1、BRITISH STANDARD BS 903-5:2004 Physical testing of rubber Part 5: Guide to the application of rubber testing to finite element analysis ICS 83.060 BS 903-5:2004 This British Standard was published under the authority of the Standards Policy and Strategy Committee on 2 August 2004 BSI 2 August 2004 T
2、he following BSI references relate to the work on this British Standard: Committee reference PRI/22 Draft for comment 03/100129 DC ISBN 0 580 43926 7 Committees responsible for this British Standard The preparation of this British Standard was entrusted to Technical Committee PRI/22, Physical testin
3、g of rubber, upon which the following bodies were represented: British Rubber Manufacturers Association Ltd. CIA Chemical Industries Association Materials Engineering Research Laboratory Ltd. RAPRA Technology Ltd. Royal Society of Chemistry SATRA Technology Centre Tun Abdul Razak Research Centre Ame
4、ndments issued since publication Amd. No. Date CommentsBS 903-5:2004 BSI 2 August 2004 i Contents Page Committees responsible Inside front cover Foreword ii 1S c o p e 1 2 Normative references 1 3T e r m s a n d d e f i n i t i o n s 1 4S y m b o l s 3 5 Introduction to FEA 3 6 Stressstrain behaviou
5、r 4 7 Mechanical failure 19 8 Friction 32 9 Thermal properties 33 10 Heat build-up 36 Annex A (informative) Stress-extension relationships in simple deformations and parameter optimization 37 Annex B (informative) Relationship between stress in simple shear and pure shear 38 Annex C (informative) An
6、 example of fitting models to experimental data 38 Bibliography 42 Figure 1 Values of I 1and I 2for different deformation modes. 10 Figure 2 Schematic diagram of apparatus for equibiaxial straining of a flat sheet 13 Figure 3 Schematic diagram of apparatus for inflation of a sheet 14 Figure 4 Schema
7、tic diagram of pure shear apparatus 15 Figure 5 Schematic diagram of apparatus for constrained compression 17 Figure C.1 Fits to experimental data in uniaxial tension 39 Figure C.2 Prediction of behaviour in pure shear based on fits to experimental data in uniaxial tension 40 Figure C.3 Prediction o
8、f behaviour in equibiaxial extension based on fits to experimental data in uniaxial tension 41 Table 1 Fracture test pieces for rubber 22 Table C.1 Constants derived from fits to uniaxial tension data 38BS 903-5:2004 ii BSI 2 August 2004 Foreword This British Standard has been prepared by Technical
9、Committee PRI/22. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front
10、 cover, an inside front cover, pages i and ii, pages 1 to 43 and a back cover. The BSI copyright notice displayed in this document indicates when the document was last issued.BS 903-5:2004 BSI 2 August 2004 1 1 Scope This part of BS 903 gives recommendations for test procedures and guidance on appro
11、priate methods for determining model parameters from test data for use in finite element analysis (FEA) of rubber. It covers stressstrain characterization, mechanical failure, friction, thermal properties and heat build-up. It is applicable to solid vulcanized rubbers of hardness 20 to 80 IRHD for w
12、hich the deformation is predominantly elastic, and to cellular materials formed using such rubbers. It might prove useful for other materials which can be deformed elastically to large strain. 2 Normative references The following referenced documents are indispensable for the application of this doc
13、ument. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. BS 903-A1, Physical testing of rubber Part A1: Determination of density. BS 903-A2, Physical testing of rubber Part A2: Method for de
14、termination of tensile stress-strain properties. BS 903-A4, Physical testing of rubber Part A4: Determination of compression stressstrain properties. BS 903-A10 (BS ISO 132), Rubber, vulcanized or thermoplastic Part A10: Determination of flex cracking and crack growth (De Mattia). BS 903-A14, Physic
15、al testing of rubber Part A14: Method for determination of modulus in shear or adhesion to rigid plates Quadruple shear method. BS 903-A21, Physical testing of rubber Part A21: Determination of rubber to metal bond strength. BS 903-A42 (BS ISO 3384), Physical testing of rubber Part A42: Determinatio
16、n of stress relaxation in compression at ambient and at elevated temperatures. BS 903-A51, Physical testing of rubber Part A51: Determination of resistance to tension fatigue. BS 903-A61 (BS ISO 15113), Physical testing of rubber Part A61: Determination of the frictional properties. BS 7608, Code of
17、 practice for fatigue design and assessment of steel structures. BS EN 12667, Thermal performance of building materials and products Determination of thermal resistance by means of guarded hot plate and heat flow meter methods Products of high and medium thermal resistance. BS ISO 34-1, Physical tes
