Abstract
An iterative analytical theory in the mechanics of layered composite systems is developed. The prehistory of the nonclassical theory of layered systems is presented. The division of this theory into two principal directions - discrete-structural and continuous-structural - is mentioned. The basic iterative Ambartsumyan theory, which belongs to the second direction, is described. The formation of the generalized iteration theory of first approximation is shown. In this theory, the disagreement between the kinematic and static models is removed, i.e., a generalization of these models is realized. The theory of second approximation is described. An iterative principle is presented for the formation of a higher-approximation nonclassical theory. Based on this principle, theories of anisotropic composite shallow shells, plates, and beams are formulated. Comparative calculation results for different layered composite systems are presented.
Similar content being viewed by others
REFERENCES
S. P. Timoshenko, "On the correction for shear of the differential equation for transverse vibrations of prismatic bars," Philos. Mag. J. Sci., 41, Ser. 6, No. 245, 744–746 (1921).
S. G. Lekhnitskii, "Calculation of stresses in composite beams," Vestn. Inzh. Tekhn., No. 9 (1935).
E. Reissner, "On the theory of bending of elastic plates," J. Math. Phys., XXXIII, 184–191 (1944).
E. Reissner, "Finite deflections of sandwich plates," J. Aero. Sci., 15, No. 17, 17–23 (1948).
A. P. Prusakov, "Basic equations of three-layer plates with a light core," Prikl. Matem. Mekh., XI, Iss. 1, 27–36 (1951).
E. I. Grigolyuk and P. P. Chulkov, "Calculation of three-layer plates with a rigid core," Izv. Akad. Nauk. SSSR. Otd. Tekhn. Nauk., No. 1, 77–84 (1957).
E. I. Grigolyuk, "Equations of three-layer shells with a light core," Izv. Akad. Nauk. SSSR. Otd. Tekhn. Nauk., No. 1, 26–34 (1958).
E. I. Grigolyuk and P. P. Chulkov, "Nonlinear equations of shallow multilayer shells of regular structure," Dokl. Akad. Nauk. SSSR. Mekh. Tverd. Tela, No. 1, 163–169 (1967).
E. I. Grigolyuk and P. P. Chulkov, Stability and Vibrations of Three-Layer Shells [in Russian], Mashinostroenie, Moscow (1973).
V. V. Bolotin, "To the theory of layered plates," Izv. Akad. Nauk SSSR. Otd. Tekhn. Nauk. Mekh. Mashinostr., No.3, 65–72 (1963).
V. V. Bolotin and Yu. N. Novichkov, Mechanics of Multilayered Structures[in Russian], Mashinostroenie, Moscow (1980).
A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, Strength of Polymer and Composite Materials [in Russian], Zinatne, Riga (1980).
A. N. Guz', Ya. M. Grigorenko, I. Yu. Babich, et al., Mechanics of Structural Elements. Mechanics of Composite Materials and Structural Elements in 3 Vols. Vol. 2 [in Russian], Naukova Dumka, Kiev (1983).
R. Christensen, Mechanics of Composite Materials, John Wiley & Sons, New York–Chichester–Brisbane–Toronto (1979).
B. E. Pobedrya, Mechanics of Composite Materials [in Russian], Nauka, Moscow (1984).
V. V. Vasil'ev, Mechanics of Structures of Composite Materials [in Russian], Mashinostroenie, Moscow (1988).
E. I. Grigolyuk and I. T. Selezov, Nonclassical Vibration Theory of Rods, Plates, and Shells. Reviews of the Science and Technology. Vol. 5 [in Russian], Nauka, Moscow (1972).
E. I. Grigolyuk and F. A. Kogan, "State of the art of the theory of multilayered shells," Prikl. Mekh., 8, No. 6, 3–17 (1972).
