Plane Elastic Body Research Articles

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

A boundary-only BEM procedure is employed to solve the transient dynamic analysis of nonhomogeneous anisotropic plane elastic bodies. The response of such bodies is governed by two coupled linear, second-order hyperbolic PDEs with spatially dependent coefficients. The lack of a reliable 2D time-domain elastodynamic fundamental solution is overcome using the principle of the Analog Equation, a method by which the equations of motion of the problem are substituted by two coupled quasi-static Poisson-type equations having as nonhomogeneous terms the components of a fictitious time-dependent load distribution in the specified domain. The standard BEM is employed for the solution of the substitute equations. To avoid the appearance of the domain integral in the integral representation of the solution, the fictitious load distribution is approximated by multiquadrics with unknown time-dependent expansion coefficients, which are calculated at discrete timepoints by collocating the equations of motion at a predefined set of domain interpolation nodes. The obtained numerical results by the proposed method demonstrate its stability and accuracy over other numerical methods.

Reconstruction of penetrable inclusions in elastic waves by boundary measurements

We use Ikehataʼs enclosure method to reconstruct penetrable unknown inclusions in a plane elastic body in time-harmonic waves. Complex geometrical optics solutions with complex polynomial phases are adopted as the probing utility. In a situation similar to ours, due to the presence of a zeroth order term in the equation, some technical assumptions need to be assumed in early researches. In a recent work of Sini and Yoshida, they succeeded in abandoning these assumptions by using a different idea to obtain a crucial estimate. In particular the boundaries of the inclusions need only to be Lipschitz. In this work we apply the same idea to our model. Itʼs interesting that, with more careful treatment, we find the boundaries of the inclusions can in fact be assumed to be only continuous.

On Special Orthotropy of Paper

In this paper it is shown that the well known special orthotropy of paper, explained by micromechanical models of complex bodies, actually corresponds to a special case of orthotropy of a plane elastic body. The existence of this special orthotropy was predicted by the author in a previous work.

Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems

Under some generic assumptions we prove the unique continuation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.

A boundary element modeling of fatigue crack growth in a plane elastic plate

This paper presents an extension of a boundary element method to fatigue growth analysis of mixed-mode cracked plane elastic bodies. The method consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity element due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the modified maximum strain energy density criterion. In numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the boundary element method. Crack growth is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. Some numerical results of fatigue growth in a plane elastic plate with a center-inclined crack under uniaxial cyclic loading are given.

An Efficient and Accurate Numerical Method of Stress Intensity Factors Calculation of a Branched Crack

Based on the analytical solution of Crouch to the problem of a constant discontinuity in displacement over a finite line segment in an infinite elastic solid, in the present paper, the crack-tip displacement discontinuity elements, which can be classified as the left and the right crack-tip elements, are presented to model the singularity of stress near a crack tip. Furthermore, the crack-tip elements together with the constant displacement discontinuity elements presented by Crouch and Starfied are used to develop a numerical approach for calculating the stress intensity factors (SIFs) of general plane cracks. In the boundary element implementation, the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called the hybrid displacement discontinuity method (HDDM). Numerical examples are given and compared with the available solutions. It can be found that the numerical approach is simple, yet very accurate for calculating the SIFs of branched cracks. As a new example, cracks emanating from a rhombus hole in an infinite plate under biaxial loads are taken into consideration. The numerical results indicate the efficiency of the present numerical approach and can reveal the effect of the biaxial load on the SIFs. In addition, the hybrid displacement discontinuity method together with the maximum circumferential stress criterion (Erdogan and Sih) becomes a very effective numerical approach for simulating the fatigue crack propagation process in plane elastic bodies under mixed-mode conditions. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the HDDM. Crack propagation is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented.

A formation problem in plane elasticity

This article discusses modification of the shape of a plane elastic body whose uncontrolled geometric configuration is an annulus, fixed along its interior boundary and free on its outer boundary. The paper discusses the use of radial and circumferential actuators, each of so-called monotropic type, to produce strictly radial deformations of the elastic structure so as to achieve target configurations, or shapes, in a specific class.

