Two-dimensional Elastic Analysis Research Articles

An EFG-SBM based partitioning algorithm for two-dimensional elastic analysis of cyclically symmetrical structures

Purpose – The purpose of this paper is to exploit the cyclic symmetry to reduce the computational expense of scaled boundary method (SBM) which may cumber its further application. Design/methodology/approach – A partitioning EFG-SB (Element-free Galerkin-SB) algorithm is proposed for the two-dimensional elastic analysis of cyclically symmetric structures. Findings – By utilizing the cyclic symmetry and partitioning algorithm, the whole computational cost can be significantly reduced. Three numerical examples are given to illustrate the advantages of the proposed algorithm. Originality/value – It is proved that the matrices of eigenvalue and system equations of EFG-SBM for cyclically symmetric structures are block-circulant so long as a kind of symmetry-adapted reference coordinate system is adopted. No matter whether displacement constraints are cyclically symmetric or not, the partition is available for the eigenvalue equations. Therefore the major computational cost can be saved via the proposed partitioning algorithm. This paper provides an efficient algorithm for the two-dimensional elastic analysis of cyclically symmetric structures using EFG-SBM. A higher computing efficiency can be expected since the proposed partitioning algorithm facilitates parallel processing.

Interface crack embedded in a bi-material plane under shear and compression

A two-dimensional elastic analysis is made for a crack lying at the interface between two bonded dissimilar homogeneous isotropic media subjected to shear and compression. The Fourier transform technique is applied to reduce the problem to dual integral equations, and the solution is obtained in closed form. Full stress fields at an arbitrary position of the whole bi-material plane induced by a closed mode-II interface crack are explicitly obtained for uniform and linearly distributed shear stress at the crack surface, respectively. The stress intensity factors are gained in closed form. Obtained results indicate that the shear stresses ahead of the crack tip exhibit a usual square-root singularity, rather than oscillatory singularity. Disturbed normal stresses are also singular at the crack surfaces behind the crack tips and depend on a Dundurs’ parameter. Using the maximum tangential stress criterion, the Dundurs’ parameter has an influence on the initiation angle of an interface crack in mixed compression–shear mode.

Influence of soft ferromagnetic substrate on magneto-elastic behavior in a superconducting coated conductor strip

Ferromagnetic materials will affect not only the electromagnetic response but also the mechanical behaviors of coated conductors. The influence of soft ferromagnetic substrate on magneto-elastic behavior in a superconductor/ferromagnetic (SC/FM) bilayer exposed to a transverse magnetic field is investigated theoretically. The ferromagnetic substrate is regarded as ideal soft magnets with high permeability and small magnetic hysteresis. Due to the composite structure of SC/FM hybrids, magneto-elastic behavior will be subjected to combined effect of equivalent force and flexural moment. Analytical expressions for internal stress and strain components are derived by virtue of a two-dimensional elasticity analysis. It is worth pointing out that the y component of strain has much larger positive value during field ascent, which may result in the delamitation at the interface. Irreversible magnetostrictive behaviors are observed both along x direction and along y direction. For the thickness dependence of magnetostriction, the flexural moment dominates when the SC thickness is small while the equivalent force plays a critical role at higher SC thickness.

A scaled boundary finite element method for cyclically symmetric two-dimensional elastic analysis

A SBFEM (scaled boundary finite element method) based approach is developed for numerical analysis of 2-D elastic systems with rotationally periodic (or cyclic) symmetry under arbitrary load conditions. It is shown that the coefficient matrices of the global SBFEM equations for the rotationally periodic system are block-circulant when a symmetry-adapted reference co-ordinate system is used. Furthermore, both the eigenproblem and stiffness matrix equation are partitioned into a series of independent sub-problems. By solving these sub-problems via a partitioning algorithm, solution of the whole problem can be obtained at low computing expense. Two numerical examples are provided to illustrate the efficiency of the proposed approach.

Stress concentration in polymer extrusion dies

A two-dimensional elastic analysis using complex variables is presented to find the stress concentration at the internal corners of a non-circular pressurized cylinder. The problem is made difficult because the region is doubly connected. An approximate solution based upon Schwarz’s alternating method, which is useful for most practical die cross sections, is obtained. The calculation presented here is shown to agree with previous work in the special case where the cross section is an elliptical ring.

