Three-dimensional Analytical Solution Research Articles

Three-dimensional analytical solution of free vibrations of a simply supported composite plate in contact with a fluid

Background:The quantification of fluid–structure interactions is critical for the design and optimization of marine structures. As composite materials gain more traction in naval construction, there is a dire need of modeling tools to detail the dynamic response of composites in contact with fluids. Current structural theories and finite-element simulations offer promising pathways, but their strong hypotheses on through-the-thickness deformations and high computational cost challenge their practical feasibility. Methods:We propose an analytical three-dimensional solution to the free vibration of a bidirectional composite, separating a fluid column from vacuum. We leverage Pagano’s method to describe the three-dimensional displacement field in each ply of the composite plate and the fluid potential in the form of a Fourier series. Results:An analytical solution to the coupled fluid–structure problem is derived by imposing the continuity of displacement and stress at the ply interfaces, as well as boundary conditions on dry and wetted plies. We specialize the solution to sandwich structures of naval interest, offering insight on how the interactions with water can dramatically affect mode shapes and through-the-thickness profiles of displacement and stress. Discussion:The proposed approach provides the foundations for the design of thick composite plates for use in marine structures.

Three-dimensional analytical elasticity solutions in isosceles-right triangular composite laminates

In this study, analytical solutions are derived for three-dimensional, transversely isotropic and multilayered isosceles-right triangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism and the dual-variable and position method. For multilayered isosceles-right triangular plates, we derive the analytical solution by the new propagating method in terms of the symmetry theory. Numerical results clearly show the influence of load distribution, stacking sequence, and mechanical and geometric parameters on the elastic fields. The proposed solutions can also serve as benchmarks for other numerical methods such as the finite element method.

Three-dimensional thermal-mechanical analytical solutions for the half-space pressed by an inclined rigid cylindrical structure

Three-dimensional thermal-mechanical analytical solutions to the elastic half-space pressed by an inclined rigid cylindrical structure are presented in this article for the first time, based on the recent thermo-elastic general solutions in terms of harmonic functions and potential theory method. The obtained solutions are proved to be closed. The adequate validations are provided by comparing contact displacement and contact pressure on the contact plane from the presented solutions with those without heat effect in literatures. The geometric distribution of temperature on the contact region is also in accordance with that by previous researcher for the composite material under uniform heat flux. In addition, for the purpose of engineering application, a numerical example which is thermal contact problem between an inclined rigid circular displacement restrictor and elastic half-space is given, and the distributions of key field variables and effects of heat flux on contact pressure and normal displacement are discussed in detail.

Three-dimensional analytical solutions of non-Darcy seepage in front of a grouted subsea tunnel face

In this paper, the impact of non-Darcy seepage on the steady seepage field in front of a grouted subsea tunnel face is analysed. Analytical solutions are proposed by combining the three-dimensional (3D) seepage continuity equation, which is expressed in spherical coordinates, with the Hansbo seepage model. These solutions, which are also suitable for calculating the hydraulic head distribution in front of the tunnel face without a grouted zone, are verified by a series of numerical simulations. Comparisons show that this study enabled the augmentation of the classical analytical model. Finally, the influence of relevant parameters, including the nonlinear parameters in the Hansbo seepage model, the relative permeability (i.e., kr/kg), and the grouted zone thickness, are investigated. The results show that the exponential parameter m and critical hydraulic gradient i1 in the Hansbo seepage model have a more significant impact on the water inflow than the water pressure on the grouted zone. Based on the non-Darcy seepage model, the water inflow is lower and the water pressure on the grouted zone is larger than that obtained by Darcy's model.

A Three-Dimensional Analytical Solution of Stress Field in Casing-Cement-Stratum System Considering Initial Stress State

Accurate stress field calculation of the casing-cement-stratum system is crucial for evaluating wellbore integrity. Previous models treated in-situ stress as boundary pressure loads, leading to unrealistic infinite displacements at infinity. This study presents a three-dimensional (3D) analytical solution for the stress field within the casing-cement-stratum system in inclined wells, considering in-situ stress and hydrostatic stress in cement as the initial stress state and taking into account stress components related to the axial direction. Assuming a plane strain condition and superimposing the in-plane plane strain problem, elastic uni-axial stress problem and anti-plane shear problem, a 3D analytical solution is obtained. Comparisons with previous models indicate that the existing model overestimates the absolute values of stress components and failure potential of casing and cement in both 2D and 3D scenarios. The presence of initial stress in cement greatly increases the absolute value of the compressive stress state but decreases the failure potential in cement, which has not been well studied. Additionally, a low Young’s modulus and high initial stress state of the cement benefits the cement’s integrity since the maximum Mises stress significantly decreases. The new 3D analytical solution can provide a benchmark for 3D numerical simulation and quick assessment for wellbore integrity.

