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TRANSMISSION AND DIFFRACTION OF IMPULSE WAVES IN FOAM MEDIA WITH CAVITIES.

UDC 539.3

Introduction. Recently foam materials have been widely used in construction, which makes it possible not only to cheapen and facilitate construction, but also they have good thermal and vibration insulation characteristics. This explains significant interest in development of research methods of stress state of such materials due to the effects of various dynamic loads caused by technological and mechanical influences. Research of waves transmission processes that arise from the action of dynamic loads enables more accurate strength evaluation of such elements' structures and the effectiveness of their use.

Works devoted to the experimental study of dimensional effects in polymer foams with open and closed cells within the Cosserat moment theory of elasticity are well-known [1-3]. Polymer foams have shown significant dimensional effects at torsion and bending: the studied samples are more rigid than expected according to classical elasticity [3]. Such dimensional effects are taken into account on the basis of the moment (micropolar) theory of elasticity, which allows the points to rotate and also transmit and include distributed moments (moment stresses). On the basis of experimental studies it has been shown that foam with cell size of 0.4 mm has isotropic characteristics.

Research of the stressed state of foam media for a plane and spatial stress state should be carried out on the basis of the equations of the Cosserat moment continuum taking into account the rotational-shear deformation [4].

Numerical methods for studying the transmission of wave processes in media, taking into account the moment stress [5], are described.

A lot of papers have been devoted to building analytic solutions for specific classes of dynamic and static problems for the Cosserat moment continuum [6].

An approach based on the combined use of the boundary integral equations method and time Fourier transform is developed, which makes it possible to investigate the dynamic stress state of media with microstructure weakened by tunnel cavities of arbitrary cross-section during the action of dynamic [7] and impulse [8] load, which is applied to the boundary of the cavity. The integral equations constructed within the framework of the Cosserat pseudo-continuum take into account the influence of rotational-shear deformations that appear in the media due to the action of dynamic loading.

Aim. The aim of the work is to develop a method for studying the transmission and diffraction of weak shock waves on tunnel cavities of arbitrary cross-section in foam media.

Materials and methods. Let us consider an infinite isotropic polymer foam medium (Fig. 1), which is weakened by a tunnel cavity of an arbitrary cross-section (Fig. 2), whose boundary is marked by L [3]. Cartesian coordinate system O[x.sub.1][x.sub.2][x.sub.3] will be placed in the gravity center of the medium.

In order to study the transmission and diffraction of elastic impulse waves in a tunnel cavity, we shall analyse the distribution of dynamic radial and circular stresses in a foam polymeric medium.

During the analysis of deformations in a foam medium it is evident that under the action of a load each microparticle of the medium carries not only translational, but also rotational displacement [9]. Each micro-rotation of the particles of the medium is associated with their translational displacement. In addition, microparticles of the foam medium can not complete micro-rotation without displacement. This is due to the structure of the foam material (Fig. 3).

Therefore the vectors of macro- and micro-rotations coincide for foam materials:

[mathematical expression not reproducible], (1)

where u--gravity center displacement vector;

[bar.[omega]]--rotation vector.

It should be noted that U and [bar.[omega]] are continuous functions.

The equation of foam media motion can be described on the basis of the Cosserat pseudo-continuum equations in the form of [5, 6]

([lambda] + 2[mu])grad divu + rot rot ([B/4][DELTA]u - [mu]u) = [rho][[partial derivative].sup.2]u/[partial derivative][t.sup.2]], (2)

where u(x,t) = {[u.sub.j](x,t)}, j = 1, 2--arbitrary point displacement vector x = {[x.sub.1], [x.sub.2]};

[rho]--material density;

[lambda], [mu]--Lame constants;

[DELTA]--Laplace operator;

B--constant that corresponds to microstructure of material (B = [gamma] + [epsilon]);

t--time.

