Benchmarkingfiniteelement simulation of rigid indenters in elastomers S.J. Jerrams, N. Reece-Pinchin

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1 Benchmarkingfiniteelement simulation of rigid indenters in elastomers S.J. Jerrams, N. Reece-Pinchin Abstract Verifications of finite element techniques applied to elastomers are difficult to achieve since few classical solutions to hyperelastic, nonlinear analyses exist. The Axisymmetric Boussinesq Problem of normal pressure applied to the boundary of an elastic half-space is considered. Deformations in compliant materials due to different rigid indenters are dependent on load and shear modulus. Since the statistical theory for rubber assumes a constant shear modulus, such indenter tests may be appropriate for benchmarking hyperelastic finite element analyses. The level of shear strain at which the rigidity modulus ceases to be linear and its value for higher strains will determine how closely an individual rubber conforms to Boussinesq behaviour. Hence comparisons are made between classical theory, physical tests and finite element analysis for flat ended, spherical and conical axisymmetric punch indenters. Coefficients of friction input are varied to correlate with displacements in tests. In conclusion, choice of material constants input to finite element analyses are discussed. The implications for the current research programme into modelling compliant materials used for tactile sensing are evaluated. 1 Introduction Finite element techniques, used to evaluate stress levels in a component of a particular material, can normally be verified by modelling and analysing a problem in the same material for which there is a known solution. Personal verification of hyperelastic analyses are less well understood than those for Hookean materials, due to the lack of nonlinear analytical solutions.

2 546 Computational Methods and Experimental Measurements A possible method for benchmarking analyses for rubber-like materials, is to simulate normal indentations in rubber test-pieces using rigid punch-like indenters having different configurations. Determining stress distribution in an elastic half-space, when it is deformed by the normal indentation of a rigid punch, is termed a Boussinesq Problem ( Sneddon 1975 ) [ 1 ]. The derived formulae for three axisymmetric Boussinesq Problems are given in Figure 1. F = 8Gad (1 ) F = ( 2 ) = 8G(Pcote 71 (3) Flat Spherical Conical Where and F = Indenting Force G = Shear Modulus a = Indenter radius d = Displacement Figure 1: Boussinesq formulae for axisymmetric indentation The formulae assume that the coefficient of friction ( i) between the punch and the surface of the indented rubber block is zero, that the rigidity modulus is constant for any shear strain and, of course, that the dimensions of the block are large when compared with the diameter of the punch (2a) and the depth of indentation ( d) ( Gent 1992 ) [ 2 ]. Rubber test-pieces and our finite element model can be sufficiently large to approximate to the elastic half-space assumption, but the coefficient of friction will increase for increases in indentation as some contact with the punch outside diameter occurs. Shear modulus is assumed to remain virtually constant for low displacements but will fall as shear strain becomes larger ( Treloar 1975 ) [ 3 ]. This characteristic is overstated for some rubbers as demonstrated in therigiditymodulus tests conducted on a 68 Shore A hardness, natural rubber by Jerrams and Bowen

3 Computational Methods and Experimental Measurements 547 (1995) [4]. These tests also confirmed the shear modulus quoted for the rubber as 1.5MPa at 25% shear strain, which is the required level of strain for determining rigidity modulus specified in BS903, Part A [ 5 ]. The indenter tests used the same rubber; Poisson's ratio , Mooney-Rivlin constants ( C^ and C^ and MPa respectively ) and hence the same initial shear modulus MPa. Punch indentation will be predominantly dependent on shear. Thus the inability of a finite element analysis based on Mooney-Rivlin constants, to predict deformations for a rubber, possesssing a constant initial shear modulus for only a small strain range should be substantiated. The ' rigid ' punches were made in mild steel ( Fig. 2 ). They are 4mm in diameter and conform to the shapes given in Fig. 1. The conical indenter has a semi-angle ( 0 ) of 45. As the formula relating force to indentation for this case does not include a term for punch diameter, analytical and test results could be expected to diverge from those predicted by the formula for depths greater than 2mm. The rubber test-pieces were 50mm cubes bonded to mild steel bases ( Fig 2). Figure 2: Punches and test-piece for indentation tests

