Abstract
Gradient enriched continua are an elegant and versatile class of material models that are able to simulate a variety of physical phenomena, ranging from singularity-free descriptions of crack tips and dislocations, via size dependent mechanical response, to dispersive wave propagation. However, the increased order of the governing partial differential equations has historically complicated analytical and numerical solution methods. Inspired by the work of Ru and Aifantis (Acta Mech. 101(1-4), 59–68 (1993)), this contribution focusses on operator split methods that allow to reduce the order of the governing equations. It will be shown that this order reduction leads to multiscale reformulations of the original equations in which the macrolevel unknowns are fully coupled to the microlevel unknowns. As a first example, gradient enriched equations of elastodynamics are considered with second order and fourth order microinertia terms. The second example concerns dynamic piezomagnetics with gradient enrichment of both the mechanical fields and the magnetic fields.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aifantis, E.: Strain gradient interpretation of size effects. Int. J. Fract. 95(1-4), 299–314 (1999). https://doi.org/10.1007/978-94-011-4659-3_16
Aifantis, E.C.: On the microstructural origin of certain inelastic models. J. Engng. Mater. Technol. 106(4), 326–330 (1984). https://doi.org/10.1115/1.3225725
Aifantis, E.C.: The physics of plastic deformation. Int. J. Plasticity 3(3), 211–247 (1987). https://doi.org/10.1016/0749-6419(87)90021-0
Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Engng Sci. 30(10), 279–1299 (1992). https://doi.org/10.1016/0020-7225(92)90141-3
Altan, S., Aifantis, E.: On the structure of the mode III crack-tip in gradient elasticity. Scripta Metallurgica et Materialia 26(2), 319–324 (1992). https://doi.org/10.1016/0956-716x(92)90194-j
Askes, H., Aifantis, E.C.: Numerical modeling of size effects with gradient elasticity – Formulation, meshless discretization and examples. Int. J. Fract. 117(4), 347–358 (2002). https://doi.org/10.1023/A:102222552
Askes, H., Aifantis, E.C.: Gradient elasticity theories in statics and dynamics – A unification of approaches. Int. J. Fract. 139(2), 297–304 (2006). https://doi.org/10.1007/s10704-006-8375-4
Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48(13), 1962–1990 (2011). https://doi.org/10.1016/j.ijsolstr.2011.03.006
Askes, H., Bennett, T., Aifantis, E.C.: A new formulation and C 0-implementation of dynamically consistent gradient elasticity. Int. J. Numer. Methods Engng. 72(1), 111–126 (2007). https://doi.org/10.1002/nme.2017
Askes, H., Bennett, T., Gitman, I., Aifantis, E.: A multi-scale formulation of gradient elasticity and its finite element implementation. In: Papadrakakis, M., Topping, B. (eds.) Trends in Engineering Computational Technology, pp. 189–208. Saxe-Coburg Publications, Stirlingshire (2008a). https://doi.org/10.4203/csets.20.10
Askes, H., Morata, I., Aifantis, E.C.: Finite element analysis with staggered gradient elasticity. Computers & Structures 86(11-12), 1266–1279 (2008b). https://doi.org/10.1016/j.compstruc.2007.11.002
Bennett, T., Askes, H.: Finite element modelling of wave dispersion with dynamically consistent gradient elasticity. Comput. Mech. 43(6), 815–825 (2009). https://doi.org/10.1007/s00466-008-0347-2
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(2), 185–193 (2007). https://doi.org/10.1007/s10704-008-9192-8
Bennett, T., Rodríguez-Ferran, A., Askes, H.: Damage regularisation with inertia gradients. Eur. J. Mech. A/Solids 31(1), 131–138 (2012). https://doi.org/10.1016/j.euromechsol.2011.08.005
