Effect of electric displacement saturation on the stress intensity factor for a crack in a ferroelectric ceramic

A crack in a ferroelectric ceramic with perfect saturation under electric loading is analyzed. The boundary of the electric displacement saturation zone ahead of the crack tip is assumed to be ellipse in shape. The shape and size of ferroelectric domain switching zone near a crack tip is determined based on the nonlinear electric theory. The stress intensity factor induced by ferroelectric domain switching under small-scale conditions is numerically obtained as a function of the electric saturation zone parameter and the ratio of the coercive electric field to the yield electric field. It is found that the stress intensity factor increases as the ratio of the semi-axes of the saturation ellipse increases.     

Effect of electrical polarization saturation on stress intensity factors in a piezoelectric ceramic

The electric-field induced stress intensity factor in a piezoelectric medium of limited electrical polarization is evaluated based on a strip-saturation model of the Dugdale-type. Particular emphasis is placed on the effect of the saturation condition on the near tip field and the stress intensity factor. To this end, the general solution is derived in terms of the (unspecified) normal electrical displacement distribution along the saturated strip. Since the saturated strip is representative of the unknown saturated zone, the normal electrical displacement may suffer discontinuity across the saturated strip. It is found that the crack-tip field and the stress intensity factors depend on the discontinuity of the normal electrical displacement across the saturated strip although this dependency disappears in some practically important cases. A crack perpendicular to the poling axis in a general poled ferroelectric is discussed in detail to illustrate the implications of the strip-saturation model for electric-field induced cracking. The results show that some discrepancies between theory and experiments, for which the classical linear piezoelectric model gives qualitatively incorrect results, can be explained clearly in terms of the stress intensity factor given by the strip-saturation model. In particular, these results are independent of the form of the saturation condition imposed on the saturated strip.     

Perturbation of a dynamic planar crack moving in a model viscoelastic solid

Weight functions, which give stress intensity factors in terms of applied loading, are constructed, for three-dimensional time-dependent loading of a semi-infinite crack, propagating at uniform speed. Both a model problem, governed by a scalar wave equation, and the full vectorial problem for Mode I loading, are considered. The medium through which the crack propagates is viscoelastic; the approach is general but explicit formulae are given when the medium is a Maxwell fluid. The weight functions are exploited to develop formulae for the first-order perturbations of stress intensity factors when the crack edge is no longer straight but becomes slightly wavy. Implications for stability, and for “crack front waves” in the case of the Mode I problem, are discussed.     

Effect of the compliant interlayer on the dynamic stress intensity factor in a piecewise-homogeneous solid with a circular crack

The dynamic behavior of a circular crack in an elastic composite consisting of two dissimilar half-spaces connected by a thin compliant interlayer is studied. One half-space contains a defect aligned perpendicular to the interlayer; the defect surfaces are loaded by normal harmonic forces, which ensures the symmetry of the stress-strain state. The thin interlayer is modeled by conditions of a nonideal contact of the half-spaces. The problem is reduced to a boundary integral equation with respect to the function of dynamic opening of the defect. The numerical solution of this equation yields frequency dependences of the mode I stress intensity factor in the vicinity of the crack for different values of interlayer thickness and relations between the moduli of elasticity of the composite components. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 197–207, May–June, 2008.     

A note on the stress singularity at a non-uniformly moving crack tip

The stress-singularity at a crack tip moving arbitrarily in an elastic plate under plane strain conditions is investigated. By formulating the wave-equations in a polar coordinate system attached to the crack-tip, it is found by an asymptotic analysis that the angle-dependence of the singularity is only dependent on the instantaneous cracktip velocity. This result is used to derive a relation between the dynamic stress-intensity factor and the energyrelease rate.     

Effect of electric boundary conditions on crack propagation in ferroelectric ceramics

In this paper, the effect of electric boundary conditions on Mode I crack propagation in ferroelectric ceramics is studied by using both linear and nonlinear piezoelectric fracture mechanics. In linear analysis, impermeable cracks under open circuit and short circuit are analyzed using the Stroh formalism and a rescaling method. It is shown that the energy release rate in short circuit is larger than that in open circuit. In nonlinear analysis, permeable crack conditions are used and the nonlinear effect of domain switching near a crack tip is considered using an energy-based switching criterion proposed by Hwang et al.（Acta Metal. Mater.,1995）. In open circuit, a large depolarization field induced by domain switching makes switching much more diffcult than that in short circuit. Analysis shows that the energy release rate in short circuit is still larger than that in open circuit, and is also larger than the linear result. Consequently,whether using linear or nonlinear fracture analysis, a crack is found easier to propagate in short circuit than in open circuit, which is consistent with the experimental observations of Kounga Njiwa et al.（Eng. Fract. Mech., 2006）.     