The boundary conditions of the problem are formulated the following way

[mathematical expression not reproducible], (3)

where [[phi].sub.2](x, t), [[phi].sub.2](x, t)--the known at cavity boundary of function, determined by incident elastic wave potential [10];

[??]--normal to cavity boundary.

In order to solve the problem the approach [7, 8] based on joint application of the boundary integral equations method and time Fourier transform is used [11]

[mathematical expression not reproducible], (4)

where [omega]--cyclic frequency.

In the sphere of Fourier-images the motion equation (2) of the moment Cosserat pseudocontinuum will have the following form

[mathematical expression not reproducible], (5)

where [[??].sub.j]--displacement Fourier-images; that are calculated on the basis of the dependencies (4);

[mathematical expression not reproducible].

The Fourier-images of boundary conditions (3) are indicated as follows

[mathematical expression not reproducible], (6)

where [[??].sub.2] (x, t), [[??].sub.2] (x, t)--images of functions given at cavity boundary [10],

According to [7, 8] the potential image of the general displacement solution of the first main problem is assumed as [12]

[mathematical expression not reproducible], (7)

where [p.sub.1], [p.sub.2]--unknown complex potential functions;

[U.sup.*.sub.ij]--fundamental tensor of dominant functions chosen in the form of [U.sup.*.sub.jk] = [U.sup.* kl.sub.jk] + [U.sup.*m.sub.jk];

[U.sup.*kl.sub.jk]--displacement fundamental tensor of classic elasticity theory [10];

[U.sup.*m.sub.jk]--fundamental tensor, that takes into account effect of rotational-shear deformations in the Cosserat pseudo-continuum [7].

Integration along the boundary is performed by variables [x.sup.0.sub.1], [x.sup.0.sub.2], while [x.sup.0] = {[x.sup.0.sub.1], [x.sup.0.sub.2]}.

In order to fulfill the image conditions (2) on the cavity boundary let us calculate stress according to the formulas similar to the Hooke law in the moment Cosserat continuum [4]

[mathematical expression not reproducible], (8)

where [[sigma].sub.kl]--force stress;

[[mu].sub.kl]--couple stress;

[[epsilon].sub.klm]--Levi-Civita symbol;

[r.sub.k] = [[epsilon].sub.klm] [[??].sub.m,l]/2;

[bar.[omega]]--micro-rotation, that is determined taking into account (1);

[alpha], [beta], [gamma], [epsilon]--elastic constants of material in the Cosserat continuum.

After completing the above transformations we obtain integral expression of the following form:

[mathematical expression not reproducible], (9)

where [f.su7b.j], [g.sub.j]--known functions containing modified Bessel functions.

Having singled out irregular components of subintegral functions and having carried out the boundary transition on the basis of the Plemel-Sokhotsky formulas in dependences (9), we obtain a system of integral equations for identifying unknown on the boundary functions

[mathematical expression not reproducible], (10)

where pds = -iqd[zeta]--unknown function, p = [p.sub.1] + i[p.sub.2], [zeta] = [x.sup.0.sub.1] + i[x.sup.0.sub.2];

[[??].sub.1] = 1 - B[[omega].sup.2]/(4[rho]) for the case of plane deformation.

Here the integrals are understood in terms of principal value.

The system of integral equations (10) was solved numerically using the method [10], which is based on consistent use of the mechanical quadratures and collocation method.

After identifying unknown functions at the boundary, the calculation of circular stress images on the boundary of the cavity and images of radial stresses in the medium was carried out numerically on the basis of the dependences obtained from the formulas (8):

[mathematical expression not reproducible], (11)

where [h.sub.j], [w.sub.j]--known functions;

[[??].sub.2] = -(1 + v) / v for the case of plane deformation.

The calculation of the originals obtained on the basis of the formulas (11) of stresses was carried out using the inverse discrete Fourier transform [11] based on the Cooley-Tukey algorithm [12].

