A new BEM for modeling and simulation of 3T MDD laser-generated ultrasound stress waves in FGA smart materials

Mohamed A. Fahmy1,2

1Jamoum University College, Umm Al-Qura University, Alshohdaa 25371, Jamoum, Makkah, Saudi Arabia.

2Faculty of Computers and Informatics, Suez Canal University, New Campus, 4.5 Km, Ring Road, El Salam District, 41522 Ismailia, Egypt.




The goal of this study is to present a new theory known as the three-temperature memory-dependent derivative (MDD) of ultrasound stress waves in functionally graded anisotropic (FGA) smart materials. It is extremely difficult to address the difficulties related to this theory analytically due to its severe nonlinearity. As a result, we suggest a new boundary element method (BEM) to solve such equations. The suggested BEM technique incorporates the benefits of both continuous and discrete descriptions. The numerical results are visually represented to demonstrate the impacts of MDD three temperatures and anisotropy on the ultrasound stress waves in FGA smart materials. The numerical findings verify the proposed methodology’s validity and accuracy. We may conclude that the offered results are useful for comprehending the FGA smart materials. As a result, our findings contribute to the advancement of the industrial applications of FGA smart materials.

Cite as:

Fahmy, M. A. (2021). A new BEM for modeling and simulation of 3T MDD laser-generated ultrasound stress waves in FGA smart materials. Computer Methods in Materials Science, 21(2), 95-104. https://doi.org/10.7494/cmms.2021.2.0739

Article (PDF):


Boundary element method, Modeling and simulation, Three-temperature, Memory-dependent derivative, Laser ultrasonics, Nonlinear thermal stress waves, Functionally graded anisotropic, Smart materials


Banerjee, P.K., & Butterfield, R. (1981). Boundary element methods in engineering science. McGraw-Hill.

Biot, M.A. (1956). Thermoelasticity and irreversible thermo-dynamics. Journal of Applied Physics, 27(3), 249–253. https://doi.org/10.1063/1.1722351.

Chandrasekharaiah, D.S. (1998). Hyperbolic thermoelasticity: a review of recent literature. Applied Mechanics Reviews, 51(12), 705–729. https://doi.org/10.1115/1.3098984.

Diettelm, K. (1997). Generalized compound quadrature formulae for finite-part integrals. IMA Journal of Numerical Analysis, 17(3), 479–493. https://doi.org/10.1093/imanum/17.3.479.

Dragoş, L. (1984). Fundamental solutions in micropolar elasticity. International Journal of Engineering Science, 22(3), 265–275. https://doi.org/10.1016/0020-7225(84)90007-7.

Duhamel, J. (1837). Some memoire sur les phenomenes thermo-mechanique. Journal de l’École Polytechnique, 15, 1–57.

Erigen, A.C. (1968). Theory of micropolar elasticity, In H. Liebowitz (Ed.). Fracture. An advanced treatise (vol. 2: Mathematical fundamentals). Academic Press.

Fahmy, M.A. (2011). A time-stepping DRBEM for magneto-thermo-viscoelastic interactions in a rotating nonhomogeneous anisotropic solid. International Journal of Applied Mechanics, 3(4), 711–734. https://doi.org/10.1142/S1758825111001202.

Fahmy, M.A. (2012a). A time-stepping DRBEM for the transient magneto-thermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid. Engineering Analysis with Boundary Elements, 36(3), 335–345. https://doi.org/10.1016/j.enganabound.2011.09.004.

Fahmy, M.A. (2012b). Transient magneto-thermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source. Applied Mathematical Modelling, 36(10), 4565–4578. https://doi.org/10.1016/j.apm.2011.11.036.

Fahmy, M.A. (2013). Implicit–Explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic functionally graded solids. Engineering Analysis with Boundary Elements, 37(1), 107–115. https://doi.org/10.1016/j.enganabound.2012.08.002.

Fahmy, M.A. (2018). Boundary element algorithm for modeling and simulation of dual-phase lag bioheat transfer and biomechanics of anisotropic soft tissues. International Journal of Applied Mechanics, 10(10), 1850108. https://doi.org/10.1142/S1758825118501089.

Fahmy, M.A. (2019). A New Boundary Element Strategy for Modeling and Simulation of Three Temperatures Nonlinear Generalized Micropolar-Magneto-Thermoelastic Wave Propagation Problems in FGA Structures. Engineering Analysis with Boundary Elements, 108, 192–200. https://doi.org/10.1016/j.enganabound.2019.08.006.