Three-dimensional analytical elasticity solution for the mechanical analysis of arbitrarily-supported, cross and angle-ply composite plates under patch loads

Patch-type discontinuous loads lead to highly localized stresses which are three-dimensional (3D) in nature and can initiate failures. Their accurate characterization is of primal importance for composite and sandwich structures. The present contribution elaborates a 3D analytical elasticity solution for the mechanical analysis of cross and angle-ply composite plates under transverse patch loads and arbitrary support conditions. For the solution, the Reissner-type, mixed variational principle is used to derive the governing equations. Analytical solutions are obtained using the multiterm, extended Kantorovich method, by dividing the plate into loaded and unloaded parts. The present analytical solution exactly satisfies inter-segment, inter-layer continuity and edge support conditions, treating stresses and displacements as primal variables. Its accuracy is established by direct comparisons with existing analytical – where available – and 3D finite element results, both for cross-ply and angle-ply designs, for different boundary conditions. Extensive analysis is performed to quantify the effect of thickness, angle-ply and patch load position on the composite mechanical performance. Through thickness stress asymmetries are identified and quantified, along with angle-ply designs that can smoothen stress concentrations in the vicinity of patch loads. The methodology can be used as a reference solution in the design of multilayer anisotropic plates under discontinuous loads.

Three-dimensional analytical solutions to the half infinite thermo-elastic body pressed by a rigid cylindrical structure in thermal environment

The novelty of this paper is to propose exact three-dimensional analytical solutions to the half infinite thermo-elastic body pressed by a rigid cylindrical structure for the first time, in which uniform heat flux flowing from rigid structure into the half infinite body is considered. Using recent general solutions expressed by four harmonic functions and extending potential theory method to consider heat effect, the governing integral equations are first derived. Then, the equations are analytically solved and displacement, stress, and temperature field components expressed by elementary functions are obtained. To validate the presented solutions, the correctness of the solutions is exactly examined, and normal contact displacement and stress without heat effect are compared with those in literatures. Furthermore, a numerical example which is the stainless steel thermo-elastic body pressed by a displacement restrictor is studied. Displacement, stress, and temperature distributions are analyzed, and the influence of heat flux on the pressed depth is also discussed in detail. The analytical solutions presented in this paper are rather important in engineering practice and can provide a basis for indentation analysis, displacement restrictor design, foundation settlement calculation, and so on.

A Design Tool for Solar Thermal Remediation Using Borehole Heat Exchangers.

Low temperature heating of the subsurface (increases of about 5-20 °C) can substantially increase rates of biotic and abiotic destruction of dissolved contaminants such as chlorinated solvents. Low-temperature heating can be sustainably and cost-effectively achieved using solar thermal collectors coupled with closed-loop borehole heat exchangers. This technology has been implemented at several sites in the United States and abroad with favorable results. The design of a solar thermal remediation system requires a quantitative understanding of heat transfer from the array of borehole heat exchangers. We present an easy-to-use design tool that is based on a transient three-dimensional analytical heat transfer solution and is programmed in Visual Basic in Excel. This tool can be used to quickly explore the effect of design variables on the forecast temperature field. Typical design variables would include the site-specific average and monthly solar insolation data, solar collector configuration, borehole heat exchanger geometry and spacing, and the effects of environmental variables such as groundwater velocity, background subsurface temperature, and thermal conductivity. The design tool has been verified by comparisons with the TOUGH2 multiphase heat transfer code for a three-dimensional multi-heater system with seasonally variable thermal power rates. The code has been validated by comparisons to observed temperatures measured at a solar thermal field remediation application at a site in Colorado.

