Plane Elasticity Problems Research Articles

Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

Equilibrium Solution of Two – Dimensional Non-Homogeneous Equations in the Theory of Elastic Mixtures

Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force – stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the particle and the stress state of the particle, which is in conformity with existing experimental results.

Improved dynamic analysis for plane elasticity problems with a boundary meshfree method and distributed sources

In this paper, we present an improved dynamic analysis method to address plane elasticity problems. Distributed sources are employed in a boundary-free mesh-based approach known as Boundary Distributed Source (BDS). This method, which builds upon a modified Method of Fundamental Solutions (MFS), aims to eliminate singularities without the reliance on fictitious boundaries. Distributed sources are strategically positioned directly on the real boundary, centered over disks. The method's efficiency and simplicity are highlighted by its ability to operate without the need for precise data concerning adjacent points. The dual reciprocity method (DRM) is utilized to approximate the non-homogeneous inertial term. Furthermore, the present formulation investigates several factors that influence the convergence of the proposed method, including the number of collocation points, internal points, the radius of the circular disk, and the time step size. Based on the numerical results, it is evident that the method exhibits remarkable accuracy and efficiency when applied to plane elasticity problems including forced, harmonic, and free analyses. As compared to conventional numerical methods for dynamic analysis, the method outperforms in both accuracy and convergence rate.

Geometrically Non-Linear Plane Elasticity Problem in the Area of an Angular Boundary Cut-Out

A relevant problem in the development and improvement of numeric analytical methods for the research of structures, buildings and construction is studying the stress–strain state of structures and construction with boundaries that have complex shapes. Deformations and stresses arise in a domain with a geometrically non-linear shape of the boundary (cut-outs and cuts). These stresses and deformations have great values and gradients. Experiments carried out using the photoelasticity method show a change in the deformation order ratios for different subareas of the boundary cut-out area depending on proximity to the apex of the angular cut-out. Areas with minor deformations are observed, and areas where linear deformations and shears are more significant than rotations are also observed. In addition, areas where section rotations are more significant than linear and shear deformations are observed. According to the experimental data, the mathematical model of the SSS in the area of the apex of the cut-out of the domain boundary should take into account non-linear deformations. Hence, it is necessary to formulate the boundary value problem of the theory of elasticity, taking into account the geometrical non-linearity. The research aim of this paper is to formulate the problem of the elasticity theory taking into account the geometrical non-linearity in furtherance of the proposed mathematical model justified by the experimental data obtained using the photoelasticity method. The obtained formulation of the elasticity theory problem allows analyzing the form of the system of equations of the boundary value problem depending on the proximity of the considered area to the irregular point of the boundary, i.e., taking into account the difference in the effect of linear and shear deformations, rotations and forced deformations on the solution to the geometrically non-linear elastic problem dealing with forced deformations in the area of an angular cut-out of the boundary of the plane domain.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

Lattice sums for double periodic polyanalytic functions

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum S_2 is equal to pi for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.

Iterative decoupling procedure for the coupled 1D problems on example of closed cylindrical shell at static loading

The general iterative method of formal decoupling of two coupled 1D problems is suggested. Its essence consists in the alternative consideration of one of two problems as the main one, the aim of which is to produce the eigenfunctions. Then the other problem is considered as an auxiliary one, which is aimed to obtain the particular solution. The latter is presented as a linear combination of already known eigenfunctions, and then is substituted again into the first problem to refine the constants of main governing differential equation and to get the corrected eigenfunctions. The convergent iterative procedure of gradual refinement of the above constants is elaborated. The accuracy and technique of application is considered in details for the closed cylindrical shell problem, which is expanded in Fourier series with respect to circumferential coordinate. This actually leads to coupled 1D problems – one is the plane elasticity problem, and other – the plate problem. The results of calculation show the high accuracy of the method, which allows to get any required number of significant digits in eigenvalues. The comparison with known theoretical and FEM results for radial displacement and axial bending moment distribution at the vicinity of point of concentrated radial force application demonstrates the efficiency and accuracy of the method.

Analysis of plane elasticity problems using the dual mesh control domain method

The dual mesh control domain method (DMCDM) proposed by Reddy combines the advantages of the finite element method (interpolation of variables) and the finite volume method (satisfaction of the global form of the governing equations). Currently, the method has been used to solve two-dimensional heat transfer, fluid and solid mechanics problems on structured meshes of linear right triangular and bilinear rectangular elements. However, such meshes are not useful from the standpoint of most practical problems which are posed on irregular domains. The present study extends the DMCDM for the analysis of plane elasticity problems involving non-rectangular domains meshed with arbitrarily shaped elements. The approach makes use of the isoparametric formulation, in which the primal mesh used for the approximation of the geometry is also used for approximating the dependent unknowns. Several boundary value problems of elasticity are solved using the DMCDM, and the results are compared with those obtained using the finite element method (FEM). The results show that the DMCDM exhibits excellent convergence and accuracy while requiring less computational cost when compared with the FEM.

