Formula omitted] finite element analysis in gradient enhanced continua

Abstract

Finite element analysis of the gradient enhanced elastic continuum requires C1 continuity across the element boundaries. However the use of C1 finite elements is usually avoided due to the complexity in their implementation and the excessive number of primary unknowns at each node. Hence except for a very few solid mechanics problems such as plate bending, the potential capabilities of C1 finite elements are not widely explored. This paper presents a series of analyses where two benchmark problems of gradient elasticity are tackled by using C1 finite elements in which displacements and their first and second derivatives are the primary nodal unknowns. It is shown that very accurate solutions are achieved with relatively coarse finite element meshes. Moreover, it is shown that C1 finite elements are very useful for elastoplastic analyses where a large number of C0 elements are usually needed to achieve accurate results. An elastoplastic analysis of bending in a deep beam shows that even a very coarse mesh can provide highly accurate results. Successful performance of C1 finite elements in the above mentioned problems suggests that C1 elements are useful alternatives for numerical investigation of the issues of size effect and strain localization in the response of elastic and elastoplastic systems.