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1
Content available On the damping intensity of the odd Fourier impulse loading the boundary of the periodic composite
Kula Dorota
Przegląd Naukowy Inżynieria i Kształtowanie Środowiska
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2019
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Vol. 28, No. 3
321--331
EN
Investigated in the paper boundary effect behaviour for a single odd amplitude which loads rectangular boundary of the two-phased periodic composite layer confirms the common view through the prism of the expected strong suppression of the boundary impulses of the physical field near the boundary of the region occupied by the composite. There is no presence of a composite reaction to the boundary loadings mentioned here different than the exponential damping effect. However, the presence in the general equation describing the boundary effect equations the component 2?ssurf pqz?q with the first space derivative responsible for suppression of the solution along the axis Oz should cause not only exponential type of boundary temperature fluctuation damping. This component disappears in principle for the boundary effect analysed for a single impulse.
2
Content available Vibrations of periodic sandwich plates with inert core
Marczak J., Jędrysiak J.
Vibrations in Physical Systems
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2016
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Vol. 27
265--272
EN
In this note a free vibration analysis of periodic three-layered sandwich structures is performed. The equations of motion of such structures, which are derived basing on Kirchhoff's thin plate theory, contain periodic, non-continuous and highly oscillating coefficients, which makes them difficult to solve. In this work, the tolerance averaging technique is applied in order to transform the mentioned system of equations into a form with constant coefficients, which takes into account the effect of the microstructure size. The differences between two modelling procedures are shown and discussed. Eventually, formulas for free vibration frequencies of an exemplary 2D structure are derived and an analysis of influence of certain varying material properties is performed.
3
Content available remote The fourier series implementation issues in the tolerance modeling of thermal conductivity of periodic composites
Kula D., Wierzbicki E.
Engineering Transactions
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2015
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Vol. 63, iss. 1
77--92
EN
The aim of this study is to propose a partially-averaged model of heat conduction in simple micro-periodic composite conductors. In this model, as in many known models of this type, the type of microstructure is represented by the single scalar parameter, which is referred to as microstructure parameter. Unlike other known averaged models of this type, the resulting model allows for the formulation exact solutions to initial-boundary value problems formulated for the parabolic heat conduction equation. If tolerance approximations will be applied to averaged temperature filed this model becomes asymptotically exact model. The term “asymptotically exact model” refers to models in the framework of which solutions coincident with exact solutions to the mentioned problems for the parabolic heat transfer equation when the microstructure parameter tends to zero.
4
Content available remote From tolerance modeling to exact description of heat conduction in biperiodic composites
Wierzbicki E., Mazewska M.
Engineering Transactions
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2015
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Vol. 63, iss. 1
133--142
EN
In the paper we study the temperature boundary effect behavior in the dividing wall made of a composite conductor in which every area parallel to the outside and the inside plane areas has identical biperiodic structure. The proposed modeling approach is restricted to the hexagonal case of bi-periodicity in which the hexagonal cell can be divided onto three rhombus parts with different thermal properties. Considerations deal with anisotropic conductors.
5
Content available On Fourier implementation of tolerance modeling of linear elastodynamics of periodic composites
Kula D.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 4
73--80
EN
The report presents a new model of linear elastodynamics of periodic composites. The study deals with a certain extension of the ideas yield tolerance modelling. In the proposed model the displacement field is determined by an infinite number of unknowns. This disappointment seems to be apparent in many cases of periodicity since the sequence of tolerance shape functions used in the modelling procedure is not unique and opens the possibility to control the model equations. It has been formulated and solved, within the framework of the resulting model, the problem of independence short-term and long-term perturbations by linearly elastic composite solid.
6
Content available Boundary effect behavior in the hexagonal-type biperiodic structures with rotational symmetry
Mazewska M., Kula D., Wierzbicki E.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 2
73--85
EN
The report presents a study of the thermal boundary effect behaviour in the hexagonal-type rigid conductors. Every hexagonal cell of the conductor is made of three rhombus parts. Some special cases of a considered conductor with rhombus parts built of two layers of different isotropic material are analyzed. The tolerance averaging approach is used as a tool of modeling. The most important result of the paper is specification of parameters which determine the character and the intensity of the boundary effect behavior. Introduced in the contribution the boundary effect intensity parameter is suggested to be treated as a certain measure of the intensity of boundary effect behavior.
