Nonlinear static isogeometric analysis of arbitrarily curved Kirchhoff-Love shells

TL;DR

This paper develops a nonlinear isogeometric analysis framework for arbitrarily curved Kirchhoff-Love shells, emphasizing the importance of full shell metric and strain distribution for accurate structural response predictions.

Contribution

It introduces a rigorous nonlinear shell analysis method incorporating the full metric and elastic relations, and compares simplified models for efficiency and accuracy.

Findings

Full metric consideration is often necessary for accurate shell response.

The proposed formulation is robust and efficient for strongly curved shells.

A simplified model balances accuracy and computational cost effectively.

Abstract

The geometrically rigorous nonlinear analysis of elastic shells is considered in the context of finite, but small, strain theory. The research is focused on the introduction of the full shell metric and examination of its influence on the nonlinear structural response. The exact relation between the reference and equidistant strains is employed and the complete analytic elastic constitutive relation between energetically conjugated forces and strains is derived via the reciprocal shift tensor. Utilizing these strict relations, the geometric stiffness matrix is derived explicitly by the variation of the unknown metric. Moreover, a compact form of this matrix is presented. Despite the linear displacement distribution due to the Kirchhoff-Love hypothesis, a nonlinear strain distribution arises along the shell thickness. This fact is sometimes disregarded for the nonlinear analysis of thin shells based on the initial geometry, thereby ignoring the strong curviness of a shell at some subsequent configuration. We show that the curviness of a shell at each configuration determines the appropriate shell formulation. For shells that become strongly curved at some configurations during deformation, the nonlinear distribution of strain throughout the thickness must be considered in order to obtain accurate results. We investigate four computational models: one based on the full analytical constitutive relation, and three simplified ones. Robustness, efficiency and accuracy of the presented formulation are examined via selected numerical experiments. Our main finding is that the employment of the full metric is often required when the complete response of the shells is sought, even for the initially thin shells. Finally, the simplified model that provided the best balance between efficiency and accuracy is suggested for the nonlinear analysis of strongly curved shells.