Elasto-plastic Solution Research Articles

Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition

In this paper, exact solutions to purely elastic, partially plastic and fully plastic deformation and stresses of rotating functionally graded (FG) cylindrical pressure vessels under thermal load are obtained. The material properties are assumed to vary nonlinearly in the radial direction of the vessel whereas the Poisson's ratio due to slight variations in engineering materials is kept constant. To the best of the researchers' knowledge, in the literature, there is no study carried out exact solution for thermal stresses in elastic-plastic rotating FGM thick cylindrical pressure vessels. The novelty of the current work is to present a complete exact elastoplastic solution for thick cylindrical pressure vessels made of FGM under thermal loading, taking into account all stress regions. Assuming that the inner surface is exposed to an airstream, and that the outer surface is exposed to a uniform heat flux. The heat conduction equation for the one-dimensional problem in cylindrical coordinates is used to obtain temperature distribution in the vessel. Tresca's yield criterion and its associated flow rule are used to formulate six different plastic regions for an ideal plastic material. All these stages are studied in detail. It is shown that the thermo-elasto-plastic response of the functionally graded pressurized cylindrical vessel is affected notably by the radial variation of material properties.

Elasto-plastic solution for cold-regional tunnels considering the compound effect of non-uniform frost heave, supporting strength and supporting time

The influence of accurate evaluation of frost heave on tunnel safety plays an important role in construction technology and measures to prevent frost damage of cold-regional tunnels. To propose new theoretical insights for the analysis and design of tunnels in cold region, a new calculation model that considers the effect of non-uniform frost heave is established. During the analysis process, according to the supporting strength and supporting time, this calculation model is subdivided into three cases: (1) rp<rL; (2) rL<rp<rf; (3) rp>rf. An engineering example is studied to derive the stresses and deformation of the lining and surrounding rock of cold-regional tunnels. By comparing the stress distribution and the plastic damage radius for the ε0 value of 0 and 0.0055, it is revealed that frost heave does not promote the development of the plastic zone but increases the stress difference in lining, which is one of the main reasons that leads to the instability of tunnels in a cold region. Finally, the influence of the supporting strength, supporting time and initial ground stress on the plastic damage radius and the stress acting on the lining is studied. The calculation method and results can provide a guide for the construction of cold-regional tunnels.

Elasto-plastic solution for shallow tunnel in semi-infinite space

• Elasto-plastic solution for shallow tunnel in a semi-infinite space was proposed. • The gravitational effect on stress in the elastic surrounding rocks is notable. • The influence of gravity on the elasto-plastic model is limited. • Calculations of plastic zone shape are consistent with the results generated by Midas-GTS. This study focuses on a novel approach to develop an elasto-plastic solution for surrounding rocks around a shallow tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. A shallow circular tunnel was excavated in continuous, isotropic, and homogeneous rock masses. The solution for the surrounding rock in semi-infinite space was regarded as a semi-infinite plane problem, and the semi-infinite plane was partitioned along the curve trajectories of the bipolar coordinate. Based on the Mohr-Coulomb failure criterion and considering the gravitational effect, the closed-form solutions for the stress distribution of the surrounding rock were proposed. Furthermore, novel approaches for the calculation of the shape of the plastic zone around the tunnel and the critical internal support pressure were obtained. These presented approaches were validated by comparing the results of finite element method and the published literature. Using the numerical method, the sensitivity of gravity and the influences of the geological strength index, internal support pressure, surface loading, and buried depth of the tunnel on the plastic radius, shape of the plastic zone, and stress distribution of the surrounding rocks around the shallow circular tunnel were discussed.

