Plane Strain Problem Research Articles

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.

Stress solution for an arbitrarily shaped tunnel with a neighboring circular cavity

This paper gives an analytical stress solution for the plane strain problem that the deep rock mass contains an arbitrarily shaped tunnel and a circular cavity subjected to in situ stresses. Based on the discovery that the mapping function for an arbitrarily shaped hole can also manage a tunnel and a circular hole mapping into two circles, the plate bounded by the two holes is conformally mapped onto an annulus. The complex variable method is used to establish the stress boundary condition, and then the two analytical functions are solved by the power series method. Comparison with ANSYS solution is made to validate the correctness of deductions and solving process. To consider practical underground engineering, the cases of different lateral stresses, distances of the two holes, sizes, and positions of the cavity are discussed.

Bearing Capacity of Karst Cave Roof under Pile Foundation Load Using Limit Analysis

Reducing the computational workload by simplifying the analysis of karst foundations into a plane strain problem can yield significant advantages. Yet, such an approach fails in reproducing the engineering situation in a rigorous manner. In this regard, this paper proposes an upper-bound method that can effectively analyze the bearing capacity of three-dimensional karst foundations. This method is utilized to investigate the impact of various pile diameters, the ratio of roof thickness to pile diameter, and the ratio of cave width to pile diameter on the stability of karst foundations. The validity of this method is established through an illustrative example. The outcomes illustrate that when subjected to both tensile and compressive horizontal stresses if the maximum horizontal stress surpasses the tensile strength of the rock mass, the roof rock mass may suffer damage. Increasing the ratio of roof thickness to pile diameter can bring down the horizontal stress value. The stability factor is largely influenced by the ratio of roof thickness to pile diameter. The most prominent growth trend of the stability factor is observed when the ratio is less than 3. If the ratio of the roof thickness to pile diameter exceeds 3, the prediction of the bearing capacity estimation for the karst foundation in three-dimensional circumstances is more conservative than that in two-dimensional circumstances.

Solving multi-material problems in solid mechanics using physics-informed neural networks based on domain decomposition technology

Physics-informed neural networks (PINNs) are widely used in the field of solid mechanics. Currently, PINNs are mainly used to solve problems involving single homogeneous materials. However, they have limited ability to handle the discontinuities that arise from multi-material, and they lack the capability to rigorously express complex material contact models. We propose a method for solving multi-material problems in solid mechanics using physics-informed neural networks. Inspired by domain decomposition technology, the calculation domain is divided according to the geometric distribution of materials, with different subnetworks applied to represent field variables. This study explains how the invariant momentum balance, kinematic relations, and different constitutive relations controlled by the material properties are incorporated into the subnetworks, and use additional regular terms to describe the contact relations between materials. Various test cases ranging from two-dimensional plane strain problems to three-dimensional stretching problems are solved using the proposed method. We introduce the concept of parameter sharing in multi-task learning (MTL) and incorporate it in the proposed method, which yields additional degrees of freedom in choosing the sharing structure and sharing mode. Compared with common physics-informed neural network algorithms, which are based on fully independent parameters, we develop a network structure with partial sharing structure and all-sharing mode that achieves higher accuracy when solving the example problems.

On the plane strain problem of dynamical flexoelectric effect in dielectrics subject to Sinc-shaped surface loading

Flexoelectricity is a nanoscale phenomenon in dielectric crystals, where the mechanical loads resulting from the first strain gradient shares producing electricity. In this work, we present a numerical investigation for the plane strain problem of the dynamical flexoelectric effect in isotropic dielectrics. A surface loading based on the Sinc function, that is, sincπ(x), with a temporal exponentially decay, is applied to the upper surface of a semi-infinite dielectric crystal. Two cases for the electric potential are considered: (1) Open circuit, in which there is zero charge on the terminals; and (2) Short-circuit, where there is a given value for the potential of the terminals. This type of surface loading is often used in applications such as seismic analysis and electrical engineering, especially signal processing and digital signal processing. Also, it can be used to model other types of dynamic loads, such as wind or wave loads. Furthermore, this setting generates a two-dimensional formulation, known as the plane strain problem. Laplace and Fourier transform techniques are employed for solving the field equations. Suitable numerical techniques are implemented to present the solution to the physical domain. We choose the Strontium Titanate material (SrTiO3) for the two-dimensional simulation. Discussion for the behavior of the dynamical polarization, electric field, electric potential, and stresses is provided in the two-dimensional setting. Effects of the model parameters on electricity generation and behavior are studied.

