Analytical Field Solutions Research Articles

Dilatational disk and finite cylindrical inclusion in elastic nanowire

For the first time, strict analytical solutions for the elastic fields and the strain energies of an infinitely thin dilatational disk (DD) and a dilatational cylindrical inclusion (CyI) of finite length coaxially embedded in an infinite elastically isotropic cylinder with free surface are given and analyzed in detail. The solutions are represented in an integral form that is suitable for further analytical use and numerical study. The screening effect of the cylinder free surface on the elastic fields and the strain energies of the DD and the CyI is discussed. It is shown that this effect is significant for both the axial displacement and stress fields when the DD and CyI radii are comparable with the cylinder radius, however it is rather weak for the DD strain energy (the energy release does not exceed ∼10%). In contrast, for the CyI strain energy, the screening effect can be very strong. It is also shown that the hydrostatic stress is inhomogeneous and exists not only inside the CyI, as is the case with this stress for inclusions in an infinite medium, but also outside it. This stress is concentrated on the free surface at the points that are closest to the CyI boundary. Inside the CyI, the hydrostatic stress is much higher in magnitude than outside it.

An Analytical Solution for the Steady Seepage of Localized Line Leakage in Tunnels

This paper proposes an analytical solution for the seepage field when a localized line leakage occurs in a tunnel by accurately considering the boundary conditions at the leakage site, which overcomes the problem of current methods, such as the equivalent method or methods improving on the existing analytical solution for fully drained tunnels, being unable to give an accurate analytical solution. First, the semi-infinite seepage region is converted into a rectangular seepage region using two conformal transformations. Subsequently, in order to accurately consider the boundary conditions at the leakage site, the rectangular seepage region with a discontinuous boundary is divided into three subregions with continuous boundaries, and the water head solution for each subregion is given by using the separated variable method. Finally, the principle of orthogonality of trigonometric functions is specially adopted to construct a non-homogeneous system of equations to solve the unknowns in the analytical solution, and through the inverse transformation of the conformal transformation, an analytical solution for the steady-state seepage field when localized line leakage occurs in a tunnel is obtained. The solution proposed is verified by its satisfactory agreement with the numerical simulation results and existing experimental results, and is much more accurate than the existing analytical solution. In addition, the proposed analytical solution is much less computationally demanding compared to numerical simulations. Finally, the capability of the proposed analytical solution is demonstrated by a parametric analysis of the tunnel burial depth, leakage location, and leakage width, and some meaningful conclusions are drawn.

Rheological study of water-based Cu nanofluid between two corrugated curved walls under constant pressure gradient

Nanofluids have captured the attention of scientists due to their superior thermophysical properties compared to ordinary liquids. Consequently, nanofluids serve as suitable cooling agents applicable to systems requiring swift response to thermal changes, such as vehicle engines. Therefore, the current article scrutinizes the characteristics of heat transfer on the pressure-driven flow of magnetized nanofluid between two curved corrugated surfaces in the presence of heat generation/absorption. Copper oxide nanoparticles (Cu) have been combined with pure water to form a nanofluid called Cu/H2O. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field, temperature field, and volumetric flow rate taking the corrugation amplitude as the perturbation parameter. The impact of dissimilar parameters such as the curvature parameter (0.5≤k≤10), wave number (1≤α≤3), magnetic parameter (1≤M≤5), and wave amplitude (0.1≤ε≤0.8) on the flow fields are analyzed through graphs and discussed in detail. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. If we increase the magnetic parameter from 1 to 4, the velocity profile at the specified point decreases by 30 %. If we increase the heat source/sink parameter from 2 to 5, the temperature profile at the specified point increases by approximately 17 %. The flow rate is increased by the corrugations for any phase difference between the corrugated curved walls depending on the corrugation wavenumber and the channel radius of curvature.

A study on the pressure‐driven flow of magnetized non‐Newtonian Casson fluid between two corrugated curved walls of an arbitrary phase difference

AbstractPressure‐driven movement is a fundamental concept with numerous applications in various industries, scientific disciplines, and fields of engineering. Its proper execution is vital for promoting revolutionary innovations and providing solutions in numerous sectors. Therefore, this article scrutinizes the pressure‐driven flow of magnetized Casson fluid between two curved corrugated walls. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field and volumetric flow rate, taking the corrugation amplitude as the perturbation parameter. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. The velocity exhibited a declining trend due to an increase in the Casson fluid parameter. For a sufficiently large corrugation wavenumber, the flow rate decreases, and the phase difference becomes irrelevant. However, the reduction in flow can be minimized by decreasing the channel radius of curvature. In general, a smooth curved channel will give the maximum flow rate for a large corrugation wavenumber. The model can be used to simulate blood flow in arteries with varying geometries and magnetic fields, aiding in the study of cardiovascular diseases and the design of medical devices like stents.

