiffness of each material system. Values of E 1 for each material system can be viewed in the 0 or first row of Table 3.5, which gives laminate longitudinal stiffness as a function of angle. Review of the test results from every material system at 90 show very nonlinear stress-strain curves. Typically, transverse stiffnesses for stiff composites are calculated from the initial slope of stress-strain curves for laminates tested at 90 to the fiber angle. The transverse stiffness, E 2, can be obtained as a nonlinear function of extensional strain for each material system by use of the following simple expression: 76

94 E 2 ( ε x ) = dσ x dε x 90 σ x ε x 90 (3.27) Transverse moduli for each material system, as a function of extensional strain, are shown in Figure The transverse stiffness for each material system starts relatively high, and decreases, leveling off at approximately 50% strain. Transverse moduli at 50% and greater strains are relatively independent of fiber type and fiber volume fraction for the material systems tested. Transverse Stiffness (MPa) Urethane/fiberglass Urethane/cotton Silicone/fiberglass Silicone/cotton Transverse Stiffness (psi) Strain (m/m) Figure 3.29 Transverse modulus, E 2, of each material system measured using a [± θ] 2 laminate, as a function of extensional strain. 77

95 3.7.7b Poisson s Ratio and Shear Modulus The shear modulus of a composite laminate can be obtained if either the transverse strain or the Poisson s ratio is known, as well as axial stress and strain, of a ±45 angle-ply laminate [15]. For a laminate with a linear stress-strain curve, the shear modulus can be defined as: G 12 = σ x ( ε x ε y ) (3.28) With a specimen in tension, the transverse strain has a negative sign. The relationship between extensional strain and transverse strain is: ε y = ν xy ε x (3.29) where ν xy is the Poisson s ratio at 45. Equation 3.28 can be rewritten as: G 12 = σ x ( + ν xy )ε x (3.30) Equation 3.30 is the more convenient way to obtain the shear modulus as a function of extensional strain when the Poisson s ratio is known. Axial and transverse strains were measured for laminates with off-axis angles of ± 45. The same best-linear-fit method used to obtain strain correction coefficients was employed to obtain the Poisson s ratios, using Equation Average Poisson s ratios for each material system are given in Table 3.9. The ±45 fiberglass-reinforced silicone rubber specimens were destroyed in previous tests, so a Poisson s ratio at ±30 was measured. That ratio compared favorably with calculated values, hence a calculated value at ±45 for the fiberglass-reinforced silicone rubber material system was used. 78

96 TABLE 3.9 Average Poisson s ratios (ν xy ) for each material system. Material System Poisson s ratios, (ν xy ) otton/urethane 1.29 otton/silicone 1.42 Fiberglass/Urethane 1.14 Fiberglass/Silicone 0.98* * alculated, based on ν xy 30 measurement. The average Poisson s ratio for each material system was used to find the instantaneous shear modulus as a function of extensional strain using the following modification of equation 3.30: G 12 ( ε x ) dσ x dε x = ( + ν xy ) σ ε x ( + ν xy ) 45 (3.31) Since equation 3.31 deals with stress at a specific value of strain, the model is valid for nonlinear shear moduli as well as linear moduli. The shear modulus, G 12, for each material system as a function of extensional strain ε x, is shown in Figure The two urethane-matrix test results show clear softening-then-level trends. The fiberglass reinforcement caused the average shear moduli for the fiberglass-reinforced material systems to be approximately three times higher than their cotton-reinforced counterparts. The silicone-matrix results showed clear softening followed by stiffening trends. 3.8 SUMMARY OF EXPERIMENTAL BEHAVIOR The current work discusses tensile stress-strain results for elastomer matrices, cotton fibers, and four sets of fiber-reinforced elastomers (FRE) with fiber volume fractions 79

97 Urethane/Fiberglass 6 Urethane/otton Silicone/Fiberglass 800 Shear Modulus (MPa) Silicone/otton Shear Modulus (psi) Strain (m/m) Figure 3.30 Shear modulus, G 12, of each material system measured using a [± θ] 2 laminate, as a function of extensional strain. ranging from 12% to 62%. Angle-ply specimens of each elastomer composite were fabricated with off-axis angles from 0 to 90 in 15 increments. Procedures for obtaining accurate strain data at high elongations were successfully demonstrated. The average extensional stiffness of individual cotton fibers increased by 74% to 128% when impregnated with an elastomer. The increase is due to additional shear resistance between individual twisted cotton strands and added constraint against radial contraction of the cotton fibers. The constraint is provided by the impregnated rubber (during axial extension). Test results show that FRE laminate stiffness and nonlinearity can vary significantly with fiber angle. The nonlinear stiffening or softening trends of the silicone and urethane rub- 80

98 bers are reflected in their respective fiber-reinforced elastomers, but FRE stiffnesses can be orders of magnitudes higher than elastomer stiffnesses. Axial stiffness at low off-axis angles is a function of the reinforcement stiffness. At high off-axis angles, laminate stiffness and strength are functions of fiber type, fiber volume fraction and elastomer. Measured axial stiffnesses for the 0 fiberglass-reinforced elastomers specimens may be low because the compliant elastomer matrix continues to deform rather than transmit shear forces from test fixture to fiber. Initial axial, transverse, and shear material properties measured for each material system were compared with predictions made using constituent properties. orrelation was reasonable for the cotton-reinforced elastomer composites but could be improved for the fiberglass-reinforced elastomer composites. The nonlinear transverse and shear material properties extracted from the test data will aid in accurately predicting the response of fiber-reinforced elastomer composite structures and will improve understanding of the fundamental response mechanisms of such FRE materials. Besides the capability to sense changes in its environment, a smart structure must also be able to change a physical characteristic such as stiffness, damping rate, or configuration. Because FRE stiffness and hence damping rates are functions of strain, and large deformations are possible, tailored flexible composites provide a new category of materials that are especially suited for smart structures. Successful use of FRE materials in smart structures, however, depends on accurate nonlinear material properties and the exploitation of specific response mechanisms. 81

99 82

100 HAPTER 4 NONLINEAR MODELING OF FIBER-REINFORED ELASTOMERS 4.1 SYNOPSIS Accurately predicting the response of fiber-reinforced elastomer or flexible composite structures can be improved by the addition of material, geometric and fiber-rotation nonlinear models to classical laminated plate theory. Material nonlinearity is included in the form of nonlinear orthotropic material properties as functions of extensional strain. Nonlinear properties were obtained from the experimental results of fiber-reinforced elastomeric (FRE) angle-ply specimens at 0, 45, and 90 discussed in hapter 3. Axial stiffness and Poisson s ratio are considered constant. Geometric nonlinearity is removed from the transverse and shear stiffnesses. A six-coefficient Ogden model was chosen to represent the nonlinear stiffnesses. Geometric nonlinearity is included through the addition of nonlinear extensional terms from the Lagrangian strain tensor. The nonlinear strain-displacement relations and the nonlinear material models were added to the code of a pre-existing composites analysis software package. The improved analytical tool will aid in understanding the behavior of FRE materials and will enable their use in stiffnessand deformation-tailored smart structures. Because fiber rotation (fiber re-orientation) is a function of geometry and boundary conditions, it is included in a simple model of a rubber muscle. 83

101 4.2 INTRODUTION Increased development of components fabricated from fiber-reinforced elastomers or flexible composites will be limited without the ability to accurately predict the response of such materials. The analysis of traditional composites uses classical lamination theory. lassical lamination theory assumes that strains are small and that orthotropic material properties are linear, however, fiber-reinforced elastomers can experience large strains and typically exhibit nonlinear stress-strain characteristics. Elastomers (rubber) can deform up to 800% and usually have very nonlinear stress-strain curves. Rubber can increase in stiffness when stretched (e.g., silicone), or can decrease in stiffness (e.g., urethane). The deformation of a FRE laminate can change significantly with fiber orientation due to extreme differences in stiffness between matrix and fiber. The difference in matrix and fiber stiffness for typical stiff composites is approximately one to two orders of magnitude. The difference in matrix and fiber stiffness for fiber-reinforced elastomeric composites is approximately five orders of magnitude. Laminate Poisson s ratios, which affect transverse deflection, and are functions of layer stiffnesses, can vary considerably with fiber angle. Because of these unusual characteristics, successful use of FRE materials depends on accurate nonlinear material properties, and the exploitation of those properties in the prediction of FRE response mechanisms. 4.3 ONTRIBUTIONS TO THE STATE OF THE ART The combination of elastomer (rubber) and directional reinforcement into materials such as fiber-reinforced elastomers or elastomer composites is not new, but has primarily used in tires, belting, or impregnated fabrics. The fabrics are constructed such that significant out-of-plane bending is possible, but inplane shear or elongation is restricted. The 84

102 current work considers FRE laminates where inplane shear and extension are not restricted. Due to the cording (twisted fibers) in tires and high concentrations of fillers in the rubber, elongation of the reinforced area in a tire is typically under 10%. Relatively little work has been published on the prediction of FRE responses at higher elongations of 20% to 200%. Because of the smaller strains in cord-rubber composites, linear strain-displacement relations and simple rubber material models, such as the Mooney-Rivlin model are assumed [1,2]. Bert [3] has developed models for cord-rubber composites with different properties in tension and compression. lark [4] used a bi-linear model of the transverse and shear stiffnesses, with strains up to 10%. His model does not always predict stiffness properly and is directed primarily towards cord-rubber applications. Woo [5] has conducted extensive characterizations of human and animal ligaments and has developed viscoelastic strain models that describe the response of ligaments very well. Woo s viscoelastic models are not incorporated into the current work but would be useful for future bio-mechanical applications. hou and Luo [6-8] have conducted perhaps the most comprehensive work on the nonlinear elastic behavior of flexible composites. Their work deals primarily with wavy fibers in an elastomeric matrix. Geometric nonlinearity is introduced through the straightening of the wavy fibers as the specimen is elongated. Material nonlinearity is incorporated through use of a third order polynomial material model. Predicted results were compared with test results from specimens with fiber volume fractions on the order of one or two percent. Total elongation of their specimens was approximately twenty percent. In a work completed after the current nonlinear model was created, Derstine [9] combines the fiber and elastomer using a micro-mechanics model, a 85

