Elastostatic Analysis Research Articles

A finite elastostatic analysis of bimaterial interface cracks

A numerical method is developed to study the bimaterial interface problem in Neo-Hookean materials under plane stress conditions. Comparison is made with the analytical predictions for the asymptotic field of the problem. The range of dominance of the asymptotic solution at different load levels is established and the amplitudes of the crack-tip asymptotic field are related to the far field loading. The numerical model is extended to analyze the experiments conducted on specimens with an edge crack at the interface between two dissimilar Solithane plates that are characterized by Mooney-Rivlin material behavior.

Boundary element analysis of stress intensity factors for an interface crack in dissimilar materials.

This paper describes the boundary element elastostatic analysis of a crack along the straight interface between two dissimilar materials. It is well known that the interface crack has an oscillation singularity at the crack tip which is quite different from a crack in homogeneous materials. Therefore, the definition of the stress intensity factor is not clarified yet and it is very difficult to analyze the interface crack by the numerical methods, such as FEM and BEM. We proposed a method to determine the stress intensity factors K1 and K2, for the interface crack by means of extrapolating the solutions at the points apart from the crack tip to avoid the oscillation singularity. This method can be widely applied to the BEM and also FEM analyses with any changes of the programmes. It is confirmed that the present results of the stress intensity factors for an interface crack in the infinite plate are completely consistent with the exact solutions. The stress intensity factors for various interface cracks in the finite plate and the adhesive joint are analyzed by the present method and BEM analyses, which is specially developed for this study.

Curved Box‐Girder Bridges

Based on the curvilinear coordinate system, the spline finite strip method is extended to the elasto-static analysis of circular and noncircular box-girder bridges. As the curvature effect cannot be ignored, the webs of the bridges have to be treated as thin shells and the flanges as flat curved plates. The shape functions for the description of the displacement field (radial, tangential, and vertical displacements) are given as products of B-3 spline functions in the longitudinal direction and piecewise polynomials in the other direction. The stress-strain and stiffness matrices can then be formed as in the standard finite element method. Three examples of box-girder bridges of different geometrical shapes are employed to demonstrate the accuracy and versatility of the method. Compared to the finite element method, this method yields considerable savings in both computer time and effort, since only a small number of unknowns are generally required in the analysis.

A boundary element formulation for natural convection problems

AbstractThis paper presents a boundary element formulation employing a penalty function technique for two‐dimensional steady thermal convection problems. By regarding the convective and buoyancy force terms in Navier‐Stokes equations as body forces, the standard elastostatics analysis can be extended to solve the Navier‐Stokes equations. In a similar manner, the standard potential analysis is extended to solve the energy transport equation. Finally, some numerical results are included, for typical natural convection problems, in order to demonstrate the efficiency of the present method.

Elastostatic analysis of problems involving complex toroidal geometries

The three-dimensional boundary integral equation (BIE) method is used to solve two elastostatic problems involving complex toroidal geometries. The first problem concerns a thick-walled 180 degree pipe bend connected to tangent pipes, subjected to in-plane bending and internal pressure. Results are presented for stresses and are compared with results from previous numerical and experimental work. The second problem concerns a thick-walled pressurized torus, with cross bores situated in the toroidal plane. Separate solutions are given for cross bores located at the intrados and extrados of the torus. Results are presented for the stress concentration factor, and are compared with numerical work for thick-walled cylindrical pipes.

Boundary element elastic analysis of layered soils by a successive stiffness method

AbstractThe elastostatic analysis of layered systems (such as a soil consisting of a set of L individually homogeneous strata) is tackled here on the basis of discretized boundary integral equations (boundary element method). By exploiting the peculiar chain‐like pattern of the system, a recursive formula is obtained which generates economically a ‘stiffness matrix’ of the first n layers (from bottom) at the upper interface with the subsequent layer (n = 1…L). The ‘successive stiffness’ method proposed is shown to imply noteworthy advantages with respect to both the standard boundary element method by zones (or subregions) and another ad hoc, earlier method resting on a boundary element approach combined with the transfer matrix concept. This conclusion is corroborated by two‐dimensional examples.

Three‐dimensional transient thermo‐elastic analyses by the BIE method

AbstractThe proposed method is the direct boundary integral equation (BIE) method applied to three‐dimensional transient heat transfer analyses and to corresponding elastostatic analyses under thermal loading. The mechanical and thermal problems are decoupled. For those not familiar with the BIE method, a short survey of the basic principles is given to help them understand the mathematical treatment, which is performed in the frame of distribution theory. The tempered elementary solution of the transient heat transfer problem contains time and space variables. The numerical treatment of the thermal equation is performed analytically with respect to time and numerically with respect to space in the same way as for steady state equations. Great care is paid to space integration in order to compute the values of the kernels precisely enough on the surface and inside the volume. The computational treatments of both problems are realized in such a way as to minimize computing time. Differences between transient and steady state analyses, on the one hand, and between elastostatic analyses with and without thermal loading, on the other, are emphasized. Finally, an analytical example shows the good behaviour of the proposed method to solve singular thermal problems and an industrial example shows the accuracy of the results obtained by using the BIE method to solve three‐dimensional thermo‐elastic problems.