Analysis of Soil-Compacting Effect Caused by Shield Tunneling Using Three-Dimensional Elastoplastic Solution of Cylindrical Cavity Expansion

Soil squeezing effect and formation disturbance caused by tunnel excavation can be simulated by cylindrical cavity expansion due to the comparability between tunneling and cavity expansion. Although most of the existing theoretical derivation is based on simple constitutive model of soil foundation, not only the relation between principal stress components was simplified in the solution process, but also the stress history, initial stress anisotropy, and stress-induced anisotropy of structural soil were neglected. The mechanical characteristics of soil are closely related to its stress history, so there is a gap between the above research and the actual engineering conditions. A three-dimensional elastoplastic solution of cylindrical cavity expansion is obtained based on the theory of critical state soil mechanics and engineering characteristics of shield tunneling. In order to fully consider the influence of initial anisotropy and induced anisotropy on the mechanical behavior of soils, the soil elastoplastic constitutive relation of cavity expansion is described in the course ofK0-based modified Cam-clay (K0-MCC) model after soil yielding. An equation with equal number of variables is obtained under the elastic-plastic boundary condition based on the Lagrange multiplier method. By solving the extreme value of the original function, the analytical solution of radial, tangential, and vertical effective stresses distribution around the circular tunnel excavation is obtained. In addition, changes of elastic deformation area and plastic deformation area for soil during the shield excavation have been analyzed. Calculation results are compared with the numerical solutions which usually consider isotropic soil behavior as the basic assumption. In this paper, a constitutive model which is more consistent with the actual mechanical behavior of the soil and the construction process of the shield tunnel is considered. Therefore, the numerical solutions are more realistic and suitable for the shield excavation analysis and can provide theoretical guidance required for design of shield tunneling.

Quick approximate elastoplastic solutions of wellbore stability problems based on numerical simulation and statistical analysis

Wellbore instability has been a chronic issue for well operators over several decades in petroleum industry. Traditional linear elastic models may sometimes fail to provide a proper mud weight window for drilling engineers. Elastoplastic models can better represent the rock behavior and, therefore, more accurately evaluate the risk of wellbore instability. However, elastoplastic models have failed to gain popularity in the industry because of the model complexity and computation cost. This work proposes an approximating method in a novel manner, incorporating both the validity of the elastoplastic constitutive model and the rapidity of the linear elastic model to predict wellbore behavior. The non-associative strain hardening Drucker-Prager elastoplastic model is used. The relationship between the yielded zone area calculated by the elastoplastic model and the pseudo-yielded zone area calibrated by the linear elastic model is statistically investigated. It is found that the two can be correlated with high confidence based on a set of common input parameters, like in-situ stresses, wellbore pressure, Young's Modulus, etc. Three correlation equations are provided according to the value range of the predicting terms and an application example is addressed at the end. In conclusion, this approach will help engineers make reliable wellbore stability decisions without resorting to sophisticated elastoplastic models. The equations can be directly used in simple spreadsheet functions or real-time data processing schemes to make faster and more efficient decision.

Elastoplastic Analysis of Circular Tunnel Based on Drucker–Prager Criterion

Based on the Drucker–Prager yield criterion, the theoretical solution of stratigraphic deformation in tunnel excavation process is deduced by the cavity expansion theory. In view of soil loosening around the tunnel caused by the tunnel excavation process, the internal friction angle of the surrounding soil is not a constant but a function of normal stress. The piecewise linearization of the nonlinear yield function is used to analyze the elastoplastic solution of the cylindrical hole shrinkage. A comparison is conducted with a plastic zone in which the internal friction angle of the soil remains unchanged. It can be concluded that the radial stress, the tangential stress, the radial strain, and the tangential strain around the inner wall calculated from the former are smaller.

A Finite Volume Formulation for the Elasto-Plastic Analysis of Rectangular Mindlin-Reissner Plates, a Non-Layered Approach

This paper extends the previous work of authors and presents a non-layered Finite Volume formulation for the elasto-plastic analysis of Mindlin-Reissner plates. The incremental algorithm of the elasto-plastic solution procedure is shown in detail. The performance of the formulation is examined by analyzing of plates with different boundary conditions and loading types. The results are illustrated and compared with the predictions of the layered approach. These several comparisons reveal that the non-layered Finite Volume approach can present accurate results with low CPU time usage despite its simplicity of the solution procedure.

