Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions

Abstract

This paper investigates the influence of crack geometry, crack-face and loading conditions, and the permittivity of a medium inside the crack gap on intensity factors of planar and non-planar cracks in linear piezoelectric media. A weakly singular boundary integral equation method together with the near-front approximation is adopted to accurately determine the intensity factors. Obtained results indicate that the non-flat crack surface, the electric field, and the permittivity of a medium inside the crack gap play a crucial role on the behavior of intensity factors. The mode-I stress intensity factors (K_{\mathrm{I}}) for two representative non-planar cracks under different crack-face conditions are found significantly different and they possess both upper and lower bounds. In addition, K_{\mathrm{I}} for impermeable and semi-permeable non-planar cracks treated depends strongly on the electric field whereas those of impermeable, permeable, and semi-permeable penny-shaped cracks are identical and independent of the electric field. The stress/electric intensity factors predicted by permeable and energetically consistent models are, respectively, independent of and dependent on the electric field for the penny-shaped crack and the two representative non-planar cracks. Also, the permittivity of a medium inside the crack gap strongly affects the intensity factors for all crack configurations considered except for K_{\mathrm{I}} of the semi-permeable penny-shaped crack.

Keywords

crack-face conditions / intensity factors / non-flat cracks / permittivity / piezoelectric media / SGBEM

Introduction

Piezoelectric materials are widely employed in various engineering and industrial applications (e.g., actuators, sensors, transducers, etc.). Extensive studies have been carried out in the past decades to gain an in-depth understanding of responses and behavior of such materials in both macroscopic and microscopic scales [¹–¹⁰]. Besides the desirable mechanical-electrical coupling feature, the piezoelectric materials possess a major drawback in that they are quite brittle in nature and susceptible to fracture initiation, during either the manufacturing process or usage. Therefore, extensive investigations are required to gain insights into the fracture behavior of piezoelectric materials. One of the most challenging tasks in the modeling of fractures in piezoelectric media is the development of a physically suitable mathematical model capable of simulating actual conditions on the crack surface to accurately mimic the real physical phenomena. Several models of crack-face conditions have been proposed, and their pros and cons were extensively discussed. For instance, an electrically permeable condition was introduced by Parton [¹¹]; an electrically impermeable condition was proposed by Deeg [¹²]; a semi-permeable condition was employed by Hao and Shen [¹³]; and an energetically consistent boundary condition was proposed by Landis [¹⁴]. Among those electrical boundary conditions, an energetically consistent boundary condition seems to represent more realistic cracks than the permeable, impermeable, and semi-permeable models (an extensive review can be found in the work of Rungamornrat et al. [¹⁵]). Nevertheless, most previous publications focused mainly on cracks with simple geometries (viz., Griffith cracks, straight cracks, kinked cracks/branched cracks, inclined cracks, edge cracks, and inclined edge cracks) in two-dimensional bodies and planar cracks of various shapes embedded in an infinite and/or finite three-dimensional domains under simple loading conditions [¹⁶–⁴⁴], while relatively few studies relevant to the investigation of non-planar and multiple cracks have been reported, whereby most of them only concerned the development of computational tools for fracture analysis [⁴⁴–⁴⁹].

Selected useful results from previous investigations can be summarized as follows. For two-dimensional boundary value problems, Xu and Rajapakse [²⁵] proposed an analytical solution for an arbitrarily oriented elliptical void embedded in a piezoelectric infinite medium. By reducing void solutions to crack solutions, they found that under varying remote electric fields and with relatively small applied remote mechanical stress, the electric intensity factors of semi-permeable and permeable cracks are nearly identical, and both results are significantly different from the solution of impermeable cracks when a crack is not parallel to the poling direction of the piezoelectric material. In contrast, when the crack is parallel to the poling direction, the electric intensity factors of impermeable and semi-permeable cracks are identical, and the solution of permeable cracks vanishes. Finally, it is interesting to note that the electric intensity factors for three types of crack-face conditions (i.e., impermeable, permeable, and semi-permeable) are nearly identical when the remote electric field is close to zero. Such finding seems valid for any orientation of the poling direction considered. Wang and Jiang [²⁶] studied a slit crack embedded in a infinite piezoelectric medium that is subjected to far-field mechanical and electrical loadings. They found that for positive applied electric fields, the electric intensity factor of the semi-permeable crack varies from the permeable solution (the lower bound) to the impermeable solution (the upper bound) as the applied tensile stress increases. Later, Wang and Jiang [³¹] studied an arbitrarily oriented crack in an infinite medium subjected to mechanical and electrical loads. They found that there exists a discrepancy between the electric intensity factors computed from semi-permeable and permeable boundary conditions when the tensile mechanical load is high and the poling direction is perpendicular to the crack surfaces. This finding contradicts the results obtained by Xu and Rajapakse [²⁵]. However, Wang and Jiang [³¹] pointed out that the discrepancy exists due to the different levels of applied remote mechanical load. This implies that the levels of applied mechanical and electrical loads play an important role in the fracture behavior of piezoelectric cracked media. In addition, Wang and Jiang [³¹] also noted that for positive applied electric fields, high mechanical loads, and when the poling direction is perpendicular to the crack surfaces, the electric intensity factor for the impermeable and permeable conditions represent the upper and lower bounds of the result for the semi-permeable condition, respectively. Conversely, if the applied electric field is negative, such solution for impermeable and permeable conditions are the lower and upper bounds of the result for the semi-permeable condition, respectively, and it can be seen that the electric intensity factor for the permeable model is independent of the electric field. Subsequently, Denda and Mansukh [⁴⁶] investigated the interaction of multiple cracks in two-dimensional bodies using the boundary element method. However, their development is still limited to multiple straight cracks under impermeable and permeable boundary conditions.

For three-dimensional boundary value problems, Li and Lee [³²] studied a penny-shaped crack embedded in a transversely isotropic piezoelectric infinite medium subjected to remote uniform tensile stress and constant electric field using three crack-face conditions (i.e., permeable, impermeable, and semi-permeable conditions). They found that the mode-I stress intensity factors for a penny-shaped crack under such three crack models (i.e., permeable, impermeable, and semi-permeable conditions) are identical and depend only on mechanical loading. However, the electric intensity factors for those three crack models are completely different. The electric intensity factor for semi-permeable cracks depends on various parameters (i.e., mechanical loading, electrical loading, material properties, and permittivity of the medium inside the crack gap), whereas the electric intensity factor for the impermeable condition depends only on the electrical loading. In contrast, the electric intensity factor for the permeable solution does not depend on the electrical loading but depends on the material properties and the mechanical loading. These useful conclusions are in agreement with the results obtained by several investigators [¹⁸,²⁰,³³,³⁵].

Chiang and Weng [³⁵] investigated a similar problem. They proposed a critical state that consists of the critical stress and applied electric displacement. They pointed out that when the effective electric displacement is positive and the remote tensile stress is less than the critical stress, the electric intensity factor for the semi-permeable model falls between the result of the impermeable model (that is the upper bound) and the result of the permeable model (that is the lower bound). However, under a positive effective electric displacement and when the remote tensile stress is greater than the critical stress, the electric intensity factor for the impermeable model becomes the lower bound whereas the results for the permeable becomes the upper bound of the results for the semi-permeable condition.

Recently, Li et al. [³⁹] found that the mode-I stress intensity factor (as well as the electric intensity factor) for a penny-shaped crack under an energetically-consistent boundary condition depends on various parameters besides the mechanical loading conditions (i.e., mechanical loading, electric loading, material properties, and the permittivity of the medium inside the crack gap). This finding differs from the results obtained from the impermeable, permeable, and semi-permeable boundary conditions where the mode-I stress intensity factor depends only on the mechanical loading. Moreover, as pointed out by several investigators (e.g., Li and Chen [⁵⁰], Motola and Banks-Sills [⁵¹], and Li et al. [³⁹]), the non-zero mechanical traction proposed by Landis [¹⁴] that is induced by the crack opening displacement, the jump in the electric potential, and the permittivity of the medium inside the crack gap, tends to close the crack.

