Application of the method of fundamental solutions for

TRECOP’14

Proceedings of the VI International Symposium on

TRENDS IN CONTINUUM PHYSICS

EDITORS

Bogdan T. Maruszewski Poznan University of Technology Poznan, Poland

Wolfgang Muschik Technische Universitaet Berlin Berlin, Germany

Andrzej Radowicz Kielce University of Technology Kielce, Poland

Poznan – Bedlewo May 4-7, 2014

Organizers: Poznan University of Technology, Institute of Applied Mechanics, Poznan Technische Universitaet Berlin, Institut fuer Theoretische Physik, Berlin Kielce University of Technology, Kielce Polish Academy of Sciences, Institute of Molecular Physics, Poznan Polish Society of Theoretical and Applied Mechanics, Poznan Division, Poznan Gdansk University of Technology, Gdansk

Scientific Committee: Prof. K.W. Wojciechowski – chairman, Prof. J. Awrejcewicz, Prof. V.I. Alshits, Prof. A. Berezovski, Prof. E. Ciancio, Prof. Y. Chaplya, Prof. J. Engelbrecht, Prof. K.H. Hoffmann, Prof. D. Jou, Prof. J.A. Kołodziej, Prof. B.T. Maruszewski, Prof. G.A. Maugin, Prof. S. Matysiak, Prof. W. Muschik, Prof. H. Petryk, Prof. A. Radowicz, Prof. L. Restuccia, Prof. J. Rushchitsky, Prof. J. Rybicki, Prof. I. Selezov, Prof. S. Sieniutycz, Prof. G. Szefer, Prof. A.A.F. van de Ven

Organizing Committee: Co-chairmen: Prof. Bogdan T. Maruszewski, Poznan University of Technology,

Institute of Applied Mechanics, Poznan, Poland, Prof. Wolfgang Muschik, Technische Universitaet Berlin, Institut fuer Theoretische Physik, Berlin, Germany, Prof. Andrzej Radowicz, Kielce University of Technology, Kielce, Poland

Secretary: Paweł Fritzkowski, PhD Members: Paulina Fopp, Jakub Grabski, MSc, Hubert Jopek, PhD,

Tomasz Machnicki, Roman Starosta, DSc, Tomasz Stręk, Assoc. Prof., Tomasz Walczak, PhD

© Institute of Applied Mechanics, Poznan University of Technology,

Poznan, Poland © TRECOP LOGO Prof. Gerard A. Maugin, Universite Pierre-et-Marie-Curie,

Paris VI, Paris, France

Cover design: Piotr Gołębniak Publisher: Agencja Reklamowa COMPRINT

ul. H. Rzepeckiej 26A, 60-465 Poznan, Poland www.comprint.com.pl

ISBN 978-83-89333-58-2

3

PREFACE

The book contains abstracts of the papers accepted for presentation at the VI International Symposium on TRENDS IN CONTINUUM PHYSICS, TRECOP’14.

The International Symposium on Trends in Continuum Physics, already the sixth one, takes place on May 4-7, 2014 in Bedlewo near Poznan, Poland, at the Conference Center of Mathematical Institute of Polish Academy of Sciences.

One of the main aims of the meeting is to bring together scientists from Eastern Europe working in various fields of widely understood continuum physics with those of Western and Central Europe for extending their existing co-operation and for creating new connections. A special emphasis is put on the representation of various concepts applied to the different physical fields.

Conference topics related to continuum physics include: Fundamentals of continuum physics, New trends in thermodynamics, New trends in electrodynamics, Relativistic physics, Physics of materials:

Ideas of defective crystals, Ferroic crystals, Liquid crystals, Molecular crystals, High-temperature superconductors, Semiconductors, Plasma, Polymers, Amorphous media, Smart materials, Anomalous phenomena in materials,

Biophysics, Multiphase systems, Multiscale problems.

On behalf of the Scientific and Organizing Committees of the VI International Symposium TRECOP’14 we wish all participants stimulating and enjoyable time in Bedlewo.

B.T. Maruszewski W. Muschik A. Radowicz

CONTENTS

INVITED LECTURES

1. Alshits V.I., Darinskaya E.V., Koldaeva M.V., Minyukov S.A., Petrzhik E.A., Belov A.Yu., Morozov V.A., Kats V.M., Lukin A.A., Naimi E.K., Radowicz A.

Resonance magnetoplasticity in the ESR scheme under super low magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Cao Long V., Leoński W., Bui Dinh T. Analytical and numerical methods for finding solitary waves . . . . . . . . . . . . . . . . . . . . . 8

3. Dudek M.R., Wojciechowski K.W., Grima J.N., Caruana-Gauci R., Dudek K., Pigłowski P., Zapotoczny B.

A mean-field model of magnetic domains growth in a magnetic-mechanical auxetic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4. Guz A., Rushchitsky J. On features of continuum description of nanocomposite materials . . . . . . . . . . . . . . . . . 10

5. Langner M. Molecular machines – new dimension of biological sciences . . . . . . . . . . . . . . . . . . . . . 11

6. Tretiakov K.V., Wojciechowski K.W. An influence of size polydispersity on elastic properties of model systems . . . . . . . . . . . 12

7. Wojciechowski K.W., Poźniak A.A. Computer simulations of planar auxetic foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

REGULAR LECTURES

8. Drabik D., Langner M., Przybyło M., Walczak T. Algorithm development of vesicle fluctuations spectroscopy technique . . . . . . . . . . . . . 14

9. Grabski J.K., Kołodziej J.A. Laminar flow of power-law fluid between corrugated plates . . . . . . . . . . . . . . . . . . . . . 16

10. Hein R. The method of modeling for discrete-continuous systems . . . . . . . . . . . . . . . . . . . . . . . . 18

11. Jankowska M.A., Kołodziej J.A. Application of the method of fundamental solutions for the study

of the stress state of the plate subjected to uniaxial tension . . . . . . . . . . . . . . . . . . . . . . 20

12. Jopek H., Tabaszewski M., Stręk T. Acoustic properties of sandwich panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

13. Kamiński H., Fritzkowski P. Auxetic quarter-space under concentrated force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

14. Maruszewski B.T., Drzewiecki A., Starosta R. The effective material coefficients and thermoelastic damping

in a rectangular auxetic plate during forced vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 28

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15. Maruszewski B.T., Sypniewska-Kamińska G., Walczak T. Three-dimensional elastic torsion of auxetic elliptic rods

using the method of fundamental solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

16. Narojczyk J.W., Wojciechowski K.W., Kowalik M. Elastic properties of static solids with n–inverse–power interactions:

negative Poisson’s ratio in presence of polydispersity . . . . . . . . . . . . . . . . . . . . . . . . . . 31

17. Nienartowicz M., Stręk T. Finite element analysis of sandwich two-phase composite . . . . . . . . . . . . . . . . . . . . . . . 33

18. Pigłowski P.M., Wojciechowski K.W. A short introduction into shockwave physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

19. Poźniak A.A., Wojciechowski K.W. Poisson’s ratio of randomly disordered anti-chiral structures

with variable anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

20. Uściłowska A., Fraska A. Implementation of HAM and meshless method for torsion

of functionally graded orthotropic bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

21. Winczewski S., Rybicki J. Highly efficient calculation method of bond order parameters . . . . . . . . . . . . . . . . . . . . 39

SUPPLEMENT: INVITED LECTURE

22. Uściłowska A. The influence of auxetic material characteristics on the soliton wave propagation

parameters – numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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V.I. Alshits1,2, E.V. Darinskaya1, M.V. Koldaeva1, S.A. Minyukov1, E.A. Petrzhik1, A.Yu. Belov1, V.A. Morozov3, V.M. Kats3, A.A. Lukin3, E.K. Naimi4, A. Radowicz5

1 A.V. Shubnikov Institute of Crystallography RAS, Moscow, 119333 Russia 2 Polish-Japanese Institute of Information Technology, Warsaw, 02-008 Poland 3 St. Petersburg State University, St. Petersburg, 199034 Russia 4 National University of Science and Technology MISiS, Moscow, 119049 Russia 5 Kielce University of Technology, Kielce, 25-314 Poland

RESONANCE MAGNETOPLASTICITY IN THE ESR SCHEME

UNDER SUPER LOW MAGNETIC FIELDS Resonance of dislocation mobility is experimentally studied in NaCl crystals exposed to the crossed superlow magnetic fields, the Earth field and the ac pump field of harmonic or pulse type. The measured peaks of mean dislocation paths form spectra unusual for ESR.