Problem of plane stress state of two-layer body in dynamic elastic-plastic formulation

Composite materials are widely used in industry and everyday life. Many different methods are used to calculate and develop composite materials. Many methods of calculation and design of such materials are successfully used. In this article, for the design of composite and reinforced materials, a technique for solving dynamic contact problems in more precise an elastic-plastic mathematical formulation is used. To consider the physical nonlinearity of the deformation process, the method of successive approximations is used, which makes it possible to reduce the nonlinear problem to a solution of the sequences of linear problems. The problem of a plane stress state (PStS) of a beam made from the composite reinforced double-layered material is being solved in dynamic elastic-plastic mathematical model. The reinforced or armed material consists of two layers: the upper (first) thin layer of solid steel and the lower (second) main layer of glass. This composite base is rigidly attached to an absolutely hard half-space. Rigid adhesion of the layers to each other is assumed. Glass is a very strong and very fragile material at the same time. The fragility of glass is due to the fact that there are many microcracks on the surface, and when a load is applied to the glass surface, these microcracks begin to grow and lead to the destruction of glass products. If we glue or immobilize the tops of microcracks on the surface, we will get a strong reinforced armed material that will be lighter, stronger and not subject to degradation of material properties such as aging, corrosion and creep. The impact process was modelled as a non-stationary plane stress state problem with an even distributed load in the contact area, which changes according to a linear law. The fields of the Odquist parameter and normal stresses were studied and compared to corresponding results of plane strain (PSS) problem with the same material of layers, same their thickness and size of the contact area. The upper reinforcing layer of metal or steel can be applied to the glass surface so that metal or steel atoms penetrate deeply, fill microcracks and bind their tops. The top layer can be quite thin.

A Three-Dimensional Analytical Solution of Stress Field in Casing-Cement-Stratum System Considering Initial Stress State

Accurate stress field calculation of the casing-cement-stratum system is crucial for evaluating wellbore integrity. Previous models treated in-situ stress as boundary pressure loads, leading to unrealistic infinite displacements at infinity. This study presents a three-dimensional (3D) analytical solution for the stress field within the casing-cement-stratum system in inclined wells, considering in-situ stress and hydrostatic stress in cement as the initial stress state and taking into account stress components related to the axial direction. Assuming a plane strain condition and superimposing the in-plane plane strain problem, elastic uni-axial stress problem and anti-plane shear problem, a 3D analytical solution is obtained. Comparisons with previous models indicate that the existing model overestimates the absolute values of stress components and failure potential of casing and cement in both 2D and 3D scenarios. The presence of initial stress in cement greatly increases the absolute value of the compressive stress state but decreases the failure potential in cement, which has not been well studied. Additionally, a low Young’s modulus and high initial stress state of the cement benefits the cement’s integrity since the maximum Mises stress significantly decreases. The new 3D analytical solution can provide a benchmark for 3D numerical simulation and quick assessment for wellbore integrity.

The effects of structure-soil-structure interaction on seismic response of high-rise buildings equipped with optimized tuned mass damper

Nowadays, various vibration control tools and methods are used to improve the behavior of structures. Tuned Mass Damper (TMD) is one of the simplest and most used passive control tools, which consists of a combination of spring, mass, and a damping mechanism. However, some factors, such as the type of soil beneath the structure and the existence of another structure, can affect the damper's performance. This study investigates the effects of Structure-Soil-Structure Interaction (SSSI) on the seismic response of two 20-story steel moment frames equipped with TMD. Assuming an elastic behavior for the soil and plane strain problem, the reduced stiffness matrix of the foundation-soil-foundation system is obtained using static analysis of the 2D finite element method in Abaqus. Then, it is applied to 2D linear models of structures using a combination of springs and a newly developed element in OpenSEES. Furthermore, particle swarm optimization algorithm is used to optimize the design parameters of TMD. The seismic responses are obtained under ten far-field seismic records from the time history analysis. The results reveal that considering SSSI can significantly increase the responses of both structures. Nevertheless, exploiting TMD with optimized parameters in the 20-story structures reduces the seismic responses.

