Transactions of the Korean Society of Mechanical Engineers A (대한기계학회논문집A)

The Korean Society of Mechanical Engineers (대한기계학회)
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Transient Linear Viscoelastic Stress Analysis Based on the Equations of Motion in Time Integral

시간적분형 운동방정식에 근거한 동점탄성 문제의 응력해석

  • 이성희 (금오공과대학교 생산기술연구소) ;
  • 심우진 (금오공과대학교)
  • Published : 2003.09.01
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Abstract

In this paper, the finite element equations for the transient linear viscoelastic stress analysis are presented in time domain, whose variational formulation is derived by using the Galerkin's method based on the equations of motion in time integral. Since the inertia terms are not included in the variational formulation, the time integration schemes such as the Newmark's method widely used in the classical dynamic analysis based on the equations of motion in time differential are not required in the development of that formulation, resulting in a computationally simple and stable numerical algorithm. The viscoelastic material is assumed to behave as a standard linear solid in shear and an elastic solid in dilatation. To show the validity of the presented method, two numerical examples are solved nuder plane strain and plane stress conditions and good results are obtained.

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References

  1. Reddy, J.N., 1993, An Introduction to the Finite Element Method (2nd edn.), McGraw-Hill, London
  2. Bathe, K.J., 1996, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs
  3. Manolis, G.D., Beskos D.E., 1988, Boundary Element Methods in Elastodynamics, Routledge London
  4. Dominguez, J., 1993, Boundary Elements Methods in Dynamics, Elsevier, Amsterdam
  5. Leitman, M.J., 'Variational Principles in the Linear Dynamic Theory of Viscoelasticity,' Q. Appl. Math., Vol. 24, No. 1, pp. 37-46, 1966 https://doi.org/10.1090/qam/197010
  6. Oden, J.T. and Reddy, J.N., 1976, Variational Methods in Theoretical Mechanics, Springer-Verlag, Berlin
  7. Christensen R.M., 1982, Theory of Viscoelasticity, Academic Press, New York
  8. Fliigge, W., 1975, Viscoelasticity, 2nd edition, Spriger- Verlag, Berlin
  9. Haddad, Y.M., Viscoelasticity of Engineering Materials, Chapman and Hall, London, 1995
  10. Nickell, R.E., 1968, Stress Wave Analysis in Layered Thermoviscoelastic Materials by the Extended Ritz Method, Technical Report S-175, VII, Rohm and Haas Co., Redstone Research Laboratories, Huntsville, Alabama
  11. Becker, E.B. and Nickell, R.E., 'Stress Wave Propagation Using the Extended Ritz Method,' AIAA/ASME 10th Structures, Structural Dynamics, and Materials Conference, pp. 336-348, 1969
  12. Goudreau, G.L., 1970, Evaluation of Discrete Methods for the Linear Dynamic Response of Elastic and Viscoelastic Solids, Structures and Material Research, Report No. 69-15, Structural Engineering Laboratory, University of California, Berkeley, California
  13. Golla, D.F. and Hughes, P.C., 'Dynamics of Viscoelastic Structures - A Time Domain Finite Element Formulation,' J. Appl. Mech., Vol. 52, pp. 897-906, 1985 https://doi.org/10.1115/1.3169166
  14. Spyrakos, C.C., 'Exact Finite Element Method Analysis of Viscoelastic Tapered Structures to Transient Loads,' Edited by W.Pilkey and B.Pilkey, NASA 58th Shock and Vibration Symposium, Vol. 1, pp. 361-375, NASA, Washington D.C., 1987
  15. Ha, T. Santos, J.E. and Sheen, D., 2002, 'Nonconforming Finite Element Methods for the Simulation of Waves in Viscoelastic Solids,' Compo Methods Appl. Mech. Engrg., Vol. 191, pp. 5647-5670 https://doi.org/10.1016/S0045-7825(02)00469-3
  16. Manolis, G.D. and Beskos, D.E., 1981, 'Dynamic Stress Concentration Studies by Boundary Integrals and Laplace Transform,' Int. J. Num. Meth. Eng., Vol. 17, pp. 573-599 https://doi.org/10.1002/nme.1620170407
  17. Luco, J.E. and De Barros, F.C.P., 'Dynamic Displacements and Stresses in the Vicinity of a Cylindrical Cavity Embedded in a Half-Space,' Earthquake engineering and Structural Dynamics, Vol. 23, pp. 321-340, 1994 https://doi.org/10.1002/eqe.4290230307
  18. Danyluk, MJ. Geubelle, P.H. and Hilton, H.H., 1998, 'Two- Dimensional Dynamic and Three-Dimensional Fracture in Viscoelastic Materials,' Int. J. Solids Structures, Vol. 35, Nos. 28-29, pp. 3831-3853 https://doi.org/10.1016/S0020-7683(97)00221-7
  19. Gaul, L. and Schanz, M., 1999, 'A Comparative Study of Three Boundary Element Approaches to Calculate the Transient Response of Viscoelastic Solids with Unbounded Domains,' Comp. Meth. Appl. Mech. Eng., Vol. 179, pp. 111-123 https://doi.org/10.1016/S0045-7825(99)00032-8
  20. Perez-Gavilan, U. and Aliabadi, M.H., 2001, 'A symmetric Galerkin Boundary Element Method for Dynamic Frequency Domain Viscoelastic Problems,' Computers and Structures, Vol. 79, pp. 2621-2633 https://doi.org/10.1016/S0045-7949(01)00090-6
  21. Eldred, L.B. Baker, W.P. and Palazotto, A.N., 1996, 'Numerical Application of Fractional Derivative Model Constitutive Relations for Viscoelastic Materials,' Computers and Structures, Vol. 60, No.6, pp. 875 - 882 https://doi.org/10.1016/0045-7949(95)00447-5
  22. Abu-Alshaikh, I. Turhan, D. and Mengi, Y., 2001, 'Propagation of Transient Out-of-Plane Shear Waves in Viscoelastic Layered Media,' Int. J. Mech. Sci., Vol. 43, pp. 2911-2928 https://doi.org/10.1016/S0020-7403(01)00063-7
  23. Biwa, S., 2001, 'independent Scattering and Wave Attenuation in Viscoelastic Composites,' Mechanics of Materials, Vol. 33, pp. 635-647 https://doi.org/10.1016/S0167-6636(01)00080-1
  24. Woo-Jin Sim and Sung-Hee Lee, 2001, 'A Time-Domain Finite Element Formulation for Transient Dynamic Linear Elasticity,' Transactions of the KSME A, Vol. 25, No.4, pp. 574 - 581
  25. Sim, W.J. and Lee, S.H., 2002, 'Transient Linear Elastodynamic Analysis in Time Domain Based on the Integra-differential Equations,' Int. J. Structural Engineering and Mechanics, Vol. 14, No. 1, pp. 71-84 https://doi.org/10.12989/sem.2002.14.1.071
  26. Woo-Jin Sim and Ho-Seop Lee, 1992, 'Time-Domain Finite Element Formulation for Linear Viscoelastic Analysis Based on a Hereditary Type Constitutive Law,' Transactions of the KSME A, Vol. 16, No.8, pp. 1429-1437
  27. Gurtin M.E., 1964, 'Variational Principles for Linear Elastodynamics,' Archive for Rational Mechanics and Analysis, Vol. 16, pp. 34 - 50 https://doi.org/10.1007/BF00248489