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Nonlinear extension of Kolosov-Muskhelishvili stress function formalism

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.
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Preprints, Working Papers, ...
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https://hal-cnrs.archives-ouvertes.fr/hal-03859510
Contributor : Mokhtar Adda-Bedia Connect in order to contact the contributor
Submitted on : Friday, November 18, 2022 - 11:43:58 AM
Last modification on : Saturday, November 19, 2022 - 3:51:28 AM

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  • HAL Id : hal-03859510, version 1
  • ARXIV : 2208.13181

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Oran Szachter, Eytan Katzav, Mokhtar Adda-Bedia, Michael Moshe. Nonlinear extension of Kolosov-Muskhelishvili stress function formalism. 2022. ⟨hal-03859510⟩

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