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1

Rayleigh wave propagation in isotropic sandy layer sliding over isotropic sandy semi-infinite medium with sliding contact

Madan Dinesh Kumar, Kumar Naveen, Rani Annu

International Journal of Applied Mechanics and Engineering

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2023

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Vol. 28, no. 1

58--70

EN

The present study aims to investigate Rayleigh wave propagation in an isotropic sandy layer overlying an isotropic sandy semi-infinite medium, with interface considered to be imperfect (slide contact and dislocation like model). Expressions for displacement components are obtained using the variable separation method. The dispersion frequency equation for the Rayleigh wave propagating in sandy media is derived using suitable boundary conditions. Particular cases, such as when the interface is in smooth contact and when sandy media are replaced by elastic media, are also discussed. Using MATLAB software, the effects of the imperfectness parameter (slide contact and dislocation like model) and sandy parameter on the Rayleigh waves’ phase velocity are investigated and compared with the already obtained results of the dislocation like model. The present study may find useful applications in geophysics, civil engineering and soil mechanics.

2

Magneto-thermo-piezo-elastic wave in an initially stressed rotating monoclinic crystal in a two-temperature theory

Yadav Anand Kumar

International Journal of Applied Mechanics and Engineering

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2023

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Vol. 28, no. 3

127--158

EN

This research problem is an investigation of wave propagation in a rotating initially stressed monoclinic piezoelectric thermo-elastic medium under with the effect of a magnetic field. A two-temperature generalized theory of thermo-elasticity in the context of Lord-Shulman’s theory is applied to study the waves under the magnetic field. The governing equations of a rotating initially stressed monoclinic piezoelectric thermo-elastic medium with a magnetic field are formulated. This research problem is solved analytically, for a two-dimensional model of the piezo-electric monoclinic solid, and concluded that there must be four piezo-thermoelastic waves, three coupled quasi waves (qP (quasi-P), qT (quasi-thermal), and qSV (quasi-SV)) and one piezoelectric potential (PE) wave propagating at different speeds. It is found that at least one of these waves is evanescent (an evanescent wave is a non-propagating wave that exists) and that there are therefore no more than three bulk waves. The speeds of different waves are calculated and the influence of the piezoelectric effect, two-temperature parameter, frequency, rotation, and magnetic field on phase velocity, attenuation coefficient, and specific loss is shown graphically. This model may be used in various fields, e.g. wireless communications, signal processing, and military defense equipment are all pertinent to this study.

3

Reflection-refraction coefficients and energy ratios in couple stress micropolar thermoviscous elastic solid

Sahrawat Ravinder Kumar, Poonam

International Journal of Applied Mechanics and Engineering

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2021

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Vol. 26, no. 2

47--69

EN

The reflection and refraction phenomenon of propagation of waves in couple stress micropolar thermoviscous elastic solid media with independent viscoelastic and micropolar properties have been studied. The structure of the model has been taken such that the plane interface is divides the given media into two half spaces in perfect contact. Here, we find that there are five waves, one of them is propagating independently while others are set of two coupled waves travelling with different speeds. Energy ratios, reflection and refraction coefficients relative to numerous reflected and refracted waves have been investigated when set of two coupled longitudinal waves and set of two coupled transverse waves strike at the interface through the solid medium. The inequality of energy ratios, refraction coefficients and reflection coefficients are evaluated numerically and presented graphically under three theories of thermoelasticity, namely, Green-Lindsay theory (GL), Lord-Shulman theory (LS), Coupled theory (CT) versus angular frequency and angle of incidence.

4

Mathematical study of Rayleigh waves in piezoelectric microstretch thermoelastic medium

Kumar Arvind, Abo-Dahab S. M., Ailawalia Praveen

Mechanics and Mechanical Engineering

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2019

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Vol. 23, nr 1

86--93

EN

This paper is concerned with the study of propagation of Rayleigh waves in a homogeneous isotropic piezo-electric microstretch-thermoelastic solid half-space. The medium is subjected to stress-free, isothermal boundary. After developing a mathematical model, the dispersion curve in the form of polynomial equation is obtained. Phase velocity and attenuation coefficient of the Rayleigh wave are computed numerically. The numerically simulated results are depicted graphically. Some special cases have also been derived from the present investigation.