Mode-III fracture of four cracks emanating from an elliptical hole in one-dimensional orthorhombic quasicrystals with piezoelectric effect

In this study, a new generalized conformal mapping is introduced using the generalized complex variable method, focusing on the investigation of the anti-plane problem of an elliptical hole with four cracks in one-dimensional (1D) orthogonal piezoelectric quasicrystals. Under the consideration of electrically impermeable and electrically permeable boundary conditions, the governing equations of the anti-plane problem in 1D orthogonal piezoelectric quasicrystals are successfully simplified into an eighth-order partial differential equation and a fourth-order partial differential equation by introducing a potential function. Based on this, analytical solutions for the stress field and electric field are obtained, and the exact solution for the stress intensity factor (SIF) is derived. In addition, formulas for the electric displacement intensity factor (EDIF) and energy release rate are provided. Subsequently, the influence of geometric parameters and external loads on the field intensity factors and energy release rate is analyzed through numerical examples, providing a theoretical basis for engineering mechanics analysis.

The plane piston problem in a weakly gravitating generalized Chaplygin gas

The present study examines the behavior of a plane piston problem as it moves at a constant velocity in a generalized Chaplygin gas in the presence of a weak gravitational field. A systematic perturbation approach is utilized to obtain an analytical solution for the flow field up to the first order of ϵ. Also, the basic state solution is derived in case of weak shock waves. Further, we analyze the distribution patterns of flow variables for different values of α graphically. Moreover, an approximate solution for the strong shock wave has been found and discussed. The acquired solutions are highly influenced by generalized Chaplygin gas and the applied gravity, which is intriguing aspect from both a physical and scientific standpoint.

Experimental Analysis Of Flow Birefringence In Jeffery-Hamel Flow

The photoelastic method, a stress field measurement method in solid mechanics, is being considered for application to fluids. In previous studies, simple shear flow and uniaxial extensional flow experiments have shown a relationship between the measured phase retardation and the velocity field. However, no clear relationship has been shown for extensional and shear combined flow fields. The objective of the present study is to clarify the relationship between the velocity field and the measured phase retardation in an extensional-shear combined flow. For this objective, photoelastic measurements were conducted in a steady flow field using the Jeffery-Hamel flow, which is an extensional-shear combined flow with an analytical solution for the velocity field. Comparison with the analytical velocity field showed that birefringence was proportional to the 0.88 and 0.92 power of the deformation in the shear or extensional-dominated region, respectively. The results show that the birefringence Δ_n followed the power law of extensional rate ε^{0.95} where it is dominant. Whereas shear is dominant, Δ_n proportional to a power-law of γ^{0.88} holds. These results are consistent with previous studies using shear flow and uniaxial extensional flow. Furthermore, it is shown that in the extensional-shear combined flow, the sum-of-squares root of two equations suggests that the theory developed mainly for solids could also be applied to fluids.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Ciliary propulsion through non-uniform flows

The classical paper by Lighthill (Commun. Pure Appl. Maths, vol. 109, 1952, p. 118) on the propulsion of ciliated microorganisms has become the reference against which many modern studies on swimming in low Reynolds number are compared. However, Lighthill's study was limited to propulsion in a uniform flow, whereas several biologically relevant microorganisms experience non-uniform flows. Here we propose a benchmark for ciliary propulsion in paraboloidal flows. We first consider the axisymmetric problem, with the microorganisms on the centreline of the background flow, and derive exact analytical solutions for the flow field. Our results reveal flow features, swimming characteristics and performance metrics markedly different from those generated in a uniform flow. In particular, the background paraboloidal flow introduces a Stokes quadrupole singularity at the leading-order flow field, generating vortices. Moreover, we determine the necessary conditions on the strength of the background flow for optimal power dissipation and swimming efficiency. We then consider the more general case of a microorganism off the centreline of the background flow. In this case, the squirmer experiences a paraboloidal, linear shear and uniform flows due to its position relative to the flow's centreline. Our findings show that while the linear shear flow does not affect the translational and rotational velocities of the squirmer, it does influence the velocity field and, therefore, the power dissipation.

