Generalized Intensity Factors Research Articles

Fracture analysis of circular hole edge arbitrary position crack with surface effects in magnetoelectroelastic materials

The mechanical properties of magnetoelectroelastic (MEE) materials are brittle, which is easy to produce nano-defects (hole or crack). The micro-fields of nanoscale cracks and holes are size dependent significantly. Nanoscale cracks and holes interact with each other, and the interference effect is very obvious. In this paper, the Model III fracture behavior of nanoscale circular hole edge arbitrary position radial crack is studied under anti-plane mechanical load, in-plane electric displacement load and in-plane magnetic induction load. Based on the surface elasticity theory, the conformal mapping technique and the MEE elasticity theory, the MEE fields expressions of the nano-hole and nano-crack are obtained, and the analytical solution of generalized intensity factors (stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor) at crack tip is given. The comparison between the present results and the existing research shows the correctness of the present method in this paper. The influences of the microstructure and coupling loads on the generalized intensity factors are discussed.

Fracture analysis of cracked magneto-electro-elastic functionally graded materials using scaled boundary finite element method

This paper develops the scaled boundary finite element method to analyse fracture of functionally graded magneto-electro-elastic materials. Polygon meshes are employed to discretize the domain. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required to calculate the intensity factors. When the material gradients of the coefficients in the constitutive matrix are expressed as a series of power functions of the scaled boundary coordinates, the stiffness matrices can be integrated analytically. The formulation enables the generalized intensity factors of stress, electric displacement and magnetic induction fields along the radial direction to be represented analytically. This permits the calculation of the generalized intensity factors directly from the scaled boundary finite element solution of the singular stress, electric displacement and magnetic induction fields by following the standard stress recovery procedures in the finite element method. Several numerical benchmarks are presented to validate the proposed technique with the results reported in the literature.

A new framework based on XFEM for cracked semipermeable piezoelectric material

In the present work, a novel load-controlled iteration scheme within the framework of extended finite element method (XFEM) is proposed to model the semipermeable crack in a piezoelectric material. To capture the stress and electric displacement singularity at the crack tip, the electromechanical 6-fold enrichment functions are used. The domain form of electromechanical interaction integral is employed to determine the generalized intensity factor. The accuracy of a proposed scheme based on XFEM is validated against the reference solutions derived from the iterative capacitor analogy (ICA) method. The effect of several crack configurations, polarization angle, far-field mechanical and electrical traction on electric displacement intensity factor (EDIF) are analyzed for cracked semipermeable piezoelectric material.

Extended isogeometric analysis for fracture in functionally graded magneto-electro-elastic material

In this work, we present an extended isogeometric analysis (XIGA) for cracked functionally graded magneto-electro-elastic (FGMEE) material. The material properties inside the FGMEE domain vary exponentially. An electrically and magnetically impermeable crack in the FGMEE domain is modeled by adding appropriate enrichment functions into the isogeometric analysis (IGA) approximation. Heaviside function is utilized to model the displacement, electric, and magnetic potential along the crack face. In contrast, conventional four-field crack tip enrichment functions are used to mimic the crack tip singularity characteristics. The generalized intensity factors (IFs), i.e., the stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF), are assessed using magneto-electro-mechanical interaction integral. A series of numerical studies over the cracked homogeneous and FGMEE material are presented. The accuracy of XIGA is examined by comparing the obtained results with the reference solutions. Various parameters such as volume fraction, crack length, loading combination parameter (both electrical and magnetic), material non-homogeneity parameter and crack orientation are taken to examine their effects on the generalized IFs.

