Analytical and numerical investigation of the equilibrium of an anisotropic nonlinear elastic solid using a minimization theory with an injectivity constraint

• Grant number: 22/07083-8
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: October 01, 2022
• End date: February 28, 2025
• Field of knowledge: Engineering - Civil Engineering - Structural Engineering
• Principal Investigator: Adair Roberto Aguiar
• Grantee: Lucas Almeida Rocha
• Host Institution: Escola de Engenharia de São Carlos (EESC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The classical theory of linear elasticity predicts spurious phenomena, such as the self-intersection of matter, in the vicinity of interior points of anisotropic solids, corners, and crack tips. The self-intersection phenomenon is associated with the violation of the kinematical condition J > 0, where J is the determinant of the deformation gradient near these points. One way to impose J > 0 combines the classical theory of linear elasticity with a Lagrange multiplier technique. The associated constrained minimization problem is highly nonlinear; it may admit more than one minimizer and, in general, requires a numerical solution. This constrained minimization theory, together with a penalty formulation, has been used in theoretical and numerical investigations. In this work, we shall extend our investigations to the context of the nonlinear elasticity theory since the violation of J > 0 is associated with finite strains. We shall study the equilibrium of an annular disk composed of a nonlinearly elastic and cylindrically anisotropic material, fixed on its inner surface, and subjected to an external uniform pressure load. The case of a solid disk will be considered as a particular case in which the inner radius tends to zero. We shall use different models for the nonlinearly elastic material. The constraint J > 0 shall be imposed numerically by penalty and augmented Lagrangian methods. This research is of interest in the investigation of solids having a stiffer response in the radial direction than in the tangential direction, such as in the case of carbon fibers with radial microstructure, certain types of woods, and fiber-reinforced composites.