Higher-order responses of three-dimensional elastic plate structures and their numerical illustration by p-FEM

The displacement of three-dimensional linearly elastic plate domains can be expanded as a compound power-series asymptotics, when the thickness parameter ε tends to zero. The leading term u° in this expansion is the well-known Kirchhoff-Love displacement field, which is the solution to the limit case when ε → 0. Herein, we focus our discussion on plate domains with either clamped or free lateral boundary conditions, and characterize the loading conditions for which the leading term vanishes. In these situations the first non-zero term u^k in the expansion appears for k = 2,3 or 4 and is denoted as higher-order response of order 2,3 or 4, respectively. We provide herein explicit loading conditions under which higher order responses in three-dimensional plate structures are visible, and demonstrate the mathematical analysis by numerical simulation using the p-version finite element method. Owing to the need for highly accurate results and 'needle elements' (having extremely large aspect ratio up to 10 000), a p-version finite element analysis is mandatory for obtaining reliable and highly accurate results.