Modelling the behaviour of rubber-like materials to obtain correlation with rigidity modulus tests S.J. Jerrams, J. Bowen School of Engineering, Coventry University, Coventry England Abstract Finite element software for analysing components made from rubber-like materials employ strain energy functions to enable the determination of stress and displacement. Most use Mooney-Rivlin constants to define the material behaviour and these constants are simply related to the rigidity modulus for a given material. The statistical theory for the stress-strain relationship for rubbers implies that the material will obey Hooke's Law in shear though not in tension or compression. Previous research suggests that the modulus ofrigidityremains constant for shear strains up to 1.0. Values forrigiditymodulus for a 60IRHD rubber are determined from standard tests on dual-lap and quad-lap specimens. These are compared with the value given by the manufacturer, that determined from the quoted Mooney-Rivlin constants and those obtained from two-dimensional plane strain and three dimensionalfiniteelement analyses conducted using ANSYS software. Increasingly, nonlinear rather than linear finite element stress analyses are required by industry. Hyperelastic materials are less well understood than Hookean materials by engineers and without commensurate testing and consequent revision of input data,finiteelement analysis could not be trusted. Finally disparities between testing and analyses are discussed and the implications for hyperelastic analyses considered with respect to ongoing research requirements. 1 Introduction Engineers lack confidence infiniteelement analyses of elastomers due to the sparsity of test results and standard analytical solutions for nonlinear materials.
512 Computational Methods and Experimental Measurements Verification of linear elastic analyses are not problematical as material behaviour and constants are well understood. This is less so for nonlinear stresses and strains, though increasingly FEA solutions are required for situations where components exhibit plastic or hyperelastic deformations. Rubber-like materials subjected to tensile and compressive loads do not possess a constant modulus of elasticity for any part of a stress-strain curve. However linearity for shear strains up to 1.0 ( 100% ) have been observed when shear stress (i ) is plotted against shear strain ( y )fromtensile test results ( Treloar 1975 [ 1 ]). The statistical theory of rubber deformation suggests a constant value of shear modulus for long-chain molecules subjected to moderate extensions, but for higher extensions this theory becomes inadequate. Typical discrepancy between theory and experimentation is shown in Fig. 1. 1,6 - THEORY/ TEST 1,2-0,8-0,4-0,0 I I 3 4 SHEAR STRAIN Figure 1: Relationship between shear stress and shear strain determined from tensile test data and from statistical theory for a typical rubber. Hyperelastic finite element analyses use strain energy ( density ) functions to predict stresses and deformations. The inclusion of more terms in a function improves accuracy, but higher-order models require more complex determination of material coefficients. The strain energy function used by ANSYS is of the form shown below. W = Cio(I-3) + Co,(H-3) + X(l/IIl2-l) + Y(III-l)2 (1) where W = work of deformation ( strain energy function ) I, II and III are invariants of the right Cauchy-Green deformation tensor and are expressed in terms of A^, A^ and A^ the principal stretch ratios CIQ, CQI, X and Y are Mooney-Rivlin constants.
Computational Methods and Experimental Measurements 513 = 1/2 1-2v where v = Poisson's ratio ( ANSYS Tutorial - Hyperelastic Elements [ 2 ] ) If the material is virtually incompressible the third and fourth terms in the righthand side of equation 1 are insignificant since III is almost unity. Thefirsttwo Mooney-Rivlin constants for an incompressible material are related to the initial Shear Modulus by the formula Shear Modulus ( Rigidity Modulus ) G - 2 ( C,o + Q,, ) ( N / mm^ ) ( MARC software also uses Mooney-Rivlin constants in a two parameter model and a five parameter model where the additional constants are determined from material tests. However, the strain energy functions determined from Mooney- Rivlin constants are derived solely on even powers of the extension ratios, which is based on expedience rather than rationality Ogden wrote a strain energy function in series form to avoid this difficulty ( Treloar 1975 [ 3 ] ). The function, given below is offered by MARC and the values of the constants c^ and Pa are determined from tensile tests... _ A,3«n-3) (3) a" n Research is underway to determine if tactile sensors, employed in Robotics, can use stresses and deformations in compliant materials to recognise the shape of components. Further, it is hoped that FEA can model and predict these deformations to provide information for the sensors. Previous research has assumed that pads holding the sensors were subject to linear deformations ( Fearing 1983 [ 4 ], Fearing and Hollerbach 1985 [5] and Speeter 1992 [6]) Initial analyses into contact stresses in elastomers suggest that component shapes can be identified at specific depths in compliant media ( Jerrams and Hookes 1993 [ 7 ]), but physical tests are needed to verify this work and to provide information for subsequent modelling. Thus it is essential that software allows the determination of material behaviour at high stretch ratios. A sample of 60IRHD medium hard rubber was provided by Metalastik, Dunlop Ltd., conforming to the specification shown below. Material: First grade natural rubber compound Ref N 19066 68 Shore A hardness
