Electro-elastic stress analysis for a screw dislocation interacting with a coated inclusion in piezoelectric solid

Summary. The electro-elastic stress investigation on the interaction problem of a piezoelectric screw dislocation near a coated inclusion in a piezoelectric material has been carried out. In our study, three dissimilar material phases are involved: the matrix, the inclusion and the coating layer. All the three materials are piezoelectric and with different material constants. Explicit closed-form analytical solutions for the stress and electric displacement fields are obtained by using the complex variable method. The image force acting on the screw dislocation is calculated by using the generalized Peach-Koehler formula. Numerical examples for different material constant combinations are performed. The influences of material properties of the inclusion and the coating layer on the image forces are examined and discussed.     

Electro-elastic stress analysis for a wedge-shaped crack interacting with a screw dislocation in piezoelectric solid

The electro-elastic stress investigation on the interaction problem of a screw dislocation near the tip of a semi-infinite wedge-shaped crack in piezoelectric material has been carried out. Explicit closed-form analytical solutions are obtained for the stress intensity factor (SIF) and the electric displacement intensity factor (EDIF) of the crack, as well as the force on dislocation. The derivation is based on the conformal mapping method and the perturbation technique. The dislocation has Burger's vector normal to the isotropic basal plane, with a line force and a line charge being applied at the core of the dislocation. The influence of the location and the wedge angle of the crack on the image force of the dislocation has been discussed in detail. At the same time the effect of the dislocation on the crack behavior has been also examined under different configurations. Two types of PZT materials are used to numerically illustrate the influences of the wedge angle and the location of the dislocation on the image force and the crack intensities. Results obtained in the current study can be fully reduced to various special cases available in the literatures.     

Electroelastic analysis for a Griffith crack interacting with a coated inclusion in piezoelectric solid

A Mode III Griffith crack interacting with a coated inclusion in piezoelectric media is investigated. The crack, the coated inclusion are embedded in an infinitely extended piezoelectric matrix media, with the crack being along the radial direction of the inclusion. In the study, three different piezoelectric material phases are involved: the inclusion, the coating layer, and the matrix. A far-field loading condition is considered. During the solution procedure, the crack is simulated as a continuous distribution of screw dislocations. By using the solution of a screw dislocation near a coated inclusion in piezoelectric media as the Green function, the problem is formulated into a set of singular integral equations, which are solved by numerical method. The stress and electric displacement intensity factors are derived in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Numerical examples are given for various material constants combinations and geometric parameters.     

Stress intensity factor for a Griffith crack interacting with a coated inclusion

Stress investigation for the interaction problem between a coated circular inclusion and a near-by line crack has been carried out. The crack and the coated inclusion (a coated fiber) are embedded in an infinitely extended isotropic matrix, with the crack being along the radial direction of the inclusion. Two loading conditions, namely, the tensile and shear loading ones are considered. During the solution procedure, the crack is treated as a continuous distribution of edge dislocations. By using the solution of an edge dislocation near a coated fiber as the Green's function, the problem is formulated into a set of singular integral equations which are solved by Erdogan and Gupta (1972) method. The expressions for the stress intensity factors of the crack are then obtained in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Several numerical examples are given for various material and geometric parameters. The solutions obtained from the integral equations have been checked and confirmed by the finite element analysis results.     

A piezoelectric screw dislocation interacting with a nonuniformly coated circular inclusion

This paper investigated the interaction of a piezoelectric screw dislocation with a nonuniformly coated circular inclusion in an unbounded piezoelectric matrix subjected to remote antiplane shear and electric fields. In addition to having a discontinuous displacement and a discontinuous electric potential across the slip plane, the dislocation is subjected to a line force and a line charge at the core. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general solutions in an explicit series form. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution for the corresponding homogeneous problem. The presented series solutions have rapid convergence which is guaranteed numerically. The image force acting on the piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. The numerical results show that the varying thickness of the interphase layer will exert a significant influence on the shear stress and electric field within the circular inclusion, and on the direction and magnitude of the image force. This solution can be used as Green’s function for the analysis of the corresponding piezoelectric matrix cracking problem.     

A line dislocation interacting with a semi-infinite crack in piezoelectric solid

Closed-form analytical solutions are presented for the physical problem of a semi-infinite crack interacting with a line dislocation under the loading of a line force and a line charge in two-dimensional infinite anisotropic piezoelectric medium. The crack can be a conventional Griffith crack or an anti-crack (a rigid line inhomogeneity). Using the extended Stroh formalism and perturbation technique, the explicit expressions of the field intensity factors and the image force on the dislocation are computed as functions of dislocation location and material constants. The results are discussed and compared with those from special cases existed in the literature. The analytical solutions obtained can be applied to studying interacting cracks and crack branching problems in piezoelectric solids.     