103 process science model to determine the local geometry of a 3-D braid, and a stiffness averaging routine for calculating the local stiffness of the material. Fiber re-orientation is calculated. The final stress-strain is used as input in a nonlinear finite element model. Total strain, for the results presented, was under ten percent urrent ontributions ontributions of the current work include the use of an improved material model, the use of geometric nonlinearity, clarification of the differences between nonlinear strain-displacement and fiber-reorientation, and the prediction of stress-strain responses up to and beyond 200%. Separation of the contributions of nonlinear material properties, geometrically nonlinear strain-displacement relations and nonlinearity due to fiber re-orientation is critical for the development of an accurate model. Modeling of material nonlinearity is improved by use of the extremely accurate Ogden model [10]. Geometric nonlinearity is removed from nonlinear orthotropic properties. This step is necessary because the instruments used to measure strain considered the measured strain to be small and linear. Fiber re-orientation is a function of specimen geometry, rather than an inherent lamina property. Fiber re-orientation is included in a model of a simple rubber muscle. Efforts were made to keep the computer model as simple as possible while retaining accuracy. Results from testing of the constituent rubbers support the conclusion that viscoelastic relations need not be included if quasi-static conditions are imposed, hence viscoelastic relations are not part of the current model. 4.4 NONLINEAR MODELING OF FIBER-REINFORED ELASTOMERS The nonlinear model is based on classical laminated plate theory with the addition of nonlinear material properties and geometric strain-displacement nonlinearity. An over- 86

104 view of classical laminated plate theory is presented in order to understand the significance of the nonlinear additions. The use of nonlinear material properties requires several supplemental steps and formulae. The actual addition of the nonlinear strain-displacement relations is quite simple but, when coupled with the nonlinear properties, changes the method of solution from closed form to a step or iterative form. The inclusion of fiber re-orientation (rotation) is geometry and boundary-specific. For example an axi-symmetric FRE cylinder with an off-axis angle of 30 degrees will experience more fiber rotation than a thin flat rectangular angle-ply specimen with the same offaxis angle. As the tube is elongated the continuous fibers in the tube must change their angle because the length of the tube will increase and the diameter will decrease. However, for a thin flat specimen, a fiber may be cut at the left and right sides of its gage section. Since the relative stiffness of the elastomer is many times lower than the fiber, the fiber will tend to stay at the same orientation and translate in the direction of deformation. If uncut fibers are placed in a zigzag or sinosoidal pattern, elongation of the article will produce more fiber rotation or re-orientation. Since fiber-reorientation is geometry specific, a simple model of a rubber muscle was created and will demonstrate the significance of fiber re-orientation. The large deformations and significant reductions in cross-sectional area require selection of the appropriate definitions for stress and strain. The Lagrangian description considers properties relative to initial positions, while the Eulerian description considers properties relative to the current position. Because rubber is highly deformable, each definition has its advantages. Standard rubber models [10], such as the Mooney-Rivlin, Ogden, and Peng use the auchy (engineering) stress, σ i, which is defined as 87

105 σ i = F i A o (4.1) where F i is the applied force in the i direction and A o is the original cross-sectional area. The previously mentioned rubber models use extension ratio (stretch) instead of strain. Extension ratio, a i can be defined as a i = 1 + ε i (4.2) and ε i = L L o i (4.3) where ε i is engineering extensional strain in the i direction, L is the change in length, and L o is the original gage length of the specimen. To maintain consistency with classical lamination theory, and to be able to use one of the above rubber material models, results are presented using the Lagrangian description (engineering stress and strain) Overview of Linear lassical Lamination Theory lassical lamination theory use the orthotropic material properties E 1, E 2, G 12, and ν 12 to describe the inplane axial modulus of elasticity, inplane transverse modulus of elasticity, inplane shear stiffness, and inplane major Poisson s ratio, respectively, in each layer in a laminate. For each layer: ν E 1 = ν E 2 (4.4) E Q 1 11 = ν 12 ν 21 (4.5) 88

106 ν 12 E Q 2 12 = ν 12 ν 21 (4.6) E Q 2 22 = ν 12 ν 21 Q 66 = G 12 (4.7) (4.8) These Q s or reduced stiffnesses can be combined in matrix form to find the stresses in an orthotropic layer: σ 1 σ 2 = Q 11 Q 12 0 Q 12 Q 22 0 ε 1 ε 2 (4.9) τ Q 66 γ 12 The values representing each layer (stress, strain or elastic constants) must be rotated from a global coordinate system to a local orientation θ: σ x σ y τ xy = 2 cos θ 2 sin θ 2 θ sin 2sinθcosθ 2 θ cos 2sinθcosθ 2sinθcosθ 2sinθcosθ cos θ sin θ 2 2 σ 1 σ 2 σ 12 (4.10) These values are rotated as depicted in figure y 1 Figure 4.1 oordinate systems for laminate and local (layer) axes. θ x If the elastic constants Qij are rotated we get the transformed stiffnesses Qij: 89

107 Q 11 = Q 11 cos 4 θ + 2(Q Q 66 )sin 2 θ cos 2 θ + Q 22 sin 4 θ (4.11) Q 12 = (Q 11 + Q 22-4Q 66 )sin 2 θ cos 2 θ + Q 12 (cos 4 θ + sin 4 θ) (4.12) Q 22 = Q 11 sin 4 θ + 2(Q Q 66 )sin 2 θ cos 2 θ + Q 22 cos 4 θ (4.13) Q 16 =(Q 11 - Q 22-2Q 66 )sinθ cos 3 θ + (Q 12 - Q Q 66 )sin 3 θ cosθ (4.14) Q 26 =(Q 11 - Q 22-2Q 66 )sin 3 θ cosθ + (Q 12 - Q Q 66 )sinθ cos 3 θ (4.15) Q 66 = (Q 11 + Q 22-2Q 12-2Q 66 )sin 2 θ cos 2 θ + Q 66 (cos 4 θ + sin 4 θ) (4.16) hou [8] suggests that the transformed stiffnesses Qij can be approximated by: Q 11 = E 2 +E 1 cos 4 θ (4.17) Q 12 = E 1 sin 2 θ cos 2 θ + E 2 /2 (4.18) Q 22 = E 2 +E 1 sin 4 θ (4.19) Q 16 =E 1 sinθ cos 3 θ (4.20) Q 26 =E 1 sin 3 θ cos 3 θ (4.21) Q 66 = E 1 sin 2 θ cos 2 θ + E 2 /4 (4.22) and are suitable for flexible composites. When compared with Equations 4.11 to 4.16, Equations 4.17 through 4.22, did not predict shear stiffnesses properly at off-axis angles near 45, were not used in the nonlinear model, and are not recommended unless the shear stiffness, G 12, is not known. Using the relations 4.11 through 4.16, we can obtain off-axis stress or strains using: σ x σ y = Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 ε x ε y (4.23) τ xy Q 16 Q 26 Q 66 γ xy 90

108 The laminate stiffnesses can be assembled by summing the contributions from each of n layers. Each layer k is t k thick, and its mid-plane is a distance z k from the mid-plane of the total laminate: A ij = n k = 1 ( Q ij ) k t k (4.24) n 1 B ij = -- ( Q 2 ij ) k = 1 k z k 2 2 z k 1 ( ) (4.25) n 1 D ij = -- ( Q 3 ij ) k = 1 (4.26) By assuming the Kirchoff hypothesis that planes will remain planar when a plate is under bending, and treating u o, v o, and w o as the mid-plane displacements of a laminated plate, the linear strains at a point (x,y,z) are k z k 3 3 z k 1 ( ) γ xy ε x ε y = = = u 0 y u 0 x v 0 y + 2 w 0 z x 2 2 w 0 z v 0 x y 2 2 w 0 2z x y (4.27) These strains can be put in the form: ε x ε y = ε x o ε y o + z κ x κ y (4.28) γ xy o γ xy κ xy 91

109 Now forces N i and moments M i (per unit length) can be obtained for the entire laminate using the laminate stiffnesses, mid-plane laminate strains and mid-plane laminate curvatures. N x N y N xy M x = A 11 A 12 A 16 B 11 B 12 B 16 A 12 A 22 A 26 B 12 B 22 B 26 A 16 A 26 A 66 B 16 B 26 B 66 B 11 B 12 B 16 D 11 D 12 D 16 ε x o ε y o o ε xy κ x o (4.29) M y M xy B 12 B 22 B 26 D 12 D 22 D 26 B 16 B 26 B 66 D 16 D 26 D 66 κ y o o κ xy Likewise, mid-plane strains and curvatures can be obtained by: ε x o ε y o o ε xy κ x o = A 11 A 12 A 16 B 11 B 12 B 16 A 12 A 22 A 26 B 12 B 22 B 26 A 16 A 26 A 66 B 16 B 26 B 66 B 11 B 12 B 16 D 11 D 12 D 16 1 N x N y N xy M x (4.30) κ y o o κ xy B 12 B 22 B 26 D 12 D 22 D 26 B 16 B 26 B 66 D 16 D 26 D 66 M y M xy At this point essentially all of the linear inplane and bending behavior of a laminate can be predicted. The theoretical development presented above is from reference [11]. Note that even though E 1, E 2, G 12, and ν 12 were considered constants, nothing stops us from treating them as functions of strain to obtain correct stresses, forces or strains. When the elastic values listed above are not constant, we have the condition of material nonlinearity. 92

110 4.4.2 Material Nonlinearity lassical lamination theory uses constants to describe the inplane axial modulus of elasticity, inplane transverse modulus of elasticity, inplane shear stiffness, and inplane major Poisson s ratio, respectively, in each layer in a laminate. For fiber-reinforced elastomeric materials these terms may not be constant but can be functions of strain (we assume that viscoelastic characteristics need not be included in the nonlinear material model if quasi-static conditions are maintained). The axial stiffness E 1 is highly dependent on axial fiber stiffness. If the reinforcing fiber stiffness is considered constant, the axial stiffness E 1 is considered constant as well. Inspection of the test results in hapter 3, from every material system at 0 show linear stress-strain curves. This indicates that a single constant, E 1, can be used for the extensional stiffness of each material system. Values of E 1 for each material system can be viewed in the 0 or first row of Table 3.5, which gives laminate longitudinal stiffness as a function of angle. FRE shear and transverse properties, however, are definitely not linear. Predicting nonlinear material properties at high elongations is sometimes more of an art than a science. Typically, no one model can match all material properties. hou [8] uses a third-order polynomial model. lark [4] treats the transverse stiffness as a bi-linear curve. Since the material models are intended to represent the response of fiber-reinforced rubber, and the response of the FRE materials are similar to their rubber matrices, it s likely that existing rubber models could be well suited to model the nonlinear transverse and shear stiffnesses. For this purpose, two rubber models are reviewed; the popular Mooney-Rivlin model, and a more accurate Ogden model, as well as lark and hou s models. 93