18、ting of rubber Determination of tear strength Part 1: Trouser, angle and crescent test pieces. BS ISO 4664, Rubber Guide to the determination of dynamic properties. BS ISO 23529, Rubber Physical test methods Preparation and conditioning of test pieces and preferred test conditions. 3 Terms and defin
19、itions For the purposes of this British Standard, the following terms and definitions apply. 3.1 finite element analysis FEA numerical method of analysing a product in which the numerical calculations are carried out for discrete, linked elements of the product NOTE FEA is usually carried out using
20、a commercial software package which allows visual simulation of the product to aid creation of the finite element model and viewing of the results of the analysis.BS 903-5:2004 2 BSI 2 August 2004 3.2 deformation mode deformation arising through a particular relationship of the principal extension r
21、atios NOTE Examples are uniaxial tension, uniaxial compression, pure shear, simple shear or equibiaxial extension. 3.3 deviatoric deformation deformation involving change of shape with no associated change of volume 3.4 volumetric deformation deformation involving change of volume with no associated
22、 change of shape 3.5 extension ratio ratio of deformed to undeformed length of a line element of material in a specified direction NOTE 1 This is also known as a stretch ratio. NOTE 2 The term extension is used to cover all types of deformation; thus a compression is defined with an extension ratio
23、less than one. 3.6 hyperelasticdeforming to high strains under the application of stresses, and returning to the original shape when the stresses are reduced, such that there is no loss of energy 3.7 incompressibledeforming without a change in volume NOTE For infinitesimal strains, incompressibility
24、 is equivalent to a Poissons ratio of 0.5. For large deformations, the condition is that 1 2 3= 1. 3.8 modelcomputer simulation of a product or structure, containing all the information necessary to carry out a finite element analysis 3.9 modeldescription, expressed mathematically, of some aspect of
25、 the behaviour of the material, which is implemented in the FEA package 3.10 principal strain axis one of three orthogonal directions within the material along which the deformation results only in changes in length of a line element of the material without any rotation of the element, other than th
26、at associated with any rigid body rotation 3.11 principal stress stress acting on a surface element of material normal to a principal stress axis 3.12 principal extension ratio i ratio of deformed to undeformed length of an element of material in the direction of one of the principal strain axes 3.1
27、3 principal stress axis one of three orthogonal directions in the material with the property that the stress acting on a surface normal to it has the same directionBS 903-5:2004 BSI 2 August 2004 3 3.14 stableshowing an increase in stress for an increase in extension NOTE This guide applies only to
28、elastic materials for which unstable models are not physically possible. 3.15 stiffness ratio of force to deflection of a test piece or product NOTE Dynamic stiffness is the ratio of the amplitude of the periodic force to that of an applied sinusoidal deformation. 3.16 strain energy function mathema
29、tical expression for the strain energy per unit unstrained volume (strain energy density) of the hyperelastic material in terms of its deformation 3.17 strain invariant I 1 , I 2 , I 3 function of the principal extension ratios which is independent of the choice of co-ordinate axes NOTE The most com
30、monly used are defined in Clause 4. 4 Symbols The following symbols are used with the following definitions throughout this part of BS 903. Other symbols are defined in the clauses in which they appear. 5 Introduction to FEA FEA has become a popular tool for design engineers and others. It provides
31、a means of obtaining a computer simulation of the behaviour of a product from which useful quantitative predictions may be made. Some of the uses of FEA for products containing rubber are: to simulate the response of the product under a system of applied forces or deflections; to optimize the design
32、 (shape) of a component; to identify regions where failure can occur; to estimate the sealing pressure of a seal; to optimize the cure time and temperature of a bulky product. The widespread use of FEA among engineers has been aided by the development of a number of commercial FEA software packages,
33、 which enable those without a specialist knowledge of mathematics or numerical analysis methods to carry out FEA. As well as the program which performs the analysis (the solver), many such software packages also contain programs which allow a visual simulation of the product on the computer screen (
