Analytical Solutions For Stresses Research Articles

Fracture mechanism and energy evolution of sandstone with a circular inclusion

Understanding the stress distribution, deformation and fracture behavior around a circular hole or inclusion is critical for the stability assessment of rock structures such as tunnels and backfilled openings. In this study, the analytical solutions of stress and displacement distribution around a circular inclusion of different elastic properties were first derived based on the theory of elasticity. Then prismatic sandstones with a circular hole in different filling modes were prepared and tested in uniaxial compression tests . Progressive deformation and fracturing behavior around holes were analyzed by digital image correlation technique . Moreover, the energy evolution characteristics of specimens were studied to reveal its deformation and failure mechanism. The results show that after filling, the radial and tangential stress and displacement of the inclusion increase, while the stress and displacement around the surrounding rock of the hole decreases, thereby improving the strength and deformation properties . In addition, the crack types and initiation positions around the unfilled and filled holes are quite different due to the stress redistribution. Based on the energy evolution characteristics and the major principal strain fields, the stress-strain behavior is divided into five stages. It is found that the inclusion not only has passive supporting and energy absorption effects on the surrounding rock, but also has a similar crack arrest effect on the initiation and propagation of new cracks. Finally, the effects of inclusion elastic properties and lateral stress coefficient on the stress concentration characteristics around the filled hole were discussed.

Analytical solution for predicting the interaction stress of axially loaded concrete-filled double-tube columns

Concrete-filled double-tube (CFDT) members are promising steel-concrete composite structural forms. Existing studies mainly focus on members’ load-deformation performance. The interaction stress between components, which is a fundamental cause for change in load-deformation performance, receives much less attention. This study presents an analytical solution for interaction stress between components of axially loaded CFDT columns. The stress–strain relationships and force equilibrium equations of the inner concrete, inner steel tube, sandwiched concrete, and outer steel tube were analysed using elasticity method first. Subsequently, the deformation compatibility equations were established concerning the interactions between the components. Based on the above analysis, an analytical solution was derived. Equivalent lateral deformation coefficients were used to substitute Poisson’s ratio of concrete to account for the nonlinear behaviour of materials during loading. The proposed model shows reasonable agreement with experimental results. Finite element analysis further shows the accuracy of the interaction stress model till an axial compressive strain of 0.01 of the composite column. The derived interaction stress model was successfully applied in predicting the load-bearing capacities of CFDT columns through validation of 39 specimens from five independent studies. The average ratio of the predicted-to-experimental bearing capacities was 1.054, with a coefficient of variation of 0.077. The accuracy of models in Eurocode 4, ACI 318-11, and GB 50936 were also evaluated. Thereby the proposed model provides insights into the interaction analysis between components and mechanism-based load-bearing capacity prediction of CFDT columns.

Generalized Thermoelastic Interaction in a Half-Space under a Nonlocal Thermoelastic Model

In the current article, the nonlocal thermoelastic theory is used to discuss the wave propagation in unbounded thermoelastic materials. Due to the inclusion of relaxation time in thermal conduction formulation and the equations of motion, this model was developed using Lord and Shulman’s generalized thermoelastic model. The theory of the nonlocal continuum proposed by Eringen is used to obtain this model. The integral transforms of the Laplace transform methods used to generate an analytical solution for physical variables are utilized to produce the analytical solutions for the thermal stress, displacement, and temperature distribution. The effects of nonlocal parameters and relaxation time on the wave propagation distributions of physical fields for material are visually shown and explored.

Efficient Analytical Model for Time-Dependent Behavior of Layered Functionally Graded Plates with Viscoelastic Interlayers

Layered functionally graded (FG) plate plays an important role in engineering constructions. In this work, an efficient analytical solution is proposed to investigate the time-dependent characters of layered FG plate with viscoelastic interlayers, in which each FG layer is described by the elasticity theory, and the interlayer viscoelasticity is simulated by the generalized Maxwell model. The constitutive equations in the interlayer are simplified, and then, the analytical solutions of stresses and displacements for the layered FG plate are solved by virtue of the efficient recursive matrix method. Some examples are analyzed to investigate the influences of geometric and material characteristics on the long-term behaviors for the FG plate.

