Critical comparison of popular hyper-elastic material models in design of anti-vibration mounts for automotive industry through FEA

Transcription

1 Critical comparison of popular hyper-elastic material models in design of anti-vibration mounts for automotive industry through FEA Shashikant Sharma Sr. CAE Engineer, Paulstra CRC 46 Fuller NE, Grand Rapids, MI, 495, USA ABSTRACT: Often a mount designer is caught between the conflicting targets of matching stiffness as well as durability in a short timeframe. This paper attempts to address challenges associated with the correlation between elastomeric mount performance and simulation results. The effects of insufficient material characterization data (type and range), calibration, chosen constitutive model and stability check on stiffness and stress/strain prediction are discussed. Coefficients are now determined using direct (least square fit) as well as weighted, using genetic algorithm based curve-fitting. A step-by-step methodology (iterative) has been developed. This method allows error to be identified and attributed to material model and/or thermal shrinkage, pre-load, mesh, or boundary conditions. Case studies are included with Neo-Hookean, -term Mooney-Rivlin, Yeoh and Rivlin material models using in-house nonlinear FEA software, ARC3D. INRODUCTION Elastomeric materials are unique by virtue of their properties, e.g. high elongation, reversibility, incompressibility and damping, and thus utilised in many engineering applications such as shock absorbers and vibration isolators (also known as mounts in automotive industry.) A mount design must meet stiffness, rate of energy absorption and durability targets defined as per the application and this starts from hand calculations (Lindley 974) using the shear modulus, shape factors etc. Unfortunately, these simplified calculations (widely published) suffer from many assumptions and are valid only for simple shapes. Finite element methods, using high speed (& cheap) computers, have transformed the product development cycle and yielded in low cost products that can be designed in a shorter time span even with complex shapes and loading environments. Nonetheless, in the case of rubber products, practicing engineers have not taken full advantage, partly due to complex continuum mechanics and partly owing to its non-linear material behaviour that differs from compound to compound mix. Concise information relevant to engine mount design is often not available in literature with respect to use of various available material models. An attempt has been made in this paper to bring out their pros and cons and a suitable FEA process to deal with correlation issues. The following paragraphs describe material models which define stress-strain relationship for rubber as hyper-elastic material which are used in finite element analyses. A step-bystep method (characterization, parameter estimation, calibration, and common mistakes) is presented with a real-life example of transmission mount in order to achieve good correlation between simulation and experimental data. MATERIAL CHARACTERIZATION Several constitutive models have been published (Mooney 94, Ogden 97, Rivlin 948, Treloar 975, Yeoh 99) for nonlinear large elastic deformations based on strain energy density functions. The other category is micro mechanics based (Arruda & Boyce 993). In this paper, strain energy based polynomial models, Rivlin, Mooney-Rivlin, Yeoh and Neo-Hookean models were evaluated due to their availability in ARC3D. The main question is What material properties are required, how to generate them and how close do they simulate a reality? The standard ASTM4 tensile test was proposed for its simplicity and as a measure of quality control. Problems of such single test usage in generating material constants for other strain states have been reported and verified (Peeters & Kussner 999). Therefore the compound in question was tested under 3 pure states namely, simple tension, planar tension and equi-bi-axial tension. Three strain levels of 8%, 34% and 7% (Sharma, unpubl., Miller ) were chosen as repre-

2 sentative amplitudes for slow cycling as low, middle and high strain ranges, since mounts are normally designed below 4% dynamic strains to meet durability criterion. A genetic algorithm (Corne et al 999) based optimization routine was introduced in curve-fitting module, to allow user to calibrate or weight a particular state over other as additional capability than regression fit (Johannknecht et al 999 and Menderes & Konter 999). Austrell PE, 997 is an excellent treatise on characterization and direct search methods. Benefits of such algorithm were however restrained by problems associated with polynomial models (Sharma, unpubl. ). Figure shows -term Mooney-Rivlin model (eq. ) with small strain assumption (ABAQUS/Standard manual 997), built from only uni-axial test data. It clearly shows the non-physical stress-strain behavior at large strains. σ u = 6ε u (C + C (C + C )ε u ) () Nominal Stress Figure. Stress-Strain variation Mooney-Rivlin model Therefore following relations were used, without small strain simplification. Neo-Hookean model: U NH = C ( I 3 ) () Mooney-Rivlin model: U MR = C ( I 3 ) + C (I 3) (3) Yeoh model: U Y = C ( I 3 ) + C ( I 3 ) + C 3 ( I 3 ) 3 (4) Rivlin model (9-term): U R = 3 ij i=, j= i j C ( I 3) ( I 3) Arruda-Boyce Model: (Arruda & Boyce 993) U AB ( I = nkt + 3) ( I 3 N C=.3, C=.6 C=.3, C=. - C=.3,C=-.6 C=.3, C=.5-3 C=.3, C=-.5 C=.3, C=-.5 C=.3, C=-. -4 U n iax ia l S tra in 3 ( I 9) + ( I 7) N 5N ) + ( I 43) N (5) (6) Where U is strain energy density function and I i are reduced strain invariants (Bonet & Wood 997). Traditionally Neo-Hookean model has been used as a first approximation (linear), and also to check mesh integrity, as this model is unconditionally stable. The single constant can be easily derived from cheap durometer and shear modulus readings. Yeoh and Arruda Boyce models are also function of I only and thus stable (Figures -3). There is a common belief that more number of terms would lead to a better fit (Yeoh 997) with nonlinear stress-strain curve, but this also results in unstable strain energy functions outside certain stretch ratio limits, as shown in Figure 4. Figure. Strain energy Vs stretch ratios Neo-Hookean model U (l,l) Figure 3. Strain energy Vs stretch ratios Arruda Boyce model U (l,l) U (l,l) Lambda Lambda NH, C =.5 AB, CR=.56, N=4 MR, C =.5, C =-.5 Lambda Lambda Lambda Lambda Figure 4. Strain energy Vs stretch ratios Mooney-Rivlin model

