New DKFT Elements for the Finite Element Analysis of Thin Viscoelastic Plates
Authors
Abstract
In this paper, finite element analysis of thin viscoelastic plates is performed by proposing new plate elements using complex Fourier shape functions. New discrete Kirchhoff Fourier Theory (DKFT) plate elements are constructed by the enrichment of quadratic function fields in a six-noded triangular plate element with complex Fourier radial basis functions. In order to illustrate the validity and accuracy of the presented approach and robustness of the proposed elements in viscoelasticity, finite element analysis of square and elliptical viscoelastic thin plates is performed and the results are compared to those of analytical solutions and with those obtained by discrete Kirchhoff Theory (DKT) elements and the commercial software ABAQUS. The results show that FE solutions using DKFT elements have an excellent agreement with the analytical solutions and ABAQUS solutions, even though noticeably fewer elements, in comparison to DKT and classic plate elements, are employed, which means that the computational costs are reduced effectively.
Keywords
1. Akoz, A. Y., Kadioglu, F., and Tekin, G., “Quasi-Static and Dynamic Analysis of Viscoelastic Plates”, Mechanics of Time-Dependent Materials, Vol. 19, pp. 483-503, 2015.
2. Kadioglu, F., and Tekin, G., “Analysis of Plates under Point Load using Zener Material Model”, International Journal of Computer Electrical Engineering, Vol. 9, pp. 484-491, 2017.
3. Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, 2nd edition, New York, USA, 1959.
4. Sadd, M. H., Elasticity: Theory, Applications and Numerics, Elsevier Academic Press, Massachusetts, USA, 2005.
5. Wang, Y. Z., and Tsai, T. J., “Static and Dynamic Analysis of a Viscoelastic Plate by the Finite Element Method”, Applied Acoustics, Vol. 25, pp. 77-94, 1988.
6. Flugge, W., Viscoelasticity, Springer, 2nd edition, Berlin, Germany, 1975.
7. Lakes, R. S., Viscoelastic Materials, Cambridge University Press, New York, USA, 2009.
8. Christensen, R. M., Theory of Viscoelasticity, Academic Press, 2nd edition, New York, USA, 1982.
9. Brinson, H. F., and Brinson, L. C., Polymer Engineering Science and Viscoelasticity: An Introduction, Springer, 2nd edition, New York, USA, 2015.
10. Mase, G. E., “Behavior of Viscoelastic Plates in Bending”, Journal of the Engineering Mechanics Division, ASCE, Vol. 86, pp. 25-39, 1960.
11. Marvin, E. L., “Viscoelastic Plate on Poroelastic Foundation”, Journal of the Engineering Mechanics Division, ASCE, Vol. 98, pp. 911-928, 1972.
12. Mase, G. E., “Transient Response of Linear Viscoelastic Plates”, Journal of Applied Mechanics, Vol. 27, pp. 589-590, 1960.
13. Sarkar, S. K., “Deflection of Viscoelastic Plates under Concentrated Impulsive Load”, Journal of Applied Mechanics, Vol. 31, pp. 708-710, 1964.
14. Nagaya, K., “Dynamics of Viscoelastic Plate with Curved Boundaries of Arbitrary Shape”, Journal of Applied Mechanics, Vol. 45, pp. 629-635, 1978.
15. Srinivas, S., and Rao, A. K., “An Exact Analysis of Free Vibrations of Simply-Supported Viscoelastic Plates”, Journal of Sound and Vibration, Vol. 19, pp. 251-259, 1971.
16. DeLeeuw, S. L., “Circular Viscoelastic Plates Subjected to In-Plane Loads”, AIAA Journal, Vol. 9, pp. 931-937, 1971.
17. Robertson, S. R., “Solving the Problem of Forced Motion of Viscoelastic Plates by Valanis’ Method with an Application to a Circular Plate”, Journal of Sound and Vibration, Vol. 14, pp. 263-278, 1971.
18. Logan, D. L., A First Course in the Finite Element Method, Cengage Learning Engineering, 5th edition, Connecticut, USA, 2012.
19. Brebbia, C. A., and Dominguez, J., Boundary Elements: An Introductory Course, WIT Press/Computational Mechanics Publications, 2nd edition, Southampton, UK, 1992.
20. White, J. L., “Finite Elements in Linear Viscoelasticity”, Proceedings of the 2nd Conference on Matrix Method in Structural Mechanics, Wright Patterson Air Force Base, Ohio, USA, pp. 489-516, 15-17 October, 1968.
21. Chen, T. M., “The Hybrid Laplace Transform/ Finite Element Method Applied to the Quasi-Static and Dynamic Analysis of Viscoelastic Timoshenko Beams”, International Journal for Numerical Methods in Engineering, Vol. 38, pp. 509-522, 1995.
22. Yi, S., and Hilton, H. H., “Dynamic Finite Element Analysis of Viscoelastic Composite Plates”, International Journal for Numerical Methods in Engineering, Vol. 37, pp. 4081-4096, 1994.
