Coupled thermomechanical analysis of viscoelastic dampers

Figure 2 shows a contour of the logarithmic shear strain in the damper at the maximum stroke of the final cycle in the coupled model using hyperelasticity, showing the large deformations present in the viscoelastic damper material. The hysteresis responses predicted using the two different material models are shown in Figure 3 for the coupled thermomechanical analysis and Figure 4 for the uncoupled stress analysis. The importance of accounting for the heat generated by dissipation of mechanical energy is immediately apparent in the way that the hysteresis loops flatten and decrease in area over time in the coupled analyses. These changes occur because the material loses stiffness as the temperature increases. No such decrease is noted in the pure stress analysis—the cyclic hysteresis response reaches a steady state after a single cycle has been completed.

Figure 3 and Figure 4 show that the forces are slightly higher when using the hyperelastic model for the damping material than when using the linear elastic model. This difference is expected due to the nonlinear nature of the stress-strain relationship obtained using hyperelasticity and the use of finite deformation theory.

Figure 5 shows the contour of the temperature in the damper, indicating that some regions of the damper experience a temperature increase of approximately 8 degrees over the 10-second duration of the analysis. This increase is due to the heat generated by mechanical energy dissipation. The generated heat accumulates in the viscoelastic material faster than it is conducted to the cooler steel plates, leading to a steady temperature increase. Figure 6 shows the time history of the temperature at the central node (node 1217) of the damper in the coupled analysis. The temperature rise is more pronounced in the model using hyperelasticity due to the higher stresses (and subsequently greater energy dissipation) experienced when using the hyperelastic material definition.

One additional aspect of interest for this material definition is how the predicted stress relaxation compares to the material’s actual response. Recall that for this viscous damper, the hysteresis response is of primary interest. Therefore, the viscoelastic Prony series was calibrated to capture that aspect of the material response accurately. Shen and Soong performed relaxation experiments on a test fixture using the material in this example. The force response of that test fixture can be approximated using the model from this example by multiplying the sum of the x-direction reaction forces in the upper nodes of the model by 5, permitting a direct comparison between experiment and simulation.

Figure 7 compares the time history of the experimentally measured and simulated reaction forces resulting from loading the damping material to a nominal engineering shear strain of 20% over one second and then holding the displacements of the test fixture constant. As seen in the figure, the force in the simulation drops more quickly than it does in the experiment, and to a much lower value. This result highlights the intrinsic limitation of linear viscoelasticity described in the introduction of this example—it is generally not possible to represent both the hysteresis response and the stress relaxation response accurately using a linear viscoelastic Prony series. Therefore, while the approaches described in this example are appropriate for modeling a viscoelastic material under cyclic loading, the approach described here is not recommended for analyses in which stress relaxation is of primary importance.