7
Content available Some remarks on the tolerance averaging of heat conduction in chessboard palisade-type periodic composites
Kula D., Mazewska M., Wierzbicki E.
Przegląd Naukowy Inżynieria i Kształtowanie Środowiska
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2012
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Vol. 21, No. 3
131--140
EN
The paper deals with periodic composites which material structure is described by the anisotropic conductivity matrix invariant with respect to the pair of orthogonal translations such that the periodic cell determined by them coincides with the shape of chess-board cell. Moreover, it will be assumed that the material structure is invariant over the π /2-rotations with the symmetry axes of any chess-board cell and orthogonal to this cell as the axes of the rotation. The main aim of this paper is to discuss the problem of the macroisotropic properties of heat conduction of the considered conductor.
PL
W pracy zaproponowano alternatywną postać tolerancyjnie uśrednionego modelu przewodnictwa ciepła kompozytu o mikroperiodycznej strukturze materialnej typu palisadowego o przekroju szachownicowym. Założono niezmienniczość tej mikrostruktury względem obrotów o kąt π /2 względem osi przechodzącej przez środek pojedynczej kostki szachownicy, a także taką samą niezmienniczość stosowanych funkcji kształtu. Ponieważ typowe równania tolerancyjne mają w takim przypadku matematyczne wady uniemożliwiające ich bezpośrednie stosowanie, zaproponowano więc nową postać modelu, który jest już wolny od tych niedogodności. W niektórych ważnych przypadkach nowa postać modelu ma współczynniki izotropowe, co wskazuje na to, że w skali makro zdarza się, że geometryczna budowa kompozytu ma decydujący wpływ na jego makrowłasności materiałowe.
8
Content available Stationary heat conduction in transversally graded laminates with uniform and non-uniform distribution of laminas
Jędrysiak J.
Fizyka Budowli w Teorii i Praktyce
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2010
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T. 5, nr 3
17-20
EN
A problem of stationary heat conduction in laminates with transversally graded structure is analysed. These composites consist of two components, non-periodically distributed as laminas. Averaged (macroscopic) properties of these laminates are slowly and continuously varied along the axis normal to laminas. In this contribution the stationary heat conduction problem is analysed in the framework of the tolerance model, proposed by Jędrysiak and Radzikowska [2,6,3].
9
Content available On the tolerance averaging of heat conduction for periodic hexagonal-type composites
Wierzbicki E.
Civil and Environmental Engineering Reports
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2010
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No. 5
95-99
EN
The paper deals periodic composites which two-dimensional material structure is determined by the anisotropic conductivity matrix invariant with respect to the pair of translations such that the periodic cell determined by them coincide with the shapely hexagon. Moreover, it will be assumed that the material structure is invariant over the [pi]/3-rotations with the centers of every hexagon as the origin of the rotation. The main aim of this paper is to prove that the tolerance model of such composite, obtained by the orthogonalization method, is isotropic.
PL
W pracy zaproponowano równoważną postać tolerancyjnie uśrednionego modelu przewodnictwa ciepła kompozytu o mikroperiodycznej strukturze materialnej typu plastra miodu. Założono niezmienniczość tej mikrostruktury względem obrotów o kąt 2 [pi] 3 a także taką samą niezmienniczość stosowanych funkcji kształtu. W otrzymanej nowej postaci model ten ma współczynniki izotropowe.
10
Content available remote On the tolerance overaging for differential operators with periodic coefficients
Borowska J., Łaciński Ł., Rychlewska J., Woźniak C.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2008
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Vol. 7, nr 1
17-21
EN
The aim of this contribution is to propose the tolerance averaging for differential operators with periodic coefficients. The averaging technique presented in this paper is based on proper limit passages with tolerance parameter to zero. This approach is a certain generalization of that presented in [1].
11
Content available Stationary heat conduction in a laminate with functionally graded macrostructure (FGM)
Jędrysiak J., Radzikowska A.
