Plane Strain Problem Research Articles

Asymptotic analysis of axisymmetric plane strain problem involving fractional order heat conduction

ABSTRACTThis work is concerned with the asymptotic solutions of the axisymmetric plane strain problem involving the fractional order heat conduction. The governing equations for the axisymmetric plane strain problem are derived by means of fractional calculus. The Laplace transform technique is used to obtain the general solutions for any set of boundary conditions in the physical domain. The asymptotic solutions for a specific problem of an infinite cylinder with the boundary subjected to a thermal shock is derived by means of the limit theorem of Laplace transform. Utilizing these solutions, the thermoelastic behavior induced by transient thermal shock can be clearly illustrated, and the jumps locating at the position of each wavefront can also be accurately captured. Some comparisons for the predictions of thermoelastic response are conducted to estimate the effect of the fractional order parameter on the thermoelastic behavior.

Debonding of confined elastomeric layer using cohesive zone model

Wavy or undulatory debonding is often observed when a confined/sandwiched elastomeric layer is pulled off from a stiff adherend. Here we analyze this debonding phenomenon using a cohesive zone model (CZM). Using stability analysis of linear equations governing plane strain quasi-static deformations of an elastomer, we find (i) a non-dimensional number relating the elastomer layer thickness, h, the long term Young׳s modulus, E∞, of the interlayer material, the peak traction, Tc, in the CZM bilinear traction-separation (TS) relation, and the fracture energy, Gc, of the interface between the adherend and the elastomer layer, and (ii) the critical value of this number that provides a necessary condition for undulations to occur during debonding. For the elastomer modeled as a linear viscoelastic material with the shear modulus given by a Prony series and a rate-independent bilinear TS relation in the CZM, the stability analysis predicts that a necessary condition for a wavy solution is that Tc2h/GcE∞ exceed 4.15. This is supported by numerically solving governing equations by the finite element method (FEM). Lastly, we use the FEM to study three-dimensional deformations of the peeling (induced by an edge displacement) of a flexible plate from a thin elastomeric layer perfectly bonded to a rigid substrate. These simulations predict progressive debonding with a fingerlike front for sufficiently confined interlayers when the TS parameters satisfy a constraint similar to that found from the stability analysis of the plane strain problem.

The sensitivity of the adaptive algorithm with a posteriori error control to marking criteria

The aim of this work is to compare different marking strategies, their influence on the work of adaptive algorithms with a posteriori error control for plane elasticity problems. The error control was performed using a functional error majorant. The implemented adaptive algorithms were based on the functional error majorant with no symmetry limitation on the free tensor, computed using the zero-order Raviart–Thomas approximations on triangular meshes. The four most commonly used element-marking criteria were used in adaptation. Numerical results for several plane-strain problems have been presented, including the case of different materials and geometry. A comprehensive analysis of the obtained results was given.

Inclined crack through a rhombic thin superconducting strip with transport current

The fracture properties of an inclined crack through a rhombic thin superconducting strip with transport current are investigated. The flux and current density distributions in the processes of both ascending and descending transport currents are given in clear forms. Adopting the plane strain problem model, the energy release rates (ERRs) of the inclined crack are calculated by the finite element method (FEM). The effects of inclined angle, crack length, maximal thickness and applied maximal transport current on the ERRs are discussed in detail in the process of descending transport current. All the obtained results should be of vital importance for the application of superconductors.

Element Free Galerkin Formulation for Problems in Composite Micromechanics

The heterogeneity and anisotropy of structural composites make the application of the standard mesh-based methods using the meshing of interfacial region between matrix and fibers a difficult task. The objective of this study is to present the EFG formulation for problems of composite micromechanics. It is expected that the tediousness and approximations involved in mesh generation, and hence inaccuracies in the results can be avoided using the new meshless techniques such as the Element Free Galerkin (EFG) method. The theoretical methodologies, computer implementations and practical application are carried out. Periodic boundary conditions of the unit cell under tensile load are set up. The method of Lagrange multipliers is introduced for the treatment of material discontinuity at the fiber-matrix interface in which both the displacement continuity and traction reciprocity are satisfied. The EFG method is formulated for the generalized plane strain problems. Examples are presented to illustrate the effectiveness of the proposed micromechanical model, and it is validated by comparing the results with available numerical solutions.

