1D HERMITE ELEMENTS FOR C1-CONTINUOUS SOLUTIONS IN SECOND GRADIENT ELASTICITY

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.

A family of triangular Hermite finite elements complementing the Bogner–Fox–Schmit rectangle

The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles whose sides are parallel to two different straight lines.We propose a new family of triangular Hermite elements that involves straight-sided elements and elements with a curved side. Such an element can be used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain or a domain with a curved part of the boundary and provides continuous differentiability of an approximate solution in thewhole domain up to the boundary.

Analysis of Euler-Bernoulli Beam with Piecewise Quadratic Hermite Finite Elements

The piecewise quadratic Hermite polynomials are employed in the finite element context to analyze the static and free vibration behaviors of Euler-Bernoulli beam. The desirable C1 continuity is achieved for the piecewise quadratic Hermite element that is required for the numerical solution of the Galerkin weak form of Euler-Bernoulli beam. In contrast to the classical cubic Hermite element, the piecewise quadratic Hermite element has a piecewise constant curvature representation within each element and thus the integration of the stiffness matrix is trivial. Several benchmark problems are shown to demonstrate the convergence properties of the piecewise quadratic Hermite element. The frequency error of the beam free vibration with this quadratic Hermite element is derived as well. Numerical examples consistently verify the analytical convergence rates.