Search

The paper deals with the thermoelastic problem of a multilayered pipe subjected to normal loadings on its inner surface and temperature differences on its internal and external surfaces. Two types of nonhomogeneous pipe materials of pipe are considered: (1) a ring-layered composite composed of two repeated thermoelastic solids with varying thickness and (2) a functionally graded ring layer. The ring-layered pipe with periodic structure is investigated by using the homogenized model with microlocal parameters. A homogenization approach is proposed for the modelling of the FGM pipe. The analysis of obtained circumferential, radial and axial stress is presented in the form of figures and discussed in detail. It was shown that the proposed approach to the homogenization allows us to correctly calculate the averaged characteristics in the representative cell (the macro-characteristics) and also the characteristics dependent on the choice of the component in the representative cell (the micro-characteristics) for both microperiodic composites and functionally graded materials.

The paper deals with the analysis of nonhomogeneous inner coatings for a homogeneous pipe with respect of heat loss from the outer pipe surface. Two kinds of the coatings in the form of ring layers are considered: (1º) with the thermal properties changing continuously along the coating thickness (called the coating A), (2º) multilayered coatings with piecewise continuous thermal properties (called the coatings B). The analysis is connected with the stationary heat conduction problems. Some special cases of the coatings A and B are investigated. The obtained analytical results and the comparison of the coatings are presented.

The study examines problems of heat conduction in a half-space with a thermal conductivity coefficient that is dependent on temperature. A boundary plane is heated locally in a circle zone at a given temperature as a function of radius. A solution is obtained for any function that describes temperature in the heating zone. Two special cases are investigated in detail, namely Case 1 with given constant temperature in the circle zone and Case 2 with temperature given as a function of radius, r. The temperature of the boundary on the exterior of the heating zone is assumed as zero. The Hankel transform method is applied to obtain a solution for the formulated problem. The effect of thermal properties on temperature distributions in the considered body is investigated. The obtained results were compared with finite element method model.

The quasi-stationary problem of heat conduction for a homogeneous half-space and composite layer being a coating is considered in the paper. Heat fluxes distributed in a strip and moving on the boundary plane of the considered body with a constant velocity are taken into consideration. The composite layer being the coating is assumed to be composed of four repeated periodical beams with rectangular cross sections. The homogenized model with microlocal parameters is used for the description of the composite layer. Some special cases of an influence of composite components and their locations on the temperature field are analysed.

The paper deals with an imaging computer tomography method based on simple image processing techniques for two phase flow analysis. Moreover, it has been presented the algorithm of 3D bubble trajectory reconstruction using a single high speed camera and the system of mirrors. In the experiment a glass tank filled with distilled water was used. The nozzle through which the bubbles were generated was placed in the center of the tank bottom. Through the use of basic image processing and analysis techniques such as noise reduction, smoothing, edge detection and few algorithms like close contour filling, tracking single bubble etc. proposed in paper it became possible to draw out 3D trajectories for gas bubble paths in liquid. In the paper the measurement error of imaging computer tomography method has been estimated. The maximum measurement error recorded for this method was within the limits ±0,65 [mm] for a certain set of parameters like: resolution, mirror angle and deviation error of z axis from the 90o vertical line. Trajectories of subsequently departing bubbles were visualized in the form of figures.

The paper deals with the plane problem of normal and tangential load for microperiodic composite with slant lamination. The boundary condition was considered as a normal and tangential load given on the surface of the composite half-plane. Microperiodic composite consisted of two components, differing in their Young’s modules. The homogenized model with microlocal parameters, which is presented by Woźniak (Int J Eng Sci 25:483–498, 1987. https://doi.org/10.1016/0020-7225(87)90102-9 ), Jedrysiak (Compos Struct 202:1253–1262, 2018. https://doi.org/10.1016/J.COMPSTRUCT.2018.05.155 ) was used to solve the considered problem. The problem was formulated and solved using the averaged boundary condition given by Perkowski et al. (Compos Sci Technol 67:2683–2690, 2007. https://doi.org/10.1016/J.COMPSCITECH.2007.02.013 ). The considered boundary value problem will be solved using the Fourier transform method. An analytical solution was obtained in a general form for any form of the function describing normal and tangential loads on the surface of a half-plane. An analytical solution was determined for the special cases of an elliptical, normal and tangential loads and for the special case of the concentrated force. The results are shown in the form of figures.

