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.     

Axisymmetric problem of a nonhomogeneous elastic layer

Summary The paper deals with a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property, which is defined by the relation , i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate by the power product form, arbitrarily. Fundamental differential equation for such nonhomogeneous medium has been already proposed in [5]. It is given by a second-order partial differential equation. However, it was found that the fundamental equation is not sufficient in general to solve several kinds of boundary-value problems. On the other hand, it is shown in the present paper making use of the fundamental equations system for a nonhomogeneous medium, which has been proposed in our previous paper [7], it is possible to solve axisymmetric problems for a thick plate (layer) subjected to an arbitrarily distributed load or a concentrated load on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacements stress and components are shown in graphical form. Accepted for publication 25 March 1997     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation

Summary A hypersingular integral equation or a differential-integral equation is used to solve the penny-shaped crack problem. It is found that, if a displacement jump (crack opening displacement COD) takes the form of (a ²−x ²−y ²)^1/2 x ^m y ⁿ , where a denotes the radius of the circular region, the relevant traction applied on the crack face can be evaluated in a closed form, and the stress intensity factor can be derived immediately. Finally, some particular solutions of the penny-shaped crack problem are presented in this paper. Received 1 July 1997; accepted for publication 13 October 1997     

Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer

The assumptions of impermeable and permeable cracks give rise to significant errors in determining electro-elastic behavior of a cracked piezoelectric material. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Considering the presence of the dielectric of an opening crack interior and the permeability of the crack surfaces for electric field, this paper analyzes electro-elastic behavior induced by a penny-shaped dielectric crack in a piezoelectric ceramic layer. In the cases of prescribed displacement or prescribed stress at the layer surfaces, the Hankel transform technique is employed to reduce the problem to Fredholm integral equations with a parameter dependent nonlinearly on the unknown functions. For an infinite piezoelectric space, a closed-form solution can be derived explicitly, while for a piezoelectric layer, an iterative technique is suggested to solve the resulting nonlinear equations. Field intensity factors are obtained in terms of the solution of the equations. Numerical results of the crack opening displacement intensity factors are presented for a cracked PZT-5H layer and the effect of applied electric field on crack growth are examined for both cases. The results indicate that the fracture toughness of a piezoelectric ceramic is affected by the direction of applied electric fields, dependent on the elastic boundary conditions.     

Eigenvalues of the antiplane-shear crack problem for a power-law material

This paper discusses the problem of finding the eigenvalue spectrum in determining the stress and strain fields at the tip of an antiplane-shear crack in a power-law material. It is shown that the perturbation method provides an analytical dependence of the eigenvalue on the material nonlinearity parameter and the eigenvalue of the linear problem. Thus, it is possible to find the entire spectrum of eigenvalues and not only the eigenvalue of the Hutchinson-Rice-Rosengren problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 173–180, January–February, 2008.     

Contact problem for a plane crack under a normally incident antiplane shear wave

The contact-interaction problem for a stationary plane crack with friction between its edges under the action of a normal (to the crack plane) harmonic shear wave is addressed. Antiplane deformation conditions are considered. The distribution of contact forces and displacement discontinuity of crack edges are studied Published in Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 138–142, May 2007.     

Green's function of the displacement boundary value problem for a heat source in an infinite plane with an arbitrary shaped rigid inclusion

Summary Green's functions of the displacement boundary value problem are derived within two-dimensional thermoelasticity for a heat source in an infinite plane with an arbitrary shaped rigid inclusion. The following two cases are considered: either rigid-body displacements and rigid-body rotations of the inclusion are allowed or no rigid-body displacements and no rigid-body rotations of the inclusion are possible. To solve these problems, fundamental solutions are developed for a point heat source, for rigid-body rotations of the inclusion, and for concentrated loads acting on the inclusion. Complex stress functions, temperature function, a rational mapping function and the thermal dislocation method are used for the analysis. In analytical examples, distributions of stresses are developed for an infinite plane with a rectangular rigid inclusion. Received 5 August 1998; accepted for publication 1 December 1998     

Stress intensity factor in a contact problem for a plane crack under an antiplane shear wave

The frictional contact interaction of the finite edges of a plane crack under the action of a normally incident harmonic shear wave that produces antiplane deformation is studied. The influence of the forces of contact interaction on the stress intensity factor is analyzed Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 115–119, September 2007.     

On use of the Galerkin method to solve the fracture mechanics problem for a linear crack under normal loading

The problem of contact interaction of the opposite faces of a linear crack under a normally incident harmonic tension-compression wave is numerically solved by the Galerkin method with piecewise-linear continuous elements. The dependence of the stress intensity factor (opening mode) on the wave number is investigated Published in Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 137–142, November 2005.     

Passages to the limit in the dynamic problem for a crack at the interface between elastic media

The paper analyzes numerically the passages to the limit in the dynamic problem for a penny-shaped crack at the interface between dissimilar linear elastic, homogeneous, isotropic materials as either the frequency of harmonic load or the difference between the properties of the materials decreases. It is shown that as the frequency decreases, the solution of the dynamic problem tends to that of the static problem, and as the physical and mechanical properties of the materials become less different, the original problem goes into the dynamic problem for a crack in a homogeneous body __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 26–34, July 2008.     

Refinement of the plastic-zone boundary in the vicinity of a crack tip for the quasiviscous and viscous types of fracture

A refined solution of the elastoplastic problem of an insulated mode I crack in a thin plate of reasonably large dimensions is obtained. Estimates of the plastic zone in the vicinity of the crack tip are given for quasiviscous and viscous types of fracture.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 126–132, January–February, 2005     

Oscillation of solutions for a class of nonlinear neutral parabolic differential equations boundary value problem

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A variational formulation for a boundary value problem considering an electro-sensitive elastomer interacting with two bodies

We present a variational formulation for an electro-elastic body in contact with two semi-infinite rigid bodies, which are electric conductors and have a distribution of free charge. These three bodies are surrounded by free space, where far away we have a given electric displacement and an electric potential on disjoint surfaces.     

Estimating the error of the beam approximation in the plane stability problem for a rectangular plate with a central crack

The plane stability problem for a rectangular, linearly elastic, isotropic plate with a central crack is solved. The dependence of the critical load on the crack length is studied using exact (the three-dimensional linearized theory of stability of elastic bodies) and approximate (beam approximation) approaches. The results produced by the beam approach are evaluated.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 117–126, November 2004.This revised version was published online in April 2005 with a corrected cover date.