Length-scale effect in stability problems for thin biperiodic cylindrical shells: extended tolerance modelling
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O R I G I NA L A RT I C L E
B. Tomczyk · M. Goła˛bczak
· A. Litawska · A. Goła˛bczak
Length-scale effect in stability problems for thin biperiodic cylindrical shells: extended tolerance modelling
Received: 20 September 2020 / Accepted: 7 October 2020 © The Author(s) 2020
Abstract Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells of periodically microinhomogeneous structure in circumferential and axial directions (biperiodic shells) are investigated. The aim of this contribution is to formulate and discuss a new averaged nonasymptotic model for the analysis of selected stability problems for these shells. This, so-called, general nonasymptotic tolerance model is derived by applying a certain extended version of the known tolerance modelling procedure. Contrary to the starting exact shell equations with highly oscillating, noncontinuous and periodic coefficients, governing equations of the tolerance model have constant coefficients depending also on a cell size. Hence, the model makes it possible to investigate the effect of a microstructure size on the global shell stability (the length-scale effect). Keywords Micro-heterogeneous biperiodic cylindrical shells · Extended tolerance modelling · Stability problems · Length-scale effect
1 Introduction Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells with a periodically micro-inhomogeneous structure in circumferential and axial directions (biperiodic shells) are objects of consideration. By periodic inhomogeneity we shall mean periodically varying thickness and/or periodically varying inertial and elastic properties of the shell material. We restrict our consideration to those biperiodic cylindrical shells, which are Communicated by Andreas Öchsner. B. Tomczyk Department of Mechanics and Building Structures, Warsaw University of Life Sciences, Nowoursynowska Str. 166, 02-787 Warsaw, Poland E-mail: [email protected] M. Goła˛bczak (B) Institute of Machine Tools and Production Engineering, Lodz University of Technology, Stefanowskiego Str. 1/15, 90-924 Lodz, Poland E-mail: [email protected] A. Litawska Department of Structural Mechanics, Lodz University of Technology, Politechniki Str. 6, 90-924 Lodz, Poland E-mail: [email protected] A. Goła˛bczak State Vocational University in Włocławek, 3 Maja 17 Str., 87-800 Włocławek, Poland E-mail: [email protected]
B. Tomczyk et al.
Fig. 1 Fragment of the shell reinforced by two families of biperiodically spaced ribbs
composed of a large number of identical elements. Moreover, every such element, called a periodicity cell, can be treated as a thin shell. Typical examples of such shells are presented in Fig. 1 (stiffened shell). The mechanical problems of periodic structures (shells, plates, beams) are described by partial differential equations with periodic, highly oscillating and discontinuous coefficients. Thus, these equations are too complicated to constitute the basis for investigations of most of the engineering problems. To obtain average
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