Resonant waves in elastic structured media: Dynamic homogenisation versus Green’s functions

We address an important issue of dynamic homogenisation in vector elasticity for a doubly periodic mass-spring elastic lattice. The notion of logarithmically growing resonant waves is used in the analysis of star-shaped wave forms induced by an oscillating point force. We note that the dispersion surfaces for Floquet–Bloch waves in the elastic lattice may contain critical points of the saddle type. Based on the local quadratic approximations of a dispersion surface, where the radian frequency is considered as a function of wave vector components, we deduce properties of a transient asymptotic solution associated with the contribution of the point source to the wave form. The notion of local Green’s functions is used to describe localised wave forms corresponding to the resonant frequency. The special feature of the problem is that, at the same resonant frequency, the Taylor quadratic approximations for different groups of the critical points on the dispersion surfaces (and hence different Floquet–Bloch vectors) are different. Thus, it is shown that for the vector case of micro-structured elastic medium there is no uniformly defined dynamic homogenisation procedure for a given resonant frequency. Instead, the continuous approximation of the wave field can be obtained through the asymptotic analysis of the lattice Green’s functions, presented in this paper.     

On the efficient representation of the half-space impedance Green’s function for the Helmholtz equation

A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green’s function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for representing this Green’s function are the Sommerfeld integral and the (closely related) method of complex images. The former is extremely efficient when the source is at some distance from the half-space boundary, but involves an unwieldy range of integration as the source gets closer and closer. Complex image-based methods, on the other hand, can be quite efficient when the source is close to the boundary, but they do not easily permit the use of the superposition principle since the selection of complex image locations depends on both the source and the target. We have developed a new, hybrid representation which uses a finite number of real images (dependent only on the source location) coupled with a rapidly converging Sommerfeld-like integral. While our method applies in both two and three dimensions, we restrict the detailed analysis and numerical experiments here to the two-dimensional case.     

Green’s functions for multi-phase isotropic laminated plates

This work develops a series of Green’s functions for multi-phase Kirchhoff isotropic laminated plates. First, we derive the Green’s functions for a composite laminated plate composed of two bonded dissimilar isotropic laminated semi-infinite plates. Second, the obtained results for bimaterials are judiciously applied to obtain the Green’s function solution for a circular elastic inclusion embedded in an infinite isotropic laminated plate. Third, Green’s functions for a composite space composed of an arbitrary number of wedges of different isotropic laminated plates are derived. Finally, we derive Green’s functions for a laminated plate with an elliptical and a parabolic boundary, respectively.     

Green’s formula and singularity at a triple contact line. Example of finite-displacement solution

The various equations at the surfaces and triple contact lines of a deformable body are obtained from a variational condition, by applying Green’s formula in the whole space and on the Riemannian surfaces. The surface equations are similar to the Cauchy’s equations for the volume, but involve a special definition of the ‘divergence’ (tensorial product of the covariant derivatives on the surface and the whole space). The normal component of the divergence equation generalizes the Laplace’s equation for a fluid–fluid interface. Assuming that Green’s formula remains valid at the contact line (despite the singularity), two equations are obtained at this line. The first one expresses that the fluid–fluid surface tension is equilibrated by the two surface stresses (and not by the volume stresses of the body) and suggests a finite displacement at this line (contrary to the infinite-displacement solution of classical elasticity, in which the surface properties are not taken into account). The second equation represents a strong modification of Young’s capillary equation. The validity of Green’s formula and the existence of a finite-displacement solution are justified with an explicit example of finite-displacement solution in the simple case of a half-space elastic solid bounded by a plane. The solution satisfies the contact line equations and its elastic energy is finite (whereas it is infinite for the classical elastic solution). The strain tensor components generally have different limits when approaching the contact line under different directions. Although Green’s formula cannot be directly applied, because the stress tensor components do not belong to the Sobolev space H¹(V)$H^1(\text{V})$, it is shown that this formula remains valid. As a consequence, there is no contribution of the volume stresses at the contact line. The validity of Green’s formula plays a central role in the theory.     

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.     

