Two-dimensional Elasticity Problems Research Articles

Mathematical modeling of elastic state in a three-component plate containing a crack due to the action of unidirectional tension

Purpose. A two-dimensional mathematical model for the problem of elasticity theory in a three-component plate containing rectilinear crack due to the action of mechanical efforts is examined. As a consequence, the intensity of stresses in the vicinity of tops of the crack increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks, we determined stress intensity factors (SIFs) at the tops of the crack, which are subsequently used to determine critical values of the tension. Therefore, the aim of present work is to determine the two-dimensional elastic state in plate containing an elastic two-component circular inclusion and crack under conditions of power load in the case of unidirectional tension of the plate perpendicular for the crack line. This makes it possible to determine the critical values of unidirectional tension in order to prevent crack growth, which will not allow the local destruction of the body. Methodology. The methods of studying two-dimensional elastic state body with crack as stress concentrators based on the function of complex variable method by which the problem of elasticity theory is reduced to singular integral equations (SIE) of the first and second kind, the numerical solution by the method of mechanical quadratures was obtained. Findings. In this paper two-dimensional mathematical model in the form of the system of two singular integral equations on closed contour (boundary of inclusion) and unclosed contour (crack) are obtained; numerical solutions of these integral equations were received by the method of mechanical quadratures; stress intensity factors at the tops of a crack are identify and explored to detect the effects of mechanical character. Graphical dependencies of SIFs, which characterize distribution of the intensity of stresses at the tops of a crack as function of elastic properties of inclusion and also as function of the distance between crack and inclusion are obtained. This makes it possible to analyze the intensity of stresses in the vicinity of a crack's tops depending on the geometrical and mechanical factors, as well as to determine the limit of permissible values of unidirectional tension of the plate perpendicular to the crack line at which the crack begins to grow and the body being locally destroyed. It is shown that the proper selection of elastic characteristics of the components of three-component plate can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIFs at the crack's tops. Originality. Scientific novelty lies in the fact that the solutions of the new two-dimensional problems of elasticity for a specified region (plate containing an elastic two- component circular inclusion and a rectilinear crack) under the action of unidirectional tension of the plate perpendicular to the crack line are obtained. Practical value. Practical value of the present work lies in the possibility of a more complete accounting of actual stressed-strained state in the piecewise-homogeneous elements of a structure with cracks that work under conditions of different mechanical loads. The results of specific studies that are given in the form of graphs could be used when designing rational operational modes of structural elements. In this case, the possibility for preventing the growth of a crack through the appropriate selection of composite's components with the corresponding mechanical characteristics is obtained.

An isogeometric-meshfree collocation approach for two-dimensional elastic fracture problems with contact loading

A strong form-based isogeometric-meshfree moving least-squares collocation (IMMLS-C) approach is developed for two-dimensional linear elastic fracture problems with contact loading. The IMMLS-C approach uses reproducing conditions to establish an equivalence between MLS shape functions and isogeometric basis functions. The advantages of this approach include the exact geometry representation, convenient crack modeling and flexible adaptive refinement. The IMMLS-C approach focuses on solving fracture problems based on strong formulations without requiring the numerical integration of Galerkin weak forms. Traction-free boundary conditions are enforced over a set of collocation points located on both sides of a crack surface. The displacement discontinuity along the crack surface is modeled by the visibility criterion and the singularity of near-tip stress fields is captured by adaptive mesh refinement without adding tip-enrichment functions, thereby reducing the degrees of freedom compared with the extended finite element method. Moreover, contact constraints are enforced by introducing a penalty algorithm to the strong formulations. The numerical results demonstrate that the adaptive refinement is able to achieve a high convergence rate at a low computational cost.

Boundary element analysis for elasticity problems using expanding element interpolation method

PurposeThis paper aims to propose a boundary element analysis of two-dimensional linear elasticity problems by a new expanding element interpolation method.Design/methodology/approachThe expanding element is made up based on a traditional discontinuous element by adding virtual nodes along the perimeter of the element. The internal nodes of the original discontinuous element are referred to as source nodes and its shape function as raw shape function. The shape functions of the expanding element constructed on both source nodes and virtual nodes are referred as fine shape functions. Boundary variables are interpolated by the fine shape functions, while the boundary integral equations are collocated on source nodes.FindingsThe expanding element inherits the advantages of both the continuous and discontinuous elements while overcomes their disadvantages. The polynomial order of fine shape functions of the expanding elements increases by two compared with their corresponding raw shape functions, while the expanding elements still keep independence to each other as the original discontinuous elements. This feature makes the expanding elements able to naturally and accurately interpolate both continuous and discontinuous fields.Originality/valueNumerical examples are presented to verify the proposed method. Results have demonstrated that the accuracy, efficiency and convergence rate of the expanding element method.