Drained Elastoplastic Solution for Cylindrical Cavity Expansion in K0-Consolidated Anisotropic Soil

This paper presents a semianalytical solution for drained expansion of a cylindrical cavity in K0-consolidated clay. The solution is derived based on the K0-based anisotropic modified Cam-clay model (K0-AMCC), which can properly consider the initial stress anisotropy and the initial stress-induced anisotropy of natural soils. Parametric studies are carried out to investigate the effects of different overconsolidation ratios on the growth of plastic zone and cavity pressure, the distributions of the stress components and the specific volume around the cavity, and the effective stress path. The results are compared with the published solution based on the modified Cam-clay model to study the effects of the initial stress anisotropy and the initial stress-induced anisotropy on the cylindrical cavity expansion. The present solution provides a more realistic interpretation for practical geotechnical engineering problems such as pressuremeter tests and cone penetration tests.

Fully coupled solution for the consolidation of poroelastic soil around geosynthetic encased stone columns

The paper presents an extension of a recently developed fully coupled elastoplastic method (Pulko and Logar, 2016) for the analysis of a poroelastic thick-walled soil cylinder around an elastoplastic end-bearing stone column to account for the influence of an elastic geosynthetic encasement. The method was developed in the framework of Biot's consolidation theory (Biot, 1941) and is based on a unit cell concept, wherein the column encasement is modeled as a thin elastic membrane, which can only sustain tension and acts in the radial direction. Analytical closed-form expressions for excess pore pressures, stresses, strains, displacements and encasement forces were derived in the Laplace domain. The final elastoplastic solution in time domain was obtained numerically by using efficient numerical scheme for the inverse Laplace transform. The validity of the solution was checked against finite element analyses and compared with previously developed analytical methods. The results showing the influence of column encasement on transient state of settlements, strains, excess pore pressures and encasement forces under instantaneous or time dependent load are presented and discussed.

Elasto-plastic analysis and finite element simulation of thick-walled functionally graded cylinder subjected to combined pressure and thermal loading

AbstractAn exact analytical elasto-plastic solution for thick-walled cylinder made of functionally graded materials (FGMs) subjected to combined pressure and thermal loading is presented in this paper. It is assumed that the cylinder is bonded at both ends, the material is radially graded and complies with the elastic perfectly plastic behavior. The relations in determining the plastic zone radius as well as the radial, circumferential, longitudinal and effective stresses in both elastic and plastic zones are obtained for any combined loading condition. Moreover, using ABAQUS/Explicit software, the functionally graded (FG) cylinder is simulated in every respect. Comparison of the present theoretical results with those from a finite element simulation illustrates the accuracy of the present analysis.

Numerical investigation of residual stresses in high temperature ZrO2 insulation coatings on Ag tape substrate for magnet technologies

In this research, the numerical investigation of residual stresses was carried out in high temperature ZrO2 insulation coatings on Ag tape substrate for magnet technologies. ZrO2 coatings were fabricated on Ag substrate using a reel-to-reel sol–gel system. The microstructural observation of ZrO2 coatings was performed through an environmental scanning electron microscope (ESEM). ESEM micrograph revealed that ZrO2 coatings with crack had mosaic structure which is suitable for high temperature insulation fields. The thermal stresses in the each layer were determined by the means of the lamination theory for both elastic solution and the finite element method (FEM) for elastoplastic solution in the temperature range of 298 to 50 K in liquid helium. The residual stresses happened throughout the production are neglected because of gradually cooling process. As a result of this, all of the residual stresses at the 298 K are accepted to be zero. Difference of thermal expansion coefficients and elasticity modulus cause the thermal stresses in the layers.

An elasto-plastic solution for channel cracking of brittle coating on polymer substrate

In this article, an elasto-plastic channel-cracking model is presented to study the open-mode fracture of a thin layer brittle coating grown on a polymer substrate. A linear elastic shear interlayer is introduced to describe the stress transfer from the elasto-plastic substrate to the brittle coating, on basis of the shear-lag principle. The channel cracking behavior involves three stages: elastic, elasto-plastic and plastic stages, which are solved in a continuous manner based on the deformation status of the substrate. Explicit solutions are derived for the mutli-stage cracking process. Corresponding experimental tests for a titanium oxide (TiO2) coating on a poly (ethylene terephthalate) substrate are conducted. The fracture toughness of the coating layer is estimated based on the crack spacing versus layer thickness relationship at certain strain levels. This method is found to be more reliable than the traditional methods using crack onset strain. Parametric studies of the fracture energy release rate for the coating and interfacial compliance of the thin film system are conducted, through which the effect of plastic deformation on the channel cracking behavior is studied extensively. The results indicate that the tangent modulus of the substrate controls the evolution curvature of crack spacing where a smaller tangent modulus corresponds to a slower saturation of crack spacing. The energy release rate also varies significantly with the properties of the interlayer. The study highlights the necessity of an elasto-plastic model for the thin film systems of brittle coating on a plastic substrate.