Study on Transverse Seismic Response Characteristics of Large Diameter Vertical Double-Layer Overlapping Pipe Jacking in the Soil-Rock Composite Stratum

The sharp change of stiffness in the soil–rock combination stratum is the weak point in the seismic design of the pipe jacking structure. To study the seismic response characteristics of vertical double-layer overlapping pipe jacking, based on the typical soil–rock combination strata in Jinan, two electric power pipe jacking tunnels of 3.6 m diameter were studied as the research objects, where the upper pipe jacking is located in the soil and the lower pipe jacking is located in the composite stratum of half soil and half rock. The soil–rock-overlapping tunnel system was deemed as a plane strain problem. By using the dynamic time history method and considering the non-linearity of material, the seismic response characteristics of a vertical overlapping pipe jacking tunnel under seismic wave were discussed from five aspects, acceleration response, displacement response, stress response, soil interlayer response and the influence of soil–rock combination stratum. The results indicate that under the action of ground motion, the peak acceleration and relative horizontal displacement of the upper pipe jacking are greater than those of the lower pipe jacking; small pipe jacking spacing will lead to the aggravating earthquake failure effect; due to the stiffness difference, the relative horizontal displacement and stress of pipe jacking structure at the soil–rock interface change abruptly. The vertical double-layer arrangement of the pipe jacking increases the buried depth, and the stratum is prone to be uneven hardness. Therefore, the seismic design and relevant structural measures of large diameter vertical overlapping pipe jacking structure should be strengthened.

A modified bond-based peridynamic model without limitations on elastic properties

This study proposes a novel modified bond-based peridynamic model based on bonds' classifications. Bonds are classified as per the equivalent hypothetical local strains into horizontal normal, vertical normal, and shear bonds. While the classical Bond-Based PeriDynamic (BB-PD) only considers the stretch of bonds, the proposed model takes into account all strain components of the bonds. A local imaginary element is considered around each bond to estimate its true strains. Constitutive relations are derived by equating the strain energies of the bonds' deformations to the strain energies of classical continuum mechanics for a generalized combined loading condition. A novel critical stretch criterion and a critical angle criterion are proposed to predict the failure of normal and shear strain bonds, respectively. Moreover, it is demonstrated that, unlike the classical BB-PD, the proposed model imposes no restrictions on the value of Poisson's ratio. The model is validated by analyzing some intact plane stress and plane strain problems subjected to mechanical and thermal loads. In addition, the deformation and damage contours, as well as the corresponding stress-strain responses, are presented and validated for various problems with pre-existing defects using the extended finite element method.

On the geometry transferability of the hybrid iterative numerical solver for differential equations

The discovery of fast numerical solvers prompted a clear and rapid shift towards iterative techniques in many applications, especially in computational mechanics, due to the increased necessity for solving very large linear systems. Most numerical solvers are highly dependent on the problem geometry and discretization, facing issues when any of these properties change. The newly developed Hybrid Iterative Numerical Transferable Solver (HINTS) combines a standard solver with a neural operator to achieve better performance, focusing on a single geometry at a time. In this work, we explore the “T” in HINTS, i.e., the geometry transferability properties of HINTS. We first propose to directly employ HINTS built for a specific geometry to a different but related geometry without any adjustments. In addition, we propose the integration of an operator level transfer learning with HINTS to even further improve the convergence of HINTS on new geometries and discretizations. We conduct numerical experiments for a Darcy flow problem and a plane-strain elasticity problem. The results show that both the direct application of HINTS and the transfer learning enhanced HINTS are able to accurately solve these problems on different geometries. In addition, using transfer learning, HINTS is able to converge to machine zero even faster than the direct application of HINTS.

Static Response of Circular Cavities With a Non-Radial Crack Subject to Antiplane Stress

Abstract Tunnels, pipelines, and other subterranean circular cavities are common components of modern infrastructure. In addition, seismic activities are common in many areas with pipelines, which may put these structures under unknown risk of fracture. A particular risk case of interest can be characterized as a plane strain problem with a circular cavity and crack in an infinite plane under antiplane stress. Antiplane, i.e., mode III, loading has seen less study relative to modes I and II due to the lower risk factor in structures that are especially vulnerable to fracture (e.g., in the automotive and aerospace industries), and the increase in complexity compared to modes I and II. The work here further explores this phenomenon on circular cavities, and particularly, the effect of non-radial cracks on the stress intensity factor via a parametric study. The study introduces a semi-analytical method and also uses commercial finite element software to further expand on the investigation.