As mentioned above, most previous publications focused on the modeling and analysis of planar cracks. A question arises as to the difference between the fracture behavior of planar and non-planar cracks (or curvilinear cracks and/or multiple cracks) in such materials under the four types of crack-face conditions. Recently, Rungamornrat and Mear [⁴⁸] used a numerical technique called a weakly singular symmetric Galerkin boundary element method (SGBEM) to determine the stress intensity factors and electric intensity factor of an arbitrarily shaped crack embedded in three-dimensional, generally anisotropic, linear piezoelectric infinite media. They found that the fracture behavior of planar and non-planar cracks under the impermeable condition are different. More specifically, while the mechanical electrical loading can cause both stress intensity factors and electric intensity factors and vice versa factor, as observed for non-planar cracks (i.e., spherical cap cracks), there is no coupling effect due to the mechanical and electrical loads found in planar cracks (i.e., penny-shaped cracks) (e.g., [²⁰,³²,³⁵]). Nevertheless, their investigation was restricted to impermeable cracks in infinite media.

Sanz et al. [⁴⁷] and Solis et al. [⁴⁹], the pioneers in the analysis of intensity factors of transversely isotropic piezoelectric finite cracked bodies, revealed that the mode-I stress intensity factor and electric intensity factor can be produced by either uniform mechanical traction or uniform electrical loading. Again, these finding are different from those in planar cracks embedded in a infinite piezoelectric domain under the impermeable assumption. However, the works of Sanz et al. [⁴⁷] and Solis et al. [⁴⁹] were limited to the impermeable crack-face condition only. Phongtinnaboot et al. [⁵²] generalized the work of Rungamornrat and Mear [⁴⁸] to investigate the stress intensity factors and electric intensity factor of cracked finite piezoelectric domains. The electrically impermeable assumption was still adopted for the crack surface.

Most recently, Rungamornrat et al. [¹⁵] successfully generalized the work of Rungamornrat and Mear [⁴⁸] to analyze cracked piezoelectric infinite domains under the four types of electrical boundary conditions (i.e., permeable, impermeable, semi-permeable, and an energetically consistent boundary condition.) While they proposed a technique that can solve cracks of arbitrary shapes and under various electrical boundary conditions, they focused only on computational aspects and a full discussion of the role of electrical boundary conditions models was not presented. There are several crucial issues that arise and pose several nontrivial research problems in this specific area, and therefore, still require rigorous investigations, such as the difference between the fracture behavior of planar and non-planar cracks under the four types of crack-face conditions and a wide range of electrical/mechanical loads; the suitability of semi-permeable and energetically consistent conditions compared to extreme crack-face conditions such as permeable and impermeable models; the influence of the dielectric permittivity of a medium inside the crack gap, etc.

The current study is motivated by these outstanding challenges to fully explore the influence of several parameters such as crack geometry, crack-face conditions, loading conditions, and the permittivity of the medium inside the crack gap on the stress and electric intensity factors along the crack front of cracked piezoelectric bodies. Three representative configurations of cracks, including a penny-shaped crack, a spherical cap crack, and a cylindrical crack, and four models of electrical boundary conditions (viz., permeable, impermeable, semi-permeable, and an energetically consistent boundary conditions) are considered in the present study.

Problem description

Let us consider an isolated crack in a three-dimensional, homogeneous, linear piezoelectric, infinite medium \Omega as shown in Fig. 1. The crack is represented by two identical surfaces S^{+} and S^{-} with their corresponding outward unit normal vectors {\mathit{n}}^{+} and {\mathit{n}}^{-}, respectively. The body is assumed to be free of body force and body charge, and can be subjected to mechanical and electrical loadings on either the crack surface or the remote boundary. The elastic constants, the piezoelectric constants, and the dielectric permittivities of the constituent material are fully prescribed and denoted in a concise form by E_{iJKl}. The boundary conditions on the crack surface can be simulated by one of the following four models: impermeable, permeable, semi-permeable, and energetically consistent models. The first two models ignore the contribution of the permittivity of the medium inside the crack gap, whereas such parameters is fully prescribed and its role is considered in both semi-permeable and energetically consistent models, and are assumed to be prescribed and defined by {\mathit{\kappa }}_{\mathrm{c}}. It should be noted here that the notations and nomenclatures utilized by Rungamornrat et al. [¹⁵] are adopted in this study and detailed explanations will not be repeated here for brevity.

For impermeable cracks, the generalized tractions on the upper and lower crack surfaces, denoted by t_I^{+} and t_I^{-}, are fully prescribed whereas the jump in the crack-face generalized displacement, defined by \mathrm{\Delta }{u}_I\equiv u_I^{+}-u_I^{-}, are unknown a priori. This implies that the permittivity of the medium inside the crack gap is negligible or in other words the medium between the upper and lower crack surfaces is treated as an insulator (e.g., Deeg [¹²]). For permeable cracks, the mechanical tractions on both crack surfaces t_i^{+},\hspace{0.17em}{t}_i^{-}, the zero jump in the crack-face electrical potential \mathrm{\Delta }{u}_{\mathrm{4}}\equiv u_{\mathrm{4}}^{+}-u_{\mathrm{4}}^{-}=\mathrm{0}, and the zero sum of the crack-face electric charges \mathrm{\Sigma }{t}_{\mathrm{4}}\equiv t_{\mathrm{4}}^{+}+t_{\mathrm{4}}^{-}=\mathrm{0} are fully prescribed, whereas the jump in the crack-face displacement, \mathrm{\Delta }{u}_i\equiv u_i^{+}-u_i^{-} and the crack-face electric charge t_{\mathrm{4}}^{+}=-t_{\mathrm{4}}^{-} are unknown a priori. For this particular case, the permittivity of a medium inside the crack gap is assumed indefinitely large (e.g., Parton [¹¹]). For semi-permeable cracks, the mechanical tractions on both crack surfaces, t_i^{+},\hspace{0.17em}{t}_i^{-} are fully prescribed and the sum of the surface electric charges vanishes (i.e., \Sigma t_{\mathrm{4}}=\mathrm{0}) whereas the jump in the crack-face generalized displacement \mathrm{\Delta }{u}_I and the crack-face electric charge t_{\mathrm{4}}^{+}=-t_{\mathrm{4}}^{-} are unknown a priori and also satisfy the following relation

(1)t_{\mathrm{4}}^{+}\mathrm{\Delta }{u}_i{n}_i^{+}=-{\mathit{\kappa }}_{\mathrm{c}}\mathrm{\Delta }{u}_{\mathrm{4}},

where {\mathit{\kappa }}_{\mathrm{c}} is the dielectric permittivity of the medium inside the crack gap (e.g., Hao and Shen [¹³]). It should be noted that if the permittivity inside the crack gap is significantly large (i.e., {\mathit{\kappa }}_c\to \infty), the electrically semi-permeable condition will converge to the conducting boundary condition (e.g., [³²,³⁵]). Here and in what follows, the lower case indices range from 1 to 3; the upper case indices range from 1 to 4, and repeated indices imply a summation over their range. Note that the superscript “+” and “-” are employed to emphasize the quantities associated with the upper crack surface S^{+} and the lower crack surface S^{-}, respectively.