Singular integral equations with two fixed singularities and applications to fractured composites

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector Riemann-Hilbert problem on a real axis with a piecewise constant matrix coefficient that has two points of discontinuity. A condition of solvability and a closed-form solution to the integral equation are derived. For the Chebyshev polynomials of the first kind in the right hand-side, the solution of the integral equation is expressed in terms of two nonorthogonal polynomials with associated weights. Based on this new generalized spectral relation for the singular operator with two fixed singularities an approximate solution to the complete singular integral equation is derived by recasting it as an infinite system of linear algebraic equations of the second kind. The method is illustrated by solving two problems of fracture mechanics, the antiplane and plane strain problems for a finite crack in a composite plane. The plane is formed by a strip and two half-planes; the elastic constants of the strip are different from those of the half-planes. The crack is orthogonal to the interfaces, and it is located in the strip with the ends lying in the interfaces. Numerical results are reported and discussed.

Numerical Simulation of Fluid-Solid Coupling in Fractured Porous Media with Discrete Fracture Model and Extended Finite Element Method

Fluid-solid coupling is ubiquitous in the process of fluid flow underground and has a significant influence on the development of oil and gas reservoirs. To investigate these phenomena, the coupled mathematical model of solid deformation and fluid flow in fractured porous media is established. In this study, the discrete fracture model (DFM) is applied to capture fluid flow in the fractured porous media, which represents fractures explicitly and avoids calculating shape factor for cross flow. In addition, the extended finite element method (XFEM) is applied to capture solid deformation due to the discontinuity caused by fractures. More importantly, this model captures the change of fractures aperture during the simulation, and then adjusts fluid flow in the fractures. The final linear equation set is derived and solved for a 2D plane strain problem. Results show that the combination of discrete fracture model and extended finite element method is suited for simulating coupled deformation and fluid flow in fractured porous media.

Upper bound analysis of forging penetration in a radial forging process

A theoretical model based on upper bound approach is proposed to study forging penetration in a radial forging process. Using the transformations for both the geometric coordinate system and the velocity field, the parabolic discontinuity boundaries in axisymmetric problem are mapped into straight lines in the transformed field. As a result, the present model is simplified and its implementation is analogous to the rigid block upper bound approach in plane strain problem. In order to quantify the penetration of plastic deformation, forging penetration depth is introduced and defined as the radial distance from the outside surface of the workpiece to the intersection point of the assumed plastic and rigid regions. This model is verified by comparing the predicted forging load with published experimental data, and by comparing the predicted forging penetration depth with that of the finite element simulation. Finally the effects of process parameters such as the radial reduction rate and the inlet angle of hammer on the forging penetration are investigated.

Determination of Contact Stresses when Upsetting a Rectangular Strip

Closed analytical solutions of a plane strain problem for contact stresses in zones with variable friction forces for upsetting the strip made of ideal rigid-plastic material are derived. It is established that the use of the Mises exact plasticity condition for the slip zone instead of the approximate one written in principal stresses allowing for the Coulomb friction law leads to an essential (by a factor of 2–4 at f > 0,2) increase in the calculated slip zone length. To match the theoretical results with the experimental ones, a new method for calculating the contact stresses is proposed, in which their distributions in the slip and stagnation zones are derived from the solution of the plane strain problem using a parabolic approximation of the plasticity condition. The sizes of slip and stagnation zones are found depending on the characteristics of contact friction and conditions of appearance of the peaks near the strip end in the normal stress diagram are determined. The possibility of occurrence of retardation zones under tangent contact stresses lower than the yield point under simple shear τs are determined. Formulas for determining the upsetting force are refined.

