Numerical method for nonlinear models of penny-shaped cracks in three-dimensional magnetoelectroelastic media

Green functions corresponding to uniformly distributed extended displacement discontinuities on an annular crack element in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium are derived. Using the obtained Green functions, an extended displacement discontinuity method is presented to analyze a penny-shaped crack under axisymmetric loadings. Using the electric and magnetic polarization saturation model and the electric and magnetic breakdown model, the electric and magnetic yielding zones, the extended displacement discontinuities, the extended stress intensity factors and the J-integral are numerically calculated. The accuracy and efficiency of the proposed method are demonstrated by comparing the numerical results with those obtained from analytical solutions.     

Moving antiplane shear crack in transversely isotropic magnetoelectroelastic media

An antiplane shear strip crack moving uniformly in transversely isotopic magnetoelectroelastic media when subjected to representative non-constant crack-face loading conditions is studied. Readily calculable explicit closed-form representations are determined and discussed for the components of the stress, electric and magnetic fields created throughout the material. Representative numerical data are presented. Alternative boundary conditions for which corresponding analyses can be derived analogously are listed.     

Dynamic crack propagation analysis of orthotropic media by the extended finite element method

Dynamic crack propagation of composites is investigated in this paper based on the recent advances and development of orthotropic enrichment functions within the framework of partition of unity and the extended finite element method (XFEM). The method allows for analysis of the whole crack propagation pattern on an unaltered finite element mesh, defined independent of the existence of any predefined crack or its propagation path. A relatively simple, though efficient formulation is implemented, which consists of using a dynamic crack initiation toughness, a crack orientation along the maximum circumferential stress, and a simple equation to presume the crack speed. Dynamic stress intensity factors (DSIFs) are evaluated by means of the domain separation integral method. The governing elastodynamics equation is first transformed into a standard weak formulation and is then discretized into an XFEM system of time dependent equations, to be solved by the unconditionally stable Newmark time integration scheme. A number of benchmark and test problems are simulated and the results are compared with available reference results.     

Fracture in magnetoelectroelastic materials using the extended finite element method

Static fracture analyses in two‐dimensional linear magnetoelectroelastic (MEE) solids is studied by means of the extended finite element method (X‐FEM). In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and the crack‐tip asymptotic functions to the standard finite element approximation using the framework of partition of unity. In this study, media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. New enrichment functions for cracks in transversely isotropic MEE materials are derived, and the computation of fracture parameters using the domain form of the contour interaction integral is presented. The convergence rates in energy for topological and geometric enrichments are studied. Excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical study of dynamic crack growth by the finite element method

Recent developments in numerical techniques for dynamic transient stress analysis have ensured that realistic models can now be employed in crack propagation studies. In this paper transient dynamic finite element solutions are undertaken for both double cantilever beam (DCB) and pipeline problems with propagation of the crack being permitted. Standard parabolic isoparametric elements are employed for spatial discretization with an explicit (central difference) scheme being employed for time integration. Both critical stress and energy balance crack propagation criteria are considered.The pressurised pipeline problem is solved for as a fully three-dimensional solid. Firstly, a stationary crack is considered and both large deformations and plasticity effects are accounted for. The transient case of a dynamically propagating crack is then modelled, employing both a stress and energy criterion. Elastic large deformation behaviour is permitted for this case.

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Résumé Des développements récents dans les techniques numériques pour l'analyse des contraintes dynamiques transitoires ont permis d'utiliser à présent des modèles réalistes dans les études de propagation des fissures. Dans ce mémoire, on envisage des solutions par éléments finis pour les transitoires dynamiques dans les cas de la poutre double cantilever et de problèmes de pipelines où l'on autorise la propagation d'une fissure. On recourt aux éléments paramétriques paraboliques standards pour réaliser une division discrète de l'espace, et l'on utilise pour l'intégration dans le temps un schéma explicite à différence centrale. On considére à la fois les critères de contraintes critiques et d'équilibre d'énergie lors de la propagation de la fissure. Le problème du pipeline pressurisé est solutionné en considérant ce dernier comme un solide tridimensionnel. En premier lieu, on considère une fissure stationnaire et l'on tient compte des effets des grandes déformations et de la plasticité. On met ensuite en équation le cas transitoire d'une fissure en propagation dynamique, en utilisant un critère de contrainte et un critère d'énergie. Ce cas permit d'envisager le comportement sous des déformations élastiques importantes.