Pullout Characteristics and Damage Softening Model of the Geogrid-Soil Interface

The mechanical properties of geogrid-soil interface is very important in design and stability analysis of reinforced soil structure. In order to study the complicated mechanism of geogrid-soil interface, a series of pullout tests of HDPE uniaxial tensile geogrids with the different transverse ribs spacing is used to investigate the interaction characteristics in the laboratory. The test results show that pullout force and displacement curves are characterized as strain softening; compared with the no-reinforced case, the case reinforced with geogrid has larger cohesion and lower friction angles. The ductility of soil is enhanced due to geogrid reinforcement. Based on the basic control equations of the interface and damage theory, trilinear shear stress-displacement damage softening model is proposed to describe the strain-softening characteristics of geogrid-soil interface. Analytical solutions of interface tension, shear stress, and displacement at different stages are derived considering strain softening based on damage, and the development of shear stress and progressive failure mode of the geogrid-soil interface at different pullout stages is revealed. Furthermore, the proposed model is verified by experimental results.

Analytical solutions for stress and displacement around inclined rectangular stopes with a large width-to-height ratio by the complex variable theory

Analytical solutions for stress and displacement around inclined rectangular stopes with a large width-to-height ratio by the complex variable theory

Methodology for Determining the Compliance Matrix and Analyzing the Skew Anisotropic Plate with an Arbitrarily Shaped Hole

The previous analytical methods mainly focused on the analysis of orthotropic, transversely isotropic, or even isotropic materials, and little analyses have been conducted on nonorthogonal and extreme anisotropic material properties. In this study, the infinite skew anisotropic plate with an arbitrary-shaped hole are considered. Initially, two local Cartesian coordinate systems are established along the skew orientations in the plate, and the equations for the elastic constants of the material are established according to the transformation formulae under rotation of the coordinate axes. On this basis, the elastic compliance matrix for the skew anisotropic material can be obtained in combination with the engineering constants (Young’s modulus, Poisson’s ratio, and shear modulus). Subsequently, the complex variable method is utilized to derive the analytical solutions for stress along the hole boundary. The elliptical, hexagonal, and square holes perforated in skew anisotropic plates are used as examples, respectively, and the stress distributions along the hole boundary are analyzed for different external loadings. Finally, the theoretical method presented in this paper is verified by means of the Finite Element Analysis of ANSYS software.

Analytical solution for stress and displacement of a finite thickness soil layer subjected to strip footing

Consider an elastic soil layer with a shallow-buried strip footing at the upper boundary and bedrock at the lower boundary, the displacement along the contact surface between the soil layer and the bedrock can be considered as zero when the bedrock is sufficiently rigid relative to the soil layer. The stress and displacement fields in the soil layer are solved on the premise that the displacement on the contact surface between the soil layer and the strip footing is given. The soil layer with the finite thickness on the physical plane is mapped onto the inner domain of the unit circle on the image plane based on the conformal transformation. Then, the complex variable method is applied to give the stress and displacement fields in the soil layer by two analytic functions, the coefficients of which can be determined by the stress and displacement boundary conditions on the upper and lower boundaries of the soil layer. In calculation, the power series method is introduced to establish a set of linear equations containing the analytic function coefficients, and then the coefficients are determined by solving the linear equations. This paper presents the analytical solutions of stress and displacement in the soil layer under two different displacement boundary conditions on the contact surface between the soil layer and the footing. The obtained results are compared with the results of the numerical method based on ANSYS. Finally, the influence of different displacement boundary conditions and soil thickness on the stresses on the surface and internal region of the soil layer is analyzed.

Traction stress-based fatigue failure mode identification of load-carrying welded cruciform joints

This paper proposes an analytical solution of traction stress at the weld root of load-carrying welded cruciform joints, incorporating the effects of weld geometry, especially the gap between the loading and intermediate plates. Additionally, this study suggests fatigue failure criteria to differentiate toe and root failures and verifies these criteria by 372 experimental data. Results indicate that: (1) the analytical solution can quantify the stress state at the weld root and predict the root cracking angle; (2) the failure criteria are adequate to judge toe and root failures with high accuracy. Finally, weld geometry effects are discussed, and fatigue design for joints is suggested.

