Papers by Keyword: Analytical Solution

Abstract: Research purpose is in establishment of dependencies for parameters of a stress-strain state of a layered composite material with an arbitrary but finite amount of layers and a local breakage in continuity of the outermost layer. Research methodology is in construction of a mathematical model of interaction of hard and soft layers considering given arbitrary amount of layers and with a damaged hard layer and analytical solution of the model using methods of mechanics of composite materials. A character of stress redistribution in layers of a composite material with a local breakage of the outermost layer is established. A breakage in continuity of a hard layer leads to distortion of cross-sections of a sample. The damaged layer has larger displacements than the other ones. The uniform distribution of external loads among the layers is disturbed. Disturbance of a stress-strain state is localized both along the sample and along its thickness. Damage to the outermost layer leads to an increase in the tensile load of the adjacent layer by more than 60 % of the average load on layers. The two layers closest to the damaged one are loaded with a force that exceeds their average total load by almost 80 %. The breakage of the middle layer leads to smaller disturbances. The nature of local stress disturbances also depends on the amount of layers in a sample. As the total amount of layers increases, the extreme forces occurring in a sample with a damaged layer decrease to almost constant values if the amount of layers is at least ten. Scientific novelty is in establishing the dependencies for stress-strain state indicators on structural parameters of a composite tractive element with a local breakage of the outermost layer. The linear formulation of the problem and the principle of superposition make it possible to use the obtained dependencies in a case of applying force to one layer and fixing others. The obtained solutions allow determining a stress-strain state of a sample of layered structure and create conditions for implementing justified solutions regarding the conditions and permissibility of using belts of layered structure with layers damaged during operation, thus ensuring the safety of their use and making the most of their technical resource.

Abstract: The deformation of semisolid slurry within a mould or die is complex in the case of semisolid forming. Understanding and improving the efficiency of such a forming process requires a systematic study of the flow of semisolid slurry under deformation. This study considers the flow characteristics of semisolid A356 alloy slurry under deformation between two parallel plates. The semisolid slurry is represented here by an apparent viscosity under deformation and cooling. The process is then modelled using momentum and energy conservation equations, comprising an analytical solution to predict related flow and deformation behaviour of the slurry. The final solution involves coupling the governing equations by developing a numerical code on the FORTRAN platform. The model then predicts the distribution of temperature, solid fraction, apparent viscosity of the semisolid slurry, and stress to deform the slurry. The deformation stress found in this study has a realistic value, which is also supported by the available research. Prediction of deformation characteristics for any semisolid slurry is possible using the present model, which is simple and appropriate. This study also found that the deformation stress increases with an increase in the plate length and decreases with an increase in the slurry deformation velocity.

Abstract: In the present study, an analytical solution for MHD flow-heat transfer highly non-linear equations of non-Newtonian third-grade nanofluid is established using the AGM method while considering the effect of the magnetic field, the radiation heat transfer, the inclination and the nanoparticles fraction. From dimensionless analysis, the main characteristic parameters are identified, specifically the viscoelastic parameter, the magnetic parameter, the gravitational parameter, the generalized pressure gradient, the thermal radiation parameter, the Brinkman number and the Hamilton number. Two classes of problems, namely, plane Couette flow and plane Poiseuille flow, are considered. Validation was conducted using results from established numerical methods, including Mathematica software, the Adomian Decomposition Method (ADM), and BVP4C solver to benchmark our findings derived via the Akbari Gangi Method. The comparative analysis reveals the reliability and accuracy of the established analytical solutions. The effect of the main parameters of water-SWCNT nanofluid on velocity and temperature profiles are graphically illustrated and discussed. The main results reveal that increasing a magnetic parameter results in a significant drop in the velocity. Furthermore, the rise in Brinkman's number and the radiation parameter affect the temperature differently. Additionally, the viscoelastic and gravitational parameters have opposite velocity and temperature effects. The results demonstrate the complex interaction between several physical characteristic parameters in the fluid dynamics and heat transfer processes. The efficient and highly accurate series-based analytical solutions for flow velocity and temperature obtained through the Akbari-Ganji Method provide valuable insights and are a powerful tool for addressing similar problems in fluid dynamics and heat transfer.

Abstract: This paper analyzes stress distribution in a solid elliptical cross-section under axial load, bending moment, and torsional load, which are common in structural and mechanical engineering. The study aims to derive an explicit formula for von Mises stress as a function of cross-sectional coordinates, providing a unified measure to assess material failure under combined loads.

Abstract: Beams find extensive applications in Nanoelectromechanical systems (NEMS) and Microelectromechanical systems (MEMS). The mechanical characteristics of these microstructures are significantly influenced by both their inherent microstructure and the forces acting at the micro/nano scales. Classical continuum theories fall short in capturing these small-scale effects due to the absence of a length scale parameter in their constitutive relations. To address this limitation, the existing literature primarily relies on the stress gradient nonlocal approach, which, however, has been found flawed and its universal applicability questioned in various scenarios. Therefore, the authors have endeavored to emphasize the strain gradient nonlocal approach, which has been relatively less explored. In this study, carbon nanotubes are modeled using the isotropic Timoshenko beam theory. To introduce the small-scale size effect into the model, the second-order negative strain gradient theory (NSGT) is employed. The Euler-Lagrange differential equations of motion and their corresponding boundary conditions are derived through Hamilton's principle. Analytical solutions are developed for static bending under uniformly distributed transverse load and free vibration problems using Navier's approach. Mathematical results are presented to validate the proposed solutions. Both analyses reveal that the nonlocal effect implemented in this study stiffens the structures, resulting in reduced static deflection and increased natural frequencies. It is noteworthy that beams with dimensions comparable to microstructural length scales exhibit a significant nonlocal effect, which diminishes as the structure's size increases. Additionally, the response obtained using the Timoshenko beam model is softer in comparison to the Euler-Bernoulli model due to the consideration of shear deformation.

