Semi-analytical solution for mechanical analysis of tunnels crossing strike-slip fault zone considering nonuniform fault displacement and uncertain fault plane position

The tunnel subjected to strike-slip fault dislocation exhibits severe and catastrophic damage.The existing analysis models frequently assume uniform fault displacement and fixed fault plane position.In contrast,post-earthquake observations indicate that the displacement near the fault zone is typically nonuniform,and the fault plane position is uncertain.In this study,we first established a series of improved governing equations to analyze the mechanical response of tunnels under strike-slip fault dislocation.The proposed methodology incorporated key factors such as nonuniform fault displacement and uncertain fault plane position into the governing equations,thereby significantly enhancing the applicability range and accuracy of the model.In contrast to previous analytical models,the maximum computational error has decreased from 57.1%to 1.1%.Subsequently,we conducted a rigorous validation of the proposed methodology by undertaking a comparative analysis with a 3D finite element numerical model,and the results from both approaches exhibited a high degree of qualitative and quantitative agreement with a maximum error of 9.9%.Finally,the proposed methodology was utilized to perform a parametric analysis to explore the effects of various parameters,such as fault displacement,fault zone width,fault zone strength,the ratio of maximum fault displacement of the hanging wall to the footwall,and fault plane position,on the response of tunnels subjected to strike-slip fault dislocation.The findings indicate a progressive increase in the peak internal forces of the tunnel with the rise in fault displacement and fault zone strength.Conversely,an augmentation in fault zone width is found to contribute to a decrease in the peak internal forces.For example,for a fault zone width of 10 m,the peak values of bending moment,shear force,and axial force are approximately 46.9%,102.4%,and 28.7% higher,respectively,compared to those observed for a fault zone width of 50 m.Furthermore,the position of the peak internal forces is influenced by variations in the ratio of maximum fault displacement of the hanging wall to footwall and the fault plane location,while the peak values of shear force and axial force always align with the fault plane.The maximum peak internal forces are observed when the footwall exclusively bears the entirety of the fault displacement,corresponding to a ratio of 0:1.The peak values of bending moment,shear force,and axial force for the ratio of 0:1 amount to approximately 123.8%,148.6%,and 111.1% of those for the ratio of 0.5:0.5,respectively.

Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary

The semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation for unsaturated soils with a semi-permeable drainage boundary are pre- seated. Two variables are introduced to transform the two coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations （PDFs）, which are easily solved by the Laplace transform method. Then, the pore-water pressure, pore-air pressure, and soil settlement are obtained in the Laplace domain. The Crump method is adopted to perform the inverse Laplace transform in order to obtain the semi-analytical solutions in the time domain. It is shown that the proposed solutions are more applicable to various types of boundary conditions and agree well with the existing solutions from the literature. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with single, double, mixed, and semi-permeable drainage boundaries. The changes in the pore-air and pore-water pres- sures and the soil settlement with the time factor at different values of the semi-permeable drainage boundary parameters are illustrated. In addition, parametric studies are con- ducted on the pore-air and pore-water pressures at different ratios （the air permeability coefficient to the water permeability coefficient） and depths.

General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

This paper presents general semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and poreair pressures into an equivalent set of partial differential equations （PDEs）, which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domMn. The Crump＇s method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.

Hamiltonian parametric element and semi-analytical solution for smart laminated plates

Based on the Hellinger-Reissner （H-R） mixed variational principle for piezoelectric material, a unified 4-node Hamiltonian isoparametric element of anisotropy piezoelectric material is established. A new semi-analytical solution for the natural vibration of smart laminated plates and the transient response of the laminated cantilever with piezoelectric patch is presented. The major steps of mathematical model are as follows： the piezoelectric layer and host layer of laminated plate are considered as unattached three-dimensional bodies and discretized by the Hamiltonian isoparametric elements. The control equation of whole structure is derived by considering the compatibility of generalized displacements and generalized stresses on the interface between layers. There is no restriction for the side-face geometrical boundaries, the thickness and the number of layers of plate by the use of the present isoparametric element. Present method has wide application area.