Three-dimensional dynamic stress analysis on a penny-shaped crack interacting with a suddenly transformed spherical inclusion

In this paper, the interaction between a penny-shaped crack and a near-by suddenly transformed spherical inclusion in 3-dimensional solid is investigated to assess the dynamic effect of the transformation. To simplify the solution procedure, the current problem is divided into two sub-problems by using the superposition principle. A time domain boundary integral equation method (BIEM) is adopted to evaluate the stress and displacement fields. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. The impact effect of the spherical inclusion when it experiences a pure dilatational eigenstrain on the penny-shaped crack is studied. The relationship between the relative location of the inclusion and its impact effect on the time history of the Mode I crack intensity factor is discussed in detail.     

The interacting stress field around a composite crack with a misfitting inclusion

The problem of a composite crack interacting with a circular misfitting inclusion in an infinite elastic medium is investigated. The close-formed solutions of the stress fields in the inclusion and the matrix are obtained by using the dislocation model of cracks and the point force method as well as the complex function technique. The stress intensity factors at two tips of the crack are calculated from the stress components.     

The stress intensity factor Green's function for a crack interacting with a circular inclusion

The Green's function is constructed for the stress intensity factor due to the unit dipole force applied to the crack surface in the presence of a circular inclusion in front of the crack tip. An explicit functional form of the Green's function is proposed in terms of dipole force location, Young's modulus ratio and the inclusion size and position with respect to the crack tip. This is achieved through a combination of the dimensional analysis and parametric studies by means of the finite element method. The purpose of this paper is to provide the basis for further studies of a crack interaction with an array of microdefects and/or inclusions.     

Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.     

A screw dislocation interacting with a coated nano-inhomogeneity incorporating interface stress

Effect of interface stress on the interaction between a screw dislocation and a coated nano-inhomogeneity is investigated in the framework of surface elasticity. By using the complex variable function method, the stress fields in heterogeneous materials and the image force acting on the screw dislocation are derived analytically. The results indicate that, when the radius of the coated inhomogeneity is reduced to nanometers, the influence of interface stress on the motion of the dislocation near the inhomogeneity becomes significant and the image force acting on the screw dislocation depends on the size of the coated inhomogeneity, which differs from the classical solution. And it is showed that the thicker coated layer relative to inclusion decreases or increases the influence of a coated inclusion on the image force of dislocation, not as expected.     

A slit-like inclusion in an orthorhombic piezoelectric solid

Summary A slit-like inclusion in an unbounded orthorhombic piezoelectric solid is considered. We demonstrate that the associated polarization tensor correct to the first order of the aspect ratio of the inclusion can be obtained in an analytical form. As an application, the results are exploited to estimate the effective moduli of a cracked piezoelectric solid. Micromechanical models of the self-consistent and Mori-Tanaka methods are implemented in the present case. The methods offer simple approaches to estimate the stiffness changes due to the presence of aligned longitudinal slit-like cracks.     

Plastic zone correction on a Zener–Stroh crack interacting with a circular inclusion in ductile materials

Elastic–plastic stress analysis on a matrix Zener–Stroh crack interacting with a circular inclusion (fibre) in fibre‐reinforced composites has been carried out. The Zener–Stroh crack is initiated near the fibre in the pure matrix. Plastic zone correction is introduced the first time for such a crack–inclusion interaction problem so that the fracture behaviour can be analysed more accurately. To determine the plastic zone sizes, a generalized Irwin model is proposed for the mixed‐mode problem where the Von Mises stress yielding criterion is employed. Different to a Griffith crack, a Zener–Stroh crack propagation always occurs from the sharp tip whose relative position to the inclusion has great effect on the elastic–plastic fracture behaviour of the crack. In our study, the plastic zone size (PZS), crack tip opening displacement (CTOD) and effective stress intensity factor (SIF) are evaluated by solving the formulated singular integral equations. Through the numerical examples, the influence of the inclusion (fibre) shear modulus, crack–inclusion distance and the crack sharp tip position on the fracture behaviour of the crack is discussed. It is found that the shear modulus ratio and the crack–inclusion distance have great effect on the normalized values of PZS and CTOD, but the effects highly depend on the crack sharp tip position.     

An approach for analysis of poled/depolarized piezoelectric materials with a crack

In this paper, the influence of dielectric medium inside a crack on crack growth, in an infinite poled or depolarized ceramic, has been studied by employing an electric boundary condition derived from the exact boundary conditions proposed by Sosa (1996). The effect of remanent polarization has also been examined. The results obtained show that electric displacement on crack surfaces is not always zero. Hence, for studying fracture problems of piezoelectric ceramics with cracks accurately, the exact boundary conditions should be implemented. In addition, the results indicate that the effect of remanent polarization is equivalent to that of a positive electric field and it cannot be neglected. It is also found that a positive electric field always has a tendency to open a crack, and a negative electric field tends to close a crack.     

A new method for calculating the stress intensity factors of a crack with a circular inclusion

Summary A new method for calculating the stress intensity factors of a crack-inclusion system is proposed. The analytical solution for the stress intensity factors of crack with a circular inclusion at an arbitrary position is obtained successfully. Several special cases are worked out further.     