The usefulness and limit of the direct regular method in boundary element elastostatic analysis.

The Direct Regular Method is applied to BEM elastostatic analysis. In this method, the domain of the problem is discretized by defining freedoms on its boundary, as usual, but the load point of the fundamental solution is located outside the domain so that the singular integral can be avoided. Stress intensity factors for a crack and stress concentration for a hole are analyzed by this method and the usefulness and limit of this method are discussed in comparison with Direct BEM. It is shown that this method can give an accurate solution for these problems if the load points are arranged very near the boundary. It is found that this method is especially useful for the problem with a corner point where the tractions on both edges are unknown. This method can be used in combination with the Direct Method.

Plane elastostatic analysis of a crack passing through a bimaterial interface.

The plane elastostatic problem for a bonded plane consisting of an isotropic and an anisotropic half planed is considered when a straight crack originating in one half-plane is passing through a bimaterial interface into an other. It is assumed that the crack is normal to the interface and that the bonded plane is subjected to a constant strain away from, and perpendicular to, the crack. A fundamental stress field due to two colinear edge dislocations existing in each half plane is firstly analyzed, and then the method of continuous distribution of dislocations reduces the problem to a system of simultaneous singular integral equations for the dislocation density. The singular behaviour of stresses at the point of intersection of the crack and the interface as well as at the crack-tips in investigated in detail.

Plane elastostatic analysis of a crack passing through a bi-material interface.

Plane elastostatic analysis of a crack passing through a bi-material interface.

Free-form shell analysis by a mixed-hybrid finite element approach

A novel finite element scheme is proposed for the linear elasto-static analysis of free-form thinwalled shells. On the basis of a modified version of the variational theorem due to Hellinger and Reissner, a doubly curved triangular element is developed with six degrees-of-freedom (three translations plus three rotations) per nodal point. For each element, separate trial functions are introduced in order to approximate the displacements and the stress resultants. It is shown that by properly choosing these interpolants, an element is obtained which (i) is capable of exactly representing the rigid body and constant strain modes; (ii) does not possess any spurious kinematic modes; (iii) fulfils C1 interelement continuity and (iv) satisfies exactly the interior domain static equilibrium conditions. Two representative examples have been solved in order to illustrate the advantages of the proposed scheme over existing ones.

Boundary element analysis of viscous flow by penalty function formulation

This paper presents a boundary element formulation employing the penalty function technique for steady viscous flow problems. By regarding the convective terms in Navier-Stokes equations as body forces in elastostatics, the standard elastostatics analysis can be extended to solve viscous flow problems. In particular, several investigations were carried out for evaluating the convective terms. These include a finite difference scheme (employing both upwind and central approximations) and a boundary integral formulation. Finally, some numerical results are shown in order to demonstrate the validity of the present method and its accuracy in the evaluation of convective terms.

The usefulness and the limit of the Direct Regular Method in boundary element elastic analysis.

The Direct Regular Method is applied to BEM elastostatic analysis. In this method, the domain of the problem is discretized by defining freedoms on its boundary, as usual ; but the load point of the foundamental solution is located outside the domain so that the singular integral can be avoided. Stress intensity factors for a crack and stress concentration for a hole are analyzed by this method and the usefulness and limit of this method are discussed in comparison with Direct BEM. It is shown that this method can give an acculate solution for these problems if the load points are arranged very near the boundary. It is found that his method is especially useful for the problem with a corner point where the tractions on both edges are unknown. This method can be used combined with the Direct Method.

Automated Computer Simulation of Various Physical Phenomena for the Application to Automated Machine Design : 2nd Report, Elastostatic Analysis of Three-dimensional Structures Composed of Curved Plates

As a basis for the automated machine design, an automated computer simulation method and a computer program for elastostatic analysis of three-dimensional structures composed of curved plates are developed in this second report. This method consists of two parts, automated mesh generation and automated stress analysis. The stress analysis is based on the Assumed Stress Hybrid Method and the Gaussian Elimination Method. The element is a triangular element and the mesh generation is carried out automatically for all curved surfaces of plates. The computer program developed in this report can determine the elastostatic stresses in three-dimensional plate structures, such that the relative error of each computed stress does not surpass the limiting error which the analyst allows.