111 4.4.2a The Bi-Linear Stress-Strain Model lark [4] assumes that the stress-strain curves for E 1, E 2 and G 12 are bi-linear, the theory is a good first approximation beyond linear stress-strain relations and is conceptually very easy to understand. lark s basic theory is depicted in Figure 4.2. Region I represents a lower moduli, usually associated with compression, while Region II is usually associated with tension. The value of strain ε 1 *, where the change occurs, is considered a material property. Although the bi-linear model does consider compressive behavior separately, it does not allow for multiple changes in stiffness. σ region I ε 1 region II ε Figure 4.2 Bi-modular or bi-linear stress-strain material model b The Mooney-Rivlin Material Model The two-coefficient Mooney-Rivlin [10] strain-energy function is the most widely used constitutive relationship in the stress analysis of elastomers. It is not the most accurate, however, if the material experiences both softening and stiffening during elongation. The model was derived by Mooney and Rivlin based on a linear relationship between 94

112 stress and strain in simple shear. For incompressible materials the strain function can be expressed as U = c 1 ( I 1 3) + c 2 ( I 2 3), (4.31) where I 1 and I 2 are the principal invariants of the strain tensor. For a special case of uniaxial tension of an incompressible Mooney-Rivlin material, the stress-strain equation can be expressed as: S = 2( a a 2 )( c 1 + c 2 a 1 ) (4.32) where S is the auchy or engineering stress (the ratio of force to original area) and a is the stretch or extension ratio (1 + ε). For both the incompressible and compressible forms, the initial shear modulus is: G = 2(c 1 + c 2 ) (4.33) If the material is incompressible, the initial tensile modulus E is calculated by: E = 6(c 1 + c 2 ) (4.34) For a compressible Mooney-Rivlin material the initial modulus is: where K is the initial bulk modulus [10]. E = (9KG)/(3K+G) (4.35) The three-coefficient Mooney-Rivlin model [12] can be expressed as S = c 2 c 1 a c a a 3 a 3 (4.36) where c 1, c 2 and c 3 can be obtained through a curve-fitting algorithm. 95

113 4.4.2c The Ogden Material Model The Ogden material model [10] relates the strain-energy density as a separable function of the principal stretches (extension ratios). For incompressible materials, the strain energy function can be expressed as, U = 3 i = 1 m c j b j j = 1 b --- ( a j i 1) (4.37) where c j and b j are the material coefficients and a i are the three principal stretch ratios (i=1 to 3). If the coefficients c j and b j are chosen correctly, this material model can provide extremely accurate representations of the mechanical response of hyperelastic materials for large ranges of deformation. Depending on the type of response needed, the coefficients c j and b j can be developed from a simple tensile stress-strain curve. For a simple tension test, the Ogden formulation can be expressed as, n σ c j a b j 1 ( a b j) = ( ) j = 1 (4.38) The number of coefficients needed to predict the stress-strain characteristics of an elastomer depends on the amount of accuracy desired by the user. Typically three sets of coefficients are sufficient to fit the data for most types of vulcanized rubbers exhibiting behavior close to the natural rubber. One methodology to calculate three sets of Ogden coefficients is given in reference [10]. A comparison of the third order polynomial, two-coefficient Mooney-Rivlin model, the three-coefficient Mooney-Rivlin model, and a six-coefficient Ogden model, as illus- 96

114 Shear Modulus (kpa) Nonlinear Models s/c 45 G12 Ogden 6 3rd order polynomial Mooney-Rivlin 2 Mooney-Rivlin Shear Modulus (psi) Strain (m/m) Figure 4.3 omparison of several material models with silicone/cotton shear modulus. trated in Figure 4.3, shows the relative accuracy of the four models. The polynomial relation worked well for some materials but not others. The Ogden model, while slightly more computationally intensive, is the most accurate d Implementation of the Ogden Material Model Implementation of the Ogden model consists of two main steps: 1) making sure the experimental data is in the correct form, and 2) obtaining the six coefficients for each property. 97

115 Using the relations explained in hapter 3 (Equations ), shear and transverse moduli were obtained as a function of strain. We must remember, however, that when stress and strain were measured, the linear definition of strain was used: ε i = L L o i (4.39) and that: ε x = u 0 x 2 w z 0 x 2 (4.40) is the definition of linear strain in the axial direction. Since the specimens saw extremely high elongations, the stress resulting from the strain included the contribution of geometric nonlinearity. In the axial direction the nonlinear definition of axial strain is: e x u u 0 2 w 0 = + x 2 x z x 2 2 (4.41) Because there was no bending in the specimens tested, the bending contributions in Equations 4.40 and 4.41 can be ignored. The contributions of geometric nonlinearity to the stress-strain curve, and hence stiffness, can not be ignored and must be removed in order to have accurate nonlinear material properties. The linear strain ε x was plotted as a function of e x. A curve fitting program was used to find an extremely accurate relation that expresses ε x as a function of e x : ε x = e x e x e x e x (4.42) The reduced strain was used to obtain a reduced stress by the following relation: σ i = σ + ( i 1 ε i ε )E i 1 inst (4.43) 98

116 where E inst is the instantaneous Young s modulus, and i represents the current strain state. The reduced stress is plotted as a function of the measured linear strain ε x, and the nonlinear shear and transverse properties are obtained from the paired values. Now that the nonlinear properties have been put in the proper form, and geometric nonlinearity has been removed, the six Ogden coefficients for each material property must be obtained. The Ogden model produces stress as a function of extension ratio, and since the extension ratio is a constant plus strain, the Ogden model can be considered as predicting stress as a function of strain. The instantaneous derivative of the predicted Ogden stress with respect to extension ratio yields: [ G 12, E 2 ] = c j ( bj 1)a b j 2 ( b j )a b j + j (4.44) which is the instantaneous shear or transverse stiffness as a function of strain. The above relation, with three coefficients for c j and three for b j provides a much cleaner and more accurate form for the nonlinear stiffnesses, than the original Ogden model. A curve fitting program, Sigma Plot, was used to obtain the six coefficients for each material property. The stiffness versus strain values were smoothed where local irregularities occurred. Also, some properties showed a small knee in the data at very small strains. Since, at very small strains, variations in stiffness would contribute little to overall stress, these knees were removed. The resulting coefficients are given in Table 4.1. Because the average elongation at 45 is approximately 40%, while elongations at 90 can exceed 200%, an axial failure strain, ε xg, for the 45 specimens is presented as well. The failure strain is discussed in the section on solution procedures. ( ) 99

117 TABLE 4.1 Ogden coefficients for each shear stiffness oefficients Silicone/fiberglass Silicone/cotton Urethane/fiberglass Urethane/cotton b b b e c c c e e xg The Ogden coefficients for the transverse strain are given in Table 4.2. Since the Ogden model was not intended to predict the stress-strain response of the four material combinations beyond the failure strains of the 90 o specimens, no failure strain is reported, or needed in the computer model. A visual depiction of how well the Ogden rubber model fits the nonlinear shear and transverse properties is shown in Figures 4.4 and 4.5, respectively. Notice that there is very good correlation with both shear and transverse stiffnesses. There is a slight divergence at the highest strains for the silicone/fiberglass shear stiffness. This divergence shows up in some of the silicone/fiberglass predictions made later. For silicone/cotton shear stiffness, the Ogden model is started at approximately 15% strain in order to optimize the material properties at higher deflections, although it still models the silicone/cotton shear property adequately at lower strains. Although the Ogden model works very well, and better than other models considered, like any curve fit, care must be taken to ensure the fit is best in the regions of most interest. 100

118 Shear Stiffness (kpa) Nonlinear Shear s/c G12 s/c Ogd-6 s/g G12 s/g Ogd-6 u/c G12 u/c Ogd-6 u/g G12 u/g Ogd Shear Stiffness (psi) Strain (m/m) Figure 4.4 Experimental and modeled nonlinear shear stiffness for each material system with a [+45/-45] 2 layup. Shear Stiffness (kpa) Transverse Stiffness s/c E2 s/c Ogd-6 s/g E2 s/g Ogd-6 u/c E2 u/c Ogd-6 u/g E2 u/g Ogd Shear Stiffness (psi) Strain (m/m) Figure 4.5 Experimental and modeled nonlinear transverse stiffness for each material system with a [+90/-90] 2 layup. 101

119 TABLE 4.2 Ogden coefficients for each transverse stiffness oefficients Silicone/fiberglass Silicone/cotton Urethane/fiberglass Urethane/cotton b b e-3 b c c c Geometric Nonlinearity The limitation of classical lamination theory that strain is assumed to be a linear function of displacement and that strains are small is now addressed. Although some very rigid elastomers may behave in a linear manner, most elastomers can strain from 25% to 800%. Linear strain-displacement theory assumes that: γ xy ε x ε y = = = u 0 y u 0 x v 0 y + 2 w z 0 x 2 2 w 0 z y 2 v 0 x 2 w 0 2z x y (4.45) The geometrically nonlinear strain-displacement theory, also known as the Lagrangian nonlinear strain tensor, is 102

120 Γ xy Implementation of Nonlinear Model The nonlinear additions to classical lamination theory were coded into an existing composites analysis program. The initial DOS-based program, called PLAM, was written by Dr. Steven L. Folkman and Larry Peel, while at Utah State University. The proe x e y = u u 0 2 v 0 2 w 0 x 2 x + x + x 2 w 0 = + z x 2 v u 0 2 v 0 2 w 0 y 2 y + y + y 2 w 0 = + z y 2 u 0 y v 0 u 0 u 0 v 0 v 0 w 0 w 0 w z x x y x y x y x y (4.46) Because we are concerned primarily with inplane deformation, with no bending (although bending is considered in Equations 4.46 and 4.47), the definitions for strain can be simplified. The following relations assume that off-axis strains are fairly small, rotations about the x and y axes are moderately small, and that rotations about the z axis are negligible. Accordingly, the strain-displacement relations are: e x e y γ xy u u 0 2 w 0 = + x 2 x z x 2 v v 0 2 w 0 = + y 2 y z y 2 = u 0 y + v 0 x w 0 2z x y (4.47) where e x, e y, and γ xy represent the axial, transverse and shear strains that are used in the modified nonlinear model. Note that the definition for shear strain remains the same. This is consistent with formulations that hou [8] has used. 103

