Nonlinear extension of Kolosov-Muskhelishvili stress function formalism

TL;DR

This paper extends the Kolosov-Muskhelishvili stress function formalism to nonlinear elasticity, enabling complex analysis methods to address large deformation problems near singularities like cracks.

Contribution

It introduces a nonlinear stress function formalism based on the classical complex representation, broadening the scope of analytical solutions in nonlinear elastic fracture mechanics.

Findings

Nonlinear solutions depend strongly on remote loads.

Universal near-tip solutions are invalid in nonlinear regimes.

Previous linear-based crack analyses may be inadequate.

Abstract

The method of stress-function in elasticity theory is a powerful analytical tool with applications to a wide range of physical systems, including defective crystals, fluctuating membranes, and more. A complex coordinates formulation of stress function, known as Kolosov-Muskhelishvili formalism, enabled the analysis of elastic problems with singular domains, particularly cracks, forming the basis for fracture mechanics. A shortcoming of this method is its limitation to linear elasticity, which assumes Hookean energy and linear strain measure. Under finite loads, the linearized strain fails to describe the deformation field adequately, reflecting the onset of geometric nonlinearity. The latter is common in materials experiencing large rotations, such as regions close to the crack tip or elastic metamaterials. While a nonlinear stress function formalism exists, Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for nonlinear stress function. The new formalism allows us to port methods from complex analysis to nonlinear elasticity and to solve nonlinear problems in singular domains. Upon implementing the method to the crack problem, we discover that nonlinear solutions strongly depend on the applied remote loads, excluding a universal form of the solution close to the crack tip and questioning the validity of previous studies of nonlinear crack analysis.