Abstract: In this article, the heat transfer and flow pattern characteristics are discussed in the proximity of convective boundary condition for three kinds of nanoparticles, namely gold, Platinum and magnetite with three different shapes, namely spherical, platelets, and lamina. Here water is taken as a base liquid. The thermal radiation impact is assumed into account. The partial differential equations are shifted into ordinary differential equations by applying an acceptable transformation and then exact solutions are acquired by promoting the Laplace transform technique. Solid volume fraction is fluctuated as 5%, 10%, 15%, and 20%. The variations of nanoliquid motion and heat transfer are displayed graphically as well as the numerical values of skin friction and rate of heat transfer at the plate are displayed in tabular pattern. In particular, the liquid motion as well as the heat transfer is least for lamina type nanoparticles, medium for platelet type nanoparticles, and greatest for spherical type nanoparticles. Moreover, the skin friction escalates and the rate of heat transfer declines for three types of nanoliquids in three distinct shapes with the progress of time. This report can be further utilized to authenticate the effectiveness of acquired mathematical results for another sophisticated nanoliquid problems.

Abstract: Fiber-reinforced cementitious matrix (FRCM) composites have been increasingly adopted as externally bonded reinforcement (EBR) of existing concrete and masonry members. Being debonding at the matrix-fiber interface one of the most frequent failure mechanisms of externally bonded FRCM, the matrix-fiber bond behavior represents a fundamental aspect for the effectiveness of the external reinforcement. A cohesive material law (CML) that describes the interface where debonding occurs can be used to model the bond behavior observed. In this paper, a rigid-trilinear CML is used to solve the differential equation that governs the bond problem at the matrix-fiber interface of an FRCM composite. The CML adopted has peculiar characteristics that entail for a finite length of the bond stress transfer zone (BSTZ). Furthermore, it allows for a simple and accurate analytical solution of the bond problem. The analytical solution obtained is compared with the results of an experimental campaign comprising single-lap direct shear tests of a polyparaphenylene benzobisoxazole (PBO) FRCM composite specifically designed for masonry substrates. Different calibrations of the rigid-trilinear CML are proposed, also considering the matrix-fiber free end slip.

Abstract: Contamination of surface water bodies by a wide range of organic and inorganic pollutants has been a serious problem in the recent time, these have an effect on human and aquatic animals. The water quality deterioration calls for regular monitoring of the water quality in order to maintain the health and sustainability of the aquatic ecosystems. Accurate monitoring of discharged pollutants into the rivers may be time taking and labour intensive. Water quality models are significant tools for simulating water quality and controlling the surface water pollution. The purpose of this study is to develop a simplified mathematical model which is hybrid cells in series model (HCIS) to simulate the spatial and temporal variation of nitrate concentration in natural rivers. The HCIS model was formulated to serve as an alternative method to the Fickian based models. Analytical solutions for the first order reaction kinetics of nitrate with the advection and dispersion process were derived using Laplace transformation technique. The model considered the effect of nitrate concentration at several points along the river downstream by considering the transformation of nitrite to nitrate through nitrification process. In addition, the uptake of nitrate by algae for its growth and conversion of nitrate to nitrogen gas due to denitrification process were considered. The HCIS-NO₃ model was applied to uMgeni River, South Africa to investigate the nitrate concentration along the river. Furthermore, the quantitative measures based on the coefficient of determination (R²) and standard errors (SE) were used to evaluate the performance of the model. The result shows that the simulated values agreed with the measured values of nitrate concentration in the river which resulted in a R² value of 0.72 and a low standard error. Analytical solutions of HCIS - NO₃ model were compared with the numerical solutions of the Fickian based ADE model for hypothetical problems. Comparison of the responses indicates that the HCIS - NO₃ and ADE- NO₃ models were in good agreement. The study shows that the hybrid model is a simple and effective tool for simulating pollutant transport in natural rivers.

Abstract: Vibration problem of variable cross-sectional nanorods have been investigated. Analytical solutions have been determined for the variable cross-sectional nanorods for a family of cross-sectional variation. Cross-sectional area variation has been assumed as power function of the axial coordinate. Nonlocal governing equation of motion has been obtained as a second order linear differential equation. Bessel functions have been used in analytical solution of the governing differential equation. Effect of nonlocal and area variation power parameters on dynamics of nanorods have been analyzed. Mode shapes of nanorod have been depicted in various cases and boundary conditions. Present results could be useful at design of atomic force microscope’s probe tip selection.

Abstract: Excess fin length results in material waste and additional weight leading to increased cost with no benefit in return. Moreover, extra fin length affects the overall performance of the fin as fluid motion is suppressed, resulting in reduced convective heat transfer coefficient. To achieve a miniaturised system with effective cooling, the determination of appropriate length of extended surfaces becomes a key performance and fabrication process factor. Therefore, the present work aims at determining the proper or effective length of a convective-radiative moving fin of functionally graded material under the influence of a magnetic field. The developed governing equation of the analysis is solved analytically with the aid of Kummer’s function. The analytical solutions are used to investigate the effects of non-homogeneity, convective, radiative and magnetic parameter on the thermal performance and the proper fin length. The present study is hoped to assist in making cost-effective decisions on designing cooling approaches for different consumer electronics and high-power systems under various operating conditions.