Crack-tip Field Intensity Factors Research Articles

A simplified evaluation of the mechanical energy release rate of kinked cracks in piezoelectric materials using the boundary element method

Based on the concept of the hoop field intensity factors of an initial crack prior to any kink, an apparent hoop mechanical (strain) energy release rate (MERR) is defined to approximate the MERR of a piezoelectric crack with an infinitesimal kink at any arbitrary angle. The validity and the efficiency of the simplified approximation are examined by numerical examples using the boundary element method (BEM). The generalized crack-opening-displacements or displacement jumps are computed by the traction boundary integral equations (BIEs). By using the displacement extrapolation method, the crack-tip field intensity factors of any arbitrarily kinked crack in linear piezoelectric materials are obtained and the BEM results are validated by comparing them with the available reference analytical results. Then, the differences between the conventional field intensity factors and MERR of an infinitesimally kinked crack and the hoop field intensity factors and hoop MERR of the main crack prior to any kink are numerically analyzed. Finally, the crack propagation in an infinite linear piezoelectric material is numerically simulated. The paths of the crack growth are predicted by adopting four different fracture criteria, namely, the maximum hoop stress intensity factor (SIF) and MERR fracture criteria for the main crack-tip before the next propagation, and the maximum KI and MERR fracture criteria for the kinked tip of the main crack with an infinitesimal branch at an arbitrary kinking angle evaluated by using a trial crack extension technique. The comparisons among these results show that the present simplified approximation can efficiently provide a sufficient accuracy for numerical simulation of crack growth in linear piezoelectric materials.

An interaction integral and a modified crack closure integral for evaluating piezoelectric crack-tip fracture parameters in BEM

To evaluate the crack-tip field intensity factors of a piezoelectric crack with any inclined angle, the current widely-used interaction integral method (I-integral) is here extended to the boundary element applications under some coordinate transformations. As well, a new modified crack closure integral method (MCCI) is proposed by considering the discontinuous quarter-point singular elements for the crack-face discretization arising from the dual boundary element method (BEM). This dual BEM involves the strongly singular displacement boundary integral equations (BIEs) for the external boundary and the hypersingular traction BIEs for the crack faces. The crack-tip fracture parameters evaluated by the I-integral and MCCI are verified by the existing analytical solutions and meanwhile, compared with those results achieved by the classical displacement extrapolation method and the J-integral. Three examples are presented to show the high accuracy of the interaction integral method and the improvement of MCCI for the piezoelectric crack problems.

Domain-independent I-integrals for force and couple stress intensity factor evaluations of a crack in micropolar thermoelastic medium

This paper establishes the domain-independent I-integrals (DII-integrals) to solve the force stress intensity factors (FSIFs) and couple stress intensity factors (CSIFs) of a crack in a two-dimensional micropolar thermoelastic solid. The DII-integrals are derived from the J-integral based on the superposition of an actual field and an auxiliary field. The auxiliary field can be designed freely, so that mixed-mode FSIFs and the in-plane CSIF can be decoupled. Analytical solutions for the near-tip asymptotic field of a crack in infinite micropolar elasticity are firstly adopted to define the auxiliary field and then, a DII-integral is established. The DII-integral is domain-dependent for thermal mismatch interfaces, which is troublesome to thermal fracture analysis of composites with complex thermal mismatch interfaces. In order to effectively solve the crack-tip field intensity factors for such composites, this paper discusses the conditions to establish a DII-integral and then, proposes a new auxiliary field which has an important feature, i.e., the mean auxiliary stress is zero. The DII-integral using this new auxiliary is proved to be domain-independent not only for interfaces with discontinuous mechanical properties, but also for interfaces with discontinuous thermal properties. Finally, these two DII-integrals are employed to investigate four micropolar plates with different thermoelastic properties. The results confirm that both DII-integrals are reliable and convenient for homogeneous, continuously nonhomogeneous micropolar materials, whereas the new DII-integral is more appropriate for micropolar materials with complex thermal mismatch interfaces.

Boundary Element Analysis for Piezoelectric Cracks by an Interaction Integral

In this paper, an interaction integral is applied to evaluate the crack-tip field intensity factors for piezoelectric cracks by using BEM. Based on this, the fracture parameters for different crack configurations and loading conditions are analyzed in details for both the center crack and edge crack problem. According to the present results, the path-independent behavior of the interaction integral is verified. The comparison of the I-integral results with those by the J-integral and the displacement interpolating methods shows a good agreement.

