We formulate the problem of quasistatic thermoelastoplastic bending of layered plates with regular structures in the geometrically linear statement. The mechanical behavior of isotropic layers is described by the deformation-type relations of thermoelastoplasticity with regard for their different tensile and compression resistances. The linearized governing relations of layered media are deduced with the help of the method of variable parameters of elasticity. The obtained equations enable us to describe, with different degrees of accuracy, the stress-strain state of these plates by taking into account their weakened resistance to transverse shears. Note that the relations of traditional nonclassical Reissner and Reddy theories follow from these equations as particular cases. Within the framework of the proposed refined theories and Reddy theory, the force boundary conditions for tangential stresses are satisfied on the front surfaces. The boundary conditions for normal stresses are not satisfied on these surfaces. The variations of deflections across the thickness of the structures are not taken into account. The threedimensional equilibrium equations and the boundary conditions imposed on the end surface of the plate are reduced to two-dimensional relations by the method of weighted residuals. As weight functions, we use homogeneous polynomials in the transverse coordinate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. A. Abrosimov and V. G. Bazhenov, Nonlinear Problems of the Dynamics of Composite Structures [in Russian], Izd. Nizhegorod. Gos. Univ., Nizhnii Novgorod (2002).
S. A. Ambartsumyan, Theory of Anisotropic Plates. Strength, Stability, and Vibrations [in Russian], Nauka, Moscow (1987).
A. N. Andreev and Yu. V. Nemirovskii, Multilayer Anisotropic Shells and Plates. Bending, Stability, and Vibrations [in Russian], Nauka, Novosibirsk (2001).
V. A. Bazhenov, O. P. Krivenko, and N. A. Solovei, Nonlinear Deformation and Stability of Elastic Shells with Inhomogeneous Structures: Models, Methods, Algorithms, and Poorly Studied and New Problems [in Russian], “Librokom,” Moscow (2013).
N. I. Bezukhov, V. L. Bazhanov, I. I. Gol’denblat, N. A. Nikolaenko, and A. M. Sinyukov, Strength, Stability, and Vibration Analyses under the Conditions of High Temperatures [in Russian], Mashinostroenie, Moscow (1965).
V. L. Biderman, Mechanics of Thin-Walled Structures. Statics [in Russian], Mashinostroenie, Moscow (1977).
V. V. Vasil’ev, Mechanics of Structures of Made of Composite Materials [in Russian], Mashinostroenie, Moscow (1988).
V. G. Zubchaninov, Mechanics of Processes of Plastic Media [in Russian], Fizmatlit, Moscow (2010).
A. A. Il’yushin, Works, Vol. 3: Theory of Thermoviscoelasticity [in Russian], Fizmatlit, Moscow (2007).
G. M. Kulikov, “Thermoelasticity of flexible multilayer anisotropic shells,“ Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 33–42 (1994).
S. G. Lekhnitskii, Anisotropic Plates [in Russian], Gostekhizdat, Moscow (1947).
N. N. Malinin, Applied Theory of Plasticity and Creep [in Russian], Mashinostroenie, Moscow (1968).
A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, Resistance of Stiff Polymeric Materials [in Russian], Zinatne, Riga (1972).
Yu. V. Nemirovskii and G. L. Gorynin, “Method of stiffness functions in the problems of numerical analyses of multilayer rods and plates,” Vestnik Nizhegorod. Univ., Mekh. Deform. Tverd. Tela, No. 4 (4), 1654–1656 (2011).
V. V. Pikul’, Mechanics of Shells [in Russian), Dal’nauka, Vladivostok (2009).
A. A. Berlin (editor), Polymeric Composite Materials: Structure, Properties, Technology: A Textbook [in Russian), Professiya, St. Petersburg (2009).
M. I. Savenkova, S. V. Sheshenin, and I. M. Zakalyukina, “Application of the averaging method in the problem of elastoplastic bending of a plate,” Vestn. Mosk. Gos. Stroit. Univ., No. 9, 156–164 (2012).
G. Lubin (editor), Handbook of Composites, Van Nostrand Reinhold, New York (1982).
A. P. Yankovskii, “Investigation of the unsteady creep of layered plates with regular structures made of nonlinear hereditary materials with regard for weakened resistance to transverse shears,” Mekh. Kompoz. Mater. Konstruk., 21, No. 3, 412–433 (2015).