121 gram is written in Lahey Fortran, which has additional graphics and text capability, and allows the easy creation of interactive DOS-based programs a The omputer ode The modified program, PFRE3, was compiled using the Lahey LF90 Fortran compiler. The editor, compiler, and compiled program operate well under Windows 3.11 and Windows 95. A series of figures (screen captures) showing the flow and output of the program are presented in Appendix A. PFRE3 code added for the current research is included in Appendix B. A linear material properties data file (MDAT2.DAT), a nonlinear material properties data file (FREDAT.DAT), and a nonlinear stress-vs.-strain output file (FREOUT.DAT) that uses input from Appendix A, are included in Appendix b Method of Solution To analyze a fiber-reinforced elastic laminate, the user must have initial linear material properties, the six Ogden coefficients for the shear and transverse stiffness, the shear failure strain, and a lay-up definition. After the material properties are entered into PFRE3, they are stored in the material data files and do not need to be re-entered. After the angleply layup is defined, a laminate ABD stiffness matrix is determined using the initial linear properties. From the laminate stiffness matrix a laminate Poisson s ratio is calculated. The user moves to the nonlinear menu and keys in the specimen width, number of calculation steps, initial length, and the increment of length that the specimen will be elongated at each calculation step. The incremental axial linear and nonlinear strains, and the total linear and nonlinear axial strains respectively, are defined by: 104

122 ε x = L L (4.48) e x = L L L L (4.49) ε x = n ( L) L (4.50) e x = n L n ( L) L L 2 (4.51) where L is the length increment, L is the initial length, and n is the number of increments calculated to that point. The transverse linear strain is calculated by the relation: ε y = ν xy ε x (4.52) where ν xy is the laminate Poisson s ratio. The same procedure is used in calculating all other incremental and total transverse strains. Because geometric nonlinearity has been removed from the material properties, linear strain is used to determine the instantaneous shear and transverse moduli. At the n th iteration the total linear strain is averaged with the (n-1) th linear strain. The resulting strain is used to obtain new stiffness values for E 2 and G 12, which are then used to recalculate the laminate stiffness (ABD) matrix. If the n th total linear strain was used to obtain the laminate stiffness matrix, the total stress is slightly over-predicted. If the strain exceeds ε xg for the material being used, the program calculates a shear stiffness based on a line tangent to the shear-strain curve at ε xg. If the predicted shear modulus is negative, a small positive value for shear stiffness is used. The total axial engineering stress at the n th iteration is predicted by: 105

123 formulation for the composite cylinder was obtained from Whitney [13], with modificaσ x r 11 e x r 12 e y ( ) n = ( σ x ) n 1 + ( A (( ) n ( e x ) n 1 ) + A (( ) n ( e y ) n 1 )) w (4.53) r r 12 A12 where A and are the recalculated inplane axial and coupling stiffness, and w is the laminate or specimen width. Since the experimental data was recorded as linear strain vs. total stress, the program outputs to the screen and to an output file, at the n th iteration, total linear strain and total stress. If bending of the laminate was expected, bending contributions could be added in a similar manner. Laminate Poisson s ratios are also predicted by the nonlinear model. The model predicted that the fiberglass-reinforced silicone and urethane specimens would exhibit extremely high Poisson s ratios of 5 to 32 from fiber angles of approximately 1 to 25. The predicted Poisson s ratios are displayed in Figure 4.6. The high Poisson s ratios provide another reason why gripping problems are noted with the low off-axis fiberglass-reinforced specimens. Poisson s ratios are functions of fiber angle, fiber stiffness, and elastomer stiffness. Further measurements of Poisson s ratios from FRE specimens should be conducted. The ability to tailor FRE laminates with extremely high Poisson s ratios, and perhaps high negative Poisson s ratios open the door for new applications such as a fastener that expands transversely when extended, or unique actuators. Discussions of general stress-strain characteristics predicted by the improved nonlinear model are presented in hapter 5 along with comparisons to measured stress-strain responses of the four fiber-reinforced elastomeric material systems. 4.5 FIBER RE-ORIENTATION AND THE RUBBER MUSLE A rubber muscle is modeled as a composite cylinder with internal pressure. Basic 106

124 35 Poisson s ratio, xy silicone/cotton silicone/glass urethane/cotton urethane/glass Off-axis angle (degrees) Figure 4.6 Predicted Poisson s ratios as a function of angle for each material system. tions made for material nonlinearity and fiber re-orientation. The model uses the nonlinear material model of PFRE3. The rubber muscle is fabricated using the same balanced angle-ply lay-up scheme as the experimental FRE specimens. Because expected strains are less than 10%, and for simplicity, nonlinear strain-displacement relations are not included in the rubber muscle model. The length and diameter of the tube are updated after each iteration, however, to provide a measure of nonlinearity. Fiber angle re-orientation is included as a function of geometry (length and diameter) changes as discussed below. The axi-symmetric composite cylinder with an angle-ply lay-up is considered to initially have a constant radius R and length l. Because of symmetry along the longitudinal 107

125 axis, the cylinder experiences only axial displacement u, radial deflection w. If the cylinder is restrained at its ends, an axial stress-resultant N x is produced instead of axial displacement. The governing equations for such a composite cylinder are: 2 u A A w + = 0 2 x R x (4.54) 2 v A 66 2 x 3 w B 16 = 0 3 x (4.55) and A R u x 3 v w i w B 16 x 3 + D 11 x 4 = p + N x x (4.56) where x represents the axial direction, and the laminate stiffnesses are as defined earlier. The laminate stiffnesses are updated after each iteration. The initial stress-resultant N x i changes as the internal pressure p o changes, and is defined as: N x i p o R 2 π p o R = = (4.57) 2πR 2 Because the cylinder is axi-symmetric the transverse or circumferential displacement v is identically zero. Therefore Equation 4.55 can be discarded, as well as the second term of Equation The circumferential strain, is not zero, however, but is a function of the out-of-plane displacement: ε s = w --- R (4.58) where s represents the circumferential direction. 108

126 At the cylinder end x = 0, the boundary conditions are: w = 0 (4.59) w x = 0 (4.60) N x = 0 u = 0, or u=0 (4.61) x At the cylinder end x = l, the boundary conditions are: w = 0 (4.62) w x = 0 (4.63) N x = 0 u = 0, or u=0 (4.64) x The rubber muscle actuator model is set up so that both initial axial force and axial displacement, u, are calculated as a function of pressure, but the force calculation assumes that axial displacement is zero, and the displacement calculation assumes that axial force is zero. To formulate the axial displacement correctly, however, an additional boundary condition was used for the axial displacement, at x = l/2: u = 0 (4.65) Because of axi-symmetry, all relations are functions of only the axial direction x. The partial differential equations become ordinary differential equations and can be solved by direct integration. Using the boundary conditions, a relation for the out-of-plane displacement w, was obtained, and used to obtain the axial displacement u. Hyperbolic and polynomial expressions were used to give a shape for the displacement w. Since both predicted shapes were 109

127 very similar, the polynomial relation was used for simplicity. Assuming a 4th-order polynomial, the out-of-plane displacement can be represented as: wx ( ) = Fl ( 2 x 2 2lx 3 + x 4 ) (4.66) where the coefficient F is an unknown coefficient. Substituting Equation 4.66 into Equation 4.54, and using the boundary conditions of 4.61 or 4.64, and 4.65, the axial displacement is: ux ( ) = A F R -- l 2 x lx A 11 + x l (4.67) The coefficient F is obtained by substituting Equations 4.66 and 4.67 into Equation 4.56, where p o is the internal pressure: F = 16p o mπx π 2 sin nm l n = 135,, m = 135,, A A 11 R 2 ( l 2 x 2 2lx 3 + x 4 p o R ) ( 2l 2 12lx + 12x 2 ) 24D 2 11 (4.68) The summations are used to represent the constant internal pressure. Knowing the displacements, the equivalent initial axial force can be determined: F x u = ( 2πR)N x = 2πRA l w + 2πRA R (4.69) This formulation assumes that the axial force is due to the radial expansion of the muscle minus the axial contractive effects. The formulation is consistent with experimental observations. Another way to understand the mechanism is to think of two people pulling tightly on the ends of a rope. If someone else pulls transversely on the middle of the rope, the two end people will be drawn together with considerable force. 110

128 After axial and radial expansions are determined, and geometry changes are calculated, fiber re-orientation can be considered. Using the instantaneous fiber angle, an effective muscle length is calculated for each unique ply: l eff = πR tanθ i (4.70) At each iteration, the incremental pressure is used to calculate an incremental change in radius and length, therefore, the new fiber angle for each ply, at each step or iteration is: 1 2πR + 2w θ i = tan u l eff + l eff -- l (4.71) At each iteration, the positive transverse strain is used to update the material properties, using the capabilities of the nonlinear material model discussed earlier. The model calculates the total cylinder contraction, and the average diameter change. At a given pressure, the contractive force will likely decrease significantly as the muscle contracts. That characteristic will be incorporated into a later version of the rubber muscle model. The current model only predicts initial contractive force. As the fiber angle of an angle-ply FRE cylinder increases, the amount of cylinder contraction decreases. After approximately 55, the cylinder begins to elongate. The current rubber muscle model does not accurately predict the force developed under such axial expansion, hence fiber angles should be kept lower than 55. The rubber muscle model was added to the code of PFRE3, the complete nonlinear fiber-reinforced elastomer and rubber muscle model is called PFRE4. The rubber muscle model is implemented as another interactive menu, similar to the nonlinear fiber-reinforced elastomer menu. The FORTRAN code for the rubber muscle model is included at 111

129 the end of Appendix B, and a screen capture showing the screen is given at the end of Appendix A. Photographs of a fabricated rubber muscle, and predictions from the rubber muscle model, using the nonlinear properties obtained from hapter 3, are presented and discussed in hapter 5. The discussion shows how contractive force and fiber re-orientation vary, based on material type, pressure, muscle length, and muscle diameter. 4.6 SUMMARY OF THE NONLINEAR MODEL An improved model for the mechanical response of fiber-reinforced elastomeric composites has been presented. The model uses classical lamination theory as its basis. Material nonlinearity is included in the form of nonlinear orthotropic material properties that are functions of linear extensional strain. Nonlinear shear and transverse properties were obtained from the experimental results of fiber-reinforced elastomeric (FRE) angle-ply specimens at 0, 45, and 90 as discussed in hapter 3. Axial stiffness and Poisson s ratio are considered constant. Geometric nonlinearity is removed from the transverse and shear stiffnesses, in order that each nonlinear contribution might be considered separately. A bi-linear model, a two-coefficient Mooney-Rivlin model, a three-coefficient Mooney- Rivlin model and a six-coefficient Ogden model were considered in attempts to model the nonlinear material properties. The highly accurate Ogden model was chosen to represent the nonlinear stiffening and softening stiffnesses. Geometric nonlinearity is included through the addition of the nonlinear extensional terms of the Lagrangian strain tensor. The nonlinear strain-displacement relations and the nonlinear material models were added to the code of a pre-existing composites analysis software package. The improved analyt- 112