BEM analysis of electrically limited permeable cracks considering Coulomb tractions in piezoelectric materials

In this paper, the electrically limited permeable piezoelectric cracks considering the induced Coulomb tractions on the crack faces are studied with respect to the dielectric constant of the crack medium by BEM. As limiting cases, the permeable and impermeable electric crack-face boundary conditions are also considered. Based on the dual boundary integral equations, the crack-tip field intensity factors and the energy release rate are evaluated by the displacement extrapolation method and the J-integral method, respectively. The crack-face part of the contour integral is then handled by two approaches, namely numerical quadrature and approximate formula. The analytical solutions for a Griffith crack are also presented for verification purpose. Three examples are presented to check the efficiency and the accuracy of these approaches. The effects of the dielectric constant of the crack medium and the induced Coulomb tractions on the crack-tip field intensity factors and the energy release rate are also discussed.

Comparison of several BEM-based approaches in evaluating crack-tip field intensity factors in piezoelectric materials

From the viewpoint of fracture mechanics, of importance is the near-tip field which can be characterized as field intensity factors. In this paper, the crack-tip field intensity factors of the stress and electric displacement in two dimensional piezoelectric solids are evaluated by using four approaches including the displacement extrapolation, the stress method, the J-integral and the modified crack closure integral method (MCCI) based on a boundary element method (BEM). The strongly singular displacement boundary integral equations (BIEs) are applied on the external boundary of the cracked solid, while the hypersingular traction BIEs are used on the crack faces. Three numerical examples are presented to show the path independence and the high accuracy of the J-integral in piezoelectric materials and to analyze the pros and cons of these approaches in evaluating the field intensity factors.

Transient electromechanical cracking of finite piezoelectric bodies

This article studies the problem of a finite crack in a finite piezoelectric medium subjected to transient electric and mechanical loads. Unlike past studies that mainly focused on the infinitesimal cracks, the current article considers the effect of the medium boundaries on the transient fracture behavior of piezoelectric materials. A system of singular integral equations is formulated in terms of the crack surface displacement and electric potential. Effects of crack length and medium size on the crack-tip field intensity factors are investigated in detail. Numerical results for the time-dependent stress and electric displacement intensity factors are obtained. Electrically impermeable crack assumption, electrically permeable crack assumption, and notch of finite height are investigated. Some new conclusions are drawn.

Thermoelastic problem of steady-state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials

Plane thermoelasticity solutions are presented for the problem of a crack in bonded materials with a graded interfacial zone. The interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermoelastic moduli between dissimilar, homogeneous half-planes. The crack is assumed to exist in one of the half-planes at an arbitrary angle to the graded interfacial zone, disturbing uniform steady-state heat flows. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic thermoelasticity equations. Formulation of the current nonisothermal crack problem lends itself to the derivation of two sets of Cauchy-type singular integral equations for heat conduction and thermal stress analyses. The heat-flux intensity factors and the thermal-stress intensity factors are defined and evaluated in order to quantify the singular characters of temperature gradients and thermal stresses, respectively, in the near-tip region. Numerical results include the variations of such crack-tip field intensity factors versus the crack orientation angle for various combinations of material and geometric parameters of the dissimilar media bonded through the thermoelastically graded interfacial zone. The dependence of the near-tip thermoelastic singular field on the degree of crack-surface partial insulation is also addressed.

Effect of Finite Cracking on the Magneto-electric Coupling Properties of Magneto-electro-elastic Composite Laminates

Magneto-electro-elastic composites can be developed in the form of a composite laminate by alternating the ferromagnetic layers and ferroelectric layers during stacking. Presence of interfacial crack may influence the magneto-electro-mechanical coupling behavior of magneto-electro-elastic materials considerably. This article establishes an analytical model for the analysis of interfacial crack problem in a layered magneto-electro-elastic medium. Unlike most existing studies, this article focuses on the magneto-electro-elastic laminate of finite size both in the thickness and the length directions. According to the boundary conditions, the mechanical, electric, and magnetic fields in the cracked magneto-electro-elastic laminate are formulated by singular integral method. The effect of interfacial cracking on the effective properties of the magneto-electro-elastic composites is studied. In addition, the crack tip field intensity factors are also given as they are not available in open literature.

An exact analysis for mode III cracks between two dissimilar magnetoelectroelastic layers

This paper develops a closed-form solution for an interface crack in a layered magnetoelectroelastic strip of finite width. The strip is subjected to anti-plane mechanical and in-plane electric and magnetic fields. Explicit expressions for the stress, electric, and magnetic fields, together with their intensity factors, are obtained for two extreme cases of an impermeable and a permeable cracks. The stress intensity factor does not depend on the electromagnetic boundary conditions assumed for the crack. However, the electrically and magnetically permeable boundary conditions on the crack profile have a significant influence on the crack-tip electromagnetic field intensity factors. Solutions for some special cases, such as a central crack, an edge crack, two symmetric collinear cracks, and a row of collinear interface cracks, are also obtained in closed forms.

