EAP - Proceedings of the Estonian Academy of Sciences

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SINCE 1952

proceedings
of the estonian academy of sciences

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Nonlinear wave motion and complexity; pp. 66–71

PDF | doi: 10.3176/proc.2010.2.02

Author

Jüri Engelbrecht

Abstract

Contemporary complexity science deals with problems which involve many variables interacting with each other in such a way that a new quality appears. An important cornestone of complex systems is nonlinearity. In this paper nonlinear wave motion in microstructured solids is analysed from the viewpoint of complexity. The basic models are derived by using the concept of internal variables which are related to dissipation inequality. The scale dependence results in wave hierarchies where dispersive effects are important. In such nonlinear models solitary wave structures may emerge – a typical sign of a new quality characteristic of complexity. In addition, examples from biophysics are presented, which demonstrate clearly the similarity to the ideas of complexity shown for waves in solids.

References

Askar, A. 1985. Lattice Dynamical Foundations of Continuum. World Scientific, Singapore.

Berezovski, A., Engelbrecht, J., and Maugin, G. A. 2009. Internal variables and generalized continuum theories. In IUTAM Symposium on Scaling in Solid Mechanics (Borodich, F. M., ed.). Springer, Heidelberg, 69–80.

Berezovski, A., Engelbrecht, J., and Maugin, G. A. 2010. Generalized thermomechanics with internal variables. Arch. Appl. Mech.,
doi:10.1007/s00419-010-0412-0

Engelbrecht, J. 1997. Nonlinear Wave Dynamics. Complexity and Simplicity. Kluwer, Dordrecht.

Engelbrecht, J. 2009. Complexity in mechanics. Rend. Sem. Mat. Univ. Pol. Torino, 67, 293–325.

Engelbrecht, J. and Vendelin, M. 2000. Microstructure described by hierarchical internal variables. J. Non-Equilib. Thermodyn., 25, 119–130.

Engelbrecht, J., Vendelin, M., and Maugin, G. A. 2000. Hierarchical internal variables reflecting microstructural properties: application to cardiac muscle contraction. Rend. Sem. Mat. Univ. Pol. Torino, 58, 83–90.

Engelbrecht, J., Berezovski, A., Pastrone, F., and Braun, M. 2005. Waves in microstructured materials and dispersion. Phil. Mag., 85, 4127–4141.
doi:10.1080/14786430500362769

Engelbrecht, J., Pastrone, F., Braun, M., and Berezovski, A. 2007a. Hierarchies of waves in nonclassical materials. In Universality in Nonclassical Nonlinearity: Applications to Nondestructive Evaluations and Ultrasonics (Delsanto, P.-P., ed.). Springer, New York, 29–47.

Engelbrecht, J., Berezovski, A., and Salupere, A. 2007b. Nonlinear deformation waves in solids and dispersion. Wave Motion, 44, 493–500.
doi:10.1016/j.wavemoti.2007.02.006

Eringen, A. C. 1999. Microcontinuum Field Theories. Springer, New York.

Eringen, A. C. and Suhubi, E. S. 1964. Nonlinear theory of simple microelastic solids I & II. Int. J. Engng. Sci., 2, 189–203, 389–404.

Hodgkin, A. L. and Huxley, A. F. 1952. Quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117, 500–544.

Holland, J. H. 1998. Emergence. From Chaos to Order. Oxford University Press, Oxford.

Janno, J. and Engelbrecht, J. 2005a. Solitary waves in nonlinear microstructured materials. J. Phys. A: Math. Gen., 38, 5159–5172.
doi:10.1088/0305-4470/38/23/006

Janno, J. and Engelbrecht, J. 2005b. Waves in microstructured solids: inverse problems. Wave Motion, 43, 1–11.
doi:10.1016/j.wavemoti.2005.04.006

Janno, J. and Engelbrecht, J. 2005c. An inverse solitary wave problem related to microstructured materials. Inverse Problems, 21, 2019–2039.
doi:10.1088/0266-5611/21/6/014

Janno, J. and Engelbrecht, J. 2008. Inverse problems related to a coupled system of microstructure. Inverse Problems, 24, 1–15.
doi:10.1088/0266-5611/24/4/045017

Kitano, H. 2001. Foundation of Systems Biology. MIT Press, Boston.

Maugin, G. A. 1990. Internal variables and dissipative structures. J. Non-Equilib. Thermodyn., 15, 173–192.

Maugin, G. A. 1993. Material Inhomogeneities in Elasticity. Chapman and Hall, London.

Maugin, G. A. 1999. Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford.

Maugin, G. A. 2006. On canonical equations of continuum thermomechanics. Mech. Res. Comm., 33, 705–710.
doi:10.1016/j.mechrescom.2005.09.004

Maugin, G. A. and Engelbrecht, J. 1994. A thermodynamical viewpoint on nerve pulse dynamics. J. Non-Equilib. Thermodyn., 19, 9–23.

Maugin, G. A. and Muschik, W. 1994. Thermodynamics with internal variables I & II. J. Non-Equilib. Thermodyn., 19, 217–249, 250–289.

Mayers, R. A. (ed.). 2009. Encyclopedia of Complexity and System Science, vols 1–10. Springer.

Mindlin, R. D. 1964. Micro-structure in linear elasticity. Arch. Rat. Mech. Anal., 16, 51–78.
doi:10.1007/BF00248490

Nicolis, C. and Nicolis, G. 2007. Foundations of Complex Systems. World Scientific, New Jersey.

Prigogine, I. and Stengers, I. 1984. Order out of Chaos. Heinemann, London.

Santosa, F. and Symes, W. W. 1991. A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math., 51, 984–1005.
doi:10.1137/0151049

Taylor, M. C. 2003. The Moment of Complexity. Chicago University Press, Chicago.

Ván, P., Berezovski, A., and Engelbrecht, J. 2008. Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn., 33, 235–254.

Vendelin, M., Bovendeerd, P. H., Arts, T., Engelbrecht, J., and van Campen, D. H. 2000. Cardiac mechanoenergetics replicated by cross-bridge model. Ann. Biomed. Eng., 28, 629–640.
doi:10.1114/1.1305910

Vendelin, M., Saks, V., and Engelbrecht, J. 2007. Principles of mathematical modeling and in silico studies of integrated cellular energetics. In Molecular System Bioenergetics: Energy for Life (Saks, V., ed.). Wiley VCH, Weinheim, 407–433.
doi:10.1002/9783527621095.ch12

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