Finite Element Model Updating Combined with Multi-Response Optimization for Hyper-Elastic Materials Characterization

The experimental stress-strain curves from the standardized tests of Tensile, Plane Stress, Compression, Volumetric Compression, and Shear, are normally used to obtain the invariant λi and constants of material C_i that will define the behavior elastomers. Obtaining these experimental curves requires the use of expensive and complex experimental equipment. For years, a direct method called model updating, which is based on the combination of parameterized finite element (FE) models and experimental force-displacement curves, which are simpler and more economical than stress-strain curves, has been used to obtain the C_i constants. Model updating has the disadvantage of requiring a high computational cost when it is used without the support of any known optimization method or when the number of standardized tests and required C_i constants is high. This paper proposes a methodology that combines the model updating method, the mentioned standardized tests and the multi-response surface method (MRS) with desirability functions to automatically determine the most appropriate C_i constants for modeling the behavior of a group of elastomers. For each standardized test, quadratic regression models were generated for modeling the error functions (ER), which represent the distance between the force-displacement curves that were obtained experimentally and those that were obtained by means of the parameterized FE models. The process of adjusting each C_i constant was carried out with desirability functions, considering the same value of importance for all of the standardized tests. As a practical example, the proposed methodology was validated with the following elastomers: nitrile butadiene rubber (NBR), ethylene-vinyl acetate (EVA), styrene butadiene rubber (SBR) and polyurethane (PUR). Mooney⁻Rivlin, Ogden, Arruda⁻Boyce and Gent were considered as the hyper-elastic models for modeling the mechanical behavior of the mentioned elastomers. The validation results, after the C_i parameters were adjusted, showed that the Mooney⁻Rivlin model was the hyper-elastic model that has the least error of all materials studied (MAEnorm = 0.054 for NBR, MAEnorm = 0.127 for NBR, MAEnorm = 0.116 for EVA and MAEnorm = 0.061 for NBR). The small error obtained in the adjustment of the C_i constants, as well as the computational cost of new materials, suggests that the methodology that this paper proposes could be a simpler and more economical alternative to use to obtain the optimal C_i constants of any type of elastomer than other more sophisticated methods.