Intensity Factors Of Stresses Research Articles

Problems of stationary heat conduction and thermoelasticity for a body with a heat permeable disk-shaped inclusion (crack)

We reduce a problem of stationary heat conduction for a body with a heat permeable disk-shaped inclusion between the surfaces of which nonideal thermal contact takes place to a hypersingular integral equation of the second kind. The exact solution of this equation for the case where its right-hand side is a polynomial of the third degree and, in the axisymmetric case, a polynomial of the n th degree, is presented. For these cases, we determine shear stresses on the inclusion generated by heat dipoles and intensity factors of tangential stresses for a heat permeable circular crack.

Singularities of an interface crack in electrostrictive materials

In the present work, the singularities of an interface crack between two dissimilar electrostrictive materials under electric loads are investigated. Within the framework of two-dimensional deformation, the problem is solved using the complex variable method. Three crack models, that is, permeable, impermeable and conducting crack models are considered individually. Complex potentials and intensity factors of total stresses are derived by considering both the Maxwell stresses in the surrounding space at infinity and inside the crack. It is found that, for the above three crack models, the singularities of total stress are the same as those in traditional bi-materials with an interface crack; however, the intensities of the total stress depend on the actual crack model used.

Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid

The problem of two collinear electromagnetically dielectric cracks in a magnetoelectroelastic material is investigated under in-plane electro-magneto-mechanical loadings. The semi-permeable crack-face boundary conditions are adopted to simulate the case of two collinear cracks full of a dielectric interior. Utilizing the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel, which then are solved explicitly. The intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD) and the energy release rates near the inner and outer crack tips are determined in closed forms for two cases of possible far-field electro-magneto-mechanical loadings respectively. Numerical results for a BaTiO 3–CoFe 2O 4 composite are carried out to show the effects of applied mechanical loadings on the crack-face electric displacement and magnetic induction, the stress intensity factor and the COD intensity factor, respectively. The obtained results reveal that when the applied mechanical loading is stress, applied electromagnetic loadings have no influences on the stress intensity factor. When the applied mechanical loadings is train, the applied positive electromagnetic loadings decrease the intensity factors of stress and COD, and the applied negative ones increase the intensity factors of stress and COD. The variations of energy release rates are also given to show the effects of the geometry of two collinear dielectric cracks.

Transient response of two collinear dielectric cracks in a piezoelectric solid under inplane impacts

The paper is focused on the dynamic analysis of two collinear dielectric cracks in a piezoelectric material under the action of in-plane electromechanical impacts. Considering the dielectric permeability of crack interior, the electric displacements at the crack surfaces are governed by the jumps of electric potential and crack opening displacement across the cracks. The permeable and impermeable crack models are the limiting cases of the general one. The Laplace and Fourier transform techniques are further utilized to solve the mixed initial-boundary-value problem, and then to obtain the singular integral equations with Cauchy kernel, which are solved numerically. Dynamic intensity factors of stress, electric displacement and crack opening displacement are determined in time domain by means of a numerical inversion of the Laplace transform. Numerical results for PZT-5H are calculated to show the effects of the dielectric permeability inside the cracks, applied electric loadings and the geometry of the cracks on the fracture parameters in graphics. The observations reveal that based on the COD intensity factor, a positive electric field enhances the dynamic dielectric crack growth and a negative one impedes the dynamic dielectric crack growth in a piezoelectric solid.

An arc-shaped crack in an electrostrictive material

The plane problem of a circular-arc crack in an infinite electrostrictive solid under remote electric fields is studied based on the complex variable method. First, explicit solutions for the complex potentials are presented in closed-form. Secondly, intensity factors of total pseudo stresses are derived by taking the Maxwell stresses around the infinite surrounding space and inside the crack into account. Then, numerical results are given to discuss the effects of electric fields on the fracture of electrostrictive materials. It is found that when the interior of the crack is filled with the same gas as that at infinity, the applied electric field has no effects on crack growth; however, when the interior of crack and the surrounding space at infinity are filled with different gases, the applied electric field may either enhance or retard crack growth, which depends on the electric boundary conditions adopted on the crack faces, the Maxwell stresses on the crack faces and at infinity, and the central angle of the crack.

Fracture of electrostrictive solids subjected to combined mechanical and electric loads

Based on the complex variable method, this paper studies the effects of electric fields on the fracture of an electrostrictive solid under combined mechanical and electrical loads at infinity. The electric field inside a deformed crack is first determined by using the semi-permeable crack model. Then, the complex potentials and the intensity factors of stresses are presented, respectively, in concise and closed forms. Numerical results are also obtained to discuss the effects of applied electric and/or mechanical loads on the induced electric fields inside the crack and the stress intensity factors when the interior of the deformed crack and the surrounding space at infinity are filled with different gases.

