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1
Content available remote Displacements caused by the temperature in multicomponent, multi-layered periodic material structures
100%
Bagdasaryan V. , Wągrowska M. , Szlachetka O.
Mechanics and Mechanical Engineering
|
2018
|
tom Vol. 22, nr 3
809--819
EN
The present study aims to analyse a two-dimensional problem of displacements in theory of thermal stresses for multicomponent, multi-layered periodic composites. The model equations are obtained within the framework of the tolerance modelling procedure. These equations allow to determine the distribution of displacements caused by the temperature field in the theory of thermal stresses. The paper presents an example of a solution of a boundary value problem.
2
Content available A study on natural frequencies of Timoshenko beam with rapidly varying stiffness
100%
Świątek M. , Domagalski Ł. , Jędrysiak J.
Vibrations in Physical Systems
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2016
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tom Vol. 27
369--376
EN
Vibrations of Timoshenko beams with properties periodically varying along the axis are under consideration. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain the governing equations with constant coefficients. The dynamics of Timoshenko beam with the effect of the cell length is described. A asymptotic model is then constructed, which is further studied in analysis of the low order natural frequencies. The proposed model is able to describe dynamics of beams made of non-slender cells.
3
Content available Modelling of thin periodic plates subjected to large deflections
100%
Domagalski Ł. , Jędrysiak J.
Civil and Environmental Engineering Reports
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2010
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tom No. 5
133-148
EN
The objects under consideration are thin linear-elastic plates with periodic structure in planes parallel to the plate midplane, subjected to large (of the order of plate thickness) deflections. The main aim is to propose a mathematical model describing geometrically nonlinear problems of such plates, which is based on the tolerance averaging technique, cf. Woźniak et al. [3]. Results calculated for a special static problem by the tolerance model are compared with results obtained within the known tolerance linear model of thin plates.
PL
W pracy rozpatrywane są cienkie, liniowo-sprężyste płyty o budowie periodycznej w płaszczyznach równoległych do płaszczyzny środkowej. Zagadnienia statyki i dynamiki tego rodzaju płyt w zakresie dużych ugięć opisane są układem równań różniczkowych nieliniowych o silnie oscylujących, periodycznych, nieciągłych współczynnikach (por. książka pod red. Woźniaka i in. [3]). W celu otrzymania równań o stałych współczynnikach zastosowano tu technikę tolerancyjnego uśredniania, omówioną w książce pod red. Woźniaka, Michalaka i Jędrysiaka [5]. Zaproponowano nieliniowy model tolerancyjny, opisujący nieliniowo-geometryczne zagadnienia cienkich płyt periodycznych. Model ten zastosowano do wyznaczenia ugięć dla danego obciążenia, a otrzymane wyniki porównano z wynikami uzyskanymi w ramach liniowego modelu tolerancyjnego.
4
Content available Free vibrations of thin microstructured plates
100%
Jędrysiak J. , Pazera E.
Vibrations in Physical Systems
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2014
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tom Vol. 26
93--98
EN
In this paper it is presented a problem of free vibrations of thin microstructured plates, which can be treated as made of functionally graded material on the macrolevel. The size of the microstructure of the plates is of an order of the plate thickness. In order to obtain averaged governing equations of these plates the tolerance modelling technique is applied, cf. [14, 15, 7]. The derived tolerance model equations have the terms dependent of the microstructure size. Hence, the tolerance model describes the effect of the microstructure size. In order to evaluate results, the asymptotic model is introduced. Obtained results can be compared to those calculated by using the finite element method.
5
On the tolerance modelling of periodically stiffened plates
100%
Nagórko W.
Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska
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2002
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tom z. 19
111-112
6
Modelowanie tolerancyjne w mechanice konstrukcji
100%
Nagórko W. , Wągrowska M.
Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska
|
2000
|
tom z. 14
111-116
PL
W mechanice zachodzi potrzeba konstrukcji teorii prostszych, przybliżonych będących z sobą w pewnej tolerancji. Przykładem takich teorii są teorie płyt i powłok. W pracy przedstawiono ogólny schemat konstrukcji teorii uwzględniający ocenę przybliżenia. Rozważania zawężono do teorii płyt.