Green’s function solution for transient heat conduction in annular fin during solidification of phase change material

The Green’s function method is applied for the transient temperature of an annular fin when a phase change material (PCM) solidifies on it. The solidification of the PCMs takes place in a cylindrical shell storage. The thickness of the solid PCM on the fin varies with time and is obtained by the Megerlin method. The models are found with the Bessel equation to form an analytical solution. Three different kinds of boundary conditions are investigated. The comparison between analytical and numerical solutions is given. The results demonstrate that the significant accuracy is obtained for the temperature distribution for the fin in all cases.     

Evaluation of Green’s function for vertical point-load excitation applied to the surface of a layered semi-infinite elastic medium

The topic of this paper is to show that the integrals of infinite extent representing the surface displacements of a layered half-space loaded by a harmonic, vertical point load can be reduced to integrals with finite integration range. The displacements are first expressed through wave potentials and the Hankel integral transform in the radial coordinate is applied to the governing equations and boundary conditions, leading to the solutions in the transformed domain. After the application of the inverse Hankel transform it is shown that the inversion integrands are symmetric/antimetric in the transformation parameter and that this characteristic is preserved for any number of layers. Based on this fact the infinite inversion integrals are reduced to integrals with finite range by choosing the suitable representation of the Bessel function and use of the fundamental rules of contour integration, permitting simpler analytical or numerical evaluation. A numerical example is presented and the results are compared to those obtained by the CLASSI program.     

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.     

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.     

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.     

Green’s functions for bending problem of a thin anisotropic plate with an elliptic hole

The Green’s functions have not been studied in open literatures for the bending problem of an anisotropic plate with an elliptic hole subjected to a normal concentrated force and a concentrated moment. In this paper, the problem is investigated and the Green’s functions are first obtained by using the complex potential approach. The techniques of conformal mapping transformation and analytic continuation are used to derive the closed-form complex stress functions. The Green’s functions obtained have some potential applications in the analysis of composite structures such as the modification of the displacement compatibility model for notched stiffened composite panels and the formulation of a new method for interlaminar stress analysis around holes of laminates.     

Inclusion of an arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric half plane

This paper presents an exact closed-form solution for the Eshelby problem of a polygonal inclusion with graded eigenstrains in an anisotropic piezoelectric half plane with traction-free on its surface. Using the line-source Green’s function, the line integral is carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. The solutions are applied to the semiconductor quantum wire (QWR) of square, triangular, and rectangular shapes, with results clearly illustrating various influencing factors on the induced fields. The exact closed-form solution should be useful to the analysis of nanoscale QWR structures where large strain and electric fields could be induced by the non-uniform misfit strain.     

Universal behaviour of a wave chaos based electromagnetic reverberation chamber

In this article, we present a numerical investigation of three-dimensional electromagnetic Sinai-like cavities. We computed around 600 eigenmodes for two different geometries: a parallelepipedic cavity with one half-sphere on one wall and a parallelepipedic cavity with one half-sphere and two spherical caps on three adjacent walls. We show that the statistical requirements of a well operating reverberation chamber are better satisfied in the more complex geometry without a mechanical mode-stirrer/tuner. This is due to the fact that our proposed cavities exhibit spatial and spectral statistical behaviours very close to those predicted by random matrix theory. More specifically, we show that in the range of frequency corresponding to the first few hundred modes, the suppression of non-generic modes (regarding their spatial statistics) can be achieved by reducing drastically the amount of parallel walls. Finally, we compare the influence of losses on the statistical complex response of the field inside a parallelepipedic and a chaotic cavity. We demonstrate that, in a chaotic cavity without any stirring process, the low frequency limit of a well operating reverberation chamber can be significantly reduced below the usual values obtained in mode-stirred reverberation chambers.     