A lumped stress method for plane elastic problems and the discrete-continuum approximation

This paper proposes a rational method to approximate a plane elastic body through a latticed structure composed of truss elements. The method is based on the introduction of a relaxed stress energy that allows an extension of the original problem to a larger space of admissible stress fields, including stresses concentrated along lines. Use is made of polyhedral approximations of the Airy stress function. The truss analogy is employed to obtain a displacement formulation. The paper includes several numerical applications of the method to sample problems, a numerical convergence study and comparisons with exact solutions and standard finite element approximations.

A new boundary element method for mixed boundary value problems involving cracks and holes: Interactions between rigid inclusions and cracks

This paper derives a new boundary integral equation (BIE) formulation for plane elastic bodies containing cracks and holes and subjected to mixed displacement/ traction boundary conditions, and proposes a new boundary element method (BEM) based upon this formulation. The basic unknown in the formulation is a complex boundary function H(t), which is a linear combination of the boundary traction and boundary displacement density. The present BIE formulation can be related directly to Muskhelishvili's formalism. Singular interpolation functions of order r−1/2 (where r is the distance measured from the crack tip) are introduced such that singular integrand involved at the element level can be integrated analytically. By applying the BEM, the interaction between a rigid circular inclusion and a crack is investigated in details. Our results for the stress intensity factor are comparable with those given by Erdogan and Gupta (1975) and Gharpuray et al. (1990) for a crack emanating from a stiff inclusion, and with those by Erdogan et al. (1974) for a crack in the neighborhood of a stiff inclusion.

Solution of the plane Hertz problem

The paper considers the problem of one‐sided frictionless compression of plane elastic bodies that are initially in contact with each other at a point. The first terms of an asymptotic solution of the problem are constructed by the method of joined asymptotic expansions. Determination of the approach of the bodies as a function of the pressing force reduces to calculating so‐called of local compliance. The problems of contact of an elastic ring and elastic circular disks with punches and an elastic disk compressed between two elastic strips are considered. An asymptotic model for the quasistatic collision of plane elastic bodies is proposed.

A new boundary integral formulation for plane elastic bodies containing cracks and holes

This paper presents a new boundary integral formulation for a plane elastic body containing an arbitrary number of cracks and holes. The body is assumed to be linear elastic and isotropic, but can be of either finite or infinite extend. The cracks inside the body can be either internal or edge crack, and either straight or curvilinear; and the holes can be of arbitrary number and shape. Starting from Somigliana formula, we obtain a system of boundary integral equations by applying integration by parts. In complex variables notation, the stress and displacement components can be expressed in terms of Muskhelishvilis analytic functions, which are in turn written as functions of boundary traction and displacement data in the form of Cauchy integral. The complex boundary integral equations for traction involve only singularity of order 1/ r, where r is the distance measured from the singular boundary points, and no hypersingular terms appear. This new boundary integral formulation provides an effective basis in solving problems both analytically and numerically. To illustrate the validity of our new integral formulation, a number of classical problems are re-examined analytically using the present formulation: (i) an infinite body containing a circular hole subject to far field biaxial stress, internal pressure, and a point force on the holes boundary respectively; and (ii) an infinite body containing a circular-arc crack under remote uniaxial tension. To illustrate the applicability of the present formulation for boundary element method analysis, two numerical examples for the interactions between two collinear cracks are considered and the results agree well with the existing solutions by Chandra et al. (1995) for the case of finite rectangular plates and with Isida (cited in p. 195 of Murakami, 1987) for the case of infinite plates.