Berezovski, A., Engelbrecht, J., Berezovski, M.: Waves in microstructured solids: a unified viewpoint of modeling. Acta Mech. 220(1–4), 349–363 (2011). https://doi.org/10.1007/s00707-011-0468-0
Berezovski, A., Engelbrecht, J., Salupere, A., Tamm, K., Peets, T., Berezovski, M.: Dispersive waves in microstructured solids. Int. J. Solids Struct. 50(11-12), 1981–1990 (2013). https://doi.org/10.1016/j.ijsolstr.2013.02.018
De Domenico, D., Askes, H.: Computational aspects of a new multi-scale dispersive gradient elasticity model with micro-inertia. Int. J. Numer. Methods Engng. 109(1), 52–72 (2017). https://doi.org/10.1002/nme.5278
De Domenico, D., Askes, H.: Nano-scale wave dispersion beyond the first Brillouin zone simulated with inertia gradient continua. J. Appl. Phys. 124(20), 205107 (2018). https://doi.org/10.1063/1.5045838
De Domenico, D., Askes, H., Aifantis, E.C.: Gradient elasticity and dispersive wave propagation: Model motivation and length scale identification procedures in concrete and composite laminates. Int. J. Solids Struct. 158, 176–190 (2019). https://doi.org/10.1016/j.ijsolstr.2018.09.007
De Domenico, D., Askes, H.: A new multi-scale dispersive gradient elasticity model with micro-inertia: Formulation and-finite element implementation. Int. J. Numer. Methods Engng. 108(5), 485–512 (2016). https://doi.org/10.1002/nme.5222
Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Systems 3(1), 43–82 (2015). https://doi.org/10.2140/memocs.2015.3.43
Engelbrecht, J., Berezovski, A., Pastrone, F., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85(33-35), 4127–4141 (2005). https://doi.org/10.1080/14786430500362769
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983). https://doi.org/10.1063/1.332803
Georgiadis, H., Vardoulakis, I., Lykotrafitis, G.: Torsional surface waves in a gradient-elastic half-space. Wave Motion 31(4), 333–348 (2000). https://doi.org/10.1016/s0165-2125(99)00035-9
Gitman, I., Askes, H., Sluys, L.: Representative volume size as a macroscopic length scale parameter. In: Proceedings of 5th International Conference on Fracture Mechanics of Concrete and Concrete Structures, vol. 1, pp. 483–491 (2004)
Gitman, I.M., Askes, H., Aifantis, E.C.: The representative volume size in static and dynamic micro-macro transitions. Int. J. Fract. 135(1-4), L3–L9 (2005). https://doi.org/10.1007/s10704-005-4389-6
Gutkin, M.Y., Aifantis, E.: Screw dislocation in gradient elasticity. Scripta Mater. 35(11), 1353–1358 (1996). https://doi.org/10.1016/1359-6462(96)00295-3
Gutkin, M.Y., Aifantis, E.: Edge dislocation in gradient elasticity. Scripta Mater. 36(1), 129–135 (1997). https://doi.org/10.1016/s1359-6462(96)00352-1
Gutkin, M.Y., Aifantis, E.: Dislocations in the theory of gradient elasticity. Scripta Mater. 5(40), 559–566 (1999). https://doi.org/10.1016/s1359-6462(98)00424-2
Huerta, A., Pijaudier-Cabot, G.: Discretization influence on regularization by two localization limiters. J. Engng. Mech. 120(6), 1198–1218 (1994). https://doi.org/10.1061/(asce)0733-9399(1994)120:6(1198)
Jirásek, M., Marfia, S.: Non-local damage model based on displacement averaging. Int. J. Numer. Methods Engng. 63(1), 77–102 (2005). https://doi.org/10.1002/nme.1262
Kouznetsova, V., Geers, M., Brekelmans, W.: Size of a representative volume element in a second-order computational homogenization framework. Int. J. Multiscale Comput. Engng. 2(4), 575–598 (2004). https://doi.org/10.1615/intjmultcompeng.v2.i4.50
Lurie, S., Belov, P., Volkov-Bogorodsky, D., Tuchkova, N.: Nanomechanical modeling of the nanostructures and dispersed composites. Comput. Mater. Sci. 28(3-4), 529–539 (2003). https://doi.org/10.1016/j.commatsci.2003.08.010
Metrikine, A., Askes, H.: An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice. Phil. Mag. 86(21-22), 3259–3286 (2006). https://doi.org/10.1080/14786430500197827