Elastoplastic Solution for Spherical Cavity Expansion in Modified Cam-Clay Soil under Drained Condition

AbstractThis paper presents an innovative semianalytical solution for the expansion of a drained spherical cavity with finite initial radius. The widely known modified Cam-clay model is adopted to ...

Effect of various coal constitutive models on coupled thermo-mechanical modeling of underground coal gasification

Underground coal gasification (UCG) is an advanced method that can address the high energy demand in future by adding new resources to the proved reserves. The high temperature nature of the gasification reactions along with the formation of cavities within a coal seam necessitates a comprehensive analysis of mechanical and thermomechanical impacts on the UCG process. On the other hand, geomechanics plays a significant role in the efficiency of the UCG process by varying the reaction rates, porosity, and permeability in a flow model. More importantly, geomechanical modeling of UCG is necessary to mitigate the associated environmental hazards such as land subsidence and groundwater contamination.In this study, the results from a reservoir simulator are coupled with a geomechanical module to solve for the displacement and stress variations. A controlled retracting injection point (CRIP) method is applied in the reservoir simulator to model reactions and geochemistry. By performing sensitivity analyses using the linear elastic, hyperbolic, and elasto-plastic constitutive models, the two compelling factors of high temperature and cavity evolution, which have the highest contributions to the geomechanical responses of the UCG process, are comprehensively investigated. At the end, a stress rebalancing technique is imposed on the linear elastic model to mimic the elasto-plastic solution by capturing the shear softening and the post-failure behavior of coal.

Investigation of thermo-elasto-plastic behavior of thick-walled spherical vessels with inner functionally graded coatings

Having high corrosion and thermal resistance characteristics, Functionally Graded Materials have been found wide range of applications in engineering fields. The analytical elasto-plastic solution for spherical vessels composed of outer homogenous part and inner FG coating under combined pressure and thermal loading is presented in this paper. Furthermore, the loading conditions resulted into onset of yielding from inner radius and fully plastic vessel are obtained. Material constituents in both homogenous and FG parts are assumed to be aluminum and SiC as metal matrix and ceramic particles, respectively. Although the amount of SiC particles in the homogenous part of vessel is assumed to be constant, it is varying in the radial direction of FG part based on the power law function. The results concerning two types of vessels with harder and softer FG coatings are also compared with those from homogenous vessel. Moreover, two dimensional finite element model was applied to simulate the process while the results were considered to verify the accuracy of the analytical solution. Generally, the results demonstrate that coating and grading behavior can really affect the stress distribution and yield pattern in the homogenous vessel with an inner FG coating.

Springback analysis of channel cross-sectioned bar of work-hardening materials under torsional loading

Purpose The purpose of this paper is to deal with the springback problems of channel cross-section bars of linear and non-linear work-hardening materials under torsional loading. Using the deformation theory of plasticity, a numerical scheme based on the finite difference approximation has been proposed. The growth of the elastic-plastic boundary and the resulting stresses while loading, and the springback and the residual stresses after unloading are calculated. Design/methodology/approach The numerical method which has been described in this paper for obtaining the solution of elasto-plastic solution can also be used for other sections. The only care that needs to be taken is to decrease the mesh size near points of stress concentration. The advantage of this technique is that it automatically takes care of all plastic zones developing over the section at different loads and gives a solution satisfying the elastic and plastic torsion equations in their respective regions. Findings As expected, elastic recovery is found to be more with decreasing values of n and λ. The difference in springback becomes more and more with increasing values of angle of twist. The material will approach an elastic ideally plastic behavior with increasing values of λ and n. Originality/value It seems that no attempt has been made to study residual stresses in elasto-plastic torsion of a work-hardening material for a channel cross-section.