130 ical tool will aid in understanding the behavior of FRE materials and will enable their use in stiffness- and deformation-tailored smart structures. A model of a rubber muscle actuator is presented. The model consists of a composite cylinder that includes the nonlinear material model and fiber rotation (fiber re-orientation). The model is useful in aiding the qualitative understanding of the mechanical behavior of inflated FRE cylinders or rubber muscles and fiber re-orientation. 113

131 114

132 HAPTER 5 OMPARISON OF PREDITED AND EXPERIMENTAL DATA 5.1 SYNOPSIS Predicting the response of fiber-reinforced elastomer or flexible composite structures is improved by the addition of material, geometric and fiber-rotation nonlinear models to classical laminated plate theory. FRE stress-strain responses predicted by the nonlinear laminated plate model of hapter 4 are compared with measured stress-strain responses of balanced angle-ply specimens with off-axis angles ranging from 0 to 90 in 15 increments. orrelation between predicted and experimental results typically ranged from good to excellent. Background information on the nonlinear model, and a review of related previous work are presented in hapter 4. The response of a rubber muscle actuator is also predicted. The rubber muscle model indicates that fiber re-orientation is a function of initial fiber angle and material type, and that very high initial contractive forces are possible. 5.2 PREDITED AND EXPERIMENTAL STRESS-STRAIN RESPONSES Predictions using PFRE3 were compared with the experimental data from hapter 3. The predicted and experimental results at 0 for the fiberglass-reinforced elastomeric specimens do not match because the test fixture did not completely load the 0 fiberglassreinforced specimens. 115

133 Predicted results are compared with average experimental results for each specimen type. The cotton-reinforced silicone rubber stress-strain results are considered first otton-reinforced Silicone Silicone rubber and cotton fibers represent the most compliant matrix and fiber components of the four combinations investigated. Predicted and average stress-strain results from the 0 to 45 balanced angle-ply specimens are shown in Figure 5.1. Specimens with fibers at 0 are the stiffest and are the left-most curve. Predicted and average stress-strain results from the 45 to 90 specimens are shown in Figure 5.2 (specimen results at 45 are repeated here to facilitate comparison between the two related graphs). orrelation between predicted and measured results are good except for 15. The predicted and measured results at 15 have approximately the same slope. At 30 and 45 the predicted results follow the same trends and stiffnesses, but are slightly low. This is expected since the initial knee in the cotton/silicone shear stiffness was not modeled by the Ogden relation. Up to approximately 50% strain, stress-strain curves for 60 to 90 specimens are very similar. This has ramifications for elastic tailoring, since at 60, a laminate has considerably more shear and transverse stiffness Fiberglass-Reinforced Silicone The fiberglass-reinforced silicone composite specimens combine a stiff fiber and a compliant matrix. The predicted and average results from the 0 to 45 off-axis angles are shown in Figure 5.3. The predicted and average results from the 45 to 90 off-axis angles are shown in Figure 5.4. orrelation is excellent except at strains higher than 25% for 60 and 75, and at 0 as noted earlier. Looking back at the Ogden model of the silicone/glass shear strain, the model diverged slightly at the shear failure strain which is approximately 116

134 Stress (MPa) s/c Predicted s/c 0 avg s/c 15 avg s/c 30 avg s/c 45 avg Stress (psi) Strain (m/m) 0 Figure 5.1 Predicted and measured cotton/silicone stress-strain behavior from [± 0 ] 2 to [± 45 ] 2. Stress (kpa) s/c Predicted s/c 45 avg s/c 60 avg s/c 75 avg s/c 90 avg Stress (psi) Strain (m/m) Figure 5.2 Predicted and measured cotton/silicone stress-strain behavior from [± 45 ] 2 to [± 90 ]

135 Stress (MPa) s/g Predicted s/g 0 avg s/g 15 avg s/g 30 avg s/g 45 avg Stress (psi) Strain (m/m) Figure 5.3 Predicted and measured fiberglass/silicone stress-strain behavior from [± 0 ] 2 to [± 45 ] 2. Stress (kpa) s/g Predicted s/g 45 avg s/g 60 avg s/g 75 avg s/g 90 avg Stress (psi) Strain (m/m) Figure 5.4 Predicted and measured fiberglass/silicone stress-strain behavior from [± 45 ] 2 to [± 90 ]

136 25%. This divergence caused the 60 and 75 results to be over-predicted at higher strains. The results at 90 are modeled better because shear has little effect at the high angle otton-reinforced Urethane Predicted and average stress-strain results from urethane/cotton specimens with 0 to 45 off-axis angles are shown in Figure 5.5. As the angle is increased, specimen stiffness and strength decrease. Figure 5.6 shows predicted and average test results for 45 through 90 specimens. The softening effect of the urethane rubber is readily apparent for 60 and greater angle-ply specimens. Similar to the silicone/cotton results, at strains less than 25%, there is little difference between the 75, the 90, and to a lesser extent, the 60 stress-strain curves, which PFRE3 accurately predicts. From 0 to 30, and 75 to 90 the predictions and average results compare very favorably. At 45 and 60 the predictions are somewhat low, but show the same trends. The reasons for this are related to the laminate Poisson s ratio used to calculate G 12, as discussed later Fiberglass-Reinforced Urethane Predicted and average stress-strain results for urethane/fiberglass specimens at 0 to 45 off-axis angles are shown in Figure 5.7. As noted earlier, the 0 experimental and predicted results do not correspond. Predicted and experimental results for 45 to 90 specimens are shown in Figure 5.8. Due to manufacturing error, the 30 specimens were actually 37, and the 60 specimens were 53. orrelation between predicted and experimental results are quite good except at 37 and 53. It s quite likely that the test data is faulty for these specimens, since correlation is very good at other angles. 119

137 Stress (MPa) u/c Predicted u/c 0 avg u/c 15 avg u/c 30 avg u/c 45 avg Stress (psi) Strain (m/m) Figure 5.5 Predicted and measured urethane/cotton stress-strain behavior from [± 0 ] 2 to [± 45 ] u/c Predicted 500 Stress (kpa) u/c 45 avg u/c 60 avg u/c 75 avg u/c 90 avg Stress (psi) Strain (m/m) Figure 5.6 Predicted and measured urethane/cotton stress-strain behavior from [± 45 ] 2 to [± 90 ]

138 Stress (MPa) u/g Predicted u/g 0 avg u/g 15 avg u/g 37 avg u/g 37 Predicted u/g 45 avg Stress (psi) Strain (m/m) Figure 5.7 Predicted and measured urethane/glass stress-strain behavior from [± 0 ] 2 to [± 45 ] 2. Stress (kpa) u/g Predicted u/g 45 avg u/g 53 avg u/g 53 Predicted u/g 75 avg u/g 90 avg Stress (psi) Strain (m/m) 0 Figure 5.8 Predicted and measured urethane/glass stress-strain behavior from [± 45 ] 2 to [± 90 ]

139 5.3 DISUSSION OF PREDITED RESULTS orrelation between experimental and predicted results at low and high off-axis angles were typically very good. This indicates that the model predicts fiber- and matrix-dominated stress-strain response very well. The predicted results at 45 and 60 followed the same trends as experimental results, and except for the silicone/glass predictions (which were very close) were slightly low. This would indicate that the shear-dominated response of the specimens is not being modeled sufficiently. The first attempt to rectify this apparent concern was to use hou s approximations (Equations ) for transformed layer stiffnesses. Unfortunately, correlations with these approximations produced worse results than those presented. Other modified forms of the transformed layer stiffnesses were developed and considered. These also produced inconsistent results. Laminate Poisson s ratios at 45 were predicted for each material system, using initial properties and classical lamination theory, and compared with measured values. They are given in Table 5.1. lassical lamination theory always predicts a 45 laminate Poisson s ratio less than those measured. In FRE material systems such as those currently considered where the fiber stiffness is several orders of magnitude higher than the matrix stiffness, the Poisson s ratio is predicted to be approximately 1, but never greater. Since excellent correlation was shown for the silicone/fiberglass material system, which used a calculated Poisson s ratio, one of several conclusions can be made: 1) the laminate Poisson s ratios need to be re-measured under more tightly controlled conditions; 2) there are additional mechanisms affecting the response of shear-dominated FRE specimens, which are not currently included in the nonlinear model; or 3) the Poisson s ratio is not constant and must be measured in a manner similar to the other nonlinear properties. It is highly 122

140 likely that all three factors contribute to the discrepancy, and should be explored in future work. For the present a better correlation can be obtained by using measured nonlinear shear stiffnesses that employ a calculated 45 laminate Poisson s ratio. TABLE 5.1 Measured and predicted Poisson s ratios (ν xy ) for each material system Material System Predicted Poisson s ratios, (ν xy ) Measured Poisson s ratios, (ν xy ) otton/urethane otton/silicone Fiberglass/Urethane Fiberglass/Silicone * * alculated, based on ν xy 30 measurement. The nonlinear trends of the experimental results are predicted quite well by the current model, even in shear dominated regions. Inconsistent correlations at a few instances show no trend and are related to Poisson s ratio and fabrication issues rather than modeling concerns. 5.4 PREDITIONS FROM THE RUBBER MUSLE MODEL An inflated rubber muscle that was fabricated as part of the current work is shown in Figure 5.9. When inflated with air, the rubber muscle actuator exhibits considerable initial contractive force. The fabricated rubber muscle use small Kevlar TM tows oriented at a ±20 lay-up in a urethane matrix. A simple model of the rubber muscle was developed and is presented in hapter 4. The rubber muscle model is an addition to, and uses the nonlinear fiber-reinforced elastomer model discussed in hapter 4. The rubber muscle model is still somewhat rudimentary, and is intended to provide qualitative answers, how- 123