Elastic fields due to dislocation arrays in anisotropic bimaterials

Based on the single-dislocation Green’s function, analytical solutions of the elastic fields due to dislocation arrays in an anisotropic bimaterial system are derived by virtue of the Cottrell summation formula. The singularity in the Peach–Koehler (P–K) force is removed by both rigorous mathematical approach and physical energy consideration. Numerical results for dislocation arrays in the Cu/Nb bimaterial with Kurdjumov–Sachs (K–S) orientation show that: (1) the traction continuity and periodic condition are both satisfied; (2) the maximum magnitude of the traction at the interface due to a mixed dislocation array is smaller than that due to a single mixed dislocation. In other words, the traction at the interface could be suppressed by the corresponding array with a relatively high density (L < 10 nm); however, the shear stress on the glide plane increases with increasing dislocation density; (3) the Cu/Nb interface attracts the mixed dislocation array in copper and repels the screw one there. This implies that the P–K force depends not only on the material properties, but also on the crystal orientation and the type of Burgers vector, among others.     

Generalized derivatives of a region function and its applications

This paper is based on some fundamental concepts im [7], Clarke’s generalizedderivatives,as well as Lasotra’s and Strauss’s definitions of differential D(x) of amultivalued function f(x).Thereby,the generalized derivatives of a region function F(x) isdefined asD_F(x)=U∩{G(x)(?)B(R), (?)x∈B(R); G(x)=F’_x=F’(x)}The existence of the generalized derivatives of a region function F(x) is discussed:thenecessary and sufficient conditions of existence of the Fréchet generalized derivatives ofsuch a function is established.     

An interpolated time-domain equivalent source method for modeling transient acoustic radiation over a mass-like plane based on the transient half-space Green’s function

Interpolated time-domain equivalent source method (ITDESM) is based on the assumption of free space, which makes it not suitable for reconstructing the transient acoustic quantities in the half space. Here, a half-space ITDESM is proposed to model the transient acoustic radiation over a mass-like plane. In this method, the free transient Green’s function existing in the conventional ITDESM is replaced by a closed-form transient half-space Green’s function for a mass-like plane. Such transient Green’s function enables one to take the reflection effect of the mass-like plane into consideration. Modeling acoustic radiation from three transient monopoles above an infinite plane with mass-like behavior is studied by numerical simulations to demonstrate the feasibility of the half-space ITDESM. The proposed method is also examined by comparing the reconstruction accuracy among a free-field model, a rigid plane model and a mass-like plane model. An experiment with an impacted steel plate lying above a table plate is conducted in the semi-anechoic room, and the results further verify the effectiveness of the proposed method.     

Nonlocal elasticity defined by Eringen’s integral model: Introduction of a boundary layer method

In this paper we consider a nonlocal elasticity theory defined by Eringen’s integral model and introduce, for the first time, a boundary layer method by presenting the exponential basis functions (EBFs) for such a class of problems. The EBFs, playing the role of the fundamental solutions, are found so that they satisfy the governing equations on an unbounded domain. Some insight to the theory is given by showing that the EBFs satisfying the Navier equations in the classical elasticity theory also satisfy the governing equations in the nonlocal theory. Some additional EBFs are particularly obtained for the nonlocal theory. In order to use the EBFs on bounded domains, the effects of the boundary conditions are taken into account by truncating the kernel/attenuation function in the constitutive equations. This leads to some residuals in the governing equations which appear near the boundaries. A weighted residual approach is employed to minimize the residuals near the boundaries. The method presented in this paper has much in common with Trefftz methods especially when the influence area of the kernel function is much smaller than the main computational domain. Several one/two dimensional problems are solved to demonstrate the way in which the EBFs can be used through the proposed boundary layer method.     

Two-dimensional time-harmonic dynamic Green’s functions in transversely isotropic piezoelectric solids

The two-dimensional time-harmonic dynamic Green’s functions in an infinite transversely isotropic piezoelectric solid are obtained. After introduction of a new function, the original problem is reduced to the determination of the Green’s function for the two-dimensional Helmholtz equation and that for the two-dimensional Laplace equation. The explicit expressions of all the field components are presented. It is verified that the obtained dynamic Green’s functions can reduce to the corresponding static ones by letting the circular frequency be zero. The asymptotic expansions for Green’s functions at far-field are also given.     

THE DISPLACEMENT FUNCTION OF QUASI-CONFORMING ELEMENT AND ITS NODE ERROR

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