On the modelling of stability problems for thin cylindrical shells with two-directional micro-periodic structure

Tomczyk, Barbara; Bagdasaryan, Vazgen; Gołąbczak, M.; Litawska, Anna

doi:10.1016/j.compstruct.2021.114495

Cited by 12 publications

(9 citation statements)

References 22 publications

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“…Hence, the tolerance models allow us to investigate the lengthscale effect in dynamical and stationary problems. From comparison of the doubly underlined terms in constitutive relations (12), (13) of the general tolerance model with the corresponding doubly underlined terms in constitutive relations (15), ( 16) of the standard model, it follows that general constitutive relations (12), ( 13) contain a bigger number of terms depending on the microstructure size than the standard constitutive relations (15), (16). Thus, from the analytical results it follows that the general model proposed here makes it possible to investigate the length-scale effect in more detail.…”

Section: Standard Tolerance Model Equationsmentioning

confidence: 99%

“…Some applications of this averaging method to the modelling of mechanical and thermomechanical problems for various periodic structures are shown in many works. We can mention here monograph by Tomczyk [10] and papers by Tomczyk et al [11,12], where the length-scale effect in mechanics of periodic cylindrical shells is investigated; papers by Baron [13], where dynamic problems of medium thickness periodic plates are studied and by Marczak and Je ˛drysiak [14], Marczak [15,16], where dynamics of periodic sandwich plates is analysed; papers by Je ˛drysiak [17][18][19], which deal with stability of thin periodic plates; papers by Łaciński and Woźniak [20], Rychlewska et al [21], Ostrowski and Je ˛drysiak [22], Kubacka and Ostrowski [23], where problems of heat conduction in conductors with periodic structure are analysed. Let us also mention papers by Bagdasaryan et al [24], Tomczyk and Goła ˛bczak [25], Tomczyk et al [26], which deal with coupled thermoelasticity problems, respectively, for multicomponent, multi-layered periodic composites, for thin cylindrical shells with micro-periodic structure in circumferential direction (uniperiodic shells) and for thin cylindrical shells with micro-periodic structure in circumferential and axial directions (biperiodic shells).…”

Section: Introductionmentioning

confidence: 99%

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Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

Tomczyk

Gołąbczak

Litawska

et al. 2022

Continuum Mech. Thermodyn.

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Dynamic problems of thin linearly elastic Kirchhoff–Love-type circular cylindrical shells having geometrical, elastic and inertial properties densely and periodically varying in circumferential direction (uniperiodic shells) are studied. In order to take into account the effect of a cell size on the global dynamic behaviour of such shells (the length-scale effect), a new mathematical averaged non-asymptotic model is formulated. This so-called the general tolerance model is derived by applying a certain extended version of the well-known tolerance modelling technique. Governing equations of this averaged model have constant coefficients depending also on a microstructure size, contrary to the starting exact shell equations with periodic, non-continuous and highly oscillating coefficients (the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells). The effect of a cell size on the transversal free vibrations of an uniperiodic shell strip is studied. It will be shown that within this general tolerance model not only fundamental cell-independent, but also the new additional cell-dependent free vibration frequencies can be derived and analysed. The obtained results will be compared with the corresponding results derived from the knownnon-asymptotic standard tolerance model and from the asymptotic one.

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Section: Standard Tolerance Model Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

Tomczyk

Gołąbczak

Litawska

et al. 2022

Continuum Mech. Thermodyn.

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“…where θ is an unknown temperature field; c and ρ specify the material properties such as a specific heat and a mass density, respectively; and k ij defines the components of the conductivity tensor. To average this equation, tolerance modelling was used [16][17][18][19][20][21][22]. Tolerance modelling, also called the tolerance averaging technique, introduces in a process of modelling a new concepts, definitions, and assumptions.…”

Section: Averaged Equationsmentioning

confidence: 99%

“…Equation (1) describing the heat conduction phenomenon with reference to biperiodic composite is an equation with noncontinuous coefficients:

where θ is an unknown temperature field; c and ρ specify the material properties such as a specific heat and a mass density, respectively; and k ij defines the components of the conductivity tensor. To average this equation, tolerance modelling was used [ 16 , 17 , 18 , 19 , 20 , 21 , 22 ].…”

Section: Averaged Equationsmentioning

confidence: 99%

A Finite Difference Algorithm Applied to the Averaged Equations of the Heat Conduction Issue in Biperiodic Composites—Robin Boundary Conditions

Kubacka

Ostrowski

2021

Materials

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This note deals with the heat conduction issue in biperiodic composites made of two different materials. To consider such a nonuniform structure, the equations describing the behavior of the composite under thermal (Robin) boundary conditions were averaged by using tolerance modelling. In this note, the process of creating an algorithm that uses the finite difference method to deal with averaged model equations is shown. This algorithm can be used to solve these equations and find out the temperature field distribution of a biperiodic composite.

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“…[ 25 ]. Within the literature, one can find multiple applications of this technique in various mechanical issues, such as stability analysis [ 26 , 27 , 28 , 29 ], dynamics [ 30 , 31 , 32 , 33 ] or even heat conduction issues [ 34 , 35 , 36 , 37 ].…”

Section: Introductionmentioning

confidence: 99%

Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core

Marczak

2022

Materials

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Sandwich structures are commonly used in many branches of modern engineering, such as aerospace or naval constructions. In this work, a vibration analysis of such structures is performed with the use of an anlytical model based on a zig-zag hypothesis. Due to the assumed periodic microstructure, which may occure in any layer of the structure, the initial governing equations describing its dynamic behaviour may contain periodic, non-continuous coefficients. The main aim of the presented paper is to show an analytical solution to the issue of the vibration analysis of the mentioned structures. With the use of the tolerance averaging technique, the initial governing equations are transformed to the form with constant coefficients, which is convenient to solve using well-known mathematical methods. The derived model is a versatile solution for any type of periodically inhomogeneous sandwich plate, including sandwich plates with a honeycomb core. Eventually, in the calculation example, the application of the derived averaged model in the analysis of vibrations of such structures is presented and discussed. The convergence of results of the tolerance model and FEM analysis proves the correctness and superiority of the proposed solution.

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On the modelling of stability problems for thin cylindrical shells with two-directional micro-periodic structure

Cited by 12 publications

References 22 publications

Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

Extended tolerance modelling of dynamic problems for thin uniperiodic cylindrical shells

A Finite Difference Algorithm Applied to the Averaged Equations of the Heat Conduction Issue in Biperiodic Composites—Robin Boundary Conditions

Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core

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