Mode-I Plane Elasticity Problem of Two Asymmetrical Edge Cracks Emanating from an Elliptical Hole in Two-Dimensional Decagonal Quasicrystals

We consider the plane elasticity problem of two asymmetrical edge cracks emanating from an elliptical hole in two-dimensional decagonal quasicrystals (QCs) under remotely uniform tensile stress. A complex variation method of two-dimensional QCs is developed to solve the plane elasticity problem of two-dimensional decagonal QCs containing complex defects. The analytical solutions for the stress field and the stress intensity factors near the crack tip are expressed by using a conformal mapping technique and complex potential theory. Some special cases of the results are also obtained, such as the T-type crack, cross crack, and Griffith crack. The effects of geometrical parameters of crack configuration on the stress intensity factors are presented graphically.

Frictional shakedown of a coupled continuous contact

The plane elastic problem of a circular disc shrink fitted into a hole in an elastically similar plane, with a frictional interface, is studied. The disc is subject to a radial force and torque both of which vary with time. In the steady-state the loads vary harmonically, and a relative phase shift is considered. Frictional slip is accounted for by edge dislocations distributed along the interface. It is shown that the steady-state response, which might, for example, be cyclic slip or full adhesion (shakedown) for a particular loading regime, depends on the loading history before the steady-state loading regime is entered. This contrasts with Melan’s theorem for elastic–plastic shakedown, and is a consequence of the normal–tangential coupling in this frictional system.

Linear barycentric rational collocation method to solve plane elasticity problems.

A linear barycentric rational collocation method for equilibrium equations with polar coordinates is considered. The discrete linear equations is changed into the matrix forms. With the help of error of barycentrix polar coordinate interpolation, the convergence rate of the linear barycentric rational collocation method for equilibrium equations can be obtained. At last, some numerical examples are given to valid the proposed theorem.

Anisotropic elastic applications in composite materials using the isogeometric boundary element method

This paper describes an isogeometric analysis of the boundary element method, called IGABEM, applied to anisotropic 2D plane elastic problems. The Lekhnitskii's anisotropic fundamental solution is used and the singular integral terms is regularized. For the weak singularity kernel, the Telles transform is used. On the other hand, in the strong singularity term, the Singularity Subtraction Technique - SST is used. The shape functions used are the Non-Uniform Rational B-Spline - NURBS. Thus, the same mathematical representation of Computer Aided Design - CAD is used in the implemented computational code. This avoids the generation of meshes and provides an exact representation of most complex geometries. As a reflection of this isoparametric concept, errors from geometric interpolation are excluded, improving numerical results. Furthermore, to increase the numerical efficiency of the code, the NURBS are decomposed into Bézier curves. To evaluate the accuracy of the formulation, complex problems using high-order isogeometric boundary elements are analyzed. Their results are compared to analytical solutions showing good agreement.

Tectonic Earthquake Source Mechanism Model Based on Moment Theory

During the construction of strategic facilities, micro zoning is required to determine the peak values of terrain acceleration due to possible earthquakes. The acceleration of the soil in this case strongly depends not only on the power of the earthquake, but also on the earthquake source mechanism and the geodynamic state of the terrain. The known dislocation models of a single dipole with a moment and a pair of dipoles without a moment of the earthquake source mechanism satisfactorily describe the observed effects of the quadrant stress distribution on the Earth’s surface during earthquakes. When carrying out calculations within the framework of the theory of elasticity, the actions of the dipoles are expressed through volumetric forces. There are two known models of replacing the moments with equivalent forces: one of them is based on the equilibrium equations for an infinitely elastic space (Landau and Lifshitz,1965; Maruyama,1963), and the other is based on the representation theorem for elastic bodies, by introducing a singular internal volume, at the boundary of which there are dislocations (Vvedenskaya, 1969; Aki and Richards, 1983). Although these models involve moment effects, they themselves are derived from the momentless theory of elasticity. In our work, we propose a double dipole effect without a moment based on the moment theory of elasticity. The proposed model of the earthquake source mechanism is applied to solve the problem of stress variations in the Earth’s crust in Central Asia using the example of a particular earthquake with a simplified orientation of the rupture plane. Stress variation is understood as the difference in stresses in problems with and without an earthquake mechanism. Static stresses are obtained by solving the inverse elasticity problem with partially unknown boundary conditions. The lithosphere is a prismatic body consisting of several homogeneous blocks, the upper surfaces of which correspond to the relief of Central Asia. Verification of the results of the numerical solution is carried out by comparing the obtained stresses with previously established empirical values. As a priori stresses for solving the inverse problem, we used the solution of the elastic plane problem, the boundary conditions in which correspond to the lateral compression of the lithosphere of the region of the Indian and Arabian plates on the one hand, and the Eurasian plate on the other hand. The obtained solutions of the problem were used to analyze the geodynamic state of Central Asia. Based on the results of laboratory experiments, the unambiguity of the conclusions about the geodynamic state of the Earth’s crust (compression, extension) according to the Lode-Nadai coefficient, which are currently used by many researchers, is questioned. It is shown, contrary to earlier statements, that the values μσ = +1 and μσ = –1 can correspond simultaneously to both tension and compression cases, depending on the spatial form (ellipsoid) of the stress state. Geodynamic analysis of the Earth’s crust is carried out according to the Anderson method.