A Generic Analytical Elastic Solution for Excavation Responses of an Arbitrarily Shaped Deep Opening under Biaxial In Situ Stresses

This paper presents a generic analytical solution for stresses and displacements developed around an arbitrarily shaped underground opening excavated in an elastic soil/rock mass with biaxial in situ stresses. A series of separable functions satisfying the governing biharmonic equation is employed as the basic stress function for the problem considered. With the basic stress function and the stress rotation equation, the normal stress and the shear stress on the opening boundary surface are expressed in terms of separable function series with unknown coefficients to make use of the boundary condition. Based on the variational theory, the unknown coefficients are determined by minimizing the difference between the real stress boundary condition and the assumed stress function on the opening surface. The excavation responses around different shapes of openings, including elliptical-shaped, ovaloid-shaped, horseshoe-shaped, and D-shaped openings, are investigated and compared with those from the finite-element simulations. The results show that both the stress field and the displacement field agree well with those from the corresponding numerical model, which verify the proposed solving technique. The proposed solution provides an easy, generic, and feasible approach to assess the stresses and displacements developed around an arbitrarily shaped opening, which is simpler and more straightforward than the complex theory–based solutions.

An analytical stress field for bi-material V-notches with end hole: New solution and effects of higher order terms

Current manuscript develops an analytical stress solution for the bi-material V-notches with an end hole (VO). To do so, based on the Kolosov-Muskhelishvili’s approach, while appropriate potential functions are employed, the boundary conditions of the problem are imposed to the solution to reduce the number of unknown parameters. Subsequently, the analytical stress field is derived as an asymptotic series solution, where each term possessing a constant coefficient and an order of singularity. The order of singularity for each term is obtained from the characteristic equations of the problem which is dependent on the notch geometry and material combinations. The so-called least square method (LSM) is then used to compute the constant coefficients of the asymptotic series for several case studies. Special attention is given to the bi-material notch stress intensity factors (BNSIFs) and the coefficients of the higher order terms (HOTs) in the stress series expansion. The accuracy of the presented stress solution is verified by benchmarking the results with numerical values obtained from finite element (FE) method. In this process, several notch opening angles and notch radii are simulated using the three-point bend (3PB) specimen. The developed asymptotic stress solution is demonstrated to be capable of accurately evaluating the stress field around bi-material VO-notched structures.

A Semi-analytical Solution of Face Extrusion Deformation Considering Nonlivnear Changes of Rock Mass Mechanical Parameters

The stress state of rock mass is redistributed after the tunnel excavation, resulting in different confining stress at different radial distances. Relevant studies demonstrate that the mechanical characteristics of rock are related with the confining stress, causing various the mechanical parameters of rock mass at different radial distances. This paper aims to take these kinds of mechanical properties of rock mass into the analytical solutions of stress and deformation around the tunnel face. Firstly, the calculation formulas of the rock mass mechanical parameters changing with the confining stress is determined based on the existing rock triaxial test data. Then, a new model considering the influence of confining stress is proposed based on Mohr-Coulomb (M-C) failure critical. At the same time, an given radial length increment divides the plastic zone into a series of concentric annuli. The finite difference approach may be used to compute consecutive stress and strain increases for each annulus. A comparative analysis of the Pierpaolo Oreste's model and the proposed model is carried out through calculation examples, and the results show that: ignoring the effect of the confining stress on the mechanical parameters of rock mass, the plastic radius of the tunnel face will be smaller than the actual value, and the face extrusion deformation value will be larger. It is emphasized that the influence of confining stress on the mechanical parameters of rock mass should be considered in actual engineering calculations, especially the high initial stress rock mass. Finally, combined with the numerical calculation model, the rationality of the application of the circular tunnel elastoplastic model in the non-circular tunnel is verified.

