Analytical Solutions For Stresses Research Articles

A coupled thermal-hydraulic-mechanical model for the kinetic dissociation of methane hydrate in a depressurizing well

Methane hydrate (MH) has significant potential to become a major resource of the future energy supply. Depressurization is one of the major methods of MH development used in MH production wells. The redistributions of pore pressure and hydrate saturation change the effective stress and mechanical properties of MH-bearing sediment, resulting in complicated thermal-hydraulic-mechanical behaviours. In this study, the transient multi-phase seepage and transient heat transfer behaviour around a wellbore were analysed. A multiannulus wellbore was obtained via the relations between the hydrate saturation and the mechanical parameters. The analytical stress and displacement solutions for a multi-annulus wellbore were deduced according to elastic mechanics. Then, a new thermal-hydraulic-mechanical coupling model was proposed to gain a better understanding of the kinetic dissociation behaviour of MH in a depressurizing well. The results showed that pore pressure diffusion transforms from a transient state to a relatively steady state in only a few minutes, while a transient state of heat diffusion exists for a relatively long time. The dissociating front moves farther at the beginning of dissociation and then shrinks back towards the wellbore wall. Thereafter, the dissociating front moves nearly linearly with the production time. Due to the flowability differences among the gas, water and MH, the area where the gas and water saturations change is larger than that where the MH saturation decreases. There exists an annulus zone surrounding the wellbore where the mechanical parameters are much lower than those of the zone far from the wellbore. The outer diameter of this annulus increases slowly with the production time. Stress transilience surrounding the wellbore is caused by the transilience of mechanical parameters after the decomposition of MH.

Adhesive behavior of transversely isotropic piezoelectric bimaterials

The adhesive contact behavior between two dissimilar transversely isotropic piezoelectric cylinders is investigated in this paper. The effect of adhesion on the contact behavior between piezoelectric bimaterials is considered within the framework of the non-slipping JKR model, which takes the coupling effect between the tangential and normal tractions inside the contact region into account. The explicit analytical solutions of stress and electric displacement fields within the contact region are obtained by solving the resulting coupling singular integral equations with the analytical function theory. The relationship between the contact size and external loadings are established by adopting the Griffith energy balance criterion at the edges of the contact zone. It is found that different types of piezoelectric bimaterials show the same adhesive contact behaviors, which is different from the case of a single layer piezoelectric material. The result obtained from the present paper should be helpful for the design of micro-electro-mechanical systems (MEMS) devices involving multi-layered smart structures.

Stress Distribution and Mechanical Behaviour of Rock Mass Containing Two Openings Underground: Analytical and Numerical Studies

In this study, stress solution for rock mass containing two rectangular openings was calculated based on the Schwarz alternating method to investigate the stress distribution in rock mass around openings with different layouts. In addition, large-scale numerical models were further established for the two-opening system by means of the PFC-FLAC coupling method, in which the stress evolution, failure patterns, and acoustic emission (AE) events were presented. With the combination of analytical and numerical solutions, the interaction mechanism between two openings under different layouts was discussed from the perspective of stress and failure. The result shows that the confining stress within a certain range contributes to relieving tensile stress concentration around openings. The stress condition within the connecting area and coalescence pattern between two adjacent openings is dominated by their layout. Compared with small-size rock specimens in laboratory tests, the failure patterns around openings show a better agreement with the stress concentration characteristics determined by analytical stress solutions.

Tensile Failure and Fracture Width of Partially Permeable Wellbores with Applications in Lost Circulation Material Design

SummaryEstimation of near-wellbore fracture widths remains central to designing the particle size distribution (PSD) and composition of lost circulation material (LCM) blends. Although elastic rock models are often used for this purpose, they fall short in capturing the substantial effect of pore fluid pressure on the fracture width. The problem is addressed in this paper by incorporating the poroelastic back stress in width estimation of axial fractures nearby an inclined wellbore. The poroelastic back stress is caused by a nonideal drilling fluid filter cake allowing for fluid pressure communication between the wellbore and pore space of the rock surrounding the wellbore. In this aspect, a proper definition of the filter-cake efficiency is made in terms of the wellbore pressure, far-field pore fluid pressure, and pore fluid pressure of the rock surrounding the wellbore. The value of this parameter is estimated from the standard drilling fluid filtration test results, as well as the formation rock permeability. The filter-cake efficiency is next used to develop the long-time, asymptotic analytical solution for the poroelastic stress of an inclined wellbore. By accounting for the obtained poroelastic back stress, an improved estimation of the wellbore tensile limit that depends on the filter-cake efficiency parameter is developed. For wellbore pressures beyond the wellbore tensile limit, the width of the near-wellbore fractures is estimated. The fracture width estimation is made through an analytical, dislocation-based fracture mechanics solution to the integral equation describing the nonlocal stress equilibrium along the fracture faces. The commonly practiced scheme for geometric design of LCM blends is enhanced by using the presented improvement in estimation of the near-wellbore fracture width. A case study is used to demonstrate the substantial effect of drilling fluid filtration properties and the resulting poroelastic back stress on the wellbore tensile limit, estimated fracture width, and consequently, composition of the recommended LCM blend.

