How to Analyse the Static and Dynamic Response of Viscoelastic Components

The purpose of this booklet is to introduce a linear static finite element analyst,including an engineering graduate with at least one year's experience of linear static finite element analysis, to the static and dynamic structural analyses of viscoelastically-damped structures. Such structures are complex in nature and consist of viscoelastic materials sandwiched between parts or components made of metallic or composite materials. Viscoelastic materials are made up of long-chain molecules. Their properties, which are determined by the degree of mobility or entanglement of the long chain molecules, encompass all the three regimes of viscoelastic behaviour, namely: the rubbery regime, the rubber-to-glass transition regime, and the glassy regime. These regimes encompass rubbers, elastomers, polymers, epoxies and plastics. The classification of a viscoelastic material as a rubber or plastic, for example, is intrinsically with respect to normal ambient temperature of about 20degreeC. A viscoelastic material which is classed as a rubber at normal ambient temperature of 20degreeC can be made to behave as a glassy or plastic material at a relatively low temperature of -40degreeC or less. Similarly, a viscoelastic material which is classed as a plastic at normal ambient temperature of 20degreeC can be made to behave as a rubbery material at a relatively high temperature of 150degreeC or more. Thus, whether a viscoelastic material behaves as a rubber or plastic depends on the temperature range of the applications. Unlike most metallic materials, whose mechanical properties are approximately constant over typical operating temperatures of say 0degreeC to 50degreeC and frequencies of say 0 to 10 kHz, the mechanical properties of viscoelastic materials are temperature, frequency and time dependent. In addition, while the mechanical properties of most metallic materials can be obtained from material data handbooks, the temperature-, frequency- and time-dependent characteristics of the mechanical properties of viscoelastic materials have to be determined experimentally. Furthermore, while the mechanical properties of metallic materials, such as elastic modulus, are presented as real numbers, the mechanical properties of viscoelastic materials are usually presented in the frequency domain as complex numbers which have real and imaginary parts. The complex number representation is due to the intrinsic or internal damping of viscoelastic materials which, in their rubber-to-glass transition regimes, can be several orders of magnitude greater than those of most metals. Consequently, the first step in the static and/or dynamic finite element analyses of viscoelastic components is the determination of their static elastic and hyperelastic characteristics and/or their frequency and time domain dynamic mechanical properties. In addition to such issues as element type, finite element mesh design, material model, and the accuracy of the finite element solution algorithm, the accuracy of the subsequent finite element analysis is influenced by the accuracy of the material data used. Hence, this document begins with an introduction to the principles of viscoelasticity. This is followed by a description of various experimental techniques for the measurement of the static elastic and hyperelastic characteristics and of the frequency and time domain dynamic mechanical properties of viscoelastic materials. Next, some guidelines for the finite element analyses of viscoelastic components are presented. Finally, some examples of static and dynamic analyses of viscoelastic components using the finite element method are presented.

Table of Contents:
1. Introduction 1 1.1 Items Covered 2 1.2 Items not Covered 2 1.3 Applications of Viscoelastic Materials 2 1.4 Layout of Booklet 3 1. Introduction 1 1.5 Items Covered 2 1.6 Items not Covered 2 1.7 Applications of Viscoelastic Materials 2 1.8 Layout of Booklet 3 2. Fundamentals of Viscoelasticity 5 2.1 Linear Viscoelasticity 5 2.2 Definitions of Stress Relaxation Modulus and Creep Compliance Function 6 2.3 Definitions of Complex Modulus and Complex Compliance 9 2.4 Constitutive Equations of Viscoelasticity 12 2.4.1 Integral Formulation of the Constitutive Equations 12 2.4.2 Differential Formulation of Constitutive Equations Based on Mechanical Models 14 2.4.3 Generalised Differential Formulation of the Constitutive Equations 17 2.4.4 Fractional Calculus Formulation of the Constitutive Equations 19 2.4.5 Formal Relation Between Relaxation and Complex Moduli 20 2.5 Generalised Hooke's Law and Interconversion of Properties 23 2.6 Hyperelastic Behaviour of Elastomers/Rubbers 25 2.6.1 Modelling the Behaviour of Solid Elastomers/Rubbers 25 2.6.2 Modelling Compressibility Effects on Behaviour of Elastomers 27 2.6.3 Modelling the Behaviour of Elastomeric Foams 2.7 Effects of Temperature and Frequency 28 3 Characterisation of Static and Dynamic Properties of Viscoelastic Materials 31 3.1 Experimental Static Test Methods 31 3.1.1 Uniaxial Tension and Compression Methods 31 3.1.2 Equibiaxial Method 32 3.1.3 Planar and Simple Shear Methods 33 3.2 Experimental Dynamic Test Methods 34 3.2.1 Choice of Test Method 34 3.2.2 Creep and Stress Relaxation Methods 35 3.2.3 Torsion Pendulum Method 38 3.2.4 Forced Vibration Non-Resonance Methods 39 3.2.5 Resonance Methods 42 3.2.6 Wave Propagation Methods 48 3.3 Characterisation of the Complex Moduli of Materials 49 3.3.1 Illustrative Experimental Procedures 49 3.3.2 Method of Reduced Variables 51 3.3.3 Models for Representing Complex Moduli 56 4 General FEA Procedures/Guidelines 59 4.1 Initial Considerations 59 4.2 Choice of Finite Elements 59 4.2.1 Linear versus Quadratic Elements 60 4.2.2 Full versus Reduced Integration 60 4.2.3 Compressible versus Incompressible Material Behaviour 61 4.3 Choice of Material Models/Properties and Type of Analysis 62 4.3.1 Instantaneous or Long-term Modulus for Static Elastic Analysis 62 4.3.2 Hyperelastic Material Models for Static Hyperelastic Analysis 63 4.3.3 Relaxation Moduli for Time Domain Dynamic Analysis 63 4.3.4 Complex Moduli for Frequency Domain Dynamic Analysis 64 5 Examples of Static Analysis of Viscoelastic Components using FEM 67 5.1 Static Stress Distribution in a Cantilevered Lap-Jointed Beam 67 5.1.1 Geometrical and material properties 67 5.1.2 Finite element analysis 67 5.1.3. Discussion of results 69 5.2 Hyperelastic Behaviour of Rubber Mount 72 5.2.1 Finite element modelling 72 5.2.2 Discussion of results 74 6 Examples of Time Domain Dynamic Analysis of Viscoelastic Components using FEM 79 6.1 Transient Response of Mechanical Filters 79 6.1.1 Derivation of Extensional Relaxation Modulus 79 6.1.2 Finite Element Model 82 6.1.3 Discussion of Results 82 6.2 Shock Loading of Rubber Mounts 85 6.2.1 Derivation of Prony Series Constants 86 6.2.2 Finite Element Model 86 6.2.3 Discussion of Results 87 7 Examples of Frequency Domain Analysis of Viscoelastic Components using FEM 91 7.1 Vibration Transmissibility of Rubber Isolators 91 7.2.1 Finite Element Modelling 91 7.2.2 Predicted and Measured Characteristics 93 7.2 Vibration Characteristics of Potted Beam and Plate 95 7.2.1 Finite Element Analysis 95 7.4.2 Discussion of Results 97 8 Concluding Remarks 101 9 References 103 Appendix: Description of ABAQUS FEA Elements Used in Examples 107