Influence of Shear Strain on the Deflection of Girders

Numerical calculations are a standard part of modern structural design. Engineers remain particularly interested in real problems where analytical and numerical solutions can be compared with experimental results. Such cases are typical examples of benchmarks because they are used to verify the assumptions introduced. This study shows in detail how shear stresses affect the deflection of a relatively short and high cantilever when the span-to-height ratio of the cross-section is less than five. Such models are frequently used in the design of cantilevers that support heavily loaded beams, for example in the cement industry (e.g., often as structural elements for a heat exchanger system) or for the assessment of short cantilever limit states that appear during excavation in rock sediments. The models are also suitable for designing the various details and joints in the industry of prefabricated elements. This work analyzes in depth the analytical solutions for the displacement field of the linear elastic plane stress theory with two displacement boundary conditions. Also, the solutions were compared with the beam, two-, and three-dimensional numerical models using SAP2000. The results highlight the fundamental principles and solutions behind plane stress and beam theories, with an insight into the advantages and limitations of such models. Doi: 10.28991/CEJ-2024-010-05-04 Full Text: PDF

Improved Common-Mode Leakage Current Measurement Method for Insulation Condition Monitoring in Distribution Grids

Common-mode (CM) leakage current is an essential insulation indicator for condition monitoring in distribution grids. However, online measurement of CM leakage current is a difficult task in the industry due to the disturbances of far larger load currents. To address this challenge, a novel CM leakage current measurement method based on multi-layer magnetic shield has been proposed in this paper. The analytical solution for the magnetic field around the shield is firstly derived and validated by finite element analysis. Comprehensive comparison with previous single-layer shield design is then conducted for a typical industrial case, proving that an improvement of about 46 dB can be achieved for CM leakage current measurement by the novel design. A sensor prototype has also been developed with optimized parameters and validated by experiments. The effectiveness of the proposed method is proved, with minor CM leakage current accurately extracted from load currents which are 3000 times larger.

Analytical Solution for the Stress Field in Surrounding Rock of a Near-Fault Tunnel Considering Tectonic Stress Boundary Conditions

Analytical Solution for the Stress Field in Surrounding Rock of a Near-Fault Tunnel Considering Tectonic Stress Boundary Conditions

Analytical Solution for the Steady Seepage Field of a Foundation Pit in an Anisotropic Layer

The soil piping and drift sand phenomenon is one of the catastrophic failure forms in foundation pit excavations in coastal buildings. Presently, there is a deficiency in the theoretical research regarding the seepage fields around foundation pits, primarily due to the complexity of theoretical solutions given the difficulty in accurately describing the distribution of the groundwater’s hydraulic head in a seepage field. This study proposes an explicit analytical solution for the steady-state seepage field surrounding a foundation pit under anisotropic conditions. A numerical model, constructed with FLAC3D 7.0 software, was utilized to validate the solution presented in this study. The effects of the foundation pit’s width, the distance between the retaining wall and the impervious layer, and anisotropic seepage conditions on the total head are studied through parameter research. The study shows that the flow behavior of a foundation pit is sensitive to parameters such as the anisotropy of the soil layer and the width of the foundation pit. Further, the study also analyzes the influence of the above parameters on the exit gradient and proposes a simplified algorithm for the exit hydraulic gradient at the base of a foundation pit, which can control the error within 5%. This method makes a certain contribution to improving seepage calculations for foundation pits and is applicable to the seepage problem of anisotropic soil layers.

An Oxide Growth-Coupled Viscoplasticity Model and Its Application to Interfacial Stress Analysis near an Air Hole within a Thermal Barrier Coating

Strength assessment for thermal barrier coatings (TBCs) is vital in the safety design of hot-section components in engines. However, several crucial factors, including thermally grown oxide (TGO) growth and creep–plasticity interaction, have been less considered in thermo-mechanical analyses for TBCs near air holes. In this study, a unified viscoplastic constitutive model incorporating TGO growth is developed and integrated into a finite element framework. The model considers multiple factors, including TGO growth, creep–plasticity interaction, interface undulation, and temperature gradient. Additionally, an analytical solution for the non-uniform temperature field of a TBC is derived. The model is then applied to calculate interfacial stresses and accumulated strain energies in the TBC near an air hole, which promote interface debonding. The obtained results can be utilized to investigate the mechanisms of hole edge delamination in TBCs, considering the combined effects of multiple complex factors. A competition for the potential failure initiation location is revealed between the first oxide layer and the evolving TGO/bond coat interface. The developed viscoplasticity model demonstrates effective capability in modelling a range of dynamic behaviors that collectively contribute to hole edge delamination failure.