Analytical Solution of Stress in a Transversely Isotropic Floor Rock Mass under Distributed Loading in an Arbitrary Direction

Rock masses with a distinct structure may present a transversely isotropic character; thus, the stress state in a transversely isotropic elastic half-plane surface is an important way to assess the behavior of the interaction between the distributed loading and the surroundings. Most previous theoretical analyses have considered a loading direction that is either vertical or horizontal, and the stress distribution that results from the effect of different loading directions remains unclear. In this paper, based on the transversely isotropic elastic half-plane surface theory, a stress solution that is applicable to distributed loading in any direction is proposed to further examine the loading effect. The consistency between the analytical solution and numerical simulations showed the effectiveness of the proposal that was introduced. Then, it was utilized to analyze the stress distribution rule by changing the Poisson’s ratio and Young’s modulus of the model. The effects of the formation dip angle on the stress state are also examined. The stress distribution, depending on the physical property parameters and relative angle, is predicted using an analytical solution, and the mechanisms associated with the transversely isotropic elastic half-plane surface subjected to the loading in any direction are clarified. Additionally, extensive analyses regarding this case study, with respect to the mechanical behavior associated with changes in the stress boundary, is available. Hence, the proposed analytical solution can more realistically account for the loading problem in many engineering practices.

Calculation Derivation and Test Verification of Indirect Tensile Strength of Asphalt Pavement Interlayers at Low Temperatures.

When the direct tensile test is adopted to determine the interlayer tensile strength of the asphalt pavements, specimen separation or internal cracking often occurs at the bonding area of the loading head, rather than at the interlaminar bonding interface. In view of the tedious and discrete data of the direct tensile test, this paper attempts to introduce an indirect tensile test to determine the interlayer bond strength of asphalt pavement to solve this problem. However, the indirect tensile test method of a binder lacks the corresponding mechanical theory. This paper deduces the calculation formula of the indirect tensile strength of a binder based on elastic theory. A mechanical model of the test was established with the finite element method. In accordance with the two-dimensional elastic theory and the Flamant solution, an analytical solution of tensile stress in the indirect tensile test is proposed through the stress superposition. On this basis, the calculation formula for the indirect tensile strength of the interlaminar bonding is derived according to Tresca’s law. A low-temperature indirect tensile test was designed and conducted to verify the correctness of the formula. By comparing the results of the indirect tensile test and direct tensile test, it is found that the interlaminar strength of the mixture measured by them is similar, and the dispersion of indirect tensile test results is small. The results show that the indirect tensile test can replace the direct tensile test to evaluate the interlaminar tensile strength.

A generalized 2D Bézier-based solution for stress analysis of notched epoxy resin plates reinforced with graphene nanoplatelets

In this study, a robust Bezier-based multi-step method is extended to accurately solve the governing fourth-order complex partial differential equation (PDE) in linear elastic fracture mechanics (LEFM) problems. The Bezier technique was first introduced by the authors to solve initial value problems in one dimension. Now, the method is further extended to simultaneously solve Boundary Value Problems (BVPs) in orthogonal directions. To examine the accuracy and performance of the present method, the first-mode normalized stress intensity factor (SIF) of a 2D epoxy resin plate having an initial edge crack and reinforced with randomly oriented graphene nanoplatelets (GnP) is determined and compared with the associated exact analytical solution using the Bayesian statistical analysis. Besides, the impact of GnP aspect ratio on the normalized crack opening displacement (COD) of the reinforced matrix is elaborated for the first time in the literature. Results of the present study suggest that GnPs with maximum aspect ratio are most effective to enhance elastic properties of the plate and potentially limit the edge crack propagation. Specifically, inclusion of 0.5 and 1.0% of needle-shaped GnPs in a notched epoxy resin plate decrease the maximum normalized COD by 33 and 50%, respectively, while for square-shaped GnPs, these reductions are limited to 20 and 37%, respectively.

