J. R. Berger

On Green's function for a three-dimensional exponentially graded elastic solid

The problem of a point force acting in an unbounded, three–dimensional, isotropic elastic solid is considered. Kelvin solved this problem for homogeneous materials. Here, the material is inhomogeneous; it is ‘functionally graded’. Specifically, the solid is ‘exponentially graded’, which means that the Lamé moduli vary exponentially in a given fixed direction. The solution for the Greens function is obtained by Fourier transforms, and consists of a singular part, given by the Kelvin solution, plus a non–singular remainder. This grading term is not obtained in simple closed form, but as the sum of single integrals over finite intervals of modified Bessel functions, and double integrals over finite regions of elementary functions. Knowledge of this new fundamental solution for graded materials permits the development of boundary–integral methods for these technologically important inhomogeneous solids.

The method of fundamental solutions for heat conduction in layered materials

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two-dimensional problems of steady-state heat conduction in isotropic and anisotropic bimaterials. Two approaches are used: a domain decomposition technique and a single-domain approach in which modified fundamental solutions are employed. The modified fundamental solutions satisfy the interface continuity conditions automatically for planar interfaces. The two approaches are tested and compared on several test problems and their relative merits and disadvantages discussed. Finally, we use the domain decomposition approach to investigate bimaterial problems where the interface is non-planar and the modified fundamental solutions cannot be used. Copyright

The method of fundamental solutions for layered elastic materials

In this paper, we investigate the application of the method of fundamental solutions to two-dimensional elasticity problems in isotropic and anisotropic single materials and bimaterials. A domain decomposition technique is employed in the bimaterial case where the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several test problems and its relative merits and disadvantages are discussed.

An overdeterministic approach for measuringK I using strain gages

An overdeterministic method for determining the opening-mode stress-intensity factor,KI, from many measurements of the radial strain,∈rr, is described. The method was verified with an experimental study of a compact-tension specimen where strains along the 0, 45 and 90-degree lines were measured using strip gages with ten strain sensors per strip gage. The results indicated errors in the range of one to three percent with three or four parameter models of the strain field in the region near the crack tip.

Waves in wood: free vibrations of a wooden pole

Elastic waves in materials with cylindrical orthotropy are considered, this being a plausible model for a wooden pole. For time-harmonic motions, the problem is reduced to some coupled ordinary differential equations. Previously, these have been solved using the method of Frobenius (power-series expansions). Here, Neumann series (expansions in Bessel functions of various orders) are used, motivated by the known classical solutions for homogeneous isotropic solids. This is shown to give an effective and natural method for wave propagation in cylindrically orthotropic materials. As an example, the frequencies of free vibration of a wooden pole are computed. The problem itself arose from a study of ultrasonic devices as used in the detection of rotten regions inside wooden telegraph (utility) poles and trees; some background to these applications is given.

A Boundary Element Regularization Method for the Boundary Determination in Potential Corrosion Damage

In this paper, we consider the inverse problem for the Laplace equation in two-dimensions which requires the determination of the location, size and shape of an unknown, or partially unknown, portion n z of the boundary z of a solution domain z R 2 from additional Cauchy data on the remaining portion of the boundary o = z m n . This problem arises in the study of quantitative non-destructive evaluation of corrosion in materials in which boundary measurements of currents and voltages are used to determine the material loss caused by corrosion. This inverse problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularization procedure. The choice of the regularization parameter is based on an L-curve type criterion although, alternatively one may use the discrepancy principle. Several examples which involve noisy Cauchy input data are thoroughly investigated showing that the numerical method produces a stable approximate solution which is also convergent to the exact solution as the data errors tend to zero.

Greens functions for boundary element analysis of anisotropic bimaterials

We present several Greens functions for anisotropic bimaterials for two-dimensional elasticity and steady-state heat transfer problems. The details of the various Greens functions for perfect, slipping, and cracked interfaces are given for mechanical loading conditions. Previously reported formulations for cubic materials are extended to materials with general anisotropy in which plane strain deformations can exist. We also give the steady-state Greens function for thermal loading of a bimaterial with a perfectly bonded interface. The Greens functions are incorporated in boundary integral formulations and method of fundamental solutions formulations for analysis of finite solids under general boundary conditions.

Time-harmonic torsional waves in a composite cylinder with an imperfect interface

In this paper, the propagation of time-harmonic torsional waves in composite elastic cylinders is investigated. An imperfect interface is considered where tractions are continuous across the interface and the displacement jump is proportional to the stress acting on the interface. A frequency equation is derived for the rod and dispersion curves of normalized frequency as a function of normalized wave number for elastic bimaterials with varying values for the interface constant F are presented. The analysis is shown to recover the dispersion curves for a bimaterial rod with a perfect (welded) interface (F = 0), and has the correct limiting behavior for large F. It is shown that the modes, at any given frequency, are orthogonal, and it is outlined how the problem of reflection of a torsional mode by a planar defect (such as a circumferential crack) can be treated.

Boundary element analysis of anisotropic bimaterials with special Green's functions☆

Abstract The boundary integral equations incorporating the Greens function for anisotropic solids containing planar interfaces are presented. The fundamental displacement and traction solutions are determined from the displacement Greens function of Tewary, Wagoner, and Hirth ( Journal of Materials Research , 1989, 4 , 113–123). The fundamental solutions numerically degenerate to the Kelvin solution in the isotropic limit. The boundary integral equations are formulated with the use of constant boundary elements. The constant boundary elements allow for analytical evaluation of the boundary integrals. The application of the method is demonstrated by analyzing a copper-solder system subjected to mechanical loading.

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.