34、the pre- and post-processor). These are very useful for generating the correct input for the analysis (the FEA model) and for viewing the results, such as the deformed shape of the product, or the temperature distribution following a thermal analysis. Relatively simple, so-called “linear”, FEA packa
35、ges are available and are suitable for carrying out analyses where the strains are small, such as stress analysis of metals. Two difficulties make such packages unsuitable for analysing rubber products: they assume that the strains are very small, and that the material is compressible (Poissons rati
36、o significantly less than 0.5). Instead, for mechanical analyses of rubber products, so called “non-linear” FEA packages are required. Non-linear packages are also required for most transient thermal analyses, such as simulating heat flow during vulcanization. I 1 is the first strain invariant, defi
37、ned such that I 1= 1 2+ 2 2+ 3 2 ; I 2 is the second strain invariant, defined such that I 2= 1 2 2 2+ 2 2 3 2+ 3 2 1 2 ; I 3 is the third strain invariant, defined such that I 3= 1 2 2 2 3 2 ; W is the strain energy density, as defined in 3.16; 1 , 2 , 3 are the three principal extension ratios.BS
38、903-5:2004 4 BSI 2 August 2004 As with any computer simulation, the accuracy of the analysis is heavily dependent on the quality of the information supplied to the computer. The skill of the FEA practitioner lies in selecting the best analysis method, and also in using sensible models for describing
39、 the behaviour of the materials, with accurately measured values of the model parameters. Adequate models for some features of rubber behaviour are not yet provided with any existing commercial FEA package, notably for: a) dynamic properties of rubber (see 6.3); b) the Mullins effect (see 6.3); c) h
40、eat build-up (see Clause 10). The purpose of this guide is to give some background information on the material models available in FEA packages for modelling rubber, and to recommend suitable test methods for measuring the model parameters. 6 Stressstrain behaviour 6.1 Solid (incompressible) rubbers
41、 6.1.1 Introduction Traditionally, rubber is modelled as a perfect hyperelastic material. This means that features such as set, hysteresis and strain softening are ignored. Commercial FEA packages are gradually introducing models which do take account of these features, but such models do not always
42、 provide a realistic description of the material behaviour, so should be used with caution. Further details are given in 6.3. 6.1.2 Hyperelastic models Commercial FEA packages normally provide a choice of hyperelastic models expressed as strain energy functions, and the user is required to input one
43、 or more parameters of the function. The strain energy function is a mathematical expression for the amount of energy stored in the material, arising from work done in deforming it. In its most general form, it may be written as: or where the symbols have the meanings given in Clause 4. Stressstrain
44、 equations may be obtained directly from equation (1) by differentiation and, conversely, experimental stressstrain curves may be used to fit the parameters of the strain energy function. The necessary equations are given in Annex A. Many FEA packages also provide a curve-fitting procedure in which
45、the user may enter tables of experimental stressstrain data and the program provides the best-fit values of the parameters for the chosen function (see 6.7). Alternatively, the function need not have an explicit form, but values required to fit a particular experimental data set may be obtained nume
46、rically (see 6.1.4.7). 6.1.3 Choosing a hyperelastic model 6.1.3.1 If a model has a large number of parameters which are fitted from a limited amount of test data it is possible that the model will be unstable and unrepresentative of any real material for strains or deformation modes outside those c
47、overed by the test data. Therefore the model with the fewest number of fitted parameters which is able to give a satisfactory description of the rubber behaviour should be used. 6.1.3.2 Hyperelastic models do not account for the effects of set, hysteresis, strain softening and frequency. However, th
48、ese features usually have a significant effect on the stressstrain behaviour of filled rubbers and thus experimental stressstrain curves used to fit the parameters of the hyperelastic model are dependent on these features. It is unnecessary and unwarranted to attempt to obtain a fit to the experimen
49、tal data which is much better than the quality of the experimental data, bearing in mind its dependence on features which cannot be modelled. Furthermore, other imperfections and approximations in the FEA can also outweigh imperfections in the material model. W = W(I 1 , I 2 , I 3 ) (1a) W = W( 1 , 2 , 3 ) (1b)BS 903-5:2004 BSI 2 August 2004 5 6.1.3.3 For solid rubbers, the bulk modulus i