Analytical Stress Solution and Numerical Mechanical Behavior of Rock Mass Containing an Opening under Different Confining Stress Conditions

In this study, the triangle interpolation method for the calculation of mapping functions of plates containing an opening with arbitrary shapes is investigated with an improved method for point adjudgment during iterations. Afterwards, four kinds of openings with typical shapes are considered and the mapping functions for them are calculated, based on which the influence of calculation parameters such as iteration time and the number of terms on the accuracy of mapping function is discussed. Finally, the stress around an inverted U-shaped opening and around an arched opening under different far-field stress conditions is calculated and the effect of opening shape and lateral pressure coefficient on stress distribution and rock mechanical behaviors is further analyzed combined with the discrete element method (DEM) numerical simulation. The result shows that the stability and failure pattern of the rock mass is correlated with the stress around the opening, which is affected by the opening shape. The existence of opening also greatly reduces the enhancing influence of confining stress on rock specimen.

Numerical and experimental modeling of geomechanical behavior of partially saturated soils

The Barcelona Basic Model (BBM) has been implemented in a finite difference-based computer program to simulate the behavior of unsaturated soils subjected to wetting. The BBM implementation was verified using analytical solutions, and the proposed model has been used to simulate the response of a compacted embankment under complete saturation and suction induced conditions. Numerical analyses indicate that considerable amount of total and differential settlements could develop at the top surface of the embankment. BBM is executed into FLAC2D extending a defined module for modified Cam Clay (MCC) and has been set up an analytical solution for suction-dependent stress and strain. Evaluating the effect of anisotropy and nonlinear apparent tensile strength in unsaturated soils, a modification to BBM formulation has been proposed and optimized by developing numerical analyses to reduce the size of elastic region of loading collapse (LC) curve. Then, an experimental study in the literature is investigated by utilizing comparative curves from BBM and modified BBM indicating well agreement with natural circumstances. As a result of the work presented in this research, finite difference codes with BBM and modified BBM has the capability of simulating the real behavior and is operational being applied to problems associated with earthen structures in unsaturated or partially saturated of expansive soils as a three-phases medium.

Analytical stress solution for rock mass containing two holes based on an improved Schwarz alternating method

In this study, the accuracy and generality of stress calculation for a plate containing double holes via Schwarz alternating method was investigated. The original Schwarz alternating method with inverse mapping function shows a good performance in terms of the stress solution for a plate containing two arched holes. However, for a plate containing two holes with corners, evitable error may be produced during the calculation because of the unsatisfying accuracy of inverse mapping function. This error is particularly obvious in the stress calculation for a plate containing two triangular holes. To address the problem, an improved Schwarz alternating method without the consideration of inverse mapping function was proposed in this study. The coordinate transformation of points around hole boundaries is realized by an optimization method instead of the inverse mapping function, which leads to little error during the calculation and thus improves the stress solution accuracy especially for plates containing holes with corners. With the employment of the improved method, the influences of connecting angle and lateral pressure coefficient on stress solutions for plates containing double U-shaped holes were discussed. In addition, DEM simulations were further conducted to study the influence of stress characteristics on the failure patterns of rock mass. The result shows that the hoop stress linearly relates to lateral pressure coefficient in the elastic plate and fractures always initiate around hole boundaries where high stress concentrates.

Asymptotically analytical solution of elastic stress for convex polygonal holes in an infinite plane under various loading conditions

Asymptotically analytical solution of elastic stress for convex polygonal holes in an infinite plane under various loading conditions

Nonlocal thermo-hydro-mechanical (THM) coupling dynamic response of saturated porous thermoelastic media with temperature-dependent physical properties

The coupled thermal-hydro-mechanical (THM) analysis of saturated porous media is of great significance and has been widely used in various engineering fields. However, the classical Biot theory assumes that the physical parameters of porous media are constant and ignores the influence of pore size on wave propagation characteristics, which cannot reflect the actual situation. Based on the nonlocal theory and Biot wave equation, a nonlocal coupled Biot THM dynamic model for porous media with variable physical properties is established by introducing the temperature-dependent factor function. Utilizing the nonlocal coupled Biot THM theory, the dynamic response of forced convection in a half-space saturated porous media under surface harmonic thermal action is investigated in this paper. The analytical solutions of temperature increment, displacement, stress, and pore water pressure of saturated porous media are obtained by means of differential operator decomposition. In addition, the effects of the temperature-dependent factor and porosity on the nonlocal model at different frequencies, and the effects of the nonlocal parameter on the distribution rules of various physical fields at different frequencies are explored in depth.

