Wyniki wyszukiwania - BazTech

1

Elastic properties of a unidirectional composite reinforced with hexagonal array of fibers

Czaplinski T., Drygaś P., Gluzman S., Mityushev V., Nawalaniec W., Zietek G.

Archives of Mechanics

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2018

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Vol. 70, nr 3

207--239

EN

It is difficult to measure the transverse shear modulus of the fibrous composites. Thus, theoretical investigations by means of analytical and numerical techniques are paramount. In particular, they are important for the regime with high-concentration of fibers. We apply general techniques to study the mechanical properties of unidirectional fibers with a circular section embedded into the matrix and organized into the hexagonal array. Our theoretical considerations are designed to include two regimes, of low and high concentrations of inclusions. The former regime is controlled by Hashin–Shtrikman lower bounds, while the latter is controlled by square-root singularity. We derived the analytical formulae for the effective shear, Young and bulk moduli in the form of the rational expressions valid up to O(f7) by the method of functional equations. The obtained formulae contains elastic constants of components in a symbolic form as well as the concentration f. The general scheme based on the asymptotically equivalent transformations is developed to extend the obtained analytical formulae to the critical concentration of touching fibers. A comparison with the numerical FEM is performed for all concentrations of inclusions. Good agreement is achieved for all available concentrations.

2

Random composite: stirred or shaken?

Gluzman S., Mityushev V., Nawalaniec W., Sokal G.

Archives of Mechanics

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2016

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Vol. 68, nr 3

229--241

EN

A James Bond’s (JB) catchphrase “shaken, not stirred" is explored for the problem of effective conductivity of composites. The superconductivity critical index s for the conductivity of random non-overlapping disks turns out to be distinctly different for shaking and stirring protocols. In the case of stirring modeled by random walks the formula s(τ) = 0.5 + 0.8 3√τ is deduced for evolution of the critical index with the normalized time 0 ≤ τ ≤ 1, which is proportional to the number of random walks and serving as the disorder measure. Strikingly, the coefficient 0.8 is very close to the critical index for shaking protocol and 0.5 is the critical index for regular lattices. The obtained formula for s is based on the analytical solution to the 2D conductivity problem of randomly distributed disks up to O(x19), where x denotes the concentration of inclusions and its extension to special 3D composites.

3

Series, index and threshold for random 2D composite

Gluzman S., Mityushev V.

Archives of Mechanics

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2015

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Vol. 67, nr 1

75--93

EN

Effective conductivity of a 2D random composite is expressed in the form of long series in the volume fraction of ideally conducting disks. The problem of a direct reconstruction of the critical index for superconductivity from the series is solved with good accuracy, for the first time. General analytical expressions for conductivity in the whole range of concentrations are derived and compared with the regular composite and existing models.

4

Cross-properties of the effective conductivity of the regular array of ideal conductors

Gluzman S., Mityushev V., Nawalaniec W.

Archives of Mechanics

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2014

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Vol. 66, nr 4

287--301

EN

We present an accurate expression for the effective conductivity of a regular square-lattice arrangement of ideally conducting cylinders, valid for arbitrary concentrations. The formula smoothly interpolates between the two asymptotic expressions derived for low and high concentrations of the cylinders. Analogy with critical phenomena is suggested and taken to the extent of calculating the superconductivity critical exponent and the particle-phase threshold from the very long expansions in concentration. The obtained formula is valid for all concentrations including touching cylinders, hence it completely solves with high accuracy the problem of the effective conductivity for the square array.

5

Simulations of representative volume elements for random 2D composites with circular non - overlapping inclusion

Czapla R., Nawalaniec W., Mityushev V.

Theoretical and Applied Informatics

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2012

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Vol. 24, No. 3

227-242

EN

Consider two-dimensional two-component periodic composite made from a collection of non-overlapping, identical, circular disks, embedded in a matrix. In accordance with a theory of the representative cells (representative volume elements), the effective conductivity of disks is expressed in terms of the generalized Eisenstein-Rayleigh sums (ER sums). Straightforward computation of the ER sums is possible only for the sums of lower orders. In the present paper, a fast algorithm to compute higher order sums worked out by use of random walks and Monte Carlo simulations. The algorithm is recurrent, i.e., an ER sum of the fixed order is expressed in terms of the ER sums of lower orders by simple formulae. Relations between the Eisenstein and Weierstrass functions and algebraic dependences between their derivatives are also used to improve the algorithm. The obtained numerical results are applied to investigation of the structure of composites.