Configurational force method enables fracture assessment in soft materials

Configurational mechanics offers a framework for quantifying the tendency of defects to alter the material configuration. When applied to fracture mechanics, configurational forces can be used to quantify the propensity of cracks to propagate. An alternative, well-established approach involves analytical solutions for crack tip displacement fields. However, these solutions typically apply to a limited range of constitutive behaviors and oftentimes to the linear small strain regime. The ease of calculating configurational forces in a numerical Finite Element implementation, along with their applicability to soft fracture at large strains, motivates the study of their performance as a standalone fracture framework. In contrast to the majority of works that remain theoretical and numerical, our study includes a robust experimental approach to configurational forces at finite strains. We report tensile experiments on a soft elastomer with pre-cuts ante fracture initiation. In a first attempt to approach the J-integral via configurational forces, we explore the performance of the linear elastic fracture mechanics solutions on Pacman-shaped domains that reproduce the crack tip vicinity. Then, we implement the entire boundary value problem with three-dimensional simulations that replicate the empirical tensile deformation of the soft elastomer samples. Subsequently, the results are benchmarked against estimations of the J-integral obtained through a bespoke finite strain analytical crack tip solution. With the successful validation of the configurational force method at finite strains, we aim to establish a pipeline for the calculation of configurational forces in a standalone manner and circumventing the need for close-form analytical solutions.

The effect of 3D surface roughness on acoustic wave propagation in a cylindrical waveguide

This paper studies the acoustic wave scattering and attenuation in a cylindrical waveguide (pipe) with wall roughness varying along all three dimensions and roughness height smaller than the acoustic wavelength. Using the decomposition of the acoustic wave field into deterministic and random components, small perturbation method and Fourier transform the analytical solution of a 3-D averaged acoustic wave field is obtained. The correction term describing the mechanism of wave attenuation caused by roughness and determined by the modal cross-talk is also derived. The solution for the plane wave is validated in the frequency range extended well beyond the second cut-off frequency, where the crosstalk between the fundamental and non-axisymmetric modes are observed. The analytical solution is compared with the numerical results obtained with the Monte-Carlo method and Finite Element solver. The numerical study results have demonstrated a close agreement with the analytical solution for the averaged sound field, dispersion curves, and the wave attenuation effect expressed as the wavenumber correction term. A key novelty of this work is a comprehensive analysis of wave dispersion and cut-off frequency changes due to the presence of 3-D wall roughness in a pipe.

An isoparametric inclusion model for determining the thermo-elastic fields produced by varying Eigen-temperature gradients

The existence of inclusions or inhomogeneities may disrupt heat flow, resulting in increased stresses and temperature fluctuations at the material interface. Analyzing the thermo-elastic fields surrounding such microstructures is crucial for unveiling the intricate failure mechanisms within materials. Based on the method of Green's function and contour integral, this work presents a complete set of explicit analytical solutions for thermal and elastic fields of an arbitrary polygonal inclusion subjected to linearly varying eigen-temperature gradients or thermal eigenstrains. In contrast to the previous studies on potentials showing analogy with anti-plane elasticity, a mathematical analogy of Green's functions between steady-state heat conduction and thermal inclusion for plane elasticity is accordingly established. By employing the linear interpolation of the nodal eigen-temperature gradients or eigenstrains, the work further extends the traditional isoparametric element idea to the thermos-elastic inclusions, and proposes a novel isoparametric inclusion model to solve arbitrarily shaped inclusions with any distributed eigen-temperature gradients. Unlike the finite element method, which requires meshing for both the matrix and inclusion, the present model exhibits efficiency, flexibility, and versatility with the mesh discretization only conducted inside the inclusion domain. The inclusion based isoparametric method may have potential applications in mechanical engineering and material science for analyzing the thermo-structural behavior of various microstructures.