Fracture analysis of a piezoelectric layer with a penny-shaped and energetically consistent crack

The problem of an eccentric penny-shaped crack embedded in a piezoelectric layer is addressed by using the energetically consistent boundary conditions. The Hankel transform technique is applied to solve the boundary-value problem. Then two coupling Fredholm integral equations are derived and solved by using the composite Simpson’s rule. The intensity factors of stress, electric displacement, crack opening displacement and electric potential together with the energy release rate are further given. The effects of the thickness of a piezoelectric layer and the discharge field inside the penny-shaped crack on the fracture parameters of concern are discussed through numerical computations. The observations reveal that an increase of the discharge field decreases the stress intensity factor and the energy release rate. An eccentric penny-shaped crack is easier to propagate than a mid-plane one in a piezoelectric layer, and the geometry of the crack along with the layer thickness have significant influences on the electrostatic traction acting on the crack faces. The solutions for a penny-shaped dielectric crack in an infinite or a semi-infinite piezoelectric material can be obtained easily.     

Energy analysis of crack interaction with an elastic inclusion

This paper gives an energy analysis of an elastic solid with a crack which penetrates an elastic inclusion. The purpose of our work is to evaluate the energy release rates (ERR) associated with crack tip extension while the inclusion is stationary, and to evaluate the ERR due to inclusion translation, rotation and expansion with respect to the crack tip. Reduction and increase in the crack ERR caused by an inclusion (shielding and amplification effects of the inclusion) are expressed in terms of the inclusion elastic properties normalized by Young's modulus of the bulk material. The variation in ERR as a crack approaches and passes through a circular inclusion is also examined.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

Exact stress distribution,crack shape and energy for a running penny-shaped crack in an infinite elastic solid

The integral solutions for an axisymmetrical crack propagating at arbitrary speed in an infinite elastic solid are obtained as sums of associated static solutions and stress-waves integrals. For a circular crack running at a constant speed, exact dynamic solutions for crack shape and stress distribution with singularities in the crack plane are obtained in closed forms easily comparable to associated static solutions. The dynamic solution reduces to the static solution at zero crack speed and deviates at speed other than zero. Deviation between dynamic and static solutions is governed by dynamic correction factors which are nondimensional functions of Poisson's ratio and the ratio between crack speed and shear-wave speed. Values of these dynamic factors are obtained for large range of crack speed and deviation can clearly be determined from the results obtained. Exact expressions for dynamic stress-intensity factor and energy functions are also obtained in terms of crack speed.

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Zusammenfassung Die Integrallösungen für einen assymetrischen Riß, der sich mit einer beliebigen Geschwindigkeit in einem unendlichen elastischen Körper ausbreitet ergeben sich als die Summe von zugehörigen statischen Lösungen und Integrale von Spannungswellen. Die exakten dynamische Lösungen für Rißform und Spannungsverteilung mit Besonderheiten in der Rißebene, im Falle eines kreisförmigen Rissesder sich mit gleicher Geschwindigkeit ausbreitet ergeben sich in geschlossener Form, leicht vergleichbar mit der zugehörigen statischen Lösung.Die dynamische Lösung stimmt mit der statischen überein wenn die Ausbreitungsgeschwindígkeit null ist und weicht davon ab wenn die Ausbreitungsgeschwindigkeit größer als null ist. Der Unterschied zwischen den dynamischen und den statische Lösungen wird durch Korrektionsfaktoren, die undimensionale Funktionen dem Poissonverhältnis und dem Verhältnis zwischen der Rißverbreitungsgeschwindigkeit und der Geschwindigkeit der Querwellen. Die Werte dieser dynamischen Faktoren können für einen großen Bereich von Rißgeschwindigkeiten aufgestellt werden und der Unterschied kan einwandfrei von diesen Resultaten bestimmt werden.Exakte Formeln für Spannungsintensitätsfaktoren und Energieformeln werden in Form von Rißgeschwindigkeit aufgestellt.

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Résumé Les solutions intégrales pour une fissure axisymétrique progressant à une vitesse quelconque dans un solide élastique infini se présentent sous la forme de sommations d'intégrales associées de solutions statiques et d'ondes de contraintes.Dans le cas d'une fissure circulaire se développant à vitesse constante, les solutions dynamiques exactes pour la forme de la fissure, la distribution des contraintes et leurs singularités dans le plan de la fissure, sont obtenues sous des formes fermées faciles à comparer aux solutions statiques correspondantes.La solution dynamique se ramène à la solution élastique lorsque la vitesse de fissuration est égale à zéro. Elle s'en distingue pour les vitesses supérieures à zéro. La divergence entre la solution statique et la solution dynamique est régie par des facteurs de correction dynamique, fonctions non dimensionnelles du rapport de Poisson et du quotient de la vitesse de la fissure par la vitesse des ondes transversales.Des valeurs de ces facteurs dynamiques ont été obtenues pour une gamme étendue de vitesses de fissuration; la divergence peut être clairement exprimée à partir des résultats obtenus.Des expressions exactes du facteur dynamique d'intensité des contraintes et des fonctions d'énergie ont également été trouvées en fonction de la vitesse de fissuration.