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Technical Paper NONLINEAR FINITE ELEMENT ANALYSIS OF ELASTOMERS Technical Paper P REFACE MSC.Software Corporation, the Appendices describe the physics and worldwide leader in rubber analysis, mechanical properties of rubber, proper would like to share some of our modeling of incompressibility in rubber experiences and expertise in analyzing FEA, and most importantly, testing elastomers with you. methods for determination of material This White Paper introduces you to the properties. Simulation issues and useful nonlinear finite element analysis (FEA) hints are found throughout the text and of rubber-like polymers generally grouped in the Case Studies. under the name “elastomers.” You may Rubber FEA is an extensive subject, which have a nonlinear rubber problem—and involves rubber chemistry, manufacturing not even know it… processes, material characterization, finite element theory, and the latest advances in computational mechanics. A selected list The Paper is primarily intended for two of Suggestions for Further Reading is types of readers: included. These references cite some of ENGINEERING MANAGERS who are the most recent research on FEA of involved in manufacturing of elastomeric elastomers and demonstrate practical components, but do not currently possess applications. They are categorized by nonlinear FEA tools, or who may have an subject for the reader’s convenience. educational/professional background other than mechanical engineering. DESIGN ENGINEERS who are perhaps familiar with linear, or even nonlinear, FEA concepts, but would like to know more about analyzing elastomers. It is assumed that the reader is familiar with basic principles in strength of materials theory. This White Paper is a complement to MSC.Software’s White Paper on Nonlinear FEA, which introduces the reader to nonlinear FEA methods and also includes metal forming applications. The contents of this White Paper are intentionally organized for the convenience of these two kinds of readers. For an “Engineering Manager,” topics of interest include an Executive Summary to get an overview of the subject, the Case Studies to see some real-world rubber FEA applications, and any other industry specific topics. The “Design Engineer,” on the other hand, can examine the significant features on analysis of elastomers (which constitute the bulk of the Paper). The 1 T ABLE O F C ONTENTS CASE STUDIES PAGE 1 Executive Summary 3 2 Material Behavior 6 2.1. Time-Independent Nonlinear Elasticity 7 2.2. Viscoelasticity 10 2.3. Composites 13 2.4. Hysteresis 15 2.5. Stability 17 2.6. Other Polymeric Materials 18 3 Automatic Determination of Material Parameters from Test Data 21 4 Damage and Failure 24 5 Dynamics, Vibrations, and Acoustics 25 6 Automated Contact Analysis Techniques 29 7 Automated Solution Strategies 32 8 Automated Remeshing and Rezoning 34 9 Current Trends and Future Research 36 10 User Conveniences and Services 37 11 Conclusion 39 CASE STUDIES A O-Ring Under Compression 9 B Car Tire 14 C Constant-Velocity Rubber Boot Compression and Bending 20 D Rubber Mount 28 E Car Door Seal: Automated Multibody Contact 31 APPENDICES A Physics of Rubber 40 B Numerical Treatment of Incompressibility 42 C Material Testing Methods 44 D Answers to Commonly Asked Questions in Rubber Product Design 56 SUGGESTIONS FOR FURTHER READING 58 ABOUT MSC.SOFTWARE CORPORATION 61 LOCATIONS 61 On the Cover The cover shows a photorealistic rendition of the strain energy density on the deformed configuration of a rubber constant velocity automotive boot. One can observe the hidden portion of the bellows in the reflections on the visible surfaces. 2 S ECTIONEXECUTIVE1 E XECUTIVE S UMMARY This white paper discusses Rubber is a very unique material. For certain foam rubber materials, the assumption the salient features During processing and shaping, it of near incompressibility is relaxed, since large regarding the mechanics behaves mostly like a highly volume change can be achieved by the application and finite element viscous fluid. After its polymer of relatively moderate stresses. analysis (FEA) of chains have been crosslinked by elastomers. Although the vulcanization (or by curing), main focus of the paper is rubber can undergo large reversible on elastomers (or rubber- elastic deformations. Unless like materials) and foams, damage occurs, it will return to its O-RING many of these concepts original shape are also applicable to the FEA of glass, plastics, and after removal of the load. biomaterials. Therefore, this White Paper should be Proper analysis of rubber of value not only to the rubber and tire industries, components requires special but also to those involved in the following: material modeling and nonlinear • Glass, plastics, finite element ceramic, and solid analysis