Stress-strain state of elastoplastic bodies with crack

A physical cut model is used to describe the changes in the stress-strain state (SSS) in elastoplastic bodies weakened by cracks. The distance between the crack edges is considered to be finite in contrast to the mathematical cut. The interactive layer with a thickness limited by the possibility of using the hypothesis of continuity is distinguished on the physical cut extension. Distribution of stresses and strains over the layer thickness is constant and does not depend on the geometry of the boundary between the cut and the interactive layer. The relationship between stresses and strains is determined by the deformation plasticity theory. The problem of plane strain or plane stress state of an arbitrary finite body weakened by a physical cut is reduced to solving a system of two variational equations for displacement fields in the body parts adjacent to the interactive layer. The proposed approach eliminates the singularity in stress distribution in contrast to the mathematical cut model. Use of local strength criteria allows us to determine the time, point and direction of the fracture initiation. Possibilities of the proposed model are illustrated by solving the problems of determining the SSS of a rectangular body weakened by a physical cut under symmetric and antisymmetric loadings.

STRESS INTENSITY FACTOR FOR DOUBLE EDGE CRACKED FINITE PLATE SUBJECTED TO TENSILE STRESS

A finite rectangular plate with double edge crack under uniaxial tension depends on the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strain problem are studied in the present paper. The effect of crack position, crack oblique and the kinked crack orientation are investigated to predict if a crack starts to grow. These problems are solved by calculating the Stress Intensity Factor SIF for mode I (KI) and II (KII) near the crack tip theoretically using mathematical equations and numerically using finite element software ANSYS R15. A good agreement is observed between the theoretical and numerical solutions. The results show that the KI increases with increasing the relative crack length and tensile stress and these values are increased when the crack position draws near the plate edge while in case of parallel cracks the mutual shielding effect reduces KI in each crack.In mixed mode, it is shown that the maximum values of KI and KII occur at crack angle β=0o and 45o, respectively and the orientation of the kinked crack have significant effects on the KI and KII.

Deformation of a poroelastic layer overlying an elastic half‐space due to dip‐slip faulting

SummaryAn analytical solution of the plane strain problem of the deformation of a homogeneous, isotropic, poroelastic layer of uniform thickness overlying a homogeneous, isotropic, elastic half‐space due to two‐dimensional seismic sources buried in the elastic half‐space has been obtained. The integral expressions for the displacements, stresses and pore pressure have been obtained using the stress function approach by applying suitable boundary conditions at the free surface and the interface. The solution obtained is in the Laplace–Fourier transform domain. The case of a vertical dip‐slip line dislocation for the oceanic crust model of Earth is studied in detail. Schapery's formula is used for the Laplace inversion and the extended Simpson's formula for the Fourier inversion. Diffusion of pore pressure in the layer is studied numerically. Contour maps showing the pore pressure in the poroelastic layer have been plotted. The effect of the compressibility of the solid and fluid constituents on pore pressure has also been studied. Copyright © 2015 John Wiley & Sons, Ltd.

Extended JKR theory on adhesive contact between elastic coatings on rigid cylinders under plane strain

This study extended the conventional JKR theory on adhesive contact between elastic cylinders to the frictionless adhesive contact between elastic films coated on rigid cylindrical rollers. The plane-strain elasticity problem of indentation on an elastic film perfectly bonded to a rigid half-plane was revisited, with which the force-depth relations for both flat and cylindrical indentations were obtained in a simple form. By following Hertz analysis and using the solution of the cylindrical indentation on an elastic film, the adhesionless Hertzian contact between elastic films coated on rigid cylinders was obtained. However, the obtained Hertzian contact state should be considered as an approximate solution, since the pressure distributions in the contact region of the two elastic films may not match perfectly each other in the case of different Poisson’s ratios or thicknesses of the elastic films. By properly superposing the linear elastic solutions of flat and cylindrical indentations, the approximate solution to the adhesive JKR contact state between elastic films coated on rigid cylinders was also obtained, of which the relations between applied force, penetration depth, and contact width were expressed in terms of the geometric parameters and elastic constants of the two elastic films. The pull-off force at the moment of debonding two elastic films was also obtained as a function of the adhesion energy and compared with the normal two-dimensional JKR result.