The Viscoelastic Solution to Geertsma’s Subsidence Problem

Abstract This paper presents the viscoelastic analytical solution for stress and displacement due to prescribed time-varying changes in the pore fluid pressure of a disk-shaped inclusion embedded within a semi-infinite, viscoelastic medium. The correspondence principle of viscoelasticity, along with Hankel–Fourier and Laplace transforms, is used to derive the solution. The instantaneous viscoelastic solution, corresponding to the response immediately after the inclusion pore pressure change, recovers the elastic solution to the same problem (Geertsma 1973). Results are presented for fractional Maxwell and Burgers models of viscoelasticity after being applied to a set of experimental data from creep tests on shale. Solution results are demonstrated and discussed for the cases of constant inclusion depletion, as well as delayed injection of fluid into a previously depleted inclusion.

Analytical Approach to the Coupled Effects of Slope Angle and Seepage on Shallow Lined Tunnel Response

The existing analytical models of shallow lined tunnels pay little attention to the coupled effects of slope angle and seepage, leading to an overestimation of tunnel stability. In order to overcome such a problem, based on the methods of coordinate transformation, complex function and conformal mapping, we attempt to provide an analytical approach to estimate the effective stress and deformation in shallow lined tunnels considering the corresponding influence factors in this study. First, the stress function is extended to the Laurent series in the complex plane, followed by Laplace transform and inverse transform methods on stress function, and the analytical solutions for stress and displacement fields around the tunnel are achieved during different stages. Then, Mein–Larson formula and G–A formula are used to obtain the pore water pressure and effective stress of surrounding rock and lining, considering the effects of rainfall and groundwater. The analytical solution provided in this study can be reduced without considering the lining support and slope angle effects. Furthermore, the rationality and effectiveness of the proposed solution are achieved by a comparison of numerical results. Finally, a comprehensive parametric investigation based on the proposed solution is performed, including buried depth, slope angle and rainfall intensity. The results show that the small change of stress and deformation of the ground surface is observed as the buried depth becomes more than 30[Formula: see text]m. If the slope angle varies within a range from 0 to 30[Formula: see text], the influence of slope angle change on the stress and deformation of surrounding rock can be generally neglected. When the rainfall intensity is greater than a certain threshold or rainfall time is shorter than a value, these two parameters are in a limited effect on the seepage change.

An alternative method of solvent‐induced stresses in an elastic thin slab: Moutier theorem

Understanding the solvent-induced stresses in polymeric materials is of profound importance for drug release and soft robotics. This work presents an alternative method to obtain an analytical solution of solvent-induced stress in an elastic slab using the Moutier theorem under the framework of equilibrium thermodynamics and the elastic solution available in the literature. Using the analytical solution and the temporal evolution of the mass of organic solvents in polymeric slabs, we analyze the spatiotemporal evolution of the solvent-induced stress in the polymeric slabs and reveal the effects of the solvent transport on the stress evolution. The approach presented in this work provides a simple method to solve the decoupling problem involving diffusion and solvent-induced stresses.

Effects of the Nonlocal Thermoelastic Model in a Thermoelastic Nanoscale Material

In this work, a novel nonlocal model without energy dissipations is presented to investigate the impacts of the nonlocal thermoelastic parameters in a nanoscale material by the eigenvalue approach. The basic equations are applied under the Green and Naghdi model without energy dissipations. To obtain this model, the theory of the non-local continuum suggested by Eringen is applied. The Laplace transformation technique is used for the basic formulations to obtain the analytical solutions of the thermal stress, the displacement, and the temperature during the nanoscale thermo-electric medium. The inverse of the Laplace transformation is used with the numerical technique to obtain the complete solutions of the studying fields in the time–space domains. The main physical fields are displayed graphically and theoretically discussed under the influence of nonlocal parameters.

Thermally Induced Flexural Vibration Problem of Inhomogeneous Rectangular Plates: When Temperature Change at the Bottom Surface Is Kept at Zero

Herein, a thermally induced flexural vibration problem of inhomogeneous rectangular plates in which the material constants are given as a power of the coordinate in the thickness direction is considered. Assuming that the top surface of the plate uniformly receives cyclic heat supply and that the bottom surface is kept at zero temperature change, a problem of 1D heat conduction in an unsteady state in the inhomogeneous plate is analyzed theoretically. Analytical solutions of temperature change, deflection, and stresses in the plate with simply supported edges in which power exponents in the material constants are free from any constraints are derived. Performing numerical calculations, an effect of inhomogeneity in the specific heat capacity on amplitude and phase shift in temperature change inside of plate is clarified. To evaluate the influence of inhomogeneity in the material constants on the inertia effect in the thermoelastic response of the plate, amplification factors are defined as ratios of maximum amplitudes in dynamic solutions of deflection and stresses to those in quasistatic solutions. It is clarified that amplification factors in deflection and stresses of the plate are significantly affected by inhomogeneity in the Young's modulus of elasticity and the mass density.

