Two-dimensional Elasticity Problems Research Articles

Asymptotic closed-form solutions including boundary-layers for orthotropic beams

In this paper, asymptotic closed-form solutions with boundary-layers are explicitly obtained. The virtual work principle is applied to the problem by employing the method of multiple-scales, which allows one to systematically separate two-dimensional elasticity problems into the interior and boundary-layer problems. From the interior problem, we calculate the warping displacements first and then construct boundary-layer displacements by employing them as a basis. Saint-Venant’s principle is applied to the boundary-layers so that they satisfy the surface boundary conditions in a weak sense. For stress prescribed boundaries, the minimization process is enforced to satisfy the conditions, which allow one to predict the free-edge boundary-layers. Cantilevers with three types of materials are taken as numerical examples. Asymptotic closed-form decay rates, displacements, and stresses are derived and compared to a two-dimensional finite element analysis. The results obtained herein clearly show that the present approach yields a reliable prediction of boundary-layers for orthotropic beams.

The Improved Interpolating Complex Variable Element-Free Galerkin Method for Two-Dimensional Elastic Problems

The Improved Interpolating Complex Variable Element-Free Galerkin Method for Two-Dimensional Elastic Problems

Solution to two-dimensional elastic problems involving functionally graded material in radial co-ordinates

Solution to two-dimensional elastic problems involving functionally graded material in radial co-ordinates

The method of fundamental solutions for two-dimensional elasticity problems based on the Airy stress function

This paper presents a new version of the method of fundamental solutions (MFS) for two-dimensional linear elasticity problems based on the stress function (Airy stress function), which is different from the MFS utilizing the fundamental solutions of displacement. The displacement compatibilities are derived by the single-valuedness of displacements in multiply-connected region. Based on the strain and rotation of the line element on the boundary, the displacement boundary conditions are deduced in forms of the Airy stress function. Furthermore, the displacement conditions exclude pure rigid body motion, and are in the derivative forms instead of the integral ones. The interpolation equations are also reconstructed by the single-valuedness conditions of the displacements and the approximate solutions in a multiply-connected region. The numerical examples show that the proposed method has good effect and keeps high accuracy in all kinds of boundary conditions.

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforcedviathe rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in theH(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in theL2-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

2D Quasi-Static Accurate Solutions for Isotropic Thermoelastic Materials with Applications

Thermally induced stress is an important scientific problem in engineering applications. In this paper, an accurate and efficient method for the two-dimensional quasi-static thermal elastic problem is established to explore the thermal stress problem. First, the compact quasi-static two-dimensional general solution is derived in terms of simple potential functions. The general solution is simple in form and can be derived for arbitrary boundary problems subjected to a line heat load. This is completely new to the literature. Second, Green’s function solutions of an infinite plane under the line pulse heat source based on the general solutions are presented to analyze the thermal stress field. Lastly, numerical results are taken into account to study the temperature and stress field induced by the dynamic heat source load. The corresponding analysis can constitute to reveal the mechanism of thermal elastic problems and provide some guidance for experiments or engineering structural design in the future work.

A wave based method for two-dimensional time-harmonic elastic wave propagation in anisotropic media

In this paper, the wave based method (WBM) is extended to solve two-dimensional anisotropic elastic wave problems. The wave functions satisfying the governing equations are employed as basis functions in the WBM. To remedy the absence of the basis functions for the general anisotropic elastic wave problems, a new method is proposed. In the method, the two governing equations are transformed into one high-order scalar equation and the basis functions are obtained by using the wave functions of the scalar equation. Two examples are carried out to validate the accuracy and efficiency of the present method.

Localized method of fundamental solutions for two-dimensional anisotropic elasticity problems

The purpose of the presented paper is to develop further the Localized Method of Fundamental Solutions (LMFS) for solving two-dimensional anisotropic elasticity problems. The computational domain is divided into overlapping subdomains. In the LMFS, the classical Method of Fundamental Solutions (MFS) is employed in each of these local subdomains to get an expression of the solution for the main node of this subdomain. The expression is structured by a linear combination of the solutions of the other nodes in this subdomain. Displacement or traction boundary conditions are satisfied at the boundary nodes. The solution is calculated from an equation set formed by the boundary conditions for the boundary nodes and expressions in the subdomain for the interior nodes. The presented three numerical examples demonstrate that the novel method is suitable for solving large-scale problems, and especially, the problems with complicated domains.

Elzaki transform method for finding solutions to two-dimensional elasticity problems in polar coordinates formulated using Airy stress functions

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ODE). The general solution for the Airy stress potential function in the Elzaki transform space was obtained by solving the ODE. By inversion, the general solution for the Airy stress potential function was obtained in the physical domain space variables in terms of four unknown integration constants. Normal stresses and shear stress fields were also determined for the general case of 2D elasticity problems. The Flamant problem was solved as a particular illustration of 2D elasticity problems. The stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces were used simultaneously to determine the four constants of integration. The Airy stress potential function and the normal and shear stress fields were thus completely determined. The principle of superposition was used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.