Resonant relaxation of dislocation structure is experimentally studied for NaCl crystals with different impurity content under the action of crossed magnetic fields: the static field of the Earth BEarth and the alternating pump field B

~ of radio-frequency bandwidth. The peaks of dislocation paths l() differed by their heights and positions of the resonance frequencies r. The maximum heights of peaks occurred for dislocations with directions L orthogonal to the plane of magnetic fields when the vectors {L, B~ , BEarth} were mutually perpendicular and belonged to the system <100>. Changes in the angle between the fields B~ and BEarth in the

plane L, as well as variations of the concentration of the Ca impurity, affected only a height of the peak. On the other hand, the sample rotation by the angle about its edge [100] relative the Earth field BEarth led to creation of a pair of resonance peaks with frequencies r1,2 dependent on : cos01 rr (Fig. 1) and

sin02 rr . Detailed study of relaxation

displacements of dislocations for the orientation = 0 in an extended frequency range ν = (0.5–7.3) MHz

of the pump field B~ and for the series of static fields В = 26÷261 Т revealed a quartet of

equidistant peaks l(). In the most part of the studied region of the fields B the frequencies of the ESR peaks were related to the Zeeman splitting of energy levels with the four g-factors close to 2 and the difference of neighboring values g = 0.09 independent of B (Fig. 2).

The other type of spectra l() form low frequency peaks of the series r2(). The study of this resonances at the frequency range = (5-440) kHz for the series of angles = 0-5 has given the spectrum of maximums described by frequencies )sin( ii A .

Fig. 1. Plot of the ratio of resonant frequencies vs. angle i for NaCl crystals from the three series studied; solid curve shows fit to the cosi function

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Fig. 3. Spectrum of the mean dislocation paths for the orientations θ = 0. The lower vertical straight line segments correspond to theoretical angles recalculated to frequencies

Within experiment errors the parameters i are independent of . In the considering model the angles i characterize the deviations of the axes of impurity centers Ca+-Cl- from the crystallographic directions <100> in the dislocation cores. Computer simulation of an edge dislocation core gives the angle i series close to the measured magnitudes (Fig. 3).

The analogous effect of resonant relaxation of dislocation structure in NaCl(Ca) crystals was found in the regime of pulse pumping. In this case dislocation displacements are

coursed by the combined actions of the magnetic field of the Earth and the pulse pumping field. The measured dependences of dislocation paths l on the pulse duration had their maximum (Fig. 4) at r 0.53 s., which relates to the usual ESR condition where g 2 and r = 1

r . The

pulse amplitude (~10 T) is 3-5 times more that of the harmonic pump field but still remains rather low (~10 T). The relaxation has an explosive character when for 0.5 s practically the same number of dislocations move on similar distances ~100 m as compared with the relaxation in a harmonic pump regime during 5 min.

The work was supported by the RFBR (no. 13_02_00341) and by the Presidium of the RAS (basic research program no. 24). References [1] Alshits V.I. et al., ESR in the Earth’s magnetic field as a cause of

dislocation motion in NaCl crystals, JETP Lett., 91 (2010) 97-101. [2] Alshits V.I. et al., Resonant dislocation motion in NaCl crystals in the

EPR scheme in the Earth’s magnetic field with pulsed pumping, Phys. Solid State, 55 (2013) 2289-2296.

[3] Alshits V.I. et al., Anisotropic resonant magnetoplasticity of NaCl crystals in the Earth’s magnetic field, Phys. Solid State, 55 (2013) 358-366.

[4] Alshits V.I. et al., Quartet of resonance peaks of dislocation displacements in NaCl crystals during their magnetic processing in the low-frequency EPR scheme, JETP Lett., 98 (2013) 28-32.

[5] Alshits V.I. et al., Determination of the positions of impurity centers in a dislocation core in a NaCl crystal from magnetoplasticity spectra, JETP Lett., 99 (2014) 82-88.

Fig. 4. The dependence of the mean dislocation pathson the impulse duration

Fig. 2. Spectrum of the mean dislocation paths for different magnetic fields: 165 (a), 50 (b) and 26 μT (c)

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V. Cao Long 1,3, W. Leoński 1, T. Bui Dinh 2

1 Quantum Optics and Engineering Division, Institute of Physics, University of Zielona Gora, Szafrana 4a, 65-516 Zielona Gora, Poland

2 Vinh University, 182 Le Duan Str., Vinh, Nghe An, Vietnam 3 Faculty of Physics, University of Warsaw, Hoża 69, 00-681 Warszawa, Poland

ANALYTICAL AND NUMERICAL METHODS

FOR FINDING SOLITARY WAVES

It is well known that a great deal of physical processes involved in a given nonlinear problem may be understood in terms of formation of spatial, temporal or spatiotemporal localized structures or solitons. Being a true high-technology application of these mathematical objects, optical solitons have been the objects of intensive theoretical and experimental studies in physics and mathematical physics in the past several decades, in particular due to their potential applications in long distance communication and all-optical ultrafast switching devices, when they are special solutions of the so-called Higher-Order Nonlinear Schroedinger Equation, which describes the propagation of light waves in nonlinear optical media, especially in optical fibers. Recently, these soliton-like solutions are the subject of interest in both theoretical and experimental studies for finding new materials as auxetics materials, the nematic liquid crystal.

The study of the above-mentioned localized waves is a difficult task, as nonlinear partial differential equations (PDEs) of a given system are usually not integrable. By investigating the intergrability of a nonlinear PDE, one gains crutial insight into the structure of the equation and nature of its solutions. With the exception of some analytical solutions obtained by well-known methods as inverse scattering method, Hirota’s method, Painleve test [1], varational method [2], Jacobi eliliptic function expansion method [3], solitary wave solutions have to be determined numerically. In some cases, one should use analytical and numerical methods simultaneously. These methoda will be reviewed in our talk, in particular the method introduced in [2] with some applications to concrete physical phenomena.

References [1] Cao Long V., Goldstein P.P., A Concise Course in Nonlinear Partial Differential

Equations, University of Zielona Gora, 2008. [2] Cao Long V., Propagation technique for ultrashort pulses, Reviews on Advanced Materials

Science, 23 (2010) 8-20. [3] El-Wakil S.A., Abdou M.A., Elhanbaly A., New solitons and periodic wave solutions for

nonlinear evolution equations, Phys. Lett. A, 353 (2006) 40-47.

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M.R. Dudek 1, K.W. Wojciechowski 2, J.N. Grima 3,4, R. Caruana-Gauci 4, K. Dudek 1,4, P. Pigłowski 2, B. Zapotoczny 2

1 Institute of Physics, University of Zielona Gora, Szafrana 4a, 65-069 Zielona Gora, Poland 2 Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan,

Poland 3 Metamaterials Unit, Faculty of Science, University of Malta, Msida MSD 2080, Malta 4 Department of Chemistry, Faculty of Science, University of Malta, Msida MSD 2080, Malta

A MEAN-FIELD MODEL OF MAGNETIC DOMAINS GROWTH

IN A MAGNETIC-MECHANICAL AUXETIC SYSTEM We propose a simple mean-field like description of magnetic components in magnetic-mechanical auxetic systems (MMA). MMAs represent a class of novel systems having magnetic insertions embedded within a non-magnetic matrix with a negative Poisson’s ratio. These systems start to be considered as magnetic-mechanical sensors or tunable filters where the change of their aperture is supported by the magnetic field. We discuss the effect of mechanical deformation on magnetic domains growth kinetics. In the model under consideration, the magnetic insertions are represented by Ising spins which experience space dependent mean field. The Metropolis Mean Field Algorithm, which has been recently introduced in [1], has been used to describe the domain kinetics. The purely mechanical description of tunable MMAs introduced in [2] is extended to include thermal properties of magnetic phase. References [1] Dudek M.R., Grima J.N., Cauchi R., Zerafa C., Gatt R., Zapotoczny B., Space Dependent

Mean Field Approximation Modelling, J. Stat. Phys., 154 (2014) 1508-1515. [2] Grima J.N., Caruana-Gauci R., Dudek M.R., Wojciechowski K.W., Gatt R., Smart

metamaterials with tunable auxetic and other properties, Smart Mater. Struct., 22 (2013) 084016.

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A. Guz 1, J. Rushchitsky 2

1 Alexander GUZ, Prof., S.P. Timoshenko Institute of Mechanics, Nesterov Str. 3, Kiev, 03680, Ukraine, e-mail: [email protected]

2 Jeremiah RUSHCHITSKY, Prof., S.P. Timoshenko Institute of Mechanics, Nesterov Str. 3, Kiev, 03680, Ukraine, e-mail: [email protected]

ON FEATURES OF CONTINUUM DESCRIPTION

OF NANOCOMPOSITE MATERIALS Two basic features of continuum description of nanocomposite materials are discussed within the frame work of structural mechanics of materials and construction units. First, a continuum as a basic notion in mechanics of materials is considered, where the principle of continualization is stated and the notion of representative volume is introduced. Then some facts from the theory of composite materials are shown and commented. Here the principle of homogenization is stated and two basic models of theory of composite materials (homogeneous body and piece-wise homogeneous body) are described. The first feature is formulated as invalidity of the principle of continualization in some types of nanoformations used as fillers in nanocomposite materials. The second feature is stated as necessity of transition from the two-component structure to the three-component structure of nanosized composite materials. Both features are commented and accompanied by examples. References [1] Guz A.N., Rushchitsky J.J., Short Introduction to Mechanics of Nanocomposites, Scientific

& Academic Publishing, Rosemead, 2013. [2] Guz A.N., Rushchitsky J.J., Some fundamental aspects of mechanics of nanocomposite

materials, Journal of Nanotechnology, 24 (2013) 1-15.