For energetically consistent cracks (e.g., Landis [¹⁴]), the mechanical tractions on both crack surfaces t_i^{+},\hspace{0.17em}{t}_i^{-} can be partitioned into two parts: {\sigma }_i^{+}+{\tau }_i^{+} and {\sigma }_i^{-}+{\tau }_i^{-} where \{{\sigma }_i^{+},\mathrm{\hspace{0.17em}}{\sigma }_i^{-}\} and \{{\tau }_i^{+},\mathrm{\hspace{0.17em}}{\tau }_i^{-}\} are the normal and shear tractions, respectively. For this crack model, the mechanical shear tractions \{{\tau }_i^{+},{\tau }_i^{-}\} are fully prescribed whereas the jump in the crack-face generalized displacement \mathrm{\Delta }{u}_I, the mechanical normal tractions \{{\sigma }_i^{+},{\sigma }_i^{-}\}, and the surface electric charge t_{\mathrm{4}}^{+}=-t_{\mathrm{4}}^{-} are unknown a priori but satisfy the condition (1) and the following relation

(2){\sigma }_j^{+}=\left(\mathrm{1}/\mathrm{2}\right){\mathit{\kappa }}_{\mathrm{c}}{n}_j^{+}{\left(\mathrm{\Delta }{u}_{\mathrm{4}}\right)}^{\mathrm{2}}/{\left(\mathrm{\Delta }{u}_i{n}_i^{+}\right)}^{\mathrm{2}}.

Extensive details of the electrical boundary conditions used in the present investigation can also be found in the work of Rungamornrat et al. [¹⁵].

Formulation

A set of basic equations for linear piezoelectric materials and the boundary integral equations governing an isolated crack of arbitrary shape embedded in a three-dimensional, generally anisotropic, linear piezoelectric infinite medium under various types of crack-face conditions are summarized below.

Basic field equations

The electro-mechanical behavior of a piezoelectric medium is governed by the classical theory of linear piezoelectricity. The basic field equations for a vanishing distributed source can be expressed in a concise form as

(3){\displaystyle \frac{\partial {\sigma }_{iJ}}{\partial x_i}}=\mathrm{0},\mathrm{}{\sigma }_{iJ}=E_{iJKl}{\displaystyle \frac{\partial u_K}{\partial x_l}},

where {\sigma }_{iJ} represents the generalized stress (i.e., the stress tensor {\sigma }_{ij} and the electric induction vector {\sigma }_{i\mathrm{4}}); u_K represents the generalized displacement (i.e., the displacement u_k and the electric potential u_{\mathrm{4}}), and E_{iJKl} denotes the generalized moduli (i.e., elastic constants E_{ijkl}, piezoelectric constants E_{i\mathrm{4}kl}, and dielectric permittivities -E_{i\mathrm{44}l}). The generalized traction at any point on a smooth surface with a unit normal \mathit{n} is given by t_J={\sigma }_{iJ}{n}_i (i.e., the mechanical traction t_j={\sigma }_{ij}{n}_i and the surface electric charge t_{\mathrm{4}}={\sigma }_{i\mathrm{4}}{n}_i).

Boundary integral equations

Boundary integral equation methods have been found efficient for modeling cracks in homogenous, linear, infinite media because in the absence of a distributed source, the key governing equation involves only unknowns on the crack surface. This positive feature renders the solution methodology computationally efficient than domain-based techniques such as the conventional finite element and finite difference methods. In the present study, the well-known, symmetric Galerkin boundary element method (SGBEM) developed by Rungamornrat and Mear [⁴⁸] and Rungamornrat et al. [¹⁵] is selected as key computational procedure in the analysis.

For isolated cracks in a linear piezoelectric, infinite medium that is free of remote loading, the generalized stress {\sigma }_{lK} at any interior point \mathit{x} can be expressed in terms of the crack-faced data via the following standard integral relation

(4){\sigma }_{lK}\left(\mathit{x}\right)=-{\displaystyle \underset{S^{+}}{\int }{S}_{lK}^J(\mathit{\xi }-\mathit{x})\Sigma t_J\left(\mathit{\xi }\right)}\mathrm{d}A\left(\mathit{\xi }\right)+{\displaystyle \underset{S^{+}}{\int }{\Sigma }_{iJ}^{lK}(\mathit{\xi }-\mathit{x})n_i\left(\mathit{\xi }\right)}\mathrm{\Delta }{u}_J\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right),

where \Sigma t_J=t_J^{+}+t_J^{-} and \mathrm{\Delta }{u}_J=u_J^{+}-u_J^{-} denote the sum of the crack-face generalized traction and the jump in the crack-face generalized displacement, respectively, and S_{iJ}^P(\mathit{\xi }-\mathit{x}) and {\Sigma }_{iJ}^{lK}(\mathit{\xi }-\mathit{x}) are known fundamental solutions for an un-cracked whole space. While Eq. (4) can be directly used to form a boundary integral equation sufficient for determining the unknown relative crack-face generalized displacement, the resulting integral equations contain both the strongly singular kernel S_{iJ}^P(\mathit{\xi }-\mathit{x}) and the hyper-singular kernel {\Sigma }_{iJ}^{lK}(\mathit{\xi }-\mathit{x}) and this, as a direct consequence, requires nontrivial numerical treatments in the solution procedure (e.g., [⁵³–⁵⁷]). To avoid such difficulties, an alternative singularity-reduced boundary integral equation for the generalized crack-face traction proposed by Rungamornrat et al. [¹⁵] and Rungamornrat and Mear [⁴⁸] can be utilized. To construct such weakly singular integral equation, they employed the following special decompositions of the kernels S_{iJ}^P(\mathit{\xi }-\mathit{x}) and {\Sigma }_{iJ}^{lK}(\mathit{\xi }-\mathit{x}):

(5)S_{iJ}^P(\mathit{\xi }-\mathit{x})=H_{iJ}^P(\mathit{\xi }-\mathit{x})+{ϵ}_{ism}{\displaystyle \frac{\partial G_{mJ}^P(\mathit{\xi }-\mathit{x})}{\partial {\xi }_s}},

(6){\Sigma }_{iJ}^{lK}(\mathit{\xi }-\mathit{x})=-E_{iJKl}\delta (\mathit{\xi }-\mathit{x})+{ϵ}_{ism}{\displaystyle \frac{\partial }{\partial {\xi }_s}}{ϵ}_{lrt}{\displaystyle \frac{\partial }{\partial {\xi }_r}}{C}_{mJ}^{tK}(\mathit{\xi }-\mathit{x}),

where {ϵ}_{ism} is the standard alternating symbol; \delta (\mathit{\xi }-\mathit{x}) is a Dirac-delta function centered at \mathit{x}; H_{iJ}^P(\mathit{\xi }-\mathit{x})=-{\delta }_{JP}({\xi }_i-x_i)/\mathrm{4}\mathrm{\pi }{r}^{\mathrm{3}} with {\delta }_{JP} denoting the generalized Kronecker-delta and r=\left|\right|\mathit{\xi }-\mathit{x}\left|\right|; and G_{mJ}^P(\mathit{\xi }-\mathit{x}) and C_{mJ}^{tK}(\mathit{\xi }-\mathit{x}) are given, for generally anisotropic piezoelectric materials, by

(7)G_{mJ}^P(\mathit{\xi }-\mathit{x})={\displaystyle \frac{{ϵ}_{mqa}{E}_{qJKl}}{\mathrm{8}{\mathrm{\pi }}^{\mathrm{2}}r}}{\displaystyle \underset{\mathit{z}·\mathit{r}=\mathrm{0}}{\propto }{(z,z)}_{KP}^{-\mathrm{1}}{z}_a{z}_l\mathrm{d}s\left(z\right)},

(8)\begin{array}{l}{C}_{mJ}^{tK}(\mathit{\xi }-\mathit{x})={\displaystyle \frac{A_{mtsl}^{KJPQ}}{\mathrm{8}{\mathrm{\pi }}^{\mathrm{2}}r}}{\displaystyle \underset{\mathit{z}·\mathit{r}=\mathrm{0}}{\propto }{(z,z)}_{PQ}^{-\mathrm{1}}{z}_s{z}_l\mathrm{d}s\left(z\right),}\hfill \\ A_{mtsl}^{KJPQ}={ϵ}_{aum}{ϵ}_{adt}\left\{E_{uKPs}{E}_{dJQl}-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_{dJKu}{E}_{lPQs}\right\},\hfill \end{array}

in which \mathit{z} is a unit vector, {(\mathit{z},\mathit{z})}_{PQ}=z_m{E}_{mPQn}{z}_n, {(\mathit{z},\mathit{z})}^{-\mathrm{1}} is the inverse of (\mathit{z},\mathit{z}); and the line integrals are carried out along a unit circle on the plane normal to the vector \mathit{r}=\mathit{\xi }-\mathit{x}. It is remarked that the two-point functions G_{mJ}^P(\mathit{\xi }-\mathit{x}) and C_{mJ}^{tK}(\mathit{\xi }-\mathit{x}) are only singular at \mathit{\xi }=\mathit{x} of \mathcal{O}(\mathrm{1}/r).