Fully coupled solution for the consolidation of poroelastic soil around elastoplastic stone column

The paper presents a new fully coupled elastoplastic solution for the response of a poroelastic thick-walled soil cylinder around an elastoplastic stone column using Biot’s (J Appl Phys 12:155–164, 1941) consolidation theory. A unit cell concept is adopted for the soil–stone column analysis, and the problem is formulated in cylindrical coordinates. Expressions for excess pore pressure, stresses and displacements in the Laplace domain are derived analytically taking into account elastic or plastic behavior of the column. The inverse of the Laplace transform is evaluated numerically using an efficient scheme to obtain the final elastoplastic solution in time domain. The validity of the new solution has been checked against finite element solution and compared with some previously developed analytical methods for the stone column analysis. The results showing settlements, change in excess pore pressures and stresses with time are presented in terms of time factor. The proposed solution can be used to calculate transient state of settlements, distribution of deformations, stresses and excess pore pressures in soil and column under instantaneous or time-dependent monotonically increasing rigid vertical load.

Drained and undrained analyses of cylindrical cavity contractions by bounding surface plasticity

This paper develops a rigorous semi-analytical approach for drained–undrained cylindrical cavity contraction problems in bounding surface elastoplastic geomaterials. For undrained situations, the effective radial, tangential, and vertical component stresses can be solved directly from the constitutive governing differential equations as an initial value problem, the excess pore pressure subsequently being determined from the radial equilibrium equation. In contrast, for the drained case the Eulerian radial equilibrium equation must be first transformed into an equivalent one in a Lagrangian description via the introduction of an auxiliary variable, then solved together with the elastoplastic constitutive relation for the three stress components as well as the specific volume. It is observed that during drained–undrained contraction processes, plastic deformations occur immediately as a direct result of employing the bounding surface model, so outside the cavity there exists no purely elastic zone. The computed stress distributions and in particular the stress path prediction through an example analysis capture well the anticipated elastoplastic and failure behaviour of the geomaterials surrounding the cavity. The validity and accuracy of the proposed semi-analytical elastoplastic solutions are justified through comparison with ABAQUS numerical results, and their applicability to the tunnel excavation and wellbore drilling problems is also demonstrated.

Opening-mode micro-cracking in brittle coatings on ductile wires/rods

A simplified two-dimensional model is presented to simulate periodically distributed micro-cracking in a thin coating fully bonded to an elastoplastic aluminum wire/rod. The alumina coating which is generated by anodic oxidation is treated as an elastic material, while the ductile aluminum wire substrate is characterized by a bi-linear elastoplastic hardening model. An elastoplastic shear lag model is applied to transfer the shearing stress from the substrate layer to the thin coating. When the coated structure is subjected to different applied load, the system will undergo different stress levels and exhibit different cracking stages. Accordingly, explicit solutions corresponding to different loading stages are presented based on a generalized axisymmetric formulation. Finite element simulation is employed to verify the present elastic solution when the applied load is relatively small and the whole system is in the elastic state. Experimental characterization is conducted to validate the present elastoplastic solution when the substrate or interlayer undergoes large deformation. A versatile high-fidelity optical microscope is utilized to check the micro structure of the coating and continuously monitor the fracture development in the coating layer, through which valuable detailed micro-cracking information, such as critical applied load corresponding to crack initiation, crack pattern and crack spacing, is obtained. It shows that the presented fracture model is able to accurately capture the stress and strain distribution in the coated structure and predict the fracture initiation, infilling, and saturation in the thin coating layer.

Elastoplastic problem for noncircular openings under Coulomb’s criterion

An elastoplastic problem is considered for elliptical, arched, square and polygonal cross-section openings on the assumption of Coulomb’s limit equilibrium and varied lateral earth pressure coefficients in intact rock mass. The problem is solved using the method of small parameters and the finite element method. The elastoplastic solutions are obtained for mutually influencing underground openings. The mechanisms of concentration of vertical stresses in a pillar established between mutually influencing openings are determined.