3 A comparison of the experimental (nominal stressstrain) and fitted (using eqn -5) data is plotted in figures 5- for all three states and from low to high strain levels. Experimental data points are from loading-path of th stabilized cycle for each state and strain range, as material model from first data cycle will be stiffer than the actual mount in service Figure 5. Low strain Neo-Hookean model (C =.485 MPa) ST stands for simple tension, EB stands for equi-bi-axial and PT stands for planar tension specimen in all the plots. Pure shear state as represented by planar tensile test appears to be over-predicted in all models, from low Nom inal Strain Figure 6. Low strain Mooney Rivlin model (C =.4394 MPa, C =-.45 MPa) Nom inal Strain Figure 7. Low strain Yeoh model (C =.4485 MPa, C = Mpa, C 3 = MPa) to high strain levels. Amplitude dependent or softening behavior is apparent from reduction in material constant value. Also it should be noticed that no material model fits accurately in all states Figure 8. Low strain Rivlin model (C i =.4433,.6686, -.96,.89998,.5847,.436,.48464,.75, MPa) Figure 9. High strain Neo-Hookean model (C =.374 MPa) Figure. High strain Mooney Rivlin model (C =.9776 MPa, C =.595 MPa) At high strain, Rivlin model (figure ) appears to have captured equi-bi-axial well, but starts to deviate sharply after 68% strain. Similarly Yeoh model (figure ) shows inflection point at a lower strain point. Neo-Hookean and Mooney Rivlin models show quite linear behavior.

4 Figure. High strain Yeoh model (C = MPa, C = Mpa, C 3 =.43 MPa) Nom inal Strain Figure. High strain Rivlin model (C i =.693,.863, , -.987,.65684,.6467, ,.7456, MPa). Model verification Different material models have been studied for stiffness with FE model using various specimen e.g. tensile, shear (with tensile, compressive pre-load and combined torsional load). Also, effect of different mesh density on stiffness was studied, coarser the stiffer (Sharma ). Only tensile case is presented (due to brevity) in Figure 3. 3 FE MODELING AND SOLUTION PROCEDURE 3. Software I-DEAS software is used for preparing mesh and ARC3D is used as nonlinear solver, which is essentially NIKE3D (Bradley et al 995) based. Femview is used as post-processor for plotting and animating user-defined, convenient iso-stress/strain contours. Stiffness is usually computed by first fitting a curve through force-deflection data and then taking first order derivative symbolically to avoid numerical noise. 3. Solution phase Once a material model is chosen as satisfactory and stability limits are derived, a good solution strategy would involve simulating actual loads on a mount alongwith process simulation e.g. thermal shrinkage due to mold cooling (causing residual strains), swaging and pre-loading in various directions. Mixed u-p elements (Sussman and Bathe 987) were used to avoid locking. Sometimes it is better to model rubber as slightly compressible and avoid numerical hitches. Full Newton method with line search has been used to iterate nonlinear solution with double precision and appropriate convergence tolerances on energy and displacement as well. Precautions were taken to account for change in boundary conditions from one process to another. 4 EXAMPLE: TRANSMOUNT A typical transmission mount shown in figure 4.8 Stiffness (N/mm) Mooney Rivlin Neo-Hookean Yeoh displacement (mm) Figure 3. Stiffness behavior check, Neo-Hookean and Mooney-Rivlin models are not able to capture stiffening effect at large strains. Figure 4. Transmission mount on a fixture consists of top and bottom metal brackets bonded to rubber legs. The mount was characterized (displacement control) for stiffness in X Y & Z directions with -8mm of pre-load in -Z. Results are tabulated in table. Half symmetry models (about YZ and XZ planes) were used to save computation time. YZ model is shown in figure 6.