Elastostatic interaction of multiple arbitrarily shaped cracks in plane inhomogeneous regions

A numerical technique has been developed for the determination of stress fields associated with multiple arbitrarily shaped cracks in plane inhomogeneous regions. The procedure allows the elastostatic analysis of cracks interacting with one or more straight bimaterial interfaces; of cracks located near, or emanating from, circular inclusions; and of cracks that emanate from single or multiple origins. The cracks may be branched or blunted, and may be subjected to arbitrarily applied stresses. The technique employs an efficient surface integral method, using distributions of edge dislocations to represent the cracks. The resulting singular integral equations are solved using a Gauss-Chebyshev integration formula; appropriate conditions are developed for closing the set of equations governing cracks intersecting inhomogeneity boundaries, based on a consideration of the stresses and displacements at the points of intersection. Crack-tip stress intensity factor results are presented for several crack configurations. The overall scheme provides a more general, direct, and convenient approach than other available schemes. A computer program has been developed to implement the various formulations in a single framework.

Graphic aids in a B.I.E. program for 3-D elastostatic analysis

An interactive graphics program has been developed which enables the analyst to check input data and examine analysis results when performing 3-D elastostatic analyses by the boundary integral element method. A description is given of the options available whereby the analyst can both request different display configurations and modify the 3-D representation. The program has been designed in such way as to exploit the video refreshing display's potential to the full; the examples in the included photographs give some indication of how it can be used.

Analysis for an Embedded Ice Pressure Sensor

A plane elastostatic analysis is presented for the problem of a rectangular, orthotropic, elastic inclusion in an isotropic elastic host, with differential thermal expansion between the inclusion and the host. The analysis assumes that bending as well as shear deflection of the inclusion boundaries can be neglected. The product of this analysis is a closed form solution applicable to an embedded ice pressure sensor of rectangular cross section having relatively rigid sensing face cover plates. The form of the solution displays several significant results concerning inclusion effects for both in-line and transverse ice stresses, Poisson’s ratio effects, and thermal expansion effects. The most striking result is that the magnitude of the error induced in ice pressure measurements by neglecting differential thermal expansion is not a function of the sensor’s stiffness in the sensing direction.

SERBA: a B.I.E. program with linear elements for 2-D elastostatics analysis

The great developments that have occurred during the last few years in the finite element method and its applications has kept hidden other options for computation. The boundary integral element method now appears as a valid alternative and, in certain cases, has significant advantages. This method deals only with the boundary of the domain, while the F.E.M. analyses the whole domain. This has the following advantages: the dimensions of the problem to be studied are reduced by one, consequently simplifying the system of equations and preparation of input data. It is also possible to analyse infinite domains without discretization errors. These simplifications have the drawbacks of having to solve a full and non-symmetric matrix and some difficulties are incurred in the imposition of boundary conditions when complicated variations of the function over the boundary are assumed. In this paper a practical treatment of these problems, in particular boundary conditions imposition, has been carried out using the computer program shown below. Program SERBA solves general elastostatics problems in 2-dimensional continua using the boundary integral equation method. The boundary of the domain is discretized by line or elements over which the functions are assumed to vary linearly. Data (stresses and/or displacements) are introduced in the local co-ordinate system (element co-ordinates). Resulting stresses are obtained in local co-ordinates and displacements in a general system. The program has been written in Fortran ASCII and implemented on a 1108 Univac Computer. For 100 elements the core requirements are about 40 Kwords. Also available is a Fortran IV version (3 segments)implemented on a 21 MX Hewlett-Packard computer,using 15 Kwords.

Axisymmetric stresses in transversely isotropic finite cylinders

The elastostatic analysis of transversely isotropic finite cylinders with stress-free lateral surfaces is considered using the formulation in terms of a displacement potential. Solution in the form of the eigenfunction expansion where each of the eigen-functions satisfies the lateral boundary conditions is obtained. The end conditions are satisfied by means of the general orthogonality relations for the radial eigenfunctions. The orthogonality technique gives closed-form expressions for the Fourier-Bessel coefficients for certain types of mixed end conditions. For other cases the technique leads to an infinite set of linear equations. The convergence of these equations are examined and solutions are obtained by suitable truncation. Both stress and displacement end conditions are considered and numerical results for two types of transversely isotropic materials are presented.

Progress in the use of boundary integral equations, illustrated by examples

Progress in the numerical treatment of boundary integral equations is reviewed, with particular reference to developments presented at the 1975 ASME Applied Mechanics Conference. The computational strategy of a program for three-dimensional elastostatic analysis is described. In this strategy the surface is approximated by curved quadrilateral and triangular elements. Displacement and traction are defined in terms of linear, quadratic or cubic shape functions of intrinsic coordinates. A system of simultaneous equations is obtained whose coefficients are integrals of kernel-shape function products. The elastic body is divided into subrogions so that the matrix is of banded form, and the coefficients are evaluated using Gaussian quadrature formulas. The program is designed to be capable of treating large problems without excessive core storage requirements. The structure of files created by the program and the order of calculations are such that input/output time is kept low. Two examples are presented: the first demonstrates the possibility of economically solving problems for the infinite space with cavities; the second is the solution of a problem of three-dimensional fracture mechanics and shows the performance of the program in practical use.