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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 20
TRENDS IN ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis, B.H.V. Topping
Chapter 10
A MultiScale Formulation of Gradient Elasticity and Its Finite Element Implementation H. Askes^{1}, T. Bennett^{1}, I.M. Gitman^{1} and E.C. Aifantis^{2,3}
^{1}Department of Civil and Structural Engineering, University of Sheffield, United Kingdom H. Askes, T. Bennett, I.M. Gitman, E.C. Aifantis, "A MultiScale Formulation of Gradient Elasticity and Its Finite Element Implementation", in M. Papadrakakis, B.H.V. Topping, (Editors), "Trends in Engineering Computational Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 10, pp 189208, 2008. doi:10.4203/csets.20.10
Keywords: gradient elasticity, higherorder continuum, multiscale methods, representative volume element, wave dispersion, wave propagation.
Summary
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 [1]
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 [2].
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 fourthorder spatial derivatives. To overcome this drawback, an operator split in the spirit of the RuAifantis theorem [3] has been developed, by which the model can be rewritten as two fully coupled sets of secondorder equations [4]. The primary unknowns of these equations can be shown to be the microscopic displacement field and the macroscopic displacement field. Thus, a true multiscale 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 gradientenriched theories can be obtained from the homogenisation of an RVE, (ii) how the fourthorder equations can be rewritten as a symmetric set of coupled secondorder 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. References
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