Solution of two-dimensional elasticity problems using a high-accuracy boundary element method

For the two-dimensional elasticity problems, we give a uniqueness and existence analysis via the single-layer potential approach leading to a system of integral equation that contains a weakly singular operator. For its numerical solutions we describe a O(h3) order quadrature method based on the specific integral formula including convergence and stability analysis. Moreover, the asymptotic expansion of errors with odd power O(h3) is got and the accuracy of numerical approximations can be improved to the order of O(h5) by the Richardson extrapolation algorithm (EA) once. The efficiency of the method is illustrated by two numerical examples.

Localized method of fundamental solutions for 2D harmonic elastic wave problems

In this paper, a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) is applied to two-dimensional (2D) harmonic elastic wave problems. Due to the dense and ill-conditioned coefficient matrices, the traditional MFS is computationally expensive and time-consuming to solve the above-mentioned problems which require a large amount number of nodes to obtain excellent numerical results. In the LMFS, the domain is divided into small subdomains, in which the traditional MFS is applied to represent the physical variables as linear combinations of the fundamental solution. A sparse and banded system of linear equations is yielded in this method, so that it is attractive to solve large-scale problems. A numerical example demonstrates the effectiveness and accuracy of the proposed LMFS.

Semi analytical cum numerical analysis of two-dimensional elasticity problem of plate with eccentric hole in bipolar coordinate system

A semi analytical cum numerical method for analyzing 2D elasticity problem of plate with eccentric hole in bipolar coordinate systems is given. Complete formulation of the plate problem is derived in bipolar coordinate system. A novel numerical integration technique is adopted to obtain the numerical results. Few parameters are identified that adversely affects the stresses and displacements of isotropic and FG plate with eccentric hole. Verification of the mathematical models with numerical results such as finite element models (COMSOL Multiphysics) and wide variety of applicability of the developed models for hypothetical/real field problems is seen from the present work.

The interpolating element-free Galerkin method for elastic large deformation problems

This paper presents an interpolating element-free Galerkin (IEFG) method for solving the two-dimensional (2D) elastic large deformation problems. By using the improved interpolating moving least-squares method to form shape function, and using the Galerkin weak form of 2D elastic large deformation problems to obtain the discrete equations, we obtain the formulae of the IEFG method for 2D elastic large deformation problems. As the displacement boundary conditions can be applied directly, the IEFG method can acquire higher computational efficiency and accuracy than the traditional element-free Galerkin (EFG) method, which is based on the moving least-squares approximation and can not apply the displacement boundary conditions directly. To analyze the influences of node distribution, scale parameter of influence domain and the loading step on the numerical solutions of the IEFG method, three numerical examples are proposed. The IEFG method has almost the same high accuracy as the EFG method, and for some 2D elastic large deformation problems the IEFG method even has higher computational accuracy.

Periodic pile-soil system as a barrier for seismic surface waves

This paper investigates the bandgap properties of two-dimensional elastic problem with pile barrier for seismic surface waves. The main objective of this proposed work is to investigate the propagation of surface waves through periodically arranged piles in a single soil medium and to study the feasibility of surface waves attenuation by finite element technique. We consider the idea of seismic metamaterials in this study. One class of seismic metamaterials is explored with a periodic arrangement of vertical pile inclusions inside of the ground. The unit-cell analysis of the pile-soil system has been discussed in this work to attain the bandgaps with the low-frequency range. We varied all geometrical and mechanical parameters with shape of the pile to display the productivity of the weakening zone. These low-frequency ranges bandgaps suggest that the planned pile-soil system is efficient in reducing surface waves. The possibility study exposes that this type of pile-soil system can be applied as seismic barricades for mitigating seismic waves to guard critical civil structures from earthquake dangers. This comes as an interesting approach to potentially shield the LIGO detectors in the low-frequency range and can have an impact on the sensitivity of existing and future ground-based detectors. The present result will open a new prospect for the wave barricades designed with a low cost of money.

Least squares weighted residual method for finding the elastic stress fields in rectangular plates under uniaxial parabolically distributed edge loads

In this work, the least squares weighted residual method is used to solve the two-dimensional (2D) elasticity problem of a rectangular plate of in-plane dimensions 2a 2b subjected to parabolic edge tensile loads applied at the two edges x = a. The problem is expressed using Beltrami–Michell stress formulation. Airy’s stress function method is applied to the stress compatibility equation, and the problem is expressed as a boundary value problem (BVP) represented by a non-homogeneous biharmonic equation. Airy’s stress functions are chosen in terms of one and three unknown parameters and coordinate functions that satisfy both the domain equations and the boundary conditions on the loaded edges. Least squares weighted residual integral formulations of the problems are solved to determine the unknown parameters and thus the Airy stress function. The normal and shear stress fields are determined for the one-parameter and the three-parameter coordinate functions. The solutions for the stress fields are found to satisfy the stress boundary conditions as well as the domain equation. The presented solutions for the Airy stress function and the normal stresses and shear stress fields are identical with solutions obtained by using variational Ritz methods, Bubnov–Galerkin methods and agree with results obtained by Timoshenko and Goodier.

