# Constitutive equation with internal damping for materials under cyclic and dynamic loadings using a fully coupled thermal-structural finite element analysis

• L Ecsi
• P Elesztos

## Abstract

In this paper a universal constitutive equation with internal damping for materials under cyclic and dynamic loading is presented using fully coupled thermal-structural finite element analysis. The equation adapts the idea of a spring dashpot system connected in parallel for continuum utilizing appropriate deformation measures, which are independent of rigid body motion, thus enabling more precise numerical simulation of real material. In this work, mathematical formulation of the problem is presented and demonstrated in numerical examples using a solid bar in cyclic tension and a cross-shaped specimen in biaxial tension. Elastic and plastic loading cases with and without heat generation rate per unit volume were studied, where the heat generation rate was defined as 80% of the dissipated energy per unit time. Although the calculation results are in good agreement with the only experiment we could find in technical literature, more detailed tests are needed to draw final conclusions.

## References

HOLZAPFEL, G. A., Nonlinear solid mechanics, A continuum approach for engineering, John Wiley & Sons LTD., Chichester, 2001.

BONET, J., WOOD, R. D., Nonlinear continuum mechanics for finite element analysis, Cambridge University Press, Cambridge, 1997.

BIRD, R. B., STEWART, W. E., LIGHTFOOT, E. N., Transport phenomena, 2nd. Ed., John Wiley & Sons INC, NY, 2003.

STEGER, H. G., SIEGHART, J., GLAUNINGER, E.: Műszaki mechanika 3, Termodinamika, szilárdságtan, rezgéstan, Műszaki Könyvkiadó, Budapest, 1995.

MAUGIN, G.A., The thermomechanics of plasticity and fracture, Cambridge University Press, Cambridge, 1992.

LEMAITRE, J., CHABOCHE, J. L., Mechanics of solid materials, Cambridge university press, Cambridge, 1994.

MARŠÍK, F., Termodynamika kontinua, Academia, Praha, 1999.

ŠILHAVÝ, M., The mechanics and thermodynamics of continuous media, Springer-Verlag, Berlin Heidelberg, 1997.

KOZÁK, I., Kontinuummechanika, Miskolci Egyetemi Kiadó, Miskolc, 1995.

BELYTSCHKO, T., LIU, W. K., MORAN, B., Nonlinear finite elements for continua and structures, John Wiley & Sons LTD, Chichester, 2000.

EVANS, L. C., Partial differential equations, American Mathematical Society, Providence, Rhode Island, 1998.

REKTORYS, K., Variačcní metody v inženýrských problémech a v problémech matematické fyziky, Academia, Praha, 1999.

ÉCSI, L.: Numerical behaviour of a solid body under various mechanical loads using finite element method with new energy balance equation for fully coupled thermal-structural analysis, In proceedings of the sixth internationale congress on THERMAL STRESSES, THERMAL STRESSES 2005, TU Wien, Vienna, Austria, Vol. 2, pp. 543-546, 26-29 May 2005. https://doi.org/10.1080/01495730500373586

ÉCSI, L., ÉLESZTŐS, P.: An attempt to simulate more precisely the behavior of a solid body using new energy conservation equation for fully coupled thermal structural analysis, In proceedings of the III. European conference on Computational mechanics: Solids, structures and coupled problems in engineering, ECCM 2006, National Laboratory of Civil Engineering, Lisbon, Portugal, 5-8. June 2006. https://doi.org/10.1007/1-4020-5370-3_88

ÉCSI, L., ÉLESZTŐS, P.: Constitutive equation with internal damping for materials under dynamic and cyclic loadings, V zborníku prednášsok medzinárodnej konferencie 46th International conference in Experimental stress analysis 2008, EAN 2008, Horní Bečva, ČR, 2.-5. 6. 2008.

ÉCSI, L.: Extended NOIHKH model usage for cyclic plasticity of metals, Engineering Mechanics, Vol. 13, No. 2, pp. 83-92. 2006.

ÉCSI, L.: Extended NoIHKH model with damage evolution for cyclic plasticity of metals, Acta Mechanica Slovaca, Roč. 9, č. 4., pp. 95-106., 2005.

LEMAITRE, J. Handbook of material behaviour models, Deformations of materials, Vol. 1, London, Academic press, 2001, 350 pp. ISBN 0-12-443342-1.

TREBUŇA, F., ŠIMČÁK, F., Príručka experimentálnej mechaniky, Edícia vedeckej a odbornej literatúry, TypoPress, Košice, 2007.

BUDó, Á., Kísérleti fizika I., Nemzeti tankönyvkiadó, Budapest, 1997.

Franke, H.: Lexikon der Physik, Franckh'sche Verlagshandlung, W. Keller & Co, Stuttgart, 1959.

2009-06-30

## How to Cite

Ecsi, L. and Elesztos, P. (2009) “Constitutive equation with internal damping for materials under cyclic and dynamic loadings using a fully coupled thermal-structural finite element analysis”, The International Journal of Multiphysics, 3(2), pp. 155-166. doi: 10.1260/175095409788837829.

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