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.     

Inplane deformation of a circular inhomogeneity with imperfect interface

The plane elastic problem of a circular inhomogeneity with an imperfect interface of spring-constant-type is reduced to the solution of a Somigliana dislocation problem, when the solution for the corresponding problem with a perfect interface is known. The Burger's vector of the Somigliana dislocation is determined so that its components satisfy two interfacial conditions involving the traction components of the corresponding problem with a perfect interface. Employing complex variables, a two-phase potential solution to the Somigliana dislocation inhomogeneity problem is developed for a general form of the Burger's vector. Detailed results are reported for a uniform eigenstrain in the inhomogeneity, and for a remote uniform heat flow in the matrix. In the latter case, the inhomogeneity behaves as a void, when it begins to slide.     

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ₂/μ₁.     

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.     

Anti-plane time-harmonic Green's functions for a circular inhomogeneity in piezoelectric medium with a spring- or membrane-type interface

The present work focuses on the two-dimensional anti-plane time-harmonic dynamic Green's functions for a circular inhomogeneity immersed in an infinite matrix with an imperfect interface, where both the inhomogeneity and matrix are assumed to be piezoelectric and transversely isotropic. Two types of imperfect interface, the spring-type interface with electromechanical coupling and the membrane-type interface, are considered. The former type is often used to model the electromechanical damage along the interface while the latter is largely employed to simulate surface/interface effect of nano-sized inhomogeneity. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in an unbounded matrix as well as the circular inhomogeneity are respectively derived. The present solutions can recover the anti-plane Green's functions for some special cases, such as the dynamic or quasi-static Green's functions of piezoelectricity with perfect interface as well as the dynamic or quasi-static Green's functions of pure elasticity with imperfect interface. For detailed discussions, selected analytical results are presented. Dependence of the electromechanical fields on circular frequency as well as interface properties is examined. The size effect related to interfacial imperfection is also discussed.     

Piezoelectric screw dislocations interacting with a circular inclusion with imperfect interface

The electroelastic coupling interaction between multiple screw dislocations and a circular inclusion with an imperfect interface in a piezoelectric solid is investigated. The appointed screw dislocation may be located either outside or inside the inclusion and is subjected to a line charge and a line force at the core. The analytic solutions of electroelastic fields are obtained by means of the complex-variable method. With the aid of the generalized Peach–Koehler formula, the explicit expressions of image forces exerted on the piezoelectric screw dislocations are derived. The motion and the equilibrium position of the appointed screw dislocation near the circular interface are discussed for variable parameters (interface imperfection, material electroelastic mismatch, and dislocation position), and the influence of the nearby parallel screw dislocations is also considered. It is found that the piezoelectric screw dislocation is always attracted by the electromechanical imperfect interface. When the interface imperfection is strong, the impact of material electroelastic mismatch on the image force and the equilibrium position of the dislocation becomes weak. Additionally, the effect of the nearby dislocations on the mobility of the appointed dislocation is very important.     

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.     

Interaction between an edge dislocation and a circular inclusion with an inhomogeneously imperfect interface

This research presents an analytical study of the interaction problem of an edge dislocation with a circular inclusion with a circumferentially inhomogeneously imperfect interface. The interface, which is modeled as a spring (interphase) layer with vanishing thickness, is characterized by that in which there is a displacement jump across the interface in the same direction as the corresponding tractions, and the same degree of imperfection is realized in both the normal and tangential directions. Furthermore, the interface parameter is nonuniform along the interface. In order to arrive at an elementary form solution, we introduce a conformal mapping function. Then the stress field as well as the Peach–Koehler force acting on the edge dislocation can be obtained from the derived complex potentials. Calculations demonstrate that the nonuniform interface parameter has a significant influence on the stress field.     

Analysis of a cracked concrete containing an inclusion with inhomogeneously imperfect interface

The distributed dislocation technique is applied to determine the behavior of a cracked concrete matrix containing an inclusion. The analysis of cracked concrete in the presence of inclusions such as steel expansions is a practical problem that needs special attention. The solution to the problem of interaction of an edge dislocation with a circular inclusion having circumferentially inhomogeneously imperfect interface is available in the literature. This analytical solution is used in the distributed dislocation technique to obtain the stress intensity factor for the cracked concrete in the presence of inclusion. The interface of the matrix and the inclusion is assumed inhomogeneously imperfect and the stress intensity factor is determined for the cracked concrete for a case of two identical cracks on diametrically opposite sides of the inclusion. Consideration of this general inhomogeneously imperfect interface is the contribution of this paper. The variation of the inhomogeneity parameters is studied and presented. Additionally, the general assumption for the interface is simplified to the special case of perfectly bonded interface. The observations for the perfect interface are coincident with the previously reported results.     

