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.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

AGUIAR, ADAIR R.; ROCHA, LUCAS A.. A minimization theory infinite elasticity to prevent self-intersection. INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, v. 310, p. 21-pg., 2025-01-13. (22/07083-8)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

ROCHA, Lucas Almeida. A constrained minimization theory to prevent self-intersection in hyperelastic solids. 2025. Doctoral Thesis - Universidade de São Paulo (USP). Escola de Engenharia de São Carlos (EESC/SBD) São Carlos.