Accurate determination of mode I and II leading coefficients of the Williams expansion by finite element analysis

Leading coefficients of the Williams expansion are evaluated by using the fractal finite-element method (FFEM). By means of the self-similarity principle, an infinite number of elements is generated at the vicinity of the crack tip to model the crack tip singularity. The Williams expansion series with higher-degree coefficients is used to capture the singular and non-singular stress behaviour around the crack tip and to condense the large amount of nodal displacements at the crack tip to a small set of unknown coefficients. New sets of coefficients up to the sixth degree for mode I and fourth degree for mode II problems are solved. The important fracture parameters such as stress intensity factors and T-stress can be obtained directly from the coefficients without employing any path independent integrals. Convergence study reveals that the present method is simple and very coarse finite element meshes with 12 leading terms in the William expansion can yield very accurate solutions. The effects of the influence of crack length on the higher-degree coefficients of some common plane crack problems are studied in detail. © 2005 Elsevier B.V. All rights reserved.postprin

Dynamic response of multiple coplanar interface cracks between two dissimilar piezoelectric materials

The linear piezoelectricity theory is applied to investigate the dynamic response of coplanar interface cracks between two dissimilar piezoelectric materials subjected to the mechanical and electrical impacts. The number of cracks is arbitrary, and the interface cracks are assumed to be permeable for electric field. Integral transforms and dislocation density function are employed to reduce the problem to Cauchy singular integral equations. Numerical examples are given to show the effects of crack relative position and material property parameters on the variations of dynamic energy release rate.postprin

Fracture analysis of an electrically conductive interface crack with a contact zone in a magnetoelectroelastic bimaterial system

An electrically conductive interface crack with a contact zone in a magnetoelectroelastic (MEE) bimaterial system is considered. The bimaterial is polarized in the direction orthogonal to the crack faces and is loaded by remote tension and shear forces as well as electrical and magnetic fields parallel to the crack faces. It is assumed that the electrical field inside the crack faces is equal to zero and the magnetic quantities are continuous across the crack faces. Using special expressions of magnetoelectromechanical quantities via sectionally-analytic functions proposed in this paper, a combined Dirichlet-Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical, electrical and magnetic parameters are presented. A simple transcendental equation is derived for the determination of the contact zone length. Stress, electric field and magnetic field intensity factors and the contact zone length are found for various loading cases. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Magnetoelectrically permeable conditions in the crack region are also investigated and comparisons of different crack models are performed. Results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic (MEE) structures and devices.postprin

Pre-fracture zone model on magnetoelectrically permeable interface crack between two dissimilar magnetoelectroelastic materials

A plane strain problem for two magnetoelectroelastic (MEE) half-planes adhered by a thin isotropic interlayer is considered. A novel crack model, i.e., a magnetoelectrically permeable interface crack with pre-fracture zones is introduced for MEE bimaterial system. The stresses in pre-fracture zones and the lengths of pre-fracture zones are unknown, which are determined by solving the corresponding Hilbert problem and solving nonlinear equations introduced by yielding condition on the pre-fracture zones. Some particular cases are further analyzed and numerically discussed. In the suggested model, any singularities connected with the crack are eliminated, and the results presented in this paper should have potential applications to the design of multilayered MEE structures and devices. Copyright © (2013) by International Conference on Fracture.postprin

Moving crack with a contact zone at interface of magnetoelectroelastic bimaterial

The plane-strain problem of a moving crack at the interface of two dissimilar magnetoelectroelastic (MEE) materials is investigated. Assuming that the crack moves at a constant speed in the subsonic regime, a fracture analysis of a finite crack under concentrated loading imposed onto the crack face is first carried out. By applying magnetoelectric (ME) permeable boundary conditions at the crack face, a combined Dirichlet-Riemann problem is formulated and solved analytically. The expressions for the fracture parameters, including the relative length of the contact zone and field intensity factors (FIFs), are obtained in the analytical form. A crack of a semi-infinite length with a contact zone under concentrated loading is further presented as a specific case examined with the obtained solution. Then a moving crack of finite length at the interface under remote mix-mode loading is also analyzed and the corresponding fracture parameters are presented in an analytical form. Finally, numerical examples are provided for the material combination of barium titanate-cobalt ferrite composites to examine the influence of the speed of the moving crack, poling direction, material volume fraction, load position and load ratio on the fracture parameters, from which some new and interesting conclusions related to the crack model in this study are drawn

The extended finite element method with new crack-tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X-FEM) with new crack-tip enrichment functions. In the X-FEM, crack modeling is facilitated by adding a discontinuous function and crack-tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, E class and class, two classes of crack-tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J-integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency.postprin

Fracture analysis of bounded magnetoelectroelastic layers with interfacial cracks under magnetoelectromechanical loads: Plane problem

Fracture behaviors of multiple interfacial cracks between dissimilar magnetoelectroelastic layers subjected to in-plane magnetoelectromechanical loads are investigated by using integral transform method and singular integral equation technique. The number of the interfacial cracks is arbitrary, and the crack surfaces are assumed to be magnetoelectrically impermeable. The field intensity factors including stress, electric displacement and magnetic induction intensity factors as well as the energy release rates (ERRs) are derived. The effects of loading combinations, crack configurations and material property parameters on the fracture behaviors are evaluated according to energy release rate criterion. Numerical results show that both negative electrical and magnetic loads inhibit crack extension, and that the material constants have different and important effects on the ERRs. The results presented here should have potential applications to the design of multilayered magnetoelectroelastic structures. © The Author(s), 2010.postprin

A LeVeque-type Lower Bound for Discrepancy

A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on Rk / Zk. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus

Integral identities based on symmetric and skew-symmetric weight functions for a semi-infinite interfacial crack in anisotropic magnetoelectroelastic bimaterials

In this paper, we address a semi-infinite interfacial crack problem in an anisotropic magnetoelectroelastic (MEE) bimaterial system subjected to a magnetoelectromechanical asymmetric load on the crack surface. First, the symmetric and skew-symmetric weight functions are derived for a two-dimensional (2-D) deformation problem. Using these weight functions and extending the Betti formula to MEE materials, the integral identities are further obtained and the present crack problem is formulated in terms of singular integral equations, which establish the relationship between the applied external load and the generalized displacement jump across the crack faces. The illustrative examples in relation to Mode III, and Mode I and Mode II problems show that the method developed in this study avoids the use of Green's function and is very convenient for the fracture analysis of MEE solids, in which a multi-field coupled effect is observed.postprin