141 Figure 5.9 Example of an inflated rubber muscle fabricated by Peel. ever the magnitudes of the initial forces, angle re-orientation, and displacements are consistent with the response of the fabricated muscle. Direct comparison of the contractive response of the fabricated rubber muscle with predicted results has proven difficult because material properties for the fabricated rubber muscle have not been obtained and the extremely high Poisson s ratios of the inflated muscle cause the metal fittings on the ends of the muscle to pull out when clamped in a test fixture. Using an angle-ply lay-up, [±θ] 2, with an initial fiber angle of 15, a ply thickness of mm (0.025 in), an initial diameter of 12.7 mm (0.5 in), and an initial muscle (actuator) length of 254 mm (10 in), results are presented for the four sets of material properties obtained in hapter 3. Figure 5.10 illustrates initial contractive force versus pressure for each material system. As shown in Figure 5.10, initial contractive force is very independent of material type, for the same pressures and geometry. Fiber angle re-orientation as a function of pressure for each material system is shown in Figure The fiber angle changes are very much a function of material properties. 124

142 Force (N) Pressure (psi) Silicone/otton 8000 Silicone/Fiberglass 7000 Urethane/otton 6000 Urethane/Fiberglass Pressure (kpa) Force (lbs) Figure 5.10 Predicted initial contractive muscle force versus pressure for each material system with a [± 15 ] 2 lay-up. The greatest fiber angle changes are evident where axial stiffness is lower. The tendency of the muscle diameter to increase, and the muscle length to decrease, since they are also measures of geometry, follow the same trends as fiber angle re-orientation. Varying the rubber muscle wall thickness has essentially no effect on initial contractive force, but did affect geometry changes such as length, diameter, and fiber angle. Increasing muscle length and / or muscle diameter increases contractive force and fiber reorientation, because of the effective increase in surface area. Evidence of nonlinearity is more prominent for longer muscle lengths, larger diameters and lower initial fiber angles. hanging initial fiber angle has a very large effect on initial contractive force, as shown in Figure 5.12, and on the amount of fiber re-orientation, shown in Figure hanging initial fiber angle is essentially the same as changing axial stiffness so the 125

143 Pressure (psi) Fiber Angle (degrees) Silicone/otton Silicone/Fiberglass Urethane/otton Urethane/Fiberglass Fiber Angle (degrees) Pressure (kpa) 0 Figure 5.11 Predicted fiber angle change as a function of pressure for each material system with a [± 15 ] 2 lay-up. results shown in Figure 5.13 are consistent with Figure The extremely large fiber angle re-orientation for the lower initial fiber angles is possible because the lower transverse stiffnesses at lower fiber angles cause the muscle to bulge out more. To understand the predicted results better, let us go back to the analogy of two people pulling on the ends of a rope, as discussed in hapter 4. Assuming the rope is strong enough that it won t break, if one were to pull transversely on the rope, the amount of axial contractive force induced is not a strong function of what the rope is made of, or its diameter. If one is to increase the length of the rope, however, the same transverse force will produce a higher axial force, but decrease axial displacement. If the rope is somewhat compliant, it s possible to see how the same force would be transmitted as in the stiff 126

144 Pressure (psi) ontractive Force (kn) s/c ang=10 s/c ang=15 s/c ang=20 s/c ang=30 s/c ang= ontractive Force (lbs) Pressure (kpa) Figure 5.12 Predicted initial contractive force as a function of pressure for different initial fiber angles. Pressure (psi) Fiber Angle (degrees) s/c ang=10 s/c ang=15 s/c ang=20 s/c ang=30 s/c ang= Fiber Angle (degrees) Pressure (kpa) 0 Figure 5.13 Predicted fiber angle change as a function of pressure for different initial fiber angles. 127

145 rope, but transverse deflections would be greater. If the rubber muscle is used as an actuator, the capabilities of an improved rubber muscle model will be useful in tailoring the force and deflection response. 5.5 HAPTER SUMMARY Stress-strain predictions from the nonlinear laminated plate model are compared with the measured stress-strain response of balanced angle-ply specimens with off-axis angles ranging from 0 to 90 in 15 increments. orrelation between predicted and test results range from fair to excellent. Fiber and matrix dominated stress-strain responses are modeled very well. The model also gives fair to good comparisons for shear dominated modes, but more investigation needs to be made into laminate Poisson s ratios. There are also a few instances where the test data appears to be flawed, hence correlation was inconclusive. omparison of predicted with mechanical responses provides additional insight into FRE response mechanisms, since one is typically able to vary parameters in a model that are not easily reproduced by experiment. Predictions from the rubber muscle model reveal significant insights about the mechanical behavior of FRE muscles or flexible composite cylinders. The initial contractive force is essentially independent of material type, but very dependent on pressure, initial fiber angle, diameter, and length. Fiber re-orientation is a function of geometry, material type, and pressure. A refined rubber muscle model will be useful in the design of rubber muscles, and will aid their use as actuators in flexible and smart structures. 128

146 HAPTER 6 LOSURE AND REOMMENDATIONS FOR FUTURE WORK 6.1 GENERAL OMMENTS Broad exploratory research into the response and analysis of fiber-reinforced elastomeric composites is the intent of the current work. Although specific niches of fiber-reinforced elastomers (FRE) have been investigated in the past, considerably more research was needed in order to fully exploit their unique characteristics. The work presented in this dissertation fills some of the gaps. Research is presented from three major areas of emphasis in order to provide solid and balanced understanding of FRE materials. These areas are fabrication and processing, experimental results, and theoretical modeling. The successful use of FRE composites, like typical advanced composites, require a solid background in all three areas in order to take advantages of specific characteristics. The current research should be considered a plateau that enables FRE researchers to see the peaks or rewards of future work, and regions where more research is needed. For example, processing and fabrication issues caused perhaps the most work for the current research. As the research has matured it became very evident that considerable more experimental data, and different types of experiments have become the most pressing needs. The nonlinear model presented in hapter 4 is simpler than initially expected, but 129

147 works quite well in most cases as shown in hapter 5. Improvements to the model could be made, based on additional test data. In the following sections, conclusions are made for each area, and suggestions are made for future work. 6.2 PROESSING AND FABRIATION ONLUSIONS In the rubber industry, most emphasis in process research is placed on chemical properties and cost. Mechanical properties are just one of many factors to consider, including chemical compatibility, thermal characteristics, UV degradation, and so on. The approach for the presented research, however, has been to use the knowledge of the rubber industry and to primarily consider mechanical properties. The successful small-batch prepreg fabrication process combines common and aerospace-type fibers with an elastomeric matrix, and uses processes similar to those used for aerospace composites. Basic chemistry issues are addressed when fiber-to-rubber adhesion concerns are resolved. The method developed is based on aerospace composites methodologies and is a non-calendering method for fabricating good quality, medium to high fiber volume fraction fiber-reinforced elastomer (FRE) prepreg. The prepreg is used to laminate specimens and FRE applications. The manufacturing quality was verified by prepreg uniformity, increased fiber volume fractions, and consistent experimental results from fabricated specimens. The fabrication method uses a combination of filament winding, standard lamination techniques, autoclave curing, and a knowledge of elastomer cure parameters to produce good quality prepreg. Fiber-to-elastomer adhesion was accomplished by a careful choice of fibers and resins, selection of autoclave cure cycle parameters, and application of a primer on the fiberglass to aid adhesion to the silicone rubber. Some of the adhesion con- 130

148 cerns were only addressed after several trial and error iterations, and should not be underestimated. Fiber parallelism and straightness were aided by the use of circumferential windings on a filament winder. Rough sensitivity studies of processing parameters such as autoclave pressure, vacuum pressure, cure stage of the elastomer matrix, and elastomer stiffness indicate that increased autoclave pressure and higher vacuum produce fewer voids, thinner prepreg, and better fiber adhesion. ure-stage parameters are discussed under Bond-line Strength, below. Fiber volume fractions can be increased by decreasing winder head advance as a function of mandrel speed, and increasing tow tension. Fiber volume fraction is also a direct function of the amount of elastomer applied, and an inverse function of the pore size of the teflon-coated peel-ply. The present approach allows a researcher with a working knowledge of advanced composites fabrication skills and common composites fabrication equipment to fabricate small batches of good quality FRE prepreg and applications. 6.3 FUTURE PROESSING AND FABRIATION Increased Fiber Volume Fractions: Adequate cotton fiber volume fractions were obtained, but fiberglass fiber volume fractions were lower than expected. The higher fiberglass fiber volume fractions can easily be obtained by increasing tow tension, which decreases tow spreading, allowing overlap of adjacent tows by decreasing the filament winder head advance relative to mandrel rotation, and carefully controlling the amount of elastomer resin applied. Lamination Preparation: Because some angle-ply specimens failed by scissoring (shear) along lamina bond-lines, prepreg layers should be roughened before lamination, thus increasing bond area and ensuring clean surfaces. A few of the prepreg layers moved 131

149 during cure due to the fluid nature of the rubber glue between layers. A fixture should be fabricated that would hold the layers in precise orientation during cure, or perhaps the viscosity of the rubber adhesive could be altered. Increased Bond-line Strength: Bond-line strength could be increased by reducing the autoclave cure time of the filament wound prepreg for the urethane composites; and increasing cure cycle times for the silicone prepreg. Inadequate mixing or incomplete curing of the silicone rubber may prevent total polymerization of the rubber constituents. The oils or constituents left can actually hinder rubber adhesion; hence, new-to-old silicone rubber adhesion is best when the old rubber is fully cured. New-to-old urethane rubber adhesion, on the other hand, is best when the old rubber is not fully cured. Waterjet cutting: The water-jet process should be the preferred procedure for cutting the whole dog-bone specimen from a laminate, rather than cutting out just the dog-boned areas. General: Although thickness variations of the prepregs and laminates were acceptable, and comparable to typical advanced composite laminates, further improvement is still possible. Thickness variations could be further reduced by using thicker caul plates, using higher autoclave pressures, and metering the elastomer resin onto the mandrel. For high-volume production of FRE prepreg, a process similar to advanced composites prepreg fabrication could be used, with the additional challenges of finding suitable elastomers that have extremely low viscosities and can be stored in a B-staged or partially cured state for weeks or months. 132