A three-dimensional (3D) analytical solution for the ultimate side shear resistance of rock-socketed shafts

This paper presents a new analytical solution to the ultimate side shear resistance τsf of rock socketed shafts with consideration of three-dimensional (3D) strength and shaft sidewall roughness. Specifically, the shaft sidewall profile is simplified as triangular asperities and micro-mechanical analysis is performed on the shear behavior of one representative asperity. A 3D version of the Hoek-Brown (HB) criterion for rock mass and the equilibrium equations in a cylindrical coordinate system are first utilized to derive the governing equations. Then the governing equations are solved by using the method of characteristics. Finally, the τsf is obtained by considering the constant normal stiffness condition and the relative displacement of the rock mass due to sliding along the shaft-rock mass interface. Twenty-four (24) model and field test shafts with measured τsf are collected for analysis to verify the proposed solution. The results indicate that the predicted τsf values are in good agreement with the measured τsf values. Extensive parametric studies are also conducted to investigate the effects of rock mass properties, shaft dimensions, and sidewall roughness parameters on the ultimate side shear resistance factor αf (the ratio of τsf to the unconfined compressive strength of intact rock σc) of rock socketed shafts. The results indicate that the αf increases with rock constant mi, geological strength index (GSI) and asperity half chord length λ, but decreases with shaft diameter B, σc and disturbance D. Furthermore, the αf first increases and then decreases with the asperity inclination angle δ, resulting in an optimum asperity inclination angle δOp. Therefore, it is important to reduce the disturbance of rock mass from construction and select a proper asperity inclination angle when designing a rock socketed shaft.

Three-Dimensional Analytical Solutions for Acoustic Transverse Modes in a Cylindrical Duct with Axial Temperature Gradient and Non-Zero Mach Number

Cylindrical ducts with axial mean temperature gradient and mean flows are typical elements in rocket engines, can combustors, and afterburners. Accurate analytical solutions for the acoustic waves of the longitudinal and transverse modes within these ducts can significantly improve the performance of low order acoustic network models for analyses of acoustic behaviours and combustion instabilities in these kinds of ducts. Here, we derive an acoustic wave equation as a function of pressure perturbation based on the linearised Euler equations (LEEs), and the modified WKB approximation method is applied to derive analytical solutions based on very few assumptions. The eigenvalue system is built based on the proposed solutions and applied to predict the resonant frequencies and growth rate for transverse modes. Validations of the proposed solutions are performed by comparing them to the numerical results directly calculated from the LEEs. Good agreements are found between analytical reconstruction and numerical results of three-dimensional transverse modes. The system with both mean temperature profile and mean flow presents a larger absolute value of the growth rate than the condition of either uniform mean temperature or no mean flow.

Three-dimensional analytical solution for fluid-saturated transversely isotropic poroelastic multilayer formation

According to the potential theory and mirror image method, the three-dimensional (3D) Green's functions of a fluid-saturated transversely isotropic multilayer formation with different thicknesses were first obtained. The exact 3D explicit closed analytical forms of the pressure increment and stress fields were expressed by simple functions which can be used to accurately calculate the full-field distribution under a fluid source. The numerical results reveal that the method had good convergence, high accuracy, and high stability. Based on this method, the effects of the modulus ratio, stratified characteristic, medium thickness, and medium stacking sequence on the 3D hydraulic coupling response of the medium were examined. Numerical results indicated that these parameters affected the behavior of the formation in different degrees. The main contributions of this study are as follows: (1) the anisotropy and stratified characteristic of the formation material are simultaneously considered; (2) the Green's functions are given as a simple explicit function for fluid-saturated multilayer transversely isotropic formation; and (3) the Green's function is a potential function that can be calculated as a boundary element to improve the calculation accuracy.

Modeling Multiphase Heat and Mass Transfer in an Agitated Filter Dryer by Integrating Experiment, Computations, and Analytical Solutions.