Semi-permeable Yoffe-type interfacial crack analysis in MEE composites based on the strip electro-magnetic polarization saturation model

Fracture analysis of a semi-permeable Yoffe-type interfacial crack propagating subsonically in magneto-electro-elastic(MEE) composites is presented based on the strip electromagnetic polarization saturation(SEMPS) model. The electro-magnetic fields inside the crack are considered under the semi-permeable boundary condition. Nonlinear effects near the interfacial crack tip are represented by different electro-magnetic saturation zones. Utilizing the extended Stroh's method, we derive the moving dislocation densities as well as intensity factor and energy release rate for Yoffe-type MEE interfacial crack. Numerical results through an iterative approach are presented to show the characteristics of fracturedominant parameters with respect to propagation velocity and boundary condition category. The fracture-dominant parameters under the semi-permeable boundary condition are lower than those under the impermeable one, which implies that the electro-magnetic fields in the crack gap can retard the propagation of MEE interfacial crack.     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

On a moving interface crack with a contact zone in a piezoelectric bimaterial

An inplane problem for a crack moving with constant subsonic speed along the interface of two piezoelectric materials is considered. A mechanically frictionless and electrically permeable contact zone is assumed at the right crack tip whilst for the open part of the crack both electrically permeable and electrically insulated conditions are considered. In the first case a moving concentrated loading is prescribed at the crack faces and in the second case an additional electrical charge at the crack faces is prescribed as well. The main attention is devoted to electrically permeable crack faces. Introducing a moving coordinate system at the leading crack tip the corresponding inhomogeneous combined Dirichlet–Riemann problem is formulated and solved exactly for this case. All electromechanical characteristics at the interface are presented in a closed form for arbitrary contact zone lengths, and further, the transcendental equation for the determination of the real contact zone length is derived. As a particular case of the obtained solution a semi-infinite crack with a contact zone is considered. The numerical analysis performed for a certain piezoelectric bimaterial showed an essential increase of the contact zone length and the associated stress intensity factor especially for the near-critical speed region. Similar investigations have been performed for an electrically insulated crack and the same behavior of the above mentioned parameters is observed.     

Effect of a flat inclusion on stress intensity factor of a semi-infinite crack

The elastic plane interaction between an arbitrarily located and oriented flat inclusion and a semi-infinite crack subjected to a remote Mode I loading is considered. The method uses distributions of edge dislocations to formulate integral expressions of flat inclusion (including crack) tractions and is shown to be very accurate by a test problem. The stress intensity factors of the main crack tip are presented for a variety of crack inclusion geometries. It is seen that the flat inclusion could either yield a stress enhancement or stress shielding effect to the main crack tip depending upon the location, orientation and thickness of the flat inclusion, and depending upon the modulus ratios of the flat inclusion to matrix.     

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behavior of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localized surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.     

Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

We present a family of phase-field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg–Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase-fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack-face boundary conditions in sharp models) into the phase-field framework. The proposed models are verified through comparisons with the corresponding sharp-crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack-face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.     

Effect of crack orientation statistics on effective stiffness of mircocracked solid

This paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics. To do so, the representative unit cell approach has been used. The microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks; a sufficient number is taken to account for the microstructure statistics. The developed method combines the superposition principle, the technique of complex potentials and certain new results in the theory of special functions. A proper choice of potentials provides reducing the boundary-value problem to an ordinary, well-posed set of linear algebraic equations. The exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields. The convergence study has been performed; the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks. Comparison has been made with the selected simple micromechanical models, namely, non-interaction approximation, differential scheme and modified differential scheme. It is found that, among these models, the differential scheme provides the best fit of the numerical data.     

Dynamic problem of elasticity theory for a space with a moving semi-infinite crack

Taganrog Pedagogical Institute, Russia. Translated from Prikladnaya Mekhanika, Vol. 28, No. 12, pp. 56–63, December, 1992.