5 Table. Experimental & FEA predicted stiffness (N/mm) mm preload Kx Ky Kz Experimental FEA %error mm preload Kx Ky Experimental FEA %error.4.5 was used to correct predicted stiffness in -8mm pre-load cases. Further to this, as the strain levels at higher pre-load were beyond the low-level Neo- Hookean model, a softening factor of C (high) /C (low).8-.9 (appropriate based on strain level interpolation) was also applied in 6-8mm pre-load cases. Corrected stiffnesses were within.75% to.6% as tabulated in Table. -4 mm preload Kx Ky Experimental FEA %error mm preload Kx Ky Experimental FEA %error mm preload Kx Ky Experimental FEA %error RESULTS AND DISCUSSION This transmount is essentially under compression (negative Z) and shear in (X&Y) directions. Neo- Hookean model has been used due to its stability and as a starting point. Also experimental curves didn t show need of inflection point to be modeled. FEA model predicted stiffness with upper limit of +3.85%, +4.7% and +3.% error in Z, Y and X directions respectively. Highest error of +4.7% was at highest pre-load of 8mm. As already discussed, all the material models over-predicted shear, thus a correction factor.6 (ratio of fitted to experimental value at final strain from figure 5) Figure 6. Half symmetry FEA model Table. Corrected stiffness (N/mm) & %error Preload Experiment FEA Corrected %error (mm) X Y X Y X Y X Y It is also interesting to note that if stiffness of top/bottom metal strips and fixture (K fixture ) are in series with rubber stiffness, measurement error can easily grow to as high as % at fixture to rubber stiffness (K rubber ) ratio of as per equation 7 and figure 7. K measured = { K rubber / [ + (K rubber / K fixture )] } (7) %error Kf / Kr Figure 5. Planar tension (pure shear) comparison Figure 7. Measurement error Vs Fixture-mount stiffness ratio

6 It is apparent from the present studies that to get good correlation in such multi-axially loaded components, it is necessary to:. characterize the compound in question for all anticipated strain levels and in all three pure states, figures 5-.. estimate constants by direct search or optimization, followed by stability checks. Check model s usability beyond experimental data range by extrapolation and tune genetic algorithm weighing factors for different strain states based on anticipated strain state in application. Figures -3 (Neo-Hookean and Arruda-Boyce models) show only one minima for strain energy density function at stretch ratio (lambda,lambda=,) and are monotonically increasing as stretch ratios increase, and thus are stable. Whereas figure 4 (Mooney-Rivlin model) shows extremas for strain energy density function and thus put a limit on the functional region of strains as linear combination of stretch ratios. Therefore, for each chosen model, the constants must be verified for their functional limits on tensile, compressive and shear strains. 3. record errors in individual states between experimental and fitted stress-strain curves for all selected models. This is also another key to achieving good correlation, as curve-fitting errors may accumulate on numerical solution error and corrupt final results. 4. carry out FEA of part with process loading followed by actual load. This step can identify, if a particular process is contributing more or less significantly on total response of component. Take strain measure into consideration while comparing Green-Lagrange strains from FEA and nominal strains from experiment, else error will accrue as nonlinear strains diverge from engineering strains a lot at large strains as shown in figure 8 (Crisfield ). Strain Engg. Green-L Almansi-E Natural Stretch Ratio Figure 8. Strain measures Vs stretch ratio 5. apply correction factors based on error or softening when appropriate. 6. design fixture preferrably 5 times stiffer than mount to keep measurement error within 5% as shown in figure to calibrate experimental stiffness data properly, in order to eliminate hysteresis, and other offsets due to in-built strains during different processes e.g. molding, swaging, assembly etc. 8. to study sensitivity of boundary conditions, bulk modulus and mesh density on results. 9. to account for heat generation due to cyclic loading in case of dynamic stiffness measurement. Usually stiffness drops and stabilises after, cycles (Sharma unpubl. 3). 6 CONCLUSIONS Strain energy density function based polynomial models, namely Neo-Hookean, Mooney-Rivlin, Yeoh and Rivlin and micro-mechanics based (Arruda-Boyce model briefly) were studied and compared for their subtle effects on final response (stiffness) of transmount. Effect of curve-fitting methods, namely direct search and optimisation on material parameter estimation was studied. In the light of multiple sources of errors which can accrue (as discussed earlier), it was difficult to substantiate effects of tuning weighing factors on progreesiveconditioned (Austrell 997) stress-strain data, though it needs to be studied for few more components under different type of loading and strain state. In summary, a suitable step-by-step process is developed to get good correlation and identify sources of errors. 7 SCOPE OF FUTURE WORK An attempt is being made to incorporate userdefined material model to account for softening (Govindjee 99 & Simo 987) and shear overprediction, where software should automatically update FE model properties based on strain levels in each element and avoid iterative manual correction later. Effect of stress relaxation during progressiveconditioning and collection of data should also be studied. 8 ACKNOWLEDGEMENT Author would like to thank Dermot Ashby, Dr. Borz Fariborzi and Dr. Pengfei Shi of Paulstra CRC USA, and Daniel Benoualid and his team at CDR Hutchinson, France for their kind support and stimulating discussions time to time.

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