Plane Strain Problem Research Articles

Elastic Fields Around Multiple Stiff Prestressed Arcs Located on a Circle

Abstract The plane strain problem of an isotropic elastic matrix subjected to uniform far-field load and containing multiple stiff prestressed arcs located on the same circle is considered. The boundary conditions for the arcs are described by those of either Gurtin–Murdoch or Steigmann–Ogden theories in which the arcs are endowed with their own elastic energies. The material parameters for each arc can in general be different. The problem is reduced to the system of real variables hypersingular boundary integral equations in terms of two scalar unknowns expressed via the components of the stress tensors of the arcs. The unknowns are approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The coefficients in series are found from the system of linear algebraic equations that are solved using the collocation method. The expressions for the stress intensity factors are derived and numerical examples are presented to illustrate the influence of governing dimensionless parameters.

A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion

We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.

A Volterra edge dislocation problem in second-order elasticity theory

A displacement function technique has been developed to study plane strain problems in second-order elasticity theory for incompressible elastic materials. The formulation culminates in the development of a single inhomogeneous biharmonic equation of the form [Formula: see text] for the second-order displacement function [Formula: see text]. In the second-order, the isotropic stress associated with constitutive behaviour is governed by an inhomogeneous harmonic equation of the form [Formula: see text]. Both functions [Formula: see text] and [Formula: see text] depend only on the solution to the analogous problem in classical elasticity. While there are similarities to the Airy stress function, the displacement function approach enables the evaluation of the first- and second-order displacement fields purely through its derivatives, thus eliminating the introduction of arbitrary rigid body terms normally associated with formulations where the strains derived from a stress function approach need to be integrated. The displacement function approach is applied to develop a second-order elasticity solution to the problem of a Volterra edge dislocation.

An elliptical compressible liquid inclusion in an infinite nonlinearly coupled thermoelectric matrix

We study the plane strain problem associated with an elliptical thermoelectrically insulated and compressible liquid inclusion embedded in an infinite homogeneous and isotropic nonlinearly coupled thermoelectric matrix under uniform remote electric current density and energy flux. With the aid of conformal mapping and analytic continuation, an exact closed-form solution to the problem is derived. In particular, an elementary expression for the internal uniform hydrostatic tension within the compressible liquid inclusion is obtained. Our result shows that the internal uniform hydrostatic tension is unaffected by the remote energy flux and is proportional to the square of the two components of the remote electric current density vector. Also presented is the hoop stress distribution along the elliptical interface on the matrix side.

Plane strain problem of an elastic matrix containing multiple Gurtin–Murdoch material surfaces along straight segments

This paper presents the study of the plane strain problem of an infinite isotropic elastic medium subjected to far-field load and containing multiple Gurtin–Murdoch material surfaces located along straight segments. Each material segment represents a membrane of vanishing thickness characterized by its own elastic stiffness and residual surface tension. The governing equations, the jump conditions, and the surface tip conditions are reviewed. The displacements in the matrix are sought as the sum of complex variable single-layer elastic potentials whose densities are equal to the jumps in complex tractions across the segments. The densities are found by solving the system of coupled hypersingular boundary integral equations. The approximations by a series of Chebyshev’s polynomials of the second kind are used with the square root weight functions chosen to satisfy the tip conditions automatically. Numerical examples are presented to illustrate the influence of dimensionless parameters and to study the effects of interactions.

Stress Analysis of Semi-covered Excavation Foundation Pit in Metro Stations based on Incremental Method

The station foundation pit of Ningbo Metro Line 2 is taken as engineering background, the support structure form of the semi covered excavation foundation pit is introduced. The standard section of the foundation pit is considered as a plane strain problem, a force calculation and analysis model for the half cover system structure is established based on the incremental method, concentrated load of a waste truck is applied within 5.0 m cover plate range outside the middle column and an impact coefficient of 0.4 is considered, overload with value of 20kPa is applied to the rest parts of the cover plate and 5.0 m outside the diaphragm wall. The excavation of the foundation pit is divided into 5 stages, the surface overload, structural self weight, soil pressure and elastic resistance increment, and support axial force are applied respectively, and the bending moment increment of the connecting wall and cover plate system at each excavation stage are calculated. The bearing capacities of the diaphragm wall and cover plate reinforcement are respectively calculated based on the rectangular bending normal section components, and they both meet the strength requirements.