EN
We need sometimes to construct some simpler, approximated theories in mechanics they have to be in a tolerance. The examples of these theories are theories of plates and shells. We present in this paper the process of constructing theory, which takes into consideration the estimation of approximation. Our considerations are reduced to the theory of plates.
7
Content available Nonlinear vibrations of periodic beams
100%
Domagalski Ł. , Jędrysiak J.
Vibrations in Physical Systems
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2014
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tom Vol. 26
73--78
EN
Geometrically nonlinear vibrations of beams with properties periodically varying along the axis are investigated. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain the governing equations with constant coefficients. The proposed model describes the dynamics of the beam with the effect of the microstructure size.
8
Content available Vibrations of thin functionally graded plates with tolerance-periodic microstructure
100%
Kaźmierczak-Sobińska M. , Jędrysiak J.
Vibrations in Physical Systems
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2014
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tom Vol. 26
99--104
EN
This paper presents a problem of vibrations of thin functionally graded plates. To describe this kind of plates three averaged models are proposed: a tolerance model, an asymptotic model and a combined asymptotictolerance model, cf. [10]. Calculational results obtained for a functionally graded plate band using the proposed models are compared to each other.
9
Content available Free vibrations of medium thickness microstructured plates
100%
Jędrysiak J.
Vibrations in Physical Systems
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2016
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tom Vol. 27
169--174
EN
A problem of free vibrations of medium thickness microstructured plates, which can be treated as made of functionally graded material on the macrolevel is presented. The size of the microstructure of these plates is of an order of the plate thickness. Averaged governing equations of these plates can be obtained using the tolerance modelling technique, cf. [18, 19, 9]. Because, the derived tolerance model equations have the terms dependent of the microstructure size, this model describes the effect of the microstructure size. Results can be evaluated introducing the asymptotic model. Calculated results can be compared to those from the finite element method or a similar tolerance model of thin plates, cf. [9].
10
Content available remote Fourier variant homogenization treatment of single impulse boundary effect behaviour
100%
Kula D. , Wierzbicki E. , Witkowska-Dobrev J. , Wodzyński Ł.
Mechanics and Mechanical Engineering
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2018
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tom Vol. 22, nr 3
683--690
EN
Boundary effect behavior understood as near-boundary suppression of boundary fluctuation loads is described in various ways depending on the mathematical representation of solutions and the type of the center. In the case of periodic composites, the homogenization method is decisive here. In the framework of the Tolerance Averaging Approach, developed by prof. Cz. Wo´zniak leading to an approximate model of phenomena related to periodic composites this effect is described by a homogeneous part of differential equation for fluctuation amplitudes and usually this approximate description of the boundary effect behavior is restricted to a single fluctuation. In this paper, contrary to the previous elaborations, the boundary effect is developed in the variant of the tolerance thermal conductivity model in which the temperature field is represented by the Fourier expansions composed by an average temperature with infinite number of Fourier terms imposed on the average temperature as tolerance fluctuation suppressed in the framework of the boundary effect.
11
Content available remote Fourier variant homogenization of the heat transfer processes in periodic composites
100%
Wierzbicki E. , Kula D. , Wodzyński Ł.
Mechanics and Mechanical Engineering
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2018
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tom Vol. 22, nr 3
719--726
EN
The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperaturę field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM-type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged.
12
On the modelling of vibrations of functionally graded plates
100%
Kaźmierczak M. , Jędrysiak J.
Zeszyty Naukowe. Budownictwo / Politechnika Łódzka
|
2012
|
tom Z. 64
61-68
EN
In this paper there are considered functionally graded plates. To describe vibrations of these plates and take into account the effect of the microstructure it is applied the tolerance method, cf. [10, 11]. There are formulated governing equations of three presented models: the tolerance model, the asymptotic model and the combined asymptotic-tolerance model.