Fizyka Budowli w Teorii i Praktyce
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2008
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T. 3
23-26
EN
In this contribution a problem of heat conuction in a laminate with functionally graded macrostructure is considered. This composite is made of two components, which re non-periodically distributed in laminae with constant thickess. Macroscopic (effective) properties of this composite are continuously varied along an axis perpendicular to laminae. In this note an averaged model of heat conduction is applied. This model is based on concepts of the tolerance averaging technique, cf. Woźniak and Wierzbicki |9|.
12
Content available Przewodnictwo ciepła w kompozycie z gradacją własności (FGM)
Radzikowska A., Jędrysiak J.
Fizyka Budowli w Teorii i Praktyce
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2007
|
T. 2
239-242
PL
W pracy rozpatrywane jest przewodnictwo ciepła w warstwie utworzonej z dwóch przewodników nieperiodycznie rozmjeszczonych w podwarstwach o stałej grubości. Własności makroskopowe warstwy zmieniają się w sposób ciągły wzdłuż jej grubości. W pracy zastosowano uśredniony model przewodnictwa ciepła, otrzymany przy wykorzystaniu zmodyfikowanej technjki tolerancyjnego uśredniania, przedstawionej w monografii Woźniaka i Wierzbickiego [6].
EN
In the note, a heat conduction in a layer made of non-periodical1y distributed laminas of two conductors is considered. Macroscopic properties of the layer are continuously graded across its thickness. The aim of the note is to apply an averaged model of heat conduction of a layer with functionally graded properties, based on the modelling approach called the tolerance averaging technique, shown in the book [6] by Woźniak and Wierzbicki.
13
Content available On continuum modelling of the dynamic behaviour of periodic lattice-type plates with a complex structure
Cielecka I., Jędrysiak J.
Journal of Theoretical and Applied Mechanics
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2002
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Vol. 40 nr 1
23-46
EN
A new continuum model for studying the dynamic problems of periodic elastic lattice-type plates of an arbitrary lay-out is proposed. The general line of approach is partly based on the tolerance averaging techniques developed by Woźniak and Wierzbicki (2000) for the termomechanics of composite solids and applied by Cielecka et al. (1998, 2000) to the modelling of dense cellular structures. The proposed model describes the microstructure length-scale effect on the dynamic plate behaviour. The obtained equations are applied to the analysis of wave propagation in a special latticed plate. It is shown that the lenght-scale effect plays an important role and cannot be neglected in the above analysis.
PL
Ciągły model zagadnień dynamicznych periodycznych płyt siatkowych o złożonej strukturze. W pracy zaproponowano nowy ciągły model do analizowania zagadnień dynamicznych sprężystych, periodycznych płyt siatkowych. W przedstawionym podejściu częściowo wykorzystano technikę tolerancyjnego uśredniania, opracowaną przez Woźniaka i Wierzbickiego (2000) dla termomechaniki kompozytów. Podejście to zastosowano w pracach Cieleckiej i in. (1998, 2000) w modelowaniu gęstych struktur komórkowych. Zaproponowany model opisuje wpływ wielkości mikrostruktury na dynamikę płyty siatkowej. Otrzymane równania zastosowano do analizy propagacji fal w pewnym szczególnym przypadku płyty. Pokazano, że efekt skali odgrywa ważną rolę i nie może być pominięty w powyższej analizie.
14
Content available A contribution to modelling of composite solids
Nagórko W., Wągrowska M.
Journal of Theoretical and Applied Mechanics
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2002
|
Vol. 40 nr 1
149-158
EN
The aim of this paper is to prove that micro-periodic composites made of linear-elastic isotropic components and having a hexagonal representative cell with the triple axis of material symmetry are also isotropic in the macro-scale. The prove of this statement will be given on the basis of the tolerance averaging method of modelling, Woźniak and Wierzbicki (2000).
PL
W pracy rozważa się kompozyty sprężyste o mikroperiodycznej strukturze, których elementy reprezentatywne mają trójosiową symetrię. Wykazuje się, że takie ciała mają w modelu otrzymanym w wyniku zastosowania techniki uśredniania tolerancyjnego własności ciał izotropowych.
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