Two circular incompressible liquid inclusions

AbstractWe apply Muskhelishvili's complex variable method to solve the plane strain problem associated with two circular incompressible liquid inclusions embedded in an infinite isotropic elastic matrix subjected to uniform remote in‐plane normal stresses. An analytical solution to the problem is derived primarily with the aid of conformal mapping and analytic continuation. To demonstrate our analytical solution, we present detailed numerical results characterizing the internal uniform hydrostatic stresses within the two liquid inclusions, the nonuniform hoop stresses along the two circular liquid‐solid interfaces on the matrix side and the nonuniform rigid body rotations within the two circular liquid inclusions.

Exact analytical solution for deep tunnels in viscoelastic–plastic rock considering the actual loading path

The rock rheology and time process of tunnel construction make the mechanical responses of rock a function of time and are closely related to the loading path, however, these factors are not properly or correctly considered in analytical studies. The analytical study for deep tunnels in rheological rock is performed in this study, strictly taking into account the viscoelastic‒plastic characteristic of surrounding rocks and the actual loading path during excavation and supporting stages.The plane-strain problem of a circular tunnel in infinite plane under hydraulic far-field stress is simplified, with the time-dependent supporting pressure exerted at a specific time to consider the effect of delay installation of yielding support. The solving procedure as well as the time-dependent analytical solution of displacement, stress, and plastic radius in the excavation and supporting stages are presented in detail when the rocks satisfy the Unified strength theory and perfect plasticity. The analytical solutions agree very well with the numerical results, where the time procedure of tunnel excavation and support is simulated in the numerical model. Furthermore, the proposed solution can predict well the long-term deformation of the Shangxinzhai tunnel, which validates its applicability in real engineering. It is found that the tunnel deformation is underestimated if the unloading stage is not correctly considered. Based on the analytical solution, the influence of the supporting time, start time of the yielding stage and viscosity coefficient on the time dependence of tunnel convergence and stresses is investigated.

A new method of solving plane-strain boundary value problems for the double slip and rotation model

Abstract A method of solving plane-strain boundary value problems for a reduced version of the double slip and rotation model is developed. It is assumed that the intrinsic spin vanishes. Elastic strains are neglected. The Mohr–Coulomb yield criterion is adopted. An analogy between the solutions for this model and classical rigid plastic solutions of pressure-independent plasticity is revealed. The method is based on introducing auxiliary variables that satisfy the equation of telegraphy in regions where both families of characteristics are curved. Therefore, Riemann's method can conveniently be applied to solving boundary value problems. The method is employed for analyzing the processes of plane-strain drawing and extrusion through a wedge-shaped die. Friction is neglected. The solution is given in terms of ordinary integrals. The effect of the angle of internal friction on processes’ parameters is revealed. The solution reduces to available solutions of pressure-independent plasticity if the angle of internal friction vanishes.

An asymptotic formulation for boundary value problems in a non-local elastic half-space

The primary purpose of this study is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. For the sake of simplicity, the non-locality is limited to the vertical direction, which is represented by a one-dimensional exponential kernel, and the problem is formulated within the framework of Eringen’s theory. The proposed asymptotic approach is based on the assumption that the internal characteristic length is significantly smaller than a typical wavelength. This assumption allows for the development of an asymptotic formulation that expresses the considered boundary value problem in terms of local stresses. Additionally, the formulation includes explicit correction terms to the classical boundary conditions, which arise from the non-local effects. As an example application of the derived formulation, the Rayleigh surface waves in a plane strain problem are considered. Finally, numerical results are presented for certain specific values of elastic parameters to illustrate the effects of non-locality on the analyzed system.