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M. Langner

Marek LANGNER, Prof., Institute of Biomedical Engineering and Measurements, Wroclaw Technical University, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland, e-mail: [email protected]

MOLECULAR MACHINES – NEW DIMENSION

OF BIOLOGICAL SCIENCES Biological sciences till end of the twentieth century were a simple extension of chemistry and physics. Such perceptions of biological matter limit the capability to explain fundamental properties of life. The biological system is characterized by the directional and precise control of matter flow and preservation of its spatial and temporal patterns. Those characteristics require the introduction of transformation of energy and information maintained by the directional movements/processes. The transformation of information was accounted for by the formulation of the central dogma of biology, which introduces concepts of digital and text information storage and handling which are correlated with structure and functions of cellular elements. The transformation of information requires directional sequences of processes which are necessary perform by functional units capable to carry metabolic processes in a synchronized and directional meaner. At the end of twentieth century the enzyme kinsin was discovered. Its only function was to move structures within the cell volume. This discovery resulted with the concept of molecular machine, which combined with the idea of emerging properties stimulated the broad range of research on molecules which are considered as functional units designed for specific tasks. This perception allow for the introduction of engineering approaches for studies of biological processes, previously inaccessible within the framework of chemistry and/or physics alone. New concepts of modern biology are discussed using the example of F0F1 ATPaze, the “enzyme” located in the inner mitochondrial membrane, which transforms the chemical potential difference into the ATP production.

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K.V. Tretiakov 1, K.W. Wojciechowski 2

1 Konstantin V. TRETIAKOV, Assoc. Prof., Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17/19, 60-179 Poznan, Poland, e-mail: [email protected]

2 Krzysztof W. WOJCIECHOWSKI, Prof., Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, Poland, e-mail: [email protected]

AN INFLUENCE OF SIZE POLYDISPERSITY

ON ELASTIC PROPERTIES OF MODEL SYSTEMS

Various real systems exhibit size polydispersity, i.e. are consisted of particles of different sizes – nanoparticles, colloidal suspensions, and granular materials constitute well known examples. Usually, the polydispersity is characterized by the ratio of the standard deviation to the mean value of the diameter [1]. It has a remarkable influence on the thermodynamic and dynamic behaviors of materials [1, 2]. In this lecture, results of numerical calculations of the elastic constants are discussed for crystalline phases of various polydisperse systems in which particles interact by different interatomic potentials [3-5]. Elastic constants and Poisson’s ratios are determined as functions of size polydispersity by Monte Carlo simulations in the NpT ensemble with variable shape of the periodic box. An influence of the potential of interaction between particles on the elastic properties is also considered.

The discussed results indicate directions of further research on the synthesis of materials with desired elastic properties. Acknowledgements This work was supported by the grant NN202 261 438 of the Polish Ministry of Science and Higher Education. Part of the calculations was performed at the Poznan Supercomputing and Networking Center (PCSS). References [1] Phan S.E., Russel W.B., Zhu J.X., Chaikin P.M., Effects of polydispersity on hard sphere

crystals, J. Chem. Phys., 108 (1998) 9789. [2] Pusey P.N., van Megen W., Phase-behavior of concentrated suspensions of nearly hard

colloidal spheres, Nature, 320 (1986) 340-342. [3] Tretiakov K.V., Wojciechowski K.W., Elasticity of two-dimensional crystals of

polydisperse hard disks near close packing: surprising behavior of the Poisson’s ratio, J. Chem. Phys., 136 (2012) 204506.

[4] Tretiakov K.V., Wojciechowski K.W., Elastic properties of fcc crystals of polydisperse soft spheres, Physica Status Solidi B, 250 (2013) 2020-2029.

[5] Tretiakov K.V., Wojciechowski K.W., Partially auxetic behavior in fcc crystals of hard-core repulsive Yukawa particles, Physica Status Solidi B, 251 (2014) 383-387.

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K.W. Wojciechowski 2, A.A. Poźniak 3


2 Artur A. POŹNIAK, MSc, Department of Technical Physics, Poznan University of Technology, Nieszawska 13A, 60-965 Poznan, Poland, e-mail: [email protected]

COMPUTER SIMULATIONS OF PLANAR AUXETIC FOAMS

Poisson’s ratio is one of quantities characterizing deformation of elastic materials. It is the

negative ratio of the relative change in the transverse dimension to the relative change in the longitudinal dimension of a sample when an infinitesimal change of the longitudinal stress is applied to the sample [1]. Typical materials show positive Poisson’s ratio, i.e. they shrink when stretched. Although the opposite behavior, i.e. an increase of the transverse dimension at stretching, is allowed by the elasticity theory, it has been believed for a long time that no such materials exist in reality [1]. A thermodynamically stable phase formed by simple model particles, which indicated a potential way to make an isotropic material of negative Poisson’s ratio, has been found by Wojciechowski [2]. The first isotropic material of negative Poisson’s ratio, in the form of a foam, was manufactured by Lakes [3]. Materials exhibiting negative Poisson’s ratio were coined auxetics by Evans [5].

In this lecture, a recent progress in theoretical and simulation modeling of planar auxetic foams will be presented [5].

Acknowledgements This work was supported by the grant NCN 2012/05/N/ST5/01476. Part of the simulations was performed at Poznan Supercomputing and Networking Center (PCSS). References [1] Landau L.D., Lifshitz E.M., Theory of Elasticity, Pergamon Press, London, 1986. [2] Wojciechowski K.W., Constant thermodynamic tension Monte Carlo studies of elastic

properties of a two-dimensional system of hard cyclic hexamers, Molecular Physics, 61 (1987) 1247-1258.

[3] Lakes R.S., Foam Structures with a Negative Poisson's Ratio, Science, 235 (1987) 1038-1040.

[4] Evans K.E., Auxetic polymers: a new range of materials, Endeavour, 15 (1991) 170-174. [5] Pozniak A.A., Smardzewski J., Wojciechowski K.W., Computer simulations of auxetic

foams in two dimensions, Smart Materials and Structures, 22 (2013) 084009.

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D. Drabik 1, M. Langner 2, M. Przybyło 3, T. Walczak 4

1 Dominik DRABIK, MSc, Department of Biomedical Engineering and Measurements, Wroclaw University of Technology, pl. Grunwaldzki 13, 50-377 Wroclaw, Poland, e-mail: [email protected]

2 Marek LANGNER, Prof., Department of Biomedical Engineering and Measurements, Wroclaw University of Technology, pl. Grunwaldzki 13, 50-377 Wroclaw, Poland, e-mail: [email protected]

3 Magdalena PRZYBYŁO, PhD, Department of Biomedical Engineering and Measurements, Wroclaw University of Technology, pl. Grunwaldzki 13, 50-377 Wroclaw, Poland, e-mail: [email protected]

4 Tomasz WALCZAK, PhD, Institute of Applied Mechanics, Poznan University of Technology, ul. Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected]

ALGORITHM DEVELOPMENT OF VESICLE FLUCTUATIONS

SPECTROSCOPY TECHNIQUE

The paper deals with fundamental biophysical problem of determining the lipid bilayer mechanical parameters. The proposed algorithm is based on vesicle fluctuations which are measured by spectroscopy technique.

The lipid bilayer, a constitutive component of biological membrane, organizes the cellular space, controls mass and information flow, participates in the energy transformation, and provides surfaces for biochemical processes. Understanding the molecular basis of those processes as well as their spatial and temporal coordination requires quantification of the lipid bilayer physical properties including its mechanics, which is believed to plays a critical role in cell functioning. The lipid bilayer elasticity, quantitated with the bending rigidity coefficient, affects biological membrane shape and dynamics, which are critical for the control of membranes trafficking and membranous structures topologies. The bending rigidity coefficient is defined as the energy cost needed to bend the lipid bilayer [1]. There are few techniques to measure the bending rigidity of the lipid bilayer including deformation caused by external force delivered by nanoparticle, aspiration and “flickering noise” techniques [2-5]. The first two techniques require some sort of membrane immobilization what combined with the small value of the bending rigidity (from 10 kBT to 100 kBT) is likely to result with a serous distortion of the final result. The only method for the bending rigidity determination of an unrestrained membrane is the “flickering noise” technique. The technique is based on the quantitative analysis of thermally induced fluctuations of the lipid vesicles or biological membrane, which is subsequently used to quantitate the membrane elasticity. Specifically, a series of microscopic images of giant unilamellar lipid vesicles are collected and analyzed. The outcome of this method is heavily dependent on the quality of collected images, image processing methodologies and mathematical model used. In order to improve the image quality we used images of fluorescently labeled membranes acquired with the fluorescence confocal microscopy instead of phase contrast images used by others. We have demonstrated that fluorescence images are superior to that obtained with the phase contrast microscopy. The high quality images make possible the critical evaluation of currently available image processing and data analysis methodologies.

15

Acknowledgments The work is supported by grant UOD-DEM-1-027/001. References [1] Faucon J.F., Mitov M.D., Meleard P., Bivas I., Bothorel P., Bending elasticity and thermal

fluctuations of lipid membranes. Theoretical and experimental requirements, J. Phys. Francel E, 50 (1989) 2289-2414.

