Open Access1D HERMITE ELEMENTS FOR C<sup>1</sup>-CONTINUOUS SOLUTIONS IN SECOND GRADIENT ELASTICITY
Author(s) -
Christian Liebold,
Wolfgang Müller
Publication year - 2016
Publication title -
acta polytechnica ctu proceedings
Language(s) - English
Resource type - Journals
ISSN - 2336-5382
DOI - 10.14311/app.2017.7.0033
Subject(s) - finite element method , elasticity (physics) , isotropy , mathematical analysis , mathematics , stiffness matrix , length scale , stiffness , hermite polynomials , bending stiffness , strain energy , linear elasticity , geometry , physics , classical mechanics , mechanics , optics , thermodynamics
We present a modified strain gradient theory of elasticity for linear isotropic materials in order to account for the so-called size effect. Additional material length scale parameters are introduced and the problem of static beam bending is analyzed. A numerical solution is derived by means of a finite element approach. A global C1-continuous displacement field is applied in finite element solutions because the higher-order strain energy density additionally depends on second gradients of displacements. So-called Hermite finite elements are used that allow for merging gradients between elements. The element stiffness matrix as well as the global stiffness matrix of the problem is developed. Convergence, C1-continuity and the size effect in the numerical solution is shown. Experiments on bending stiffnesses of different sized micro beams made of the polymer SU-8 are performed by using an atomic force microscope and the results are compared to the numerical solution.