Dynamic Finite Element Modeling of Elastomers

Transcription

1 Dynamic Finite Element Modeling of Elastomers Jörgen S. Bergström, Ph.D. Veryst Engineering, LLC, 47A Kearney Rd, Needham, MA Abstract: In many applications, elastomers are used as a load-carrying component that is exposed to small strain dynamic loads. In order to perform accurate finite element simulations of these elastomer components it is important to use a material model that captures the non-linear dynamic response of the material. This paper summarizes the strength and limitations of a set of material models, including linear viscoelasticity, the Bergstrom-Boyce (BB) model, and the new dynamic Bergstrom-Boyce (DBB) model. It is shown that of these models the new DBB model can most accurately predicting the dynamic response of commonly used elastomers such as natural rubber and silicone rubber. Keywords: Constitutive Model, Vibration, Rubber, Elastomer, Dynamic Loading, User material, UMAT. 1. Introduction The performance of many products in the tire and automotive industries critically hinges on the response of elastomers. Elastomers are also used as critical components in many consumer goods and medical device products. In some of these applications the elastomer is exposed to large deformations that probe the non-linear viscoelastic response of the material. In these applications the Bergstrom-Boyce model (Bergstrom, 1998, 1999, 2000, 2001), which is available as a built-in feature in Abaqus/Standard through the *Hysteresis command, is often very accurate and useful. There are also many important applications where elastomers are mainly exposed to small strain dynamic (or vibrational) loads. The goal of this study is to examine how common material models (linear viscoelasticity and the Bergstrom-Boyce model) and a new material model called the dynamic Bergstrom-Boyce (DBB) can be used to predict the material behavior of this important class of dynamic problems. 2. Experimental Data To evaluate the performance of different material models it is important to have accurate experimental data. This study is based on storage and loss moduli data for natural rubber filled with 75 phr carbon black (CB) tested at room temperature (Chazeau, 2000), and for silica filled silicone rubber tested at room temperature. (Chazeau, 2000). In the experiments, test specimens were deformed using a sinusoidal strain history: Abaqus Users Conference

2 (1) The resulting stress response can be decomposed into a sum of a sin and a cosine term: The part of the stress that is in-phase with the strain gives the storage modulus (E ), and the part of the stress that is out of sync with the strain gives the loss modulus. (E ). Figure 1 shows the storage and loss moduli as a function of applied strain amplitude for a natural rubber filled with 75 phr carbon black (CB). The figure shows that the storage modulus decreases with increasing strain amplitude. This behavior is typical for elastomers and is called the Payne effect. The loss modulus is also a function of the strain amplitude as is shown in Figure 1. (2) Figure 1. Experimental storage and loss moduli as a function of strain amplitude for a natural rubber. A second example illustrating the behavior of a silicone rubber is shown in Figures 2 and 3. The experimentally determined storage and loss moduli for the silicone rubber is qualitatively similar to the natural rubber Abaqus Users Conference

3 Figure 2. Experimental storage and loss moduli as a function of strain amplitude for a silicone rubber. Figure 3. Experimental storage modulus as a function of strain amplitude for two frequencies for a silicone rubber Abaqus Users Conference

4 3. Linear Viscoelasticity Linear viscoelasticity is an extension of hyperelasticity that incorporates time-dependence and viscous flow. The predictive performance of the linear viscoelasticity model is here exemplified with a one-term Prony series. This model can be represented using the rheological model shown in Figure 4. Figure 4. Rheological representation of a linear viscoelasticity model with one Prony series term. In this figure the two spring elements are neo-hookean, and the dashpot has the following flow rule: Due to the linear nature of linear viscoelasticity, the predicted storage and loss moduli are independent of the applied strain amplitude, as shown in Figure 5. The linear viscoelasticity model, however, does have a frequency-dependent response as illustrated in Figure 6. (3) Abaqus Users Conference

5 Figure 5. Predicted behavior of a linear viscoelastic model when exposed to an amplitude sweep. Figure 6. Predicted behavior of a linear viscoelastic model when exposed to a frequency sweep Abaqus Users Conference

6 The best fit of a one-term linear viscoelastic model to the experimental data for the natural rubber is shown in Figure 7. This figure shows that since the model cannot capture the strain amplitude dependence it cannot be considered an accurate model for predicting the dynamic behavior of elastomers. Also note that using a higher order Prony series does not help the predictions at different strain amplitudes. Figure 7. Best fit of a linear viscoelasticity model to the experimental data for natural rubber. 4. Bergstrom-Boyce Model The Bergstrom-Boyce (BB) model is a non-linear viscoelastic extension of linear viscoelasticity. The BB model has been shown to work well for predicting the large deformation behavior of elastomers (Bergstrom, 1998, 2001). The model is based on the same rheological representation as was shown in Figure 4. The only difference between the BB model and a linear viscoelasticity model with one Prony series term is the viscoelastic flow equation. The BB model is based no the following flow equation: (4) Abaqus Users Conference

7 where λ v is the chain stretch, τ is the effective shear stress, and [ξ, C, τ B, B m] are material parameters. For small strain dynamic loading the strain dependence of the flow is not important and C=0, making the flow rule the same as for linear viscoelasticity except that the stress τ is raised to a power m. This modification significantly improves to the model predictions, as is shown in Figures 8 to 10. These figures illustrate that the BB model qualitatively captures the main features of the dynamic response of elastomers, including the Payne effect. Figure 8. Predictions from the BB model when exposed to an amplitude sweep Abaqus Users Conference

8 Figure 9. Predictions from the BB model when exposed to a frequency sweep. Figure 10. Comparison between dynamic data for a natural rubber and the best fit of the BB model. This figure shows that although the BB model has the same qualitative features as the experimental data, it cannot be made to accurately capture both the Payne effect and the Abaqus Users Conference

