Lobachevskii Journal of Mathematics

, Volume 39, Issue 3, pp 448–457

# Geometrically Nonlinear Problem of Longitudinal and Transverse Bending of a Sandwich Plate with Transversally Soft Core

Article

## Abstract

The stress-strain state of sandwich plates with a transversally soft core is determined in one-dimensional geometrically nonlinear formulation. It is supposed that the edges of carrier layers in the right end section are rigidly clamped and the core is not adhesively bound with the support element. The edges of carrier layers in the left end section are assumed to be hinged on diaphragms that are absolutely rigid in the transverse direction, glued to the end section of the core. A load is applied to the median surface of the first carrier layer from the left end section. On the basis of the generalized Lagrange principle, the general statement is formulated as an operator equation in the Sobolev space. The operator is shown to be pseudo-monotonic and coercive. This makes it possible to prove a theorem that there exists a solution. A two-layer iterative method is proposed for solving the problem. The convergence of the method is examined using the additional properties of the operator (i.e., quasi-potentiality and bounded Lipschitz continuity). The iteration parameter variation limits ensuring the method convergence are found. A software package has been developed to conduct numerical experiments for the problem of longitudinal–transverse bending of a sandwich plate. Tabulation is performed with respect to both longitudinal and transverse loads. The results indicate that in terms of weight sophistication and for the given form of loading, the sandwich plate of an asymmetric structure with unequal thicknesses of carrier layers is the most rational and equally stressed plate.

## Keywords

Sandwich plate transversely soft core generalized statement solvability theorem iterative method convergence theorem numerical experiment

