Abstract
Dispersive wave propagation is simulated with a continuum elasticity theory that incorporates gradients of strain and inertia. The additional parameters are the Representative Volume Element (RVE) sizes in statics and dynamics, respectively. For the special case of a periodic laminate, expressions for these two RVE sizes can be provided based on the properties of the two components. The fourth-order governing equations are rewritten in two sets of coupled second-order equations, whereby the two sets of unknowns are the macroscopic displacements and the microscopic displacements. The resulting formulation is thus a true multi-scale continuum. In a numerical wave propagation example it is shown that the higher-order continuum model provides an excellent approximation of an explicit model of the heterogeneous laminate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aifantis E.C. (1992). On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science 30, 1279-1299
Askes H., Aifantis E.C. (2006). Gradient elasticity theories in statics and dynamics - a unification of approaches. International Journal of Fracture 139, 297-304
Askes H. Bennett T., Aifantis E.C. (2007). A new formulation and C 0 -implementation of dynamically consistent gradient elasticity. International Journal for Numerical Methods in Engineering 72, 111-126
Chen W., Fish J. (2001). A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales. ASME Journal of Applied Mechanics 68, 153-161
Engelbrecht J., Berezovski A., Pastrone F., Braun M. (2005). Waves in microstructured materials and dispersion. Philosophical Magazine 85, 4127-4141
Georgiadis H.G., Vardoulakis I., Lykotrafitis G. (2000). Torsional surface waves in a gradient-elastic half-space. Wave Motion 31, 333-348
Gitman I.M., Askes H., Aifantis E.C. (2005). The representative volume size in static and dynamic micro-macro transitions. International Journal of Fracture 135, L3-L9
Metrikine A.V., Askes H. (2002). One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation. European Journal of Mechanics A/Solids 21, 555-572
Mindlin R.D. (1964). Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 52-78
Peerlings R.H.J., de Borst R. Brekelmans W.A.M., de Vree J.H.V., Spee I. (1996). Some observations on localisation in non-local and gradient damage models. European Journal of Mechanics A/Solids 15, 937-953
Ru C.Q., Aifantis E.C. (1993). A simple approach to solve boundary-value problems in gradient elasticity. Acta Mechanica 101, 59-68
Rubin M.B., Rosenau P., Gottlieb O. (1995). Continuum model of dispersion caused by an inherent material characteristic length. Journal of Applied Physics 70, 4054-4063
Wang Z.-P., Sun C.T. (2002). Modeling micro-inertia in heterogeneous materials under dynamic loading. Wave Motion 36, 473-485
Zervos A. (2008). Finite elements for elasticity with microstructure and gradient elasticity. International Journal for Numerical Methods in Engineering 72, 564-595
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bennett, T., Gitman, I.M. & Askes, H. Elasticity Theories with Higher-order Gradients of Inertia and Stiffness for the Modelling of Wave Dispersion in Laminates. Int J Fract 148, 185–193 (2007). https://doi.org/10.1007/s10704-008-9192-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-008-9192-8