Isight is a powerful tool for material calibration. In this post, I will provide an example of using Isight to calibrate nonlinear viscoelastic material.
Parallel rheological framework in Abaqus 6.13 can be used to model non-linear viscoelasticity. The frame work consists of an arbitrary number of viscoelastic networks connected in parallel. In addition, an equilibrium network can be added with purely hyperelastic behavior, or elasto-plastic (*HYPERELASTIC with *PLASTIC) behavior. But plasticity is not considered for this calibration.
The challenge of this calibration is the large number of material parameters. For power-law strain hardening model, each viscous network requires four parameters (A, n, m and S), plus the parameters for the elastic network, the total design parameters could easily exceed ten.
We first calibrated the hyperelastic parameters using the ramp data with fast strain rate, and then we calibrated the viscous parameters using two strategies. In Strategy 1, we performed a multi-step calibration with each step optimizing a subset of parameters using direct optimization approach in Isight. In Strategy 2, we performed a single step calibration optimizing all parameters using pointer 2 approach in Isight.
- Test data
To characterize the viscoelastic response, ramp-hold test data were used. During ramp phase, the specimen was uniaxially stretched with the same strain rate (25) to three strain magnitudes (10%, 25% and 100%) (shown in a). During hold phase, the same strain magnitudes were maintained, while stress relaxation responses were recorded (b). Real time is converted to log time to better match the ramp curve and stress peak (c).
2. Hyperelastic calibration
We evaluate hyperelastic parameters using the stress-strain curve of the fast ramp data in Abaqus/CAE.
3. Viscoelastic calibration: strategy one (multi-step)
A traditional way of calibrating the viscoelastic parameters takes a multi-step approach. First the short term relaxation response of one curve is matched by optimizing one viscous network parameters. Second, by keeping the first network parameters at their optimal values, the second network is added and optimized by matching the long term relaxation response of the same curve. At the end of this step, usually good match could be achieved for one data set. However, large errors still exist for matching other data sets. Therefore, thirdly, the parameters for two networks were adjusted systematically by matching all three curves. If the matches are still not satisfactory, the third network is added and optimized. Additional network can be added until good matches achieved for all data sets.
During this optimization, one may find that hyperelastic parameters require adjustment as well, because the ramp data, allowing specimen relaxation, does not represent pure instantaneous response of the material.
Since each step only optimizes a small number of parameters (2-4), direct optimization approach (e.g. down-hill simplex) was used and convergences were usually achieved within 100 iterations for each step. However, this approach is prone to error due to multiple adjustments of the workflow. It also requires some knowledge of the material model.
4. Viscoelastic calibration strategy two (single step)
Pointer 2 is a powerful exploration approach that combined multiple optimization approaches, it is ideal for complicated unknown design space with potentially multiple optimums. In the second strategy, we tested the capability of pointer 2 in optimizing all the viscoelastic parameters, and one hyperelastic parameter (to correct the relaxation occurred during ramp). We set the viscoelastic parameters at some arbitrary starting points.
5. Results
Both approaches could find optimal solution for hyperelastic and viscoelastic parameters that matches all three test curves. Using pointer 2 approaches reduced number of Abaqus runs and the computational time to half of those of the multi-step approach.
The optimized parameters and their initial guesses are listed here:
This movie shows the optimization result of using Pointer2 approach.
Using pointer 2 allows a single step optimization of material parameters as large as 13 to match three stress relaxation responses within one hour. Further testing of using parallel execution may further reduce the calibration time.