PL
W pracy rozpatrywane są płyty o funkcyjnej gradacji własności. Aby opisać drgania tych płyt, wykorzystano technikę tolerancyjnego modelowania [10, 11]. Równania zostały wyprowadzone w ramach trzech zaproponowanych modeli: modelu tolerancyjnego, modelu asymptotycznego oraz modelu asymptotyczno-tolerancyjnego.
13
Length-scale effect in stability problems for biperiodically stiffened cylindrical shells
100%
Tomczyk B.
Zeszyty Naukowe. Budownictwo / Politechnika Łódzka
|
2012
|
tom Z. 64
25-47
EN
Thin linear-elastic cylindrical shells having a micro-periodic structure along two directions tangent to the shell midsurface (biperiodic shells) are object of considerations. The aim of this paper is to investigate the effect of a periodicity cell size on the stationary stability of such shells. In order to take into account the length-scale effect in special stability problems, a new averaged non-asymptotic model of biperiodic shells, proposed in [Tomczyk B.: Thin cylindrical shells, in: Thermomechanics of Microheterogeneous Solids and Structures. Tolerance Averaging Approach. Ed. by Woźniak C, Michalak B., Jędrysiak J., Lodz Technical University Press, Lodz 2008, pp. 165-175] is applied. In the framework of this model not only the fundamental "classical" critical forces but also the new additional higher-order critical forces depending on the period of heterogeneity will be derived and discussed. These critical forces cannot be obtained from the asymptotic models commonly used for investigations of the shell stability. The differences and similarities between results derived from the aforementioned non-asymptotic biperiodic shell model and a certain asymptotic one as well as from the non-asymptotic model for shells with a micro-periodic structure along one direction tangent to the shell midsurface (uniperiodic shells) will be discussed.
PL
Przedmiotem rozważań są cienkie, liniowo-sprężyste powłoki walcowe mające periodycznie mikro-niejednorodną strukturę w dwóch kierunkach stycznych do powierzchni środkowej powłoki (powłoki biperiodyczne). Celem pracy jest zbadanie wpływu wielkości komórki periodyczności na stacjonarną stateczność takich powłok. Aby uwzględnić efekt skali w zagadnieniach stateczności, zastosowano nowy, uśredniony, nieasymptotyczny model służący do analizy dynamiki i stateczności biperiodycznie użebrowanych powłok. Model ten zaproponowano w pracy [Tomczyk B.: Thin cylindrical shells, in: Thermomechanics of micro-heterogeneous solids and structures. Tolerance averaging approach, Part II: Model equations. Ed. by C. Woźniak, B. Michalak, J. Jędrysiak, Lodz Technical University Press, Lodz 2008, pp. 165-175]. Równania modelu wyprowadzone z wykorzystaniem techniki tolerancyjnego modelowania mają stałe współczynniki i wiele z nich zależy od długości okresu periodyczności struktury. Wzięcie pod uwagę efektu skali pozwala wyznaczać i analizować nowe, dodatkowe, wyższego rzędu siły krytyczne, zależne od wielkości mikrostruktury. Siły te nie mogą być wyprowadzone w ramach modeli asymptotycznych, powszechnie stosowanych do badania stateczności powłok. Różnice i podobieństwa między wynikami otrzymanymi z modelu tolerancyjnego dla powłok biperiodycznych oraz wynikami uzyskanymi z modelu asymptotycznego a także z modelu tolerancyjnego dla powłok z periodyczną strukturą w jednym kierunku stycznym do powierzchni środkowej są dyskutowane.
14
Content available remote Asymptotic-tolerance modelling on vibrations of functionally graded thin plates
100%
Kaźmierczak M. , Jędrysiak J.
Vibrations in Physical Systems
|
2012
|
tom Vol. 25
217--222
EN
In this note there are considered functionally graded plates. To describe vibrations of these plates and take into account the effect of the microstructure, the tolerance averaging method is applied, cf. [7, 8]. There are formulated governing equations of the asymptotic-tolerance model, cf. [8]. Calculational results obtained for a functionally graded plate band using the proposed model, are compared to results by the known – tolerance and asymptotic models.
