Two-dimensional Transient Dynamic Analysis of Decagonal Quasicrystals subjected to Shock Loading using Meshless Generalized Finite Difference (GFD) Method

علوم کاربردی و محاسباتی در مکانیک

Article 7, Volume 30, Issue 1 - Serial Number 19, June 2018, Pages 97-118   

Document Type: Original Article

DOI: 10.22067/fum-mech.v30i1.68345

Author

.Seyed Mahmoud .Hosseini^*

Ferdowsi University of Mashhad

Abstract

In this article, the transient dynamic analysis of decagonal quasicrystal (QC) is carried out using the meshless generalized finite difference (GFD) method. The transient behaviors of phonon and phason displacements in these types of QCs, when they are subjected to mechanical shock loading are studied. Also, the meshless GFD method is developed to solve the dynamic problems of QCs, considering coupling between phonon and phason displacements. The governing equations of the problem are obtained in the coupled system of PDEs by applying the coupling coefficient in the governing equations to simulate the interaction between phonon and phason displacements. A 2D rectangular domain made from decagonal (Al-Ni-Co) QCs is assumed for the problem to show the transient behaviors of phonon and phason displacements, when one side of 2D domain is excited by phonon displacement shock loading. Based on the coupling between phonon and phason displacements, any disturbance in phonon field causes some variations in phason field. To study the dynamic behaviors of phonon and phason fields, the coupled governing equations are transferred to Laplace domain. After analysis the problem by GFD method, the obtained results in Laplace domain are transferred to time domain using the Laplace inversion technique. The interactions between phonon and phason fields are studied in details. Also, the effects of coupling parameter and the phason friction coefficient on the dynamic and transient behaviors of phonon and phason displacement fields are obtained and discussed in details.

Keywords

transient dynamic analysis; Meshless method; Generalized finite difference (GFD) method; Quasicrystals; Phonon displacement; Phason displacement

Supplementary Files

EA-7-Hosseini.pdf

References

1. Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W., "Metallic Phase with Long-range Orientational Order and No Translational Symmetry", Phys. Rev. Lett., 53(20), pp. 1951-1953, (1984).

2. Fan, T.Y. and Mai, Y.W., "Elasticity Theory, Fracture Mechanics, and some Relevant Thermal Properties of Quasi – Crystalline Materials", Appl. Mech. Rev., 57(5), pp. 325-343, (2004).

3. Shi, W., "Conservation Laws of a Decagonal Quasicrystal in Elastodynamics", Eur. J. Mech. A Solids, 24, pp. 217-226, (2005).

4. Fan, T.Y., "Mathematical Theory of Elasticity of Quasicrystals and its Applications", Science Press, Beijing and Springer – Verlag, Berlin, Heidelberg.

5. Fan, T.Y., Wang, X.F., Li, W. and Zhu, A.Y., "Elasto-hydrodynamics of Quasicrystals", Philos. Mag., 89(6), pp. 501-512, (2009).

6. Bak, P., "Phenomenological Theory of Icosahedral Incommensurate (Quasiperiodic) Order in Mn-Al alloys", Phys. Rev. Lett., 54, pp. 1517-1519, (1985).

7. Rochal, S.B. and Lorman, V.L., "Anisotropy of Acoustic-phonon Properties of an Icosahedral Quasicrystal at High Temperature due to Phonon-phason Coupling", Phy. Rev. B, 62 (2), pp. 874-879, (2000).

8. Rochal, S.B. and Lorman, V.L., "Minimal Model of the Phonon-phason Dynamics in Icosahedral Quasicrystals and its Application to the Problem of Internal Friction in the J-AlPdMn alloy", Phy. Rev. B, 66, pp. 1442041-1442049, (2002).

9. Kozinkina, Y.A., Lorman, V.L. and Rochal, S.B., "Anisotropy of the Phonon-phason Dynamics and the Pinning Effect in Icosahedral AlPdMn Quasicrystals", Phy. Solid State, 45(7), pp. 1315-1321, (2003).

10. Wang, X.F. and Fan, T.Y., "Study on the Dynamics of the Double Cantilever-beam Specimen of Decagonal Al–Ni–Co Quasicrystals", Appl. Math. Comput., 211(2), pp. 336–346, (2009).

11. Agiasofitou, E. and Lazar, M., "The Elastodynamic Model of Wave– Telegraph Type for Quasicrystals", Int. J. Solids Struc., 51, pp. 923-929, (2014).