Soil-Structure Interaction of Tunnels in Soft Clay Soil

The construction and use of tunnels can be considered one of the most important features of civilization in developed nations. However, it is found that tunnels effectively helped in the consistency and fluency of traffic as means to carry passengers from one place to another, particularly in over-populated cities. But, it is very difficult to construct tunnels in soft clay soil. So, the objectives of the present study are: i) identify the most suitable shape [Circular, Elliptical and Rectangular with Arch Circular (like horseshoe)] tunnels in soft clay soil. ii) the effect of change shape of tunnels [circular, elliptical and rectangular with arch circular (like horseshoe)] on the internal forces in the tunnel and the internal stresses as well as settlement in soft clay soil. The design of tunnel cross section depends on the values of internal forces and deformations induced in the lining. These values are influenced by many factors such as type of soil and dimensions of the tunnel cross section. In the present study, a numerical analysis using finite element software ADINA version 8.5 (a finite element computer program) was used. A numerical model was presented as a plane strain problem. Two dimensions triangle element of 3-node was used to present the soft clay soil and 2 -node beam element to simulate the lining of tunnels. The material of the lining was chosen as a reinforcement concrete. The Mohr-Coulomb model was used to simulate the non linear soft clay soil with different modulus of elasticity (Es) 1000, 1500 and 1900 kN/m 2 and different Poisson’s ratio () 0.4, 0.45and 0.49. The effect of changing the thickness–radius ratio (t/r) of (0.1, 0.15, 0.2) on the internal forces and the displacement as well as the internal stresses, under the ground surface at a depth 6 times radius of the tunnel were analyzed. The values of internal forces and the displacement at the upper crown (Cr) and the spring (SL) points at the tunnel as well as internal stresses in the soft clay soil were investigated. The analysis showed that the circular shape for tunnels is the most suitable shape in soft clay soil. As the internal forces and the vertical displacement as well as the horizontal stresses for the lining of circular tunnels is less than for the two adjacent elliptical as well as rectangular with arch circular (like horseshoe) tunnels. The numerical results showed that in soft clay soil, the internal forces and stresses are slightly affected by the stiffness of circular tunnel while it is more affected as (Es) of soil increases. As the internal forces decreased up to 75%, the horizontal stresses increased to 35% and the shear stresses decreased to 70%, as well as the displacement decreased to 70% at some cases

Investigation of wave propagation in double cylindrical rods considering the effect of prestress

This paper presents the investigation of guided wave propagation in prestressed double-cylinder structure. Based on Hertzian contact theory, the interaction between two rods is treated as a plane strain problem and the stress state in the waveguide under the static load is obtained. The stress state is considered as a prestressed configuration for elastic wave propagation analysis in double cylindrical rods. The elastodynamic equation of the prestressed structure is established with the updated Lagrangian formulation and the wave finite element (WFE) method. Firstly, the equation is verified by the application on an aluminum rod compared with the Euler–Bernoulli beam theory. Then, dispersion curves for single rod and double cylindrical rods without prestress are computed. Besides, at the dimensionless frequency 0.25 the propagating modes in double cylindrical rods are identified with mode shapes and displacement vectors. The guided waves in double rods consist of the modes different from single rod. Particularly, there exist two kinds of torsional-like modes, one of which has a twist center and the other has two. The latter changes obviously with the increase of prestress in the waveguide. At low frequencies, torsional-like modes are very sensitive to the variation of prestress; in addition, the prestress configuration has little influence on propagating modes at mid-frequencies but some at high frequencies.

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

SummaryThis work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner–Mindlin thick plates in shear‐bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner–Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second‐order cone programming problem and is solved using the industrial software package MOSEK. A comprehensive study of the shear‐locking phenomenon is also conducted, and it is shown that discontinuous elements avoid such a phenomenon quite naturally whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so‐obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates. Copyright © 2015 John Wiley & Sons, Ltd.

Evaluation of wellbore integrity for HTHP gas wells under solid-temperature coupling using a new analytical model

