A minimization theory in finite elasticity to prevent self-intersection

Abstract

The theory of classical linear elasticity predicts self-intersection in the neighborhood of interior points of anisotropic solids, crack tips, and corners. This physically unrealistic behavior is characterized by the violation of the local injectivity condition, according to which, the determinant of the deformation gradient, $J\triangleq detF$, must be positive. One way to impose this condition in elasticity consists of minimizing the total potential energy subjected to the condition $J\ge \varepsilon >0$, where $\varepsilon$ is a small positive parameter.

We present a minimization theory constrained by $J\ge \varepsilon >0$ for hyperelastic solids undergoing finite deformations and derive necessary conditions for a deformation field to be a minimizer, which include both continuity of traction and dissipation-free conditions across a surface of discontinuity. We then apply this theory in the analysis of equilibrium of an annular disk made of an orthotropic St Venant-Kirchhoff material. This material is a natural constitutive extension of its classical linear counterpart. The disk is fixed on its inner surface and compressed by a constant pressure on its outer surface.

The disk problem is formulated as both a boundary value problem (disk BVP) and a minimization problem (disk MP), which are solved in the context of both the classical and the constrained ($J\ge \varepsilon$) nonlinear theories. These formulations yield non-smooth solutions for large enough pressure, which pose numerical difficulties. To address these difficulties, we use a phase-plane technique to construct a trajectory of solution for the disk BVP and the finite element method together with nonlinear programming tools to find a minimizer for the disk MP.

In the classical nonlinear theory, we find that there is a critical pressure $\bar{p}$, which tends to zero as the inner radius of the disk tends to zero, above which a solution of either the disk BVP or the disk MP becomes non-smooth and predicts $J\le 0$. In addition, $\bar{p}$ is smaller than its counterpart predicted by the classical linear theory and, therefore, serves as an upper bound below which the linear theory is valid.

In the constrained nonlinear theory, the solutions of both the disk BVP and the disk MP agree very well and satisfy all the necessary conditions for an admissible minimizer, including the injectivity condition. Analytical and numerical results show that, for an annular disk, the Lagrange multiplier field associated with the imposition of the local injectivity constraint remains bounded as $\varepsilon$ tends to zero. This behavior is different from the one reported in the literature for the disk problem formulated in the context of a constrained linear theory. In this case, the Lagrange multiplier becomes unbounded as $\varepsilon$ tends to zero.