Consolidation of partially saturated ground improved by impervious column inclusion:Governing equations and semi-analytical solutions

This study focuses on the consolidation behavior and mathematical interpretation of partially-saturated ground improved by impervious column inclusion.The constitutive relations for soil skeleton,pore air and pore water for partially saturated soils are proposed in the context of partially-saturated ground improved by impervious column inclusion.Settlement equation and dissipation equations of excess pore air/water pressures for a partially saturated improved ground are then derived.The semi-analytical solutions for ground settlement and pore pressure dissipation are then obtained through the Laplace transform and validated by the existing solutions for two special cases in the literature and the numerical results obtained from the finite difference method.A series of parametric studies is finally conducted to investigate the influence of some key factors on consolidation of partially saturated ground improved by impervious column inclusion.Based on the parametric study,it can be found that a higher value of the area replacement ratio or modulus of the pile results in a longer dissipation time of excess pore air pressure(PAP),a shorter dissipation time of excess pore water pressure(PWP),and a lower normalized settlement.

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

In this paper, a series of semi-analytical solutions to one-dimensional consolidation in unsaturated soils are obtained. The air governing equation by Fredlund for unsaturated soils consolidation is simplified. By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified governing equations of water and air, Darcy＇s law, and Fick＇s law, the transfer function between the state vectors at top and at any depth is then constructed. Finally, by the boundary conditions, the excess pore-water pressure, the excess pore-air pressure, and the soil settlement are obtained under several kinds of boundary conditions with the large-area uniform instantaneous loading. By the Crump method, the inverse Laplace transform is performed, and the semi-analytical solutions to the excess pore-water pressure, the excess pore-air pressure, and the soils settlement are obtained in the time domain. In the case of one surface which is permeable to air and water, comparisons between the semi-analytical solutions and the analytical solutions indicate that the semi-analytical solutions are correct. In the case of one surface which is permeable to air but impermeable to water, comparisons between the semi-analytical solutions and the results of the finite difference method are made, indicating that the semi-analytical solution is also correct.

A semi-analytical solution for electric double layers near an elliptical cylinder

A theoretical analysis on the electric double layer formed near the surface of an infinite cylinder with an elliptical cross section and a prescribed electric potential in an ionic conductor was performed using the linearized Gouy–Chapman theory. A semi-analytical solution in terms of the Mathieu functions was obtained. The distributions of the electric potential, cations, anions, and electric field were calculated. The effects of various physical and geometric parameters were examined. The fields vary rapidly near the elliptical boundary and are nearly uniform at far field. Electric field concentrations were found at the ends of the semi-major and semi-minor axes of the ellipse. These concentrations are sensitive to the physical and geometric parameters.

A semi-analytical solution for frost heave prediction of clay soil

Frost heave is one of the main freezing problems for construction in permafrost regions.The Konrad-Morgenstern segregation potential（SP） model is being used in practice for frost heave using numerical techniques.However,the heat release from in-situ and migrated water in the freezing zone could result in some numerical instability,so the simulation of frost fringe is not ideal.In this study,a semi-analytical solution is developed for frost heave prediction of clay soil.The prediction results to the two tests with different freezing mode with clay soil agree well with the tested behavior,which indicates the feasibility of the solution.

Semi-analytical solution for internal forces of tunnel lining with multiple longitudinal cracks

Longitudinal cracks on the tunnel lining significantly influence the performance of tunnels in operation.In this study,we propose a semi-analytical method that provides a simple and effective way to calculate the internal forces of tunnel linings with multiple cracks.The semi-analytical solution is obtained using structural analysis considering the flexural rigidity for the cracked longitudinal section of the tunnel lining.Then the proposed solution is verified numerically.Using the proposed method,the influences of the crack depth and the number of cracks on the bending moment and modified crack tip stress are investigated.With the increase in crack depth,the bending moment of lining scetion adjacent to the crack decreases,while the bending moment of lining scetion far away from the crack increases slightly.The more the number of cracks in a tunnel lining,the easier the new cracks initiated.

Undrained semi-analytical solution for cylindrical cavity expansion in anisotropic soils under biaxial stress conditions

This paper presents an undrained semi-analytical elastoplastic solution for cylindrical cavity expansion in anisotropic soil under the biaxial stress conditions.The advanced simplified SANICLAY model is used to simulate the elastoplastic behavior of soil.The cavity expansion is treated as an initial value problem and solved as a system of eight first-order ordinary differential equations including four stress components and four anisotropic parameters.The results are validated by comparing the new solutions with existing ones.The distributions of stress components and anisotropic parameters around the cavity wall,the expansion process,the stress yield trajectory of a soil element and the shape and size of elastoplastic boundary are further investigated to explore the cavity expansion response of soils under biaxial in situ stresses.The results of extensive parameters analysis demonstrate that the circumferential position of the soil element and the anisotropy of the soils have noticeable impacts on the expansion response under biaxial in situ stresses.Since the present solution not only considers the anisotropy and anisotropy evolution of natural soil,but also eliminates the conventional assumption of uniform radial pressure,the solution is better than other theoretical solutions to explain the pressure test and pile installation effect of shallow saturated soil.