A. A. Dudchenko, S. A. Lur'e, and I. F. Obraztsov, "Anisotropic multilayered plates and shells," Itogi Nauki Tekhn. Ser. Mekh. Deformir. Tverd. Tela, 15, 3–68, VINITI, Moscow (1983).
J. N. Reddy, "An evaluation of equivalent-single-layer and layerwise theories of composite laminates," Compos. Struct., 25, 21–35 (1993).
S. G. Lekhnitskii, Anisotropic Plates [in Russian], Moscow (1957).
S. A. Ambartsumyan, "Some problems of the theory of anisotropic shells," Izv. Akad. Nauk. SSSR, No. 9 (1947).
S. A. Ambartsumyan, "Calculation of layered shells of revolution," Dokl. Akad. Nauk. Arm SSR, 11, No. 2, (1949).
S. A. Ambartsumyan, "To the calculation of two-layer orthotropic shells," Izv. Akad. Nauk. SSSR, No. 7, (1957).
S. A. Ambartsumyan, "A new refined theory of anisotropic shells," Polym. Mech., 6, No. 5, 766–776 (1970).
S. A. Ambartsumyan, General Theory of Anisotropic Shells [in Russian], Nauka, Moscow (1974).
S. A. Ambartsumyan, Theory of Anisotropic Plates [in Russian], Nauka, Moscow (1987).
A. L. Gol'denveizer, "On the theory of bending of Reissner plates," Izv. Akad. Nauk. SSSR. Otd. Tekhn. Nauk., No. 4, 102–109 (1958).
E. S. Osternik, "Anisotropic layered plates of medium thickness," Izv. Akad. Nauk. Arm SSR. Mekh., 20, No. 5, 48–57 (1967).
O. F. Ryabov, Elaboration of Multilayer Shells [in Ukrainian], Budivel'nik, Kiev (1968).
A. G. Teregulov, "To the theory of multilayer anisotropic shells," Issled. Teor. Plast. Obol., Iss. VI-VII, 762–767 (1970).
A. P. Prusakov, "Nonlinear equations of bending of multilayer shells," Prikl. Mekh., 7, No. 3, 3–8 (1971).
Ya. M. Grigorenko, Isotropic and Anisotropic Layered Shells of Revolution of Variable Stiffness [in Russian], Naukova Dumka, Kiev (1973).
A. O. Rasskazov, "To the theory of multilayer orthotropic shallow shells," Prikl. Mekh., 12, No. 11, 50–56 (1976).
V. G. Piskunov, "On a variant of the nonclassical theory of multilayer shallow shells and plates," Prikl. Mekh., 15, No. 11, 76–81 (1979).
A. N. Andreev and V. V. Nemirovskii, "On a variant of the theory of elastic multilayer anisotropic plates," Prikl. Mekh., 14, No. 7, 55–62 (1978).
A. O. Rasskazov, I. I. Sokolovskaya, and N. A. Shul'ga, Theory and Calculation of Layered Orthotropic Plates and Shells [in Russian], Vishcha Shkola, Kiev (1986).
V. G. Piskunov and V. E. Verizhenko, Linear and Nonlinear Problems of Calculation of Laminate Structures [in Russian], Budivel'nik, Kiev (1986).
V. G. Piskunov, V. E. Verizhenko, V. K. Prisyazhnyuk, V. G. Sipetov, and V. S. Karpilovskii, Calculation of Inhomogeneous Shallow Shells and Plates by the Finite-Element Method [in Russian], Vishcha Shkola, Kiev (1987).
V. G. Piskunov, A. G. Gurtovoi, and V. S. Sipetov, "To the refinement of the theory of anisotropic plates," Prikl. Mekh., 20, No. 5, 82–87 (1984).
V. G. Piskunov, V. E. Verijenko, S. Adali, P. Y. Tabakov, V. K. Prisyazhnyouk, and S. G. Buryhin, "Rational higher-order theory of transverse shear deformation of anisotropic laminated plates and shells," Int. J. Solids Struct., 38, 6491–6523 (2001).