By applying Eqs. (5) and (6) to Eq. (4) together with carrying out the integration by parts via Stokes’ theorem, it results in the singularity-reduced integral relation for the generalized stress

(9)\begin{array}{l}{\sigma }_{lK}\left(\mathit{x}\right)={ϵ}_{lrt}{\displaystyle \frac{\partial }{\partial x_r}}{\displaystyle \underset{S^{+}}{\int }{G}_{tK}^J(\mathit{\xi }-\mathit{x})\Sigma t_J\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right)}\hfill \\ \mathrm{}+{ϵ}_{lrt}{\displaystyle \frac{\partial }{\partial x_r}}{\displaystyle \underset{S^{+}}{\int }{C}_{mJ}^{tK}(\mathit{\xi }-\mathit{x})}{D}_m\mathrm{\Delta }{u}_J\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right)\hfill \\ \mathrm{}-{\displaystyle \underset{S^{+}}{\int }{H}_{lK}^J(\mathit{\xi }-\mathit{x})\Sigma t_J}\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right),\hfill \end{array}

Finally, a weakly singular, weak-form boundary integral equation for the generalized traction reported in Rungamornrat et al. [¹⁵] and Rungamornrat and Mear [⁴⁸] can be obtained from Eq. (9) by a limiting process, a standard weighted residual technique, and the integration by parts via Stokes’ theorem. Its final explicit form, in terms of the crack-face data, is given by

- {\displaystyle \underset{S^{+}}{\int }{D}_t{\tilde{\nu }}_K\left(\mathit{y}\right){\displaystyle \underset{S^{+}}{\int }{C}_{mJ}^{tK}(\mathit{\xi }-\mathit{y})D_m\mathrm{\Delta }{u}_J\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{y}\right)}}

={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \underset{S^{+}}{\int }{\tilde{\nu }}_K\left(\mathit{y}\right)\mathrm{\Delta }{t}_K\left(\mathit{y}\right)\mathrm{d}A\left(\mathit{y}\right)}

+{\displaystyle \underset{S^{+}}{\int }{D}_t{\tilde{\nu }}_K\left(\mathit{y}\right){\displaystyle \underset{S^{+}}{\int }{G}_{tK}^J(\mathit{\xi }-\mathit{y})\Sigma t_J\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{y}\right)}}

(10)+{\displaystyle \underset{S^{+}}{\int }{\tilde{\nu }}_K\left(\mathit{y}\right){\displaystyle \underset{S^{+}}{\int }{H}_{iK}^J(\mathit{\xi }-\mathit{y})n_i\left(\mathit{y}\right)\Sigma t_J\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{\xi }\right)\mathrm{d}A\left(\mathit{y}\right)}},

where {\tilde{\nu }}_K is a sufficiently smooth test function; D_m(·)=n_i{ϵ}_{ism}\partial (·)/\partial {\xi }_s is a surface differential operator; \Sigma t_K=t_K^{+}+t_K^{-} and \mathrm{\Delta }{t}_K=t_K^{+}-t_K^{-} are the sum of and the jump in the generalized tractions across the crack surface; and \{C_{mJ}^{tK},G_{tK}^J,n_i{H}_{iK}^J\} are known kernels. It should be noted that the development of the boundary integral equation in Eq. (10) is independent of the crack configurations and the mechanical and electrical boundary conditions on the crack surface and, as a result, it is valid for the four types of cracks described above. In addition, since it contains complete information on the crack surface (i.e., \mathrm{\Delta }{t}_K,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\Sigma t_K, and \mathrm{\Delta }{u}_K), the boundary integral equation in Eq. (10) is sufficient for solving the primary unknowns on the crack surface when all prescribed data and known relations associated with each type of cracks are properly incorporated. From a theoretical and computational point of view, all the double-surface integrals in Eq. (10) exist in an ordinary sense since all involved kernels \{C_{mJ}^{tK},\mathrm{\hspace{0.17em}}{G}_{tK}^J,\mathrm{\hspace{0.17em}}{n}_i{H}_{iK}^J\} are only weakly singular and of \mathcal{O}(\mathrm{1}/r). This feature allows unknown data on the crack surface to be approximated by continuous interpolation functions. Although Eq. (10) is applicable only to an infinite domain that is free from the electro-mechanical remote loading, the integration of the remote loading can readily be handled using the principle of superposition (see clear explanations in Rungamornrat et al. [¹⁵]).

Solution technique

By adopting a standard discretization procedure commonly employed in the weakly singular SGBEM, the boundary integral equation for the generalized traction can be transformed into a discrete system of linear algebraic equations

(11)\mathit{C}\mathrm{\Delta }\mathit{U}+(\mathit{G}+\mathit{H})\Sigma \mathit{T}+\mathit{L}\mathrm{\Delta }\mathit{T}=\mathit{0},

where matrices \mathit{C}, \mathit{G}, and \mathit{H} result directly from the discretization of the double surface integrals containing the kernels C_{mJ}^{tK}, G_{tK}^J, and H_{iK}^J, respectively; the matrix \mathit{L} corresponds to a single surface integral containing the jump in the crack-face generalized traction; \mathrm{\Delta }\mathit{U} is a vector of nodal quantities associated with the jump in the crack-face generalized displacement, and \Sigma \mathit{T} and \mathrm{\Delta }\mathit{T} are vectors of nodal quantities of the sum of and the jump in the crack-face generalized tractions, respectively. The construction of all matrices \{\mathit{C}\mathrm{,}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathit{G},\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathit{H},\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathit{L}\} follows directly the work of [¹⁵,⁴⁸]. The approximation of the near-front, jump in the crack-face generalized displacement is enhanced by using special crack-tip elements along the crack front [¹⁵,⁴⁸,⁵⁸] and the interpolation technique is employed to efficiently evaluate involved kernels for general anisotropic materials [¹⁵,⁴⁸]. In addition to allowing the use of relatively coarse meshes in the discretization while yielding sufficiently accurate results, the former also provides the direct means to extract the intensity factors in terms of extra nodal data solved along the crack front. Details of such relevant solution procedure can be found in the work of Rungamornrat and Mear [⁴⁸] and Rungamornrat et al. [¹⁵].

To obtain the unknown crack-face data, the system of linear Eq. (11) is solved together with the prescribed crack-face boundary conditions. For both electrically permeable and impermeable cracks, the final system of governing equations after incorporating the crack-face conditions is linear and its solution can be readily obtained via a selected efficient linear solver. On the contrary, the final governing equations for the case of electrically semi-permeable and energetically consistent cracks are nonlinear due to the nonlinear crack-face conditions and they are efficiently solved via standard Newton iterative scheme. Details of the solution procedure can be found in the work of Rungamornrat et al. [¹⁵]. Once a set of nodal data \mathrm{\Delta }\mathit{U} is determined, it is directly used to post-process for both the stress and electric intensity factors along the crack front.