Metrikine, A.V., Askes, H.: One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 1: Generic formulation. Eur. J. Mech. A/Solids 21(4), 555–572 (2002). https://doi.org/10.1016/s0997-7538(02)01218-4
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16(1), 51–78 (1964). https://doi.org/10.1007/bf00248490
Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965). https://doi.org/10.1016/0020-7683(65)90006-5
Mühlhaus, H.B., Alfantis, E.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28(7), 845–857 (1991). https://doi.org/10.1016/0020-7683(91)90004-y
Papargyri-Beskou, S., Polyzos, D., Beskos, D.: Wave dispersion in gradient elastic solids and structures: a unified treatment. Int. J. Solids Struct. 46(21), 3751–3759 (2009). https://doi.org/10.1016/j.ijsolstr.2009.05.002
Peerlings, R., Geers, M., De Borst, R., Brekelmans, W.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38(44-45), 7723–7746 (2001). https://doi.org/10.1016/s0020-7683(01)00087-7
Peerlings, R.H., De Borst, R., Brekelmans, W.A., Vree, J.H., Spee, I.: Some observations on localization in non-local and gradient damage models. Eur. J. Mech. A/Solids 15(6), 937–953 (1996)
Pichugin, A., Askes, H., Tyas, A.: Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories. J. Sound Vibr. 313(3-5), 858–874 (2008). https://doi.org/10.1016/j.jsv.2007.12.005
Rodríguez-Ferran, A., Bennett, T., Askes, H., Tamayo-Mas, E.: A general framework for softening regularisation based on gradient elasticity. Int. J. Solids Struct. 48(9), 1382–1394 (2011). https://doi.org/10.1016/j.ijsolstr.2011.01.022
Ru, C., Aifantis, E.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101(1-4), 59–68 (1993). https://doi.org/10.1007/bf01175597
Tenek, L.T., Aifantis, E.: A two-dimensional finite element implementation of a special form of gradient elasticity. Comput. Modeling Engng. Sci. 3(6), 731–742 (2002)
Unger, D.J., Aifantis, E.C.: The asymptotic solution of gradient elasticity for mode III. Int. J. Fract. 71(2), R27–R32 (1995). https://doi.org/10.1007/bf00033757
Vardoulakis, I., Aifantis, E.: On the role of microstructure in the behavior of soils: effects of higher order gradients and internal inertia. Mech. Mater. 18(2), 151–158 (1994). https://doi.org/10.1016/0167-6636(94)00002-6
Xu, M., Gitman, I.M., Askes, H.: A gradient-enriched continuum model for magneto-elastic coupling: Formulation, finite element implementation and in-plane problems. Comput. Struct. 212, 275–288 (2019). https://doi.org/10.1016/j.compstruc.2018.11.004
Yarnell, J.L., Warren, J.L., Koenig, S.H.: Experimental dispersion curves for phonons in aluminum. In: Wallis, R. (ed.) Lattice Dynamics, pp. 57–61. Pergamon, Oxford (1965). https://doi.org/10.1016/b978-1-4831-9838-5.50014-5
Yue, Y., Xu, K., Aifantis, E.: Microscale size effects on the electromechanical coupling in piezoelectric material for anti-plane problem. Smart Mater. Struct. 23(12), 125043 (2014). https://doi.org/10.1088/0964-1726/23/12/125043
Acknowledgements
HA, IMG and ECA gratefully acknowledge support of the EU RISE project FRAMED-734485. MX gratefully acknowledges financial support from the China Scholarship Council and the Fundamental Research Funds for the Central Universities (FRFBR-16-017A).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Askes, H., De Domenico, D., Xu, M., Gitman, I.M., Bennett, T., Aifantis, E.C. (2019). Operator Splits and Multiscale Methods in Computational Dynamics. In: Berezovski, A., Soomere, T. (eds) Applied Wave Mathematics II. Mathematics of Planet Earth, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-29951-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-29951-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-29950-7
Online ISBN: 978-3-030-29951-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)