Abstract
The basic ideas for describing the dispersive wave motion in microstructured solids are discussed in the one-dimensional setting because then the differences between various microstructure models are clearly visible. An overview of models demonstrates a variety of approaches, but the consistent structure of the theory is best considered from the unified viewpoint of internal variables. It is shown that the unification of microstructure models can be achieved using the concept of dual internal variables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brillouin L.: Wave Propagation and Group Velocity. Academic Press, New York (1960)
Maugin G.A.: On some generalizations of Boussinesq and KdV systems. Proc. Estonian Acad. Sci. Phys. Mat. 44, 40–55 (1995)
Santosa F., Symes W.W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51, 984–1005 (1991)
Fish J., Chen W., Nagai G.: Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case. Int. J. Numer. Meth. Engng. 54, 331–346 (2002)
Askes H., Metrikine A.V.: One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure part 1: generic formulation. Eur. J. Mech. A/Solids 21, 555–572 (2002)
Erofeyev V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore (2003)
Love A.E.H.: Mathematical Theory of Elasticity. Dover, New York (1944)
Graff K.F.: Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975)
Maugin G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford (1999)
Wang Z.-P., Sun C.T.: Modeling micro-inertia in heterogeneous materials under dynamic loading. Wave Motion 36, 473–485 (2002)
Wang L.-L.: Foundations of Stress Waves. Elsevier, Amsterdam (2007)
Metrikine A.V.: On causality of the gradient elasticity models. J. Sound Vibr. 297, 727–742 (2006)
Papargyri-Beskou S., Polyzos D., Beskos D.E.: Wave dispersion in gradient elastic solids and structures: a unified treatment. Int. J. Solids Struct. 46, 3751–3759 (2009)
Engelbrecht J., Pastrone F.: Waves in microstructured solids with nonlinearities in microscale. Proc. Estonian Acad. Sci. Phys. Mat. 52, 12–20 (2003)
Mindlin R.D.: Microstructure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)
Engelbrecht J., Berezovski A., Pastrone F., Braun M.: Waves in microstructured materials and dispersion. Phil. Mag. 85, 4127–4141 (2005)
Horstemeyer M.F., Bammann D.J.: Historical review of internal state variable theory for inelasticity. Int. J. Plasticity 26, 1310–1334 (2010)
Coleman B.D., Gurtin M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)
Maugin G.A., Muschik W.: Thermodynamics with internal variables. J. Non-Equilib. Thermodyn. 19, 217–249 (1994)
Maugin G.A.: Internal variables and dissipative structures. J. Non-Equilib. Thermodyn. 15, 173–192 (1990)
Maugin G.A.: On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch. Appl. Mech. 75, 723–738 (2006)
Ván P., Berezovski A., Engelbrecht J.: Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn. 33, 235–254 (2008)
Maugin G.A.: Material Inhomogeneities in Elasticity. Chapman and Hall, London (1993)
Berezovski A., Engelbrecht J., Maugin G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, Singapore (2008)
Rice J.R.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)
Muschik W.: Aspects of Non-Equilibrium Thermodynamics. World Scientific, Singapore (1990)
Askes H., Aifantis E.C.: Gradient elasticity theories in statics and dynamics—a unification of approaches. Int. J. Fract. 139, 297–304 (2006)
Askes H., Metrikine A.V., Pichugin A.V., Bennett T.: Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Phil. Mag. 88/28, 3415–3443 (2008)
Engelbrecht J., Cermelli P., Pastrone F.: Wave hierarchy in microstructured solids. In: Maugin, G.A. (eds) Geometry, Continua and Microstructure, pp. 99–111. Hermann Publ., Paris (1999)
Berezovski A., Engelbrecht J., Maugin G.A.: One-dimensional microstructure dynamics. In: Ganghoffer, J.-F., Pastrone, F. (eds) Mechanics of Microstructured Solids: Cellular Materials, Fibre Reinforced Solids and Soft Tissues. Series: Lecture Notes in Applied and Computational Mechanics, pp. 21–28. Springer, Berlin (2009)
Berezovski A., Engelbrecht J., Maugin G.A.: Generalized thermomechanics with internal variables. Arch. Appl. Mech. 81, 229–240 (2011)
Askes H., Metrikine A.V.: Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int. J. Solids Struct. 42, 187–202 (2005)
Berezovski A., Engelbrecht J., Peets T.: Multiscale modelling of microstructured solids. Mech. Res. Commun. 37, 531–534 (2010)
Pastrone F., Cermelli P., Porubov A.: Nonlinear waves in 1-D solids with microstructure. Mater. Phys. Mech. 7, 9–16 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berezovski, A., Engelbrecht, J. & Berezovski, M. Waves in microstructured solids: a unified viewpoint of modeling. Acta Mech 220, 349–363 (2011). https://doi.org/10.1007/s00707-011-0468-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-011-0468-0