A quasi-optimal non-overlapping domain decomposition method for two-dimensional time-harmonic elastic wave problems

This article presents the construction of a new non-overlapping domain decomposition method (DDM) for two-dimensional elastic scattering problems. The method relies on a high-order Transmission Boundary Condition (TBC) between sub-domains, which accurately approximates the exact Dirichlet-to-Neumann map. First, we explain the derivation of this new TBC in the context of a non-overlapping DDM. Next, a mode-by-mode convergence study for a model problem is presented, which shows the new method to be quasi-optimal, i.e. with an optimal convergence rate for evanescent modes and an improved convergence rate for the other modes compared to the standard low-order Lysmer-Kuhlemeyer TBC. Finally, the effectiveness of the new DDM is demonstrated in a finite element context by analyzing the behavior of the method on high-frequency elastodynamic simulations.

A Comparative Study of Mesh-Free Radial Point Interpolation Method and Moving Least Squares Method-Based Error Estimation in Elastic Finite Element Analysis

The mesh-free interpolation method-based recovery of finite element discretization error is presented in this study. Two interpolation schemes, namely radial point interpolation method and moving least squares method, are taken for the recovery of finite element solution error. The global and elemental errors of finite element solution are evaluated in energy norm. The two-dimensional benchmark elastic problems are analysed using triangular/quadrilateral meshing schemes to prove the proposed mesh-free error estimation techniques validity and the effectiveness. The results of the mesh-less interpolation recovery-based error estimation are also compared with mesh-dependent least squares interpolation method-based error estimation. The mesh-free recovery is based on the fitting of a higher-order polynomial to the field variable over a mesh-less patch (radial support domain) using RPI and MLS interpolation method, while the mesh-dependent recovery is based on recovery of the field variable over a patch of nodes surrounding the particular given node using RPI, MLS and LS interpolation method. The study presents the effect of polynomial basis function (LS), polynomial basis function along with radial basis function (RPI)-based interpolation method and weighted polynomial basis function (MLS) on the recovery of finite element solution error. The quality of error estimation under different interpolation schemes is compared in terms of convergence characteristics, local/global effectivity, error distribution patterns and adaptively refined meshes. It can be concluded that weighting function of least squares interpolation affects considerably the mesh-free error estimation.

A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces

Abstract The purpose of this paper is to develop a reduced Crouzeix–Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia–Stenberg finite element method (Kouhia et al. 1995) and Crouzeix–Raviart IFEM (CRIFEM) (Kwak et al. 2017). We use a P 1 {P_{1}} -conforming like element for one of the components of the displacement vector, and a P 1 {P_{1}} -nonconforming like element for the other component. The number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM. Furthermore, we can choose penalty parameters independent of the Poisson ratio. One of the penalty parameters depends on Lamé’s second constant μ, and the other penalty parameter is independent of both μ and λ. We prove the optimal order error estimates in piecewise H 1 {H^{1}} -norm, which is independent of the Poisson ratio. Numerical experiments show optimal order of convergence both in L 2 {L^{2}} and piecewise H 1 {H^{1}} -norms for all problems including nearly incompressible cases.

Boundary Moving Least Square method for numerical evaluation of two-dimensional elastic membrane and plate dynamics problems

This paper discusses the Boundary Moving Least Square (BMLS) method for numerical implementation of various 2-D elasticity membrane and plate dynamics problems. The proposed technique is an amalgamation of boundary collocation meshless method and Moving Least Square (MLS) method. BMLS employs discrete boundary nodes in the axial direction to generate tensor product nodes in the entire solution domain, thereby simplifying complex multi-dimensional functions into manageable one-dimensional functions, as well as improving the overall computational and programming efficiency. Several numerical experiments were carried out and the result of BMLS was compared with that of Finite Element Method (FEM), which revealed that the BMLS numerical results are smoother and more continuous compared to the FEM, and exhibit distinctive characteristics of regularity and stability. BMLS is a practical numerical method with good prospects for solving membrane and plate dynamics problems.

A dispersive homogenization model for composites and its RVE existence

An asymptotic homogenization model considering wave dispersion in composites is investigated. In this approach, the effect of the microstructure through heterogeneity-induced wave dispersion is characterised by an acceleration gradient term scaled by a “dispersion tensor”. This dispersion tensor is computed within a statistically equivalent representative volume element (RVE). One-dimensional and two-dimensional elastic wave propagation problems are studied. It is found that the dispersive multiscale model shows a considerable improvement over the non-dispersive model in capturing the dynamic response of heterogeneous materials. To test the existence of an RVE for a realistic microstructure for unidirectional fiber-reinforced composites, a statistics study is performed to calculate the homogenized properties with increasing microstructure size. It is found that the convergence of the dispersion tensor is sensitive to the spatial distribution pattern. A calibration study on a composite microstructure with realistic spatial distribution shows that convergence is found although only with a relatively large micromodel.

