Completeness of representation of solutions for stationary homogeneous isotropic elastic/viscoelastic systems
Authors:
Junyong Eom and Gen Nakamura
Journal:
Quart. Appl. Math. 77 (2019), 497-506
MSC (2010):
Primary 74J05; Secondary 35E99
DOI:
https://doi.org/10.1090/qam/1536
Published electronically:
February 20, 2019
MathSciNet review:
3962578
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Abstract: The usual Helmholtz decomposition gives a decomposition of any vector valued function into a sum of a gradient of a scalar function and a rotation of a vector valued function under some mild condition. In this paper we show that the vector valued function of the second term i.e., the divergence free part of this decomposition can be further decomposed into a sum of a vector valued function polarized in one component and the rotation of a vector valued function also polarized in the same component. Hence the divergence free part only depends on two scalar functions. We refer to this as a special Helmholtz decomposition. Further we show the so-called completeness of representation associated to this decomposition for the stationary wave field of a homogeneous, isotropic viscoelastic medium. That is, this wave field can be expressed as a special Helmholtz decomposition and each of its scalar functions satisfies a Helmholtz equation. Our completeness of representation is useful for solving boundary value problems in a cylindrical domain for several partial differential equations of systems in mathematical physics such as stationary isotropic homogeneous elastic/viscoelastic equations of a system.
References
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References
- J. D. Achenbach, Wave propagation in elastic solids, 1st ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 16, North-Holland Publishing Co., Amsterdam, 1976. MR 3444825
- C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI https://doi.org/10.1002/%28SICI%291099-1476%28199806%2921%3A9%24%5Clangle%24823%3A%3AAID-MMA976%24%5Crangle%243.0.CO%3B2-B
- P. Chadwick and E. A. Trowbridge, Elastic wave fields generated by scalar wave functions, Proc. Cambridge Philos. Soc. 63 (1967), 1177–1187. MR 0218047
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology: Spectral theory and applications, Vol. 3, with the collaboration of Michel Artola and Michel Cessenat, translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. MR 1064315
- M. do Carmo, Differential Forms and Applications, Springer-Verlag, Berlin, Heidelberg, 1994. MR 1301070
- A. Eringen and S. Suhubi, Elastodynamics. Vol. II. Linear Theory, (Academic Press, New York, 1975).
- Junyong Eom, Hyeonbae Kang, Gen Nakamura, and Yun-Che Wang, Reconstruction of the shear modulus of viscoelastic systems in a thin cylinder: an inversion scheme and experiments, Inverse Problems 32 (2016), no. 9, 095007, 19. MR 3543339, DOI https://doi.org/10.1088/0266-5611/32/9/095007
- Peter Hähner, A periodic Faddeev-type solution operator, J. Differential Equations 128 (1996), no. 1, 300–308. MR 1392403, DOI https://doi.org/10.1006/jdeq.1996.0096
- Sorin Mardare, On Poincaré and de Rham’s theorems, Rev. Roumaine Math. Pures Appl. 53 (2008), no. 5-6, 523–541. MR 2474500
- V. D. Kupradze and T. V. Burchuladze, The dynamical problems of the theory of elasticity and thermoelasticity, Journal of Soviet Mathematics, 7 (1977) no.3, 415-500.
- Ronald Y. S. Pak and Morteza Eskandari-Ghadi, On the completeness of a method of potentials in elastodynamics, Quart. Appl. Math. 65 (2007), no. 4, 789–797. MR 2370361, DOI https://doi.org/10.1090/S0033-569X-07-01074-X
- Michael Renardy and Robert C. Rogers, An introduction to partial differential equations, 2nd ed., Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 2004. MR 2028503
- Shirley Llamado Yap, The Poincaré lemma and an elementary construction of vector potentials, Amer. Math. Monthly 116 (2009), no. 3, 261–267. MR 2491982, DOI https://doi.org/10.4169/193009709X470100
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Additional Information
Junyong Eom
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
MR Author ID:
1175917
Email:
eom.junyong.r2@dc.tohoku.ac.jp
Gen Nakamura
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
MR Author ID:
190160
Email:
gnaka@math.sci.hokudai.ac.jp
Received by editor(s):
April 26, 2018
Published electronically:
February 20, 2019
Additional Notes:
The second author was partially supported by grant-in-aid for Scientific Research (15K21766 and 15H05740) of the Japan Society for the Promotion of Science during the research of this paper.
Article copyright:
© Copyright 2019
Brown University