     

Dynamic crack interactions in magnetoelectroelastic composite materials

In this paper, the dynamic interactions among cracks embedded in a two-dimensional (2-D) piezoelectric-piezomagnetic composite material are analyzed by means of a hypersingular formulation of the boundary element method. In the numerical solution procedure, extended crack opening displacements and extended traction jumps across the crack are considered as basic unknowns, so that only the traction boundary integral equations are needed on the crack surfaces. Quadratic discontinuous boundary elements are implemented together with discontinuous quarter-point elements placed next to the crack tips to ensure a proper representation of the square root asymptotic behavior. Several impermeable cracks configurations subjected to time-harmonic incident L-waves are analyzed in order to characterize the effects of the magnetoelectromechanical coupling on the dynamic crack interactions and to illustrate the dependence on such coupling of the fracture parameters: stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor.     

Fracture analysis of a penny-shaped magnetically dielectric crack in a magnetoelectroelastic material

In this paper we investigate the magnetoelectroelastic behavior induced by a penny-shaped crack in a magnetoelectroelastic material. The crack is assumed to be magnetically dielectric. A closed-form solution is derived by virtue of Hankel transform technique with the introduction of certain auxiliary functions. Field intensity factors are obtained and analyzed. The results indicate that the stress intensity factor depends only on the mechanical loads. However, all the other field intensity factors depend directly on both the magnetic and dielectric permeabilities inside the crack as well as on the applied magnetoelectromechanical loads and the material properties of the magnetoelectroelastic material. Several special cases are further discussed, with the reduced results being in agreement with those from literature. Finally, according to the maximum crack opening displacement (COD) criterion, the effects of the magnetoelectromechanical loads and the crack surface conditions on the crack propagation and growth are evaluated.     

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

In this paper, a novel geometric method combined with the piecewise linear function method is introduced into the extended finite element method (XFEM) to determine the crack tip element and crack surface element. Then, by combining with the advanced mesh technique, a novel method is proposed to improve the modelling of crack propagation in triangular 2D structure with the XFEM. The numerical tests show that the accuracy, the convergence, and the stability of the XFEM can be improved using the proposed method. Moreover, the applicability of the conventional multiple enrichment scheme is discussed. Compared with the proposed method, the conventional multiple enrichment scheme has deficiency in mixed mode I and II crack. Finally, a comparative study shows that the performance of the XFEM by using the proposed method to model the crack propagation can be greatly improved.     

Numerical studies on crack problems in three-layered elastic media using an image method

Two-dimensional crack problems in a three-layered material are analyzed numerically under the conditions of plane strain. An image method is adopted to obtain fundamental solutions for dislocation dipoles in trilayered media. The governing equations for equilibrium cracks can be constructed by distributed dislocation technique and their solutions are sought in terms of the displacement discontinuity method (DDM). Comparisons are made with available analytical or reference solutions for several examples at various contrasts of material constants, and good agreements are found. A crack within a brittle adhesive layer joining two semi-infinite blocks can propagate in a variety of ways. In particular, crack paths in the form of sigmoidal waves within the adhesive layer are revisited to reveal the sensitivities of cracking paths to initial crack locations and directions and residual stresses. In addition, Z-shape and H-shape cracks alternating from interface to interface are re-examined to highlight the transition of failure modes and the role of the interlayer thickness.     

Numerical analysis of 2-D crack propagation problems using the numerical manifold method

The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.     

A moving crack in a rectangular magnetoelectroelastic body

The singular stress, electric fields and magnetic fields in a rectangular magnetoelectroelastic body containing a moving crack under longitudinal shear are obtained. Fourier transforms and Fourier sine series are used to reduce the mixed boundary value problems of the crack, which is assumed to be permeable or impermeable, to dual integral equations, and then expressed in terms of Fredholm integral equations of the second kind. Results show that the stress intensity factors are influenced by the material constants, the geometry size ratio and the velocity of the crack, and the propagation of the crack possibly brings about branching phenomena.     

有限高磁电弹性体中双半动态与静态裂纹分析

狭长体中的裂纹是断裂力学中经常采用的研究模型。含有共线无限长裂纹的条形磁电弹性体,当面内的力电磁和反平面的剪应力作用在左边裂纹尖端附近的一段裂纹面上时,往往会产生动态断裂。利用复变函数法中的拱形变换公式,导出了磁电全非渗透型边界条件下左裂纹尖端动态的应力强度因子以及机械应变能释放率的解析解。当运动速度趋于零时退化为静止状态下的解。通过数值算例分析了断裂机理,讨论了静止状态下狭长体和裂纹的几何尺寸、外力、电场和磁场分别对能量释放率的影响,为相关器件的设计与制造提供了帮助。     

Stress analysis around crack tips in finite strain problems using the eXtended finite element method

Fracture of rubber‐like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X‐FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works. Copyright © 2005 John Wiley & Sons, Ltd.     