23. Nguyen, S. N., Lee, J., and Cho, M., “A Triangular Finite Element using Laplace Transform for Viscoelastic Laminated Composite Plates Based on Efficient Higher-Order Zigzag Theory”, Composite Structures, Vol. 155, pp. 223-244, 2016.
24. Temel, B., and Sahan, M. F., “An Alternative Solution Method for the Damped Response of Laminated Mindlin Plates”, Composites Part B-Engineering, Vol. 47, pp. 107-117, 2013.
25. Attia, M. A., El-Shafei, A. G., and Mahmoud, F. F., “Nonlinear Analysis of Frictional Thermo-Viscoelastic Contact Problems using FEM”, International Journal of Applied Mechanics, Vol. 6, p. 1450028, 2014.
26. Wang, J. G., and Liu, G. R., “A Point Interpolation Meshless Method Based on Radial Basis Functions”, International Journal for Numerical Methods in Engineering, Vol. 54, pp. 1623-1648, 2002.
27. Khaji, N., and Hamzehei Javaran, S., “New Complex Fourier Shape Functions for the Analysis of Two-Dimensional Potential Problems using Boundary Element Method”, Engineering Analysis with Boundary Elements, Vol. 37, pp. 260-272, 2013.
28. Hamzehei-Javaran, S., “Approximation of the State Variables of Navier’s Differential Equation in Transient Dynamic Problems using Finite Element Method Based on Complex Fourier Shape Functions”, Asian Journal of Civil Engineering, Vol. 19, pp. 431-450, 2018.
29. Batoz, J. L., Bathe, K. J., and Ho, L. W., “A Study of Three-Node Triangular Plate Bending Elements”, International Journal for Numerical Methods in Engineering, Vol. 15, pp. 1771-1812, 1980.
30. Zocher, M. A., Groves, S. E., and Allen, D. H., “A Three-Dimensional Finite Element Formulation for Thermoviscoelastic Orthotropic Media”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 2267-2288, 1997.
31. Kansara, K., “Development of Membrane, Plate and Flat Shell Elements in Java”, M.Sc. Thesis, Virginia Polytechnic Institute & State University, Blacksburg, Virginia, USA, 2004.
32. Sorvari, J., and Hamalainen, J., “Time Integration in Linear Viscoelasticity- a Comparative Study”, Mechanics of Time-Dependent Materials, Vol. 14, pp. 307-328, 2010.
33. Zienkiewicz, O. C., Watson, M., and King, I. P., “A Numerical Method of Visco-Elastic Stress Analysis”, International Journal of Mechanical Sciences, Vol. 10, pp. 807-827, 1968.
34. Simo, J. C., and Hughes, T. J. R., Computational Inelasticity, Springer, New York, USA, 1998.
35. Feng, W. W., “A Recurrence Formula for Viscoelastic Constitutive Equations”, International Journal of Non-Linear Mechanics, Vol. 27, pp. 675-678, 1992.
36. Taylor, R. L., Pister, K. S., and Goudreau, G. L., “Thermo-Mechanical Analysis of Viscoelastic Solids”, International Journal for Numerical Methods in Engineering, Vol. 2, pp. 45-59, 1970.
2. Kadioglu, F., and Tekin, G., “Analysis of Plates under Point Load using Zener Material Model”, International Journal of Computer Electrical Engineering, Vol. 9, pp. 484-491, 2017.
3. Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, 2nd edition, New York, USA, 1959.
4. Sadd, M. H., Elasticity: Theory, Applications and Numerics, Elsevier Academic Press, Massachusetts, USA, 2005.
5. Wang, Y. Z., and Tsai, T. J., “Static and Dynamic Analysis of a Viscoelastic Plate by the Finite Element Method”, Applied Acoustics, Vol. 25, pp. 77-94, 1988.
6. Flugge, W., Viscoelasticity, Springer, 2nd edition, Berlin, Germany, 1975.
7. Lakes, R. S., Viscoelastic Materials, Cambridge University Press, New York, USA, 2009.
8. Christensen, R. M., Theory of Viscoelasticity, Academic Press, 2nd edition, New York, USA, 1982.
9. Brinson, H. F., and Brinson, L. C., Polymer Engineering Science and Viscoelasticity: An Introduction, Springer, 2nd edition, New York, USA, 2015.
10. Mase, G. E., “Behavior of Viscoelastic Plates in Bending”, Journal of the Engineering Mechanics Division, ASCE, Vol. 86, pp. 25-39, 1960.
11. Marvin, E. L., “Viscoelastic Plate on Poroelastic Foundation”, Journal of the Engineering Mechanics Division, ASCE, Vol. 98, pp. 911-928, 1972.
12. Mase, G. E., “Transient Response of Linear Viscoelastic Plates”, Journal of Applied Mechanics, Vol. 27, pp. 589-590, 1960.
13. Sarkar, S. K., “Deflection of Viscoelastic Plates under Concentrated Impulsive Load”, Journal of Applied Mechanics, Vol. 31, pp. 708-710, 1964.
14. Nagaya, K., “Dynamics of Viscoelastic Plate with Curved Boundaries of Arbitrary Shape”, Journal of Applied Mechanics, Vol. 45, pp. 629-635, 1978.
