High order 1D and 2D CUF models for transversely isotropic compressible and nearly-incompressible soft materials and structures

In the last decade, anisotropic materials have been the subjects of numerous studies due to their wide range of applications in mechanical, aeronautical, and civil engineering. More recent studies have shown also the great importance of the mechanical features possessed by biological–mechanical systems: micro-fluidics problems, bio-inspired material, soft rubber-like cross-ply, or biological tissue deal with these enhanced elastic properties. In this framework, the anisotropic behavior of soft material plays a crucial role: fiber-reinforced elastomeric materials, collagen fibers, muscular tissue, and blood vessels are classical examples of transversely isotropic hyperelastic materials, for which direction-dependent mechanical properties are evidenced. Constitutive equations for isotropic and transversely isotropic hyperelastic materials to model anisotropy are well established, both geometrical and material nonlinearities are taken into account, embedded in the strain energy function approach to hyperelasticity. Due to the limitations of the few available analytical solutions, nowadays finite elements procedures are the most common approach since they allow a wide range of investigations in terms of material properties and topology. The mathematical modeling of an efficient finite element formulation is a current challenging topic due to the almost-incompressible nature of hyperelastic materials: stabilized finite elements are required to contrast volumetric locking that prevents the computation of accurate stress predictions. This work proposes a new finite element formulation for the analysis of transversely isotropic (or continuous fiber-reinforced) hyperelastic materials based on Carrera Unified Formulation. The first part is devoted to the mathematical description of continuum mechanics governing equations for hyperelastic materials: the strain energy functions approach is described and strain and stress measures are here presented. The constitutive law is written in terms of invariants of the right CauchyGreen tensor, by introducing the dependence on the fiber-reinforcement direction with two additional pseudo-invariants depending on the deformation tensor. The analytical expression of the tangent elasticity tensor is carried out independently on the hyperelastic model considered. The second part is devoted to the description of refined CUF models for transversely isotropic hyperelastic materials. In our displacement-based finite element models, Carrera Unified Formulation is adopted: the primary unknown variables are discretized by adopting a recursive index notation, by coupling the classical FEM kinematic expansion with arbitrary expansion functions. The weak form of the governing equation is exploited by the Principle of Virtual Displacement in a Total Lagrangian Formulation and final equations are written in terms of fundamental nuclei, each of them independent of the chosen polynomial expansion of the displacement field, allowing rapid implementation of higher-order refined fully-nonlinear beam and plate models. The numerical solution is computed employing the Newton-Raphson linearization scheme coupled with an arc-length constraint: thus tangent stiffness matrix and internal forces vector are defined and characterized subsequently for 1D beam and 2D plate models. The last part is devoted to the validation of the numerical models by analyzing different problems in hyperelasticity, models calibration, and assessment of strain energy function models for transversely isotropic materials, establishing the capabilities of the present CUF-models in the case of highly nonlinear problems, both for the case of compressible and nearly-incompressible beams and plates, obtaining accurate results.