514 Computational Methods and Experimental Measurements Poisson's ratio ( v ) = 0.4997 Mooney-Rivlin constants CJQ = 0.916 MPa CQI = 0.0647 MPa. Hence the theoretical value of G, obtainedfromthe relationship G = [ 2 ( C^ + C<>i ) ] is 1.9614 MPa As afirststep toward confirming the accuracy of analyses, dual-lap and quad-lap rigidity modulus tests to BS 903: Part A14 ( or ISO 1827: 1991(E) which is identical) [ 8 ] have been modelled and analysed. The results are compared with those obtained from tests on specimens produced from the sample. Recommendations for the verification of shear modulus and hence material constants for rubber-like materials are made. This allows for improved understanding of FEA modelling of elastomers when applying the technique to more complex problems. 2 Rigidity modulus tests and analyses Before conducting the shear modulus tests, a simple tensile test was carried out to verify the Mooney-Rivlin constants. The gauge length for the test-piece was 20mm, the cross-section 19 x 5.85mm and the rate of traverse 500mm/min. The test-piece was extended to a stretch ratio (1) of approximately six. If a graph of ( > versus 1/X is plotted ( Figure 2 ), the slope of the straight line portion yields the constant CQI and the intercept on the vertical axis at 1 A, = 1 gives the value of CIQ + Qi- Normally CQI is small, but the test results gave a slope close to unity, suggesting that the supplied values of Co * and CJQ could be transposed. MARC UK also analysed the tensile test results and determined C^ and CQI for different values of K. Their evaluation confirmed that for the sample tested the Mooney-Rivlin constants appeared transposed ( CQI > C^ ), though the constants calculated resulted in an under stiff model. MARC also determined Ogden constants from the test results which gave an overstiff model for high strains. To improve this model more points would need to befittedto the curve and this will form the basis of future work. An evaluation of the suppliers method of determining the constants appeared to show that they miscalculated the value ofc^ which would also have a small effect on the determination of the initial shear modulus. The test-piece was cut from a block 300mm square by 50mm deep, unlike a conventional test-piece which would be pressed from a rolled strip, so some change in properties, surface damage and small dimensional variations were unavoidable. However, the test should still be broadly representative of material behaviour. Hyperelastic constants should not be assumed when undertaking afiniteelement stress analysis of a rubber component. Of course, typical components are not
Computational Methods and Experimental Measurements 515 manufactured from test grade rubber, reinforcing the need to tensile test a piece of rubber taken from an actual component. To consider the disparity between quoted and determined constants a finite element model of the tensile test was carried out and the simulations of shear modulus tests were made with Mooney- Rivlin constants entered as quoted and transposed. The simulations of the tensile test proved of little value because predicted displacements were far smaller for both 2D and 3D finite element models. The 2D model could be expected to be overstiff because a plane stress option is not available for ANSYS 2D hyperelastic analyses and the plane strain option used would not simulate a Poisson's effect in the third plane. The 3D model, inexplicably gave even smaller displacements. This suggests that hyperelastic elements configured to give a constant shear modulus are unable to model other modes of deformation accurately. Dual-lap and quad-lap test-pieces were manufactured as shown in Figure 3 to the British and international standards. The rubber was bonded to therigidmetal plates using Methyl-2-cyanoacrylate adhesive. Normally when conducting dual and quad-lap shear modulus tests, this bonding takes place during vulcanisation. Since the rubber parallelepipeds were cut from the same large block this was not possible. Consequently, failure of the bond could be expected at a lower shear strain than would usually obtain Slope C\, * = L = c10 (4) where: A, = stretch ratio = strained length unstrained length f = tensile stress based on unstrained cross- sectional area ( N / Figure 2: Tensile test result for rubber sample To avoid errors induced by stress relaxation, the test-pieces were pulled apart at a constant rate of 5mm per minute and it was observed that for each type of test the bonds initially failed at shear strains between 50% and 60%. Though these