Numerical results and discussions

First, computed results for certain simple cases such as a penny-shaped crack under simple electrical/mechanical loads are benchmarked with available reference solutions to validate the formulation of the governing integral equations and numerical implementations of the weakly singular SGBEM and the post-process for the stress intensity factors and the electric intensity factor. After verifying the proposed technique, we then thoroughly investigated the influence of non-flat geometry of the crack surface, remote electrical load, and permittivity of the medium inside the crack gap on the intensity factors along the crack front of representative non-flat cracks.

In the numerical study, a series of meshes is constructed first to investigate the convergence of the computed intensity factors and a sufficiently fine mesh is then chosen to ensure the convergence of the solution. In the discretization, the majority of the crack surface is discretized by standard 6-node and 8-node isoparametric elements, whereas the region adjacent to the crack boundary is discretized by 9-node special crack-tip elements [¹⁵,⁴⁸,⁵⁸]. The representative linear piezoelectric materials selected in the analysis are transversely isotropic and all material constants are taken to be those of PZT-4, as shown in Table 1.

Verification

To verify the implemented weakly singular SGBEM, consider first a penny-shaped crack of radius a embedded in a piezoelectric infinite medium made of PZT-4, as shown in Fig. 2(a). The medium is loaded by a uniform remote tension {\sigma }_{\mathrm{0}} that is varied from 10 to 100 \mathrm{MPa}, whereas the electric field E_{\mathrm{0}} is fixed at 2.5 \mathrm{MV/m}, and the dielectric permittivity of the medium inside the crack gap is chosen as {\mathit{\kappa }}_{\mathrm{c}}=\mathrm{5}{\mathit{\kappa }}_{\mathrm{0}} where {\mathit{\kappa }}_{\mathrm{0}}=\mathrm{8.85}\times {\mathrm{10}}^{-\mathrm{12}}\mathrm{\hspace{0.17em}}\mathrm{C/(Vm)} is the permittivity of the air. The poling direction and the axis of material symmetry are taken along the x_{\mathrm{3}}-axis. The mesh shown in Fig. 2(b), after the convergence is confirmed, is used in the verification. Numerical results for the normalized mode-I stress intensity factor and electric intensity factor are reported in Figs. 3(a) and 3(b), respectively, along with the analytical solutions [³²,³⁹].

It is shown that the numerical results obtained are in good agreement with the analytical solution for all four crack-face condition considered. In addition, it can be inferred from Fig. 3(a) that the mode-I stress intensity factors K_{\mathrm{I}} for the permeable, impermeable, and semi-permeable crack models are identical for the entire range of the applied uniform remote tension {\sigma }_{\mathrm{0}} treated. These findings are in agreement with the work of previous investigators (e.g., Li and Lee [³²] and Chiang and Weng [³⁵]) who pointed out that the mode-I stress intensity factors for the three crack-face conditions depend only on the mechanical loading but are independent of electrical loading and the permittivity of the medium inside the crack gap.

In addition, it can be concluded from Fig. 3(a) that the results predicted by the energetically consistent model is less than those predicted by the other three crack models for the entire range of {\sigma }_{\mathrm{0}}. This is because, as pointed out by several investigators [⁵⁰,⁵¹], the unknown mechanical traction acting on both the crack surfaces that is induced by the opening displacement of the crack, the jump in the electric potential, and the permittivity of the medium inside the crack gap, try to close the crack. However, when the uniform remote tension increases, the mode-I stress intensity factor predicted by the energetically consistent model tends to approach the results of the other three crack models.

In contrast to K_{\mathrm{I}}, the electric intensity factor K_{\mathrm{IV}} strongly depends on the electrical boundary conditions adopted on the crack surface. Results in Fig. 3(b) show that the electric intensity factors of the impermeable and permeable cracks serve, respectively, as the upper and lower bounds of the results generated by the semi-permeable and energetically consistent models for almost the entire range of {\sigma }_{\mathrm{0}} considered. However, when the uniform remote tension {\sigma }_{\mathrm{0}} is relatively small, the electric intensity factor K_{\mathrm{IV}} for the energetically consistent model is lower than that for the permeable model. This implies that the level of applied mechanical load has a significant influence on the lower bound of K_{\mathrm{IV}} for the penny-shaped crack subjected to uniform loading. As a result, one can see that the lower bound of K_{\mathrm{IV}} switches from the energetically consistent solution to the permeable solution as the applied mechanical loading increases. It is remarked also that the electric intensity factor K_{\mathrm{IV}} for the semi-permeable and energetically consistent cracks are nearly identical and both results approach the result tend to for the impermeable model when the applied remote stress increases.

Influence of crack geometry

To explore the behavior of the intensity factors along the boundary of non-flat cracks, two representative cracks, one for a spherical cap crack (shown in Fig. 4) and the other for a cylindrical crack (shown in Fig. 5), are considered. The latter and the former are chosen to represent cracks possessing the curvature in a single direction and two directions, respectively. In the numerical study, following parameters are employed: (i) various half-subtended angles of the spherical cap crack and the cylindrical crack (defined further below in Sections 5.2.1 and 5.2.2) ranging from \mathrm{5}° to \mathrm{90}° are considered; (ii) the uniform remote tension and the uniform electric field are taken as 50 MPa and 2.5 MV/m, respectively; (iii) the permittivity of the medium inside the crack gap for both semi-permeable and energetically consistent crack-face conditions are taken as {\mathit{\kappa }}_{\mathrm{c}}=\mathrm{5}{\mathit{\kappa }}_{\mathrm{0}} where {\mathit{\kappa }}_{\mathrm{0}}=\mathrm{8.85}\times {\mathrm{10}}^{-\mathrm{12}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{C/(Vm)} denotes the permittivity of the air.

Spherical cap crack

Consider a spherical cap crack embedded in a transversely isotropic piezoelectric infinite medium of PZT-4, as shown schematically in Fig. 4(a). The medium is subjected to a uniform remote tension {\sigma }_{\mathrm{0}} and a uniform electric field E_{\mathrm{0}} along the x_{\mathrm{3}}-axis. The axis of material symmetry and the poling direction are chosen to direct along the x_{\mathrm{3}}-axis. The geometry of the crack surface is defined by

(12)x_{\mathrm{1}}=R\sin \gamma \cos \beta ,\mathrm{\hspace{0.17em}}\hspace{0.17em}{x}_{\mathrm{2}}=R\sin \gamma \sin \beta ,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{x}_{\mathrm{3}}=R\cos \gamma ,

where R denotes the radius of the spherical cap crack, \beta \in \left[\mathrm{0},\mathrm{2}\mathrm{\pi }\right], and \gamma \in \left[\mathrm{0},\theta \right] with \theta denoting the half subtended angle of the spherical surface. It is remarked that the spherical cap crack considered here can be viewed as a penny-shaped crack of radius a=R\theta bended over the surface of a sphere of radius R. In the following investigation, a mesh of the crack surface containing 144 elements is adopted, as shown in Fig. 4(b).

The normalized non-zero stress intensity factors K_{\mathrm{I}} and K_{\mathrm{II}}, and the electric intensity factor K_{\mathrm{IV}} under the four crack-face conditions are shown in Figs. 6(a)–6(c), respectively. It is found that when the half subtended angle of the crack surface is relatively small, the behavior of the intensity factors of the spherical cap crack is similar to that of the penny-shaped crack. However, when the half subtended angle is sufficiently large, such behavior of the special cap crack is significantly different from that of the penny-shaped crack.

