Asymptotic dimension reduction of a Robin-type elasticity boundary value problem in thin beams

Abstract

We consider a linear elasticity boundary value problem in a beam with Robin boundary condition at an end and on a segment of the lateral boundary in the middle of the beam. The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard asymptotic approach with an additional expansion suggested to fulfil the Robin conditions. The 3D Robin conditions result into 1D Robin boundary conditions for corresponding ODEs. The asymptotic error is estimated and illustrated by a numerical comparison of the 3D and 1D solutions.

Keywords:

• linear elasticity
• robin type boundary condition
• asymptotic dimension reduction
• classical 1D rod- and beam models

AMS Subject Classifications:

• 34E05
• 34E13
• 74B05
• 74K10
• 74M15

Acknowledgments

The collaboration was financially supported by the Research Federative Structures MODMAD FED 4169 and FR CNRS 3490, by the French-German grant PROCOPE EGIDE 28481WB “Homogenization based optimization for elasticity on the network of beams”. By the Ministry of Education and Science of Russian Federation: grant “Construction, analysis and application of the methods for the boundary value and initial boundary value problems with multiple scales” No. 14.B37.21.0869. By LABEX MILYON (ANR-10-LABX-0070) of University of Lyon, within the programme “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Particularly by the project OR 190/4-1 “Mehrskalenmodellierung und -simulation der Mechanik gewebter Strukturen” funded by the German Research Foundation (DFG).