[2] Meleard P., Pott T., Bouvrais H., Ipsen J.H., Advantages of statistical analysis of giant vesicle flickering for bending elasticity measurements, The European Physical Jounal E, 34 (2011) 116.

[3] Pecreaux J., Dobereiner H.G., Prost J., Joanny J.F., Bassereau P., Refined contour analysis of giant unilamellar vesicles, The European Physical Jounal E, 13 (2004) 277-290.

[4] Henriksen J., Rowat A.C., Ipsen J.H., Vesicle fluctuation analysis of the effects of sterols on membrane bending rigidity, Euro. Biophys. J., 33 (2004) 732-741.

[5] Henriksen J.R., Ipsen J.H., Thermal undulations of quasi-spherical vesicles stabilized by gravity, The European Physical Jounal E, 9 (2002) 365-374.

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J.K. Grabski 1, J.A. Kołodziej 2

1 Jakub Krzysztof GRABSKI, MSc, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

2 Jan Adam KOŁODZIEJ, Prof., Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

LAMINAR FLOW OF POWER-LAW FLUID

BETWEEN CORRUGATED PLATES

In the paper fully developed laminar flow of power-law fluid between corrugated plates with non-slip was considered using the method of fundamental solutions and the radial basis functions.

1. Introduction The problem of laminar flow between the corrugated plated was firstly considered probably

by Wang [1]. The flow of Darcy-Birkman fluid between corrugated plates was considered by Ng and Wang [2].

In the paper characteristics of power-law fluid flow between corrugated plates with non-slip condition was obtained using the method of fundamental solutions (MFS) and the radial basis functions (RBF) [3]. 2. Mathematical formulation of the problem

For steady, fully developed and laminar flow between the corrugated plates of the incompressible viscous power-law fluid the following equation can be written in Cartesian coordinates system (x, y, z):

dzdp

yy,xw

yxy,xw

x

(1)

where w(x,y) is dimensionless velocity in z direction, dp/dz is constant pressure gradient and η(γ) is viscosity function which for the power-law fluid is defined as:

1 mK (2) where K is consistency factor, m is the power-law index and

22

yy,xw

xy,xw (3)

The no-slip condition was formulated: on 0)y,x(w (4)

where Γ denotes the wall (corrugated plate) of the considered canal. After introducing dimensionless variables:

dzdpay,xwY,XW

ayY

axX

0

2 , ,

(5)

17

and additional function:

0

E (6)

the problem described by Eq. (1) and condition (4) takes the form:

1

YY,XWE

YXY,XWE

X (7)

on 0)Y,X(W (8) After some mathematical transformations Eq. (7) takes finally the form:

YY,XW

YE

XY,XW

XE

EEYY,XW

XY,XW

11

2

2

2

2 (9)

3. The proposed method of solution

The nonlinear Eq. (9) can be solved using the Picard iteration method. In each iteration step velocity in the right-hand side of the equation is taken from previous iteration step. Thus the nonlinear problem described by Eq. (9) with boundary condition (8) is transformed into the sequence of the non-homogenous problems. On each iteration step the right-hand side is interpolated using the RBF. The approximate solution of the non-homogenous problem on each iteration step consists of the general solution and the particular solution. The particular solution is known after the RBF interpolation. The general solution is the solution of homogenous equation:

02

2

2

2

Y

Y,XWX

Y,XW hh (10)

and can be easily solved using the MFS. In the method the approximate solution is assumed as the linear combination of fundamental solutions:

Ns

iiih )r(cY,XW

1ln (11)

where ci (i = 1,2,...,Ns) are unknown coefficients which are calculated using the boundary collocation technique [4], Ns is number of source points which are located outside of the considered region in distance S and ri is distance between the point (X,Y) and the i-th source point. Acknowledgments This work was supported by the MNiSW grant 21-429/2013 DS-MK (first author) and the NCN grant 2012/07/B/ST8/03449 (second author).

References [1] Wang C.-Y., Parallel flow between corrugated plates, Journal of Engineering Mechanics –

ASCE, 102 (1976) 1088-1099. [2] Ng C.-O., Wang C.-Y., Darcy-Brinkman flow through a corrugated channel, Transport in

Porus Media, 85 (2010) 605-618. [3] Chen C.S., Karageorghis A., Smyrlis T.S. (ed.), The method of fundamental solutions –

a meshless method, Dynamic Publishers, Atlanta, 2008. [4] Kołodziej J.A., Zieliński A.P., The boundary collocation technique and their applications

in engineering, WIT Press, Southampton, 2009.

18


R. Hein Rafał HEIN, PhD, Gdansk University of Technology, Mechanical Engineering Department, Narutowicza 11/12, 80-233 Gdansk, e-mail: [email protected]

THE METHOD OF MODELING

FOR DISCRETE-CONTINUOUS SYSTEMS

The paper describes a method of discrete-continuous systems modelling. Presented method is a hybrid one. It combines the advantages of spatial discretization methods with the advantages of continuous systems modelling methods.

In the classical finite element method [3], the body is divided into all three spatial directions

(Fig. 1a, 1c). In the proposed method, the same body is divided into one (Fig. 1b) or two (Fig. 1d) spatial directions. In one of the directions the body remains continuous. Such a division results in finite elements with parameters distributed along one of the axes. Two-dimensional elements are called strips (Fig. 1b) and three-dimensional elements are called prisms (Fig. 1d). Both these elements are one-dimensional distributed systems and are described by second order partial differential equations. However, these equations also have terms related to interactions between elements. Hence, the given system is described by coupled second order partial differential equations.

Fig. 1. Spatial discretization of 2D and 3D body: a), c) conventional finite element method, b), d) proposed hybrid method

The obtained equations are next solved by using the distributed transfer function method [8, 9]. This method enables to obtain analytical or semi-analytical solutions for 1D or 2D and 3D systems respectively. The proposed method was applied to modeling 1D as well as 2D or 3D systems [1, 2, 4, 5, 6]. Many numerical calculations and computer simulations proved that the proposed method is efficient and can be applied to modeling of complex dynamic systems.

a) finite element b) strip

d)

prism finite element

c)

19

References [1] Hein R., Orlikowski C., Hybrid reduced model of rotor, The Archive of Mechanical

Engineering, LX (2013) 319-333. [2] Hein R., Orlikowski C., Application of the distributed transfer function method and the

rigid finite element method for modeling of 2-D and 3-D systems, Modelowanie Inżynierskie, 39 (2010) 97-102.

[3] Kruszewski J., Gawroński W., Wittbrodt E., Najbar F., Grabowski S., Rigid finite element method [in Polish], Arkady, 1975.

[4] Orlikowski C., Hein R., Modelling of geared multi-rotor system, Solid State Phenomena, 198 (2013) 669-674.

[5] Orlikowski C., Hein R., A simplified model of 3-D pipe system conveying flowing liquid, Solid State Phenomena, 198 (2013) 621-626.

[6] Orlikowski C., Hein R., Modelling and analysis of beam/bar structure by application of bond graphs, Journal of Theoretical and Applied Mechanics, 49 (2011) 1003-1017.

[7] Orlikowski C., Hein R., Port-based modeling of distributed-lumped parameter systems, Solid State Phenomena, 164 (2010) 183-188.

[8] Yang B., Tan C.A., Transfer functions of one-dimensional distributed parameter systems, ASME Journal of Applied Mechanics, 59 (1992) 1009-1014.

[9] Yang B., Distributed transfer function analysis of complex distributed parameter systems, ASME Journal of Applied Mechanics, 61 (1994) 84-92.

20


M.A. Jankowska 1, J.A. Kołodziej 2

1 Małgorzata A. JANKOWSKA, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

2 Jan A. KOŁODZIEJ, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

APPLICATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS

FOR THE STUDY OF THE STRESS STATE OF THE PLATE SUBJECTED TO UNIAXIAL TENSION

The paper concerns an application of the method of fundamental solutions (MFS) for the analysis of the stress in the plate with some kind of narrowing subjected to uniaxial extension. In general the method considered enables to solve linear and nonlinear nonhomogenoues boundary-value elastoplastic problems defined by the governing equation with e.g. biharmonic operator involved. A sequence of such nonhomogenoues boundary-value problems appears in the iteration procedure applied in the paper. Such approach is designated here for the identification of the regions of elastic and plastic behaviour of the material.

The considerations given in the paper deal with some plane elastoplastic problem formulated for a finite plate subjected to external loads. Our aim is to present an application of the method of fundamental solutions (MFS) to the iteration procedure (first presented by Mendelson in [5]) that enable the identification of the regions of elastic and plastic behaviour of the material. The MFS belongs to a class of so-called mesh free methods proposed by Kupradze and Aleksidze in e.g. [4]. It is used for solving some nonhomogenoues boundary-value problem in each iteration step. In contrast to the well-known mesh methods, the MFS is easy to implement and can be applied in the case of complicated geometry because there is no mesh involved. Such approach was previously used for solving the nonlinear elastoplastic problems in e.g. [2, 3].