The drying of a wet cake consisting of an active pharmaceutical ingredient and solvent in an agitated filter-dryer is a critical and challenging unit operation in the pharmaceutical industry. The complexity of this operation is attributed to the constraints on product quality in terms of its physical properties in addition to the residual solvent content. In this manuscript, a better understanding of the drying mechanism is gained by integrating insights from three-dimensional analytical solutions and computational fluid dynamics simulations into a zero-dimensional model to explain experimental data. The approach provides the time evolution of the mass flow rate of solvent from the wet cake and the center-point temperature of the cake with good accuracy. Further investigation of the zero-dimensional model reveals important parameters such as the mass transfer rate number that predicts whether the process is convection-controlled or diffusion-controlled, and the thermal load of vaporization that estimates the fraction of solvent vaporized per unit time. These parameters can be useful in devising a drying protocol for agitated-filter dryers.

Influence of permeability anisotropy of seepage flow on the tunnel face stability

This paper focuses on the influence of permeability anisotropy of seepage flow on the face stability for a shied tunnel. An analytical model has been proposed to present the hydraulic head distribution around the tunnel face in the anisotropic ground, considering the difference of permeability coefficient in the horizontal direction and the vertical direction. The rationality of the proposed model is verified by a series of numerical simulations. Then, an analytical model of face stability for a tunnel under the anisotropic seepage has been established based on the limit analysis upper bound method. Comparisons of the analytical solutions and the numerical simulations are conducted, and the limit support pressure of the two methods is consistent. The effect of permeability anisotropy and water pressure on the stability of the tunnel face is analyzed through the three-dimensional analytical solution. Anisotropy of permeability has a significant impact on the stability of the tunnel face, and its impact gradually decreases. It can also be found that the water pressure coefficient of the tunnel face has a significant effect on the limit support pressure and the failure area when the ratio of the horizontal permeability to the vertical permeability is large.

Three-Dimensional Analytical Solution of the Fractional Atmospheric Pollutant Dispersion Equation Considering Caputo and Conformable Derivatives

Three-Dimensional Analytical Solution of the Fractional Atmospheric Pollutant Dispersion Equation Considering Caputo and Conformable Derivatives

Three-dimensional analytical solution of the advection-diffusion equation for air pollution dispersion

We develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance. doi:https://doi.org/10.1017/S1446181122000037

Three-Dimensional Acoustic Analysis of a Rectangular Duct with Gradient Cross-Sections in High-Speed Trains: A Theoretical Derivation

Rectangular ducts used in the air-conditioning system of a high-speed train should be carefully designed to achieve optimal acoustic and flow performance. However, the theoretical analysis of the rectangular ducts with gradient cross-sections (RDGC) at frequencies higher than the one-dimensional cut-off frequency is rarely published. This paper has developed the three-dimensional analytical solutions to the wave equations of the expanding and shrinking RDGCs. Firstly, a homogeneous second-order variable coefficient differential equation is derived from the wave equations. Two coefficients of the solution to the differential equation are set to zero to ensure convergence. Secondly, the transfer matrices of the duct systems composed of multiple RDGCs are derived from the three-dimensional solutions. The transmission losses of the duct systems are then calculated from the transfer matrices and validated with the measurement. Finally, the acoustic performance and flow efficiency of the RDGCs with different geometries are discussed. The results show that the REC with double baffles distributed transversely has good performance in both acoustic attenuation and flow efficiency. This study shall provide a helpful guide for designing rectangular ducts used in high-speed trains.

Fate and transport modelling framework for assessing risks to soil and groundwater from chemicals accidentally released during surface operations: An Australian example application from shale gas developments

Shale and tight gas developments in the Beetaloo (28,000 km2) and Cooper (139,000 km2) basins of Australia are subject to stringent State and Federal Government controls and assessments. Several scientific investigations are ongoing to improve the scientific basis of the risks from unconventional gas developments to water and the environment. In this study a framework was developed to derive estimates of chemical dilution associated with leakage to groundwater from accidental release of chemicals used for shale and tight gas extraction in Australia. The quantitative assessment accounted for key landscape parameters that determine natural attenuation: soil type, depth to groundwater and groundwater velocity. Both basins were discretised into 1000 × 1000 m2 grids for which the unsaturated zone and groundwater dilution factors were derived. Migration of chemicals through deep unsaturated zones was calculated with the HYDRUS-1D simulator, taking account of best-available hydraulic properties from a digital soil database. A three-dimensional analytical solution of the advection–dispersion equation provided estimates of dilution in groundwater after solutes travelled 500 m from the centre (source location) to the edge of every grid cell. The combined vadose zone-groundwater dilution factors were used to determine under which conditions concentrations of hydraulic fracturing chemicals or flowback water accidentally released into the environment would decrease to levels that are no longer considered harmful to the environment. When the method was applied to 39 hydraulic fracturing chemicals scheduled for stimulation of a shale gas well, ecotoxicological risk quotients (RQ) were calculated to indicate which chemicals were of no environmental concern. This work contributes to increasing the efficiency of quantitative impact assessments and provides a framework to develop dedicated monitoring and management practices to support regulation and management of the gas industry in Australia.