## References

1. 1.
V. N. Kobelev, Calculation of Three-Layer Structures (Mashinostroenie, Moscow, 1984) [in Russian].Google Scholar
2. 2.
I. B. Badriev, M. V. Makarov, and V. N. Paimushin, “On the interaction of composite plate having a vibration-absorbing covering with incident acoustic wave,” Russ. Math. 59 (3), 66–71 (2015). doi 10.3103/S1066369X1503007X
3. 3.
Y. Frostig, “Elastica of sandwich panels with a transversely flexible core—A high-order theory approach,” Int. J. Solids Struct. 46, 2043–2059 (2009). doi 10.1016/j.ijsolstr.2008.05.007
4. 4.
S. V. Dyatchenko and A. P. Ivanov, Technology forManufacturing Ship Hulls out of Polymer Composites (Kaliningr. Gos. Tekh. Univ., Kaliningrad, 2007) [in Russian].Google Scholar
5. 5.
B. F. Prokhorov and V. N. Kobelev, Three-Layer Constructions in Ship Building (Sudostroenie, Leningrad, 1972) [in Russian].Google Scholar
6. 6.
V. V. Vasil’ev, A. A. Dobryakov, and A. A. Dudchenko, Principles of Design and Production of Aircraft Constructions from CompositionMaterials (Mosk. Aviats. Inst., Moscow, 1985) [in Russian].Google Scholar
7. 7.
V. N. Krysin, Layered Glued Construction in Aircraft Building (Mashinostroenie, Moscow, 1980) [in Russian].Google Scholar
8. 8.
N. A. Pavlov, Construction of Rockets and Spacecrafts (Mashinostroenie, Moscow, 1993) [in Russian].Google Scholar
9. 9.
V. A. Ivanov, V. N. Paimushin, and T. V. Polyakova, “Refined theory of the stability of three-layer structures (linearized equations of neutral equilibrium and simplest one-dimensional problems),” Izv. Vyssh. Uchebn. Zaved., Ser.Mat., No. 3, 15–24 (1995).
10. 10.
V. N. Paimushin and S. N. Bobrov, “Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms,” Mech. Compos.Mater. 36, 59–66 (2000).
11. 11.
H. Brezis, “équations et inéquations non-linéaires dans les espaces vectoriels en dualité,” Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968).
12. 12.
J.-L. Lions, Quelque problèmes méthodes de résolution des problèmes aux limites nonlinéaires (Dunod, Paris, 1969) [in French].
13. 13.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).
14. 14.
I. B. Badriev, V. V. Banderov, M. V. Makarov, and V. N. Paimushin, “Determination of stressstrain state of geometrically nonlinear sandwich plate,” Appl. Math. Sci. 9, 3887–3895 (2015). doi 10.12988/ams.2015.54354Google Scholar
15. 15.
I. B. Badriev, M. V. Makarov, and V. N. Paimushin, “Solvability of physically and geometrically nonlinear problem of the theory of sandwich plates with transversally-soft core,” Russ. Math. 59 (10), 57–60 (2015). doi 10.3103/S1066369X15100072.
16. 16.
I. B. Badriev, M. V. Makarov, and V. N. Paimushin, “Mathematical simulation of nonlinear problem of three-point composite sample bending test,” Proc. Eng. 150, 1056–1062 (2016). doi 10.1016/j.proeng.2016.07.214
17. 17.
I. B. Badriev, G. Z. Garipova, M. V. Makarov, and V. N. Paimushin, “Numerical solution of the issue about geometrically nonlinear behavior of sandwich plate with transversal soft filler,” Res. J. Appl. Sci. 10, 428–435 (2015). doi 10.3923/rjasci.2015.428.435Google Scholar
18. 18.
I. B. Badriev, V. V. Banderov, M. V. Makarov, and V. N. Paimushin, “Solving nonlinear problems of the theory of multilayer shells with transversal-soft aggregate,” in Proceedings of the 10th International Conference on Grid Methods for Boundary Problems and Applications (Kazan. Gos. Univ., Kazan’, 2014), pp. 103–107.Google Scholar
19. 19.
I. B. Badriev, V. V. Banderov, G. Z. Garipova, M. V. Makarov, and R. R. Shagidullin, “On the solvability of geometrically nonlinear problem of sandwich plate theory,” Appl. Math. Sci. 9, 4095–4102 (2015). doi 10.12988/ams.2015.54358Google Scholar
20. 20.
I. B. Badriev, V. V. Banderov, G. Z. Garipova, and M. V. Makarov, “On the solvability of a nonlinear problem for sandwich plates,” Vestn. Tambov. Univ., Ser. Estestv. Tekh. Nauki 20, 1034–1037 (2015).Google Scholar
21. 21.
I. B. Badriev, G. Z. Garipova, M. V. Makarov, V. N. Paimushin, and R. F. Khabibullin, “Solving physically nonlinear equilibrium problems for sandwich plates with a transversally soft core,” Lobachevskii J.Math. 36, 474–481 (2015). doi 10.1134/S1995080215040216
22. 22.
I. B. Badriev, M. V. Makarov, and V. N. Paimushin, “Numerical investigation of physically nonlinear problem of sandwich plate bending,” Proc. Eng. 150, 1050–1055 (2016). doi 10.1016/j.proeng.2016.07.213
