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TRENDS IN ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: M. Papadrakakis, B.H.V. Topping
A Multi-Scale Formulation of Gradient Elasticity and Its Finite Element Implementation
H. Askes1, T. Bennett1, I.M. Gitman1 and E.C. Aifantis2,3
1Department of Civil and Structural Engineering, University of Sheffield, United Kingdom
H. Askes, T. Bennett, I.M. Gitman, E.C. Aifantis, "A Multi-Scale Formulation of Gradient Elasticity and Its Finite Element Implementation", in M. Papadrakakis, B.H.V. Topping, (Editors), "Trends in Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 10, pp 189-208, 2008. doi:10.4203/csets.20.10
Keywords: gradient elasticity, higher-order continuum, multi-scale methods, representative volume element, wave dispersion, wave propagation.
Dispersive wave propagation occurs when the different harmonic components of a wave propagate with different velocities. Heterogeneity of the structure or the material causes dispersion, and for a physically relevant modelling of wave dispersion this heterogeneity must be accounted for. Rather than modelling every heterogeneity individually, it is more efficient to formulate enriched continuum models, such as for instance gradient elasticity theories. In the gradient elasticity theory formulated by Aifantis  the stress depends not only on the strain but also on the second derivative of the strain. This format of gradient elasticity has successfully been applied to remove strain singularities from sharp crack tips. An extension of this particular gradient elasticity theory has more recently been suggested for use in dynamics; this theory not only incorporates strain gradients but also inertia gradients . The formulation of this particular theory was based on the homogenisation of the response within a representative volume element (RVE).
The numerical implementation of such theories is hampered by the continuity requirements imposed on the interpolation functions, as dictated by the fourth-order spatial derivatives. To overcome this drawback, an operator split in the spirit of the Ru-Aifantis theorem  has been developed, by which the model can be rewritten as two fully coupled sets of second-order equations . The primary unknowns of these equations can be shown to be the microscopic displacement field and the macroscopic displacement field. Thus, a true multi-scale model is obtained in which the two displacement fields are fully interacting. We will discuss the physical and mathematical backgrounds of this model, in particular (i) how gradient-enriched theories can be obtained from the homogenisation of an RVE, (ii) how the fourth-order equations can be rewritten as a symmetric set of coupled second-order equations, (iii) the identification of the microscopic and macroscopic displacements, and (iv) the finite element implementation. The model will then be used to simulate dispersive wave propagation in heterogeneous media.
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