An Analytical Elastic Solution for Right-Angle Trapezoidal Opening in Steeply Inclined Coal Seam

In the process of underground mining, steeply inclined rocks or coal seams are often encountered, forming the openings of right-angle trapezoid. According to the geological conditions of a mining project in China, an analytical elastic solution of stress and displacement around right-angle trapezoidal opening in a homogeneous, isotropic, and linear elastic geomaterial is presented, which is based on the evaluation of the conformal mapping representation by an appropriate numerical calculation and the complex potential functions. The different results from other shaped openings are shown as follows. In a right-angle trapezoidal opening, the maximum displacements of roof falling occur on the low side, while the most horizontal displacements on the low side are around the roof and the most horizontal displacements on the high side are around the middle of the high side in this opening. These results are also compared with the numerical calculations in FLAC software, illustrating that the solution may be easily applied to rock mechanics or rock engineering for understanding the deformation of floor heave and roof falling down. The solution is also suitable for optimum design of bolt supporting in a right-angle trapezoidal opening, which is different from the traditional concept of symmetrical bolt supporting. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of an opening without symmetrical axis.

Обобщенные модельные представления теории теплового удара для локально-неравновесных процессов теплообмена

An open problem of the theory of thermal shock in terms of a generalized model of dynamic thermoelasticity under conditions of a locally nonequilibrium heat transfer process was considered. The model simultaneously includes three coordinate systems: Cartesian coordinates - a massive body bounded by a flat surface; spherical coordinates - a massive body with an internal spherical cavity; cylindrical coordinates - a massive body with an internal cylindrical cavity. Three types of intensive heating and cooling are considered: temperature, thermal, medium. The task is set: to obtain an analytical solution, to carry out numerical experiments and to give their physical analysis. As a result, generalized model representations of thermal shock in terms of dynamic thermoelasticity have been developed for locally nonequilibrium heat transfer processes simultaneously in three coordinate systems: Cartesian, spherical, and cylindrical. The presence of curvature of the boundary surface of the thermal shock area substantiates the initial statement of the dynamic problem in displacements using the proposed corresponding "compatibility" equation. The latter made it possible to propose a generalized dynamic model of the thermal reaction of massive bodies with internal cavities simultaneously in Cartesian, spherical and cylindrical coordinate systems under conditions of intense thermal heating and cooling, thermal heating and cooling, heating and cooling by the medium. The model is considered in displacements on the basis of local non-equilibrium heat transfer. An analytical solution for stresses is obtained, a numerical experiment is carried out; the wave nature of the propagation of a thermoelastic wave is described. A comparison with the classical solution is made without taking into account local non-equilibrium. On the basis of the operational solution of the problem, design engineering relations important in practical terms for the upper estimate of the maximum thermal stresses are proposed.

Analytical and Modelling Study on the Spatial Stress Distribution and Failure Process of Disc Specimen in the Brazilian Splitting Test

To correctly obtain the spatial stress distribution and failure process of disc specimen in the Brazilian splitting test, an analytical solution of three-dimensional stress is deduced. Then, the effects of height-diameter ratio and clamp radian on the spatial stress distribution and failure process are analyzed and studied combined with numerical modelling. At last, the influence of spatial effect on the tensile strength of disc specimen is discussed. The results show that the cracks firstly generate at the two ends of the specimen in the axial direction and then extend due to the nonuniform distribution of tensile stress. The macrocracks coalescence does not mean the capacity loss of radial bearing. The maximum radial bearing capacity of the disc specimen decreases with the increase of height-diameter ratio due to the spatial effect. The tensile strength obtained by the two-dimensional calculation formula is significantly smaller. Therefore, when the commonly-used height-diameter ratio of 0.5 is used in the Brazilian splitting test, a correction factor k = 1.15 − 1.25 is suggested.