About Influence of Choosing the Force Method Principal System on the Accuracy of Calculations in Solving a Stress-Based Plane Elasticity Problem

Introduction. It is known that the solution of a problem in elasticity theory is possible based on displacements, stresses, as well as mixed method. The most common is the displacement-based finite element method (FEM), however, it has its imperfections. The aim of this article is to study the features of numerical solution of a stress-based plane elasticity problem using the finite difference method (FDM) as a possible alternative to the FEM. The options for choosing the force method principal system are considered aiming to obtain the boundary conditions in calculating a rectangular beamwall by the grid method along with the influence thereof on the final result during stresses calculation.Materials and methods. The calculation of a rectangular beam-wall is made in Balka-Stenka software developed in DSTU. Comparison between a solution made by the finite difference method and a solution made by the displacementbased finite element method is made in the LIRA-SAPR software. The accuracy of calculations is determined by performing the static balance tests of the cut-off part of a structure. For the structure under consideration, the choice of five options of the force method principal system (FMРS) is analysed.Results. For the first time the conclusion was made that in contrast to the calculation of bar systems, the choice of FMРS does not significantly affect the final stress values in solving a plane elasticity problem.Discussion and conclusion. The comparison of calculation results made by the displacement-based finite element method and by the finite difference method showed some advantages of the latter. The finite difference method can act as an alternative to the displacement-based finite element method in solving a plane elasticity problem on condition of automation of the process of building the internal forces epures in a wall beam-wall contour.

Stabilization-free serendipity virtual element method for plane elasticity

We present a higher order stabilization-free virtual element method applied to plane elasticity problems. We utilize a serendipity approach to reduce the total number of degrees of freedom from the corresponding high-order approximations. The well-posedness of the problem is numerically studied via an eigenanalysis. The method is then applied to several benchmark problems from linear elasticity and we show that the method delivers optimal convergence rates in L2 norm and energy seminorm that match theoretical estimates as well as the convergence rates from higher order virtual element methods.

Expansive behavior of high-strength self-stressing and self-compacting concrete: Experimental study and analytical model

This paper presents an experimental investigation on the expansive behavior of high-strength self-compacting concrete (HSCC) with various HCSA expansive agent (EA) contents ranging from 0 % to 12 %. The fresh properties, compressive strength and expansive ratios at free and restrained conditions are studied. The expansive process and expansive mechanism of self-stressing HSCC are analyzed with a combination of X-ray diffraction (XRD) and scanning electron microscopy (SEM) analysis. Moreover, the self-stress of expansive HSCC under the confinement of steel-tube is measured. The results show that the compressive strength of HSCC decreases with increasing EA contents, which should not be over 12 % due to the large expansions damage the concrete matrix seriously. The expansive ratios increase significantly with increasing EA contents, as well as increasing amplitudes. The free expansive ratios of concrete can reach 1.08 × 10−4, 3.03 × 10−4, 5.13 × 10−4 and 7.85 × 10−4 with the EA content ranging from 6 % to 12 % at 28 curing days, respectively. Moreover, the self-stresses of expansive HSCC with EA contents of 8–12 % are 3.2–3.7 MPa. Finally, under the framework of plane strain model in elasticity theory, a calculation analytical method is developed to predict the restrained expansive ratios of HSCC confined by steel tubes. The computational formula is also established to calculate the self-stress of expansive HSCC using the axisymmetric plane problems in elasticity. The model formula provides a good description of expansive responses of self-stressing HSCC.

Memory-free loading paths for a coupled continuous contact problem with friction

The plane elastic problem of a circular disk shrink fitted into a hole in an elastically similar plane, with a frictional interface, is studied subject to the application of a radial force and torque both of which vary with time. This is done by the use of edge dislocation distributions to represent slip. The evolution of extent and position of a slip zone as well as the magnitude of slip displacements is studied. Conditions for continuous slip, leading a history or load path independent solution are derived. The results derived prepare for the study of the shakedown behaviour of a continuous coupled contact.

Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function

A solution of an orthotropic elastic plane problem is derived for a frictional rigid flat punch on a half plane with an oblique edge crack. A mapping function is used for arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped half plane problems can be solved by exchanging the mapping function. Therefore, once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, the stress distributions are shown for a frictional coefficient and an oblique edge crack angle. Four elastic constants are expressed by two independent parameters.

Application of the Boundary Element Method to Solving the Plane Elastic Problems

Application of the Boundary Element Method to Solving the Plane Elastic Problems

Mixed Smoothed Particle Hydrodynamics Method for Planar Elasticity Problems

Mixed Smoothed Particle Hydrodynamics Method for Planar Elasticity Problems