We consider the stress state of a plate with a narrowing, subjected to uniaxial tension related to the stress σB. We assume that the material is homogenous, isotropic and strains hardens isotropically. As in the monograph [5], we choose some loading path to a given state of stress and total plastic strains εij

p. The load is increased by a small amount and it produces additional plastic strains Δεij

p. Hence, the total strains can be written as

,pij

pij

eijij (1)

where εije is the elastic component of the strain, εij

p is the accumulated plastic strain up to (but not including) the current increment of load and Δεij

p is the increment of plastic strain due to the current increment of load. We divide the total loading path into N increments of load and we assume that the plastic strains have been computed for the first i – 1 increments of load. The total strains at the end of the i-th increment of load can be given as follows

21

,21

,1,1

,

,,

pixy

pxyxyxy

piyy

pyyxxyyyy

pixx

pxxyyxxxx

G

EE

(2)

where

.,,1

1,

1

1,

1

1,

i

k

pkxy

pxy

i

k

pkyy

pyy

i

k

pkxx

pxx (3)

For the formulas (2) and (3) we use the stress-strain relation of Chakrabarty [1]. Finally, we choose a yield criterion and the associated flow rule. For the von Mises and Tresca yield criteria and the Prandtl-Reuss equations we have

,3 222xyyyxxyyxxeq (4)

,3

2 2/1222

p

xypyy

pxx

pyy

pxx

peq (5)

where

.23

,22

,22 xy

eq

peqp

xyxxyyeq

peqp

yyyyxxeq

peqp

xx

(6)

Based on the equilibrium and compatibility equations and the previous relations together with the stress function ψ = ψ(x,y), such as

,,2

2yx

xyy

,,2

2yx

yxx

,,

2yx

yxxy

(7)

we can formulate the governing equation given as follows

,,,,4 yxgyxgyx (8) where

,2,2

2

2

2

2

yxxy

Eyxgpxy

pyy

pxx

.2,2

2

2

2

2

yxxy

Eyxgpxy

pyy

pxx (9)

Note that for the boundary-value problem formulation, the equation (7) with (8) and (9) is supplemented with the appropriate boundary conditions required.

Subsequently, we present some numerical procedure (see also [5]) that can be used to identify the stress state in the plate with some kind of narrowing subjected to uniaxial tension. We assume that the total loading path is divided into N increments of load. For each successive increment of load, we use the method of fundamental solutions to solve the boundary-value problem considered. With the stress function ψ, we can compute values of the equivalent stresses at some selected points (regularly located inside of the plate) at the end of each iteration step. Values of the right hand function in the equation (7) are taken to be equal to zero for the points that correspond to elastic behaviour of the material (note we make such an assumption for all the points when the computations starts). In the case of the points such that σeq > σ0, where σ0 is the initial yield stress, some distribution for the plastic strain increments is proposed (and we also take the total plastic strains p

xypyy

pxx ,, all equal to zero). After that we

apply another iteration procedure to calculate the final values of pxy

pyy

pxx ,, . When the

first set of increments of the plastic strains is found, we choose the next value of loading,

22

compute values of the previous total plastic strains (3) and repeat the procedure proposed above.

The numerical algorithm considered can be successfully applied to examine the stress state in the plate subjected to uniaxial tension. For a given value of loading we can identify the regions of elastic as well as plastic behaviour of the material. The method of fundamental solutions let us obtain the solution (i.e. the stress function approximated as a continuous function with continuous derivatives) in a quite simple way also in the case of complicated geometries. References [1] Chakrabarty J., Theory of Plasticity, McGraw-Hill Book Company, 1987. [2] Jankowska M.A., Kolodziej J.A., Application of the method of fundamental solutions for

the plane elastoplastic problem, Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), September 10-14, 2012, Vienna, Austria. Eberhardsteiner J., Böhm H.J., Rammerstorfer F.G. (ed.), Vienna University of Technology, Austria, 2012, pp. 1-17.

[3] Kolodziej J.A., Jankowska M.A., Mierzwiczak M., Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment, International Journal of Solids and Structures, 50 (2013) 4217-4225.

[4] Kupradze V.D., Aleksidze M.A., The method of functional equations for the approximate solution of certain boundary value problems, USSR Computational Mathematics and Mathematical Physics, 4 (1964) 82-126.

[5] Mendelson A., Plasticity: Theory and Application, McMillan Company, New York, 1968.

23


H. Jopek 1, M. Tabaszewski 2, T. Stręk 3

1 Hubert JOPEK, PhD, Institute of Applied Mechanics, Poznan University of Technology,

Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected] 2 Maciej TABASZEWSKI, DSc, Institute of Applied Mechanics, Poznan University of Technology,

Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected] 3 Tomasz STRĘK, Assoc. Prof., Institute of Applied Mechanics, Poznan University of Technology,

Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected]

ACOUSTIC PROPERTIES OF SANDWICH PANELS

Objective of the research is a rectangular plate composed of a corrugated main core and two three-layer facings with polyurethane cores. Compliance of the numerical and experimental results should enable to apply the FEM numerical methods in order to determine acoustic insulating power of the sandwich panel.

1. Introduction One of the most important parameter when it comes to acoustic properties of materials is the

sound absorption coefficient [2]. The considered composite material might be a resonant structure and so it could absorb sound within a certain frequency band. Numerical model might be used for simulation for a wide range of effective properties of analysed composite. Experimental research would be a significant verification of proposed numerical model. Objective of the research is a rectangular plate composed of a corrugated main core and two three-layer facings with polyurethane cores (see Figure 1). Acoustic foam (polyurethane) requires a relatively large thickness in order to provide sufficient damping of low frequency sound vibrations. However, polyurethane is well suited to the reduction of higher frequencies, so a good balance between porous sound absorbing properties and flexural absorbing properties could theoretically provide a wide spectrum of frequency damping while minimizing material requirements. Classical three-layer structures, known since the mid of the 20th century, are composed of a core and two thin facings. The structures have been researched and described in detail in many publications and monographs [1]. Nevertheless, they are always the objective of to-day research. The proposed seven-layer plate is a generalization of the classical one and its significantly better physical properties are expected. 2. Experimental research

Research of sound absorption of materials might be conducted in the reverberant sound field (in reverberation chamber ) by measuring reverberation time in the empty space as well as in the space with the investigated sample. Another possibility is to use standing wave technique in the Kundt’s tube. Despite its simplicity it is a very precise technique although its practical application is restricted mainly to porous media. It is to be shown that the simulation of a composite material fixed in the Kundt’s tube could be also a reliable model and so it allows to extrapolate its results on wider class of composites. The basis of the technique used is the measurement of maximums and minimums of acoustic pressure. Acoustic wave is generated by the speaker placed inside the tube. There is a long pipe of very small cross-sectional size provided through the centre of the speaker which is connected to a microphone. The wave propagates along the tube and reflects from the sample place on the other side of the tube and

24

the standing wave occur. Having the maximum and minimum of acoustic pressure measured one can determine the value of absorption coefficient. Moreover, the distance between consecutive minimums and the distance from sample surface to the first minimum is sufficient to determine acoustic impedance. During the experiment both mentioned values have been determined but also the shape of acoustic pressure inside the tube has been found.

3. Numerical research The FEM modeling [3] includes formulation of numerical model of impedance tube (see Figure 2), with harmonic excitation (acoustic pressure) in the form of a plane or spherical wave, and determination of distribution of acoustic pressure in the tube axis, including the values of maximum value of acoustic pressure pmax and minimal value pmin (see Figure 3). Experimental tests in the impedance tube (Kundt’s tube), comparison of the results of numerical and experimental investigation, in order to check compliance of pmax and pmin. Compliance of the numerical and experimental results should enable to apply the FEM numerical methods in order to determine acoustic insulating power of the plate wall and, to carry out numerical simulation of a coupled “structure – air” system and vibroacoustic phenomena arising inside the plate. The results so obtained for the considered plate are expected to make an original contribution to the theory of layered plates, due to analytical modelling of the plate structure, and theoretical and experimental research of the problems of strength, stability and thermal and acoustic insulation power.

A

B

Fig. 1. Steel part of sandwich panel (A) and polyurethane foam part in sandwich panel (B) A

B

C

Fig. 2. Mesh of: sandwich panel (A), details of sandwich panel in pipe (B) and sandwich panel in pipe (C)

25

A

B

Fig. 3. Total acoustic pressure field in pipe with sandwich panel inside with 1425 Hz acoustic cylindrical wave at pipe's inlet (A) and sound transmission loss of sandwich panel inside one-meter pipe calculated for frequency range from 25 to 1500 Hz (B) Acknowledgements This work was supported by the MNiSW grants 02/21/DSPB/3453 and 02/21/DSPB/3454. The simulations were carried out at the Poznan University of Technology in Institute of Applied Mechanics. References [1] Magnucki K., Jasion P., Kruś M., Kuligowski P. and Wittenbeck L., Strength and buckling

of sandwich beams with corrugated core, Journal of Theoretical and Applied Mechanics, 51 (2013) 15-24.

[2] Rossing T.D. (ed.), Handbook of Acoustic, Springer, New York, 2007. [3] Strek T., Finite Element Modelling of Sound Transmission Loss in Reflective Pipe, Finite

Element Analysis, David Moratal (ed.), Sciyo, 2010, pp. 663-684.

