Rotation is antisymmetric and therefore is not a coherent element of the classical elastic theory, which is characterized by symmetry. A new theory of linear elasticity is developed from the concept of asymmetric strain, which is defined as the transpose of the deformation gradient tensor to involve rotation as well as symmetric strain. The new theory basically differs from the prevailing micropolar theory or couple stress theory in that it maintains the same basis as the classical theory of linear elasticity and does not need extra concepts, such as “microrotation” and “couple stresses”. The constitutive relation of the new theory, the three-parameter Hooke’s law, comes from the theorem about isotropic asymmetric linear elastic materials. Concise differential equations of translational motion are derived consequently giving the same velocity formula for P-wave and a different one for S-wave. Differential equations of rotational motion are derived with the introduction of spin, which has an intrinsic connection with rotation. According to the new theory, S-wave essentially has rotation as large as deviatoric strain and should be referred to as “shear wave” in the context of asymmetric strain. There are nine partial differential equations for the deformation harmony condition in the new theory; these are given with the first spatial differentiations of asymmetric strain. Formulas for rotation energy, in addition to those for (symmetric) strain energy, are derived to form a complete set of formulas for the total mechanical energy.
References
[1]
Timoshenko, S. and Goodier, J.N. (1951) Theory of Elasticity. 2nd Edition, McGraw-Hill, New York.
[2]
Bullen, K.E. (1963) An Introduction to the Theory of Seismology. 3rd Edition, Cambridge Univ. Press, Cambridge.
[3]
Aki, K. and Richards, P.G. (1980) Quantitative Seismology, Theory and Methods. Vol. I, W.H. Freeman and Co., San Francisco.
[4]
Lee, W.H.K., Celebi, M., Todorovska, M.I. and Igel, H. (2009) Introduction to the Special Issue on Rotational Seismology and Engineering Applications. Bulletin of the Seismological Society of America, 99, 945-957. https://doi.org/10.1785/0120080344
[5]
Lee, W.H.K., Igel, H. and Trifunac, M.D. (2009) Recent Advances in Rotational Seismology. Seismological Research Letters, 80, 479-490. https://doi.org/10.1785/gssrl.80.3.479
[6]
Teisseyre, R.M. (2011) Why Rotation Seismology: Confrontation between Classic and Asymmetric Theories. Bulletin of the Seismological Society of America, 101, 1683-1691. https://doi.org/10.1785/0120100078
[7]
Teisseyre, R. (2009) Tutorial on New Development in Physics of Rotation Motions. Bulletin of the Seismological Society of America, 99, 1028-1039. https://doi.org/10.1785/0120080089
[8]
Trifunac, M.D. (1979) A Note on Surface Strains Associated with Incident Body Waves. Bulletin of the European Association of Earthquake Engineering, 5, 85-95.
[9]
Nigbor, R. (1994) Six Degree of Freedom Ground Motion Measurement. Bulletin of the Seismological Society of America, 84, 1665-1669.
[10]
Spudich, P., Steck, L.K., Hellweg, M., Fletcher, J.B. and Baker, L.M. (1995) Transient Stresses at Parkfield, California, Produced by the M7.4 Landers Earthquake of June 28, 1992: Observations from the UPSAR Dense Seismograph Array. Journal of Geophysical Research, 100, 675-690. https://doi.org/10.1029/94JB02477
[11]
Takeo, M. (1998) Ground Rotational Motions Recorded in Near-Source Region. Geophysical Research Letters, 25, 789-792. https://doi.org/10.1029/98GL00511
[12]
Igel, H., Schreiber, U., Flaws, A., Schuberth, B., Velikoseltsev, A. and Cochard, A. (2005) Rotational Motions Induced by the M8.1 Tokachi-Oki Earthquake, September 25, 2003. Geophysical Research Letters, 32, L08309. https://doi.org/10.1029/2004GL022336
[13]
Cosserat, E. and Cosserat, F. (1909) Théorie des Corps Déformables. Hermann, Paris.
[14]
Eringen, A. (1999) Microcontinuum Field Theories, I. Foundation and Solids. Springer, New York. https://doi.org/10.1007/978-1-4612-0555-5
[15]
Malvern, L.E. (1969) Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Upper Saddle River.
[16]
Toupin, R. (1964) Theories of Elasticity with Couple Stress. Archive for Rational Mechanics and Analysis, 17, 85-112. https://doi.org/10.1007/BF00253050
[17]
Eringen, A. (1966) Linear Theory of Micropolar Elasticity. Journal of Mathematics and Mechanics, 15, 909-923. https://doi.org/10.1512/iumj.1966.15.15060
[18]
Cowin, S. (1970) Stress Functions for Cosserat Elasticity. International Journal of Solids and Structures, 6, 389-398. https://doi.org/10.1016/0020-7683(70)90091-0
[19]
Nowacki, W. (1986) Theory of Asymmetric Elasticity. PWN and Pergamon Press, Warszawa, Oxford.
[20]
Dyszlewicz, J. (2004) Micropolar Theory of Elasticity. Springer, Berlin. https://doi.org/10.1007/978-3-540-45286-7
[21]
Pujol, J. (2009) Tutorial on Rotations in the Theories of Finite Deformation and Micropolar (Cosserat) Elasticity. Bulletin of the Seismological Society of America, 99, 1011-1027. https://doi.org/10.1785/0120080161
[22]
Ieşan, D. (1981) Some Applications of Micropolar Mechanics to Earthquake Problems. International Journal of Engineering Science, 19, 855-864. https://doi.org/10.1016/0020-7225(81)90119-1
[23]
Stein, S. and Wysession, M. (2003) An Introduction to Seismology, Earthquake, and Earth Structure. Blackwell Publishing, Oxford.
[24]
Aris, R. (1962) Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover, Mineola.
[25]
Fung, Y.C. (1965) Foundation of Solid Mechanics. Prentice-Hall, Englewood Cliffs.
[26]
Muskhelishvili, N.I. (1975) Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff Intern. Publ., Leyden.
[27]
Slaughter, W.S. (2003) The Linearized Theory of Elasticity. Birkhauser, Basel. https://doi.org/10.1007/978-1-4612-0093-2
[28]
Richter, C.F. (1958) Elementary Seismology. W. H. Freeman, San Francisco.
[29]
Zisman, W.A. (1933) A Comparison of the Statically and Dynamically Determined Elastic Constants of Rocks. Proceedings of the National Academy of Sciences of the United States of America, 19, 680-686. https://doi.org/10.1073/pnas.19.7.680
[30]
Lama, R.D. and Vutukuri, V.S. (1978) Handbook on Mechanical Properties of Rock, Vol. II. Trans Tech Publications, Clausthal.