Analysis of bimaterial singular regions for orthotropic and isotropic materials under thermal loading

A method to formulate an asymptotic elasticity solution for bimaterial singular regions in orthotropic materials is presented for the thermal case. In the vicinity of material and geometric singularities, approximate methods such as finite elements fail to converge and procedures to improve these solutions become necessary. The results presented herein can be used in conjunction with approximate numerical techniques for the determination of stresses and stress intensity factors in composite bonded joints, cracked bimaterial interfaces, or a junction of dissimilar materials under thermal loading. The thermal displacement and stress fields at the tip of a bimaterial wedge are developed by means of eigenfunction expansions including both singular and nonsingular terms. The thermal elasticity solution for isotropic materials as the limiting case of the orthotropic solution is also obtained. The formulations obtained are applicable to bonded joints with interfacial or interlaminar cracks, geometric and material discontinuities, and uniform thermal loading. The detailed formulation presented provides an essential component in the application of finite elements for the determination of generalized intensity factors and stresses in composites, bonded joints, or other dissimilar materials under thermal loading. Formulation results for orthotropic materials, including the limiting case of isotropic materials, are compared with published and closed form results that use different formulations, and excellent agreement is observed. Furthermore, the existence of logarithmic stress singularities of the form (k⋅ΔT⋅lnr) is investigated and discussed.

Fracture analysis of piezoelectromagnetic medium with axisymmetric cracks

The present study evaluates, by the distributed dislocation technique, the generalized intensity factors for the multiple axisymmetric cracks in a transversely isotropic piezoelectromagnetic medium under in-plane magneto-electro-mechanical loading. To this end, the solution to the problem of generalized dislocations is first obtained with the aid of the Hankel transform. The generalized dislocation in the piezoelectromagnetic medium contains Volterra prismatic and Somigliana radial dislocations in the displacement components and jump in the electric and magnetic potentials. The distributed dislocation technique is used to formulate integral equations for a system of coaxial axisymmetric cracks, including penny-shaped and annular cracks. These integral equations are of Cauchy singular type, which are solved numerically to obtain the dislocation densities on the faces of the cracks and the results are used to determine generalized intensity factors for cracks. Finally, several examples to study the crack type/location on the ensuing generalized intensity factors at tips of cracks and also the interaction between the cracks are presented.

Fracture analysis of magnetoelectroelastic bimaterials with imperfect interfaces by symplectic expansion

A Hamiltonian-based analytical method is used to study the mode III interface cracks in magnetoelectroelastic bimaterials with an imperfect interface. By introducing an unknown vector, the governing equations are reformulated in sets of first-order ordinary differential equations. Using separation of variables, eigensolutions in the symplectic space are obtained. An exact solution of the unknown vector is obtained and expressed in terms of symplectic eigensolutions. Singularities of mechanical, electric, and magnetic fields are evaluated with the generalized intensity factors. Comparisons are made to verify accuracy and stability of the proposed method. Numerical examples including mixed boundary conditions are given.

A new coupled method for high-accuracy determination of fracture parameters of an interface V-notch in magneto-electro-elastic bimaterial

A simple coupled method, called “finite element discretized symplectic method” is introduced into the mode III fracture analysis of a V-notched magneto-electro-elastic (MEE) bimaterial. High-accuracy generalized intensity factors and energy release rate are computed to evaluate the singularities of mechanical, electric and magnetic fields. The present method is carried out in two steps. In the first step, the physical domain is meshed by the conventional element for MEE media and is divided into a finite size singular region near the notch tip and a regular region far away from the notch tip. In the second step, analytical symplectic eigenfunctions are employed to transform the large number of nodal unknowns in the singular region into a small set of undetermined coefficients of a symplectic series; the nodal unknowns in the regular region remain as usual. Consequently, the computational cost is enormously reduced and fracture parameters are actually the first three coefficients of the series. Explicit expressions in the singular fields are obtained simultaneously. Numerical results are compared with the existing solutions and found to be in good agreement. Some new results are given also.

A Dual BEM Formulation for Thermo-Magneto-Piezo-Electric 2D Fracture Problems

A hypersingular approach of the boundary element formulation -BEM- is presented for the analysis of the temperature in fracture problems of multifield materials. A fundamental solution based on the extended Stroh formalism is considered. Hypersingular integrals arising in the traction boundary and heat flux integral equations are computed through a regularization technique causing that only regular integrals are computed numerically. The generality of the method allows the study of 2D solids with no shape restrictions. Generalized intensity factors -IFs- are accurately computed from nodal values next to crack tip. Finally, the possibilities of the presented approach are discussed in order to apply it for the assessment of the influence of temperature in fracture problems for materialspresenting mechanical, magnetic and/or electrical coupling