A global RBF-QR collocation technique for solving two-dimensional elliptic problems involving arbitrary interface

In recent years, several numerical methods have been developed for differential equations with discontinuous coefficients, which are known as interface problems. Our aim is to introduce an efficient global meshless method for solving elliptic problems with arbitrary interface. Meshless methods based on infinitely smooth radial basis functions (RBFs) can achieve spectral convergence rate for solving partial differential equations. RBF-QR is a newly developed method which allows numerically stable computations with RBFs for all values of the free shape parameter $$\varepsilon$$. In this paper, an appropriate RBF-QR collocation technique is applied for two-dimensional elliptic interface problems, including Poisson, Helmholtz and elasticity problems. The Helmholtz interface problem is studied, by taking into account the fact that the solutions of such problems might be oscillatory. The elasticity interface problem is discussed as an application of elliptic interface problem in vector form. Several problems involving complex interface geometries and discontinuous variable coefficients are presented to confirm the efficiency and stability of RBF-QR method for this class of problems in the case of arbitrary small values of $$\varepsilon$$.

Geometric charges and nonlinear elasticity of two-dimensional elastic metamaterials

Problems of flexible mechanical metamaterials, and highly deformable porous solids in general, are rich and complex due to their nonlinear mechanics and the presence of nontrivial geometrical effects. While numeric approaches are successful, analytic tools and conceptual frameworks are largely lacking. Using an analogy with electrostatics, and building on recent developments in a nonlinear geometric formulation of elasticity, we develop a formalism that maps the two-dimensional (2D) elastic problem into that of nonlinear interaction of elastic charges. This approach offers an intuitive conceptual framework, qualitatively explaining the linear response, the onset of mechanical instability, and aspects of the postinstability state. Apart from intuition, the formalism also quantitatively reproduces full numeric simulations of several prototypical 2D structures. Possible applications of the tools developed in this work for the study of ordered and disordered 2D porous elastic metamaterials are discussed.

Using New Hp-Cloud Approximate Function in Meshless Finite Volume Method for Solving 2D Elasticity Problems

In this paper, a novel Hp-Cloud approximate function with Kronecker delta property, named herein as HPCK, is utilized in a meshless finite volume method for two-dimensional elasticity problems. Using the enrichment parts in the HPCK approximate functions leads to excessive unknowns corresponding to nodal points. Therefore, the obtained equations from the ordinary finite volumes method (FVM) are underdetermined. We tackled this problem by proposing an algorithm that considers more control volumes (CVs), known as enrichment CVs (ECVs), inside the main CVs. Following the fulfillment of the equilibrium equations on these ECVs, sufficient equations are obtained to solve the unknowns. In the proposed FVM with HPCK approximate function, the boundary conditions are easily and directly enforced due to Kronecker delta property of HPCK. Moreover, the proposed method leads to a fast convergence and highly accurate results for the studied test problems. In addition, unlike approximation functions like MLS, this method does not involve matrix inversions, which leads to low computational costs and CPU usage while achieving desirable accuracy levels.

Quasi-conforming analysis method for trimmed CAD surfaces

In this paper, the quasi-conforming method is introduced for the analysis of trimmed computer aided design (CAD) surfaces. The main benefit of the proposed method is that the boundary curves of elements are adopted for the numerical integration directly. In the quasi-conforming technique, the strains are approximated by using polynomials, and the weighted test functions are used to weaken the strain-displacement equations. The interpolation functions are introduced for strain integration. An appropriate choice of initial strain approximation and weighted test function ensures that inner-field functions are not required for strain integration, and this is used for the analysis of trimmed CAD surfaces. For example, the assumed stress quasi-conforming method is applied for the two-dimensional linear elastic problem. All the element edges are approximated by using quadratic Bézier curves for the conciseness, and this is easily incorporated into existing finite element codes and applied to Dirichlet boundary conditions. Numerical examples indicate the effectiveness and accuracy of the method.

The localized radial basis functions for parameterized level set based structural optimization

The current work is mainly about the implementation of the local radial basis functions (LRBFs) within the level set framework for structural optimization of two-dimensional linear elastic problems. The implicit representation of the structural geometry is accomplished through the LRBFs based level set function (LSF), and the geometry modification during the optimization process is carried out through an update of the LSF using a system of coupled ordinary differential equations (ODEs) instead of the Hamilton-Jacobi (HJ) type equation. This new implementation of LRBFs and level set method (LSM) allows essential topological changes automatically, i.e hole nucleation, hole merging with each other, and with the boundary to obtain the optimal structures. The coefficient matrix of the LRBFs is less sensitive to the values of shape parameter as well as the proposed method is mesh independent as explored during the numerical experiments performed. This new implementation is applied to distinct benchmark problems for minimum compliance with single and multiple load cases. The numerical experiments performed reveal significant insightful of the use of radial basis functions (RBFs) within the LSM framework, which may be considered as a step forward for further exploration of their intrinsic capabilities and improved performance for the solution of structural optimization problems.