A three‐dimensional <i>C</i>¹ finite element for gradient elasticity

Abstract

Abstract In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C 1 continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C 1 finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C 1 elements is of particular concern. In this paper we present a new C 1 hexahedral element which, to the best of our knowledge, is the first three‐dimensional C 1 element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C 1 elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley &amp; Sons, Ltd.