Fracture angle of cracks of the two cracks interaction problem in a unidirectional fibre reinforced composite material

In this paper, we obtain the values of the fracture angle using minimum strain energy density theory developed by Sih [2]. When the values of the minimum strain energy density factor (Sin) is below that of the criticai strain energy density factor (So), the crack will propagate. It is noted that for the anisotropic case, So is no longer constant as it is for the isotropic. Here we only consider such a case as shown in Fig. 1. Consider two collinear cracks of length 2a which are situated in a generally anisotropic plane elastic body. The preferred directions are denoted by 1 and 2, the references axes are denoted by x and y. Generally they are not coincident with each other which makes angle @. A uniform tensile stress of magnitude t~ is applied in the y-direction. From the work in [3], the strain energy density factor is given by

The B.E.M. in plane elastic bodies with cracks and/or boundaries of fractal geometry

The B.E.M. in plane elastic bodies with cracks and/or boundaries of fractal geometry

The B.E.M. in plane elastic bodies with cracks and/or boundaries of fractal geometry

The scope of the present paper is to present a method which examines the influence of the fractal geometry on the stress and strain fields in cracked plane elastic bodies through a B.E. scheme combined with an iterative approximation procedure. The method proposed here is based on the description of the fractal as the attractor of a deterministic or a random iterated function system. It is an iterative method which approximates the fractal boundary by classical C 1-curves in order to avoid additional singularities. The method proposed may be seen as an extension of the classical B.I.E.M. to the case of bodies having cracks and/or boundaries of fractal geometry.

Non‐linear analysis of plane elastic bodies with inclusions by a BEM‐FEM approach

AbstractIn the present study a hybrid boundary‐element and finite‐element method is developed for the non‐linear analysis of two‐dimensional elastic media with inclusions, and it is utilized in defining the micro‐mechanical stress field in composite materials. The linear elastic, isotropic and homogeneous matrix is treated with the BEM, while the non‐linear elastic and orthotropic inclusions (fibres) are treated with the FEM.

The stress singularity coefficient at a finite rigid flat inclusion in an orthotropic plane elastic body

The analysis of a finite flat inclusion in an orthotropic plane elastic body with purely imaginary characteristic roots under inclined uniform loading at infinity is performed using Lekhnitskii's theory. The major features of the problem are exhibited and discussed.

Fundamental solutions to Laplace's potential operator and to Navier's elasticity operator for Riemann surfaces with finite and infinite numbers of sheets

Fundamental solutions of the plane potential operator for Riemann surfaces are established by conformal mapping. Insertion of these into the Neuber-Papkowitsch formula yields the corresponding fundamental solutions to Navier's operator. Fundamental solutions for Riemann surfaces can be used as kernels of integral equations with the aid of which it is possible to advantageously treat domains (e.g. plane elastic bodies) with slits or overlapping parts. Numerical results demonstrate the practicability of the method. An extensive survey of the paper is given in Section 1.

Fundamental solutions of the Laplace operator and of Navier's elasticity operator for Riemann surfaces with two branch points

Abstract The solutions are established using the methods of reciprocal radii and of conformai mapping and Neuber's and Papkowitsch's formula. They are suited as kernels of boundary integral equations. The equations can be applied to determine the displacements and stresses in plane elastic bodies having overlapping parts or bodies containing slits or cracks which begin and end in the interior of the body. Numerical results confirm that the proposed concept is practicable.

Interactive design based on the p-version of the finite element method

A case study, concerned with shape optimization of a plane elastic body, is presented. The study focuses on the assurance of reliability and accuracy of computed data on which design decisions are based. The design procedure is interactive in the sense that the designer chooses the design parameters on the basis of information generated in the analysis cycle.

Hemivariational inequalities for linear and non linear elastic materials

In the present paper we study hemivariational inequalities arising in linear and nonlinear elastic bodies which are subjected to boundary conditions of nonmonotone multivalued type. After the formulation of the boundary conditions and the derivation of the variational principles we study a hemivariational inequality for a plane elastic body concerning the approximation of its solution. The proposed method is illustrated by an example dealing with the adhesive contact problem of two cylinders and another one concerning an adhesively supported beam.