On Thermal Shock Behavior of Functionally Graded Materials

Cracking is a common phenomenon for many media subjected to a rapid temperature change. Cracks in a medium may be in the form of either a single dominant crack or an array of multiple cracks. In this article a review is given of our work on this subject. Finite element method is used to obtain the solution of the transient temperature field in functionally graded materials (FGMs). A multiple surface cracking analysis is performed to study the thermal shock resistance behavior of a FGM, which is a ceramic/metal composite with its gradation characterized by the local volume fractions of the metal and ceramic phrases. Crack initiation behavior is discussed; and the crack tip field intensity factors and stresses are computed as functions of crack spacing, normalized time and crack depth. It is found that the thermal shock resistance of the FGM is significantly enhanced by multiple cracking. That is, thermal shock resistance is a monotonously increasing function of crack density, which is defined as the number of cracks per unit length and it increases with decreasing crack spacing. The single crack case represents the lower bound solution for the thermal shock resistance of FGMs. It is also found that the gradation of the FGM has little influence on the crack tip thermal stress intensity factors but has a strong influence on the thermal shock resistance. Multi-scale thermal shock resistance behavior of FGMs is discussed. Two critical size parameters, which control the applicability of the stress-based criterion and the fracture mechanics-based criterion for determination of thermal shock resistance of FGMs are investigated.

Multiple Surface Cracking of a Piezoelectric Layer Bonded to an Elastic Substrate Under Dynamic Anti-plane Electromechanical Impacts

In engineering applications, piezoelectric materials are usually bonded to elastic substrates to form so-called smart materials and structures. Such materials and structures can respond to external electromechanical environments, which are often transient. This article studies a periodic array of surface cracks in a piezoelectric layer bonded to an elastic substrate subjected to transient electromechanical loads. The anti-plane mechanical and in-plane electrical loads are investigated. A system of singular integral equations is formulated in terms of the crack surface anti-plane displacement and in-plane electrical potential. Effects of crack spacing and electromechanical coupling on the time-dependent crack tip field intensity factors are investigated in detail. Both a single piezoelectric layer and a piezoelectric layer bonded to an elastic substrate are analyzed for electrically impermeable and permeable crack assumptions.

Multiple surface cracks in a piezoelectric layer bonded to an elastic substrate under transient electromechanical loads

Piezoelectric materials are usually bonded to elastic substrates to form so-called smart material structures that are capable of sensing and actuating in a controlled manner in response to a stimulus. Such smart structures are often operated at transient electromechanical environments. In this paper, the problem of a periodic array of surface cracks in a piezoelectric layer bonded to an elastic substrate subjected to transient electromechanical loads is studied. A system of singular integral equations is formulated in terms of the crack surface displacement and electric potential. Numerical results include the time-dependent stress and electric displacement intensity factors. Effects of crack spacing and electromechanical coupling on the crack tip field intensity factors are investigated in details. Numerical examples are given for a single piezoelectric layer and a piezoelectric layer bonded to an elastic substrate, subjected to independent mechanical and electrical impact loads. Some useful conclusions are given.

Thermal shock resistance enhancement of functionally graded materials by multiple cracking

Multiple cracking is a common phenomenon for a medium subjected to a sudden temperature change. In this paper, a multiple surface cracking analysis is carried out to investigate the thermal shock resistance behavior of functionally graded materials (FGMs). The FGM is taken to be a ceramic/metal composite with its gradation characterized by the local volume fractions of metal and ceramic phases. The finite element method is used to obtain the solution of the crack problem. The crack tip field intensity factors as well as the stress are computed as functions of the crack spacing, the normalized time, and the crack depth. Crack initiation behavior is discussed. It is found that the thermal shock resistance of the FGM is significantly enhanced by multiple cracking. That is, the thermal shock resistance is a monotonically increasing function of the crack density (crack density is the number of cracks per unit length; crack density increases with decreasing crack spacing). A single crack represents the lower bound solution for the thermal shock resistance of FGMs. It is also found that the gradation of the FGM has little influence on the crack tip thermal stress intensity factors but has a strong influence on the thermal shock resistance. Multiscale behavior of the thermal shock resistance of FGMs is discussed. Two critical size parameters, which control the applicability of the stress-based criterion and the fracture mechanics-based criterion to the determination of the thermal shock resistance of FGMs, are explored.