Results. On the basis of the developed approach the transmission and diffraction of a weak shock wave in foam media with a tunnel cavity of a circular and elliptic cross-section was studied. The potential of an incident elastic wave was given in the following form [10]

[mathematical expression not reproducible],

where [[phi].sub.0]--constant;

a--arbitrary characteristic size;

[mathematical expression not reproducible]--impulse modulation with time;

[tau] = t x [c.sub.1] / a--non-dimensional time parameter;

Let us carry out numerical calculations for the foam with open cells size of 0.4 mm, density [rho] = 30 kg/[cm.sup.3], Young's modulus E=81 kPa, Poisson's ratio v=0.3; size factor l=1.6 mm [3].

In order to prove the validity of the developed approach, let us move the center of the cavity on a distance of 8a from the origin of coordinates. In accordance with the basic principles of wave mechanics, until a wave reaches the corresponding cross-section the value of the dynamic radial stresses must be zero.

The results of numerical calculations of relative radial stresses for the cross-sections separated from the origin of coordinates by the distance 3a (curve 1, point [A.sub.1]), 6a (curve 2, point [A.sub.2]), 11a (curve 3, point [A.sub.3]), 12a (curve 4, point [A.sub.4]) are shown in Fig. 4 for cases of pulse duration [tau] = 1 ([[alpha].sub.*] = 10, [p.sub.*] = 185, [n.sub.*] = 2)--Fig. 4, a, [tau] = 2 ([[alpha].sub.*] = 2, [p.sub.*] = 30, [n.sub.*] = 2)--Fig. 4, b and [tau] = 8 ([[alpha].sub.*] = 1, [p.sub.*] = 1.85, [n.sub.*] = 2)--Fig. 4, c respectively. During calculations it was assumed that [[sigma].sub.0] = 1 kPa.

Fig. 4 shows that when the pulse duration of elastic incident wave decreases, there is an increase in values of the maximum relative radial stresses: there is an inverse proportion on the coefficient of proportionality, which is equal to pulse duration. Also at [tau] = 1 the distribution of dynamic radial stresses has a more significant oscillatory character due to the influence of rotational-shear deformations.

After the passage of the wave through the corresponding cross-section, the further stressed state of the medium is determined by the waves reflected from the boundary of the cavity.

Numerical calculations have confirmed the implementation of basic principles of wave mechanics for foam media: relative radial stresses are practically zero until a wave reaches the corresponding cross-section.

In order to study the effect of the cross-section shape of the cavity on the dynamic stress state of the medium we shall calculate the relative radial stresses for the case of the elliptic cross-section cavities with the ratio of the semiaxis 2 (Fig. 5, a) and 5 (Fig. 5, b) for [tau] = 8.

From Fig. 5 it is evident that the maximum values of the dynamic radial stresses in a foam medium with a tunnel elliptic cross-section cavity are 2.7 (for the ratio of semiaxis of the ellipse 2) and 4.97 (for the ratio of semiaxis of the ellipse 5) times less than the corresponding values for the case of a circular cross-section cavity. This is due to the basic properties of the diffraction phenomenon: the wave is more likely to be able to avoid the barrier in the form of an elliptical tunnel cavity, if the major semiaxis of the cross-section coincides with the direction of wave transmission.

To fully assess the stress state of the medium we investigate the distribution of dynamic circular stresses at the cavity boundary under the action of a weak shock wave at [tau] = 8. The results of numerical calculations are shown in Fig. 6. for circular (a), elliptic with a ratio of semiaxes 2 (b) and elliptic with a ratio of semiaxes 5 (c) tunnel cavities in a foam medium [3]. Here [theta] is the polar angle.

Fig. 6 proves that the maximum values of dynamic circular stresses for the case of a tunnel circular cross-section cavity are 1,5[[sigma].sub.0], for the elliptic cross-section cavities with the ratio of semiaxis 2 and 5 1,44[[sigma].sub.0] and 3,34[[sigma].sub.0] respectively. During the action of a weak shock wave the maximum dynamic stresses on the boundary of circular cross-section cavity appear at vertical points, for the elliptical cross-section --at the points of the major semiaxis. Numerical calculations confirm the validity of basic principles of wave mechanics for foam media: stresses are zero until the wave reaches the cavity.

