Stability of thin micro-periodic cylindrical shells; extended tolerance modelling

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

doi:10.1016/j.compstruct.2020.112743

Cited by 11 publications

(15 citation statements)

References 15 publications

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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%

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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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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“…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 [11] and papers by Tomczyk and Litawska [12][13][14], Tomczyk et al [15][16][17], where the length-scale effect in mechanics of periodic cylindrical shells is investigated; papers by Baron [18], where dynamic problems of medium thickness periodic plates are studied and by Marczak and Je ˛drysiak [19], Marczak [20,21], where dynamics of periodic sandwich plates is analysed; papers by Je ˛drysiak [22][23][24], which deal with stability of thin periodic plates; papers by Łaciński and Woźniak [25], Rychlewska et al [26], Ostrowski and Je ˛drysiak [27], Kubacka and Ostrowski [28], where problems of heat conduction in conductors with periodic structure are analysed. Let us also mention papers by Bagdasaryan et al [29], Tomczyk and Goła ˛bczak [30], which deal with coupled thermoelasticity problems, respectively, for multicomponent, multi-layered periodic composites and for thin cylindrical shells with microperiodic structure in circumferential direction (uniperiodic shells).…”

Section: Introductionmentioning

confidence: 99%

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

Tomczyk

Gołąbczak

Litawska

et al. 2021

Continuum Mech. Thermodyn.

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The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss two new averaged mathematical models for the analysis of selected dynamic thermoelasticity problems for the shells under consideration: the non-asymptotictolerance and the consistent asymptotic models. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation in which the heat sources are neglected. For the microperiodic shells under consideration, the starting equations mentioned above have highly oscillating, non-continuous and periodic coefficients. The tolerance model is derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. It has constant coefficients depending also on a cell size. Hence, this model makes it possible to study the effect of a microstructure size on the global shell thermoelasticity (the length-scale effect). The consistent asymptotic model is obtained using the consistent asymptotic approach. It has constant coefficients being independent of the period lengths. Moreover, the comparison between the tolerance model for biperiodic shells proposed here and the known tolerance model for cylindrical shells with a periodic structure in the circumferential direction only (uniperiodic shells) is presented.

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“…Jędrysiak [13,14]), stability of cylindrical shells (cf. Tomczyk and Szczerba [15], Tomczyk et al [16]), dynamics of beams (cf. Domagalski [17], Domagalski et al [18], Domagalski and Jędrysiak [19]), statics of plates with a dense system of ribs (cf.…”

Section: Introductionmentioning

confidence: 99%

The Stability Analysis of Periodic Beams Interacting with Periodic Elastic Foundation with the Use of the Tolerance Averaging Technique

Marczak

Jędrysiak

2021

Materials

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In this paper a stability analysis of microperiodic beams resting on the periodic inhomogeneous foundation is carried out. The main issue of this considerations, which is the analytical solution to the governing equations characterised by periodic, highly oscillating and non-continuous coefficients, is overwhelmed by the application of the tolerance averaging technique. As a result of such application, the governing equation is transformed into a form with constant coefficients which can be solved using well-known mathematical methods. In several calculation examples, the convergence of the results of two derived averaged models is examined, as well as the convergence of the lowest value of the critical force parameter derived from the averaged models with the FEM model. The results prove the superiority of the presented analytical solution over the FEM analysis in the optimisation process.

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Stability of thin micro-periodic cylindrical shells; extended tolerance modelling

Cited by 11 publications

References 15 publications

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

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

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

The Stability Analysis of Periodic Beams Interacting with Periodic Elastic Foundation with the Use of the Tolerance Averaging Technique

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