Nonsingular term effect on the fracture quantities of a crack in a piezoelectric medium (original) (raw)
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ZAMM, 2008
Treated is an arbitrarily shaped anti-plane shear crack in a finite inhomogeneous piezoelectric domain under time-harmonic loading. Within a unified scheme different types of inhomogeneity are considered for which the material parameters may vary in two directions. The problem is solved by using a numerically efficient non-hypersingular traction based boundary equation method (BIEM). The fundamental solutions for the different inhomogeneity types are derived in closed form. This is done by an appropriate functional transformation of the displacement vector in order to obtain a wave equation with constant coefficients and subsequently by application of the Radon transform. Numerical results for the stress intensity factors (SIF) are discussed for several examples. They show the effect of the material inhomogeneity and the efficiency of the method. / www.zamm-journal.org 87 T. Rangelov et al.: Dynamic behavior of cracked inhomogeneous piezoelectric solids boundaries S∪S cr is derived. For this purpose the displacements and the tractions are written as u J
BIEM solution of piezoelectric cracked finite solids under time-harmonic loading
Engineering Analysis With Boundary Elements, 2007
The time-harmonic behavior of cracked finite piezoelectric 2D solids of arbitrary shape is studied by the nonhypersingular traction boundary integral equation method (BIEM). Plane strain and generalized traction free boundary conditions along the crack are assumed. The system may be loaded at the external boundary by arbitrary mechanical or electrical loads. As numerical example a center cracked rectangular piezoelectric plate under time-harmonic tension and electrical displacement is investigated in detail. The accuracy of the proposed numerical algorithm is checked by comparison with available results obtained by other methods for special cases. Parametric studies revealing the sensitivity of the stress intensity factors (SIFs) on the frequency of the applied mechanical and electrical load, on its coupled and uncoupled character and on the piezoelectric properties of the material are presented.
General solutions of a penny-shaped crack in a piezoelectric material under opening mode loading
The Quarterly Journal of Mechanics and Applied Mathematics, 2004
The effect of a penny-shaped crack on the deformation of an infinite piezoelectric material of the hexagonal crystal class 6 mm subjected to mode I electrical and mechanical loading has been studied using the theory of linear piezoelectricity and applying appropriate boundary conditions. Depending on material properties of piezoelectric materials, four different closed-form solutions to the fields of displacement, stresses, electric field and electric displacement are obtained, from which the dependence of stress intensity factor and electrical displacement intensity factor on the external loading is calculated. By including the contribution of electrostatic energy to the calculation of energy release rate, it is found that the energy release rate is a nonlinear function of the external loading if the electric field inside the crack is not zero at the crack tip. For piezoelectric materials subjected to tensile stress and electric field parallel to the poled direction, the stress intensity factor and electrical displacement intensity factor are independent of the applied electric field, while the electric field intensity inside the crack is a function of the applied stress and electric field. Depending on the direction of the applied electric field related to the poled direction, the electric field can display the shielding effect on the propagation of the crack. Such a result may be used to explain some nonlinear phenomena observed in the fracture mechanics of piezoelectric materials.
A first order perturbation analysis of a non-ideal crack in a piezoelectric material
International Journal of Solids and Structures, 2001
In this paper a ®rst order perturbation analysis is carried out on a symmetrically perturbed non-ideal crack for three kinds of electric boundary conditions, namely permeable, impermeable and conducting crack boundary condition. By using the extended Stroh formula, the two-domain problems are reduced to standard Riemann±Hilbert problems, and the singular integral equations of the internal electric ®eld inside the permeable crack are solved. The stress and electric intensity factors (SEIFs) are determined to the ®rst order of accuracy. The results indicate that for a symmetrically perturbed non-ideal crack the electro-mechanical loading at in®nity does not aect the ®rst order solution for the mode I intensity factor for general piezoelectric materials. The energy release rate and the SEIFs are determined by remote mechanical loads only and the perturbation eect on the SEIFs and energy release rate is small. The electric ®eld distribution inside crack is constant for the zeroth order solution and quadratic for the ®rst order solution, which is dierent from the constant electric ®eld distribution for an ideal permeable crack. The internal electric concentration near the crack tip caused by the perturbation reveals that the dielectric inside the crack probably breaks down before the matrix does when the matrix is subjected to a not too high electro-mechanical load at in®nity. The SEIFs and the energy release rate are also given for the non-ideal crack under the impermeable and conducting electric boundary condition respectively. For all three kinds of electric boundary conditions, the lateral stresses r I 11 , r I 13 have no contribution to the SEIFs to the ®rst order of accuracy.
An Asymptotic Approach of The Crack Extension In Linear Piezoelectricity
As a result of a theoretical technique for elucidating the fracture mechanics of piezoelectric materials, this paper provides, on the basis of the three-dimensional model of thin plates, an asymptotic behavior in the Griffith's criterion for a weakly anisotropic thin plate with symmetry of order six, through a mathematical analysis of perturbations due to the presence of a crack. It is particularly established, in this work, the effects of both electric field and singularity of the in-plane mechanical displacement on the piezoelectric energy.
Theoretical and Applied Mechanics, 2008
A non-hypersingular traction boundary integral equation method (BIEM) is proposed for the treatment of crack systems in piezoelectric or anisotropic plane domains loaded by time-harmonic waves. The solution is based on the frequency dependent fundamental solution obtained by Radon transform. The proposed method is flexible, numerically efficient and has virtually no limitations regarding the material type, crack geometry and type of wave loading.
A mixed electric boundary value problem for a two-dimensional piezoelectric crack
International Journal of Solids and Structures, 2003
In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.
International Journal of Solids and Structures, 2000
The singular stress and electric ®elds in a rectangular piezoelectric ceramic body containing a Grith center crack under anti-plane shear loading are obtained by the theory of linear piezoelectricity. Fourier transforms and Fourier sine series are used to reduce the problem to a pair of dual integral equations, which is expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate are obtained to show the in¯uence of the electric ®eld. 7
Dynamic behaviour of a cracked inhomogeneous piezoelectric solid. In-plane case
Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles
Dynamic behaviour for certain classes of inhomogeneous piezoelectric solids with a finite anti-plane shear crack is studied by boundary integral equation method (BIEM). Fundamental solution is derived in a closed form and the asymptotic behaviour of the dynamic displacement and stress field near the crack tips is evaluated. The authors acknowledge the support of DFG grant No GR596/33-1 and of Bilateral project between BAS and IAS. 2 Zhang [ 13 ] for the homogeneous PEM case, see also Zhang and Gross [ 14 ], for the homogeneous elasto-isotropic case and Gross, Rangelov, Dineva [ 15 ] for the homogeneous piezoelectric case, the non-hypersingular BIE on S ∪ S cr is obtained. Let us represent the displacement and the traction in the form u J
Evaluation of fracture parameters in continuously nonhomogeneous piezoelectric solids
International Journal of Fracture, 2007
A contour integral method is developed for computation of stress intensity and electric intensity factors for cracks in continuously nonhomogeneous piezoelectric body under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium is the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for computation of physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weakform. This leads to local integral equations (LBIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the stress intensity factors (SIF) and electrical