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.     

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.     

Crack analysis in semi-infinite transversely isotropic piezoelectric solid. II. Penny-shaped crack near the surface

Making use of the Somigliana identity, the boundary integral equations are obtained for a planar crack of arbitrary shape in an elastic half space. The material is piezoelectric with transversal isotropy. The solution is given for a penny-shaped crack parallel to the free boundary while the loading is axially symmetric.     

An electric point charge moving along the poling direction of a transversely isotropic piezoelectric solid

The problem of an electric point charge moving constantly along the poling direction of a transversely isotropic piezoelectric solid is considered in a moving coordinate system, which moves together with the electric point charge. A general solution in the moving coordinate system is given, and all the field components, such as displacements, electric potential, stresses and electric displacements, can be concisely expressed in terms of four quasi-harmonic functions. We also present two examples to demonstrate the effect of the moving velocity on the values of _i. Once the general solution is given, the axisymmetric problem of a moving electric point charge can be easily solved. The explicit expressions of all the field components caused by the moving electric charge are presented, and the effect of the moving velocity on these field components is numerically investigated.     

A conductive rigid spheroidal inclusion in a transversely isotropic piezoelectric body subjected to an axial pull

The exact axisymmetric solution is derived for an infinite transversely isotropic piezoelectric body containing an electrically conductive, rigid spheroidal inclusion under an axial pull. A simple general solution is employed in which three quasi-harmonic functions are involved and can be assumed in a closed form. The arbitrary constants are determined from the continuity conditions at the surface of the inclusion. The load-deflection and load-potential relations are derived, especially for two degenerated cases that are very important in the strength analysis of composite piezoelectric materials.     

A penny-shaped crack in a transversely isotropic piezoelectric solid: Modes II and III problems

An exact analysis of the modes II and III problems of a pennyshaped crack in a transversely isotropic piezoelectric medium is performed in this paper. The potential theory method is employed based on the general solution of three-dimensional piezoelasticity and the four harmonics involved are represented by one complex potential. Previous results in potential theory are then utilized to obtain the exact solution that is expressed in terms of elementary functions. Comparison is made between the current results with those published and good agreement is obtained. The project supported by the National Natural Science Foundation of China (No. 19872060)     

Interface crack-tip generalized stress field and stress intensity factors in transversely isotropic piezoelectric bimaterials

Transversely isotropic piezoelectric (TIP) bimaterials with an impermeable interface crack have been classified [Int. J. Frac. 119 (2003) L41] into two classes corresponding to the vanishing of the two singularity parameters or κ. It is shown in the present paper that the related eigenvalue problems for either =0 or κ=0 are not degenerate. The crack-tip generalized stress fields are obtained subsequently. A new definition of crack-tip intensity factors is presented for interface cracks in practical TIP bimaterial of practical interest. Such defined intensity factors are real numbers, which dominate the maximum crack-tip stress singularity and do not generate any phase angle change under any dimension system transformation for physical quantities.     

Penny-shaped and half-plane cracks in a transversely isotropic piezoelectric solid under arbitrary loading

Summary The problem of a penny-shaped crack in a transversely isotropic piezoelectric material loaded by both normal and tangential tractions and by electric charges is analyzed. Closed-form solutions are obtained for the full electroelastic fields as well as for the stress and electric displacement intensity factors. Solutions are also obtained for the (non-trivial) limiting case of a half-plane crack. The results are illustrated on the example of piezoceramics PZT-6B. Received 12 July 1999; accepted for publication 20 July 1999     

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.     

Thermal stress analysis of a three-dimensional anticrack in a transversely isotropic solid

A three-dimensional analysis is performed for an infinite transversely isotropic elastic body containing an insulated rigid sheet-like inclusion (an anticrack) in the isotropy plane under a remote perpendicularly uniform heat flow. A general solution scheme is presented for the resulting boundary-value problems. Accurate results are obtained by constructing suitable potential solutions and reducing the thermal problem to a mechanical analog for the corresponding isotropic problem. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and analyzed. This solution is compared with that corresponding to a penny-shaped crack problem.     

Ward identities and the retrieval of Green’s functions in the correlations of a diffuse field

That diffuse field correlations are essentially identical to Green’s functions is becoming widely accepted. Here, a new derivation is provided for that identity, a derivation that generalizes to a larger variety of wave systems and furthermore permits derivation of other, related but distinct, identities that may be useful.     

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.     