23. 23.
I. B. Badriev and V. V. Banderov, “Numericalmethod for solving variation problems in mathematical physics,” Appl. Mech. Mater. 668–669, 1094–1097 (2014).
24. 24.
I. B. Badriev and V. V. Banderov, “Iterative methods for solving variational inequalities of the theory of soft shells,” Lobachevskii J.Math. 35, 354–365 (2014). doi 10.1134/S1995080214040015
25. 25.
I. B. Badriev, V. V. Banderov, and O. A. Zadvornov, “On the equilibrium problem of a soft network shell in the presence of several point loads,” Appl. Mech. Mater. 392, 188–190 (2013). doi 10.4028/www.scientific.net/AMM.392.188
26. 26.
I. B. Badriev and O. A. Zadvornov, “Investigation of the solvability of an axisymmetric problem of determining the equilibrium position of a soft shell of revolution,” Russ. Math. (Iz. VUZ) 49, 21–26 (2005).
27. 27.
I. B. Badriev and V. V. Banderov, “Numerical solution of the equilibrium of axisymmetric soft shells,” Vestn. Tamb. Tekh. Univ. 21 (1), 29–35 (2015).Google Scholar
28. 28.
V. N. Paimushin, “To variational solution methods of spatial problems of deformed body interfacing,” Dokl. Akad. Nauk SSSR 273, 1083–1086 (1983).
29. 29.
V. N. Paimushin, “Generalized Reissner variational principle in nonlinear mechanics of three-dimensional composite solids, with applications to the theory of multilayer shells,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 171–180 (1987).Google Scholar
30. 30.
R. A. Adams, Sobolev Spaces (Academic, New York, San Francisco, London, 1975).
31. 31.
M. M. Vainberg, Variational Method and Method of Monotone Operators (Nauka,Moscow, 1972; Wiley, New York, 1974).Google Scholar
32. 32.
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934).
33. 33.
I. B. Badriev and O. A. Zadvornov, Iteration Solution Methods of Variational Inequalities in Hilbert Spaces (Kazan. Gos. Univ., Kazan’, 2007) [in Russian].
34. 34.
F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
35. 35.
H. Gajewskii, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974).
36. 36.
S. A. Telyakovskii, Course of Lectures onMathematical Analysis 1 (MIAN, Moscow, 2009) [in Russian].Google Scholar
37. 37.
M. M. Karchevskii, “Solvability of variational problems in the nonlinear theory of shallow shells,” Differ. Equations 27, 841–847 (1991).
38. 38.
M. M. Karchevskii and V. N. Paimushin, “Variational problems in the theory of three-layer shallow shells,” Differ. Equations 30, 1126–1130 (1994).
39. 39.
I. B. Badriev, O. A. Zadvornov, and A. M. Saddek, “Convergence analysis of iterative methods for some variational inequalities with pseudomonotone operators,” Differ. Equations 37, 934–942 (2001). 10.1023/A:1011901503460.
40. 40.
M. M. Vainberg, Variational Methods for Nonlinear Operator Study (Gostekhizdat, Moscow, 1956) [in Russian].
41. 41.
M. M. Vainberg and I. M. Lavrent’ev, “Nonlinear quasipotential operators,” Dokl. Akad. Nauk SSSR 205, 1022–1024 (1972).
42. 42.
I. B. Badriev, M. M. Karchevskii, “Convergence of an iterative process in a Banach space,” J.Math. Sci. 71, 2727–2735 (1994). doi 10.1007/bf02110578
43. 43.
I. B. Badriev, “On the solving of variational inequalities of stationary problems of two-phase flow in porous media,” Appl.Mech. Mater. 392, 183–187 (2013). doi 10.4028/www.scientific.net/AMM.392.183
44. 44.
I. B. Badriev, O. A. Zadvornov, and A. D. Lyashko, “A study of variable step iterative methods for variational inequalities of the second kind,” Differ. Equations 40, 971–983 (2004). doi 10.1023/B:DIEQ.0000047028.07714.df
45. 45.
I. B. Badriev, “Mathematical simulation of stationary seepage problemwith multivalued law,” Vestn. Tambov. Univ., Ser. Estestv. Tekh. Nauki 18, 2444–2446 (2013).Google Scholar
46. 46.
I. B. Badriev and O. A. Zadvornov, “Iterative methods for solving variational second type inequalities with inversely strongly monotone operators,” Izv.Vyssh. Uchebn.Zaved., Mat. 1, 20–28 (2003).
47. 47.
A. A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, New York, Basel, 2001).
48. 48.
V. N. Paimushin, “Stability theory of three-layer plates and shells (Development stages, modern state and direction of further research),” Izv. Akad. Nauk,Mekh. Tverd. Tela, No. 2, 148–162 (2001).Google Scholar