The results reported in Fig. 6(a) indicate that when the half subtended angle of the crack surface is relatively small (e.g., \theta =\mathrm{5}°,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{10}°,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{15}°), the distribution of K_{\mathrm{I}} under the four crack models is similar to that of K_{\mathrm{I}} for the penny-shaped crack. In particular, K_{\mathrm{I}} of the impermeable, permeable, and semi-permeable cracks are nearly identical and serve as the upper bound, whereas K_{\mathrm{I}} of the energetically consistent model is less than those of the other three crack models and, therefore, serves as the lower bound solution. However, when the half subtended angle is sufficiently large, the distributions of K_{\mathrm{I}} for the impermeable, permeable, and semi-permeable crack models are obviously different and the solution predicted by the permeable model forms the upper bound, whereas K_{\mathrm{I}} for the impermeable model becomes the lower bound instead of the result generated by the energetically consistent model. It can be concluded that the lower bound switches from the energetically consistent solution to the impermeable solution as the half subtended angle increases.

It is important to remark that while the mode-I stress intensity factors of the penny-shaped crack under the impermeable, permeable, and semi-permeable crack-face conditions are identical as indicated in Refs. [³²,³⁹], such results for the spherical cap crack are not identical when the half subtended angle is sufficiently large. The discrepancy depends primarily on the half subtended angle. This implies that the non-planar feature of the crack surface has a significant influence on the mode-I stress intensity factor of the three crack models. According to the numerical results presented later in Section 5.3, it was also found that the non-flat crack surface leads to the dependence of the mode-I stress intensity factor K_{\mathrm{I}} on the electric field for both the impermeable and semi-permeable models, while the electric field has no effect on K_{\mathrm{I}} of the penny-shaped crack under the impermeable, permeable, and semi-permeable crack-face conditions. This conclusion is supported by the significant difference among results generated by the three crack models (i.e., impermeable, permeable, and semi-permeable models) when the half subtended angle of the spherical cap crack is sufficiently large.

While the lower bound of the mode-I stress intensity factor for the spherical cap crack switches when the half subtended angle is sufficiently large, the upper and lower bounds of the mode-II stress intensity factor for the spherical cap crack do not. Results in Fig. 6(b) indicates that the mode-II stress intensity factor K_{\mathrm{II}} for the impermeable and energetically consistent cracks serve, respectively, as the upper and lower bounds for (nearly) the entire range of the normalized half subtended angle \overline{\theta }, except when \theta is either very small or large (i.e., \theta =\mathrm{5}° and \theta =\mathrm{90}°). In particular, the numerical result indicates that when the half-subtended angle is relatively small (e.g., \theta =\mathrm{5}°), results generated by the four crack-face models are nearly identical and almost zero. This finding is similar to that for the penny-shaped crack subjected to uniform remote loading where the mode-II stress intensity factors for the four crack-face models vanish. However, such result becomes clearly distinct when the half subtended angles become larger (e.g., \theta =\mathrm{40}°,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{45}°,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{50}°,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{55}°) for the four crack-face models, and it can be seen that the impermeable solution is significantly different from those of the other three crack-face models. Finally, it should be noted that when the half subtended angle is relatively large (i.e., \theta =\mathrm{90}°), the mode-II stress intensity factor K_{\mathrm{II}} for the impermeable, permeable, and semi-permeable cracks are nearly identical.

Similar to the mode-I stress intensity factor but unlike the mode-II stress intensity factor, the lower bound of the electric intensity factor K_{\mathrm{IV}} for the spherical cap crack switches as the half subtended angle increases. Results in Fig. 6(c) indicate that the electric intensity factors K_{\mathrm{IV}} for the four crack-face models are different, and that the permeable solution represents the lower bound when the half subtended angle is relatively small. However, when the half subtended angle becomes larger (e.g., \theta =\mathrm{65}°,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{70}°), the energetically consistent solution becomes the lower bound instead for the permeable solution. Moreover, it is obvious that when the half subtended angle of the spherical cap crack is relatively small, K_{\mathrm{IV}} of the spherical cap crack for the four crack-face models possesses a similar behavior to that of the penny-shaped crack, where the permeable solution of K_{\mathrm{IV}} serves as the lower bound. However, K_{\mathrm{IV}} of the spherical cap crack and the penny-shaped crack obviously exhibit different behavior when the half subtended angle is sufficiently large, and it can be seen that the lower bound switches from the permeable solution to the energetically consistent solution. Finally, it can be inferred from Figs. 3(b) and 6(c) that K_{\mathrm{IV}} for the impermeable model is still the upper bound for both the penny-shaped crack and the spherical cap crack for the entire range of the half subtended angle \theta considered.

Cylindrical crack

Consider a cylindrical crack with the radius of curvature R embedded in a transversely isotropic piezoelectric infinite medium made of PZT-4, as illustrated in Fig. 5(a). The piezoelectric medium is subjected to a uniform remote tension {\sigma }_{\mathrm{0}} and a uniform electric field E_{\mathrm{0}} along the x_{\mathrm{3}}-axis. The axis of the material symmetry and the poling direction coincide with the x_{\mathrm{3}}-axis.

The geometry of the crack surface is defined by

(13)x_{\mathrm{1}}=R\sin \gamma ,\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{x}_{\mathrm{2}}\in [-d,d],\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}{x}_{\mathrm{3}}=R\cos \gamma ,

where \gamma \in \left[\mathrm{0},\theta \right], \theta denotes the half subtended angle of the crack surface, and d=R\sqrt{{\theta }^{\mathrm{2}}-{\gamma }^{\mathrm{2}}}. It is worth noting that the cylindrical crack considered here can be obtained directly by folding a penny-shaped crack of radius a=R\theta over the surface of a cylinder of radius R. In the present study, a mesh of the cylindrical crack containing 144 elements is used, as shown in Fig. 5(b). The normalized stress intensity factors \{K_{\mathrm{I}},K_{\mathrm{II}},K_{\mathrm{III}}\} and the electric intensity factor K_{\mathrm{IV}} at three different points (i.e., top, middle, and bottom points) located on the cylindrical crack as shown in Fig. 5(c) are reported in Figs. 7(a)–7(d) for four crack-face models.

Results in Fig. 7(a) indicate that the mode-I stress intensity factor K_{\mathrm{I}} for the top point is greater than those for the middle and bottom points of the cylindrical crack, when the half-subtended angle is relatively large. Moreover, it can be inferred from Fig. 7(a) that K_{\mathrm{I}} under the four types of crack-face conditions obtained from the top and bottom points of the cylindrical crack are similar to the results for the penny-shaped crack and spherical cap crack, whereas such intensity factors obtained for the middle point of the cylindrical crack are different from those for the penny-shaped crack and the spherical cap crack. In particular, it was found that for the top point, K_{\mathrm{I}} for the impermeable, permeable, and semi-permeable cracks are nearly identical, whereas that for the energetically consistent model is less than those for the other three crack-face models and, therefore, serves as the lower bound for the entire range of the normalized half-subtended angle \overline{\theta }. While K_{\mathrm{I}} at the top point of the cylindrical crack obtained from the four crack models are similar to that of the penny-shaped crack, the results at the bottom point has the similar behavior to that of the spherical cap crack. It was also found that K_{\mathrm{I}} at the bottom point of the cylindrical crack obtained from the impermeable, permeable and semi-permeable models are not identical, and that the permeable solution serves as the upper bound, whereas the lower bound of K_{\mathrm{I}} switches from the energetically consistent model to the impermeable model when the subtended angle of the cylindrical crack increases. In contrast to the top and bottom points of the cylindrical crack, K_{\mathrm{I}} at the middle point under the four crack-face conditions are different from the results obtained for the penny-shaped crack and the spherical cap crack. In particular, it can be seen that results for the impermeable, permeable, and semi-permeable crack-face models are not nearly identical as those observed at the top point and that the permeable solution serves as the upper bound, whereas the energetically consistent solution represents the lower bound for the entire range of the normalized half-subtended angle \overline{\theta }.