Thickness effects for a penny-shaped dielectric crack in a magnetoelectroelastic layer

The problem of a penny-shaped dielectric crack in a magnetoelectroelastic layer is considered within the theory of linear magnetoelectroelasticity. Under the assumption of mode-I magnetoelectromechanical loadings, the semipermeable crack-face electromagnetic boundary conditions are applied to describe the case of an opening penny-shaped crack. In order to solve the boundary-value problem, the Hankel transform technique is utilized and three coupled Fredholm integral equations are further derived. The intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) are further determined by the composite Simpson's rule. Thickness effects of a magnetoelectroelastic layer on the electric displacement and magnetic induction of the crack interior, and the field intensity factors are illustrated through numerical computations for a BaTiO3-CoFe2O4 composite. The obtained results reveal that an increase of the ratio of the layer thickness and the crack radius h/a increases the electric displacement and magnetic induction of a dielectric crack interior, and decreases the field intensity factors regardless of the permeability of the crack interior. Based on the COD intensity factor, the influences of applied electric and magnetic fields on the growth of a dielectric crack are further investigated and presented graphically.

Electroelastic analysis of an electrically dielectric Griffith crack in a piezoelectric layer

The fracture analysis of an electrically dielectric Griffith crack embedded in a piezoelectric layer is made under in-plane electro-mechanical loadings. To simulate an opening crack full of a dielectric interior, the energetically consistent crack-face boundary conditions are utilized. Applying the Fourier transform technique, the boundary-value problem is reduced to solving two coupling singular integral equations. The intensity factors of stress, electric displacement, crack opening displacement (COD) and electric potential are further determined by the Lobatto–Chebyshev collocation method. The variations of the electric displacement at the crack surfaces are investigated by using the energetically consistent and semi-permeable boundary conditions respectively. The observations show that the electric displacement inside the crack is decreasing with an increase of the ratio between the crack length and piezoelectric layer width. Numerical computations are further carried out to compare the intensity factors of stress and electric potential, and the energy release rate using the energetically consistent boundary conditions with those using the semi-permeable boundary conditions. The obtained results reveal that the stress induced by a dielectric inside a crack has great effects on the stress intensity factor and energy release rate, but little influence on the electric potential difference across the crack.

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

Abstract The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO 3 -CoFe 2 O 4 composite with a penny-shaped vacuum crack are discussed in detail.

Analysis of a dielectric crack in a magnetoelectroelastic layer

Within the theory of linear magnetoelectroelasticity, the fracture analysis of a magneto-electrically dielectric Griffith crack embedded in a magnetoelectroelastic layer is investigated under in-plane magneto-electro-mechanical loadings. The semi-permeable crack-face magnetoelectric boundary conditions are utilized to simulate the case of an opening dielectric crack. Applying the Fourier transform technique, the boundary-value problem is reduced to solving three coupling singular integral equations. Field intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) are further determined by the Lobatto-Chebyshev collocation method. The electric displacement and magnetic induction of crack interior are discussed in detail. The obtained results reveal that the magnetoelectric field inside the crack is dependent on the material properties, applied loadings, the dielectric permeability of crack interior, and the ratio of the crack length and the layer width. Numerical computations are carried out to present that the volume fraction of piezoelectric phase in BaTiO 3 – CoFe 2 O 4 composites has great influences on the magnetoelectric field inside the dielectric crack. Based on the COD intensity factor, the analysis for the growth of a dielectric crack in a magnetoelectroelastic layer is examined. The proposed procedure can be reduced to dealing with the case of a dielectric crack in a piezoelectric layer and the obtained results can be verified by the experimental observations.

Transient response of a magnetoelectroelastic solid with two collinear dielectric cracks under impacts

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.

Transient response of the crack-tip field in a magnetoelectroelastic half-space with a functionally graded coating under impacts

This paper has twofold aims. One is to study the dynamic response of a magnetoelectroelastic half-space with functionally graded coating containing crack at the interface when subjected to sudden impacts. Two different loading positions, where the material and crack surfaces are loaded respectively, are considered. By using the integral transform method, the problem is reduced to solving singular integral equations. Obtained numerical results show that the overshoots of dynamic fracture parameters are strongly amplified or reduced depending on negative or positive gradient, respectively for the case of the material surface being loaded suddenly. This implies that a functionally graded coating with a positive gradient index is preferable in engineering design due to its capability of preventing the structure from cracking. The second objective is to give a comparison of relevant dynamic parameters such as the intensity factors of stress and strain, energy release rate, and energy density factor, and their features are elucidated under dynamic combined loadings. It indicates that the strain intensity factor can overcome the drawbacks of the rest parameters, and may be chosen as an effective fracture parameter, while three others cannot be adopted as fracture criteria to describe the feature of onset of crack growth.