Fahmy, M.A. (2020). Boundary element algorithm for nonlinear modeling and simulation of three temperature anisotropic generalized micropolar piezothermoelasticity with memory-dependent derivative. International Journal of Applied Mechanics, 12(3), 2050027. https://doi.org/10.1142/S1758825120500271.

Fahmy, M.A. (2021a). A novel BEM for modeling and simulation of 3T nonlinear generalized anisotropic micropolar-thermoelasticity theory with memory dependent derivative. CMES – Computer Modeling in Engineering & Sciences, 126(1), 175–199. https://doi.org/10.32604/cmes.2021.012218.

Fahmy, M.A. (2021b). A new boundary element algorithm for a general solution of nonlinear space-time fractional dualphase-lag bio-heat transfer problems during electromagnetic radiation. Case Studies in Thermal Engineering, 25, 100918. https://doi.org/10.1016/j.csite.2021.100918.

Fahmy, M.A. (2021c). A new boundary element formulation for modeling and simulation of three-temperature distributions in carbon nanotube fiber reinforced composites with inclusions. Mathematical Methods in the Applied Science. https://doi.org/10.1002/mma.7312 [in progress].

Fahmy, M.A. (2021d). A new boundary element algorithm for modeling and simulation of nonlinear thermal stresses in micropolar FGA composites with temperature‑dependent properties. Advanced Modeling and Simulation in Engineering Sciences, 8, 1–23. https://doi.org/10.1186/s40323-021-00193-6.

Fahmy, M.A. (2021e). A New BEM for Fractional Nonlinear Generalized Porothermoelastic Wave Propagation Problems. Computers, Materials and Continua, 68(1), 59–76. https://doi.org/10.32604/cmc.2021.015115.

Fahmy, M.A., Shaw, S., Mondal, S., Abouelregal, A.E., Lotfy, K., Kudinov, I.V., & Soliman, A.H. (2021). Boundary Element Modeling for Simulation and Optimization of Three-Temperature Anisotropic Micropolar Magneto-thermoviscoelastic Problems in Porous Smart Structures Using NURBS and Genetic Algorithm. International Journal of Thermophysics, 42, https://doi.org/10.1007/s10765-020-02777-7.

Green, A.E., & Lindsay, K.A. (1972). Thermoelasticity. Journal of Elasticity, 2(1), 1–7. https://doi.org/10.1007/BF00045689.

Green, A.E., & Naghdi, P.M. (1992). On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 15(2), 253–264. https://doi.org/10.1080/01495739208946136.

Green, A.E., & Naghdi, P.M. (1993). Thermoelasticity without energy dissipation. Journal of Elasticity, 31(3), 189–208. https://doi.org/10.1007/BF00044969.

Hetnarski, R.B., & Ignaczak, J. (1996). Soliton-like Waves in a Low-temperature Nonlinear Thermoelastic Solid. International Journal of Engineering Science, 34(15), 1767–1787. https://doi.org/10.1016/S0020-7225(96)00046-8.

Huang, F.Y., & Liang, K.Z. (1996). Boundary element method for micropolar thermoelasticity. Engineering Analysis with Boundary Elements, 17(1), 19–26. https://doi.org/10.1016/0955-7997(95)00086-0.

Lord, H.W., & Shulman, Y. (1967). A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids, 15(5), 299–309. https://doi.org/10.1016/0022-5096(67)90024-5.

Neumann, F. (1885). Vorlesungen über die Theorie der Elasticität. Brestau.

Roy Choudhuri, S.K. (2007). On A Thermoelastic Three-Phase-Lag Model. Journal of Thermal Stresses, 30(3), 231–238.

Shakeriaski, F., & Ghodrat, M. (2020). The nonlinear response of Cattaneo-type thermal loading of a laser pulse on a medium using the generalized thermoelastic model. Theoretical and Applied Mechanics Letters, 10(4), 286–297. https://doi.org/10.1016/j.taml.2020.01.030.

Tzou, D.Y. (1997). Macro-to microscale heat transfer. The lagging behavior. Taylor & Francis.

Wang, J.L., & Li, H.F. (2011). Surpassing the fractional derivative: Concept of the memory-dependent derivative. Computers and Mathematics with Applications, 62(3), 1562–1567. https://doi.org/10.1016/j.camwa.2011.04.028.

Wrobel, L.C., & Brebbia, C.A. (1987). The Dual Reciprocity Boundary Element formulation for nonlinear diffusion problems. Computer Methods in Applied Mechanics and Engineering, 65(2), 147–164. https://doi.org/10.1016/0045-

Youssef, H. (2006). Theory of two-temperature-generalized thermoelasticity. IMA Journal of Applied Mathematics, 71(3), 383–390. https://doi.org/10.1093/imamat/hxh101.