In the periodic composite materials temperature or displacement fluctuations suppressed in directions perpendicular to the periodicity surfaces should expect a damping reaction from the composite. This phenomenon, known as the boundary effect behavior has been investigated only in the framework of approximated models. In this paper extended tolerance model of heat transfer in periodic composites is used as a tool allows analytical investigations of highly oscillating boundary thermal loadings. It has been shown that mentioned reaction is dual - different for even and odd fluctuations.
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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.
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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.
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According to the classical theory of Weiss, Landau, and Lifshitz, in a ferromagnetic body there is a spontaneous magnetization field m, such that ∥m∥ = τ0 = const in all points of this material Ω. In any stationary configuration, this ferromagnetic body consists of areas (Weiss domains) in which the magnetization is uniform (i.e. m = const) separated by thin transition layers (Bloch walls). Such stationary configuration corresponds to the minimum point of the magnetostrictive free energy E. We are considering an elastic magnetostrictive body in our paper. The elastic magnetostrictive free energy Eδ depends on a small parameter δ such that δ → 0. As usual, the displacement field is denoted by u. We will show that each sequence of minimizers (ui,mi) contains a subsequence that converges to a couple of fields (u0,m0). By means of a Γ-limit procedure we will show that this couple (u0,m0) is a minimizer of the new functional E0. This new functional E0 describes the magnetic-elastic properties of the body with microstructure.
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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.
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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.
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.
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.
W omawianym opracowaniu podjęto zadanie wyznaczenia ruchów pionowych obszaru Warszawy i najbliższych okolic na podstawie wyników pomiarów krajowej sieci niwelacji precyzyjnej I klasy, wykonanych w ramach II, III i IV kampanii niwelacyjnej. Jako że badany fragment krajowej sieci wysokościowej stanowi sieć wiszącą, nie dowiązaną do jakiegokolwiek poziomu odniesienia w sposób bezpośredni, wyznaczone tutaj wartości ruchów pionowych mają charakter względny i opisują jedynie wewnętrzne deformacje obiektu. Mimo to, jak się okazuje, już jako takie stanowią solidne podłoże do prowadzenia analiz i interpretacji pod kątem działania czynników fizycznych. Przedmiotem badań ruchów pionowych był fragment krajowej podstawowej sieci wysokościowej obejmujący teren Warszawy wraz z terenami przyległymi. Granice badanego obszaru, o rozpiętości około 100 km, stanowiły linie niwelacyjne łączące miejscowości Sochaczew, Wyszogród, Płońsk, Pułtusk, Wyszków, Niegów, Mińsk Mazowiecki, Stara Wieś, Góra Kalwaria, Grójec, Mszczonów. W opracowaniu przedstawiono omówienie materiału źródłowego, algorytm i wyniki poszukiwania bazy punktów odniesienia oraz zarys interpretacji przyczyn otrzymanych ruchów pionowych. Wyniki, przedstawione w postaci map ruchów pionowych, wskazują istotne (rzędu kilku milimetrów na rok) przemieszczenia punktów. Tło tych przemieszczeń wskazuje ich głównie endogeniczny charakter, ale w wynikach widać również echo lokalizacji badanego obszaru w sąsiedztwie strefy T-T. Przedstawiono także analizę przemieszczeń w kontekście stabilizacji znaków wysokościowych, wykazując, że charakter otrzymanych zmian wysokości wyraźnie koresponduje z typem stabilizacji znaków.
EN
The aim of the research was to establish vertical movements in the Warsaw region and surrounding regions, based on measurements of the 1st class national precision leveling network, conducted as part of the 2nd, 3rd and 4th leveling campaigns. As the examined fragment of the national leveling network is not a networked linked directly with any level of reference, the indicated vertical movement values are relative in nature and describe only structure deformations (Grzepkowski, Cacoń 2003). The boundaries of the region, with a span of approx. 100 km, were the leveling lines of the towns of Sochaczew, Wyszogród, Płońsk, Pułtusk, Wyszków, Niegów, Mińsk Mazowiecki, Stara Wieś, Góra Kalwaria, Grójec and Mszczonów. The paper includes an outline of source materials, algorithms and results of the reference point database search as well as a summary of the interpretation of the causes of vertical movements. The technogenic causes of height changes are a dominating factor, but other attempts of interpretation in geological context is also presented. Computed vertical movements are presented in comparison with types of leveling benchmarks stabilization.
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.
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In the note wave-type solutions to the hyperbolic heat conduction in the microperiodic composites are investigated. The considerations are related to the tolerance averaged model of the hyperbolic heat transfer problems in microperiodic composites. The Lapunov exponent notation is applied. The open form of these solutions are formulated in one shape function case.
The object of the contribution is the analysis of a certain aspects of the boundary effect behaviour in elastodynamics of FGM-type multiphase laminated composites. The aim of the paper is to answer the question what kind of assumptions are enough to be taken into account in the framework of the simplified tolerance model for the elastodynamics of the considered laminates the boundary effect will be observed. The answer to this question is obtained for a certain class of two-phased FGM-type laminates and for tolerance description based on the exclusively one arbitrary shape function.
PL
Przedmiotem rozważań jest analiza zjawisk typu warstwy brzegowej w elastodynamice wielofazowych laminatów z gradacją własności efektywnych. Celem pracy jest odpowiedź na pytanie, przy jakich założeniach przyjętych w ramach uproszczonego tolerancyjnego modelu elastodynamiki takich laminatów efekt brzegowy jest obserwowany. Odpowiedź na to pytanie uzyskano dla pewnej klasy dwufazowych laminatów typu FGM i tolerancyjnego modelu z jedną funkcją kształtu, której postać nie musi być precyzowana dla realizacji celu pracy.
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In the note a certain aspects of the interrelation between initial temperature fluctuations and the averaged temperature field in the microperiodic laminated rigid conductors are investigated. The tolerance averaging technique is t into account as a tool of modelling. It was shown that there exist initial values of initial temperature fluctuations that the heat transfer process is independent of them.
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The object of the contribution is the analysis of a certain initial-boundary problems in elastodynamics of FGM-type multiphased laminated composites. The aim of the paper is to formulate an answer to the question what kind of assumptions are enough to be taken into account in the framework of the simplified tolerance model for the elastodynamics of the considered laminates the boundary effect will be observed.
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In the note an interrelation between tolerance and homogenized effective moduli for two-constituent elastic microperiodic composite are examined. It is shown that, in the case in which constituents are isotropic, effective modulae are identical. The proof is based on the probabilistic analogy formulated for both kinds of modulae.
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In the note the tolerance averaged technique is applied to investigate a certain aspects of the interrelation between initial temperature fluctuations and the averaged temperature field in two-constituent microperiodic laminated rigid conductors. This influence is measured by proposed in the note influence functions. It was shown that the influence of initial temperature fluctuations onto a heat transfer process is observed only for very small initial time interval named in the note the initial layer interval.
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A new macroscopic model for the non-stationary heat transfer processes in a periodic hexagonal-type anisotropic rigid conductor is formulated. The main aim of this contribution is to show that the macroscopic properties of this conductor are transversally isotropic. The tolerance averaging technique as a tool of macroscopic modelling is taken into account [1].
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In the paper an approximate description of the evolution of boundary traction fluctuations in the two-component linear-elastic laminates is discussed. In the paper [1] it has been shown that in the framework of the tolerance averaged model of linear elastodynamics of the aforementioned laminates, cf. [2], in the first approximation such evolution is independent of the mean (averaged) displacement. The aim of this contribution is to discuss solutions to the boundary problem formulated in [1].
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