The paper deals with the problem of heat conduction in a nonhomogeneous rigid body which is a homogeneous conductive layer with a periodic composite coating. The coating layer is a composite in which every unit cell in general case is composed of four rigid beams with rectangular cross-sections. The assumed structure permits to analyse some special cases of coatings: laminated composites with layer parallel or perpendicular to the boundary, composites with periodically distributed inclusions with rectangular cross-section and a homogeneous layer. The distributions of temperature and heat flux distributions on the interface between the coating and the homogenous layer are investigated.

Three-dimensional trajectories of gas bubbles were the subject of study in this article. Authors carried out experiments in two glass tanks with dimensions of 300 × 300 × 1500 mm and 300 × 150 × 500 mm, respectively, filled with distilled water with a temperature equal to T = 21 ± 1 °C. The tests were carried out for an air volume flow rate, values from 0.00492 l/min to 0.0645 l/min. Basic characteristics of recorded measurement data like: analysis of the bubble detachment frequency from the nozzle, gas bubble size analysis have been shown. Based on the conclusions from those characteristics the multifractal algorithm analysis has been presented as one that allows comprehensive analysis of such a process. The results as a comparison of the main points on the air volume flow rate showed that: the trajectories of gas bubbles in liquid are multifractal and the width of the multifractal spectrum, and thus the complexity of the process being examined, depends on the air flow, the diameter of the bubble generating nozzle and the height of the liquid column in which the phenomenon is examined. The results of the analysis show that the key impact on the overall observed phenomenon is the deformation of the bubble surface, oscillation of the bubble column and observation of the two-phase flow spatially (three-dimensional).

The paper deals with the axisymmetric problem of the thermoelastic half-space with temperature dependent properties. The thermal coefficients: heat conductivity and coefficient of linear expansion are assumed to be functions of temperature. The mechanical properties: Young modulus and Poisson ratio are taken into account as constants. Two cases of boundary conditions are considered: a normal heat flux acting on a circle with given radius and two variants of the boundary conditions on the outside of the heated region: (1) a thermal insulation, or (2) a constant temperature, taken as reference. The boundary is assumed to be free of mechanical loadings. The linear dependences of thermal properties on temperature is considered as a special case. The obtained exact results are presented in the forms of multiple integrals and the detailed analysis are derived for linear dependences of the thermal properties on temperature.

The trajectory of the virtual mass centers of sequentially departing gas bubbles in water flow has been analyzed in this study. The bubbles were generated from a brass nozzle of 1 mm inner diameter in a glass tank (300 × 150 × 500 mm3), filled with distilled water maintained at 21 ± 1 °C. A three-mirror system was used to obtain two frames of the workspace in a single shot of a Phantom v1610 high-speed camera. A three-dimensional computer reconstruction algorithm was developed, which consisted of a blur filter, canny edge filter, region of interest (ROI), and Kalman filter. This algorithm was used to obtain the coordinates of the center of mass of each individual gas bubble. We analyzed the bubble trajectories obtained for three different air volumetric flow rates. A multifractal analysis was used to evaluate the gas bubble dynamics. The wavelet transform modulus maxima method was used to obtain the singularity spectra of the bubble paths. The local Holder exponent value proved that the gas bubble motion trajectories tended to behave in a deterministic and chaotic manner. In addition, the sensitivity of the chaotic bubble paths to small- and large-scale fluctuations of the lateral bubble displacement was investigated. In conclusion, the proposed approach allows us to distinguish between deterministic chaotic dynamics and the stochastic evolution of bubble paths.

This paper deals with the plane frictional contact problem for the microperiodic composite half-plane with slant layering. Problem was considered as a contact of a non-deformable punch pressed into the composite half-plane with normal and tangential forces at the state of limit equilibrium. Microperiodic composite consists of small periodic layers made of two components, differing in their stiffness’ modules. To describe macro-properties of the considered material the homogenized model with microlocal parameters, presented by [ 1 , 2 ] was used. A mathematical problem was formulated and solved using solutions obtained by [3] . Solution to presented problem was received in a general form for any load parameters and material properties. The paper shows contact pressures and stress distributions considered separately for both components of the composite material and results are presented in the form of graphs.