Effect of imperfect bonding on axisymmetric elastodynamic response of a lined circular tunnel in poroelastic soil due to a moving ring load

A two-dimensional elasticity analysis for steady-state axisymmetric dynamic response of an arbitrarily thick elastic homogeneous hollow cylinder of infinite length, which is imperfectly bonded to the surrounding fluid-saturated permeable formation, subject to an axially moving ring load, is presented. The problem solution is derived by using Biot’s dynamic theory of poroelasticity in conjunction with double Fourier transformation with respect to time (frequency) and axial coordinate (axial wave number). The analytical results are illustrated with numerical examples in which a concrete tunnel lining of uniform wall thickness is imperfectly bonded to a surrounding water-saturated poroelastic formation of soft/stiff frame characteristic. Numerical solutions for the radial shell mid-plane and formation displacements are calculated by analytical (numerical) inversion of the Fourier transformation with respect to the frequency (axial wave number). Primary attention is focused on the influence of bonding condition at the liner/soil interface, formation material type, and load velocity on the system’s dynamic response. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.

A radial boundary node method for two-dimensional elastic analysis

The boundary node method (BNM) takes the advantages of both the boundary integral equation in dimension reduction and the moving least-square (MLS) approximation in elements elimination. However, the BNM inherits the deficiency of the MLS approximation, in which the shape functions lack the delta function property. As a result boundary conditions could not be exactly implemented for the BNM. In this paper, a radial boundary node method (RBNM) is proposed. The RBNM uses radial basis functions (RBFs) instead of the MLS to construct its interpolation. Consequently, the interpolation function could pass through nodes exactly, and the shape functions are of the delta function property. The exponential (EXP) and the multiquadric (MQ) RBFs are used in the current RBNM, and their shape parameters are studied in detail through some analyses of two-dimensional elastic problems. The suitable ranges of the shape parameters are proposed for both the EXP and the MQ basis functions. It is found that the RBNM is as accurate as or even more precise than the BEM. This suggests that the current RBNM could be robust and applicable.

Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Two-dimensional elastic analysis of doubly periodic circular holes in an infinite plane is given in this paper. Two cases of loading, remote tension and remote shear, are considered. A rectangular cell is cut from the infinite plane. In both cases, the boundary value problem can be reduced to a complex mixed one. It is found that the eigenfunction expansion variational method is efficient to solve the problem. Based on the deformation response under certain loading, the notched medium could be modeled by an orthotropic medium without holes. Elastic properties for the equivalent orthotropic medium are investigated, and the stress concentration along the hole contour is studied. Finally, numerical examples and results are given.

Two-Dimensional Elastic Analyses of Splitting Test and Flexural Test of Brittle Material

Two-Dimensional Elastic Analyses of Splitting Test and Flexural Test of Brittle Material

The strength of hard-rock pillars

Abstract Observations of pillar failures in Canadian hard-rock mines indicate that the dominant mode of failure is progressive slabbing and spalling. Empirical formulas developed for the stability of hard-rock pillars suggest that the pillar strength is directly related to the pillar width-to-height ratio and that failure is seldom observed in pillars where the width-to-height ratio is greater than 2. Two-dimensional finite element analyses using conventional Hoek–Brown parameters for typical hard-rock pillars (Geological Strength Index of 40, 60 and 80) predicted rib-pillar failure envelopes that did not agree with the empirical pillar-failure envelopes. It is suggested that the conventional Hoek–Brown failure envelopes over predict the strength of hard-rock pillars because the failure process is fundamentally controlled by a cohesion-loss process in which the frictional strength component is not mobilized. Two-dimensional elastic analyses were carried out using the Hoek-Brown brittle parameters which only relies on the cohesive strength of the rock mass. The predicted pillar strength curves were generally found to be in agreement with the observed empirical failure envelopes.

Calculation of Two-Dimensional Reflection Energy of SH Waves from a Gradient Inhomogeneous Layer

In a previous paper, considering the reflection energy of SH waves from a gradient inhomogeneous layer, we offered a new rationalized layer element for a two-dimensional elastic analysis. We confirmed that the reflectance is exactly estimated using this new elements without any complexities. In this paper, we ex panded this analysis to the case of a layer with an arbitrary distribution of the acoustic impedance. As an example, when the acoustic impedance and the phase velocity vary sinusoidally with the thickness, and the mass density is constant through the layer, the resulting reflectance is in very close agreement with the exact value, and shows rapid convergence properties. Furthermore, it is shown that the existence of the maximum value of the acoustic impedance in the layer gives rise to the total reflection, even if the acoustic impedance values of the reflected and transmitted side of the layer are equal.

Analytical integrations for two-dimensional elastic analysis by the symmetric Galerkin boundary element method

In the context of linear elasticity, this paper presents a way to perform hypersingular integrals arising from the symmetric Galerkin Boundary Element Method (BEM). In contrast to the existing integration techniques, the one here proposed does not need any regularization or limit processes: in fact it works directly on the final form of the hypersingular double integrals without any previous manipulation. The present method is applied to 2D linear elastic problems, using straight elements with continuous piecewise-linear displacements and piecewise-constant tractions. Numerical tests are presented for the validation of the obtained analytic results.

