Summary
This paper discusses the propagation of plane body waves through a second-gradient micropolar elastic continuum. In an accompanying paper, this macroscopic constitutive law has been derived from the micro-level particle characteristics, which are the inter-particle stiffness, the particle size and the package density. As a result of incorporating the micro-scale effects, the body waves propagate in a dispersive manner, where dispersion becomes more prominent when the wavelength of the generated body waves reaches the order of magnitude of the particle size. After successively deriving the equations of motion and the dispersion relations for plane body wave propagation, the compressional wave properties for the second-gradient micro-polar model are compared to those for the Born-Karman lattice structure. Furthermore, distinguished features of the second-gradient micro-polar model are exhibited by comparing the dispersion relations of the coupled propagation of the shear wave and the micro-rotational wave with those of more simple constitutive models. The paper ends with a parameter study, where the effect by the translational particle contact stiffness and the rotational particle contact stiffness is examined.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Eringen, A. C.: Theory of micro-polar elasticity. In: Fracture—an advanced treatise, vol. II (Liebowitz, H., ed.), pp. 621–693. New York: Academic Press 1968.
Sluys, L. J.: Wave propagation, localisation and dispersion in softening solids. PhD thesis, Delft University of Technology, The Netherlands: 1992.
Mühlhaus, H.-B., Oka, F.: Dispersion and wave propagation in decrete and continuous models for granular materials. Int. J. Solids Structures33, 2841–2858 (1996).
Ewing, M. E., Jardetzky, W. S., Press, F.: Elastic waves in layered media. New York: McGraw-Hill 1957.
Chang, C. S., Gao, J.: Nonlinear dispersion of plane wave in granular media. Int. J. Non-Linear Mech30, 111–128 (1995).
Brillouin, L.: Wave propagation in periodic structures. New York: Dover 1946.
Selig, E. T., Waters, J. M.: Tract geotechnology and substructure management. London: Thomas Telford 1994.
Suiker, A. S. J., Chang, C. S., De Borst, R., Esveld, C.: Surface waves in a stratified half space with enhanced continuum properties. Part I: Formulation of the boundary value problem. Eur. J. Mech. A/Solids18, 749–768 (1999).
Suiker, A. S. J., Chang, C. S., De Borst, R. Esveld, C.: Surface waves in a stratified half space with enhanced continuum properties. Part II: Analysis of wave characteristics in regard to high-speed railway tracks. Eur. J. Mech. A/Solids18, 769–784 (1999).
Chang, C. S., Gao, J.: Second-gradient constitutive theory for granular material with random packing structure. Int. J. Solids Structures16, 2279–2293 (1995).
Cosserat, E., Cosserat, F.: Theorie des corps deformables. Paris: Herman et fils 1909.
Mindlin, R. D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal.16, 51–78 (1964).
Toupin, R. A.: Theory of elasticity with couple-stress. Arch. Ration. Mech. Anal.17, 85–112 (1964).
Mühlhaus, H.-B., Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique37, 271–283 (1987).
Mühlhaus, H.-B.: Application of Cosserat theory in numerical solutions of limit load problems. Ing.-Arch.59, 124–137 (1989).
De Borst, R.: Simulation of strain localisation: A reappraisal of the Cosserrat continuum. Eng. Comp.8, 317–332 (1991).
De Borst, R., Sluys, L. J.: Localisation in a Cosserat continuum under static and dynamic loading conditions. Comp. Meth. Appl. Mech. Eng.90, 805–827 (1992).
Chang, C. S., Ma, L.: Elastic material constants for isotropic granular solids with particle rotation. Int. J. Solids Structures29, 1001–1018 (1992).
Groen, A. E.: Three-dimensional elasto-plastic analysis of soils. PhD thesis, Delft University of Technology, The Netherlands: 1997.
Chang, C. S., Chao, S. J., Chang, Y.: Estimates of elastic moduli for granular material with anisotropic random packing structure. Int. J. Solids Structures32, 1989–2008 (1995).
Suiker, A. S. J., Chang, C. S., De Borst, R.: Micro-mechanical modelling of granular material. Part I: Derivation of a second-gradient micro-polar constitutive theory. Acta Mech. (this issue), pp. 161–180.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Suiker, A.S.J., de Borst, R. & Chang, C.S. Micro-mechanical modelling of granular material. Part 2: Plane wave propagation in infinite media. Acta Mechanica 149, 181–200 (2001). https://doi.org/10.1007/BF01261671
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01261671