Conclusions. The developed approach based of the combined use of time Fourier transform and the boundary integral equations method enables the investigation of transmission and diffraction of elastic impulse waves on tunnel cavities in a foam medium. The use of the equations of the moment Cosserat continuum made it possible to take into account the effect of rotational-shear deformations that take place in the medium due to the action of dynamic load.

References

[1.] Anderson W.B., & Lakes R.S. (1994). Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J. of Mater. Sci., 29, 6413-6419.

[2.] Lakes R. (1995). Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. Continuum models for materials with microstructure, 1, 1-22.

[3.] Rueger, Z., & Lakes R.S. (2016). Experimental Cosserat elasticity in open-cell polymer foam. Philosophical Magazine, 96 (2), 93-111.

[4.] Eringen A.K. (1975). Theory of micropolar elasticity. Destruction. (Vol. 2). Moscow.

[5.] Erofeev V.I. (1999). Wave processes in solids with microstructure. Moscow.

[6.] Savin G.N., & Shulga N.A. (1967). Dynamic plane problem of the moment theory of elasticity, Applied mechanics, 3(6), 216-221.

[7.] Shvabyuk V.I., Mikulich O.A., & Shvabyuk V.V. (2017). The stress state of foam media with tunnel cavities under the non-stationary dynamic loads. J. of Strength Materials, 6, 99-113.

[8.] Mikulich O.A. (2017). Calculation of Stress State of Foam Materials by Action of Dynamic Loads, Naukowi Notatky, 58, 243-247.

[9.] Wang Y., Gioia G., Cuitino A. (2000). The Deformation Habits of Compressed Open-Cell Solid Foams, J. of Engineering Materials and Technology, 122, 376-378.

[10.] Mikulich O.A. & Shvabjuk V.I. (2016). Interaction of weak shock waves with rectangular meshes in plates. Odes'kyi Politehnichnyi Universytet. PRATSI, 2, 49, 104-110.

[11.] Ufljand Ya.S. (1968). Integral transformations in the problems of the theory of elasticity. Leningrad: Nauka.

[12.] Bonnet M. (1995). Integral equations and boundary elements. Mechanical application of solids and fluids. (Equations integrales et elements de frontiure. Application en mecanique des solider et des fluids), Paris, 5CNRS Editions / Editions EYROLLES.

[13.] Sidorova T.V., Zykova T.V., & Safonov K.V. (2015) On the modification of the fast one-dimensional Fourier transform by the Cooley-Tukey algorithm, Vestnikof SibGAU, 16, 2, 360-363.

[phrase omitted]; Mikulich Olena, ORCID: https://orcid.org/0000-0003-4522-596X

[phrase omitted]; Shvabyuk Vasyl, ORCID: https://orcid.org/0000-0002-1156-4405

Received January 26, 2018

Accepted February 08, 2018

O. Mikulich, PhD, Assoc. Prof., V. Shvabjuk, DSc, Prof. Lutsk National Technical University, 75 Lvivska Str., Lutsk, Volyn region, Ukraine, 43018; e-mail: shypra@ukr.net

DOI: 10.15276/opu.1.54.2018.03

Caption: Fig. 1. Open-cell polymer foam

Caption: Fig. 2. Foam media model

Caption: Fig. 3. Deformation analysis in foam media

Caption: Fig. 4. Distribution of dynamic radial stresses in a foam medium with a cylindrical cavity

Caption: Fig. 5. Distribution of dynamic radial stresses in a foam medium with an elliptical cavity

Caption: Fig. 6. Distribution of dynamic circular stresses on the boundary of circular (a) and elliptic (b, c) cavities in a foam medium

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Author:Mikulich, O.; Shvabjuk, V.
Publication:Odes'kyi Politechnichnyi Universytet. Pratsi
Article Type:Report
Date:Mar 1, 2018
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