A mixed-mode crack analysis of rotating disk using finite element method

This paper presents a rigorous elastodynamic hybrid-displacement finite element procedure for a safety analysis of fast rotating disks with mixed-mode cracks. Based on a modified Hamilton's principle, the finite element model is derived such that the proper crack-tip singularities are taken into consideration and the interelement displacement compatibility conditions are still satisfied. Thus, the specimen can be represented by a finite element assemblage in which “singular” elements are used around the crack-tip and high-order isoparametric “regular” elements are taken elsewhere.To determine the mixed-mode stress intensity factors, the modified $\text{J}\text{?}{}_{\mathrm{k}}$ integrals for rotating cracked disks have been established taking into account the effect of centrifugal force. Using the “strain-energy-density factor” concept, the direction of crack growth of a rotating disk with an arbitrary internal crack is predicted. To provide a method of non-destructive testing in evaluating the integrity of structures, natural vibrations of cracked disk are then studied. Lastly, the influence of inertia effects due to rotating speed changes in determining the dynamic stress intensity factors is examined.For verification purposes, the simple case of a rotating disk with radial cracks is first solved. Excellent correlations between the computed results and available referenced solutions are drawn. New solutions for the circular disk with circumferential or arbitrarily-oriented cracks are also presented.     

Elastodynamic analysis of crack by finite element method using singular element

The finite element method using a singular element near the crack tip is extended to the elastodynamic problems of cracks where the displacement function of the singular element is taken from the solution of a propagating crack. The dynamic stress intensity factor for cracks of mode III or mode I deformations in a finite plate is determined.The results of computation for stationary cracks or propagating cracks under dynamic loadings are compared with the analytical solutions of other authors. It is shown that the present method satisfactorily describes the time variation of the stress intensity factor in dynamic crack problems.

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Résumé La méthode des éléments finis utilisant un élément singulier au voisinage de l'extrémité d'une fissure a été étendue aux problèmes élastodynamiques des fissures tels qu'ils se posent lorsque la fonction de déplacement d'un élément singulier est prise à partir de la solution d'une fissure en cours de propagation. Le facteur d'intensité des contraintes dynamiques correspondant à des fissures de mode III ou des déformations de mode I dans une plaque finie a été déterminé. Les résultats des calculs correspondant à des fissures stationnaires ou des fissures en cours de propagation sous des charges dynamiques sont comparées aux solutions analytiques obtenues par d'autres auteurs. On montre que la méthode présentée décrit de façon satisfaisante la variation en fonction du temps du facteur d'intensité des contraintes dans les problèmes de fissuration dynamique.

     

A dislocation interacts with a finite crack in piezoelectric media

Analytical solutions for the generalized two-dimensional problem of a dislocation interacting with a finite crack in piezoelectric media are formulated via Stroh formalism. Two kinds of electrically boundary conditions along the crack surface, namely, electrically impermeable and electrically permeable are studied in the analysis. Based on the complex variable method and the perturbation technique, the closed form solutions are obtained. The field intensity factors at the crack tip and the image forces on the dislocation due to the crack are computed and discussed.     

Parallel crack near the interface of magnetoelectroelastic bimaterials

A parallel crack near the interface of magnetoelectroelastic bimaterials is considered. The crack is modelled by using the continuously distributed edge dislocations, which are described by the density functions defined on the crack line. With the aid of the fundamental solution for the edge dislocation, the present problem is reduced to a system of singular integral equations, which can be numerically solved by using the Chebyshev numerical integration technique. Then, the stress intensity factor (SIF), the magnetic induction intensity factor (MIIF) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Using these fracture criteria, the cracking behaviour of magnetoelectroelastic bimaterials is investigated. Numerical examples demonstrate that the interface, mechanical load, magnetic load and material mismatch condition are all important factors affecting the fracture toughness of the magnetoelectroelastic bimaterials.     

A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to fracture problems in anisotropic media using a specially designed five-node singular crack-tip (T5) element. In the formulation of singular ES-FEM, only the assumed displacement values (not the derivatives) on the boundaries of the smoothing domains are needed. Thus, a layer of T5 crack-tip element is devised to construct “singular” shape functions via a simple point interpolation with a fractional order basis, without mapping procedure. The effectiveness of the present singular ES-FEM is demonstrated by intensive examples for a wide range of degrees of anisotropy.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.