A mode III moving interfacial crack based on strip magneto-electric polarization saturation model

This paper deals with a mode III Yoffe-type interfacial crack propagating subsonically under the moving strip magneto-electric saturation model. Nonlinear effects are characterized by different magnetic and electric saturation strips around the crack tip. Employing the extended Stroh method, we obtain generalized moving interfacial dislocation densities analytically under impermeable magneto-electric crack boundary conditions. The generalized intensity factor and local energy release rate with nonlinear effects are derived as fracture parameters for the moving magneto-electro-elastic (MEE) interfacial crack. Numerical results are presented to show the characteristics of fracture dominant parameters with respect to the loading as well as the propagation velocity. In addition, a dimensionless parameter defined by the ratio of the volume fraction of the composite constituents is proposed to evaluate the influences of the MEE bimaterial properties. This research will give us ideas on material selection for optimizing the fracture toughness of MEE composites.

Analysis of functionally graded magneto-electro-elastic layer with multiple cracks

In this paper, the distributed dislocation technique (DDT) is developed to be utilized for the analysis of a cracked functionally graded piezoelectric–piezomagnetic (FGPP) layer under anti-plane mechanical and in-plane electric and magnetic fields. By using the Fourier transformation, the closed-form expressions for the shear stress, electric displacement and magnetic displacement components are obtained for a generalized Volterra-type screw dislocation. The generalized dislocation in FGPP layer contains dislocation in the displacement component and jump in the electric and magnetic potentials. The expressions of generalized stress intensity factor are derived in the DDT. The solution of the dislocation problem is utilized in the DDT to solve the problem of arbitrary configurations of multiple embedded and edge cracks. The generalized intensity factors of the cracked layer are obtained. Numerical results for generalized intensity factors of straight and curved cracks are presented. The DDT is proved to be useful in the analysis of the interaction of the embedded and edge cracks in an FGPP layer.

Coupled 2D electric, magnetic, and mechanical fields in dielectrics with cracks and thin inclusions

We apply methods of the theory of thermoelastic deformation of bodies with thin inclusions to study magnetoelectroelastic media with thin inhomogeneities. We construct the integral equations of the problem and propose an efficient numerical procedure of the boundary-element method for their solution. It is established that the fields of mechanical stresses, electric displacements, and magnetic induction near the tip (front) of a thin inclusion, the mathematical model of which can be based on the coupling principle for continua of different dimensions, have a square-root singularity. For the complete description of asymptotic relations for stresses, displacements, and other quantities near the front of an inhomogeneity, we introduce 10 generalized intensity factors for stresses, electric displacements, and magnetic induction. We obtain analytical solutions of the problem for the limiting cases of a permeable crack and a rigid electroconductive inclusion in a magnetoelectroelastic medium. Results of the numerical analysis of intensity factors for an elastic isotropic inclusion in an anisotropic magnetoelectroelastic medium are presented.

Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Dynamic anti-plane fracture problem of an exponentially graded linear magnetoelectroelastic plane with a finite impermeable crack subjected to time-harmonic SH-waves is solved. Directions of wave propagation and material inhomogeneity are chosen in an arbitrary way. The fundamental solution for the coupled system of partial differential equations with variable coefficients is derived in a closed form by the hybrid usage of both an appropriate algebraic transformation for the displacement vector and the Radon transform. The formulated boundary-value problem is solved by a nonhypersingular traction boundary integral equation method (BIEM). The collocation method and parabolic approximation for the unknown generalized crack opening displacements are used for the numerical solution of the posed problem. Quarter point elements placed next to the crack-tips ensure properly modeling the singular behavior of the field variables around the crack tip. Fracture parameters as stress intensity factor, electric field intensity factor and magnetic field intensity factor are computed. Intensive simulations reveal the sensitivity of the generalized intensity factors (GIF) at the crack-tips to the material inhomogeneity, characteristics of the incident wave, coupling effects, wave-material and wave-crack interaction phenomena.