516 Computational Methods and Experimental Measurements failures did not allow the shear modulus to be determined at strains above 1.0, Figures 4 and 5 show clearly that the modulus ceases to be approximately constant for relatively low strains, linearity ceasing in the range of 6-8% shear strain. The initial slopes ( shear stress / shear strain ) achieved in the tests give Figure 3: Dual-lap and Quad-lap Test-pieces for Shear Modulus Tests 1.4 Test Shear Mode. ~~ A - 1: Ansys 2D Model, Ansys 3D Model, Theory with G @ 1.9614 MPa. 2: Theory with G @ 1.5 MPa. 3: Test. 0 0.08 0.2 0.25 0.4 Y 0.6 0.8 G @ 25% Shear Strain = T / y = 0.3745 / 0.25 = 1.498 MPa Figure 4: Shear Stress versus Shear Strain from Dual-lap Test 0 0.06 1: Ansys 2D Model, Ansys 3D Model Theory with G @ 1.9614 MPa 2: Theory with G @ 1.5 MPa. 3: Test G @ 25% Shear Strain = %/y = 0.372/0.25 = 1.49 MPa Figure 5: Shear Stress versus Shear Strain from Quad-lap Tests
Computational Methods and Experimental Measurements 517 values of G averaging 1.9642 MPa, which is very similar to 1.9614MPa, that determined from the Mooney-Rivlin constants supplied by the manufacturer. The mean value of rigidity modulus from each set of tests at the shear strain stipulated by the standards ( 25% ) is 1.494 MPa, which is virtually that quoted by Metalastik ( 1.5MPa ). To interrogate the performance of hyperelastic elements supported by ANSYS, each test was modelled in two and three dimensions. The dual-lap tests distort the parallelepipeds in simple shear as shown in Figure 4, whilst the quad-lap tests simulate pure shear (Figure 5 ). Thus both 3D and a 2D plane strain, half model analysis of the dual-lap test could be considered accurate representations, whilst a 2D plane strain, quarter model of the quad-lap test would have small errors due to the presence of strain in the Z direction. The 2D and 3D models comprised eight noded quadrilateral and brick elements respectively, though a 2D element without midside nodes is available and each element type has degenerate forms. Each model was load stepped to a maximum shear strain of 150% and the stress/strain curves obtained are shown in figures 4 and 5. Mooney-Rivlin constants were varied for all the models to alter the tensile properties of the rubber. Their sum was unaltered to keep the value of the initial shear modulus predicted by statistical theory unchanged ( Table 1 ). C,o 0.916 0. 4904 0. 0647 Q, 0. 0647 0. 4904 0.916 Assumed G 2(C,0 + C01) 1.9614 1.9614 1.9614 (MPa) Table 1 Mooney-Rivlin constants input forrigiditymodulus test simulations Shear stresses plotted against shear strains predictably give straight lines virtually sharing the same slope for each analysis and variations in constants. So analyses and test diverge at values of y of 6 to 8% and the predicted values of i are 59% in error at 50% shear strain (y = 0.5 ). Verification of hyperelastic analyses It is clear that for any large displacement hyperelastic finite element analysis, the use of strain energy ( density) functions based on Mooney-Rivlin constants will only give satisfactory results if the component is predominantly subjected to shear and then only if the shear modulus is constant for a large strain range. The value of either Mooney-Rivlin constant ( C^ or C<>i ) is largely irrelevant to an analysis predicting deformation in shear, providing the sum of the constants results in the same initial shear modulus. Thus it is obvious that a model using
518 Computational Methods and Experimental Measurements Mooney-Rivlin constants that accurately predicts behaviour in shear over a small strain range may prove utterly inadequate for other modes of deformation and for greater strain ranges. Manufacturers quoted material constants can be unreliable and tests on material taken from a component should be conducted whenever possible. Since the phenomenological theory of the behaviour of rubber assumes a constant shear modulus, benchmark testing of punch problems should indicate whether indentation can be accurately modelled and if so, over what strain ranges for individual rubbers. Unfortunately, it is evident that the requirement to correctly simulate the physical behaviour of rubber in all modes and working environments remains unrealised. References 1. Treloar L.R.G. 7%f Physics of Rubber Elasticity ( 3^ edition) pp 80-95. Claredon Press, Oxford. 1975. 2. Marx F J ANSYSRevision 4.3 Tutorial, Hyerelastic Elements pp 2.10-2.11. Swanson Analysis Systems Inc. 1987 3. Treloar L.R.G. The Physics of Rubber Elasticity ( 3>'<* edition) pp 233-234. Claredon Press, Oxford. 1975. 4. Fearing R.S. Touch processing for determining a stable grasp. Master's thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science. 1983. 5. Fearing R.S. and Hollerbach JM Basic Solid Mechanics of tactile sensing. The International Journal of Robotics Research, Vol 4, No. 3. 1985. 6. Specter T.H. Three-dimensional finite element analysis of elastic continua for tactile sensing. The International Journal of Robotics Research, vol 11, No. 1. 1992. 7. Jerrams S. J and Hookes D. The computer simulation of the indentation of compliant surfaces in production engineering. CAD, CAM, Robotics and Autonomous Factories, Vol. 4. pp 10-23. New Delhi 1993. 8. British and international standards. BS903; part A14: 1992 and ISO 1827: 1991 ( E ) Rubber, \wlcanisedor thermoplastic - Determination of modulus in shear or adhesion to rigid plates - Quadruple shear method.