tools that propellant industries are quite different • Biomechanics and the than those used for medical/dental metallic parts. The GASKET professions—implantable unique properties of surgery devices, rubber are such that: prostheses, orthopedics, 1. It can undergo large orthodontics, dental deformations under load, CAR DOOR SEAL implants, artificial limbs, sustaining strains of up to 500 artificial organs, percent in wheelchairs and beds, monitoring equipment engineering applications. • Highway safety and flight safety—seat belt 2. Its load-extension behavior is design, impact analysis, seat and padding design, markedly nonlinear. passenger protection 3. Because it is • Packaging industry viscoelastic, it • Sports and consumer exhibits industries—helmet significant damping design, shoe design, RUBBER BOOT athletic protection gear, properties. Its sports equipment safety. behavior is time- and temperature-dependent, Elastomers are used SHOCK MOUNT making it similar to glass and extensively in many plastics in industries because of their this respect. wide availability and low cost. They are also used because of their excellent damping and energy 4. It is nearly incompressible. absorption characteristics, flexibility, resiliency, This means its volume does not long service life, ability to seal against moisture, change appreciably with stress. It heat, and pressure, non-toxic properties, cannot be compressed significantly moldability, and variable stiffness. under hydrostatic load. CAR TIRE 3 E XECUTIVE S UMMARY The nonlinear FEA program, MSC.Marc (Marc) 7. Automated computers. In possesses specially-formulated elements, material and Remeshing and structural analysis, friction models, and automated contact analysis Rezoning - for the unknowns are procedures to model elastomers. Capabilities and effective solution of the nodal degrees of uniqueness of Marc in analyzing large, industry-scale problems involving freedom, like problems are highlighted throughout this White Paper. heavily distorted displacements, meshes which rotations, or the Efficient and realistic analysis for design of can lead to hydrostratic pressure. elastomeric products relies on several important 1970S: GAP ELEMENTS premature termination concepts outlined below: of analysis. 1. Nonlinear material behavior— MSC.Software offers a well-balanced combination of HISTORY OF NONLINEAR AND compressible or incompressible material sophisticated analysis code integrated seamlessly with RUBBER FEA models, time and temperature effects, easy-to-use Graphical User Interface (GUI) MSC.Marc A recent National Research Council report on presence of anisotropy due to fillers or fibers, Mentat and increasingly with MSC.Patran, for the computational mechanics research needs in the 1990s hysteresis due to cyclic loading and simulation of elastomeric products. This makes Marc [Oden, 1991] emphasized that the “materials” field is a manifestation of instabilities. uniquely suitable for the simulation of complex physics national critical technology for the United States, and 2. Automatic of rubber, foam, that areas such as damage, crack initiation and determination of plastics, and propagation, nonlinear analysis, and coupled field material parameters biomaterials. The problems still require extensive research. from test data— following sections Before embarking on the issues related to the material perhaps the single most briefly explains the behavior, it is interesting to review how the finite troublesome step for most ‘insides’ of a element method has matured in the past sixty engineers in analyzing nonlinear FEA code years–paying special attention to recent improvements elastomers; that is, (and its differences in nonlinear FEA techniques for handling rubber how to "curve fit" from a linear FEA contact problems: test data and derive program) along with parameters necessary to the accompanying 1943 Applied mathematician Courant used triangular characterize material. 1956: TRIANGULAR ELEMENT GUI capabilities. elements to solve a torsion problem. 3. Failure—causes and 1947 Prager and Synge used triangular elements analysis of failure resulting due to to solve a 2-D elasticity problem using the THE FINITE ELEMENT METHOD material damage and degradation, “hypercircle method.” cracking, and debonding. The finite element method is a computer-aided 1954-55 Argyris published work on energy methods 4. Dynamics—shock and vibration isolation engineering technique for obtaining approximate in structural analysis (creating the “Force Method” concerns, damping, harmonic analysis of numerical solutions to boundary value problems of FEA). viscoelastic materials, time versus frequency which predict the response of physical systems domain viscoelastic analysis, and implicit subjected to external
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