Система колінеарних електрично та магнітно проникних тріщин між двома п’єзоелектромагнітними матеріалами

The plane strain problem for the bimaterial space consisting of two piezoelectromagnetic half-spaces with a finite set of cracks along the material interface is considered. It is assumed that the edges of the crack are electrically and magnetically permeable, and a mixed mechanical load and electric and magnetic fields can be set at infinity. The presentations of electro-magneto-mechanical quantities through a piecewise analytical vector function are constructed with the use of constitutive equations for piezoelectromagnetic materials. By means of these presentations the problem of linear relationship is formulated and solved analytically. Using the obtained solution, all the necessary mechanical, electrical, and magnetic components along the material interface as well as the energy release rate for all crack tips are found analytically. The asymptotic presentations of all fields around the crack tips and the crack closure integral method were used. Numerical results are obtained for different loads, location of the cracks, their number and lengths. A significant increase in energy release rates was demonstrated because of cracks converge, especially for the tips located closest to each other.

Towards improving the 2D-MITC4 element for analysis of plane stress and strain problems

In this paper, we improve the performance of the 2D-MITC4 element for analysis of plane stress and plane strain problems. While the element shows excellent convergence behavior in regular meshes, it exhibits accuracy loss in distorted meshes, as many other elements do. We focus on improving performance in distorted meshes for general use in engineering practice. We simplify the assumed strain field of the original 2D-MITC4 element and introduce the geometry dependent Gauss integration scheme, in which integration points vary according to element geometry distortion. Consequently, a new 2D-MITC4 element is developed. The new 2D-MITC4 element passes the basic tests: patch, zero energy mode, and isotropy tests. Through various numerical examples, improved convergence behaviors of the new element are demonstrated, especially in distorted meshes.

A Novel Model of Semiconductor Porosity Medium According to Photo-Thermoelasticity Excitation with Initial Stress

Investigated is a novel model in the photo-thermoelasticity theory that takes into account the impact of porosity and initial stress. A generalized photo-thermoelastic that is initially stressed and has voids is taken into consideration for the general plane strain problem. The solutions for the fundamental variables in two dimensions are obtained using the Laplace–Fourier transforms method in two dimensions (2D). Physical fields such as temperature, carrier concentration, normal displacement, and change in volume fraction field can all be solved analytically. The plasma of electrons, thermal load, and mechanical boundary conditions at the porosity medium’s free surface are used to show certain illustrations. The context of the Laplace–Fourier transformation inversion operations yields complete solutions. To complete the numerical simulation and compare several thermal memories under the influence of the porosity parameters, silicon (Si), a semiconductor porosity material, is used. The main physical variables are described and graphically displayed with the new parameters.

Semianalytical Model of Unloading-Induced Pore-Water Pressure and Consequent Consolidation Settlement of Surcharge Preloading Railway Subgrade

Surcharge preloading is a suitable foundation treatment for high-speed railways in soft soil regions. As a temporary preload on the foundation, the surcharge load must be removed before track construction. In this process, rebound deformation of the foundation and variation in the pore-water pressure have been widely observed. These phenomena significantly influence the further evolution process of foundation consolidation. Although Biot’s consolidation theory is suitable for unloading-induced soil consolidation, the conventional Laplace transform method for solving Biot’s equations entails a time-consuming procedure and cannot reflect the time-dependent characteristics of the preloading and unloading process. To address these issues, a time difference method is proposed in this work. The ordinary differential equation in the transformation domain is obtained using the backward two-step difference method, and a physical analytical solution of the plane strain problem of overloaded subgrade is obtained via the Fourier inverse transform method. The proposed solution is validated through a comparison with existing literature data and actual engineering practices. Finally, based on this calculation method, the influence of the surcharge preloading amount, loading time, and soil properties on pore-water pressure dissipation and distribution after surcharge preloading removal is investigated.