26


H. Kamiński 1, P. Fritzkowski 2

1 Henryk KAMIŃSKI, PhD, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

2 Paweł FRITZKOWSKI, PhD, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

AUXETIC QUARTER-SPACE UNDER CONCENTRATED FORCE

An elastostatic problem for a quarter-infinite auxetic solid is considered. The resulting boundary value problem is solved with a use of the Mellin integral transform and numerical evaluation of residues.

1. Introduction Elasticity problems formulated for infinite, semi-infinite or wedge domains have been

widely considered by many scientists [2-4]. In such cases, very often the stress formulation can be used along with an Airy stress function. One of the classical examples is the Flamant problem: half-space under concentrated surface force system [3].

In this paper, we consider a problem involving a quarter-infinite domain occupied by auxetic solid. Because of the mixed boundary conditions, the displacement formulation is used. Lamé equations are solved by applying the integral transform technique. It allows us to analyze deformation of the loaded surface. 2. Mathematical formulation

Consider a domain with semi-infinite boundaries: 0 ≤ x < ∞, 0 ≤ y < ∞. One of the edges (x = 0) is fixed, while the other one is free. Only a vertical upward concentrated force P0 is applied to the free surface, at some distance a from the origin. The solid material is assumed to be linear isotropic, described by shear modulus G and Poisson's ratio . Typically for auxetics, the Poisson's ratio is negative. Since the loading and constraints are independent on the variable z, one can assume the plane strain state.

In the mathematical formulation the polar coordinates (r, φ) are used. From the classical theory of elasticity, Lamé equations with zero body forces are applied:

01)(2

urr

ur

ur

u rrr , (1a)

011)(2

urr

ur

ur

u rr , (1b)

where ur and uφ denote the radial and tangential displacements, respectively; and, in turn, are Lamé constants:

21

2

G , G . (2)

Moreover, the standard strain-displacement relations can be used, and the allowable stress components can be specified from the generalized Hooke’s law. The unknown function u(r, φ) = [ur, uφ]T included in the equilibrium equations (1) must fulfil the following mixed boundary conditions:

27

0)0()0( ,ru,rur (displacement conditions) (3a) )()2( 0 arP/,r (traction condition) (3b)

0)2( /,rr (traction condition) (3c) where δ denotes the Dirac delta.

3. Solution procedure

The resulting boundary value problem is not of a simple nature. To solve it analytically, we employ operational calculus, particularly the Mellin integral transform. The Mellin transform of a real function f(x) of a real variable x is defined by:

0

1d)()()]([ xxxfsf~xf sM , (4)

where s is a complex variable. Basic properties of M are given in [1]. They can be easily applied to the entire boundary value problem. The governing equations multiplied by r2 can be transformed to the form that involves the Mellin transforms of the displacements with respect to r: )( ,su~r and )( ,su~ . Similarly, the stress components, multiplied by r, can be transformed, which enables us to reformulate boundary conditions (3).

In order to solve the new problem, it is supposed that kr eCu~ 1 and

keCu~ 2 . Next,

the roots k1, …, k4 of the pair of characteristic equations can be found. General solutions are assumed in the form of a linear combination of sines and cosines of appropriate arguments. Some requirements for the real constants result purely from the differential equations. However, the coefficients can be fully determined from the transformed boundary conditions.

Since this paper is focused on deformation of the loaded surface, especially the forms of solutions ru~ and u~ for 2/ are crucial. Their inverse Mellin transforms can be obtained by numerical evaluation of residues based on the Cauchy’s residue theorem. Certain rules, that are frequently used in computational practice for rapid calculation of residues, can be found in [2]. References [1] Bateman H., Tables of integral transforms, vol. 1, McGraw-Hill, New York, 1954. [2] Bronsztejn I.N., Siemiendiajew K.A., Mathematics. Encyclopaedic Handbook [in Polish],

PWN, Warszawa, 1999. [3] Nowacki W., Theory of Elasticity [in Polish], PWN, Warszawa, 1970. [4] Sadd M.H., Elasticity: Theory, Applications, and Numerics, Elsevier, Oxford, 2005. [5] Timoshenko S., Goodier J.N., Theory of Elasticity, McGraw-Hill, New York, 1951.

28


B.T. Maruszewski 1, A. Drzewiecki 2, R. Starosta 3

1 Bogdan T. MARUSZEWSKI, Prof., Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

2 Andrzej DRZEWIECKI, PhD, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

3 Roman STAROSTA, DSc, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

THE EFFECTIVE MATERIAL COEFFICIENTS

AND THERMOELASTIC DAMPING IN A RECTANGULAR AUXETIC PLATE DURING FORCED VIBRATIONS

The paper deals with an influence of the forcing angular frequency and the dimensions of the free supported plate on the effective Poisson’s ratio and the effective Young’s modulus. Both of those parameters in such a situation are not the elastic material constants. The thermoelastic damping occurring during vibrations of the plate because of the upper and lower fibers are arternatively extended and compressed has been analyzed within the extended thermodynamical model. So, the effective coefficients are dependent also on the thermal relaxation time. All the above investigations have been done both for the classical – the positive Poisson’s ratio materials and the auxetic ones characterized by the negative Poisson’s ratio.

29


B.T. Maruszewski 1, G. Sypniewska-Kamińska 2, T. Walczak 4

1 Bogdan T. MARUSZEWSKI, Prof., Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected]

2 Grażyna SYPNIEWSKA-KAMIŃSKA, PhD, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected]

3 Tomasz WALCZAK, PhD, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, e-mail: [email protected]

THREE-DIMENSIONAL ELASTIC TORSION

OF AUXETIC ELLIPTIC RODS USING THE METHOD OF FUNDAMENTAL SOLUTIONS

The paper deals with a basic elastostatic problem of torsion of the elliptic rods. The method of fundamental solution is implemented to make numerical simulations for rods made from auxetic material.

Modern technologies often require materials of peculiar mechanical properties, which are crucial for many practical applications. Recently, an increasing interest is observed in materials exhibiting anomalous (negative) Poisson’s ratio. Young’s modulus together with Poisson’s ratio, constitute a set of quantities entirely describing mechanical properties of isotropic linear elastic bodies. Poisson’s ratio measures the negative ratio of transverse to longitudinal response due to longitudinal stress acting. The negative sign is used to make this quantity positive for common materials, as they usually shrink transversally when stretched. It means that if Poisson’s ratio is negative, the body will shrink transversally when compressed and expand when stretched [1, 2, 5]. Investigations at various models together with numerical methods to simulate the auxetic bodies behaviour are particularly useful in that context.

Numerical simulations of distributions of displacements, strains and stresses for elastic body are usually provided by the Finite Elements Method (FEM) as the most popular numerical method. These circumstances and large accuracy and efficiency of FEM confirmed in many applications cause FEM well suited as a reference to the other numerical methods designed for the same purpose. In this study, the meshless Method of Fundamental Solutions (MFS) has been applied to approximate displacements field and stresses field in auxetic media. Presented method is numerical method used for solving the boundary value problems of differential equations. It is easy to implement and gives satisfactory results for many elastostatic problems [3, 4, 6, 7]. To illustrate the difference between classical and auxetic material behaviour during torsion some numerical simulations were done.

Acknowledgments This paper has been supported by grant 21-418/2014 DS-PB.

30

References [1] Almgren R.F., An isotropic three dimensional structure with Poisson's ratio = - 1, Journal

of Elasticity, 15 (1985) 427-430. [2] Ting T.C.T., Chen T., Poisson's ratio for anisotropic elastic materials can have no bounds,

Quarterly Journal of Mechanics and Applied Mechanics, 58 (2005) 55-71. [3] Goldberg M.A., The method of fundamental solutions for Poisson’s equation, Engineering

Analysis with Boundary Elements, 16 (1995) 205-213. [4] Karageorghis A., Fairweather G., The Method of Fundamental Solutions for axisymmetric

elasticity problems, Computational Mechanics, 25 (2000) 524-532. [5] Wojciechowski K.W., Brańka A.C., Negative Poisson ratio in a two-dimensional ‘isotropic’

solid, Phys Rev. A, 40 (1989) 7222-7225. [6] Poullikkas A., The method of fundamental solutions for three-dimensional elastostatics

problems, Computers and Structures, 80 (1998) 100-107. [7] Poullikkas A., Karageorhis A., Georgiou G., The method of fundamental solutions for

three-dimensional elastostatics problems, Computer and Structures, 80 (2002) 365-370.

31


J.W. Narojczyk 1, K.W. Wojciechowski 2, M. Kowalik 3

1 Jakub W. NAROJCZYK, PhD, Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, Poland, e-mail: [email protected]


3 Mikołaj KOWALIK, PhD, Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, Poland, e-mail: [email protected]

ELASTIC PROPERTIES OF STATIC SOLIDS

WITH n–INVERSE–POWER INTERACTIONS: NEGATIVE POISSON’S RATIO IN PRESENCE OF POLYDISPERSITY

Elastic properties constitute an important characteristic of mechanical properties of

materials [1]. Computational studies of model systems constitute an efficient way to understand the role of various microscopic mechanisms responsible for macroscopic elasticity of materials. Recent computer simulations of elastic properties of two- and three-dimensional static models with n–inverse–power interactions are reviewed [2–9]. In particular, based on the study of atomic and molecular models, the influence of various forms of structural disorder on Poisson’s ratio [1] is discussed.