9 magnitude of the peak in the loss modulus. The BB model is an improvement of the linear viscoelasticity model but is still not sufficiently for accurate predictions. 5. Dynamic Bergstrom-Boyce Model The predictions from the BB model can be improved by modifying the hyperelastic network A in Figure 4. Specifically, the shear modulus of the neo-hookean element is made strain dependent: where ε eff is the effective Mises strain, and [μ Ai μ Af, ε h ] are material parameters. This modified version of the BB model is called the dynamic Bergstrom-Boyce (DBB) model. It is illustrative to perform a parametric study of the DBB to demonstrate its performance. Figures 11 to 14 depict how changing one material parameter at a time influences how the storage and loss moduli vary with strain amplitude. (5) Figure 11. Dependence of E and E on the material parameter μ A for the DBB model when exposed to strain amplitude sweeps Abaqus Users Conference

10 Figure 12. Dependence of E and E on the material parameter μ B B for the BB model when exposed to strain amplitude sweeps. Figure 13. Dependence of E and E on the material parameter τ B B for the DBB model when exposed to strain amplitude sweeps Abaqus Users Conference

11 Figure 14. Dependence of E and E on the material parameter m for the DBB model when exposed to strain amplitude sweeps. In summary, the DBB model takes the following 6 parameters: [μ Ai, μ Af, ε h, μ B, B τbb, m]. These parameters can be estimated from strain amplitude data using the following procedure: 1. μ B B = max(e ) 2. μ Af = min(e ) / 2 3. μ Ai = max(e ) / 2 - μ B B 4. ε h = strain at max(e ) 5. select m to get the right width of the E curve 6. select τ B B from: (6) 7. optimize all material parameters to get the best overall fit This procedure was used to fit the model to the experimental data for the natural rubber. A comparison between the experimental and predicted data is shown in Figure 15. The material parameters that were used to create the predictions are listed in Table Abaqus Users Conference

12 Table 1. Material parameters for the DBB model used to represent the behavior of the natural rubber. Material Parameter Name μ Ai μ Af Material Parameter Value 1.81 MPa 0.65 MPa ε h 0.02 μ BB 0.46 MPa τ BB 0.15 MPa m 1.69 Figure 15 shows that the DBB model does a good job at predicting the dynamic response of the rubber at different strain amplitudes. Figure 15. Comparison between dynamic data for a natural rubber and the best fit of the DBB model. A second example illustrating the usefulness of the model to predict the behavior of silicone rubber is given in Figure 16. This figure shows the storage and loss moduli as a function of strain amplitude. The best fit of the material parameters that were used to create the predictions are summarized in Table Abaqus Users Conference

13 Table 2. Material parameters for the DBB model used to represent the behavior of the silicone rubber. Material Parameter Name μ Ai μ Af Material Parameter Value 1.34 MPa 0.25 MPa ε h μ BB 1.14 MPa τ BB 0.14 MPa m 2.0 Figure 16. Comparison between dynamic data for a silicone rubber and the best fig of the DBB model. The calibrated material model for the silicone rubber was then used to predict the response at a lower frequency of 0.1 Hz. The results from the comparison are shown in Figure 17 which illustrates a reasonable agreement between the experimental data and the predicted data Abaqus Users Conference

14 Figure 17. Comparison between dynamic data for a silicone rubber and the best fit of the DBB model. 6. Conclusions It is clear that hyperelasticity is not suitable for predicting the dynamic response of elastomers. It is more interesting that linear viscoelasticity is not capable of predicting the strain amplitude dependence. This significant limitation is caused by the linear superposition principle that is the foundation of linear viscoelasticity. The Bergstrom-Boyce (BB) model, which is available as a built-in feature of Abaqus/Standard, provides more accurate predictions than linear viscoelasticity. The main reason for the improvement is that the BB model is based on a flow rule in which the rate of viscoelastic flow depends on the driving stress raised to a power m. The BB model has the limitation that the peak magnitude of the loss modulus is determined by the reduction in storage modulus with increasing strain amplitude, and cannot be independently specified. This limitation can be overcome by making the equilibrium response of the material strain dependent. This modified model, called the dynamic Bergstrom-Boyce (DBB) model, has been implemented as an Abaqus user-material (UMAT and VUMAT) model. The new DBB model is shown to capture the main features of the dynamic response of elastomers Abaqus Users Conference

15 7. References 1. Bergstrom, J. S., and M. C. Boyce, Constitutive Modeling of the Large Strain Time- Dependent Behavior of Elastomers," J. Mech. Phys. Solids, vol. 46, pp , Bergstrom, J. S., and M. C. Boyce, "Mechanical Behavior of Particle Filled Elastomers," Rubber Chem. Tech., vol. 72, pp , Bergstrom, J. S., and M. C. Boyce, "Large Strain Time-Dependent Behavior of Filled Elastomers," Mechanics of Materials, vol. 32, pp , Bergstrom, J. S., and M. C. Boyce, "Constitutive Modeling of the time-dependent and cyclic loading of elastomers and application to soft biological tissues," Mechanics of Materials, vol. 33, pp , Chazeau, L., J. D. Brown, L. C. Yanyo and S. S. Sternstein, Modulus Recovery Kinetics and Other Insights Into the Payne Effect for Filled Elastomers, Polymer Composites, vol. 21, No. 2, Acknowledgment Special thanks are due to Will Mars of Cooper Tire & Rubber Company, Dinesh Panneerselvam and Tod Dalrymple of SIMULIA Great Lakes Region, for helpful discussions. This work was supported by PolymerFEM.com Abaqus Users Conference