4 548 Computational Methods and Experimental Measurements Physical testing of rubber-like materials normally employs high rates of load application (BS 903, ISO 1827 ) [ 5 ]. Deformation of test-pieces using stepped loading will cause load reductions at constant extension / compression during intervals between load applications. This is due to stress relaxation; the measurement of change of stress with time under constant strain. In the short term this is a function of the viscoelastic behaviour of rubber-like materials and response to a stress or strain are not instantaneous but develop with time ( Brown 1986 ) [ 6 ]. Consequently indentation to a required depth must occur at a constant feed rate. Previous work suggests hyperelastic finite element software employing strain energy (density) functions based on even powers of extension ratios (e.g. ANS YS ) are inadequate for modelling high strains in rubber. ( Treloar 1975 [ 7 ], Jerrams and Bowen 1995 [ 4 ]). Current research into elastomer deformation for tactile sensing applications and the need to successfully model problems where rubber seals are required, call for confidence in finite element software. What follows seeks to clarify some of the issues involved and assist in the understanding of modelling assumptions and predicting behaviour of hyperelastic components. 2 Indenter tests and modelling When rubber test-pieces are subjected to successive load applications they exhibit stress softening, a phenomena referred to as the Mullins Effect [ 8 ]. Rubber, particularly when filled with reinforcing carbon black, softens after each deformation. This is most pronounced after the first load cycle, but is still detectable after many cycles. Though part of this modulus change is recoverable with prolonged storage, it requires that the test-pieces are indented only once. If not, the material constants quoted by manufacturers or determined in independent tests on a sample taken from the same piece of rubber, would be invalid. The three indenters were each displaced 4mm into two test-pieces respectively at a feed rate of 7mm / minute, using an Instron 8501 Dynamic Testing System. Both tests for each indenter gave similar results and graphs of load against indentation are shown for the three cases in figures 3-5. The Boussinesq curves for G = MPa and G = 1.5 MPa are superimposed on each figure. The ANS YS finite element analyses employed an axisymmetric option to model therigidpunches and 2D, eight noded hyperelastic elements. Load steps had to be varied for each indenter case to achieve a simulated displacement of 4mm. Coefficients offrictionwere varied between p. = 0 and 0.3 but this made very little difference to the calculated reaction forces on the punches. The analyses

5 Computational Methods and Experimental Measurements 549 showing zerofrictionare superimposed on figures 3-5. Figure 3 also shows the analysis for the flat indenter with Mooney-Rivlin constants reversed, since the work of Jerrams and Bowen [ 4 ] cast doubt on their individual values and method of calculation. The analyses were repeated by MARC UK, using a two parameter Mooney-Rivlin function and the forces for a displacement of 4mm gave similar results to ANSYS. They ran the analyses with the Mooney-Rivlin constants transposed for the flat and conical indenter and both as quoted and transposed for the spherical indenter. The forces at 4mm displacements are shown on the respective graphs. Fig. 3 suggests that it may be feasible to benchmark an hyperelastic finite element analysis if the component is predominantly loaded in shear using the Boussinesq equation for a flat ended indenter. The test results fall between the Boussinesq lines for initialrigiditymodulus and that for the stipulated 25% shear strain. The analysis results are outside this band and this indicates that software based on Mooney-Rivlin constants cannot predict the behaviour in shear for rubbers having a small linear shear stress / shear strain range. Contact problems at the edge of the indenter, causing an inability to model local shrinkage, should theoretically make this model less reliable than the spherical indenter model, since the surface description of the spherical ended indenter represents a continuum. The error in predicted load at 4mm indentation when compared with the test is approximately 18%. The MARC and ANSYS models with Mooney-Rivlin constants transposed, predictably gave a stiffer model and as a result correlated less well with test data. 180-p X MARC analysis QI > C^ ANSYS analysis Q, > ANSYS analysis C,Q > C^ 3 Boussinesq equation, G = MPa Tests F 0 100! * Boussinesq equation, G = 1.5 MPa R : so N Figure 3: Flat rigid punch indenter test and simulation