While the mode-II stress intensity factor K_{\mathrm{II}} for the penny-shaped crack subjected to uniform loading identically vanishes along the crack front, it is nonzero and completely different for the top, middle, and bottom points of the cylindrical crack. Results in Fig. 7(b) indicate that K_{\mathrm{II}} obtained at the bottom point is greater than those at the middle and top points of the cylindrical crack. In addition, it is obvious that when the half-subtended angle is relatively small (e.g., \theta =\mathrm{5}°), the mode-II stress intensity factors at all three points of the cylindrical crack for all four crack-face models are nearly identical and almost vanish. However, when the half-subtended angle increases, the stress intensity factors K_{\mathrm{II}} for the four crack-face models are different and their lower and upper bounds can be obviously seen. More specifically, it can be inferred from Fig. 7(b) that when the half-subtended angle is sufficiently large, the impermeable crack model yields the upper bound for K_{\mathrm{II}} at all three points of the cylindrical crack, whereas the permeable and energetically consistent crack models provide the lower bound of K_{\mathrm{II}} at the top point and the remaining two points (i.e., middle and bottom points), respectively. This indicates that the non-planar feature of the crack surface has the significant influence on the lower bound and the value of the mode-II stress intensity factor for the cylindrical crack. As a result, it can be seen that K_{\mathrm{II}} obtained from the four crack models are different and that the upper and lower bounds exist when the half-subtended angle becomes larger. In addition, the lower bounds of K_{\mathrm{II}} are not the same for the top point and the remaining two points of the cylindrical crack. Finally, it can be observed that when the half-subtended angle of the cylindrical crack is sufficiently large, K_{\mathrm{II}} at the top point of the cylindrical crack obtained from the impermeable model is positive and quite different from those obtained from the other three models which clearly attain the negative values.

While the mode-III stress intensity factors K_{\mathrm{III}} at the top and bottom points of the cylindrical crack for the four crack-face conditions vanish due to the symmetry, it does not vanish at the middle point of the cylindrical crack and possesses the same negative sign. Results in Fig. 7(c) also indicate that K_{\mathrm{III}} obtained from the semi-permeable and energetically consistent crack-face models fall between the impermeable and permeable solutions which constitute the upper and lower bounds, respectively. Moreover, it was also observed that when the half-subtended angle is relatively small (i.e., \theta =\mathrm{5}°), K_{\mathrm{III}} for the four crack-face conditions tends to zero. This finding is expected since the mode-III stress intensity factor vanishes for any planar cracks (e.g., penny-shaped cracks) subjected to uniform loading. Finally, it can be pointed out that, similar to the mode-II stress intensity factor K_{\mathrm{II}}, K_{\mathrm{III}} for the impermeable model is significantly different from those generated by the other three crack-face models when the half-subtended angle becomes larger.

The electric intensity factor K_{\mathrm{IV}} at the top, middle, and bottom points of the cylindrical crack are shown in Fig. 7(d). It was found that K_{\mathrm{IV}} at the top point is greater than those at the middle and bottom points. Moreover, obtained numerical results also indicate that the behavior of K_{\mathrm{IV}} at the top and bottom points of the cylindrical crack is similar to that for the penny-shaped crack and the spherical cap crack, respectively, whereas K_{\mathrm{IV}} at the middle point is significantly different from that of the penny-shaped and spherical cap cracks. More specifically, it was found that at the top point of the cylindrical crack, the electric intensity factor K_{\mathrm{IV}} for the four crack models are completely different according to the electrical boundary conditions adopted on the crack surface. It can be seen that the impermeable and permeable crack models yield, respectively, the upper and lower bounds of K_{\mathrm{IV}} to the solutions from the semi-permeable and energetically consistent models. This finding is similar to the results obtained for the penny-shaped crack. Obtained results also indicate that the electric intensity factor K_{\mathrm{IV}} at the bottom point of the cylindrical crack obtained from the four crack models is different from that of the penny-shaped crack, but is similar to that of the spherical cap crack. In addition, the impermeable crack model yields the upper bound for K_{\mathrm{IV}} for the entire range of the normalized half-subtended angle \overline{\theta }, whereas the lower bound fluctuates between solutions generated by the permeable and energetically consistent crack models as \overline{\theta } increases. Finally, it can be seen that when the half-subtended angle is large (e.g., \theta =\mathrm{75}°,\mathrm{\hspace{0.17em}}\mathrm{80}°,\mathrm{\hspace{0.17em}}\mathrm{85}°,\mathrm{\hspace{0.17em}}\mathrm{90}°), the electric intensity factor K_{\mathrm{IV}} of the permeable crack is nearly identical to that of the semi-permeable crack. In contrast to the top and bottom points, K_{\mathrm{IV}} obtained at the middle point of the cylindrical crack is different from those of the penny-shaped and spherical cap cracks. It was found that the permeable crack provides the lower bound solution for K_{\mathrm{IV}} for almost the entire range of the normalized half-subtended angle \overline{\theta } except when the subtended angle is significantly large (e.g., \theta =\mathrm{85}°,\mathrm{\hspace{0.17em}}\mathrm{90}°). Solutions generated by the permeable and energetically consistent crack models are nearly identical and serve as the lower bound. It is interesting to note that though the lower bound of K_{\mathrm{IV}} for all three points of the cylindrical crack are completely different; the impermeable crack model always yields the upper bound of K_{\mathrm{IV}} for all those three points.

Influence of remote electrical loading

To explore the influence of remote electrical loading on the intensity factors along the crack front of non-flat cracks, the same spherical cap crack and cylindrical crack embedded in a transversely isotropic piezoelectric infinite medium made of PZT-4 as shown in Figs. 4 and 5, respectively, are considered. In the numerical study, following parameters are employed: (i) the half-subtended angle for both cracks is taken as \theta =\mathrm{45}°; (ii) the uniform remote tension {\sigma }_{\mathrm{0}} is taken to be 50 MPa whereas the uniform electric field E_{\mathrm{0}} varies from -4.5 to 4.5 MV/m; (iii) the permeable, impermeable, semi-permeable, and energetically consistent crack models are considered; (iv) the permittivity of the medium inside the crack gap for both semi-permeable and energetically consistent crack models are taken to be {\mathit{\kappa }}_{\mathrm{c}}=\mathrm{5}{\mathit{\kappa }}_{\mathrm{0}} where {\mathit{\kappa }}_{\mathrm{0}}=\mathrm{8.85}\times {\mathrm{10}}^{-\mathrm{12}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{C/(Vm)}. In the analysis of the spherical cap crack and cylindrical crack, the meshes shown in Figs. 4(b) and 5(b) are employed, respectively. In addition, results for a penny-shaped crack of radius a=R\theta are also obtained and used in the comparison and discussion.

Spherical cap crack

The normalized non-zero stress intensity factors K_{\mathrm{I}}, K_{\mathrm{II}}, and the electric intensity factor K_{\mathrm{IV}} under four types of crack-face conditions are shown in Figs. 8(a)–8(c), respectively. It can be seen from the results reported in Fig. 8(a) that the mode-I stress intensity factor K_{\mathrm{I}} for the impermeable and semi-permeable spherical cap cracks depends strongly on the electric field E_{\mathrm{0}}. These findings are in contrast with those observed for planar cracks (i.e., a penny-shaped crack) where K_{\mathrm{I}} for impermeable, permeable, and semi-permeable cracks are identical and independent of the electric field. Moreover, it was also observed that K_{\mathrm{I}} for permeable planar and non-planar cracks are independent of the electric field. This implies that the curvature of non-planar cracks leads to the dependence on the electric field for K_{\mathrm{I}} for both the impermeable and semi-permeable crack-face conditions. However, such effect plays no role on the value of K_{\mathrm{I}} for the permeable condition. Moreover, it can be inferred from Fig. 8(a) that the mode-I stress intensity factor generated from the energetically consistent model strongly depends on the electric field for both the penny-shaped crack and the spherical cap crack. In addition, obtained numerical results also indicate that the energetically consistent crack-face model yields the lower bound of K_{\mathrm{I}} for the penny-shaped crack subjected to the uniform loading for the entire range of the electric field considered, while the lower bound of the spherical cap crack for the positive electric field fluctuates between the impermeable and energetically consistent solutions by showing a turning point at about E_{\mathrm{0}}=\mathrm{2.5} MV/m. Moreover, it was also observed that K_{\mathrm{I}} for all four crack-face conditions are nearly identical when the electric field vanishes, i.e., E_{\mathrm{0}}=\mathrm{0} MV/m. This finding is equally true for both the penny-shaped and spherical cap cracks, as shown in Fig. 8(a). Similar to the mode-I stress intensity factor, Fig. 8(b) indicates that the mode-II stress intensity factor K_{\mathrm{II}} for the permeable spherical cap crack is independent of the electric field, whereas K_{\mathrm{II}} of the impermeable, semi-permeable, and energetically consistent spherical cap cracks depend strongly on the electric field.