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.     

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.     

Green's functions for anisotropic piezoelectric bimaterials with a slipping interface

An explicit full-field expression of the Green's functions for anisotropic piezoelectric bimaterials with a slipping interface is derived. When the electro-elastic singularity reduces to a pure dislocation in displacement and electric potential, interaction energy between the dislocation and the bimaterials is obtained explicitly while the generalized force on the dislocation is given in a real form which is also valid for degenerate materials. The investigation demonstrates that the boundary conditions at the slipping interface between two piezoelectric materials will exert a prominent influence on the mobility of the dislocation. Project supported by the National Natural Science Foundation of China (No. 59635140).     

The two-dimensional free-space Green’s function and its derivatives in a tangent-normal system

The two-dimensional free-space Green’s function, G⁽²⁾$G^{\left(2\right)}$, and its derivatives, are used extensively in the formulation of scattering and diffraction problems through its presence in single- and double-layer potentials, and their use in integral equations. The vast majority of the results from elementary classical mathematical physics for G⁽²⁾$G^{\left(2\right)}$ is based on Cartesian coordinate-space, either directly as a Hankel function in coordinate-space or through a transform, such as the Weyl transform, also based on Cartesian coordinate-space. However, if the geometry of the problem is not Cartesian, for example in scattering from a rough surface, there are difficulties in using a transform representation for G⁽²⁾$G^{\left(2\right)}$ which depends on Cartesian geometry, as the standard Weyl transform does. Here we formulate transform-space representations using a tangent-normal coordinate system. The result for G⁽²⁾$G^{\left(2\right)}$ is a new Weyl-type tangent-normal transform representation from which the results for the vector derivatives of the single-layer potential, the double-layer potential, and the vector derivatives of the double-layer potential follow quite simply. The latter three results can be expressed in terms of two new spectral functions in tangent-normal space, _S1$S_1$ and _S2$S_2$. The overall results are new representations for G⁽²⁾$G^{\left(2\right)}$ and its derivatives which may be useful in integral equation formulations of scattering problems for non-Cartesian geometries.     

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.     

An application of Curie’s principle to elastoplastic dynamics

This paper deals with the stability and the dynamics of a harmonically excited elastic–perfectly plastic unsymmetrical oscillator. Stability of the periodic orbits is analytically investigated with a perturbation approach. The occurrence of ratcheting effect is discussed for this system, and is related to the loss of symmetry of the periodic orbit in the phase space. Curie’s principle of symmetry is numerically verified for the symmetrical system with positive damping. Therefore, the observation of ratcheting phenomenon is necessarily associated to a breaking of symmetry in the constitutive behaviour, or in the forcing term. However, the generalized version of Curie’s principle has to be considered when a negative damping is introduced.     

Explicit approximations of the indentation modulus of elastically orthotropic solids for conical indenters

The elastic problem of the contact between an axisymmetric indenter and a general anisotropic (21 independent elastic constants) half space has not been solved explicitly in closed form. Implicit methods to determine the indentation modulus originate from the work of Willis [J. Mech. Phys. Solids 14 (1966) 163]; and are now available for conical, parabolic and spherical indenters [Philos. Mag. A 81 (2001) 447; J. Mech. Phys. Solids 51 (2003) 1701]. The particular case of orthotropy has also been investigated [ASME J. Tribol. 115 (1193) 650, 125 (2003) 223]. This paper proposes an explicit solution for the indentation moduli of a transversely isotropic medium and a general orthotropic medium under rigid conical indentation in the three principal material symmetry directions. The half-space Green’s functions are interpolated from their exact extreme values, then integrated and finally simplified. The proposed closed form expressions are in very good agreement with the implicit solution schemes of [Philos. Mag. A 81 (2001) 447; J. Mech. Phys. Solids 51 (2003) 1701].     

Fer’s expansion for generalized Hamiltonian system based on Lie transformation technique

In the present paper, we present a new method for integrating the ordinary differential equation, especially for the ordinary differential equation derived from explicitly time-dependent generalized Hamiltonian dynamic system, which is based on taking a factorization of the evolution operator as an infinite product of the exponentials of Lie operators. The above process is a Lie group (algebraic) method that retains the structural intrinsic properties of the exact solution when truncated and is used to analyze the main features of the so-called Fer’s expansion. The numerical examples are presented at the end of this paper.