5

Plane waves in thermo-viscoelastic material with voids under different theories of thermoelasticity

Tomar S. K., Goyal N., Szekeres A.

International Journal of Applied Mechanics and Engineering

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2019

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Vol. 24, no. 3

691--708

EN

Propagation of time harmonic plane waves in an infinite thermo-viscoelastic material with voids has been investigated within the context of different theories of thermoelasticity. The equations of motion developed by Iesan [1] have been extended to incorporate the Lord-Shulman theory (LST) and Green-Lindsay theory (GLT) of thermoelasticity. It has been shown that there exist three coupled dilatational waves and an uncoupled shear wave propagating with distinct speeds. The presence of thermal, viscosity and voids parameters is responsible for the coupling among dilatational waves. All the existing waves are found to be dispersive and attenuated in nature. The phase speeds and attenuation coefficients of propagating waves are computed numerically for a copper material and compared under different theories of thermo-elasticity. The expressions of energies carried along each wave have also been derived. All the computed numerical results have been depicted through graphs. It is found that the influence of CT and GLT is almost same on wave propagation, while LST influences the wave propagation differently.

6

Response of corrugated interface on incident qSV-wave in monoclinic elastic half-spaces

Singh S. S., Lalvohbika J.

International Journal of Applied Mechanics and Engineering

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2018

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Vol. 23, no. 3

727--750

EN

This paper is concerned with the problem of reflection and transmission of elastic waves due to an incident plane qSV-wave at a corrugated interface between two dissimilar monoclinic elastic half-spaces. Due to the corrugated nature of the interface, there exist regularly and irregularly reflected and transmitted elastic waves. Using Rayleigh’s method of approximation, the reflection and transmission coefficients of regular and irregular waves are obtained for the first order of approximation. We have found that these coefficients are functions of the angle of incidence, elastic constants, corrugation and the frequency parameter. These coefficients are obtained for a special type of interface, z =dcos py. We have computed these coefficients for a particular model and discussed the effects of corrugation and frequency parameter.

7

Seismic crust structure beneath the Aegean region in southwest Turkey from radial anisotropic inversion of Rayleigh and Love surface waves

Çakır Ö.

Acta Geophysica

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2018

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Vol. 66, no. 6

1303--1340

EN

The Turkish plate is covered by hundreds of accelerometer and broadband seismic stations with less than 50 km inter-station distance providing high-quality earthquake recordings within the last decade. We utilize part of these stations to extract the fundamental mode Rayleigh and Love surface wave phase and group velocity data in the period range 5–20 s to determine the crust structure beneath the Aegean region in southwest Turkey. The observed surface wave signals are interpreted using both single-station and two-station techniques. A tomographic inversion technique is employed to obtain the two-dimensional group velocity maps from the single-station group velocities. One-dimensional velocity–depth profiles under each twodimensional mesh point, which are jointly interpreted to acquire the three-dimensional image of the shear-wave velocities underneath the study area, are attained by utilizing the least-squares inversion technique, which is repeated for both Rayleigh and Love surface waves. The isotropic crust structure cannot jointly invert the observed Rayleigh and Love surface waves where the radial anisotropic crust better describes the observed surface wave data. The intrusive magmatic activity related to the northward subducting African plate under the Turkish plate results the crust structure deformations, which we think, causing the observed radial anisotropy throughout complex pattern of dykes and sills. The magma flow resulting in the mineral alignment within dykes and sills contributes to the observed anisotropy. Due to the existence of dykes, the radial anisotropy in the upper crust is generally negative, i.e., vertically polarized S-waves (Vsv) are faster than horizontally polarized S-waves (Vsh). Due to the existence of sills, the radial anisotropy in the middle-to-lower crust is generally positive, i.e., horizontally polarized S-waves (Vsh) are faster than vertically polarized S-waves (Vsv). Similar radial anisotropic results to those of the single-station analyses are obtained by the two-station analyses utilizing the cross-correlograms. The widespread volcanic and plutonic rocks in the region are consistent with the current seismic interpretations of the crustal deformations.

8

Simultaneous velocity measurement of phases in a liquid-gas system

Musoski R., Stelmach J.

