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Materials (Basel). 2019 Mar 27;12(7). pii: E1019. doi: 10.3390/ma12071019.

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

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Department of Mechanical Engineering, University of La Rioja, 26004 Logroño, La Rioja, Spain.
Department of Mechanical Engineering, University of La Rioja, 26004 Logroño, La Rioja, Spain.
IK4-LORTEK, 20240 Ordizia, 20240 Guipuzcoa, Spain.
Department of Mechanical Engineering, University of La Rioja, 26004 Logroño, La Rioja, Spain.


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 Ci 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 Ci 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 Ci 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 Ci 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 Ci 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 Ci 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 Ci 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 Ci constants of any type of elastomer than other more sophisticated methods.


finite element method; hyperelastic materials; model updating; multi-response optimization

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