The chaotic properties of bubble paths in bubble chain have been analysed. The bubbles have been generated from a brass cylindrical nozzle with inner diameter equal to 1 mm and placed at the centre of the bottom of rectangular tank (300 × 150 × 500 mm), which was filled with distilled water. Bubble departure diameter was about 5.5 mm. The mirrors system and high speed camera have been used for determining the 3D paths of subsequently departing bubbles. The chaotic changes of bubble path shapes have been observed. Bubble chains obtained for three different air volume flow rates have been analysed. The bubble departure frequencies were adequately 9.4 Hz, 11.6 Hz and 12.7 Hz. The method of Hurst exponent estimation for chaotic bubble paths has been proposed. The Hurst exponent has been calculated for bubble path projections on two orthogonal planes–parallel to the tank walls (shorter and wider). Obtained Hurst exponents confirm that bubble paths have deterministic chaos character. Also, sensitivity of the chaotic bubble paths on initial conditions and the structure of liquid flow around the bubble chain has been discovered. Proposed method allows us to detect anomalies of bubble paths. There has been discovered that predictability of bubble path projection varies depending on the plane which has been chosen for the projection. We can conclude that the method proposed in the paper is suitable for measurement of chaotic bubble path properties.

The paper deals with the modelling of heat conduction in periodically stratified composites with slant layering to boundaries. The relations of the homogenized model with microlocal parameters for the case of layering parallel to the boundary are transformed to conform to requirements of the composite geometry. The plane problem of periodically stratified half-space with slant layering heated by given boundary temperature is solved analytically. The obtained results are analyzed and compared with the solutions for special cases: (1) homogenized half-space, (2) half-space with parallel layering to the boundary, (3) half-space with vertical layering to the boundary.

The paper deals with the problem of heat conduction in the FGM layer with heat conductivity coefficient dependent on depth from boundary surface. The non-homogeneous layer is ideally bounded with homogeneous half-space with constant heat conductivity coefficient. The boundary plane is heated by: a) given temperature as a function of radius r, or b) given heat flux as a function of radius r. The Hankel transform method is applied to obtain a solution of formulated problem. The influence of thermal and geometric properties of FGM layer on temperature distributions in the considered bodies was investigated.

The paper deals with the problem of SH harmonic wave propagation in an elastic layer with temperature dependent properties. The shear modulus and mass density are linearly dependent on temperature. The layer is rested on a rigid foundation and the upper boundary is free of loadings. The boundary planes are kept at different constant temperatures. The wave velocity and amplitude of stresses are analysed.

The paper deals with the two-dimensional stationary temperature distribution in a composite layer. The nonhomogenous body is assumed to be composed of periodically repeated two-layered laminae. The layering is inclined with an arbitrary angle to the boundary planes. The lower and upper boundary planes are assumed to be kept at given temperatures. The considered problem is solved within the framework of the homogenized model with microlocal parameters, where the continuity thermal conditions on interfaces are satisfied.

The paper deals with the stationary problems of diffusion in a homogeneous isotropic layer with a cylindrical hole. The hole is partially filled by diffusing substance. The considered problem is described within the framework of the Fick's law and the integral Weber–Orr transform method is applied to solve it. The obtained exact results expressed by integrals will be presented in the form of figures.

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ1(z) = μ0(1 + αz)β, where μ0, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.

The paper deals with a contact problem of a moving infinitely long punch on the boundary plane of vertically laminated half-space. The problem is solved within the homogenized model with microlocal parameters and under some physical assumptions. By using Fourier transforms the problem is reduced to a integral equation, which is solved numerically. The analysis of temperature and stress distributions and Barber’s contact parameter is presented.

The distributions of temperature and heat fluxes in a periodically laminated layer with a vertically located cylindrical hole are obtained approximately within the homogenized model with microlocal parameters. The method of solution is based on the Weber–Orr integral transforms. The effects of geometrical and mechanical parameters of the composite structure on the thermal fields are discussed and presented in graphical form.

The paper deals with the heat conduction in a semi-infinite laminated layer. The nonhomogeneous body is assumed to be composed of periodically repeated two-layered laminae. The boundary perpendicular to the layering is subjected to the acting of normal heat flux with parabolic distribution. The free exchange of heat with the surroundings is assumed on the remaining parts of the boundary. The problem is solved by using two approaches: (1°) the formulation basing on the classical model of heat conduction with the continuity conditions on interfaces, and (2°) the homogenized model with microlocal parameters (cf. Matysiak, Woźniak, Acta Mech. (1986)). The results obtained on both ways are compared and some adequate conclusions connected with the possibilities of applications of the homogenized model with microlocal parameters are presented.

The technique of integral Hankel transform to find the solution of heat conduction in half- -space coated by a multilayered package of homogenous laminae is applied. The half-space is heated by the given heat flux on the boundary surface. The temperature and heat flux distribution in the radial direction is analyzed for two types of coatings: 1) when the heat conductivity coefficient is described by a power or exponential function of the distance to the boundary surface; 2) multilayered coating has a periodic structure.