150 6.4 ONLUSIONS FROM EXPERIMENTAL WORK hapter 3 presents wide-ranging tensile test results from elastomer matrices, cotton fibers, and four sets of fiber-reinforced elastomers. Necessity dictated developing procedures for obtaining accurate strain data at high elongations from test machine head displacement. The successful strain calibration procedure was used to obtain the elastomer, fiber, and fiber-reinforced elastomer experimental results. Because measured 0 cotton-reinforced elastomer stiffnesses were higher than expected, pairs of individual rubber-impregnated fibers were tested. The average extensional stiffness of individual cotton fibers increased 74% to 128% when impregnated with an elastomer. The unexpected increase is due to additional shear resistance between individual twisted cotton strands and added constraint against radial contraction of the cotton fibers. onstraint is provided by the impregnated rubber during axial extension. As expected, the nonlinear stiffening or softening trends of the silicone and urethane rubbers are reflected in their respective fiber-reinforced elastomer composite specimens. FRE stiffnesses can be orders of magnitudes higher than elastomer stiffnesses. Axial stiffness at low off-axis angles is a function of reinforcement stiffness. FRE shear stiffnesses are strong functions of elastomer and fiber shear properties and weaker functions of fiber volume fractions. At high off-axis angles, laminate stiffness and strength are complex functions of fiber stiffness, fiber volume fraction, and elastomer stiffness. Several orders of magnitude difference in fiber and rubber properties necessitate careful fabrication of low-angle FRE specimens in order to avoid suspect experimental results. Initial orthotropic material properties obtained for each material system were compared with predictions made using constituent properties. orrelation was reasonable for 133

151 the cotton-reinforced elastomer composites but could be improved for the fiberglass-reinforced elastomer composites. The nonlinear orthotropic material properties extracted from the test data were needed to accurately predict the mechanical response of fiber-reinforced elastomer composite structures and will improve understanding of the fundamental response mechanisms of such FRE materials. 6.5 FUTURE EXPERIMENTAL WORK Extremely High Elongation Strain Gages: Researchers in Japan created a crude high elongation strain gage. That strain gage or another should be developed that would enable cheap, direct, and accurate collection of strains up to 200%. The current method is workable and adequate for the tests undertaken, but future, more sophisticated tests will require increased accuracy. The use of small strain gages on a specimen will also enable the collection of axial and transverse strain simultaneously. Poisson s Ratios: New experimental results that would be most beneficial to the current work are the examination of Poisson s ratios as a function of angle. Experimentation to determine if Poisson s ratios are nonlinear (a function of extensional strain) should also be undertaken. The nonlinearity may also be a function of angle, and would be a function of elastomer type. Twisted Fibers: Twisted fibers are used as tire-cords in cord-rubber composites to enhance three-dimensional and dynamic characteristics. Tire researchers also report increases in axial stiffness, like those found with the impregnated cotton fibers, but indicate that increases are on the order of one or two percent. Experimental data obtained from several types of rubber-impregnated twisted fibers and wires would be useful to the rope and cabling industries, and could open up new reinforcing mechanisms. 134

152 Moderate Modulus Elastomers: The elastomer matrices used in the current work have very low moduli of elasticity, and are suitable for certain flexible applications. Many applications, however, such as many aerospace applications, may not require as much deformation. Experimental data from aerospace fibers combined with moderate modulus elastomers would fill a needed gap and would be useful where moderate elongation is desired. Uniaxial FRE Specimens and Fixtures: More accurate stress-strain results are needed for uniaxial fiber-reinforced elastomers. A fixture should be developed that will successfully hold such specimens, and similarly-constructed FRE components. ompressive Data: Because the current tests were conducted only in tension, compressive experiments would aid in understanding the role of the reinforcing fiber in the compressive response. Typical raw rubbers show a more compliant response when compressed. 6.6 NONLINEAR MODELING ONLUSIONS A number of different researchers have made nonlinear additions to classical laminated plate theory. The current nonlinear model is not intended to be all-inclusive, but does model expected responses. The current nonlinear classical laminated plate model is unique in that contributions to overall nonlinearity are carefully delineated. A researcher that is familiar with the analysis of conventional composites should be able to understand and use the nonlinear additions. Traditionally, the prediction of rubber-based materials, such as cord-rubber composites has been considered good if the difference between modeled and experimental results were less than 20%. The nonlinear trends of the experimental results are predicted very 135

153 well by the current model. Inconclusive correlation for a few specimens exhibits no trend and is related to specimen defects and extreme Poisson s ratios rather than modeling concerns. Accurately predicting the response of fiber-reinforced elastomer or flexible composite structures was improved over cord-rubber composite models by the careful addition of robust material, geometric, and fiber-rotation nonlinear models to classical laminated plate theory. Material nonlinearity is included in the form of nonlinear orthotropic material properties as functions of linear extensional strain. The procedure of removing geometric nonlinearity from the nonlinear material properties is somewhat unconventional. It enables the user to include layers in the analysis that have different material types, and allows the use of linear properties for specific layers. Geometric nonlinearity is also removed from the transverse and shear stiffnesses to allow each nonlinear contribution to be considered separately. The six-coefficient Ogden model was chosen as the best compromise to represent the nonlinear stiffening and softening characteristics. Geometric nonlinearity is included by the addition of the nonlinear extensional terms of the Lagrangian strain-displacement tensor. The nonlinear strain-displacement relations and the nonlinear material models were added to the FORTRAN code of an interactive DOS-based composites analysis software package. orrelation between modeled and test results ranged from fair to excellent with a few exceptions where the test data appears to be inaccurate: Fiber and matrix dominated stress-strain responses were modeled very well. The model also gave fair to excellent comparisons for shear dominated modes, but more investigation needs to be made into laminate Poisson s ratios. 136

154 lassical lamination theory always predicts a 45 laminate Poisson s ratio less than those measured. For FRE material systems such as those currently considered where the fiber stiffness is several orders of magnitude higher than the matrix stiffness, the predicted Poisson s ratio will be approximately 1, but never greater. The measured Poisson s ratios, however, were all greater than 1. Since excellent shear correlation was shown for the silicone/fiberglass material system, which used a calculated Poisson s ratio, better correlation can be obtained by using measured nonlinear shear stiffnesses that employ a calculated 45 laminate Poisson s ratio. Predictions from the rubber muscle model reveal meaningful insights into the mechanical behavior of the FRE muscles / composite cylinders. ontractive force is independent of material type, but very dependent on pressure, initial fiber angle, diameter, and length. Fiber re-orientation is a function of geometry, material type, and pressure. 6.7 FUTURE NONLINEAR MODEL ENHANEMENTS Incorporation of the Nonlinear Model into Models of FRE Structures: The current model will accurately predict the response of many FRE-based structures, if accurate material properties are known, and fiber-reorientation is properly defined. The current model is only set up to analyze flat plates and tubes. To make the model more useful, other geometric capabilities should be added. Nonlinear Poisson s Ratio: Based on suggested future experimental data, the accuracy of the nonlinear model might be enhanced by including a model for a nonlinear Poisson s ratio. Initial Material Effects: The Ogden relation modeled properties better than any of the other models, but doesn t always include minor secondary trends such as small initial 137

155 knees in the data. These smaller effects might be included by adding more terms to the Ogden model, or using separate material models for the shear and transverse properties. Viscoelastic Model: The inclusion of viscoelastic effects on a material s properties would enable the nonlinear model to more accurately predict the response of highly dynamic FRE structures. ombined ompressive and Tensile Response: The current model should be able to model a FRE structure that is completely in compression, if the proper material properties are used. The prediction of combined compressive and tensile modes, however, would enable a more accurate analysis of complex structures in bending. 138

156 REFERENES References Found in hapter 1 [1] Kikuchi, N., Kota, S., Novel ompliant Mechanisms ould Simplify, Improve Aircraft, (University of Michigan) aftrh04.htm, AFOSR research highlights, (Oct. 1998). [2] Lee, B.L., Medzorian, J.P., Hippo, P.K., Liu, D.S., Ulrich, P.., Fatigue Lifetime Prediction of Angle-Plied Fiber-Reinforced Elastomer omposites as Pneumatic Tire Materials, Advances in Fatigue Lifetime Predictive Techniques: Second Volume, ASTM STP 1211, American Society for Testing and Materials, Philadelphia, , (1993). [3] lark, S.K., omposite Theory Applied to Elastomers, NASA R , N , (Sept. 1986). [4] Woo S. L-Y., Johnson, G.A., Livesay, Rajagopal, A Single Integral Finite Strain Viscoelastic Model of Ligaments and Tendons, Journal of Biomechanical Engineering, , (May 1996). [5] Kuo,.-M., Takahashi, K., hou T.-W., Effect of Fiber Waviness on the Nonlinear Elastic Behavior of Flexible omposites, Journal of omposite Materials, 22, , (Nov. 1988). [6] Luo, Shen-Yi, hou, Tsu-Wei, Finite Deformation and Nonlinear Elastic Behavior of Flexible omposites, ASME Winter Annual Meeting, hicago IL, (1988). [7] Luo, Shen-Yi, hou, Tsu-Wei, Modeling of Nonlinear Elastic Behavior of Elastomeric Flexible omposites Under Finite Deformation, hapter 2 in omposite Applications, ed. by Vigo and Kinzig, VH Publishers Inc., 31-50, (1992). [8] Philpot, R.J., Buckmiller, D.K., Barber, R.T., Filament winding of Thermoplastic fibers with an Elastomeric Resin Matrix, SAMPE Journal, 25 (5), 9-13, (Sept.- Oct. 1989). [9] Epstein, M. and Shishoo, R.L., Fabrication Methods for Latex-based Elastomer omposites Reinforced with Long Discontinuous Fibers, Journal of Applied Polymer Science, 44, , (1992). [10] Ibarra, L., hamorro,. Short Fiber-Elastomer omposites. Effects of Matrix and Fiber Level on Swelling and Mechanical and Dynamic Properties, Journal of Applied Polymer Science, 43, , (1991). 139