15
Content available Vibrations of non-periodic thermoelastic laminates
84%
Jędrysiak J. , Pazera E.
Vibrations in Physical Systems
|
2016
|
tom Vol. 27
175--180
EN
Vibrations of non-periodic thermoelastic laminates, which can be treated as made of functionally graded material with macroscopic properties changing continuously along direction, x1, perpendicular to the laminas on the macrolevel are considered. Three models of these laminates are presented: the tolerance, the asymptotic-tolerance and the asymptotic. Governing equations of two first of them involve terms dependent of the microstructure size. Hence, these models (the tolerance, the asymptotic-tolerance) describe the effect of the microstructure. Averaged governing equations of these laminates can be obtained using the tolerance modelling technique, cf. Jędrysiak [1]. Because the model equations have functional, but slowly-varying coefficients calculations for examples can be made numerically or by using approximated methods.
16
Content available remote Vibrations of microstructured functionally graded plates
84%
Jędrysiak J.
Vibrations in Physical Systems
|
2012
|
tom Vol. 25
193--198
EN
Functionally graded plates with a microstructure are considered. The size of the microstructure is assumed to be of an order of the plate thickness. To take into account the effect of the microstructure on vibrations of these plates the tolerance modelling method is applied. Using this method we obtain model equations with smooth functional coefficients involving terms dependent of the microstructure size.
17
Content available Vibrations of microstructured beams with axial force
84%
Jędrysiak J.
Vibrations in Physical Systems
|
2020
|
tom Vol. 31, nr 2
art. no. 2020208
EN
In this contribution there are considered vibrations of microstructured periodic slender beams, with axial force. In order to analyse the effect of the microstructure size of the beams on their vibrations the tolerance modelling method is applied. Using this method there are derived governing equations of two tolerance models - general and standard, base on two various concepts - weakly-slowly-varying functions and slowly-varying functions. These models are applied to obtain formulas of lower order and higher order frequencies with influence of the axial force. To evaluate these results of the modelling the formula of lower order frequencies in the framework of the asymptotic model (neglecting the effect of the microstructure size) is also derived.
18
Content available On the combined asymptotic-tolerance modelling of dynamic problems for thin biperiodic cylindrical shells
84%
Tomczyk B. , Litawska A.
Vibrations in Physical Systems
|
2018
|
tom Vol. 29
art. no. 2018020
EN
The objects of consideration are thin linearly elastic 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 a new averaged general asymptotic-tolerance model for the analysis of selected dynamic problems for the shells under consideration. This model is derived by applying the combined modelling which includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. The differences between the general combined model proposed here and the corresponding known standard combined model derived by means of the more restrictive concept of slowly-varying functions are discussed.
19
Content available Effect of fluctuation shape functions on vibrations of laminated structures
84%
Kubacka E.
Vibrations in Physical Systems
|
2020
|
tom Vol. 31, nr 2
art. no. 2020213
EN
In this note, the influence of the fluctuation shape functions on vibrations of the periodic laminate is analysed. The structure, used to show this impact, is the composite, consisting of the layers made of components differ in material properties like a specific heat or a thermal conductivity. The periodic laminate is microscopically heterogenous and to analyse this laminate, the tolerance averaging technique is used, therefore the influence of the thickness of the layer can be allow. One of the concepts introduced by tolerance modelling, is the fluctuation shape function, affecting on the results. The fluctuation shape function is assumed a priori and the character of vibrations is dependent on this function.
20
Content available Higher order vibrations of thin periodic plate bands with various boundary conditions
84%
Jędrysiak J. , Wyrwa P.
Vibrations in Physical Systems
|
2016
|
tom Vol. 27
181--186
EN
In this contribution there are considered thin periodic plates. The tolerance averaging method, cf. [12, 13, 4], is applied to model problems of vibrations of these plates. Hence, the effect of the microstructure size is taken into account in model equations of the tolerance model. Calculations are made for periodic plate bands using this model and the asymptotic model for various boundary conditions.
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