The paper deals with the plane problem of stress distribution near crack tips in an elastic periodically stratified space. The crack is normal to the layering in a two-constituent laminated composite. The lower and upper crack surfaces are loaded symmetrically by arbitrary normal loads. The problem is approximated using the homogenized model with micro-local parameters [C. Woźniak, A nonstandard method of modelling of thermoelastic periodic composites, Int. J. Eng. Sci. 25 (1987) 483–499; S.J. Matysiak, C. Woźniak, Micromorphic effects in a modeling of periodic multilayered elastic composites, Int. J. Eng. Sci. 25 (1987) 549–559]. The stress intensity factors are obtained and the influence of geometrical and mechanical parameters and the location of crack tips on the SIF’s values are investigated.

The paper deals with the problem of surface waves in a vertically layered elastic half-space. The transmitting medium is composed of periodically repeated two-constituent laminae, and its boundary is assumed to be normal to the layering. The problem is solved using an approximate model proposed by Woźniak (Int J Eng Sci 25:483–499, 1987) and Matysiak and Woźniak (Int J Eng Sci 25:549–559, 1987), which is referred to as the homogenized model with microlocal parameters. The velocity of the surface wave is obtained as a function of geometric and dynamic properties of the laminae. The variation of the surface wave velocity with respect to composite constituents is illustrated by numerical examples.

This paper deals with the plane contact problem of an infinite long cylinder pressured into an elastic laminated semi-space. The nonhomogeneous body is composed of periodically repeated two constituent laminae and its boundary is assumed to be normal to the layering. The homogenized model with microlocal parameters given by Woźniak [Woźniak C. A nonstandard method of modelling of thermoelastic periodic composites. Int J Eng Sci 1987;25:483–99]. Matysiak and Woźniak [Matysiak SJ, Woźniak C. Micromorphic effects in a modeling of periodic multilayered elastic composites. Int J Eng Sci 1987;25:549–59.] are applied to find an approximate solution to the problem. The problem is reduced to well-known dual integral equations and the stresses are obtained in the form of Fourier integrals. Numerical results, which show the influence of geometrical and mechanical properties of composite constituents on stress distributions, are presented in figures.

This paper deals with some comparison results for displacements and stresses in a periodically stratified elastic semi-infinite layer determined within the framework of two approaches: (1) based on the homogenized model with microlocal parameters [Woźniak (1987) Int J Eng Sci 25: 483–499; Woźniak (1987) Bull Ac Pol Tech Sci 35: 143–151]; (2) obtained directly from the theory of elasticity. A body is assumed to be composed of n elastic two-component periodically repeated laminae. The perfect mechanical bonding between the layers is assumed. The normal displacements and zero shear stresses on the boundary being a cross-section of the composite component are taken into account. The lateral boundary surfaces are assumed to be rigid fixed. The obtained results from the two approaches are compared and presented in the form of figures.

The paper deals with the propagation of shear horizontal (SH) waves in an nonhomogeneous elastic half-space composed of a layer whose shear modulus and mass density have a power dependence on the distance from the lower plane and the periodically stratified half-space. The equation which relates the wave speed to the wave-number and functions of the shear modulus and mass density is derived. The wave velocity is analyzed numerically. Especially, the influence of mechanical properties of the coating layer and the stratified foundation on the wave velocity is presented in the form of figures.

The identification of the shape of the bubble trajectory is crucial for understanding the mechanism of bubble motion in liquid. In the paper it has been presented the technique of 3D bubble trajectory reconstruction using a single high speed camera and the system of mirrors. In the experiment a glass tank filled with distilled water was used. The nozzle through which the bubbles were generated was placed in the centre of the tank. The movement of the bubbles was recorded with a high speed camera, the Phantom v1610 at a 600 fps. The techniques of image analysis has been applied to determine the coordinates of mass centre of each bubble image. The 3D trajectory of bubble can be obtained by using triangulation methods. In the paper the measurement error of imaging computer tomography has been estimated. The maximum measurement error was equal to ±0,65 [mm]. Trajectories of subsequently departing bubbles were visualized.

The stationary problems of heat conduction in a periodically stratified layer with a vertical cylindrical hole is considered. The lateral surface of the hole is kept at a given piece-wise constant temperature. The boundary planes at zero temperature are taken into account, and the ideal thermal contact between the composite constituents is assumed. The problem within the homogenized model with microlocal parameters is solved.