Chemical and Process Engineering

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2017

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Vol. 38, nr 4

577--585

EN

Results of velocity measurements of liquid and gas bubbles in a tank with a self-aspirating disk impeller are analysed. Studies were carried out using a fluorescent dye tracer in the measuring system with two cameras (simultaneous phase velocity measurement) and with one camera (sequential measurement of phase velocity). Based on a comparative analysis of the acquired data it was found that when differences in the phase velocities were small the simultaneous velocity measurement gave good results. However, sequential measurement gives greater possibilities for setting the measuring system and if the analysis of instantaneous velocities is not necessary, it seems to be a better solution.

9

Propagation of Love waves in a void medium over a sandy half space under gravity

Patra P., Gupta A. K., Kundu S.

Acta Geophysica

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2017

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Vol. 65, no. 2

269--274

EN

The present study investigates the propagation of Love wave in a void layer resting over a sandy half space under the effect of gravitational force. The equations of motion have been gathered separately for different layers, and the boundary conditions have been introduced for two different layers at their interface. The mathematical analysis of the problem has been dealt with the help of Whittaker’s function by expanding it asymptotically up to linear terms. The study reveals that in such a situation there exist two different wave fronts for the two abovementioned layers: one is for the effects of gravity and sandy parameters, whereas other is for the effect of void parameter.

10

Propagation of waves in the layer of a thermo-viscoelastic transversely isotropic medium

Gupta R. R., Gupta R. R.

International Journal of Applied Mechanics and Engineering

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2016

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Vol. 21, no. 1

21--35

EN

The article is presented to enhance our knowledge about the propagation of Lamb waves in the layer of a viscoelastic transversely isotropic medium in the context of thermoelasticity with GN theory of type-II and III. Secular equations for symmetric and skew-symmetric modes of wave propagation in completely separate terms are derived. The amplitudes of displacements and temperature distribution were also obtained. Finally, the numerical solution was carried out for cobalt and the dispersion curves, amplitudes of displacements and temperature distribution for symmetric and skew-symmetric wave modes are presented to evince the effect of anisotropy. Some particular cases are also deduced.

11

Dispersion relations for composite structures. Pt. 2, Methods of determining dispersion curves

Barski M., Stawiarski A., Chwał M.

Composites Theory and Practice

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2016

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R. 16, nr 3

147--153

EN

In the first part of the current review, the fundamental assumptions of the theoretical model of elastic waves propagation in multilayered composite material are presented. Next, the equations which describe elastic wave motion in the case of single orthotropic lamina are derived. In the second part of this work, the most commonly used method of determining dispersion curves for multilayered composite material are discussed, namely: the transfer matrix method (TMM), global matrix method (GMM), stiffness matrix method (SMM) and finally the semi-analytical finite element method (SAFE). The first three methods are based on the relationships which are derived in the first part of this review. Moreover, TMM and GMM should be considered numerically unstable in the case of a relatively large product value of wave frequency and the total thickness of the composite plate. However, SMM seems to be unconditionally stable. The last method is based on the finite element approach and it can be used in order to confirm the results obtained using the analytical method. Finally, exemplary dispersion curves are presented. The dispersion curves are determined for the 8-th layer of the composite material, which is made of carbon fiber and epoxy resin. It is assumed that the wave front travels in an arbitrary direction.

PL

W części pierwszej pracy omówiono założenia dotyczące teoretycznego modelu propagacji fal sprężystych w wielowarstwowych materiałach kompozytowych. Następnie wyprowadzono równania opisujące zjawisko propagacji fal sprężystych w pojedynczej warstwie o ortotropowych własnościach mechanicznych. W części drugiej przedstawiono podstawy najczęściej wykorzystywanych metod wyznaczania krzywych dyspersji dla ośrodków wielowarstwowych, a mianowicie: transfer matrix method (TMM), global matrix method (GMM), stiffness matrix method (SMM), a także semi-analytical finite element method (SAFE). Pierwsze trzy podejścia oparte są bezpośrednio na równaniach wyprowadzonych w części pierwszej. Metody TMM oraz GMM uważane są za numerycznie niestabilne w przypadku odpowiednio dużych wartości iloczynu częstotliwości i całkowitej grubości płyty kompozytowej. Natomiast wydaje się, że podejście SMM jest numerycznie bezwarunkowo stabilne. Ostatnia z wymienionych metod oparta jest na metodzie elementów skończonych i można ją efektywnie wykorzystać w celu potwierdzenia wyników otrzymanych przy użyciu poprzednio wymienionych algorytmów. Jako przykład pokazano krzywe dyspersji wyznaczone dla 8-warstwowego materiału kompozytowego wykonanego z włókna węglowego, przy czym założono, że czoło fali porusza się w dowolnie założonym kierunku.

