The constitutive behavior of a hyperelastic material is described in terms of a strain energy potential, which defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point in the material. Four strain energy potential forms were used in the finite element model: Arruda-Boyce, Marlow, Mooney-Rivlin, and the reduced polynomial form (N=2).
To help choose the most appropriate strain energy potential form for a given material, uniaxial and volumetric test data was directly input into the material calibration tool. For each strain energy potential, the response for each loading mode (uniaxial, biaxial, planar, and simple shear) was computed and plotted. In addition, the strain ranges for numerical stability of each potential were determined. Based on the stability check and a visual match with the test data, the appropriate hyperelastic material model was chosen.
The next step was to calibrate the strain rate effects in the hyperelastic material. The viscoelastic part of the material response was modeled using a Prony series expansion of the dimensionless relaxation modulus. The Prony series coefficients were calibrated using a simulation process automation and design optimization software package that provides a suite of visual and flexible tools for creating simulation process flows (sim-flows) to automate the exploration of design alternatives and to identify optimal performance parameters.
An optimization sim-flow was constructed using the Prony terms as design variables. Different FEA analyses representing various loading rates both in tension and compression were used in the same optimization sim-flow for calibration. The resulting simulation responses were matched against the test data using the Data Matching component of the sim-flow.
Various statistical measures, such as sum of the squared difference, correlation factor, and area between the two curves, were used to quantify the deviation between the simulation and experimental curves. Various exploratory design space techniques, as well as the Pointer technique, were used in this multi-scenario, multi-objective optimization problem to find the best combination of Prony series coefficients that would minimize the difference between the simulation and experimental results for all loading types and rates considered.
The exploratory techniques (various types of genetic algorithms, adaptive simulated annealing, etc.) were selected because of their ability to cover the entire design space in an economical fashion, while offering the best practical chance of identifying solutions within reasonable tolerance of the global optimum, even for highly nonlinear design spaces. The Pointer technique uses a proprietary algorithm that allows it to switch between a set of complimentary optimization algorithms, enabling it to efficiently solve a wide range of problems in a fully automatic manner.
Figure 2 shows the optimization simflow used for the Prony series calibration for two loading rates. In Figure 3, the simulated test results are compared using the optimized properties against the experimental data for the material used in the molded rubber neck. The test specimens were subjected to dynamic compression with strain rates of 20/s and 100/s.
A variety of component-level experimental tests was simulated using the calibrated material models. These include sled-impulse tests (for neck and lumbar spine assemblies) and pendulum tests (for arm, iliac wings, and thorax ribs assemblies). Each test was repeated for various configurations of impact speed, impact location, impact angle, and impactor weight. The objective of the tests was to validate the calibrated material parameters and the subassembly models. All the component tests were designed by PDB to be representative of the loading conditions observed during a typical crash event.