Extracting true stresses and strains from nominal stresses and strains in tensile testing

AbstractSeemingly a simple task, the extraction of the flow curve (true stress vs. true plastic strain) from nominal stresses and strains in standard tensile testing still has its unsolved points. This study addresses two of them: (i) in materials without yield point phenomenon (or generally in the region of homogeneous plastic deformation), the true stress is typically calculated assuming constant volume, ignoring the elastic volume changes. Here, we derive a set of exact analytical solutions for true stresses and strains with remarkable simplicity and beauty that fully account for the elastic volume changes. This set of exact solutions is cross‐checked by finite element simulations as well as zeroth‐ and first‐order approximations; perfect agreement has been found. (ii) In materials with a pronounced yield point phenomenon, a complicated three‐dimensional stress state inevitably arises at the edge of the Lüders bands, which masks the real (or inherent) material behaviour. To determine the real material behaviour in the Lüders region, here we use a new macroscopic analytical approach characterised by a high true upper yield point, a typical strain hardening behaviour common for many materials, and the triaxiality of the stress state that inevitably develops at the edges of the Lüders bands and that determines the stress level at the observed lower yield point. This approach is verified by experiments (including video observations as well as digital image correlation (DIC) strain distribution measurements) and finite element simulations with very good agreement.

Two-Dimensional Stress Analysis of Thick Hollow Functionally Graded Sphere Under Non-Axisymmetric Mechanical Loading

In this paper, a functionally graded thick hollow sphere is considered for the analysis of two-dimensional steady state mechanical stress in the radial and circumferential directions under mechanical loading. Modulus of elasticity is varying with continuous nonlinear variation along the radial direction and Poisson’s ratio is kept as constant. The Legendre series and Euler differential equation are used to solve Navier equations. Geometry of the sphere is assumed in spherical coordinate system. Applying mechanical boundary conditions at inner and outer radii, we have carried out the analytical solutions for stresses, strains and displacements. In the numerical example, only internal pressure is varying along circumferential direction and external pressure is kept as zero. Displacements and mechanical stresses are presented graphically and the results are discussed numerically.

Influence of the thermal insulation layer on radial stress and collapse resistance of subsea wet insulation pipe

The analytical solutions of stress and displacement at the bond between two adjacent layers of composite pipe (including wet insulation pipe) and a new design equation for the external pressure of wet insulation pipe are proposed in this paper. The stress distribution and bearing capacity of wet insulation pipe have been exploded. A series of 3D models are developed in commercial finite element software to discuss the influence factors of radial stress and compare numerical results with analytical solutions. Results showed that the distribution of radial stress and displacement is influenced by wall thickness, modulus of elasticity and Poisson's ratio of each layer. When other conditions remain unchanged, the radial stress at the bond between two adjacent layers increases with the increase of internal pressure and outer wall thickness; the radial stress decreases with the increase of inner wall thickness, modulus of elasticity and Poisson's ratio of outer layer. Under external pressure, the radial stress will first increases and then decreases along radial direction from outside to inside, and it will reach the maximum at the bond between insulation layer and steel pipe. This phenomenon will increase pressure loaded on the outer surface of steel pipe, which makes wet insulation pipe more prone to collapse. Consequently, in some cases, for the same size of steel pipe, the collapse pressure of wet insulation pipe is smaller than that of internal steel pipe.

A mechanical bond model for reinforcing bar in concrete subjected to monotonic and reversed cyclic loading

The reinforcing bar with diameters of 10 mm and 12 mm are widely used in shear walls, whose bond-slip performance has great influence on the seismic performance of the shear walls. Besides, the bond stress-slip model plays an important role in the theoretical analysis of the seismic performance of shear walls. Therefore, an analytical solution for the maximum bond stress was carried out and modified based on the thick-walled cylinder model and the tensile stress-strain model of concrete. The bond-slip tests of reinforcing bars in concrete were conducted on 48 specimens under monotonic and reversed cyclic loading. The effects of concrete strength, diameter of bar and bonding length on the bond-slip performance were examined and analyzed. The test results showed both the maximum bond stress and residual bond stress increased with increasing concrete strength, but decreased with increasing diameter of bar or bonding length. The reversed cyclic loading can cause the degeneration of the maximum bond stress and the bond stiffness. According to the bond-slip performance, the bond stress-slip model was established based on the analytical solution for the maximum bond stress, which is suitable for calculating both monotonic and reversed cyclic loading. A good agreement was observed by comparing the proposed models with the test curves in the study.