Boundary moving least square method for 2D elasticity problems

This paper applies the boundary moving least square (BMLS) method for numerical implementation of two-dimensional (2D) elasticity problems. Compared with traditional mesh-free methods, the new discrete BMLS method is adopted to simplify shape functions. The discrete equations for elasticity problems are established in each of coordinate system. Then the 2D shape functions are simplified to one-dimensional (1D) shape functions in the BMLS. The introduction of moving least square (MLS) ensures BMLS maintains high accuracy with moderate computation cost. Furthermore, the advantages of BMLS are illustrated by four examples.

The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity

In this study, the generalized finite difference method (GFDM) was used to stably and accurately solve two-dimensional (2D) inverse Cauchy problems in linear elasticity by using the Navier equations. In Cauchy problems, overdetermined boundary conditions are imposed on parts of the boundary, whereas there are missing boundary conditions on some parts of the boundary. In Cauchy problems, conventional numerical methods generally generate highly ill-conditioned matrices and thus provide unstable numerical solutions. Moreover, even if a slight noise is added in the boundary conditions, numerical errors are evidently magnified. The GFDM, one of the most promising meshless methods and an extension of the classical finite difference method, can avoid time-consuming tasks of mesh generation and numerical quadrature. The GFDM was applied in this study to stably solve the 2D Cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme. Moreover, the stability of the presented scheme for inverse Cauchy problems was proved by adding noise into the boundary conditions.

Study of stress–strain state of elastic body with hyperbolic notch

The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two-dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems are presented.

A note on Kalandiya's method for a crack problem

We discuss the problem of the instability of the numerical method proposed by Kalandiya[10] for the approximate solution of the two-dimensional elasticity problem of a crack at a circular cavity surface. Furthermore, we describe a method to overcome this problem.

A polygonal XFEM with new numerical integration for linear elastic fracture mechanics

The extended finite element method (XFEM) has become a powerful and effective technique for modeling fracture problems without remeshing. In this paper, we introduce a novel and effective computational approach that is based on polygonal XFEM (named as PolyXFEM) for the analysis of two-dimensional (2D) linear elastic fracture mechanics problems. The PolyXFEM is equipped with a new numerical integration technique that uses the concept of Cartesian transformation method (CTM) over polygonal domains. The underlying idea of the CTM is to transform a domain integral into a boundary integral and a one-dimensional integral. This is computationally more efficient compared to the two-level mapping integration commonly used on polygons. Furthermore, the PolyXFEM is effective in modeling crack problems due to the local enrichment based on the partition of unity method. Our numerical results show that improvements in accuracy are reached thanks to the higher-order of the polygonal shape functions of the PolyXFEM. In addition, efficiency in computational time is achieved because of the usage of the CTM integration scheme, which appears to be quite suitable for polygonal domains. To demonstrate the accuracy and performance of the developed PolyXFEM approach, several numerical examples for 2D linear elastic fracture problems are considered, in which static stress intensity factors and crack propagation are investigated and compared with reference solutions. The convergence and mesh independence of the PolyXFEM for crack analysis is also analyzed.

An engineering perspective to the virtual element method and its interplay with the standard finite element method

In this manuscript we thoroughly study the behavior of the virtual element method (VEM) in the context of two-dimensional linear elasticity problems for an engineering audience versed in standard FEM. Through detailed convergence studies we show the accuracy and the convergence rates recovered by VEM, and we compare them to those obtained by the h- and p-versions of the finite element method (FEM). We also demonstrate the mixability between FEM and VEM; in particular, applications of VEM for coupling non-conforming discretizations and for local refinement are presented, showing higher versatility compared to FEM. Computer implementation aspects in displacement-based finite element codes are thoroughly explained, remarking on the main differences with respect to standard FEM.

Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems

The traditional method of fundamental solutions (MFS) based on the “global” boundary discretization leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. In this study, a localized version of the MFS (LMFS) is proposed for the large-scale modeling of two-dimensional (2D) elasticity problems. In the LMFS, the whole analyzed domain can be divided into small subdomains with a simple geometry. To each of the subdomain, the traditional MFS formulation is applied and the unknown coefficients on the local geometric boundary can be calculated by the moving least square method. The new method yields a sparse and banded matrix system which makes the method very attractive for large-scale simulations. Numerical examples with up to 200,000 unknowns are solved successfully using the developed LMFS code.