Determining the intensity factors for stresses, electric-flux density, and electric-field strength in multiply connected electroelastic anisotropic media

Formulas are derived to evaluate the intensity factors of stresses, electric-flux density, and electric-field strength in two-dimensional problems of elasticity, thermoelasticity, electroelasticity, magnetoelasticity, thermoelectroelasticity, and thermomagnetoelasticity for anisotropic bodies with holes and plane (rectilinear) cracks

An Internal Crack Subjected to a Thermal Flow in Magnetoelectroelastic Solids: Exact Fundamental Solution

The analysis of intensity factors for cracks under thermal, mechanical, electrical and magnetic boundary conditions has become a very important topic in fracture mechanics. In this paper, an exact and complete fundamental solution is derived for the problem of a penny-shaped crack in magneto-electro-thermo-elastic material under prescribed thermal flux on the crack faces. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. Electro-magneto-thermo-elasticity includes the effective elastic, piezoelectric, piezomagnetic, dielectric permittivity, magnetic permeability and electromagnetic coupling moduli, as well as the effective thermal expansion coefficients and the associated pyroelectric and pyromagnetic constants. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential and the magnetic potential and temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of Hankel integral transform. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for impermeable and permeable cracks. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.

A generalized screw dislocation interacting with interfacial cracks along a circular inhomogeneity in magnetoelectroelastic solids

The interaction of a generalized screw dislocation with circular arc interfacial cracks under remote antiplane shear stresses, in-plane electric and magnetic loads in transversely isotropic magnetoelectroelastic solids is dealt with. By using the complex variable method, the general solutions to the problem are presented. The closed-form expressions of complex potentials in both the inhomogeneity and the matrix are derived for a single circular-arc interfacial crack. The intensity factors of stress, electric displacement and magnetic induction are provided explicitly. The image forces acting on the dislocation are also calculated by using the generalized Peach–Koehler formula. For the case of piezoelectric matrix and piezomagnetic inclusion, the shielding and anti-shielding effect of the dislocation upon the stress intensity factors is evaluated in detail. The results indicate that if the distance between the dislocation and the crack tip remains constant, the dislocation in the interface will have a largest shielding effect which retards the crack propagation. In addition, the influence of the interfacial crack geometry and materials magnetoelectroelastic mismatch upon the image force is discussed. Numerical computations show that the perturbation effect of the above parameters upon the image force is significant. The main result shows that a stable or unstable equilibrium point may be found when a screw dislocation approaches the surface of the crack from infinity which differs from the perfect bonded case under the same conditions. The present solutions contain a number of previously known results which can be shown to be special cases.

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.

Antiplane electro-mechanical field of a piezoelectric finite wedge under shear loading and at fixed-grounded boundary conditions

This paper presents the general solutions of antiplane electro-mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as fixed and grounded. Employing the finite Mellin transform method, the stress and electrical displacement at all fields of the piezoelectric finite wedge are derived analytically. In addition, the singularity orders and intensity factors of stress and electrical displacement can also be obtained. These parameters can be applied to examine the fracture behavior of the wedge structure. After being reduced to the problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.

Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO 3–CoFe 2O 4 composite wedge with an interface crack

The antiplane analysis is made for a bimaterial BaTiO 3–CoFe 2O 4 composite wedge containing an interface crack. The coupled magneto-electro-elastic field is induced by the piezoelectric/piezomagnetic BaTiO 3–CoFe 2O 4 composite materials. For the crack problems, the intensity factors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field at crack tips are derived analytically. Also, the energy density criterion is applied to predict the fracture behavior of the interface crack. The numerical results also show that the energy release rate for a crack in a single wedge is negative.

A Piezoelectric Screw Dislocation Interacting with a Dielectric Crack in a Hexagonal Piezoelectric Material

This paper is concerned with the interaction of a piezoelectric screw dislocation with a semi-infinite dielectric crack in a piezoelectric medium with hexagonal symmetry. The solution of the considered problem is obtained from the dislocation solution of a piezoelectric half-plane adjoining a gas medium of dielectric constant ε0 by using the conformal mapping method. The intensity factors of stress, electric displacement and electric field and the image force on the dislocation are given explicitly. The effect of electric boundary conditions on the dislocation-crack interaction is analyzed and discussed in detail. The results show that ε0 only influences the electric displacement and electric field intensity factors and the image force produced by the electric potential jump.

The strip dielectric breakdown model

This paper reports on the analysis of the strip dielectric breakdown (DB) model for an electrically impermeable crack in a piezoelectric medium based on the general linear constitutive equations. The DB model assumes that the electric field in a strip ahead of the crack tip is equal to the dielectric breakdown strength, which is in analogy with the classical Dugdale model for plastic yielding. Using the Stroh formalism and the dislocation modeling of a crack, we derived the relationship between the DB strip size and applied mechanical and electrical loads, the intensity factors of stresses and electric displacement, and the local energy release rate. Based on the results, we discussed the effect of electric fields on fracture of a transversely isotropic piezoelectric ceramic by applying the local energy release rate as a failure criterion. It is shown that for an impermeable crack perpendicular to the poling direction, a positive electric field will assist an applied mechanical stress to propagate the crack, while a negative electric field will retard crack propagation. However, for an impermeable crack parallel to the poling direction, it is found that the applied electric field does not change the mode I stress intensity factor and the local energy release rate, i.e., the applied electric field has no effect on the crack growth.