Estudo sobre diferentes medidas de deformação e apresentação da família seno-hiperbólica generalizada, com aplicação na modelagem de polímeros, biomateriais e tecidos biológicos moles

Abstract

This work aims to contribute to the study of large deformations by examining the Seth-Hill strain measures and their associated Hookean-type hyperelastic models. Additionally, it introduces a new family of strain measures called Generalized Hyperbolic Sine (GHS): simple, objective, and versatile. The research consisted of two branches. In the first branch, (i) the distinctions in the computational implementation of different strain measures in the Positional Formulation of the Finite Element Method (PFFEM) for trusses were investigated. Specific mathematical expressions for each strain measure were developed and applied in the analysis of four benchmark structures. Examining the obtained responses, the resulting variations were discussed with a focus on computational efficiency. In the second branch, (ii) an analytical study of solids was conducted, investigating Hookean-type hyperelastic models derived from the studied strain measures (GHS and Seth-Hill), using pure deformation modes to examine the behavior of the GHS family and assess the physical coherence of all responses obtained. The applicability of the GHS Hookean-type models in the mechanical description of polymers, soft biological tissues, and biomaterials was tested for various materials with broad application in engineering, using experimental data obtained from the literature. The calibration of material parameters was performed through a discrete numerical method of least squares minimization by enumeration. In the first branch, (i) the results indicate that the use of different strain measures impacts the computational cost in structural analyses of trusses, both when comparing different Hookean-type constitutive models and when comparing the same model represented by formulations generated by different strain measures. In the second branch, (ii) the results show that the GHS family improves the physical coherence of the Seth-Hill strain measures and offers versatility with the addition of a single material constant with clear physical meaning. The proposed Hookean-type hyperelastic models are applicable to both incompressible and compressible materials and have proven capable of accurately representing polymers, soft biological tissues, and biomaterials.