External work rate and dissipation during crack growth in a viscoelastic material

We analyse crack growth in viscoelastic material by use of a correspondence principle that allows elasticity solutions for traction boundary value problems in plane strain to be converted to viscoelasticity solutions. We consider an edge cracked strip in tension and assess 2 limiting cases of the geometry. In one case, we allow the component to become a geometry that is an infinite body with a semi-infinite crack, or a large body with a long crack. In the other case, we consider a component of finite width with a remaining ligament that is very small compared to the width of the strip. In these geometries we compute the work done by the applied load per unit area of crack growth and a measure that we consider to represent the dissipation per unit area of crack growth. We do so for a standard viscoelastic material with a single retardation time and for a Maxwell material with a single viscous element. In the latter case, our computation of the dissipation per unit area of crack growth is definite. We consider various aspects of the behaviour of the work done per unit area of viscoelastic crack propagation and the dissipation per unit area of crack advance. These include the extent to which these parameters depend on the rate of crack propagation, the extent to which these parameters are independent or dependant on component geometry, and the extent to which these parameters exhibit transient behaviour during crack growth at a steady rate under a constant applied stress intensity factor. A motivation is the common insight that a parameter is probably more useful as a measure of material behaviour if it is relatively insensitive to component geometry and if, during steady state response in terms of crack growth rate and applied stress intensity factor, the parameter also exhibits a steady state. We find outcomes that vary quite considerably depending on the case that we consider. We provide our results for the reader to use when considering the various models that have been proposed in the literature for viscoelastic crack propagation. We observe, however, that crack propagation models based on a rupture process zone interacting with material viscoelasticity are much simpler to implement in bodies with finite geometry than those based on quantifying the dissipation per unit area of crack growth. In this regard, we conclude that viscoelastic crack growth models that are based on quantifying the dissipation per unit area of crack growth are, by themselves, not a promising concept as we see no obvious way to extend them to provide unique results in components having a finite geometry. We further conclude that a reliable viscoelastic crack growth model should include a crack tip rupture process zone at the crack tip.

Assessment of an Assumed Strain-based Quadrilateral Membrane Element

This paper describes the development of a simple quadrilateral strain-based element for plane stress and strain problems. This element has five nodes, four located at its corners and one at the center. Each of the four corner nodes had two essential external degrees of freedom (u, v), while the center node had three degrees of freedom (u, v, ɵ); the static condensation method was used for the internal node. This element was used for both linear and dynamic analysis. Its performance was assessed using a variety of membrane and axisymmetric analysis problems. The obtained results demonstrated the good performance and accuracy of the proposed element.

Analytical Solutions for a Deep-Buried Tunnel with Noncircular Cross Sections in Orthotropic Rock Mass

For the generalized plane strain problem of a deep-buried tunnel excavated in orthotropic rock mass under far-field biaxial compression, the stress and displacement are the most important basis for determining tunnel support design. First, by using the complex variable method, the conformal mapping and affine transformation functions are represented by the same variable. Two stress functions are obtained by the integral method, and thus, the analytical solutions for stress and displacement in orthotropic rock mass are obtained. Second, taking a horseshoe-shaped tunnel as an example, the stress and displacement distributions around the tunnel are given by using the analytical and the finite element method software ANSYS, respectively. The analytical and numerical results are in good agreement, especially at the tunnel edge. Finally, compared with isotropic rock mass, the results show that material constants of the rock mass have little effect on the stress and significant effect on the displacement.

Indentation of a free beam resting on an elastic substrate with an internal lengthscale

The plane strain problem of a slender and weightless beam-plate loaded by a transversal point force in unilateral contact with a couple stress elastic foundation is investigated. The study aims to explore the consequences of the material internal lengthscale on the contact mechanics. In particular, compatibility between the beam and the foundation surface demands that both displacement and rotation match along the contact line. To this aim, couple tractions are exchanged besides the traditional contact pressure until separation between the beam and the foundation occurs. The problem is formulated making use of the Green's functions for a point force and a point couple acting atop of a couple stress elastic half-plane. A pair of coupled integral equations is thus derived, that governs the distribution of contact pressure and couple tractions, with one of them being immediately solved to provide an explicit relation between the two unknowns. In this sense, we retrieve the concept of a mechanically equivalent action, as it is the case of the Kirchhoff shear for plates. The remaining integral equation sets a cubic eigenvalue problem, whose linear term accounts for the microstructure. Its numerical solution is sought by expanding the equivalent contact pressure in series of Chebyshev polynomials vanishing at the contact region ends points, namely the lift-off points, and then applying a collocation strategy. The contact length, the distributions of contact pressure and couple tractions under the beam and the shearing force and bending moment along the beam are then obtained as a function of the material characteristic length. Results clearly indicate that accounting for the material internal lengthscale is mainly realized through exchange of the couple tractions, in the lack of which results much resemble those of the classical solution. Specifically, greater contact lengths and a stronger focusing effect about the loading point are encountered, which become very significant when the contact length approaches the internal lengthscale.