12

Dispersion relations for composite structures. Pt. 1, Basic assumptions and relationships for monoclinic lamina

Barski M., Stawiarski A., Chwał M.

Composites Theory and Practice

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2016

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R. 16, nr 3

125--131

EN

Nowadays, the propagation of elastic waves, particularly Lamb waves, is very often used in detecting damages in different kinds of composite materials. These systems are known as structural health monitoring (SHM). However, the phenomenon of Lamb wave propagation is very complex, especially in the case of thin-walled composite structures. Generally, three types of Lamb waves are observed, namely: longitudinal or pressure waves (L), shear vertical (SV) and shear horizontal (SH). The phase and group velocities of the mentioned waves depend on the thickness of the structure and the frequency of the excited signal. This fact makes proper interpretation of the received dynamic response of the structure difficult or even impossible. Therefore, determining the appropriate dispersion curves for different materials is a very important issue. In the present review, the most commonly used analytical approaches for determining dispersion curves in the case of multilayered composite plates are presented. At the very beginning of this work the solution for single isotropic plates is presented. Next, the fundamental assumptions of the theoretical model, which describe the elastic wave propagation phenomenon in multilayered materials, are discussed. In the first part, the relationships describing the elastic wave propagation for single orthotropic lamina are presented. There are two studied cases: namely when the wave front of the elastic wave travels along the principal directions of the material and when the wave front of the elastic wave travels in any arbitrary direction.

PL

Obecnie zjawisko propagacji fal sprężystszych, a w szczególności fal Lamba jest często wykorzystywane przy projektowaniu różnych systemów wykrywania uszkodzeń w wielowarstwowych materiałach kompozytowych. Systemy te są ogólnie znane pod skrótem SHM (Structural Health Monitoring). Jednakże, zjawisko propagacji fal Lamba w kompozytowych konstrukcjach cienkościennych posiada bardzo skomplikowany charakter. W ogólnym przypadku w zależności od płaszczyzny polaryzacji drgań cząstek rozróżniamy trzy rodzaje fal Lamba, a mianowicie: falę podłużną (L) oraz fale poprzeczne spolaryzowane w kierunku pionowym (SV) oraz poziomym (SH). Dodatkowo, każda z wymienionych fal w zależności od grubości materiału oraz częstotliwości generowanego sygnału posiada odpowiednie mody. Mody te propagują się z różną prędkością zarówno fazową, jak i grupową. Zjawisko to znacznie utrudnia interpretację zarejestrowanej dynamicznej odpowiedzi konstrukcji. W pracy szczegółowo opisano najczęściej wykorzystywane analityczne metody wyznaczania krzywych dyspersji. Na początku przedstawiono rozwiązanie dla jednowarstwowej płyty izotropowej. Następnie omówiono podstawowe założenia teoretycznego modelu propagacji fal sprężystych w materiałach wielowarstwowych. W części pierwszej zaprezentowano równania opisujące zjawisko propagacji fal sprężystych w jednowarstwowych płytach o własnościach ortotropowych. Rozważano dwa przypadki, a mianowicie kiedy czoło fal sprężystych porusza się wzdłuż osi głównych materiału oraz kiedy czoło fali porusza się w dowolnym kierunku.

13

Elastic properties of some clay minerals

Vyzhva S. A., Prodayvoda G. T., Vyzhva A. S.

Nafta-Gaz

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2014

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R. 70, nr 11

743--756

EN

A new method for the quantitative estimation of the elastic properties of clay minerals based on the principles of crystal-chemical systematization of clay minerals elastic properties was developed. An accurate mathematical method has been devised to approximate elastic properties of a random symmetry medium to those of a similar transversely isotropic elastic medium. A first successful attempt was made at a quantitative assessment of the elastic constants of all clay mineral constituents in transversely isotropic approximation, which fully agrees with experimental data on these minerals. Extensive research has been carried out into the anisotropy parameters of clay mineral elastic waves and their correlation with the structural features of minerals. This method can be used for approximating the elastic constants of other rock-forming minerals.