Study on the Seepage Force-Induced Stress and Poroelastic Stress by Flow Through Porous Media Around a Vertical Wellbore

Fluid flowing through reservoir pores not only generates poroelastic stress but also exerts seepage force on rock skeleton. However, the mechanism of seepage force is not clear. Traditional methods of analyzing wellbore stability and hydraulic fracture initiation are mainly focused on the poroelastic stress without the effects of seepage force. Based on the linear elasticity and consolidation theory, this paper analyzed the mechanism of seepage force and poroelastic stress, and presented an analytical solution for seepage force-induced stress around a vertical wellbore. It also introduced how to calculate poroelastic stress by exerting hypothetical body force and surface force. Through comparison and superposition of stress fields, this paper studied the change characteristics of the poroelastic and seepage force-induced stress under different borehole pressures and the effects of seepage force on the wellbore tensile failure. Numerical simulation results show that when fluid flows through the rock, using traditional models without considering, the effect of seepage force to calculate the borehole pressure-induced stress will result in lower calculation results. Compared with the traditional model, seepage force-induced circumferential tensile stress is larger, and the seepage force significantly reduces the formation breakdown pressure. Rocks near the borehole wall with lower permeability and larger Poisson’s ratio have a greater action of seepage force. When fluid flows through the reservoir, the effects of seepage forces cannot be ignored in the analysis of hydraulic fracturing and wellbore stability.

Analysis of stress transfer in short fiber-reinforced composites with a partial damage interface by a shear-lag model

The stress transfer mechanism in fiber-reinforced composites with a partial damage interface was investigated by a newly developed shear-lag model with a cohesive interface. The analytical solutions of the interfacial shear stress and the normal stress in the fiber at a damage stage were obtained. Analytical expressions of the critical stress corresponding to the initiation of damage or debonding of the interface were derived. The analytical solutions agreed well with the numerical results and existing experimental results.

Numerical modeling of thermo‐mechanical failure processes in granitic rock with polygonal finite elements

AbstractThis paper considers numerical modeling of intensive heating induced thermo‐mechanical failure processes in granitic rock. For this end, a numerical method based on polygonal finite elements and a damage‐plasticity model is developed. A staggered scheme is employed to solve the global thermo‐mechanical problem. The rock failure is described by a Rankine‐Mohr‐Coulomb plasticity model with separate scalar damage variables for tension and compression. Consistent tangent operator is derived for this model. Special attention is given to the temperature dependence of the thermo‐mechanical material properties of heterogeneous rock. In the numerical examples, the method is first verified with an analytical solution of thermal stresses in a hollow cylinder, and then qualitatively validated with the problems of thermal cracking of concentric cylinders and uniaxial compressive test on rock under elevated temperatures. Finally, the method is applied in novel simulations inspired by the degradation of sauna stones under slow heating‐rapid cooling and the comminution by rapid heating‐cooling cycles.

Analytical solution of stress in functionally graded cylindrical/spherical pressure vessel

For the exact solution of the stress in the functionally graded (FG) cylindrical/spherical pressure vessel, this paper presents a unified form of the basic equations of the FG hollow cylinder/spherical shell by introducing parameter, and then, the axial/spherical symmetry mechanical problems of FG hollow shells are studied under three different boundary conditions, respectively. Assume that the Young's modulus of the material changes with thickness distribution of cylinder/shell in the form of power functions and the value of Poisson’s ratio is constant, the analytical solutions of the displacement and stress of the FG hollow cylindrical/spherical pressure vessel are derived under different boundary conditions. By comparing the analytical solutions of the FG hollow cylinder/spherical shell obtained in this paper with the existing classical theoretical solutions and numerical solutions, the correctness of the analytical solution given in this paper is verified. In the numerical discussion, the distributions of displacement, radial stress and circumferential stress of the FG hollow cylinder/spherical shell under different gradient parameters and different size conditions are given, respectively. Finally, based on the difference numerical method, the stress distribution of FG hollow cylinder/spherical shell with the Young's modulus and Poisson's ratio along thickness distribution in the form of power functions is analyzed numerically. The results show that gradient parameter and geometric size have a great influence on the mechanical response of FG structure under different boundary conditions.