Semi-analytical Solutions for Wellbores with Graded Skin Zones in Poroelastic Media

In analyzing the complex interaction between the wellbore and the reservoir formation,the hydromechanical properties of the region proximal to the wellbore,referred to as the“wellbore skin zone”,play a pivotal role in determining flow dynamics and the resulting formation deformation.Existing models of the wellbore skin zone generally assume a constant permeability throughout,resulting in a sharp permeability discontinuity at the skin-reservoir interface.This paper introduces a model for a wellbore with a continuously graded skin zone of finite thickness within a poroelastic medium.Analytical solutions are derived using the Laplace transform method,addressing both positive and negative skin zones.Numerical results are presented to illustrate the effects of graded permeability/skin zone thickness on pore pressures and stresses around a wellbore.The results highlight a distinct divergence in stress and pore pressure fields when comparing wellbores with negative skin zones to those with positive skin zones or no skin at all.

A Semi-Analytical Potential Solution for Wave Resonance in Gap Between Floating Box and Vertical Wall

Based on potential flow theory, a dissipative semi-analytical solution is developed for the wave resonance in the narrow gap between a fixed floating box and a vertical wall by using velocity potential decompositions and matched eigenfunction expansions. The energy dissipation near the box is modelled in the potential flow solution by introducing a quadratic pressure loss condition on the gap entrance. Such a treatment is inspired by the classical local head loss formula for the sudden change of cross section in channel flow, where the energy dissipation is assumed to be proportional to the square of local velocity for high Reynolds number flows. The dimensionless energy loss coefficient is calibrated based on experimental data. And it is found to be insensitive to the incident wave height and wave frequency. With the calibrated energy loss coefficient, the resonant wave height in gap and the reflection coefficient are calculated by the present dissipative semi-analytical solution. The predictions are in good agreement with experimental data. Case studies suggest that the maximum relative energy dissipation occurs near the resonant frequency, which leads to the minimum reflection coefficient. The horizontal wave forces on the box and the vertical wall attain also maximum values near the resonant frequency, while the vertical wave force on the box decreases abruptly there to a small value.

Kangfuxin solution alleviates esophageal stenosis after endoscopic submucosal dissection:A natural ingredient strategy

BACKGROUND Esophageal stricture ranks among the most significant complications following endoscopic submucosal dissection(ESD).Excessive fibrotic repair is a typical pathological feature leading to stenosis after ESD.AIM To examine the effectiveness and underlying mechanism of Kangfuxin solution(KFX)in mitigating excessive fibrotic repair of the esophagus post-ESD.METHODS Pigs received KFX at 0.74 mL/kg/d for 21 days after esophageal full circumferential ESD.Endoscopic examinations occurred on days 7 and 21 post-ESD.In vitro,recombinant transforming growth factor(TGF)-β1(5 ng/mL)induced a fibrotic microenvironment in primary esophageal fibroblasts(pEsF).After 24 hours of KFX treatment(at 1.5%,1%,and 0.5%),expression ofα-smooth muscle actin-2(ACTA2),fibronectin(FN),and type collagen I was assessed.Profibrotic signaling was analyzed,including TGF-β1,Smad2/3,and phosphor-smad2/3(p-Smad2/3).RESULTS Compared to the Control group,the groups treated with KFX and prednisolone exhibited reduced esophageal stenosis,lower weight loss rates,and improved food tolerance 21 d after ESD.After treatment,Masson staining revealed thinner and less dense collagen fibers in the submucosal layer.Additionally,the expression of fibrotic effector molecules was notably inhibited.Mechanistically,KFX downregulated the transduction levels of fibrotic functional molecules such as TGF-β1,Smad2/3,and p-Smad2/3.In vitro,pEsF exposed to TGF-β1-induced fibrotic microenvironment displayed increased fibrotic activity,which was reversed by KFX treatment,leading to reduced activation of ACTA2,FN,and collagen I.The 1.5%KFX treatment group showed decreased expression of p-Smad 2/3 in TGF-β1-activated pEsF.CONCLUSION KFX showed promise as a therapeutic option for post-full circumferential esophageal ESD strictures,potentially by suppressing fibroblast fibrotic activity through modulation of the TGF-β1/Smads signaling pathway.

Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading

Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load,which can be expanded in terms of sinusoidal series.For plane stress problems,the stress function is assumed to consist of two parts,one being a product of a trigonometric function of the longitudinal coordinate(x) and an undetermined function of the thickness coordinate(y),and the other a linear polynomial of x with unknown coefficients depending on y.The governing equations satisfied by these y-dependent functions are derived.The expressions for stresses,resultant forces and displacements are then deduced,with integral constants determinable from the boundary conditions.While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness,the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness.The present analysis is applicable to beams with various boundary conditions at the two ends.Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.

Semi-analytical solution for fully developed forced convection in metal-foam filled tube with uniform wall temperature

Fully developed flow and heat transfer in metal-foam filled tube with uniform wall temperature(UWT) is semi-analytically investigated based on the Brinkman–Darcy model and the two-equation model, in which the inertia term, axial conduction, and thermal dispersion are ignored. A two-dimensional numerical simulation that adopts the full governing equations is also conducted to analyze the effects of neglected terms on flow and thermal transport performance by comparing with the semi-analytical solution. The effects of the relevant parameters and thermal boundary conditions including UWT and uniform heat flux(UHF) on the heat transfer characteristics are discussed based on the semi-analytical solution. The results show that the inertia term has a significant effect on the prediction of pressure drop, but has a relatively mild effect on Nusselt number. The axial conduction has significant effect on the Nusselt number at lower Reynolds number, and the effects of thermal dispersion can be neglected when the thermal conductivity ratio between fluid and solid is remarkably smaller for air/metal foam as example(kf/ks<3×10-3). The predicted Nusselt number of the semi-analytical solution is about 8% to 15% lower than that of the numerical solution with full model in the range of 4×10-5<kf/ks<3×10-3. Moreover, the temperature profile of solid is more sensitive to pore density and porosity than that of fluid under UWT condition. The Nusselt number under UWT is about 7% to 25% lower than that under UHF, and the difference is mainly determined by interfacial convection rather than solid conduction.

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff(CBS)equation.This model describes the(2+1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave.The extended simplest equation(ESE)method is applied to the model,and a variety of novel solitarywave solutions is given.These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma.The accuracy of the obtained solution is verified using a variational iteration(VI)semi-analytical scheme.The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained.The adopted scheme’s performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.

A Semi-Analytical Method for Solutions of a Certain Class of Second Order Ordinary Differential Equations

This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.

A Semi-Analytical Solution to Classic Yang–Mills Equations with Both Asymptotical Freedom and Confining Features

It is well known that confinings and asymptotic freedom are properties of quantum chromo-dynamics （QCD）. But hints of these features can also be observed at purely classic levels. For this purpose we need to find solutions to the colorly-sourceful Yang-Mills equations with both confining and asymptotic freedom features. We provide such a solution in this paper which at the near-source region is of serial form, while at the far-away region is approximately expressed through simple elementary functions. From the solution, we derive out a classically non-perturbative beta function describing the running of effective coupling constant, which is linear in the couplings both in the infrared and ultraviolet region.

Solution approximations for a mathematical model of relativistic electrons with beta derivative

The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative.For this purpose,both the modified extended tanh-function(mETF)method and the homotopy analysis method(HAM)are used.While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution.The advantage of this procedure is that discretization or normalization is not required.By applying the mETF,the exact solutions are obtained.Also,by applying the HAM semi-analytical results for the considered equation are acquired.In HAM?curve gives us a chance to find the suitable value of the for the convergence of the solution series.Also,comparative graphical representations are given to show the effectiveness,reliability of the methods.The results show that the m ETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative.This study can bring a new perspective for studies on fractional differential equations.On the other hand,it can be said that scientists can apply the considered methods for different mathematical models arising in physics,chemistry,engineering,social sciences and etc.which involves fractional differentiation.Briefly the results may cause a new insight who studies on relativistic electron modelling.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.