Stress analysis of borehole subjected to fluid injection in transversely isotropic poroelastic medium

This paper proposes an analytical solution, using the combined Laplace–Fourier integral transform technique, for a borehole drilled in transversely isotropic porous medium and subjected to a fluid discharge over a finite length of its surface. Especially, the coupled boundary condition between the total radial stress and injection-induced pore pressure at the borehole surface is addressed in a rigorous fashion, which leads essentially to a set of dual integral equations that can be solved through standard numerical procedure. The study focuses on the calculation of stress fields around the borehole with particular attention given to the time-dependent effective tangential stress and pore pressure distributions. Numerical solutions are presented for verification with those recently derived for the limiting case of an isotropic medium and, more importantly, to investigate the influences of material anisotropy on the stress responses of the borehole and porous medium.     

Comments on ‘‘Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution’’ by Shaofan Li, Peter A. Mataga [J. Mech. Phys. Solids 44 (1996) 1799-1830]

In this comment, it is pointed out that the paper [Li and Mataga, 1996. J. Mech. Phys. Solids 44, 1799-1830], which presents original and valid solution strategy for an important problem of dynamic crack propagation in piezoelectric materials, contains ultimate quantitatively and qualitatively incorrect expressions, conclusions and plots due calculation errors. The correct calculations and corresponding correct conclusions and plots are presented.     

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.     

Electroelastic analysis of a penny-shaped crack in a piezoelectric ceramic under mode I loading

Following the theory of linear piezoelectricity, we consider the electroelastic problem for a piezoelectric ceramic with a penny-shaped crack under mode I loading. The problem is formulated by means of Hankel transform and the solution is solved exactly. The stress intensity factor, energy release rate and energy density factor for the exact and impermeable crack models are expressed in closed form and compared for a P-7 piezoelectric ceramic. Based on current findings, we suggest that the energy release rate and energy density factor criteria for the exact crack model are superior to fracture criteria for the impermeable crack model.     

On the use of Csanady’s formulae in a turbulent gas–solid channel flow

The paper examines the use of expressions proposed by Csanady to predict the influence of the crossing trajectory and continuity effects on the decorrelation time scales of the fluid along solid particle trajectories in horizontal and downward vertical channel flows. The model is evaluated using data provided by a direct numerical simulation (DNS) of the carrier phase combined with a Lagrangian simulation of discrete particle (LS). Two particle relaxation times and two values of the gravity acceleration are considered. The results show the possibility of using Csanady’s expressions in a turbulent channel flow provided that the spatial and temporal correlations anisotropy is included in the model. As in isotropic homogeneous turbulence, a decrease of the decorrelation time scales is found to be more important in the directions perpendicular to the mean relative velocity.     

Comments on “Explicit transient solutions for a mode III crack subjected to dynamic concentrated loading in a piezoelectric material” by Yi-Shyong Ing, Mau-Jung Wang [Int. J. Solids Struct. 41 (2004) 3849–3864]

In this comment it is pointed out that the analysis of the dynamic stress intensity factor, dynamic electric displacement intensity factor and dynamic energy release rate conducted by Ing and Wang [Ing, Y.S., Wang, M.J., 2004. Explicit transient solutions for a mode III crack subjected to dynamic concentrated loading in a piezoelectric material. International Journal of Solids and Structures 41, 3849–3864] is incorrect. The correct analysis and corresponding correct plots are presented.     

<Emphasis Type="Italic">T</Emphasis>-stress analysis for a Griffith crack in a magnetoelectroelastic solid

T-stress as an important parameter characterizing the stress field around a cracked tip has attracted much attention. This paper concerns the T-stress near a cracked tip in a magnetoelectroelastic solid. By applying the Fourier transform, we solve the associated mixed boundary-value problem. Adopting crack-faces electromagnetic boundary conditions nonlinearly dependent on the crack opening displacement, coupled dual integral equations are derived. Then, the closed-form solution for the T-stress is obtained. A comparison of the T stresses for a cracked magnetoelectroelastic solid and for a cracked purely elastic material is made. Obtained results reveal that in addition to applied mechanical loading, the T-stress is dependent on electric and magnetic loadings for a vacuum crack.     

Évaluation de l'influence des changements de géométrie sur les déformations de séchage d'un milieu poreux fissuréInfluence of geometry changes on drying-induced strains in a cracked solid

The macroscopic response of a cracked solid subjected to drying is investigated within the framework of micromechanics. The originality of this contribution lies in the fact that the variations of the aspect ratios of cracks induced by the capillary pressure increase are accounted for. When the initial aspect ratio is small enough, it is shown that neglecting the geometrical changes yields an erroneous prediction of the sign of the macroscopic volume strain rate. To cite this article: X. Chateau et al., C. R. Mecanique 331 (2003).