On Thermoelasticity in FGL – Tolerance Averaging Technique

Pazera, Ewelina; Ostrowski, Piotr; Jędrysiak, Jarosław

doi:10.2478/mme-2018-0055

Cited by 10 publications

(16 citation statements)

References 19 publications

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“…f 1 , f 2 , h 1 , and h 2 are depicted in Figure 2. By considering the micro-macro decomposition assumption, and the other assumptions and definitions of the tolerance modelling discussed in [23][24][25], Equation (1) describing the heat conduction issue was averaged, leading to the tolerance model equations:…”

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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“…with \L hg [ given by (19) and with averaged constitutive equations for functions p ab ; r _ expressed by (20), is called the tolerance averaging of starting action functional Aðu a ; w; p ab ; r _ Þ, cf. formula (6), under decomposition (17).…”

Section: Basic Concepts and Assumptions Of The Asymptotic Modelling Pmentioning

confidence: 99%

“…In the second step, we apply the extended principle of stationary action to averaged Lagrange function (19). In this step, expressions (20) are treated as nonvariational parameters.…”

Section: Basic Concepts and Assumptions Of The Asymptotic Modelling Pmentioning

confidence: 99%

“…Combining (22) with (19) and (20) we obtain finally the explicit form of the tolerance model equations. These equations can be written in the form of…”

Section: Basic Concepts and Assumptions Of The Asymptotic Modelling Pmentioning

confidence: 99%

“…The tolerance averaging technique was also adopted to formulate mathematical models for analysis of various mechanical and thermomechanical problems for functionally graded solids, e.g. for heat conduction in longitudinally graded hollow cylinder by Ostrowski and Michalak [17,18], for thermoelasticity of transversally graded laminates by Pazera and Jędrysiak [19], Pazera et al [20], for dynamics of transversally and longitudinally graded thin cylindrical shells by Tomczyk and Szczerba [21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

Tomczyk

Gołąbczak

2020

Meccanica

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The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

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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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On Thermoelasticity in FGL – Tolerance Averaging Technique

Cited by 10 publications

References 19 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

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

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