A closed form solution to the antiplane problem of doubly periodic cracks of unequal size in piezoelectric materials

The problem of doubly periodic cracks in piezoelectric materials under far-field antiplane mechanical load and inplane electric load is investigated, where the fundamental cell contains four parallel and unequal cracks. By combining the elliptical function theory and conformal mapping technique, an exact solution in closed form to this problem is obtained. Based on this solution, the exact formulae for the crack tip field intensity factors and the effective electroelastic moduli of such cracked piezoelectric materials are derived. Numerical examples are graphically presented to show the interesting mechanical and electrical coupling phenomena induced by multicrack interactions, especially to reveal the shielding and intensifying effects of small cracks on the main cracks and to quantify the loss of electroelastic properties due to the presence of such arrays of microcracks. New and existing solutions, such as those for doubly periodic cracks in purely elastic materials and a single row or a single stack of periodical unequal cracks in piezoelectric materials are obtained as special cases of the present solution. The present solution can provide benchmark results for other numerical and approximate methods.

Thermal stress and intensity release in ferroelectric materials by multiple cracking

Release of transient thermal stress and intensity by periodic cracking in ferroelectric materials is studied. The problem is idealized to a plate containing a row of periodic surface cracks or internal cracks. The stress and electric displacement histories in an un-cracked plate are first calculated. These stresses and the electric displacements, with opposite signs, are used as the crack surface traction and electric displacement load to formulate the mixed boundary-value problem. Then an electromechanical model is developed for the solution of the mixed boundary-value problem. Both electrically impermeable cracks and electrically contacted cracks are considered. Results for the crack tip field intensity factors as well as the stress and electric displacement are computed as functions of the normalized time, the crack size, and crack spacing. Crack initiation and growth behavior are discussed. Lower bound solutions are obtained for the maximum thermal shock that the material can sustain without failure according to the stress-based criterion and the fracture mechanics-based criterion. (In this paper, a thermal shock condition is defined as a rapid temperature change in a medium. It is not necessary for the temperature to be very high.)

Exact electroelastic solutions for penny-shaped cracks under prescribed temperature or thermal flow

This letter provides an exact solution for a thermal penny-shaped crack in piezoelectric media. An arbitrarily distributed temperature field applied on the crack faces is considered first. Closed-form expressions for the stress and electric displacement intensity factors at the crack border are given. Electrically impermeable cracks, electrically permeable cracks, and the permittivity of the medium inside the cracks are considered. Then, crack tip field intensity factors for a prescribed thermal flow on the crack faces are obtained. The solutions are mainly provided for transversely isotropic materials, but results for isotropic materials and dielectrics are also given.

On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics

This paper investigates a cracked piezoelectric ceramic under remote electro-mechanical loads. The ideal crack boundary conditions for electrically impermeable and permeable crack assumptions, and the deformed crack with a yet-to-be-determined crack shape are considered. The last is referred to as the “natural boundary condition (NBC)”. Closed-form solutions to the crack-tip field intensity factors are obtained. The analysis shows that traditional approaches to the electric boundary conditions on the crack faces, that is, either the impermeable crack assumption or the permeable crack assumption, produce significantly different results for the crack-tip quantities such as electric displacement intensity factor, energy release rate and crack opening displacement. There are also considerable differences between the results obtained from traditional impermeable and permeable crack analyses and those obtained from the proposed NBC. The difference increases with applied electric loads.

FRACTURE OF A PIEZOELECTRIC MATERIAL STRIP UNDER STEADY STATE THERMAL LOAD

Thermal effects become important when the piezoelectric material has to be operated in either extremely hot or cold temperature environments. It is essential to know the interaction of mechanical defects with thermal changes. In this article, we examine the piezothermoelastic problem for a Griffith crack that is located in a piezoelectric material strip. The strip is infinite along the x-direction and has finite thickness in the y-direction. The crack plane is parallel to the boundary of the strip. The polarized axis of the piezoelectric material is either normal or parallel to the y-direction. The basic entities are the Fourier transform and singular integral equation techniques. The crack-tip fields are obtained. The variation in crack-tip field intensity factors due to changes of the crack size and location is studied for different poling directions.

Electroelastic properties of cracked piezoelectric materials under longitudinal shear

Abstract Analytical solutions are obtained to quantify the influence of cracks on electroelastic properties of piezoelectric materials containing doubly-periodic arrays of cracks. Both the rectangular and diamond-shaped arrays of cracks are considered. Solutions are obtained for the case of an antiplane shear load coupled with an in-plane electrical load. This study makes it possible to understand the multicrack interactions in piezoelectric solids and their effects on the fracture and electroelastic properties. The crack tip field intensity factors and the change in stored electroelastic energy due to the presence of many microcracks are calculated. These calculations enable the prediction of the effective elastic, piezoelectric and dielectric constants of a damaged piezoelectric material. The results of this work can be useful in developing a technique to determine the state of mechanical and electrical damage in piezoelectric materials.