Thermo-Magneto-Electro-Elastic Nonlinear Bending of Anisotropic Plates of Complex Shape
Keywords:
R-function method, anisotropic composite plate, thermo-magneto-elasticity, geometric nonlinearity, Duhamel-Neumann law, Bubnov-Galerkin method, Newmark schemeAbstract
This paper proposes a mathematical model and a computational algorithm for the nonlinear deformation of anisotropic plates of complex shape subjected to coupled thermal, electric, magnetic, and mechanical fields. The model is based on the Hamilton-
Ostrogradsky variational principle and the Kirchhoff- Love hypothesis, together with the von Karman geometric nonlinearity describing large deflections; the coupled effects of the temperature, electric, and magnetic fields in the anisotropic medium are incorporated into the constitutive equations through the Duhamel-Neumann law, the Lorentz force, and Maxwell's equations. An arbitrary multiply connected plate domain and mixed boundary conditions are described by a single analytical
expression by means of the R-function method, and a solution structure that exactly satisfies the boundary
conditions is constructed. Spatial discretization is performed using the Bubnov-Galerkin method and
Gaussian quadrature, while time integration is carried out using the implicit Newmark scheme combined with an
iterative algorithm based on an equivalent stiffness matrix for the nonlinear terms. Numerical experiments were
performed for a rigidly clamped graphite-epoxy plate under a distributed mechanical load, a magnetic field, and
a temperature change. The results show that linear theory overestimates the maximum deflection by 16.46 percent
compared with the nonlinear solution, which converged in eight iterations, owing to the membrane stiffening effect.
The maximum deflection is inversely proportional to the cube of the plate thickness, with a correlation coefficient of
0.996. The proposed approach is suitable for engineering calculations of composite structural elements.
References
S. A. Ambartsumyan, Theory of Anisotropic Plates:
Strength, Stability, and Vibrations. Stamford, CT, USA:
Technomic, 1970.
J. N. Reddy, Mechanics of Laminated Composite Plates
and Shells: Theory and Analysis, 2nd ed. Boca Raton, FL,
USA: CRC Press, 2004.
R. D. Mindlin, “Influence of rotatory inertia and shear on
flexural motions of isotropic, elastic plates,” J. Appl.
Mech., vol. 18, no. 1, pp. 31–38, 1951.
S. Volmir, Flexible Plates and Shells. Moscow, USSR:
Gostekhizdat, 1956 (in Russian).
S. Volmir, Nonlinear Dynamics of Plates and Shells.
Moscow, USSR: Nauka, 1972 (in Russian).
W. Nowacki, Thermoelasticity, 2nd ed. Oxford, U.K.:
Pergamon Press, 1986.
S. A. Ambartsumyan, G. E. Bagdasaryan, and M. V.
Belubekyan, Magnetoelasticity of Thin Shells and Plates.
Moscow, USSR: Nauka, 1977 (in Russian).
F. C. Moon, Magneto-Solid Mechanics. New York, NY,
USA: Wiley, 1984.
H. Parkus, Magneto-Thermoelasticity (CISM Courses and
Lectures, no. 118). Vienna, Austria: Springer, 1972.
N. M. Faleh, I. K. Abboud, and A. F. Nori, “Nonlinear
stability of smart nonlocal magneto-electro-thermo-elastic
beams with geometric imperfection and piezoelectric
phase effects,” Smart Struct. Syst., vol. 25, no. 6, pp. 707–
, 2020, doi: 10.12989/sss.2020.25.6.707.
Z. Cao, X. Liang, Y. Deng, X. Zha, R. Zhu, and J. Leng,
“Novel semi-analytical solutions for the transient
behaviors of functionally graded material plates in the
thermal environment,” Materials, vol. 12, no. 24, Art. no.
, 2019, doi: 10.3390/ma12244084.
Y. Li, Y. Li, Y. Han, and C. Wang, “Propagation of
coupled waves across a magneto-electro-thermo-elastic
interface with consideration of fractional order
thermoelasticity and microstructural effect,” Acta Mech.,
, doi: 10.1007/s00707-024-04000-2.
D. A. Shlyakhin, E. V. Savinova, and V. A. Yurin,
“Dynamic problem of thermoelectroelasticity for round
rigidly fixed plate,” FEFU: School of Engineering Bulletin,
no. 1(50), pp. 3–16, 2022, doi: 10.24866/2227-6858/2022-
/3-16.
Q. Li and P. Hou, “Three-dimensional quasi-static general
solution for isotropic thermoelastic medium with
applications,” Case Stud. Therm. Eng., vol. 25, Art. no.
, 2021, doi: 10.1016/j.csite.2021.100897.
V. L. Rvachev, Theory of R-Functions and Some of Its
Applications. Kiev, USSR: Naukova Dumka, 1982 (in
Russian).
V. L. Rvachev and L. V. Kurpa, R-Functions in Problems
of Bending of Plates of Complex Shape. Kiev, USSR:
Naukova Dumka, 1988 (in Russian).
L. V. Kurpa, “Method of R-functions for investigation of
parametric vibrations of orthotropic plates of complex
shape,” J. Math. Sci., vol. 174, no. 3, 2011.
L. Kurpa, G. Timchenko, and A. Osetrov, “Research of
nonlinear vibrations of laminated shallow shells with
cutouts by R-functions method,” Bull. NTU “KhPI”, Ser.
Dynamics and Strength of Machines, no. 46, pp. 86–89,
, doi: 10.20998/2078-9130.2016.46.88056.
L. Kurpa, J. Awrejcewicz, O. Mazur, and I. Morachkovska,
“Free vibrations of small-scale plates with complex shape
based on the nonlocal elasticity theory,” Acta Mech., 2022,
doi: 10.1007/s00707-022-03361-w.
V. K. Kabulov, Algorithmization in the Theory of
Elasticity and the Deformation Theory of Plasticity.
Tashkent, USSR: Fan, 1966 (in Russian).
F. Nuraliev, N. Tojiev, and B. Tahirov, “Mathematical
model and computational analysis of the nonlinear stress–
strain processes in anisotropic thermo-electro-
magnetoelastic plates with complex shapes,” Development
of Science, vol. 1, no. 3, pp. 18–23, 2024.
F. Nuraliev, M. Mirzaakhmedov, N. Tojiyev, and O.
Abdullayev, “Independent solution of the magnetoelastics
problem of thin compound-shaped isotropic plates,”
Modern Problems of Applied Mathematics and
Information Technology – Al-Khwarizmi, no. 1, 2024.
M. Artikbayev and A. Orazbayev, “Results of solving the
magnetoelastic problem of thin anisotropic plates with
complex form,” AIP Conf. Proc., vol. 3244, no. 1, Art. no.
, 2024, doi: 10.1063/5.0242124.
Sh. A. Nazirov and F. M. Nuraliev, “Mathematical
modeling of processes of electro-magnetic fields’ effects
on thin conducting plates by complex form,” in Proc. 4th
Int. Conf. Application of Information and Communication
Technologies (AICT), Tashkent, Uzbekistan, 2010, pp. 1–
, doi: 10.1109/ICAICT.2010.5612049.
F. M. Nuraliev, “Mathematical models of
magnetoelasticity of thin shells and plates,” Problems of
Computational and Applied Mathematics, no. 128, pp. 53–
, 2012 (in Russian).
Downloads
Published
How to Cite
Issue
Section
Categories
License
Copyright (c) 2026 Nuriddin Tojiyev, Shohruh Safarov, Behzod Tahirov, Otabek Abdullayev
This work is licensed under a Creative Commons Attribution 4.0 International License.