157 [11] J. Krey, K. Friedrich, Variably Flexible Aramid Fibre omposites with Elastomeric Matrices, Plastics and Rubber Processing and Applications, 11 (2), (1989). [12] Shonaike, G.O., Matsuo, T., Fabrication and Mechanical Properties of Glass Fibre Reinforced Thermoplastic Elastomer omposite, omposite Structures, 32 (1-4), , (1995). [13] Gabrys,.W., Bakis,.E., Filament Wound Elastomeric Matrix omposites for Flywheel Energy Storage Systems, Manufacturing Processes for Polymeric omposites, , (1996). [14] R.V. Harrowell, Elastomer Flywheel Energy Store, International Journal of Mechanical Science, 36 (2), , (1994). [15] Sharpless, G.., Brown, G.J., urved, Inflated, Tubular Beam, U.S. Patent , (June 6, 1995). [16] Vasiliev, V.V., Mechanics of omposites Structures, Taylor and Francis, Washington D.., (1993). [17] Tanaka, Y. Study of Artificial Rubber Muscle, Mechatronics, 3 (1), 59-75, (1993). [18] Tanaka, Y. Gofuku, A. Fujino, Y. Development of a Tactile Sensing Flexible Actuator, Advanced Motion ontrol 96, Mie University, Tsu-ity, Japan, [19] Dohta, Shujiro Kameda, Masakazu Matsushita, Hisashi, Study on a Pneumatic Rubber Hand with Flexible Strain Sensors, Fifth Triennial International Symposium on Fluid ontrol, Measurement and Visualization, Hayama, Japan, , (Sept., 1997). [20] Suzumori, K., Iikura, S., Tanaka, H., Applying a Flexible Microactuator to Robotic Mechanisms, 1992 IEEE ontrol Systems, 21-27, (Feb. 1992). [21] Suzumori, K., Asaad, S. A Novel Pneumatic Rubber Actuator for Mobile Robot Bases IEEE/RSJ International onference on Intelligent Robots and Systems, 2, , (Nov. 1996). [22] Suzumori, K. Elastic Materials Producing ompliant Robots, Robots and Autonomous Systems, 18, , (1996). [23] Suzumori, K., Kondo, F., Tanaka, H., Micro-Walking Robot Driven by Flexible Microactuator, Journal of Robotics and Mechatronics, 5 (6), , (1993). [24] Suzumori, K., Koga, A., Kondo, F., Haneda, R., Integrated Flexible Microactuator Systems, Robotica, ambridge University Press, , (1996). [25] Suzumori, K., Abe, A., Applying Flexible Microactuators to Pipeline Inspection Robots, Transactions of the IMAS/SIE International Symposium on Robotics, Mechatronics and Manufacturing Systems, Kobe Japan, , (Sept. 1992). [26] Tanaka, H., Suzumori, K., Iikura, S., Flexible Microactuator for Miniature Robots, IEEE MEMS 91, Nara, Japan, , (February 1991). [27] Tanaka, Y., Gofuku, A., Development and Analysis of an ERF Pressure ontrol Valve, Mechatronics, 7 (4), , (1997). 140

158 [28] McKibben muscles, index.html, (1997) [29] AFAS ROBOT SYSTEM, Bridgestone, rubbertuator manuals and documentation. in Japanese, (1989) [30] Peel, L.D. and Jensen, D.W., Fiber-Reinforced Elastomers - Flexible omposites in Japan, Asian Technical Information Program ATIP , (1998). References Found in hapter 2 [1] Suzumori, K., Elastic Materials Producing ompliant Robots, Robots and Autonomous Systems, 18, 135, (1996). [2] Suzumori, K. and Abe, A., Applying Flexible Micro-Actuators to Pipeline Inspection Robots, Transactions of the IMAS/SIE International Symposium on Robotics, Mechatronics and Manufacturing Systems, Kobe, Japan, 515, (1992). [3] Suzumori, K., Iikura, S., Tanaka, H., Applying a Flexible Microactuator to Robotic Mechanisms, 1992 IEEE ontrol Systems, 21, (1992) [4] Suzumori, K., Asaad, S. A Novel Pneumatic Rubber Actuator for Mobile Robot Bases IEEE/RSJ International onference on Intelligent Robots and Systems, 2, 1001, (1996). [5] Dohta, Shujiro Kameda, Masakazu Matsushita, Hisashi, Study on a Pneumatic Rubber Hand with Flexible Strain Sensors, Fifth Triennial International Symposium on Fluid ontrol, Measurement and Visualization, Hayama, Japan, 509, (1997). [6] Peel, L.D. and Jensen, D.W., Fiber-Reinforced Elastomers - Flexible omposites in Japan, Asian Technical Information Program ATIP , (1998). [7] Krey, J. and Friedrich, K., Variably Flexible Aramid Fibre omposites with Elastomeric Matrices, Plastics and Rubber Processing and Applications, 11 (2), (1989). [8] Shonaike, G.O. and Matsuo, T., Fabrication and Mechanical Properties of Glass Fibre Reinforced Thermoplastic Elastomer omposite, omposite Structures, 32 (1-4), 445, (1995). [9] Kuo,.-M., Takahashi, K. and hou T.-W., Effect of Fiber Waviness on the Nonlinear Elastic Behavior of Flexible omposites, Journal of omposite Materials, 22, 1004, (1988). [10] Luo, Shen-Yi, hou and Tsu-Wei, Finite Deformation and Nonlinear Elastic Behavior of Flexible omposites, ASME Winter Annual Meeting, hicago IL, (1988). [11] Philpot, R.J., Buckmiller, D.K. and Barber, R.T., Filament Winding of Thermoplastic Fibers With an Elastomeric Resin Matrix, SAMPE Journal, 25 (5), 9, (1989). 141

159 [12] Peel, L.D. and Jensen, D.W., On the Fabrication of Fiber-Reinforced Elastomers, Fifth International onference on omposites Engineering, Las Vegas, Nevada, (1998). [13] Peel, L.D. and Jensen, D.W., The Response of Fiber-Reinforced Elastomers Under Simple Tension, enter for Advanced Structural omposites, Brigham Young University, submitted to the Journal of omposite Materials, (1998). References Found in hapter 3 [1] Peel, L.D., Jensen, D.W. and Suzumori, K., Batch Fabrication of Fiber-Reinforced Elastomer Prepreg, SAMPE Journal of Advanced Materials, 30-3, (July - Sept. 1998). [2] Peel, L.D. and Jensen, D.W., On the Fabrication of Fiber-Reinforced Elastomers, Fifth International onference on omposites Engineering, Las Vegas, Nevada, (1998). [3] Krey, J. and Friedrich, K., Variably Flexible Aramid Fibre omposites with Elastomeric Matrices, Plastics and Rubber Processing and Applications, 11-2,(1989). [4] Kuo, -M., Takahashi, K. and hou T-W., Effect of Fiber Waviness on the Nonlinear Elastic Behavior of Flexible omposites, Journal of omposite Materials, 22, , (1988). [5] Luo, S-Y., hou and T-W., Finite Deformation and Nonlinear Elastic Behavior of Flexible omposites, ASME Winter Annual Meeting, hicago IL, (1988). [6] lark, S.K., omposite Theory Applied to Elastomers, NASA R , N , (1987). [7] Properties of Fiberglass Yarn, Filament Winding Rovings, fiberglass/filament.html, (1998). [8] Wake, W., Wootton, D., Textile Reinforcement of Elastomers, Applied Science Publishers LTD, London, (1982). [9] Hamed, G.R., Rubber hemistry and Technology, Development of Material onstants for Nonlinear Finite-Element Analysis, 61, [10] Bauman, J.T., A Theory of the Elastomer Stress-Strain urve, Fall Technical Mtg. of the Rubber Division, American hemical Society, Fig. 9, (Sept Oct. 2, 1998). [11] Brown, R., Physical Testing of Rubber, hapman and Hill, Shawbury UK, 51-53, (1996). [12] iba Specialty hemicals, Ren-co-thane Elastomer Selection Guide, KR Anderson o, (1997). [13] Hyer, M.W., Stress Analysis of Fiber-Reinforced omposite Materials, WB/ McGraw-Hill, , (1997). 142

160 [14] hou, T-W., Microstructural Design of Fiber omposites, ambridge University Press, 448, (1992). [15] Peel, L.D., ompression Failure of Angle-Ply Laminates, Masters Thesis, Virginia Polytechnic Institute and State University, Blacksburg VA, 43-44, (1991). [16] Bogdanovich, A.E., and Pastore,.M., Mechanics of Textile and Laminated omposites, hapmand & Hall, , (1996). References Found in hapter 4 [1] Lee, B.L., Medzorian, J.P., Hippo, P.K., Liu, D.S., Ulrich, P.., Fatigue Lifetime Prediction of Angle-Plied Fiber-Reinforced Elastomer omposites as Pneumatic Tire Materials, Advances in Fatigue Lifetime Predictive Techniques: Second Volume, ASTM STP 1211, American Society for Testing and Materials, Philadelphia, , (1993). [2] Lee, B.L., Smith, J.A., et. al., Fracture Behavior of Fiber-Reinforced Elastomer omposites Under Fatigue Loading, Proceedings of the Ninth International onference on omposite Materials (IM/9), Madrid Spain, (July 1993). [3] Bert,.W., Models for Fibrous omposites with Different Properties in Tension and ompression, Journal of Engineerig Materials and Technology, , (Oct. 1977). [4] lark, S.K., omposite Theory Applied to Elastomers, NASA R , N , (1987). [5] Woo S. L-Y., Johnson, G.A., Livesay, Rajagopal, A Single Integral Finite Strain Viscoelastic Model of Ligaments and Tendons, Journal of Biomechanical Engineering, , (May 1996). [6] Kuo, -M., Takahashi, K. and hou T-W., Effect of Fiber Waviness on the Nonlinear Elastic Behavior of Flexible omposites, Journal of omposite Materials, 22, , (Nov. 1988). [7] Luo, S-Y., hou and T-W., Finite Deformation and Nonlinear Elastic Behavior of Flexible omposites, ASME Winter Annual Meeting, hicago IL, (Nov. 27-Dec. 2, 1988). [8] hou, T-W., Microstructural Design of Fiber omposites, ambridge University Press, 448, (1992). [9] Derstine, M.S., Brown, R.T., Pastore,., rane, R. and Singletary, J., Modeling the Mechanical Behavior of a Urethane Matrix omposite, Fifth International onference on omposites Engineering, Las Vegas, Nevada, (July 1998). [10] Hamed, G.R., Rubber hemistry and Technology, Development of Material onstants for Nonlinear Finite-Element Analysis, 61, [11] Hyer, M.W., Stress Analysis of Fiber-Reinforced omposite Materials, WB/ McGraw-Hill, , (1997). 143

161 [12] Brown, R., Physical Testing of Rubber, hapman and Hall, London, 3rd ed., 92-96, (1996). [13] Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing o., , (1987). 144

162 APPENDIX A PFRE3 SREEN OUTPUT AND FLOW 145

163 146

164 ***** this screen picture is for reference purpose only, the actual values from the program may be different from what is presented here. 147