ELASTIC WAVES WITH FINITE ROTATIONS

ELASTIC WAVES WITH FINITE ROTATIONS

Kotchergenko, I. D.

Full Article:

Waves are represented through the superposition of the four fundamental modes of the plane areolar strain theory. This theory was presented at references[10] to[13], with application to finite rotations, orthotropic materials and finite element models, but a summary review is here presented in order to make this paper self contained and also to disclose the areolar strain concept, which although divulged for two decades still faces with poor ac-ceptance by the scientific community. It is shown that the areolar strain approach does not distinguish finite from infinitesimal strain due to the fact that in addition to the traditional “forward” strain it incorporates the “sidelong” strain into its imaginary part. Instead of comparing the change in distance between two contiguous points, the areolar strains presents the complete state of finite strains on an areola that surrounds a given point. Only first deriv-atives are used, as expected due to the physical meaning of strain and having in mind that the relative displacement between two arbitrary points of the plane should be obtained through a single line integration of the strain, along any path of integration joining these points. The areolar strain fulfils these conditions. Now, an approach for tackling with equivoluminal waves of finite rotation is presented, assuming that the gain of volume due the mathematical finite rotation is neutralized by the mathematical shrinkages due to the complex shear strain.

Waves are represented through the superposition of the four fundamental modes of the plane areolar strain theory. This theory was presented at references[10] to[13], with application to finite rotations, orthotropic materials and finite element models, but a summary review is here presented in order to make this paper self contained and also to disclose the areolar strain concept, which although divulged for two decades still faces with poor ac-ceptance by the scientific community. It is shown that the areolar strain approach does not distinguish finite from infinitesimal strain due to the fact that in addition to the traditional “forward” strain it incorporates the “sidelong” strain into its imaginary part. Instead of comparing the change in distance between two contiguous points, the areolar strains presents the complete state of finite strains on an areola that surrounds a given point. Only first deriv-atives are used, as expected due to the physical meaning of strain and having in mind that the relative displacement between two arbitrary points of the plane should be obtained through a single line integration of the strain, along any path of integration joining these points. The areolar strain fulfils these conditions. Now, an approach for tackling with equivoluminal waves of finite rotation is presented, assuming that the gain of volume due the mathematical finite rotation is neutralized by the mathematical shrinkages due to the complex shear strain.

Palavras-chave:

DOI: 10.5151/meceng-wccm2012-16727

Referências bibliográficas

• [1] Achenbach, J. D., Wave Propagation In Elastic Solids, Elsevier, pp. 187-194, 1999.
• [2] Courant R. Andamp; Hilbert, D., Methods of Mathematical Physics, Vol. II, Wiley: New York, pp. 350-351, 1989.
• [3] Elmore, W. C. and Heald, M. A., Physics of Waves, Dover Publications,1985.
• [4] England, A. H., Complex Variable Methods in Elasticity, Wiley-Interscience, pp. 28-49, 1971.
• [5] Graff, K.L., Wave Motion in Elastic Solids, Oxford. The Clarendon Press, 1975.
• [6] Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, pp. 204-220, 1927.
• [7] Muskhelishvili, N. I., Some Basic Problems of the Theory of Elasticity, Noordhoff, Gro-ningen, 1963.
• [8] Novozhilov, V.V., Foundations of the Nonlinear Theory of Elasticity, Dover Publications, Inc.: Mineola , New York, pp. 83-84, 1999.
• [9] Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, 1956.
• [10] Kotchergenko, I.D., The Areolar Strain Concept, IMECE2008 ASME International Me-chanical Engineering Congress and Exposition, Boston, October 2008.
• [11] Kotchergenko, I.D., Kolosov-Mushkhelishvili Formulas Revisited, 11thInternational Con-ference on Fracture, Turin, March 2005.
• [12] Kotchergenko, I.D., Applications of Generalized Analytic Functions to Elasticity, CMM-2005-Computer Methods in Mechanics, Polish Academy of Sciences, June 2005.
• [13] Kotchergenko, I.D., The Areolar Strain Concept Applied to Elasticity, WIT Transactions on Modelling and Simulation, Vol. 46, 2007, WIT Press, (free).
• [14] Mitrinovic, D.S., Keckic, J.D., From the History of Nonanalitic Functions, Série: Mathématiques e Physique, No. 274-371, Publications de La Faculté D’Electrotechnique de L’Université à Belgrade, 1969, (free).

Como citar:

Kotchergenko, I. D.; "ELASTIC WAVES WITH FINITE ROTATIONS", p-290-302. In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 23580828, DOI 10.5151/meceng-wccm2012-16727