Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H, M, and N. With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P_i (i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.     

Antiplane time-harmonic green’s functions for a circular inhomogeneity with an imperfect interface

Two-dimensional antiplane time-harmonic Green’s functions for a circular inhomogeneity with an imperfect interface are derived. Here the linear spring model with vanishing thickness is employed to characterize the imperfect interface. Explicit expressions for the displacement and the stress fields induced by time-harmonic antiplane line forces located both in the unbounded matrix and in the circular inhomogeneity are presented. When the circular frequency approaches zero, our results reduce to those for the static case. Numerical results are presented to show the influence of the frequency and the imperfection of the interface on the stress and displacement fields.     

Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a linear elastic transversely isotropic bi-material full-space is presented. Three-dimensional point-load Green’s functions for stresses and displacements are given in complex-plane line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, within the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration. As illustrations, the present Green’s functions are analytically degenerated to the special cases such as half-space, surface and full-space Green’s functions. Some typical numerical examples are also given to show the general features of the bi-material Green’s functions.     

Solving the elastic bending problem for a plate with mixed boundary conditions

The bending problem for an arbitrarily outlined thin plane with mixed boundary conditions is solved. A technique based on the methods of potentials and balancing loads is proposed for constructing Green’s function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi’s solution for rectangular plates __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 104–112, May 2006.     

Green’s-function method for modeling surface acoustic wave dispersion in anisotropic material systems and determination of material parameters

A Green’s-function method for modeling propagation of surface acoustic waves in anisotropic nanolayered materials is reviewed. The mathematical model, developed at NIST, provides a computationally efficient inversion algorithm for determination of the material parameters of the film, such as its elastic constants and density, from observed dispersion of the surface acoustic waves. The application of the method to a 306 nm thick TiN film having transverse isotropy on a single crystal Si substrate is discussed as an example. The errors in the values of the parameters determined by using the inversion algorithm and the question of uniqueness of the values of the parameters are discussed in detail. It is suggested that, at least in the example considered in this paper, the SAW dispersion can be used to determine any two parameters of the film, provided other parameters are known by independent measurements. In particular, the values of c₁₁ and the density of the film, obtained from the measured SAW dispersion, are the most reliable and the value of c₄₄ is the least reliable. The method is extended to account for defective bonding between the film and the interface and the effect of an intermediate layer of silica between the film and the substrate.     

Green's function of a point dislocation for the bending of a composite infinite plate with an elliptical hole at interface

Summary  The problem of a hole at bimaterial interface is of practical importance in providing a good understanding of the debonding phenomenon and for determining factors that affect the mechanical properties of composite elements of structures. The problem of a point dislocation in bending bonded dissimilar semi-infinite plates with an elliptical hole at interface is tackled in this paper. Based on the method of analytic continuation and the rational mapping function technique, the problem of obtaining the stress functions in the upper and lower plates is decoupled, and reduced to two Riemann–Hilbert problems. The closed-form solution is obtained. The stress distributions at the bimaterial interface, as well as the debonding at both vertices of the elliptical hole are studied. The stress intensities of debonding are depicted for various parameters. Received 16 March 2000; accepted for publication 12 July 2000     

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

Based on the extended Stroh formalism, we first derive the extended Green’s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi-material. These include Green’s functions of the extended dislocation, displacement discontinuities within a finite interval and the concentrated displacement discontinuities, all on the interface. The Green’s functions are then applied to obtain the integro-differential equation governing the interfacial crack. To eliminate the oscillating singularities associated with the delta function in the Green’s functions, we represent the delta function in terms of the Gaussian distribution function. In so doing, the integro-differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi-material with the extended displacement discontinuities being the unknowns. A simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities, along with the asymptotic expressions of the extended intensity factors and J-integral in terms of the discontinuities near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, and the influence of different extended loadings on the interfacial crack behaviors is further investigated.