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1

The vibrations of non-symmetric periodic sandwich beams

Marczak Jakub

Vibrations in Physical Systems

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2020

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Vol. 31, nr 2

art. no. 2020218

EN

Sandwich structures are certain specific type of composites, which are widely used in modern engineering. In this paper the vibration analysis of a specific type of sandwich beam is performed. The considered structure is not only non-symmetric towards its midplane, but also made of periodically varying isotropic materials. As a result, the governing equations of such complicated structure is characterised by periodic, non-continuous and highly oscillating coefficients. With the use of the tolerance averaging technique those equations are transformed into the form with constant coefficients. Eventually, a comparative simulations of free vibration analysis of several sandwich beams were conducted to verify the effectiveness and superiority of proposed calculation method over the FEM.

2

On the modeling of periodic sandwich structures with the use of the broken line hypothesis

Marczak J.

Mechanics and Mechanical Engineering

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2018

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Vol. 22, nr 3

761--773

EN

In this paper a dynamic analysis of sandwich plate with a certain periodic microstructure is considered. The initial system of governing equations is derived basing on the classic broken line hypothesis. As a result of transformations one can obtain a system of three differential equations of motion with periodic, highly oscillating and non-continuous coefficients. In order to derive a system of equations with constant coefficients tolerance averaging technique is applied. Eventually, in the calculation example a free vibration analysis of certain periodic plate strip is performed with the use of both the derived model and a FEM model. It can be observed that the consistency of obtained results is highly dependent on the calculation assumptions.

3

Efficiency of the needle probe test for evaluation of thermal conductivity of composite materials: two-scale analysis

Łydżba D., Różanski A., Rajczakowska M., Stefaniuk D.

Studia Geotechnica et Mechanica

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2014

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Vol. 36, nr 1

55--62

EN

The needle probe test, as a thermal conductivity measurement method, has become very popular in recent years. In the present study, the efficiency of this methodology, for the case of composite materials, is investigated based on the numerical simulations. The material under study is a two-phase composite with periodic microstructure of “matrix-inclusion” type. Two-scale analysis, incorporating micromechanics approach, is performed. First, the effective thermal conductivity of the composite considered is found by the solution of the appropriate boundary value problem stated for the single unit cell. Next, numerical simulations of the needle probe test are carried out. In this case, two different locations of the measuring sensor are considered. It is shown that the “equivalent” conductivity, derived from the probe test, is strongly affected by the location of the sensor. Moreover, comparing the results obtained for different scales, one can notice that the “equivalent” conductivity cannot be interpreted as the effective one for the composites considered. Hence, a crude approximation of the effective property is proposed based on the volume fractions of constituents and the equivalent conductivities derived from different sensor locations.

4

Modelling of static recrystallization kinetics by coupling crystal plasticity FEM and multiphase field calculations

Güvenci O., Henke T., Laschet G., Böttger B., Apel M., Bambach M., Hirt G.

Computer Methods in Materials Science

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2013

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Vol. 13, No. 2

368--374

EN

In multi-step hot forming processes, static recrystallization (SRX), which occurs in interpass times, influences the microstructure evolution, the flow stress and the final product properties. Static recrystallization is often simply modeled based on Johnson-Mehl-Avrami-Kolmogorov (JMAK) equations which are linked to the visco-plastic flow behavior of the material. Such semi-empirical models are not able to predict the SRX grain microstructure. In this paper, an approach for the simulation of static recrystallization of austenitic grains is presented which is based on the coupling of a crystal plasticity method with a multiphase field approach. The microstructure is modeled by a representative volume element (RVE) of a homogeneous austenitic grain structure with periodic boundary conditions. The grain microstructure is generated via a Voronoi tessellation. The deformation of the RVE, considering the evolution of grain orientations and dislocation density, is calculated using a crystal plasticity finite element (CP-FEM) formulation, whose material parameters have been calibrated using experimental flow curves of the considered 25MoCrS4 steel. The deformed grain structure (dislocation density, orientation) is transferred to the FDM grid used in the multiphase field approach by a dedicated interpolation scheme. In the phase field calculation, driving forces for static recrystallization are calculated based on the mean energy per grain and the curvature of the grain boundaries. A simplified nucleation model at the grain level is used to initiate the recrystallization process. Under these assumptions, it is possible to approximate the SRX kinetics obtained from the stress relaxation test, but the grain morphology predicted by the 2d model still differs from experimental findings.

PL

W wielostopniowych procesach obróbki plastycznej, rekrystalizacja statyczna (ang. static recrystallization - SRX) występująca w czasach przerw między odkształceniami, wpływa na rozwój mikrostruktury, naprężenie uplastyczniające oraz właściwości gotowego produktu. Statyczna rekrystalizacja jest często modelowana korzystając z równania Johnson-Mehl- Avrami-Kolmogorov (JMAK), które jest powiązane z lepkoplastycznym płynięciem materiału. Taki pół-empiryczny model nie jest w stanie przewidzieć mikrostruktury ziaren dla SRX. W niniejszym artykule przedstawiono podejście do symulacji statycznej rekrystalizacji austenitu wykorzystujące połączenie plastyczności kryształów z metodą pola wielofazowego. Mikrostruktura jest modelowana za pomocą reprezentatywnych elementów objętości (ang: Representative Volume Element - RVE) jednorodnej struktury ziaren austenitu z okresowymi warunkami brzegowymi. Mikrostruktura jest generowana za pomocą wieloboków Voronoi. Obliczenia odkształcenia RVE są prowadzone połączonymi metodami plastyczności kryształów i MES, z uwzględnieniem rozwoju orientacji ziaren oraz gęstości dyslokacji. Parametry modelu materiału wyznaczono na podstawie doświadczalnych krzywych płynięcia dla stali 25MoCrS4. Odkształcona struktura ziaren (gęstość dyslokacji, orientacja) jest przekazywana do siatki różnic skończonych w modelu pola wielofazowego stosując metodę interpolacji. W obliczeniach pola faz, siły pędne dla statycznej rekrystalizacji są obliczane na podstawie średniej energii w ziarnie i krzywizny granic ziaren. W celu zainicjowania rekrystalizacji stosowany jest uproszczony model zarodkowania na poziomie ziarna. Przy tych założeniach możliwe było oszacowanie kinetyki SRX na podstawie badań relaksacji naprężeń. Z drugiej strony przewidywana w modelu 2D morfologia ziaren wciąż odbiega od wyników doświadczalnych.

5

Simplicial modelling of dynamic problems in a micro-periodic composite material

Rychlewska J., Woźniak C.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2003

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Vol. 2, nr 1

159--168

EN

The aim of this paper is to propose a certain new 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 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. By applying smoothing operation the continuum models are derived directly from the discrete ones.