Determination of Fiber-Structure for the Effective Reinforcement of Planar Composite-Material Constructions

To describe the behaviour of a composite material reinforced by fibers along curvilinear trajectories, a resolving system of differential equations has been obtained. I developed a structural model within the framework of the two-dimensional non-uniform linear elasticity problem. The reinforcement is performed on the basis of three approaches: according to the grid lines of the orthogonal coordinate system determined by the given conformal representation; according to trajectories isogonal to the given curves; according to the helical trajectories in the axisymmetric formulation of the problem. Particular solutions of the corresponding boundary problems have been obtained. Recommendations for the effective reinforcement of planar constructions have been presented.

A local mesh free method for linear elasticity and fracture mechanics

The paper presents recent developments on mesh free numerical methods, implemented at the Department of Civil Engineering and Environment of the University of Brasília.The concern is a local mesh free numerical model developed to solve two-dimensional problems in elasticity and linear elastic fracture mechanics.The model formulation is based in the work theorem of the theory of structures, kinematically formulated with a simple rigid-body displacement.The discretization considers the approximation of the elastic field with moving least squares (MLS) and implements a reduced numerical integration which improves the model accuracy. Both regular and irregular nodal distributions can be considered, which makes it a reliable model.The arbitrary discretization parameters, which determine the accuracy and efficiency of local mesh free methods, can be generated automatically, in a multi objective optimization environment, with genetic algorithms (GA), which makes it a robust model.Applications of linear elastic fracture mechanics implement the singularity subtraction technique (SST) to regularize the elastic field, before the numerical solution, introducing the stress intensity factors (SIF) as additional unknowns of the problem. These features make it an efficient model.Benchmark problems were solved for an assessment of the accuracy and efficiency of these techniques.

Stress Analysis of Bilayer Composite Materials with Various Passion Ratio

Composites materials are needed because of the widely used in structures and designs. In this work, the stress analysis of two-dimensional bilayer composite materials has a different passion ratio have studied. The materials under consideration are assumed to be perfectly bonded together. Finite difference method is used for the solution of two-dimensional elastic problems. In each layer of the composite, the mechanical properties are isotropic. The results are observed that the results agree well within the acceptable limit, which also confirms the reliability of the finite difference method. Changing in Poisson’s ratio in any layer has significant effects on the results of all layers of the bilayer composite. Due to the mathematical expressions of stresses and displacements for two-dimensional elastic problems, the study of the effects of Poisson’s ratio has a significant influence.

8-node unsymmetric distortion-immune element based on Airy stress solutions for plane orthotropic problems

In this work, a simple but robust 8-node 16-DOF quadrilateral membrane element with exceptional tolerances to mesh distortion is proposed for efficiently analyzing the two-dimensional orthotropic elasticity problems, within the improved framework of the unsymmetric FEM. This unsymmetric element model adopts a self-equilibrium metric stress field, which is formulated at the basis of Airy stress solutions of the plane problem, to be the trial functions. Meanwhile, to ensure the interelement compatibility, the standard isoparametric displacement field is employed to be the element’s test function. Numerical tests reveal that this quadratic element is always capable of producing exact solutions in constant and linear stress/strain problems and delivering quite satisfactory results in other higher-order benchmark problems. In particular, it can still work very well even when the element shape deteriorates severely into a concave quadrangle or degenerated triangle.

A dual interpolation boundary face method for elasticity problems

A dual interpolation boundary face method (DiBFM) is proposed to unify the conforming and nonconforming elements in boundary element method (BEM) implementation. In the DiBFM, the nodes of a conventional conforming element are sorted into two groups: the nodes on the boundary (called virtual nodes) and the internal nodes (called source nodes). Without virtual nodes, the conforming element turns to be a conventional nonconforming element of a lower order. Physical variables are interpolated using the conforming elements, the same way as conforming BEM. Boundary integral equations are collocated at source nodes, the same way as nonconforming BEM. To make the final system of linear equations solvable, additional constraint equations are required to condense the degrees of freedom for all the virtual nodes. These constraints are constructed using the moving least-squares (MLS). Besides, both boundary integration and MLS are performed in the parametric spaces of curves, namely, the geometric data, such as coordinates, out normals and Jacobians, are calculated directly from curves rather than from elements. Thus, no geometric errors are introduced no matter how coarse the discretization is. The method has been implemented successfully for solving two-dimensional elasticity problems. A number of numerical examples with real engineering background have demonstrated the accuracy and efficiency of the new method.