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1

On the mass discretization approach to elastodynamics of periodically stratified composites

Szlachetka O., Wągrowska M., Witkowska-Dobrev J., Woźniak C.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 4

143-150

EN

The object of analysis are two component periodically stratified linear elastic multilayered composites. The aim of contribution is elastodynamics of these composites subjected to the mass discretization on the interfaces between adjacent homogeneous layers. Two specific cases of the mass discretizations are taken into account. In the first case the mass is assigned uniformly to every interface. It is shown that this kind of mass discretization leads to physically completely wrong results. In the second case, the mass is discretized uniformly but on the every second interface. In this case it is shown that the obtained elastodynamics equations have a physical sense provided that the wave are sufficiently long.

2

Root finding method for problems of elastodynamics

Krushynska A.

Computer Assisted Mechanics and Engineering Sciences

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2010

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Vol. 17, no. 1

3-11

EN

This paper presents a simple and efficient method for finding complex roots of dispersion equations occurring in many problems of elastodynamics. The method is characterized by high accuracy in root finding and absence of restrictions on function representation. The essence of the method is explained geometrically; initial guesses are found as the solutions to the appropriate problems of elastostatics. Numerical solutions to dispersion equations are obtained for two elastic isotropic waveguides: a plate of infinite cross-section and a rod of rectangular cross-section. For an infinite plate, the calculated results are in full conformity with those obtained by Newton-Raphson and bisection methods. For a waveguide of rectangular cross-section, the earlier unsolved problem of finding complex roots of dispersion equations is solved by the proposed method.

3

Stiffness and mass matrices of FEM applicable dynamic infinite element with united shape basis

Kazakov K.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 1

189-200

EN

This paper is devoted to the construction and evaluation of mass and stiffness matrices of elastodynamic four and five node infinite elements with united shape functions (EIEUSF), recently proposed by the author. Such elements can be treated as family elastodynamic infinite elements appropriate for multi-wave soil-structure interaction problems. The common characteristic of the proposed infinite elements is the so-called united shape function, based on finite number of wave shape functions. The idea and the construction of the united shape basis are described in brief. This element belongs to decay class infinite elements. It is shown that by appropriate mapping functions the formulation of such an element can be easily transformed to a mapped form. The results obtained using the proposed infinite elements are in a good agreement with the superposed results obtained by series of standard computational models. The continuity along the finite/infinite element (artificial boundary) line in two-dimensional substructure models is also discussed in brief. In this type of computational models such a line marks the artificial boundary between the near and the far field of the model.

4

On the boundary effect in elastodynamics of laminated solids

Wierzbicki E., Łacińska L.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Nauki Techniczne

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2007

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Z. 6

45-54

EN

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.

5

On tolerance averaged model of the propagation of boundary displacement fluctuations in FGM-type laminated composite

Łacińska L., Wierzbicki E.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2007

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Vol. 6, nr 1

133-140

EN

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.

6

Boundary and initial fluctuation effect on dynamic behaviour of a laminated solid

Łacińska L., Wierzbicki E.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2005

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Vol. 4, nr 1

110-122

EN

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].

7

Discrete and continuum models in elastodynamics of micro-periodic solids

Rychlewska J.

Vibrations in Physical Systems

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2004

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Vol. 21

319--322

EN

The aim of this paper is to propose a new alternative approach to the formulation of both discrete and continuum models for the analysis of dynamic problems in elastic composite solids with a periodic microstructure. The proposed approach is based on a periodic simplicial division of the unit cell, [1], and on the assumption of a uniform strain in every simplex. The main feature of the obtained discrete model is the finite-difference form of the governing equations. The proposed discrete model can be formulated on different levels of accuracy. Considerations are restricted to problems in which the typical wavelength of the macroscopic deformation pattern is sufficiently large when compared to the unit cell diameter By applying smoothing operation the continuum models are derived directly from the discrete ones. The general equations obtained in the models proposed here are applied in order to investigate dispersion phenomena in the wave propagation problems.

8

Simplified models for the vibration analysis of micro-periodic elastic media

Rychlewska J., Szymczyk J., Woźniak C.