Plane strain problem of flexoelectric cylindrical inhomogeneities

In nanotechnology, flexoelectric solids exhibit notable electrical polarization induced by internal strain gradients, rendering them promising for various applications. However, inherent material imperfections are inevitable. Particularly within flexoelectric solids, substantial strain gradients exist in proximity to internal defects, resulting in localized concentration of electrical polarization and potential structural failure. Among the various defect types, circular-shaped inhomogeneity is prevalent. This paper comprehensively investigates the plane strain problem on cylindrical inhomogeneities within flexoelectric solids. The full-field analytical solution is derived for this problem for the first time. Given that flexoelectric theory encompasses pure strain gradient elasticity theory, the strain gradient elasticity solution for plane strain cylindrical inhomogeneities is also established for the first time. This study reveals that the stiffness and size of the inhomogeneity, along with the loading ratio in two directions, exert a noteworthy influence on the local electromechanical coupling behavior near the inhomogeneity. Finally, the mixed finite element method is utilized to approximate the solution numerically, and the close agreement between the finite element results and the analytical solution demonstrates this study’s reliability and rigor. Therefore, this investigation imparts valuable insights into examining defects in flexoelectric solids and serves as a foundation for studying more intricate defect typologies.

An incompressible liquid slit between dissimilar anisotropic elastic media

We use the Stroh sextic formalism to solve the generalized plane strain problem associated with a finite incompressible liquid slit lying between dissimilar anisotropic elastic half-planes subjected to a system of uniform remote stresses. The liquid slit admits an internal uniform hydrostatic stress field and is perfectly bonded to the surrounding media. By enforcing the interface conditions on the solid-solid and liquid-solid interfaces, a Riemann-Hilbert problem of vector form is obtained and is solved analytically using a decoupling procedure. An application of the residue theorem following the imposition of the incompressibility condition of the liquid slit leads to the determination of the internal uniform hydrostatic tension within the liquid slit. When the elastic bimaterial is orthotropic, the internal uniform hydrostatic tension is equal to the remote normal stress component perpendicular to the surfaces of the liquid slit. The presence of the liquid slit will inhibit crack growth along the anisotropic elastic bimaterial interface.

Two-dimensional problem of an infinite matrix reinforced with a Steigmann–Ogden cylindrical surface of circular arc cross-section

The plane strain problem of an elastic matrix subjected to uniform far-field load and containing a Steigmann–Ogden material surface with circular arc cross-section is considered. The governing equations and the boundary conditions for the problem are reviewed. Exact complex integral representations for the elastic fields everywhere in the material are provided. The problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the first component of the surface stress tensor (surface stress) and the remaining component of that tensor and its second derivative, along with various problem parameters. The two unknowns are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters. The connection of the problem with that of rigid circular arc is discussed.

A hierarchy of asymptotic models for a fluid-loaded elastic layer

A hierarchy of asymptotic models governing long-wave low-frequency in-plane motion of a fluid-loaded elastic layer is established. In contrast to a layer with traction-free faces, modelled by Neumann boundary conditions, a fluid-loaded one assumes more involved conditions along the interfaces, dictating a special asymptotic scaling. The latter corresponds to a fluid-borne bending wave, controlled by elastic stiffness of the layer and fluid inertia. In this case, the transverse inertia of the layer and fluid compressibility do not appear at zero-order approximation. The first-order approximation is associated with a Kirchhoff plate, immersed into incompressible fluid. In the studied free vibration setup, the fluid compressibility has to be taken into account only at third order, along with elastic rotary inertia. Transverse shear deformation enters the second-order approximation along with a few other corrections. The conventional impenetrability condition has to be also refined at second order. Dispersion relations corresponding to the developed asymptotic models are compared with the polynomial expansions of the full dispersion relation, obtained from the plane-strain problem of linear elasticity.

Fracture analysis in 2D plane strain problems for composite materials containing hard inclusions and voids using an extended consecutive-interpolation quadrilateral element

This paper investigates fracture mechanics in particle-reinforced composites by using the extended finite element method enhanced by the consecutive-interpolation quadrilateral element. These composite materials have discontinuous boundaries such as cracks, voids, holes, and soft inclusions. And the extended consecutive-interpolation quadrilateral element (XCQ4) is employed to model these boundaries in two-dimensional linear elastic deformation problems. XCQ4 combines the enrichment functions in the traditional extended finite element method with the consecutive interpolation on a 4-node quadrilateral element. This element uses both nodal values and averaged nodal gradients as interpolated conditions. In fracture analysis, the stress intensity factors (SIFs) are important parameters that must be defined. In this study, the values of SIFs at the crack tips are evaluated with the help of the interaction integrals approach. The obtained numerical results are compared with other reliable results showing high accuracy and convergence rate of the XCQ4 element.

