Homogenized equation of second-order accuracy for conductivity of laminates

References

• Cosserat E, Cosserat F. Théorie des corps Déformables. Paris: Hermanns; 1909.
   Google Scholar
• Mindlin RD. Microstructure in linear elasticity. Arch Ration Mech Anal. 1965;16:51–78.
   Web of Science ®Google Scholar
• Tupin RA. Elastic materials with couple stresses. Arch Ration Mech Anal. 1962;11:385–414.
   Web of Science ®Google Scholar
• Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech. 1951;18(1):31–38.
   Web of Science ®Google Scholar
• Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech. 1945;12(2):A69–A77.
   Web of Science ®Google Scholar
• Goldenveizer AL. On Reissner's plate theory. Izvestia Akademii Nauk SSSR OTN. 1958;4:102–109.
   Google Scholar
• Bakhvalov NS. Averaging partial differential equations with rapidly oscillating coefficients. Dokl Acad Nauk SSSR. 1975;221:516–519.
   Google Scholar
• Bakhvalov NS, Panasenko GP. Homogenization: Averaging Processes in Periodic Media. Nauka, Moscow: (in Russian). 1984. English translation Dordrecht–Boston–London: Kluwer Academic Publishers; 1989. (Mathematics and Its Applications (Soviet Series); 36).
   Google Scholar
• Friedrichs KO, Dressler RF. A boundary-layer theory for elastic plates. Commun Pure Appl Math. 1961;14(1):1–33.
   Web of Science ®Google Scholar
• Goldenveizer AL. The approximation of the shell theory by the asymptotic analysis of elasticity theory. Pr Math Mech. 1963;27(4):593–608.
   Google Scholar
• Goldenveizer AL. The principles of reducing three-dimensional problems of elasticity to two-dimensional problems of the theory of plates and shells. In: Görtler H, editor. Proceedings, Eleventh International Congress of Theoretical and Applied Mechanics; Springer-Verlag; Berlin: 1964. p. 306–311.
   Google Scholar
• Ciarlet PG, Destuynder P. A justification of the two-dimensional plate model. J Mécanique. 1979;18:315–344.
   Google Scholar
• Gilbert RP, Hsiao GC, Schneider M. The two-dimensional, linear orthotropic plate. Appl Anal. 1983;15(1–4):147–169.
   Web of Science ®Google Scholar
• Goldenveizer AL, Kaplunov JD, Nolde EV. Asymptotic analysis and refinement of Timoshenko–Reisner-type theories of plates and shells. Trans Acad Sci USSR Mekhanika Tverd Tela. 1990;25(6):126–139.
   Google Scholar
• Goldenveizer AL, Kaplunov JD, Nolde EV. On Timoshenko–Reissner type theories of plates and shells. Int J Solids Struct. 1993;30(5):675–694.
   Web of Science ®Google Scholar
• Nazarov SA, Zorin AS. Two-term asymptotics of the problem on longitudinal deformation of a plate with clamped edge. Computer Mech Solids. 1991;2:10–21. (Russian).
   Google Scholar
• Nazarov SA. On the accuracy of asymptotic approximations for longitudinal deformation of a thin plate. Math Model Numer Anal. 1996;30(2):185–213.
   Web of Science ®Google Scholar
• Goldenweiser AL. The theory of elastically thin shells. Moscow: Science; 1976.
   Google Scholar
• Gregory RD, Wan FYM. On plate theories and saint-Venant's principle. Int J Solids Struct. 1985;21(10):1005–1024.
   Web of Science ®Google Scholar
• Panasenko GP. Boundary layer in homogenization problems for non-homogeneous media. In: Godunov SK, Miller JJH, Novikov VA, editors. Proceedings of the Fourth International Conference on Boundary and Interior Layers-Computational and Asymptotic Methods(BAIL 4); Dublin: Boole Press; 1986. p. 398–402.
   Google Scholar
• Panasenko GP, Reztsov MV. Averaging the 3-D elasticity problem in non homogeneous plates. Dokl Akad Nauk SSSR. 1987;294(5):1061–1065. (in Russian); English transl. in Soviet Math Dokl. 2987;35(3):630–636.
   Google Scholar
• Panasenko GP. Multi-scale modelling for structures and composites. Dordrecht: Springer; 2005.
   Google Scholar
• Bakhvalov NS, Eglit ME. Variational properties of averaged equations for periodic media I. Proc Steklov Inst Math (Trudy Mat Inst Steklov). 1992;192:3–18. 1990;192. English version.
   Google Scholar
• Kaplunov JD, Kossovitch LYu., Nolde EV. Dynamics of thin walled elastic bodies. San-Diego: Academic Press; 1998.
   Google Scholar
• Craster RV, Joseph LM, Kaplunov J. Long-wave asymptotic theories: the connection between functionally graded waveguides and periodic media. Wave Motion. 2014;51(4):581–588.
   Web of Science ®Google Scholar
• Bakhvalov NS, Eglit ME. Investigation of the effective equations with dispersion for wave propagation in stratified media and thin plates. Doklady RAN. 2002;383(6):749–753.
   Google Scholar
• Bakhvalov NS, Bogachev KYu., Eglit ME. Investigation of the effective equations with dispersion for wave propagation in heterogeneous thin rods. Doklady RAN. 2002;387(6):749–753.
   Google Scholar
• Panfilov MB. Averaged models of flow processes with heterogeneous internal structure [Doctor Sci. Thesis]. 1992.
   Google Scholar
• Smyshlyaev VP, Cherednichenko KD. On derivation of ‘strain gradient’ effects in the overall behaviour of periodic heterogeneous media. J Mech Phys Solids. 2000;48:1325–1357.
   Web of Science ®Google Scholar
• Panasenko GP. Boundary conditions for the high order homogenized equation: laminated rods, plates and composites. CR Mécanique. 2009;337(1):8–14.
   Web of Science ®Google Scholar
• Kaplunov JD, Pichugin AV. On rational boundary conditions for higher-order long-wave models. IUTAM Symposium on Scaling in Solid Mechanics; Dordrecht: Springer; 2009. p. 81–90.
   Google Scholar
• Prikazchikova L, Ece Aydin Y, Erbas B, et al. Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate. Math Mech Solids. 2020;25(1):3–16.
   Web of Science ®Google Scholar
• Kaplunov J, Prikazchikova L, Alkinidri M. Antiplane shear of an asymmetric sandwich plate. Contin Mech Thermodyn. 2021;33:1247–1262.
   Web of Science ®Google Scholar
• Kaplunov JD, Nolde EV, Shorr BF. A perturbation approach for evaluating natural frequencies of moderately thick elliptic plates. J Sound Vib. 2005;281(3–5):905–919.
   Web of Science ®Google Scholar
• Panasenko GP. Asymptotic partial decomposition of variational problems. C R Acad Sci Paris. 1999;327(11):1185–1190.
   Google Scholar
• Andrianov IV, Awrejcewicz J, Danishevskyy VV. Linear and nonlinear waves in microstructured solids: homogenization and asymptotic approaches. Boca Raton (FL): CRC Press, Taylor & Francis; 2021.
   Google Scholar