An analytical model of stress analysis around wellbore under the coupling between in-situ stresses and temperature loads is established, taking casing-cement sheath-formation as the generalized plane strain problem into account. To evaluate the failure of interfaces of wellbore, the interfaces of casing-cement sheath and cement sheath-formation are simulated as homogeneous isotropic materials, with very small wall thicknesses of 1% of casing wall thickness. According to the continuity conditions and boundary conditions, the stress distribution around the cased wellbore is determined exactly based on elastic mechanics, assuming casing, cement sheath and the interfaces of casing-cement sheath and cement sheath-formation as thin wall cylinders and formation as a thick wall cylinder, respectively. Using criteria of Mises, Drucker–Prager and JRC-JCS, the integrity of cased wellbore, including casing, cement sheath, the interfaces of casing-cement sheath and cement sheath-formation, is evaluated by defining failure functions. The proposed model is verified by comparison of the obtained results from proposed analytical method and finite element method, with minor deviations which do not exceed around 9.7%. Furthermore, the effects of parameters on the wellbore integrity are investigated. These parameters include Young's modulus of cement sheath, Poisson's ratio of cement sheath, formation temperature, the ratio of the horizontal maximum in-situ stress to the horizontal minimum in-situ stress and the casing inner pressure. The results show that the maximum value of the equivalent stress of casing decreases firstly and then increases with the increasing of casing inner pressures. Also, the inner pressure variation can lead to the location change of the maximum equivalent stress of casing.

Analysis of Magneto-Piezoelastic Anisotropic Materials

The paper is concerned with the analysis of magneto-piezoelastic anistropic materials. Analytical modeling of magneto-piezoelastic materials is essential for the design and applications in the smart composite structures incorporating them as actuating and sensing constituents. It is shown that Green’s function method is applicable to time harmonic magneto-elastic-piezoelectricity problems using the boundary integral technique, and the exact analytical solutions are obtained. As an application, a two-dimensional static plane-strain problem is considered to investigate the effect of magnetic field on piezoelectric materials. The closed-form analytical solutions are obtained for a number of boundary conditions for all components of the magneto-piezoelectric field. As a special case, numerical results are presented for two-dimensional static magneto-electroelastic field of a piezoelectric solid subjected to a concentrated line load and an electric charge. The numerical solutions are obtained for three different piezoelectric materials and they demonstrate a substantial dependence of the stress and electric field distribution on the constitutive properties and magnetic flux.

Cylindrical Bending and Vibration of Polar Material Laminates

This article considers a plane strain problem, which is known in conventional linear elasticity as cylindrical bending of simply supported plates and cross-ply laminates. By considering fibrous composites containing fibers resistant to bending, it formulates and solves corresponding polar elasticity equations governing the static and dynamic behavior of beam-like components made of a homogeneous or layered transversely isotropic material; each layer has embedded a single family of fibers. Fiber bending stiffness is accounted for through involvement of an extra elastic modulus, which, unlike its conventional elasticity counterparts that have dimensions of stress, has dimensions of force. Its involvement in the analysis implies existence of some intrinsic material area or length parameter, which may be associated, for instance, with fiber thickness of fiber spacing. A considerable amount of relevant numerical results are presented for thick beam components made of either homogeneous or two-layered transversely isotropic material. For the static bending problem, these include a detailed presentation of through-thickness distributions of displacements, stresses, as well as couple-stress. For the dynamic problem, attention is focused on the influence that fiber bending stiffness exerts on fundamental frequency parameters.

On the direct numerical simulation of plane-strain fracture in a class of strain-limiting anisotropic elastic bodies

In this paper, using a special class of nonlinear response relations between linearized strain and Cauchy stress tensors, we study a quasi-static mixed-mode (combination of mode-I and mode-II) fracture boundary value problem. A special subclass of such relations are strain-limiting in which the strains are guaranteed to be uniformly bounded in the neighborhood of crack-tip. In this article, we consider a finite element method based direct numerical simulation (DNS) of plane-strain mixed-mode fracture in a class of strain-limiting, transversely isotropic elastic bodies. We study the numerical solution of a nonlinear fracture boundary value problem with displacement as the primary unknown. The linearized version of the strong form was derived using damped Newton’s method and the corresponding weak formulation was solved using a conforming finite element method. The results of this DNS indicate that even very near the crack tip, both stress and strain remain much smaller in magnitude than the corresponding predictions from the linearized elastic fracture mechanics (LEFM). While away from strain concentrating crack-tip region, the solution to the nonlinear, strain-limiting model agrees closely with the LEFM solution. This supports recent asymptotic theoretical studies of anti-plane and plane-strain fracture boundary value problems. Further, we also study the behavior of cleavage stress in the neighborhood of crack-tip. The numerical results indicate that the cleavage stress is largest along a line directly ahead of the crack-tip in agreement with the classical LEFM solution for pure mode I loading.