12. Wang, X. and Pan, E., "Analytical Solutions for some Defect Problems in 1D Hexagonal and 2D Octagonal Quasicrystals", Pranama-J. Phy., 70, pp. 911-933, (2008).

13. Li, W. and Fan, T., "Exact Solutions of the Generalized Dugdale Model of Two-dimensional Decagonal Quasicrystals", Appl. Math. Comput., 218(7), pp. 3068–3071, (2011).

14. Sladek, J., Sladek, V. and Pan, E., "Bending aAnalyses of 1D Orthorhombic Quasicrystal Plates", Int. J. Solid. Struct., 50, pp. 3975-3983, (2013).

15. Sladek, J., Sladek, V., Krahulec, S., Zhang, Ch. and Wunsche, M., "Crack Analysis in Decagonal Quasicrystals by the MLPG", Int. J. Fract., 181, pp. 115-126, (2013).

16. Li, L.H., "Complex Potential Theory for the Plane Elasticity Problem of Decagonal Quasicrystals and its Application", Appl. Math. Comput., 219 (19), pp. 10105–10111, (2013).

17. Guo, J.-H., Yu, J. and Si, R., "A Semi-inverse Method of a Griffith Crack in One-dimensional Hexagonal Quasicrystals", Appl. Math. Comput., 219(14), pp. 7445-7449, (2013).

18. Çerdik Yaslan, H., "Equations of Anisotropic Elastodynamics in 3D Quasicrystals as a Symmetric Hyperbolic System: Deriving the Time-dependent Fundamental Solutions", Appl. Math. Model., 37(18-19), pp. 8409-8418, (2013).

19. Hosseini, S.M., Sladek, J. and Sladek, V., "Elastodynamic Analysis of a Hollow Cylinder with Decagonal Quasicrystal Properties: Meshless Implementation of Local Integral Equations", Crystals, 6 (2016), Paper no. 94.

20. Benito, J.J., Urena, F. and Gavete, L., "Influence of Several Factors in the Generalized Finite Difference Method", Appl. Math. Model. , 25, pp. 1039–1053, (2001).

21. Benito, J.J., Urena, F., Gavete, L. and Alvarez, R., "An H-adaptive Method in the Generalized Finite Differences", Comput. Meth. Appl. Mech. Eng., 192, pp. 735–759, (2003).

22. Gavete, L., Gavete, M.L. and Benito, J.J., "Improvements of Generalized Finite Difference Method and Comparison with Other Meshless Method", Appl. Math. Model., 27, pp. 831–847, (2003).

23. Benito, J.J., Urena, F. and Gavete, L., "Solving Parabolic and Hyperbolic Equations by the Generalized Finite Difference Method", J. Comput. Appl. Math., 209, pp. 208–233, (2007).

24. Benito, J.J., Urena, F., Gavete, L., Salete, E. and Muelas, A., "A GFDM with PML for Seismic Wave Equations in Heterogeneous Media", J. Comput. Appl. Math., 252, pp. 40–51, (2013).

25. Gavete, L., Urena, F., Benito, J.J. and Salete, E., "A Note on the Dynamic Analysis Using the Generalized Finite Difference Method", J. Comput. Appl. Math., 252, pp. 132–147, (2013).

26. Hosseini, S.M., "Elastic Wave Propagation and Time History Analysis in Functionally Graded Nanocomposite Cylinders Reinforced by Carbon Nanotubes Using a Hybrid Mesh-free Method", Eng. Comput , 31(7), pp. 1261–1282, (2014).

27. Gu, Y., Wang, L., Chen, W., Zhang, C. and He, X., "Application of the Meshless Generalized Finite Difference Method to Inverse Heat Source Problems", Eng. Anal. Bound. Elem. Method., 108, pp. 721-729, (2017).

28. Hosseini, S.M., "Application of a Hybrid Mesh-free Method for Shock-induced Thermoelastic Wave Propagation Analysis in a Layered Functionally Graded Thick Hollow Cylinder with Nonlinear Grading Patterns", Eng. Anal. Bound. Elem. Method., 43, pp. 56–66, (2014).

29. Hosseini, S.M., "Shock-induced Two-dimensional Coupled Non-Fickian Diffusion-elasticity Analysis Using Meshless Generalized Finite Difference (GFD) Method", Eng. Anal. Bound. Elem. Method, 61, pp. 232-240, (2015).

30. Cohen, A.M., "Numerical Methods for Laplace Transform Inversion", Springer-Verlag US, USA, (2007).

31. Bedford, A. and Drumheller, D.S., "Introduction to Elastic Wave Propagation", John Wiley & Sons Ltd., Chichester, England (1994).