15. Srinivas, S., and Rao, A. K., “An Exact Analysis of Free Vibrations of Simply-Supported Viscoelastic Plates”, Journal of Sound and Vibration, Vol. 19, pp. 251-259, 1971.
16. DeLeeuw, S. L., “Circular Viscoelastic Plates Subjected to In-Plane Loads”, AIAA Journal, Vol. 9, pp. 931-937, 1971.
17. Robertson, S. R., “Solving the Problem of Forced Motion of Viscoelastic Plates by Valanis’ Method with an Application to a Circular Plate”, Journal of Sound and Vibration, Vol. 14, pp. 263-278, 1971.
18. Logan, D. L., A First Course in the Finite Element Method, Cengage Learning Engineering, 5th edition, Connecticut, USA, 2012.
19. Brebbia, C. A., and Dominguez, J., Boundary Elements: An Introductory Course, WIT Press/Computational Mechanics Publications, 2nd edition, Southampton, UK, 1992.
20. White, J. L., “Finite Elements in Linear Viscoelasticity”, Proceedings of the 2nd Conference on Matrix Method in Structural Mechanics, Wright Patterson Air Force Base, Ohio, USA, pp. 489-516, 15-17 October, 1968.
21. Chen, T. M., “The Hybrid Laplace Transform/ Finite Element Method Applied to the Quasi-Static and Dynamic Analysis of Viscoelastic Timoshenko Beams”, International Journal for Numerical Methods in Engineering, Vol. 38, pp. 509-522, 1995.
22. Yi, S., and Hilton, H. H., “Dynamic Finite Element Analysis of Viscoelastic Composite Plates”, International Journal for Numerical Methods in Engineering, Vol. 37, pp. 4081-4096, 1994.
23. Nguyen, S. N., Lee, J., and Cho, M., “A Triangular Finite Element using Laplace Transform for Viscoelastic Laminated Composite Plates Based on Efficient Higher-Order Zigzag Theory”, Composite Structures, Vol. 155, pp. 223-244, 2016.
24. Temel, B., and Sahan, M. F., “An Alternative Solution Method for the Damped Response of Laminated Mindlin Plates”, Composites Part B-Engineering, Vol. 47, pp. 107-117, 2013.
25. Attia, M. A., El-Shafei, A. G., and Mahmoud, F. F., “Nonlinear Analysis of Frictional Thermo-Viscoelastic Contact Problems using FEM”, International Journal of Applied Mechanics, Vol. 6, p. 1450028, 2014.
26. Wang, J. G., and Liu, G. R., “A Point Interpolation Meshless Method Based on Radial Basis Functions”, International Journal for Numerical Methods in Engineering, Vol. 54, pp. 1623-1648, 2002.
27. Khaji, N., and Hamzehei Javaran, S., “New Complex Fourier Shape Functions for the Analysis of Two-Dimensional Potential Problems using Boundary Element Method”, Engineering Analysis with Boundary Elements, Vol. 37, pp. 260-272, 2013.
28. Hamzehei-Javaran, S., “Approximation of the State Variables of Navier’s Differential Equation in Transient Dynamic Problems using Finite Element Method Based on Complex Fourier Shape Functions”, Asian Journal of Civil Engineering, Vol. 19, pp. 431-450, 2018.
29. Batoz, J. L., Bathe, K. J., and Ho, L. W., “A Study of Three-Node Triangular Plate Bending Elements”, International Journal for Numerical Methods in Engineering, Vol. 15, pp. 1771-1812, 1980.
30. Zocher, M. A., Groves, S. E., and Allen, D. H., “A Three-Dimensional Finite Element Formulation for Thermoviscoelastic Orthotropic Media”, International Journal for Numerical Methods in Engineering, Vol. 40, pp. 2267-2288, 1997.
31. Kansara, K., “Development of Membrane, Plate and Flat Shell Elements in Java”, M.Sc. Thesis, Virginia Polytechnic Institute & State University, Blacksburg, Virginia, USA, 2004.
32. Sorvari, J., and Hamalainen, J., “Time Integration in Linear Viscoelasticity- a Comparative Study”, Mechanics of Time-Dependent Materials, Vol. 14, pp. 307-328, 2010.
33. Zienkiewicz, O. C., Watson, M., and King, I. P., “A Numerical Method of Visco-Elastic Stress Analysis”, International Journal of Mechanical Sciences, Vol. 10, pp. 807-827, 1968.
34. Simo, J. C., and Hughes, T. J. R., Computational Inelasticity, Springer, New York, USA, 1998.
35. Feng, W. W., “A Recurrence Formula for Viscoelastic Constitutive Equations”, International Journal of Non-Linear Mechanics, Vol. 27, pp. 675-678, 1992.
36. Taylor, R. L., Pister, K. S., and Goudreau, G. L., “Thermo-Mechanical Analysis of Viscoelastic Solids”, International Journal for Numerical Methods in Engineering, Vol. 2, pp. 45-59, 1970.
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