Mathematical Aspects of the Theory of Viscoelasticity
It is very common to speak about applications of mathematics, We have the ISIMM symposia “Trends in Applications of Mathematics to Mechanics”. We often forget about an influence and applications of other sciences to mathematics. Polish mathematician MAURIN /1/ writes: Analysis arised for “requirements” of mechanics and geometry. Many ideas of mathematical analysis originated from mechanics. Here it is necessary to mention variational principles-theorems and methods. The powerful Galerkin method is in fact the application of the principle of virtual work. Also modern numerical methods such as the finite element method and the boundary element method have been proposed by engineers. The basic idea of such a theoretical part of analysis, as the theory of distribution is has its origin in mechanics. In fact, in solutions of the influence of concentrated forces in mechanics using Fourier transform or an expansion into Fourier series there is an idea of application of functionals to such a “function”. NADAI /2/ published in 1922 his solution of a circular plate supported at the boundary by concentrated forces expressed in the form of an expansion into Fourier series, what was incorrect from the point of view of mathematics of that time.
KeywordsBoundary Element Method Tensor Operator Exceptional Point Viscoelastic Problem Anisotropic Sobolev Space
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- 1.K. Maurin: Analiza., Vol. 1(PWN, Warszawa 1971)Google Scholar
- 2.A. Nadai: Physikalische Zeitschrift 23 (1922)Google Scholar
- 3.J. Brilla: Numer. Math. 4l, 1 (1983)Google Scholar
- 4.J. Brilla: In Thin Shell Theory New Trends and App1ications, ed. by W. Olszak, CISM Courses and Lectures., Vol. 240 (Springer, Wien, New York 1980) p. 243Google Scholar
- 5.J. Brilla: In Equadiff 5, ed. by M. Greguš, Teubner-Texte zur Math., Vol. 47 (Teubner, Leipzig 1982) p. 64Google Scholar
- 6.J. Brilla: In Boundary Elements IX, Vol. 3 ed. by C.A. Brebbia, W.L. Wendland, G. Kuhn, Computational Mechanics Publications, Southampton, Boston (Springer, Berlin, Heidelberg, New York, London, Paris, Tokyo 1987) p.83Google Scholar
- 10.W. Olszak, P. Perzyna: Ing.Arch. 28 (1959)Google Scholar
- 11.J. Brilla: In Proceedings of the Eleventh International Congress of Applied Mechanics, ed. by H. Görtier (Springer, Berlin, Heidelberg, New York 1964) p. 403Google Scholar
- 13.R.M. Christensen: Theory of Viscoelasticity (Academic Press, New York, London 1971)Google Scholar
- 14.J. Brilla: In Mechnics of Visco-E1astic Media and Bodies, ed. J. Hult (Springer, Berlin, Heidelberg, New York 1975) p. 215Google Scholar
- 15.I. Babuška, A.K. Aziz: In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, ed. by A.K. Azis (Academic Press, New York 1972) P. 5Google Scholar
- 17.I.C. Gokhberg, M.G. Krein: Vvedenie v teoriju linejnych nesamosoprjaznennych operatorov (Nauka, Moskva 1965).Google Scholar