Electroelastic analysis of a piezoelectric finite wedge with mixed type boundary conditions under a pair of concentrated shear forces and free charges

The antiplane problem of a piezoelectric finite wedge with mixed type boundary conditions (i.e. traction free-electrically grounded or clamped-electrically insulated) assumed on the circular edge is studied in this paper. Using the first and second kinds of the finite Mellin transforms, the stress and electrical displacement in the piezoelectric finite wedge subjected to a pair of concentrated shear forces and free charges are derived analytically. The singularity orders and all generalized intensity factors can also be obtained. After being reduced to the crack problem, the energy density theory is applied to study the effects of the boundary conditions on the circular edge and the wedge radius on the fracture behaviors of the cracked piezoelectric medium. The problem can be further degenerated to several simple problems and the results agree well with those of previous studies.

Singularity analysis and fracture energy-release rate for composites: Piecewise homogeneous-anisotropic materials

We present a formulation for the computation of singularity solution and the fracture energy release rate for piecewise homogeneous-anisotropic materials. A new approach of matched asymptotic analysis is proposed to derive the expression of the fracture energy release rate in terms of the critical singularity exponent and the generalized intensity factors. For the computation of the singular exponents, 2-D elasticity and conductivity problems are first transformed into polar coordinates, then reduced to 1-D problems in terms of the angular variable via the Mellin transform, prior to a finite-element discretization. The resulting 1-D problem is a quadratic eigenvalue problem, which can be transformed into a linear eigenvalue problem for the computation of the eigenpairs. The generalized intensity factors, which are obtainable using contour-independent integrals, are given in terms of equivalent domain integrals, which are numerically easier to implement, more accurate and stable than the contour integrals. A better and concise proof of a fundamental proposition concerning the refracted flux across a bi-material interface in the conductivity problem is also provided. The present work provides a new and clearer exposition of the asymptotic analysis presented in [D. Leguillon, E. Sanchez-Palencia, Crack phenomena in heterogenous media, in: B.W. Schulze, H. Triebel (Eds.), Proceedings of Symposium on Analysis on Manifolds with Singularities, Breitenbrunn, 1990, B.G. Teubner, Stuttgart, 1992], and is thus more accessible to the general engineering readers, e.g., by drawing a parallel between the present formulation with traditional practices in structural dynamics. Further, the present formulation—which is more amenable to a direct implementation in a finite element code—sets the stage for a future generalization to functionally-graded materials. Several numerical examples involving bi-material and tri-material interfaces, including carbon-nanotube reinforced composites, together with validation with available analytical results, are given to illustrate the accuracy and effectiveness of the methodology.

A New Boundary Integral Equation Method for Cracked Piezoelectric Bodies

A novel integral equation method is developed in this paper for the analysis of two-dimensional general piezoelectric cracked bodies. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh’s formalism for anisotropic elasticity in conjunction with Cauchy’s integral formula. The proposed boundary integral equations contain generalized boundary displacement (displacements and electric potential) gradients and generalized tractions (tractions and electric displacement) on the non-crack boundary, and the generalized dislocations on the crack lines. The boundary integral equations can be solved using Gaussian-type integration formulas without dividing the boundary into discrete elements. The crack-tip singularity is explicitly incorporated and the generalized intensity factors can be computed directly. Numerical examples of generalized stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

Crack onset at a v-notch. Influence of the notch tip radius

A criterion to predict crack onset at a sharp notch in homogeneous brittle materials has been presented in a previous paper of one of the authors. It is reviewed and improved herein. It fulfils both the energy and the strength criteria and takes an Irwin-like form involving the generalized intensity factor of the singularity governing the elastic behaviour in the vicinity of a notch tip. The prediction agrees fairly well with the experiments although it slightly underestimates the experimental measures. A cause of this discrepancy can be that a small notch tip radius blunts the sharp corner. It is analysed in this paper by means of matched asymptotics involving 2 small parameters: a micro-crack increment length and the notch tip radius. A correction is brought to the initial prediction and a better agreement is obtained with experiments on PMMA notched specimens. Experiments performed on a stiffer material (Alumina/Zirconia) show that it is less sensitive to small notch tip radii. A remaining small discrepancy between experiments and predictions can be due to some non linear behaviour of the materials near the notch tip. In addition, without new developments, the method allows to determine the stress intensity factor at the tip of a short crack emanating from a sharp or a rounded v-notch.