6 550 Computational Methods and Experimental Measurements 120i Boussinesq equation, G = MPa X MARC analysis Q, > C,o 2 ANSYS analysis Cj<> > «3 Tests 4 Boussinesq equation, G = 1.5 MPa O MARC analysis CJQ > Qi Figure 4: Spherical rigid punch indenter test and simulation 1 Tests 2 ANSYS analysis CH> > CQI 3 Boussinesq equation, G = MPa X MARC analysis Qi > C 4 Boussinesq equation, G = 1.5 MPa DISPLACEMENT i Figure 5: Conical rigid punch indenter test and simulation The tests on the spherical indenter gave a suprisingly linear relationship between load and displacement for nearly all of the indentation. If this is true for other rubbers the Boussinesq equation for this case would prove an unreliable basis for benchmarking analyses. The test load at a displacement of 4mm was virtually midway between the values obtained in each of the analyses transposing the Mooney-Rivlin constants conducted by MARC and in the range suggested by Boussinesq theory. The ANSYS analysis gave a similar load to the test load at

7 Computational Methods and Experimental Measurements 551 4mm indentation but convergence for this analysis was poor, with displacements for indenter and rubber imperfectly matched. There is no correlation with either test or theory at other points on this curve and hence analyses based on Mooney- Rivlin constants for this case can be considered unreliable. There was also poor correlation between the tests using the conical indenter and the curves resulting from the analyses and Boussinesq equations. This could have been anticipated since the indentation would cease to be proportional to the punch semi-angle after a depth of 2mm had been reached. However attaining this depth appeared to have no influence on the behaviour of the material in tests. An extrapolation of the curves would result in the test load eventually falling below that predicted by the analyses and equation. This could be expected as the 4mm 'point' would not encounter the resistance of an increasingly large semi-angle. The line given by the equation using the materials modulus derived from the Mooney-Rivlin constants is very close to the analysis using the initial modulus, confirming the tendency of the software to represent a constant shear modulus irrespective of the degree of deformation. Hyperelastic analysis to facilitate tactile sensor design Many concerns continue to prevent realistic finite element analysis of compliant materials for general design problems and tactile sensor modelling in particular. Firstly, there is a need for a tactile sensor to be inexpensive, but a rubber necessarily having a low Mullin's effect would probably contain very little filler and this would increase its cost. To model stress relaxation, we must know the time dependent changes in material properties. Predicted modification to material constants for given conditions and load cases can prove inapplicable to other circumstances. Other time dependent characteristics such as set and creep are similarly problematical. Research shows that models based on Mooney- Rivlin constants may only prove reliable for components subjected to shear and then only in a relatively small strain range. Software offering strain energy functions based on Ogden's series may well give improved initial assessment of material behaviour but it is questionable that changes in material properties in a component subjected to complex loading can be anticipated and consequently modelled. Flat indenter simulation assuming zero coefficient of friction may prove a satisfactory method of benchmarking an analysis where shear is the predominant mode of deformation. For this to be the case it is vital that shear modulus tests

8 552 Computational Methods and Experimental Measurements are conducted on the material to establish the initial modulus and the modulus at 25% shear strain. In Rivlin's paper ' Forty years of non-linear continuum mechanics ' 1984 [ 9 ] he says of 1944 "...I embarked on the task of constructing a theory which would play the same role for vulcanised rubber, which may undergo large deformations, as does classical theory for metals..." and "...many of the pundits assured me at the time that this was an impossible task." The contribution made by Rivlin and others to understanding hyperelastic material behaviour cannot be overstated, but the problems remain so diverse that a further decade on our ability to model the deformation of rubber-like materials remains limited. References 1. Sneddon I. N. Applications of integral transforms in the theory of elasticity. Chapter A, Section III, pp 70-97, Springer-Verlag, New York Gent A. N. Ed. Engineering with rubber. How to design rubber components, pp 46-47, Hanser Treloar L.R.G. The Physics of Rubber Elasticity (3rd Ed ), pp 80-95, Claredon Press, Oxford Jerrams S. J. and Bowen J. Modelling the behaviour of rubber-like materials to obtain correlation with rigidity modulus tests. CMEM 95, Capri British Standard. BS903: Physical testing of rubber. 6. Brown R. P. Physical testing of rubber ( 2^ edition ), pp , Elsivier Applied Science Treloar L.R.G. The Physics of Rubber Elasticity (3rd. Ed. ), pp , Claredon Press, Oxford Mullins L. Rubber chemical technology, p 339, Rivlin R.S. Forty years of non-linear continuum mechanics, Proceedings of the 10* International Congress on Rheology, Mexico, 1984.