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Keywords: gradient elasticity, higher-order continuum, length scale, representative volume, wave dispersion, C⁰-continuity.

Summary

Elastic wave propagation through heterogeneous media is generally dispersive, that is, each individual harmonic travels with a different velocity. Modelling dispersive wave propagation is not trivial, since the classical theories of elasticity do not predict any dispersive effects. It is of course possible to model each individual microstructural component individually, but this would put enormous requirements on the computational resources in terms of memory and CPU times. As an alternative, enriched continuum theories can be used on the macrolevel. Gradient elasticity theories are equipped with additional spatial gradients of the displacements, and it is well known that gradient elasticity is dispersive. However, several issues need to be addressed before gradient elasticity can be used successfully in the simulation of dispersive wave propagation:

the model must be unconditionally stable for all wave numbers and the dispersion must be realistic -- this implies that all phase velocities must be real and finite;
the additional material parameters must be linked to microstructural properties for reasons of identification and experimental quantification;
the model must be suitable for implementation in a standard finite element package.

Stable and realistic wave dispersion can be achieved by so-called dynamically consistent gradient elasticity. The term dynamic consistency reflects the simultaneous appearance of higher gradients in inertia and stiffness contributions [1]. Several procedures have been derived to link the appearing length scale parameters to the microstructural properties. In this contribution, the focus will be on the homogenisation of a Representative Volume Element (RVE). If a so-called second-order homogenisation scheme is applied, the length scale parameters are found to be proportional to the size of the RVE [2]. Finite element implementations are not trivial due to the appearance of fourth-order spatial derivatives of the displacements, which would demand C¹-continuity of the interpolation functions. To avoid these stringent continuity requirements, an operator split has been developed [3] by which the original equations are rewritten as a series of second-order equations in terms of the true displacements as well as a set of auxiliary displacements. As such, dispersive wave propagation can be simulated with the commonly used low order, C⁰-continuous finite elements.

References

1: A.V. Metrikine and H. Askes, "An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice", Philosophical Magazine 86, 3259-3286, 2006. doi:10.1080/14786430500197827
2: I.M. Gitman, H. Askes and E.C. Aifantis, "The representative volume size in static and dynamic micro-macro transitions", International Journal of Fracture 135, L3-L9, 2005. doi:10.1007/s10704-005-4389-6
3: H. Askes, T. Bennett and E.C. Aifantis, "A new formulation and C⁰-implementation of dynamically consistent gradient elasticity", International Journal for Numerical Methods in Engineering (accepted for publication).

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 85 PROCEEDINGS OF THE FIFTEENTH UK CONFERENCE OF THE ASSOCIATION OF COMPUTATIONAL MECHANICS IN ENGINEERING Edited by: B.H.V. Topping Paper 5 Formulation and Finite Element Implementation of Dynamically Consistent Gradient Elasticity H. Askes¹, I.M. Gitman² and T. Bennett¹ ¹Department of Civil and Structural Engineering, University of Sheffield, United Kingdom ²School of Mechanical, Aerospace & Civil Engineering, University of Manchester, United Kingdom doi:10.4203/ccp.85.5 purchase the full-text of this paper Full Bibliographic Reference for this paper H. Askes, I.M. Gitman, T. Bennett, "Formulation and Finite Element Implementation of Dynamically Consistent Gradient Elasticity", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 5, 2007. doi:10.4203/ccp.85.5 Keywords: gradient elasticity, higher-order continuum, length scale, representative volume, wave dispersion, C⁰-continuity. Summary Elastic wave propagation through heterogeneous media is generally dispersive, that is, each individual harmonic travels with a different velocity. Modelling dispersive wave propagation is not trivial, since the classical theories of elasticity do not predict any dispersive effects. It is of course possible to model each individual microstructural component individually, but this would put enormous requirements on the computational resources in terms of memory and CPU times. As an alternative, enriched continuum theories can be used on the macrolevel. Gradient elasticity theories are equipped with additional spatial gradients of the displacements, and it is well known that gradient elasticity is dispersive. However, several issues need to be addressed before gradient elasticity can be used successfully in the simulation of dispersive wave propagation: the model must be unconditionally stable for all wave numbers and the dispersion must be realistic -- this implies that all phase velocities must be real and finite; the additional material parameters must be linked to microstructural properties for reasons of identification and experimental quantification; the model must be suitable for implementation in a standard finite element package. Stable and realistic wave dispersion can be achieved by so-called dynamically consistent gradient elasticity. The term dynamic consistency reflects the simultaneous appearance of higher gradients in inertia and stiffness contributions [1]. Several procedures have been derived to link the appearing length scale parameters to the microstructural properties. In this contribution, the focus will be on the homogenisation of a Representative Volume Element (RVE). If a so-called second-order homogenisation scheme is applied, the length scale parameters are found to be proportional to the size of the RVE [2]. Finite element implementations are not trivial due to the appearance of fourth-order spatial derivatives of the displacements, which would demand C¹-continuity of the interpolation functions. To avoid these stringent continuity requirements, an operator split has been developed [3] by which the original equations are rewritten as a series of second-order equations in terms of the true displacements as well as a set of auxiliary displacements. As such, dispersive wave propagation can be simulated with the commonly used low order, C⁰-continuous finite elements. References 1 A.V. Metrikine and H. Askes, "An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice", Philosophical Magazine 86, 3259-3286, 2006. doi:10.1080/14786430500197827 2 I.M. Gitman, H. Askes and E.C. Aifantis, "The representative volume size in static and dynamic micro-macro transitions", International Journal of Fracture 135, L3-L9, 2005. doi:10.1007/s10704-005-4389-6 3 H. Askes, T. Bennett and E.C. Aifantis, "A new formulation and C⁰-implementation of dynamically consistent gradient elasticity", International Journal for Numerical Methods in Engineering (accepted for publication). purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £75 +P&P)
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