Modeling of anisotropic hyperelastic material

Abstract

This master’s thesis presents a general study of isotropic and anisotropic hyperelastic models. Different ap- proaches are proposed to model anisotropic materials in the literature. In this work, only anisotropic materials with one and two fiber families are considered. The chosen approach considers an additive decomposition of the strain energy density into an isotropic contribution and an anisotropic contribution for each fiber family. It neglects the coupling effects between the fibers and the matrix and between the fiber families. The materi- als are tested under moderate and large deformation although they may not represent practical feasible cases. This work contains four parts: (i) a presentation of the important notions and concepts, followed by the different approaches found in the literature to model hyperelastic materials; (ii) a study of isotropic material models; (iii) a study of anisotropic material models with one fiber family; and (iv) a study of anisotropic material models with two fiber families. Firstly, the principles of continuum mechanics are reminded to apply them to the hyperelastic materials. Then, the expression of the stresses as a function of the strain energy density is introduced. It is expressed thanks to a representation theorem for the isotropic materials but different approaches coexist for anisotropic materials in the literature. In this work, coupling effects are neglected. Therefore, two anisotropic invariants are required to model each fiber family. Based on it, several isotropic and anisotropic constitutive models are presented such as the generalized Neo-Hookean model, the Yeoh model, the Gent-Thomas model, and the Holzapfel-Gasser-Ogden (HGO) model. Secondly, three isotropic models are studied and compared in detail: the generalized Neo-Hookean model, the Yeoh model, and the Gent-Thomas model. They behave distinctly. On the one hand, the stresses obtained with the generalized Neo-Hookean and Yeoh models are similar in small deformation while they differ with the Gent-Thomas model; on the other hand, in large deformations, the ones obtained with the generalized Neo-Hookean and Gent-Thomas models are similar while they differ with the Yeoh model. Then, parameters found in the literature enable to compare the stresses for rubber and porcine gray matter. They are qualita- tively identical, the difference is only quantitative. Finally, the HGO model is chosen to simulate the behavior of anisotropic materials with one and two fiber families. Simple deformations are first studied analytically and with a Python procedure to study the influ- ence of the fiber orientation and the addition of a second fiber family. For a simple shear, the maximum shear stresses are obtained for different fiber orientations during the deformation as the fiber orientation changes. However, for an isochoric tension, the components 11 of the stresses are always maximum for fibers in the direction of the tension. Then, the HGO model is implemented in the software Metafor as a user-defined material. The previous simple deformations are tested to validate the implementation. Afterwards, more complex deformations in 2D and 3D are considered, highlighting the effect of the fibers and their orientations on the deformations and the stresses.