M. Savoia and J. N. Reddy, "A variational approach to three-dimensional elasticity solutions of laminated composite plates," J. Appl. Mech. ASME, 59, 166–175 (1992).
V. S. Sipetov, "Accurate analysis of the behavior of laminated composite structures on the thermal and static effects," Mech. Compos. Mater., 25, No. 1, 123–129 (1989).
A. V. Marchuk, "Generalization of discrete and continuous structural approaches to elaboration of a mathematical model of laminated plates and masses," Mech. Compos. Mater., 32, No. 3, 261–269 (1996).
A. O. Rasskazov and A. V. Burygina, "To the refinement of the shear theory of layered shallow orthotropic shells," Prikl. Mekh., 24, No. 4, 31–37 (1988).
V. G. Piskunov, A. V. Burygina, and A. O. Rasskazov, "Shear effects of the stress state in transversely isotropic plates. 1. Vortex effect," Probl. Prochn., No. 1, 56–62 (1998).
V. G. Piskunov, A. V. Burygina, and A. O. Rasskazov, "Shear effects of the stress state in transversely isotropic plates. 2. Potential effect," Probl. Prochn., No. 1, 63–71 (1998).
V. G. Piskunov and A. O. Rasskazov, "Investigation of the stress-strain state of shallow shells and plates, based on the shear theory of second approximation," Prikl. Mekh., 34, No. 8, 103–110 (1998).
A. G. Gurtovyi, "High-precision modeling of deformation of laminated structures," Mech. Compos. Mater., 35, No. 1, 7–18 (1999).
V. V. Vasiliev, "Modern conceptions of plate theory," Compos. Struct., 48, 39–48 (2000).
R. B. Rikards and G. A. Teters, Stability of Shells of Composite Materials [in Russian], Zinatne, Riga (1974).
E. I. Grigolyuk and G. M. Kulikov, Multilayer Reinforced Shells. Calculation of Pneumatic Tires [in Russian], Mashinostroenie, Moscow (1988).
A. S. Sakharov, A. L. Kozak, A. V. Gondlyakh, and S. L. Mel'nikov, "Mathematical model of deformation of multilayered composite shell systems," Soprot. Mater. Teor. Soor., No. 44, 13–16 (1984).
V. A. Bazhenov, A. S. Sakharov, A. V. Gondlyakh, and S. L. Mel'nikov, Nonlinear Problems of the Mechanics of Multilayered shells [in Russian], Kiev (1994).
V. G. Piskunov, "The problem of constructing models for physicomechanical processes in layered structural systems," in: Mechanics of Heterogeneous Structures. Vol. 1 [in Russian], L'vov (1987), pp. 211–212.
O. N. Demchuk, "Comparison of approximate refined and exact 3D thermoelasticity solutions for laminated anisotropic plates," Mech. Compos. Mater., 31, No. 2, 154–162 (1995).
V. G. Piskunov, A. V. Goryk, and V. N. Cherednikov, "Modeling of transverse shears of piecewise homogeneous composite bars using an iterative process with account of tangential loads. 1. Construction of a model," Mech. Compos. Mater, No. 4, 287–296 (2000).
V. G. Piskunov, A. V. Goryk, and V. N. Cherednikov, "Modeling of transverse shears of piecewise homogeneous composite bars using an iterative process with account of tangential loads. 2. Resolving equations and results," Mech. Compos. Mater., 36, No. 6, 445–452 (2000).
E. S. Osternik, "Experimental investigation of deformation of the normal in layered plates," in: Theory of Shells and Plates [in Russian], Nauka, Moscow (1973), pp. 735–739.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Piskunov, V.G. An Iterative Analytical Theory in the Mechanics of Layered Composite Systems. Mechanics of Composite Materials 39, 1–16 (2003). https://doi.org/10.1023/A:1022979003150
Issue Date:
DOI: https://doi.org/10.1023/A:1022979003150