PL

Opracowano nową metodę oceny ilościowej sprężystych własności minerałów ilastych w oparciu o zasady krystaliczno-chemicznej systematyzacji tych własności. Nowa, dokładna matematyczna metoda ma na celu przybliżenie własności sprężystych ośrodka o losowej symetrii do tych o podobnym sprężystym ośrodku poprzecznie izotropowym. Pierwszej udanej próby dokonano przy ilościowej ocenie stałych sprężystych wszystkich komponentów minerałów ilastych w poprzecznie izotropowym przybliżeniu, co jest w pełni zgodne z danymi eksperymentalnymi dotyczącymi tych minerałów. Przeprowadzono wszechstronne badanie parametrów anizotropowych fal sprężystych minerałów ilastych i ich korelacji ze strukturalnymi cechami minerałów. Metoda ta może zostać wykorzystana do przybliżenia stałych sprężystych innych minerałów w formacjach skalnych.

14

Wave propagation in a micropolar transversely isotropic generalized thermoelastic half-space

Gupta R. R.

International Journal of Applied Mechanics and Engineering

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2014

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Vol. 19, no. 2

247--257

EN

Rayleigh waves in a half-space exhibiting microplar transversely isotropic generalized thermoelastic properties based on the Lord-Shulman (L-S), Green and Lindsay (G-L) and Coupled thermoelasticty (C-T) theories are discussed. The phase velocity and attenuation coefficient in the previous three different theories have been obtained. A comparison is carried out of the phase velocity, attenuation coefficient and specific loss as calculated from the different theories of generalized thermoelasticity along with the comparison of anisotropy. The amplitudes of displacements, microrotation, stresses and temperature distribution were also obtained. The results obtained and the conclusions drawn are discussed numerically and illustrated graphically. Relevant results of previous investigations are deduced as special cases.

15

Phase velocity and attenuation of longitudinal shear vibrations of hollow poroelastic cylinders

Ahmed Shah S., Javvad Hussaini S.

International Journal of Applied Mechanics and Engineering

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2014

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Vol. 19, no. 2

337--346

EN

The present paper is devoted to the study of phase velocity and attenuation of longitudinal shear vibrations of hollow poroelastic circular cylinders in the presence of dissipation. The explicit expressions for phase velocity and attenuation of longitudinal shear vibrations are derived. The frequency equation of longitudinal shear vibrations and modes obtained in a previous paper are used to compute the phase velocity and attenuation for different dissipations for thin and thick poroelastic cylindrical shells and poroelastic solid cylinder. The physical parameters of sandstone saturated with kerosene and sandstone saturated with water are used for the purpose of computation. It is found that the phase velocity is linear beyond certain frequency. Phase velocity is smaller for a typical anti-symmetric mode compared to the flexural mode. It is greater for the second mode than that of the first mode. Also the phase velocity is larger for a thin poroelastic cylindrical shell than that of a thick poroelastic cylindrical shell. The same is true for attenuation also. Attenuation is very high for the considered dissipations and it increases with the increase in dissipation.

16

Vibrations in a fluid-loaded poroelastic hollow cylinder surrounded by a fluid in plane-strain form

Shanker B., Nath C. N., Shah S. A., Reddy P. M.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 1

189--216

EN

Plane-strain vibrations in a fluid-loaded poroelastic hollow cylinder surrounded by a fluid are investigated employing Biot’s theory of wave propagation in poroelastic media. The poroelastic hollow cylinder is homogeneous and isotropic, while the inner and outer fluids are homogeneous, isotropic and inviscid. The frequency equation of the fluid-loaded poroelastic cylinder surrounded by a fluid is obtained along with several particular cases, namely, fluid-loaded poroelastic cylinder, fluid-loaded bore, poroelastic cylinder surrounded by a fluid and poroelastic solid cylinder submerged in a fluid. The frequency equations are obtained for axially symmetric, flexural and anti-symmetric vibrations each for a pervious and an impervious surface. Nondimensional frequency for propagating modes is computed as a function of the ratio of thickness to the inner radius of the core. The results are presented graphically for two types of poroelastic cylinders and then discussed.

