Kirchhoff–Love shells within strain gradient elasticity: Weak and strong formulations and an [formula omitted]-conforming isogeometric implementation

Abstract

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 H3 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 Cp−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.