PL

Rozważmy dwuwymiarowy, dwufazowy okresowy materiał kompozytowy, złożony ze zbioru nienakładających sie na siebie identycznych wtrąceń kołowych zanurzonych w osnowie. Zgodnie z teorią komórki reprezentatywnej, efektywna przewodność badanego materiału wyraża się za pomocą uogólnionych sum Eisensteina-Rayleigha (zwanych dalej sumami ER). Bezpośrednie obliczenie sum ER jest możliwe tylko w przypadku sum niższych rzędów. W artykule przedstawiono szybki algorytm obliczający sumy ER wyższych rzędów, opracowany z wykorzystaniem błądzenia losowego oraz metody Monte Carlo. Algorytm ten jest rekurencyjny, tzn. ustalona suma ER wyrażona jest za pomocą sum niższego rzędu. W celu usprawnienia działania algorytmu wykorzystano algebraiczne zależności między funkcjami Eisensteina i Wieierstrassa oraz między ich pochodnymi. Uzyskane wyniki numeryczne zastosowano do badania struktury materiałów kompozytowych.

6

Exponent in one of the variables

Mityushev V.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2007

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Vol. 12

77-79

EN

A periodicity functional equation of one complex variable which characterizes the exponential function is discussed. This functional equation can be generalized to equation for functions depending on two complex variables. It is conjectured that the second functional equation also characterizes the exponent. Applications to representations of complex continuous elementary functions are discussed.

7

Compaction of the diamond-Ti3SiC2 graded material by the high-speed centrifugal compaction process

Mityushev V., Jaworska L., Rozmus M., Królicka B.

Archives of Materials Science and Engineering

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2007

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Vol. 28, nr 11

677-682

EN

Purpose: Sedimentation of particles in a viscous fluid is a main physical problem in fluid mechanics. Sedimentation is benchmark of one of the technica; methods to produce the functionally graded materials (FGM) with a continuous spatial change of mechanical properties. The aim of the research was execution of mathematical calculations of the phases distribution for the phase graded diamond-Ti3SiC2 compacts which were verified with phases distribution in compacts after the high pressure-high temperature sintering process. Design/methodology/approach: In this paper, we construct a mathematical model of FGM basing on the modifications of the Stokes formula. We proposed an algorithm to describe sedimentation of the group of spherical particles of different sizes and different materials. Main calculations for this system and real conditions of the highspeed centrifugal compaction process are made using the Barnea-Mizrahi equation. Deposition process was carried out using the ultra centrifuge UP 65M with rotational speed of 15000 to 25000 rpm. Particle size distribution for the diamond and Ti3SiC2 powders were measured using Shimadzu apparatus. Findings: The results of calculations and microscopic analysis are compared. The obtained results of mathematical calculations demonstrate that for the considered diamond-Ti3SiC2 suspensions the obtained compact has the structure of the laminate what confirmed microscopic analysis. Practical implications: The mathematical simulations using our algorithm show that it is possible to obtain continuous concentrations of the both materials with appropriate initial suspensions. Thus our method allows to obtain graded materials. Originality/value: This mathematical model gives possibility of use to describe sedimentation of the group of spherical particles different materials and different sizes.

8

Zagadnienie brzegowe dla krzywoliniowego trójkata i njego zastosowanie w materiałach kompozytowych

Mityushev V., Reszka K.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2001

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z. 25 [190]

131-138

EN

Let G+and G- be two uniconnected domains of a plane with a cammon boundary curve L. In a paper we determine a function u(x,y) harmonic in every domains G+ G-~ and satisfied some boundary condition on L. Some applications of such functions are also presented.

9

Thermoelastic plane problem for material with circular inclusions

Mityushev V.

Archives of Mechanics

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2000

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Vol. 52, nr 6

915-932

EN

We consider two-dimensional thermoelastic composite materials in the case when the temperature is constant. Using complex potentials and applying a method of functional equations, we construct a simple algorithm to solve the corresponding boundary value problem. The stress tensor is written with the accuracy of up to the term 0 (R^2), where R = max[k,m]r[k]d[[km]^-1, r[k] is the radius of the k-th inclusion, d[km] is the distance between centers of the k-th and m-th inclusion (k is not equal to m). The effective elastic constants and the coefficient of thermal expansion are written in analytic form up to 0 (R^4).