17

Stoneley waves at swelling porous elastic media

Kumar R., Taneja D., Kumar K.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 4

1177-1192

EN

A frequency equation for Stoneley waves at a bonded interface between two swelling porous elastic half spaces [SP/SP] is derived. It is found that Stoneley waves in a swelling porous elastic medium are dispersive in nature. Numerical computations are performed to study the variation of phase velocity and attenuation coefficient with respect to the wave number. Amplitude ratios are obtained and also represented graphically. Some particular cases are also discussed.

18

Wykorzystanie sygnału wibroakustycznego w diagnostyce struktur sprężonych

Radkowski S., Szczurowski K.

Eksploatacja i Niezawodność

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2012

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Vol. 14, No. 1

84-91

PL

W pracy przedstawiono zagadnienie oceny stanu technicznego struktury sprężonej ze szczególnym uwypukleniem możliwości wykorzystania efektów modulacji amplitudowej występujących w obserwowanym sygnale wibroakustycznym. Podstawą takiego podejścia jest teza że zmianie rozkładu naprężeń w przekroju poprzecznym struktury sprężonej towarzyszy mierzalna zmiana parametrów sygnału wibroakustycznego.Wynika ona z założenia, że wraz z wywołaniem stanu sprężenia wstępnego w zginanej konstrukcji zachodzi zjawisko dyspersji, a tym samym zmiana parametrów propagacji fali, w szczególności występowanie mierzalnej różnicy wartości prędkości fazowej i grupowej.Analiza relacji między stanem naprężeń a wartościami prędkości fazowej i grupowej stwarza możliwość budowania diagnostycznych modeli odwrotnych i wyznaczania ilościowych zmian takich parametrów stanu technicznego, jak siły sprężające, moduł Younga czy naprężenia panujące w konstrukcji. Praca powstała na podstawie rozprawy doktorskiej autora.

EN

The paper presents the issue of assessment of the technical condition of a prestressed structure while particularly underscoring the possibilities offered by amplitude modulation effects which are found in the observed vibroacoustic signal. The basis for such an approach is the thesis that change of distribution of stress in the cross-section of a prestressed structure is accompanied by a measurable change of the parameters of a vibroacoustic signal.The thesis stems from the assumption that the condition of prestressing of a structure, as it is being bent, is accompanied by the phenomenon of dispersion and hence of the change of wave propagation parameters, especially the occurrence of a measurable difference between the values of phase and group velocities.Analysis of the relations between the state of stress and the values of phase and group velocities creates the possibilities of developing the reverse diagnostic models and determining the quantitative changes of such parameters of technical condition as compressive forces, Young's modul or the stress in the structure. The paper has been developed on the basis of the author's Ph.D. thesis.

19

Wave propagation in swelling porous elastic layer

Kumar R., Taneja D., Kumar K.

International Journal of Applied Mechanics and Engineering

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2011

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Vol. 16, no 2

379-398

EN

The present investigation deals with the propagation of straight and circularly crested Lamb waves in a swelling porous elastic layer subjected to stress free boundary. A Helmholtz decomposition technique has been used to simplify the mathematical model. The secular equations for different mechanical situations are obtained. Numerical computations are performed to compute the symmetric and skew-symmetric phase velocity and attenuation coefficient in Swelling Porous (SP) and without Swelling Porous (elastic) (WSP) media. At short wavelength limits, the secular equations for symmetric and skew-symmetric waves in the stress free swelling porous and without swelling porous elastic layer reduce to the Rayleigh surface wave frequency equation. The amplitudes of displacements and stresses are obtained and are presented graphically. Some special cases have been deduced from the present investigation. The present study has immense applications to geophysical problems and structure problems.

20

Wave propagation at the boundary surface of orthotropic porous material with rotation and isotropic elastic half-space

Kumar R., Kumar R.

International Journal of Applied Mechanics and Engineering

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2011

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Vol. 16, no 3

683-708

EN

The purpose of this research is to study the surface wave propagation in a layer of an orthotropic porous material with rotation lying over an isotropic elastic half-space. The frequency equation is derived after developing the mathematical model. The dispersion curves giving the phase velocity and attenuation coefficient versus wave number are plotted graphically to depict the effects of rotation and anisotropy for (I) welded contact and (II) smooth contact boundary conditions. The amplitudes of normal displacement, normal stress, volume fraction field and gradient of volume fraction field for the welded contact are obtained and are shown graphically for a particular model to depict the rotation and anisotropy effects. Some special cases are also deduced from the present investigation.