Size polydispersity of the particles forming a solid is typically responsible for an increase of the (average) Poisson’s ratio [2–9]. Moreover, for isotropic systems as polydisperse 2D discs [2, 3], 2D dimes [3, 4] and 2D trimers [5], and for cubic symmetry 3D spheres [6] and 3D dimers [7, 8], the (average) Poisson’s ratio tends to its extreme positive value in the limit of hard interactions. In the case of anisotropic models, it is possible to decrease the value of the Poisson’s ratio in certain directions even below −1 (which is the lower limit for isotropic systems [3]), by proper selection of particle’s shape. In such systems the negative value of Poisson’s ratio can be retained even in the presence of atomic size polydispersity [9].

Although the atomic size polydispersity causes the increase of the Poisson’s ratio in the hard interaction’s limit, the anisotropic systems of cubic symmetry have been found that exhibit, in some directions, the decrease of the Poisson’s ratio driven by the increase of structural disorder. The three-dimensional sphere [6] and dimer [7, 8] models are the examples of such systems. In the case of the latter system, i.e. the degenerate crystalline structure of cubic symmetry [7, 8], the size polydispersity causes a decrease of Poisson’s ratio in a certain direction for large values of the exponent n, which was not observed previously in any of studied models. This is a promising result in the context of manufacturing mechanically stable (partial) auxetics of cubic symmetry.

Acknowledgements This work was partially supported by the Polish National Centre for Science grant OPUS–3 nr 2012/05/B/ST3/03255. The authors acknowledge the use of computing facilities at the Poznan Supercomputing and Networking Center.

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References [1] Landau L.D., Lifshitz E.M., Theory of Elasticity, Pergamon Press, London, 1986. [2] Narojczyk J.W., Wojciechowski K.W., Elastic properties of two-dimensional soft discs of

various diameters at zero temperature, Journal of Non-Crystalline Solids, 352 (2006) 4292–4298.

[3] Wojciechowski K.W., Narojczyk J., Influence of disorder on the Poisson’s ratio of static solids in two dimensions, Reviews on Advanced Materials Science, 12 (2006) 120–126.

[4] Narojczyk J.W., Wojciechowski K.W., Elasticity of periodic, aperiodic structures of poly-disperse dimers in two dimensions at zero temperature, Physica Status Solidi (b), 245 (2008) 2463–2468.

[5] Narojczyk J.W., Wojciechowski K.W., Elastic properties of two-dimensional soft polydisperse trimers at zero temperature, Physica Status Solidi (b), 244 (2007) 943–954.

[6] Narojczyk J.W., Wojciechowski K.W., Elastic properties of the fcc crystals of soft spheres with size dispersion at zero temperature, Physica Status Solidi (b), 245 (2008) 606–613.

[7] Narojczyk J.W., Wojciechowski K.W., Elastic properties of degenerate f.c.c. crystal of polydisperse soft dimers at zero temperature, Journal of Non-Crystalline Solids, 356 (2010) 2026–2032.

[8] Narojczyk J.W., Wojciechowski K.W., Kowalik M., Partially auxetic behavior in static systems of soft polydisperse dimers, 2014 (in preparation).

[9] Narojczyk J.W., Alderson A., Imre A.R., Scarpa F., Wojciechowski K.W., Negative Poisson’s ratio behavior in the planar model of asymmetric trimers at zero temperature, Journal of Non-Crystalline Solids, 354 (2008) 4242–4248.

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M. Nienartowicz 1, T. Stręk 2

1 Maria NIENARTOWICZ, MSc, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

2 Tomasz STRĘK, Assoc. Prof., Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965 Poznan, Poland, e-mail: [email protected]

FINITE ELEMENT ANALYSIS

OF SANDWICH TWO-PHASE COMPOSITE

Papers present finite element analysis of sandwich two-phase composite, where topology optimization was performed. Results of three types of minimization of energy are compared.

1. Introduction Energy is one of the basic quantitative properties describing a physical system or objective's

state. Energy exists in many forms. On the purpose of this article only two of them will be presented: the thermal energy 푬풕 and the strain energy 푬풔.

The thermal energy [2], from a macroscopic thermodynamic description, full fill formula: 푬풕 = 풄풑푻 (1)

where 푐 is the heat capacity [J/kgK] and 푇 is the temperature. The temperature can be calculated by means of Fourier's equation for stationary heat

conduction problems [2]: −∇ ∙ (푘(푟)∇푇) = 푄 (2)

When external forces are applied to a body, the mechanical work done by the forces is converted, in general, into a combination of kinetic and potential energies. In the case of an elastic body constrained to prevent motion, all the work is stored in the body as elastic potential energy, which is also commonly referred to as a strain energy [1]. Total strain energy of a body subjected to load is then:

퐸 =12 휺푻푫휺푑푉 (3)

where 휺 is the small strain tensor 휺 = (∇풖 + (∇풖)퐓)/2 (the superscript T denotes transpose of a matrix or vector), 푫 is the stiffness matrix. 2. Finite Element Analysis of sandwich two-phase structure

Calculations were provided for 2D models of two-phase composite with applied boundary conditions. To find a solution the Finite Element Method and SNOPT algorithm were used with the Solid Isotropic Material with Penalization (SIMP) model [3]. In Figure 1 the boundary conditions for topology optimization are presented. A fraction of the domain to use for the distribution of the second material is equal to Afrac and takes value of 0.6.

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Fig. 1. Boundary conditions for minimization of a) the thermal energy (left), b) the strain energy (right)

In Figures 2-4 the distribution of control variable is presented for three types of

optimization. In the first type only minimization of the average strain energy was performed, in the second type only minimization of the average thermal energy was carried out and for the third type minimization of the average strain energy and the average thermal energy were simultaneously performed.

Fig. 2. Distribution of control variable for minimization of the average strain energy

Fig. 3. Distribution of control variable for minimization of the average thermal energy

Fig. 4. Distribution of control variable for minimization of the average strain energy and the average

thermal energy References [1] Hutton D. V., Fundamentals of finite element analysis, McGraw-Hill, New York, 2004. [2] Wiśniewski S., Wiśniewski T.S., Wymiana ciepła [in Polish], WNT, Warszawa, 2000. [3] Bendsøe M.P., Sigmund O., Topology Optimization Theory, Methods and Applications,

Springer, 2003.

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P.M. Pigłowski 1, K.W. Wojciechowski 2

1 Paweł M. PIGŁOWSKI, MSc, Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, Poland, e-mail: [email protected]


A SHORT INTRODUCTION INTO SHOCKWAVE PHYSICS

A simple model of a plane, stationary shockwave is presented. Some basic properties of shockwaves travelling through a gas are analytically developed. Similarities between sound waves and weak shock waves are indicated. Some results of Non-Equilibrium Molecular Dynamics (NEMD) simulations for strong shockwaves are also presented. The NEMD simulations show an unusual behaviour of temperature profiles through the front of the shock waves. They demonstrate that, in some far-from-equilibrium states, one can consider temperature as a second rank tensor quantity.

36


A.A. Poźniak 1, K.W. Wojciechowski 2

1 Artur A. POŹNIAK, MSc, Department of Technical Physics, Poznan University of Technology, Nieszawska 13A, 60-965 Poznan, Poland, e-mail: [email protected]


POISSON’S RATIO OF RANDOMLY DISORDERED

ANTI-CHIRAL STRUCTURES WITH VARIABLE ANISOTROPY

Poisson’s ratio (PR) characterizes the geometrical aspect of the elastic body deformation. In the case of three-dimensional isotropic bodies PR values fit the range 2

1;1 [1]. The lower limit does not depend on the dimensionality and holds for planar (two-dimensional) continua as well [2]. Negative value of PR indicates counterintuitive behavior of elastic body. Namely, the specimen expands (contracts) its transversal dimension when stretched (compressed) uni-axially. Bodies which PR < 0 are most commonly known as auxetics [3].

In 1987 Lakes published a paper [4] revealing a recipe for polymeric foam modification leading to negative PR. Several mechanisms responsible for the negative values of PR value have been already described at various scales (from nano to macro). Wojciechowski in 1987 showed that hard cyclic hexamers can spontaneously form an auxetic phase [5] (at molecular level). Lakes in 1991 [6] transferred this idea to the macroscopic scale by replacing hexamer molecules with circles and the interatomic interactions by ribs). This structure (or mechanism) is referred to as a chiral one. Another and qualitatively different mechanism – the anti-chiral one is of our interest. This phenomenon was discussed for the first time in the paper by Sigmund [7]. It assumes the existence of a structure consisting of rigid units and elastic ligaments. These structural elements are connected in the particular manner to constitute an anti-chiral network [7] for which PR is negative. Moreover, the value of PR can be easily tuned by a simple manipulation of the anisotropy of the structure. In fact, any negative PR is possible for anisotropic structures [8].

