Proceedings of the Institution of Civil Engineers Engineering and Computational Mechanics 163 June 2010 Issue EM2 Pages 73–82 doi: 10.1680/eacm.2010.163.2.73 Paper 900012 Received 21/02/2009 Accepted 21/04/2009 Keywords: mathematical modelling/strength & testing of materials/stress analysis Stefanos-Aldo Papanicolopulos Postdoctoral Researcher, Department of Mechanics, National Technical University of Athens, Greece Antonis Zervos Senior Lecturer, School of Civil Engineering and the Environment, University of Southampton, UK Numerical solution of crack problems in gradient elasticity S.-A. Papanicolopulos MSc, PhD and A. Zervos PhD Gradient elasticity is a constitutive framework that takes into account the microstructure of an elastic material. It considers that, in addition to the strains, second-order derivatives of the displacement also affect the energy stored in the medium. Three different yet equivalent forms of gradient elasticity can be found in the literature, reflecting the different ways in which the second-order derivatives can be grouped to form other physically meaningful quantities. This paper presents a general discretisation of gradient elasticity that can be applied to all three forms, based on the finite-element displacement formulation. The presence of higher order terms requires C 1 -continuous interpolation, and some appropriate two- and three-dimensional elements are presented. Numerical results for the displacement, stress and strain fields around cracks are shown and compared with available solutions, demonstrating the robustness and accuracy of the numerical scheme and investigating the effect of microstructure in the context of fracture mechanics. 1. INTRODUCTION Classical continuum mechanics considers constitutive models that do not take into account the microstructure of materials, since its direct effect is negligible in most cases. In cases where the effect of microstructure becomes more pronounced, as happens for example in very small-scale structures or in problems of localisation of deformation, it is necessary to resort to generalised continuum models that do take account for microstructure. These models include characteristic length parameters that enable them to consider the relative size of the geometry of the problem compared with the size of the material microstructure. One generalised model that has received considerable attention is gradient elasticity, as first formulated by Toupin (1962), Mindlin (1964) and Mindlin and Eshel (1968). Various strain gradient theories, which are either proper subsets of Mindlin’s gradient elasticity or similar to it, have been used to model problems of fracture mechanics, considering different modes of crack loading (Altan and Aifantis, 1992; Exadaktylos, 1998; Exadaktylos and Aifantis, 1996; Exadaktylos and Vardoulakis, 2001; Georgiadis, 2003; Unger and Aifantis, 1995; Vardoulakis et al., 1996; Zhang et al., 1998). Very recently, additional results were obtained for mode I and mode II cracks (Aravas and Giannakopoulos, 2009; Gourgiotis and Georgiadis, 2009). Numerical results for crack problems have been obtained by various authors (Akarapu and Zbib, 2006; Amanatidou and Aravas, 2002; Askes and Gutierrez, 2006; Karlis et al., 2007), often as a way of validating specific elements developed for solving gradient elasticity problems. However, only a limited number of results for one specific crack mode is usually provided. Thus, not only is it difficult to compare different numerical formulations but also the opportunity to systematically visualise various qualities of the solution is lost. This paper presents a displacement-only finite-element discretisation of gradient elasticity and the resulting C 1 elements that have been successfully introduced for use with this discretisation. These elements are then used to provide numerical solutions for problems of mode I, II and III cracks for different values of the characteristic length. The numerical results prove the appropriateness of using C 1 elements for fracture problems in gradient elasticity, while at the same time they present various interesting properties of the solution. Section 2 presents the theoretical background of gradient elasticity. Section 3 presents a general description of the displacement-only finite-element discretisation, detailing the requirements that arise when this discretisation is applied to gradient elasticity. Three different finite elements based on this discretisation are presented in Section 4, which are then employed for solving crack problems. The details of the problems considered and the respective results are given in Section 5. Finally, Section 6 briefly presents the main conclusions. 2. THEORY Only the basic equations of gradient elasticity are presented here, as they have been detailed by Mindlin (1964) and Mindlin and Eshel (1968). Only the static small-strain case is considered, in Cartesian coordinates. Index notation is used throughout, where single indices range from one to three and repeated indices are summed from one to three. Gradient elasticity is a hyperelasticity where the potential energy density W is a function of the first and second derivatives of the displacement; this is in contrast to classical elasticity where W is only a function of the first derivatives of displacements. To obtain independence of W from the choice of coordinate system, only the symmetric part of the displacement gradient should be considered, so that Engineering and Computational Mechanics 163 Issue EM2 Numerical solution of crack problems Papanicolopulos • Zervos 73