Kirchhoff–Love shells within strain gradient elasticity : Weak and strong formulations and an H3-conforming isogeometric implementation

Viacheslav Balobanov*, Josef Kiendl, Sergei Khakalo, Jarkko Niiranen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

35 Citations (Scopus)
111 Downloads (Pure)


A strain gradient elasticity model for shells of arbitrary geometry is derived for the first time. The Kirchhoff-Love shell kinematics is employed in the context of a one-parameter modification of Mindlin's strain gradient elasticity theory. The weak form of the static boundary value problem of the generalized shell model is formulated within an H-3 Sobolev space setting incorporating first-, second- and third-order derivatives of the displacement variables. The strong form governing equations with a complete set of boundary conditions are derived via the principle of virtual work. A detailed description focusing on the non-standard features of the implementation of the corresponding Galerkin discretizations is provided. The numerical computations are accomplished with a conforming isogeometric method by adopting C-P(-1)-continuous NURBS basis functions of order p >= 3. Convergence studies and comparisons to the corresponding three-dimensional solid element simulation verify the shell element implementation. Numerical results demonstrate the crucial capabilities of the non-standard shell model: capturing size effects and smoothening stress singularities. (C) 2018 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)837-857
Number of pages21
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 1 Feb 2019
MoE publication typeA1 Journal article-refereed


  • Convergence
  • Isogeometric analysis
  • Kirchhoff–Love shell
  • Size effects
  • Strain gradient elasticity
  • Stress singularities


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