Recently, the Finite Element Method was employed to investigate the planar anti-chiral structures with the rectangular symmetry and varying anisotropy [9]. The structures were modified to introduce the particular disorder which revealed in the dispersion of the circular nodes (rotating units). It has been shown that the discussed disorder had negligible or minor impact on the PR increase. Thinner ribs result in a lower PR, but also lower stiffness. Finally, it has been shown that for relatively thin ribs the Timoshenko beam type finite elements (one-dimensional) are suitable to perform simulations of the discussed model drastically saving computational time and resources.

Acknowledgements This work was partially supported by the (Polish) National Centre for Science under the grant NCN 2012/05/N/ST5/01476. Part of the simulations was performed at the Poznan Supercomputing and Networking Center (PCSS).

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References [1] Landau L.D., Lifshitz E.M., Theory of Elasticity, Pergamon Press, London, 1986. [2] Wojciechowski K.W., Negative Poisson Ratios at Negative Pressures, Molecular Physics

Reports, 10 (1995) 129-136. [3] Evans K.E., Auxetic polymers: a new range of materials, Endeavour, 15 (1991) 170-174. [4] Lakes R.S., Foam Structures with a Negative Poisson's Ratio, Science, 235 (1987)

1038-1040. [5] Wojciechowski K.W., Constant thermodynamic tension Monte Carlo studies of elastic


[6] Lakes R.S., Deformation mechanisms in negative Poisson’s ratio materials: structural aspects, Journal of Materials Science, 26 (1991) 2287-2292.

[7] Sigmund O., Torquato S., Aksay I.A., On the design of 1-3 piezocomposites using topology optimization, Journal of Materials Research, 13 (1998) 1038-1048.

[8] Chen Y.J., Scarpa, F., Liu Y.J., Leng J.S., Elasticity of anti-tetrachiral anisotropic lattices, International Journal of Solids and Structures, 50 (2013) 996-1004.

[9] Pozniak A.A., Wojciechowski K.W., Poisson’s ratio of rectangular anti-chiral structures with size dispersion of circular nodes, Physica Status Solidi (B), 251 (2014) 367-374.

38


A. Uściłowska 1, A. Fraska 2

1 Anita UŚCIŁOWSKA, Assoc. Prof., Institute of Materials Technology, Poznan University of Technology, Piotrowo 3, 60-965 Poznan, Poland, e-mail: [email protected]

2 Agnieszka FRASKA, PhD, Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-695 Poznan, Poland, e-mail: [email protected]

IMPLEMENTATION OF HAM AND MESHLESS METHOD FOR

TORSION OF FUNCTIONALLY GRADED ORTHOTROPIC BARS

The aim of this study is implementation of the Homotopy Analysis Method (HAM) and the Method of Fundamental Solution (MFS) for solving a torsion problem of functionally graded orthotropic bars.

The torsion problem of bars is an important issue in engineering science. Especially problem of homogeneous and isotropic twisted bars, has been undertaken by many authors. In the last time, the case of inhomogeneous and/or anisotropic material is more often discussed in literature [1, 2]. It is related to the research on functionally graded materials (FGMs), designed for special engineering applications. These modern materials are characterized by a continuous change of their properties at least in one direction.

In this work the torsion problem of linear elastic, orthotropic, prismatic bars made with FGMs is investigated. This is a boundary value problem, described by partial differential equation of second order with variable coefficients and appropriate boundary conditions. The problem is formulated for the Prandtl’s stress function. We propose the Homotopy Analysis Method combined with the meshless method to solve considered problem. Used meshfree method is the Method of Fundamental Solutions supported by Radial Basis Functions and Monomials. The HAM is a very useful tool for solving nonlinear problems [3, 4]. Moreover applying HAM with auxiliary parameter h, allows to control the convergence of the iteration process. It is undoubted advantage of HAM compared with another method based on Picard iteration often used to adapt MFS for solving nonlinear problems, because in method of Picard iteration the process of iteration may be divergent [4]. References [1] Horgan C.O., Chan A.M., Torsion of functionally graded isotropic linearly elastic bars,

Journal of Elasticity, 52 (1999) 181-199. [2] Rongqiao X., Jiansheng H., Weiqiu Ch., Saint-Venant torsion of orthotropic bars with

inhomogeneous rectangular cross section, Composite Structures, 92 (2010) 1449-1457. [3] Tsai C.-C., Homotopy method of fundamental solutions for solving certain nonlinear partial

differential equations, Engineering Analysis with Boundary Elements, 36 (2012) 1226-1234.

[4] Uściłowska A., Rozwiązywanie wybranych zagadnień nieliniowych mechaniki metodą rozwiązań podstawowych, Wydawnictwo Politechniki Poznańskiej, Poznań, 2008.

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S. Winczewski 1, J. Rybicki 2

1 Szymon WINCZEWSKI, MSc, Department of Solid State Physics, TASK Computer Center, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland, e-mail: [email protected]

2 Jarosław RYBICKI, Prof., Department of Solid State Physics, TASK Computer Center, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland, e-mail: [email protected]

HIGHLY EFFICIENT CALCULATION METHOD

OF BOND ORDER PARAMETERS Bond order parameters method [1] is widely utilized throughout the physical sciences to characterize local particle structure of computer simulated materials [2, 3, 4]. The computing process of bond order parameters involves a very frequent evaluation of spherical harmonics, which makes the method computationally extremely expensive. A new computational scheme for the evaluation of bond order parameters has been proposed and implemented. The numerical experiments that have been carried out showed that the developed new algorithm increases the efficiency of bond order parameters evaluation by 60-100 times, thus making the method applicable to the characterization of the structure of large-scale atomic systems. Acknowledgements This work was co-financed by the European Union within European Regional Development Fund, through grant Innovative Economy (POIG.02.03.00-00-096/10). References [1] Steinhardt P.J., Nelson D.R., Ronchetti M., Bond-orientational order in liquids and glasses,

Phys. Rev. B, 28 (1983) 784. [2] Wang Y., Teitel S., Dellago Ch., Melting of icosahedral gold nanoclusters from molecular

dynamics simulations, J. Chem. Phys., 122 (2005) 214722. [3] Moroni D., ten Wolde P.R., Bolhuis P.G., Interplay between structure and size in a critical

crystal nucleus, Phys. Rev. Lett., 94 (2005) 235703. [4] Desgranges C., Delhommelle J., Crystallization mechanisms for supercooled liquid Xe at

high pressure and temperature: Hybrid Monte Carlo molecular simulations, Phys. Rev. B, 77 (2008) 054201.

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A. Uściłowska

Anita UŚCIŁOWSKA, Prof., Institute of Materials Technology, Poznan University of Technology, Piotrowo 3 Street, 60-965 Poznan, Poland, e-mail: [email protected]

THE INFLUENCE OF AUXETIC MATERIAL CHARACTERISTICS

ON THE SOLITON WAVE PROPAGATION PARAMETERS – – NUMERICAL EXPERIMENT

The aim of this study is investigation of the influence of material characteristics on the soliton wave propagation parameters. The study is focused on auxetic materials and their ‘anomalous’ behaviour. The research is based on numerical experiment.

Auxetics are materials and structures of a special class. The common materials narrow when they are stretched, while the auxetics expand laterally when stretched longitudinally. This behaviour is described by negative value of Poisson’s ratio. Some researches maintain that the natural auxetic material exist. Such natural auxetics are e.g. living cat skin, cow teat skin, cancellous bones, some natural minerals. The existence of natural auxetics is still discussed by researchers, but there are a lot of examples of man-made auxetic materials. There are some papers which include the review of the auxetic material [1, 5] and the recent one [2]. The theoretical models of auxetics have been proposed by [3, 4]. The proposal of this paper is the numerical investigation of solitary wave propagation in a plate made of auxetic material. The research is based on numerical experiment. Therefore, the meshless method has been implemented to solve the problem under consideration. The main goal of the paper is to compare the soliton propagation parameters in auxetic and non-auxetic materials. The results of the numerical investigations shows dependency of the soliton wave crest width on the Poisson ratio. For classical materials the width of the wave crest increases with increasing of Poisson ratio in the range (0,0.5). These results agree with classical strength theory and confirms that the proposed in this paper numerical algorithm yields the correct solutions of the considered problem. The other situation appears for auxetic materials. During the numerical experiment it was observed that increase of Poisson ratio in the range (-1,0.76) yield the increase of the crest width. And the increase of Poisson ratio in the range (-0.75,0) causes the decrease of the crest width. The influences of the material characteristics on another parameters of the soliton wave propagation have been investigated, as well. References [1] Alderson A., A triumph of lateral thought, Chem. Ind. (Lond.), 10 (1999) X–391. [2] Prawoto Y., Seeing auxetic materials from the mechanics point of view: A structural

review on the negative Poisson’s ratio, Comp. Mat. Sci., 58 (2012) 140-153. [3] Wojciechowski K.W., Constant thermodynamic tension Monte Carlo studies of elastic


[4] Wojciechowski K.W., Two-dimensional isotropic system with a negative Poisson ratio, Phys. Lett. A, 137 (1989) 60-64.

[5] Yang W., Li Z.-M., Shi W., Xie B.-H., Yang M.-B., On auxetic materials, J. Mat. Sci., 39 (2004) 3269-3279.

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