Static Solutions for Plane Strain Problem of Coupled Diffusion and Deformation

Static Solutions for Plane Strain Problem of Coupled Diffusion and Deformation

Modeling layered composite rock with excavated tunnels subjected to ground loads

Tunnels excavation in layered composite rock subjected to ground loads to make the best of underground space is an important engineering problem in more and more mega-cities. Such a three-dimensional case, layered composite rock with excavated tunnels, is usually simplified as a two-dimensional plane strain problem in mechanics. The present study reports a Semi-Analytical Method (SAM) to predict elastic field of an excavated layered composite rock with the assistance of the Equivalent Inclusion Method (EIM) that can handle elastic field of the layers and tunnels simultaneously. In the EIM, the layer and tunnel are considered as inhomogeneities of different elastic moduli from the base rock, and the inhomogeneities are further equivalent to inclusions of identical elastic modulus to the base rock but including eigenstrains. Effectiveness of the SAM is examined by the Finite Element Method (FEM) through comparing their results. Moreover, a yield approximate index (YAI) is introduced to evaluate safety of excavated layered composite rock and acoustic emission (AE) event is used to count the number of plastic yield elements. Finally influences of depth of and distance between the tunnels, thickness, elastic modulus, number of the layer, range of ground load on principal stress, YAI, plastic zones are discussed, which may have potentials to provide guidance for practical engineering applications of tunnel excavation.

Stress Relaxation in Bended Viscoelastic Plate with Tension-Compression Asymmetry

The paper presents closed-form analytical solution to the plane-strain problem of stress relaxation in a bended plate with tension-compression asymmetry (TCA) in viscous properties. Reversible and irreversible strains are assumed to be finite. We utilize a linear viscous model with equivalent stress that is piecewise linear function of the principal stresses with TCA parameter. The specific features of the solution are discussed.

Hollow cylinder with thermally induced gradient of the yield strength: The gain in loading capacity under pressure

Plane strain problem of a thick-walled cylindrical pipe under internal and external pressures, made of elastic–perfectly plastic material with a radial dependence of the yield stress (produced, e.g., by one-sided accelerated cooling) described by a polynomial function is considered in the framework of small deformations theory, using the von Mises yield criterion. It is found that, at the fixed wall thickness and strength gradient, the gain in the ultimate loading capacity depends on the magnitude of the gradient but its dependence on the gradient direction, as well as pipe radii is rather weak. At the same time, the onset of yielding strongly depends on all the mentioned parameters. It is also shown that one-sided accelerated cooling enhances the loading capacity of the pipes, as compared to the normalized state whereas comparison with the steel that has experienced quenching with tempering may produce diverse results.

Finite elements limit analysis formulation using the power type yield criterion for plane strain and axisymmetric stability problems

The current research employs a power type yield criterion which accounts for the dependency of the shear strength parameters, namely, cohesion and internal friction, on stress level which becomes quite effective in obtaining very accurately the solutions of different stability problems in geomechanics involving soil and rock media. The present manuscript introduces plane strain and axisymmetric formulations which deal with the power type yield criterion and are based on the lower and upper bound finite elements limit analysis in conjunction with the power cone programming (PCP). To demonstrate the efficacy of the proposed formulations, the problems of the determination of the bearing capacity of strip and circular foundations in a cohesive-frictional soil medium have been addressed. In the case of a strip footing, the earthquake inertial forces in a horizontal direction have also been incorporated. The adaptive mesh pattern has been employed to improve the accuracy of the solution at the expense of minimal computational effort. The results have been thoroughly validated by making the comparison of the obtained solutions with that reported in literature. The proposed formulations are expected to be quite useful in solving different stability problems in geomechanics.