Analysis of oblique wave interaction with perforated caisson breakwaters with partial wave absorption parts

This article presents a three-dimensional (3D) iterative analytical solution for water wave interactions with perforated caisson breakwaters with partial wave absorption parts based on linear potential flow theory. A periodic boundary condition is adopted for developing the series solutions of the velocity potentials by the separation of variables method. A quadratic pressure drop condition is imposed on the perforated walls to introduce wave energy dissipation. The unknown expansion coefficients in the velocity potentials are determined by matching the boundary conditions between adjacent fluid domains combined with iterative calculations. The hydrodynamic quantities, including the reflection coefficient, wave forces, and free surface elevation, are estimated. The convergences of the series solutions and the iterative calculation process are examined. The analytical solution calculations agree well with the known analytical and experimental results for special cases in the literature. Variations in the hydrodynamic quantities versus the main influencing factors are clarified. Partial wave absorption concept design enables the hydrodynamic performance of the present caisson breakwater with a smaller perforated wall area to be similar to that of a fully perforated caisson breakwater. To achieve a lower reflection coefficient and wave forces, the ratio of the wave chamber length to the caisson length is suggested to be 0.3–0.5, and the front wave chamber width should be less than half of the total wave chamber width.

Three-dimensional biomechanical modeling of cylindrical bone-like porous materials subject to acoustic waves

This paper describes a three-dimensional (3D) analytical solution for the acoustic response of cancellous bone-like porous materials saturated with a viscous fluid. The effect of dynamic tortuosity, especially in clinically relevant ultrasound frequency ranges, is considered to investigate the effect of viscous interaction between the fluid and solid phases. The solution includes the effects of both fast and slow longitudinal waves as well as transverse waves propagating through the medium. The scattering operators and radial displacements are derived in terms of ultrasonic waveforms by applying the Helmholtz decomposition. The effect of different porosities, wall thickness ratios, and frequencies of incident waves on the radial displacement and scattering operators are investigated by considering various incident wave angles at forward and sideward directions. The results demonstrate that the incident wave angle has a significant effect on the radial displacement and scattering operators regardless of the porosity, wall thickness ratio, and viscosity of pore fluid. Furthermore, the distribution pattern of the radial displacement and scattering operators in relatively low frequency ranges is almost symmetric while asymmetric in relatively high frequency ranges. It is shown that the bone characterization using ultrasonic techniques is not only based on the mineral density, as used currently by electromagnetic wave-based tools, but also other biomechanical factors such as the porosity, viscosity of pore fluid, and wall thickness ratio of a cancellous bone structure. Also, the pattern of the reflected pressure can be an indicator of the state of a cancellous bone (healthy versus osteoporosis).

Three-dimensional exact analytical solutions of transversely isotropic plate under heat sources

An accurate and efficient calculation method for the transversely isotropic thermoelastic plates plays an important role in exact design and quality control of the engineering structures. In this study, the three-dimensional (3D) thermal stress field of transversely isotropic plates under arbitrary heat source is obtained using Green’s function method. To achieve the purpose, the 3D Green’s function solutions of transversely isotropic plate under a point heat source are firstly derived based on the general solutions which are presented in potential functions. The concise harmonic potential functions are constructed in simple functions with undetermined constants by mirror image principle. Then, the Green’s functions are achieved by substituting the harmonic functions into the general solution. All temperature and stress components can be analytically expressed using simple functions which are convenient to use. The present method without simplifying assumption of dimensions has the applicability and reliability to analysis plate structures with arbitrarily thickness. Numerical calculations show the good convergence and high efficiency of the method. This study presents a high precision and efficiency predesign method for the thermos elastic plate structures. The numerical results provide theoretical guideline for the accurate design and quality control of plate structures.