A. P. Yankovskii, “Modeling of elastoplastic bending of metal-composite layered plates with regular structures. 1. Structural model,” Vestn. Yakovlev Chuvash. Gos. Ped. Univ., Ser. Mekh. Predel. Sost., No. 2 (12), 102–111 (2012).
A. P. Yankovskii, “Modeling of the elastoplastic bending of metal-composite layered plates with regular structures. 2. Refined model of deformation,” Vestn. Yakovlev Chuvash. Gos. Ped. Univ., Ser. Mekh. Predel. Sost., No. 3 (13), 38–56 (2012).
A. P. Yankovskii, “Refined model of the bending deformation of layered beams-walls with regular structures made of nonlinear elastic materials,” Konstr. Kompozit. Mater., No. 1, 18–29 (2016).
K. Belkaid, A. Tati, and R. Boumaraf, “A simple finite element with five degrees of freedom based on Reddy’s third-order shear deformation theory,” Mech. Composit. Mater., 52, No. 2, 257–270 (2016).
F. A. Fazzolari, “A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells,” Composit., Part B: Eng., 62, 143–158 (2014).
S. K. Gill, M. Gupta, and P. S. Satsangi, “Prediction of cutting forces in machining of unidirectional glass fiber reinforced plastic composite,” Front. Mech. Eng., 8, No. 2, 187–200 (2013).
C. J. Lissenden, “Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites,” Int. J. Plasticity, 26, No. 11, 1606–1628 (2010).
K. H. Lo, R. M. Christensen, and E. M. Wu, “A high-order theory of plate deformation—Part 1: Homogeneous plates,” Trans. ASME, J. Appl. Mech., 44, No. 4, 663–668 (1977).
S. T. Mau, “A refined laminated plates theory,” Trans. ASME, J. Appl. Mech., 40, No. 2, 606–607 (1973).
R. D. Mindlin, “Thickness-shear and flexural vibrations of crystal plates,” J. Appl. Phys., 22, No. 3, 316–323 (1951). https://doi.org/10.1063/1.1699948.
A. P. Mouritz, E. Gellert, P. Burchill, and K. Challis, “Review of advanced composite structures for naval ships and submarines,” Compos. Struct., 53, No. 1, 21–42 (2001).
J. N. Reddy, “A refined nonlinear theory of plates with transverse shear deformation,” Int. J. Solids Struct., 20, No. 9–10, 881–896 (1984).
J. N. Reddy, “A simple higher-order theory for laminated composite plates,” Trans. ASME, J. Appl. Mech., 51, No. 4, 745–752 (1984).
J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, Wiley, New York (1984).
E. Reissner, “A consistent treatment of transverse shear deformations in laminated anisotropic plates,” AIAA Journal, 10, No. 5, 716–718 (1972).
E. Reissner, “On the theory of bending of elastic plates,” J. Math. & Phys., 23, No. 1–4, 184–191 (1944).
C. H. Thai, L. V. Tran, D. T. Tran, T. Nguyen-Thoi, and U. Nguyen-Xuan, “Analysis of laminated composite plates using higherorder shear deformation plate theory and mode-based smoother discrete shear gap method,” Appl. Math. Model., 36, No. 11, 5657–5677 (2012).
G. Verchery, “Design rules for the laminate stiffness,” Mech. Compos. Mater., 47, No. 1, 47–58 (2011).
J. M. Whitney, “The effect of transverse shear deformation on the bending of laminated plates,” J. Composit. Mater., 3, No. 3, 534–547 (1969).
J. M. Whitney and A. W. Leissa, “Analysis of heterogeneous anisotropic plates,” Trans. ASME, J. Appl. Mech., 36, No. 2, 261–266 (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 61, No. 1, pp. 116–129, January–March, 2018.
Rights and permissions
About this article
Cite this article
Yankovskii, А.P. Refined Model of Thermoelastoplastic Bending of Layered Plates with Regular Structures. I. Statement of the Problem. J Math Sci 249, 446–461 (2020). https://doi.org/10.1007/s10958-020-04952-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04952-5