Journal of Technical Physics

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2004

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Vol. 45, no 2

91-104

EN

The aim of this paper is to propose and apply a new approach to the formation of simplified models for the vibration analysis of a micro-periodic linear-elastic composite medium. First, we propose a discrete model which can be used to investigations of both long and short waves. The main feature of the obtained discrete model is the finite-difference form of governing equations. The continuum models are derived directly from the discrete one under the assumption that the macroscopic wavelength is large when compared to the length dimension of a periodicity cell. The proposed models describe both the low-frequency and high-frequency vibration problems and can be formulated on different levels of accuracy. The reliability of the procedure leading from discrete to continuum models is discussed on the illustrative example.

9

Trefftz polynomials reciprocity based boundary element formulations for elastodynamics

Konkol F., Kompiš V.

Computer Assisted Mechanics and Engineering Sciences

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2004

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Vol. 11, No. 2-3

239-245

EN

In this paper Trefftz polynomials are used for the BEM (Boundary Element Method) based on the reciprocity relations. BEM provides a powerful tool for the calculation of dynamic structural response in the frequency and time domains. Field equations of motion and boundary conditions are cast into boundary integral equations (BIE), which are discretized only on the boundary [1]. Trefftz polynomials or other non-singular (e.g. harmonic), Trefftz functions [2] (i.e. functions satisfying all governing differential equations but not the boundary conditions) used in the Betti's reciprocity relations lead to corresponding BIE that do not contain any (weak, strong, hyper) singularities. Fundamental solutions are not needed and evaluation of the field variables inside the domain is simpler.

10

Two approaches to the formation of a tolerance averaged equations for elastodynamics of periodic solids

Augustowska L., Wierzbicki E.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2002

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Vol. 1, nr 1

5--13

EN

The aim of this contribution is to formulate two different approaches to the computation of averaged moduli in the equations of the tolerance averaged model of microperiodic linear-elastic composites. In the first approach, which will be called basis function approach, we shall use some of the FEM concepts in order to establish the basis functions in the Galerkin approximation to the non-stationary periodic cell problem of elastodynamics. In the second approach, which will be called mode shape function approach, we shall also use FEM concepts in order to determine functions describing the character of free vibrations of the periodicity cell. This approach makes it possible to reduce a large number of unknowns occuring in the first approach.

11

Ciągły model dynamiki płyt siatkowych

Cielecka I.

Vibrations in Physical Systems

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2000

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Vol. 19

82--83

12

Różnicowe i ciągłe modele elastodynamiki dyskretnych układów periodycznych o złożonej strukturze

Lewandowska J., Szymczyk J., Woźniak C.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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1999

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z. 9

147-152

PL

W pracy przedstawiono nową, różnicową postać równań elastodynamiki dyskretnych układów periodycznych o złożonej strukturze. Pokazano sposób bezpośredniego przejścia od opisu dyskretnego do modelu ciągłego. Sformułowano odpowiednie równania dynamiki w asymptotycznym modelu deformacji długofalowych.

EN

A new finite difference formulation of elastodynamics of polyatomic lattices is presented. A way of transition from a nonlocal model to a continuum one is shown. Finite difference and continuum formulations of the asymptotic model for the long wave deformations are formulated.

13

Drilling rotations in elastodynamics of thin-walled periodic structures

Wierzbicki E., Woźniak C.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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1999

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z. 9

273-278

EN

The aim of this contribution is to obtain an averaged continuum model of linear-elastic spatial periodic thin-walled structures, cf. Figure 1. Considerations are restricted to problems independent on z-coordinate; hence the plain stress in plates and the cylindrical bending of walls in 0x1x2-plane are postulated. Under assumption that all joints interconnecting wall elements are rigid the concept of drilling rotations in 0x1x2-plane as independent dynamic variables was introduced.

PL

Celem pracy jest otrzymanie uśrednionego modelu ciągłego liniowo-sprężystych przestrzennych cienkościennych konstrukcji periodycznych (por. rys. 1). Rozważania ograniczono do zagadnień niezależnych od zmiennej z, zakładając płaski stan naprężenia w płytach oraz walcowe zginanie ścianek w płaszczyźnie 0x1x2. Przyjmując, że wszystkie połączenia ścianek są sztywne, wprowadzono pojęcie obrotów owinięcia w płaszczyźnie 0x1x2 jako niezależnych zmiennych dynamicznych.