TWO DIMENSIONAL ELASTIC ANALYSIS FOR ASPHALT MIXTURE ON STEEL PLATE

This paper is intended to report the results of the two-dimensional elastic analysis for test specimens consisting of two layers: an asphalt mixture to which the beam theory is inapplicable due to its low stiffness, and a thin steel plate on which the mixture is placed. Then, numerical computation has provided the authors with valuable data and information, such as the sizes and shapes of test specimens, friction between the upper and the lower layers, and the stiffness ratio between the two layers. All of these are essential factors in determining the mechanical property of a mixture by using the results of displacement measurement for the purpose of evaluating the elasticity of a pavement mixture.

Calculation of Two-Dimensional Reflection Energy of SH Waves from a Gradient Inhomogeneous Layer. Stacking Approximation by New Rationalized Layer Elements.

In a previous paper, considering the reflection energy of SH waves from a gradient inhomogeneous layer, we offered a new rationalized layer element for a two-dimensional elastic analysis. We confirmed that the reflectance is exactly estimated using this new elements without any complexities. In this paper, we expanded this analysis to the case of a layer with an arbitrary distribution of the acoustic impedance. As an example, when the acoustic impedance and the phase velocity vary sinusoidally with the thickness, and the mass density is constant through the layer, the ressiting reflectance is in very close agreement with the exact value, and shows rapid convergence properties. Furthermore, it is shown that the existence of the maximum value of the acoustic impedance in the layer gives rise to the total rellection, even if the acoustic impedance values of the refected and transmitted side of the layer are equal.

TWO-DIMENSIONAL ELASTIC ANALYSIS FOR A CIRCULAR TUNNEL LINING IN THE PRE-DEFORMED GROUND DUE TO INITIAL STRESSES

トンネルにおけるライニングの挙動は, 地盤とライニングの連成問題であり, ライニングの剛性と地盤の剛性および初期応力状態の三者の相対的な大小関係によって様々に変化する. 本論文では, まず, ライニングと地盤の境界面の粗さを考慮した最も単純で基本となる2次元弾性厳密解を誘導し, さらにライニング時には既に完了している初期地盤変位を考慮する弾性解へと展開した.ライニングの応力, 変形における地盤との連成関係を示す解析結果を例示するとともに, それをより一般的な形で表現するため, 著者らが提案するライニング支保剛性, 地山支保剛性および総支保剛性でライニング支保特性で定量化して論じた.

Rational Two Dimensional Analysis of Reflecting SH Waves from an Gradient Inhomogeneous Layer.

Introducing a rationalized layer element for a two-dimensional elastic analysis of a gradient inhomogeneous material, the reflectance is successfully estimated without any complexities. In the reflection and transmission analysis of elastic waves impinging on a gradient inhomogeneous layer, the difficulty comes out in the evaluation of modified Bessel functions of imaginary order depending on the inhomogeneous parameter. Then the main subject is discussion of a strategic tackle to overcome the difficulty. Rational analysis presented here is the use of a rationalized inhomogeneous layer element in stead of the previously developed element so called LILE. Development of the rationalized new elements and their availability have been explained. As the results, the method by using the new element is very convenient to the analysis of gradient inhomogeneous materials, because the fundamental solutions in the rational analysis are expressed by elementary functions, and the reflectance calculation can be carried out without any use of the modified Bessel functions for any inhomogeneous parameters, as far as it concerns in this study. The trial numerical results by using only one new element, coincide well with the exact numerical results obtained by multistacking layer model of many homogeneous elements.

Initial Post-buckling Behavior of Thick Rings Under Uniform External Hydrostatic Pressure

The initial post-buckling behavior of thick rings under external uniform hydrostatic pressure is investigated. In the analysis, no assumptions are placed upon the relative magnitudes of the elongations and rotations, and the ring is assumed to be elastic and extensional. The importance of including certain nonlinear terms in the initial post-buckling stability analysis and the effects of nonzero shearing strains on the buckling load and the initial post-buckling stability are examined. It is shown that the classical Kirchhoff assumptions, when employed for a ring lead to nonvanishing through thickness strains, εzz and εzθ, with the latter being proportional to the through thickness coordinate z. An approximate first order shear deformation analysis and a two-dimensional elasticity analysis (without beam-type kinematical assumptions) of the initial post-buckling behavior of thick rings are presented and the thickness effects on the buckling load and the initial post-buckling behavior are examined. The formulation for the composite ring was reduced to that of an isotropic ring and the results thus obtained were compared with published one-dimensional results in the literature. It is found from both the shear deformation and the two-dimensional analysis that the initial post-buckling behavior of the isotropic ring and the composite rings studied are stable. The influence of thickness on the degree of stability in the immediate post-buckling response is characterized.