Flow Characterization in a Partially Liquefied Vitreous Humor

The purpose of this study is to systematically examine the basic fluid dynamics associated with a fully liquid region within a porous material. This work has come about as a result of our investigation on the ocular fluid dynamics and transport process in a partially liquefied vitreous humor. The liquid is modeled as a sphere with Stokes flow while the surrounding infinite porous region is described by Brinkman flow. The development here provides basic three-dimensional axisymmetric results on flow characterization and also serves to evaluate the limits of validity of Darcy flow analysis for the same geometry. In the Darcy flow model, the liquid region is also treated as a porous region with a much higher permeability. Therefore, both liquid and porous regions are modeled by Darcy’s law. Besides the analytical results from Brinkman–Stokes model, the simpler case of Darcy–Darcy flow for the same geometry has been provided. The results of both cases are compared and the differences between the two sets of results provide the range of validity of our computational model (Khoobyar et al. in J Heat Transf 144:031208, 2022). Some interesting fluid-dynamical aspects of the system are observed through the analysis. For the Darcy–Darcy system, the liquid region velocity is uniform throughout, as expected for potential flow. With the Brinkman–Stokes model, the liquid region has a paraboloidal profile with the maximum possible peak value of six times the far-field velocity in the porous medium. With the liquid region having a lower resistance, the flow tends to converge there for both models as it seeks the path of least resistance. As for the validation of the Darcy–Darcy model, it is a good approximation as far as the exterior flow is concerned. However, the liquid region flow profiles for the two models are different as noted. The current Brinkman–Stokes model has led to explicit analytical solutions for the flow field for both regions. This has permitted an asymptotic analysis giving deeper insight into the flow characterization.

CO2-brine interface evolution and corresponding overpressure in high-gravity CO2 storage complex

Technical and economic feasibility of large-scale CO2 geological storage projects (GCS) requires storing the maximum possible amounts of CO2 within a given pore space. However, one limitation is attributed to the overpressure that accompanies CO2 injection. Specifically, reliable estimation of the bottom-hole pressure during CO2 injection is essential to optimize the storage potential of the formation while ensuring the integrity of the rock. Several studies presented analytical/semi-analytical solutions to predict the overpressure accompanying CO2 injection. Nevertheless, they neglect the effects of gravity override on the temporal evolution of the plume and/or the bottom-hole pressure. Effects of gravity can be quantified by the dimensionless group “gravity number” which measures the relative importance of the gravitational-to-viscous forces within system. A lower gravity number expresses a more uniform/cylindrical displacement of CO2. Conversely, biasness of CO2 flow towards the top of the injection zone translates to a larger gravity number. The main objective of this work is to develop an analytical model able to predict the bottom-hole pressure during CO2 injection through a vertical well centered in the middle of a high-gravity thick saline aquifer. While accounting for the effects of gravity, the proposed solution will be developed assuming vertical equilibrium of pressure with a sharp interface separating the injected CO2 and the in-situ brine. First, we develop a closed-form analytical solution to estimate the evolution of CO2/brine interface considering strong gravity effects. The closed-form solution is based on extending the semi-analytical and/or the iterative expressions previously derived in the literature to predict the evolution of the plume during the injection period. Then, the interface model will be coupled with the assumption of the vertical equilibrium to obtain an analytical solution for the pressure field during injection. Next, the proposed solution for the evolution of pressure will be validated against numerical simulations for cases covering wide and practical ranges of gravity numbers and mobility ratios. The solution will be also validated against real field data to determine its robustness. Predictions of the overpressure from our analytical solution indicate a reasonable agreement with the simulations performed using a more complicated multi-phase flow simulator.

A surface-tunnel frequency domain electromagnetic method for mineral exploration in Tajikistan area

For old mines, numerous mining tunnels exist, which can potentially bring us closer to the deep ore-bearing structures. Therefore, we propose a surface-tunnel frequency domain electromagnetic (EM) method to utilize these mining tunnels for geophysical exploration. In this method, the transmitter is placed on the Earth’s surface, while the receiver is positioned within the tunnel space, providing higher resolution due to its proximity to the target body. This paper presents the derived analytical solution for the electric field in the surface-tunnel configuration. We also propose a survey system for our surface-tunnel frequency domain electromagnetic method and provide a field example of lead-zinc deposits in Tajikistan. Synthetic cases demonstrate a significant enhancement of the EM signal when the receiver is moved into the tunnel. The field application validates the practicality of our surface-tunnel frequency domain EM method for deep mineral exploration in active mines.