While the fracture behavior of planar cracks (i.e., penny-shaped cracks) and non-planar cracks (i.e., spherical cap cracks) are different for the mode-I stress intensity factor when using the impermeable and semi-permeable conditions, Fig. 8(c) indicates that the electric intensity factor K_{\mathrm{IV}} for the spherical cap crack is similar to K_{\mathrm{IV}} of the penny-shaped crack under all four crack-face conditions. More specifically, K_{\mathrm{IV}} is independent of the electric field for both the permeable penny-shaped and spherical cap cracks, whereas K_{\mathrm{IV}} for the impermeable, semi-permeable, and energetically consistent crack-face models depend strongly on the electric field for both crack configurations.

According to the numerical results for the penny-shaped and spherical cap cracks subjected to remote uniform loading, the non-zero intensity factors \{K_{\mathrm{I}},K_{\mathrm{II}},K_{\mathrm{IV}}\} for the spherical cap crack and the non-zero intensity factors \{K_{\mathrm{I}},K_{\mathrm{IV}}\} for the penny-shaped crack are all independent of the electric field under the permeable condition.

Cylindrical crack

The normalized stress intensity factors K_{\mathrm{I}}, K_{\mathrm{II}}, and K_{\mathrm{III}} and the electric intensity factor K_{\mathrm{IV}} under four types of crack-face conditions are shown in Figs. 9(a)–9(d), respectively. It can be seen from the results reported in Fig. 9(a) that the mode-I stress intensity factor K_{\mathrm{I}} at the top point is greater than those at the middle and bottom points of the cylindrical crack. Moreover, it can be inferred from Fig. 9(a) that K_{\mathrm{I}} at the top point of the impermeable, permeable, and semi-permeable cracks are nearly identical, whereas that of the energetically consistent crack strongly depends on the electric field. These findings are similar to the case of the penny-shaped crack subjected to uniform loading as shown in Fig. 8(a). In contrast, the behavior of K_{\mathrm{I}} obtained from the three crack models for the remaining two points (i.e., middle and the bottom points) of the cylindrical crack is different from that of the penny-shaped crack but similar to that of the spherical cap crack. Figure 9(a) indicates that the impermeable, permeable, and semi-permeable models yield slightly different K_{\mathrm{I}} at the middle point but clearly different at the bottom point of the cylindrical crack. In addition, it can be seen that K_{\mathrm{I}} for the impermeable and semi-permeable cases depend on the electric field whereas K_{\mathrm{I}} for the permeable case is independent of the electric field. Finally, it should be noted that K_{\mathrm{I}} generated from the energetically consistent model strongly depends on the electric field for all three points of the cylindrical crack, similar to that of the penny-shaped crack.

While the mode-II stress intensity factor K_{\mathrm{II}} vanishes identically for the penny-shaped crack subjected to uniform loading, it is non-zero for the top, middle, and bottom points of the cylindrical crack. Figure 9(b) indicates that K_{\mathrm{II}} at the bottom point is greater than those at the middle and top points. Moreover, it can be inferred from Fig. 9(b) that K_{\mathrm{II}} for the impermeable, semi-permeable, and energetically consistent models depend on the electric field and vary according to the electrical boundary condition adopted on the crack surface (except when the electric field is equal to zero where K_{\mathrm{II}} for the four crack models are nearly identical), whereas K_{\mathrm{II}} for the permeable model is independent of the electric field. These findings are similar for all three points of the cylindrical crack, as shown in Fig. 9(b). While the mode-III stress intensity factor K_{\mathrm{III}} at the top and bottom points of the cylindrical crack vanish, Fig. 9(c) shows that it does not vanish at the middle point. It was observed that, similar to the mode-II stress intensity factor, K_{\mathrm{III}} at the middle point of the cylindrical crack for the impermeable, semi-permeable, and energetically consistent models depend on the electric field, whereas that for the permeable model is clearly independent of the electric field.

The electric intensity factor K_{\mathrm{IV}} obtained at the top, middle, and bottom points of the cylindrical crack are shown in Fig. 9(d). It was observed that, for the positive electric field, K_{\mathrm{IV}} at the top point is greater than those at the middle and bottom points. Moreover, it was found that the behavior of K_{\mathrm{IV}} at all three points of the cylindrical crack is similar to that of the spherical cap crack and the penny-shaped crack. More specifically, the numerical results indicate that K_{\mathrm{IV}} for the impermeable, semi-permeable, and energetically consistent models depend strongly on the electric field, whereas K_{\mathrm{IV}} at all three points of the cylindrical crack for the permeable model is independent of the electric field.

Finally, it is important to note that the intensity factors \{K_{\mathrm{I}},K_{\mathrm{II}},K_{\mathrm{III}},K_{\mathrm{IV}}\} at the top, middle, and bottom points of the cylindrical crack under the permeable crack-face condition are independent of the electric field. This similar conclusion was also observed for the case of a penny-shaped crack and a spherical cap crack.

Influence of permittivity of the medium inside the crack gap

To explore the influence of the permittivity of the medium inside the crack gap on the intensity factors along the crack front for both the spherical cap crack and the cylindrical crack (see Figs. 4 and 5) under semi-permeable and energetically consistent crack-face conditions, following parameters are used in the investigation: (i) the variation in the permittivity of the medium inside the crack gap, defined by {\mathit{\kappa }}_{\mathrm{c}}=\alpha {\mathit{\kappa }}_{\mathrm{0}} where \alpha is the relative permittivity and {\mathit{\kappa }}_{\mathrm{0}}=\mathrm{8.85}\times {\mathrm{10}}^{-\mathrm{12}}\mathrm{\hspace{0.17em}}\mathrm{\hspace{0.17em}}\mathrm{C/(Vm)} is the permittivity of the air, is considered by varying \alpha; (ii) the half-subtended angle is taken as \theta =\mathrm{45}°; (iii) the uniform remote tension and the uniform electric field are taken as 50 MPa and 2.5 MV/m, respectively; and (iv) the impermeable and permeable crack-face conditions are also considered in addition to the semi-permeable and energetically consistent cases to represent the cases \alpha =\mathrm{0} and \alpha =\infty, respectively. Again, the meshes shown in Figs. 4(b) and 5(b) are employed in the analysis. Results for a penny-shaped crack of radius a=R\theta are also obtained for the comparison purpose.

Spherical cap crack

The influence of the permittivity of the medium inside the spherical cap crack on the non-zero intensity factors, \{K_{\mathrm{I}},K_{\mathrm{II}},K_{\mathrm{IV}}\} is demenstrated in Figs. 10(a)–10(c), respectively. It was observed that the mode-I stress intensity factors K_{\mathrm{I}} of the penny-shaped crack under the semi-permeable crack-face condition is identical and clearly independent of the medium inside the crack gap and identical to those for the permeable and impermeable cases, while K_{\mathrm{I}} of the same crack predicted by the energetically consistent model strongly depends on the medium inside the crack gap and it serves as the lower bound solution for the entire range of the relative permittivity \alpha.

Unlike the penny-shaped crack, Fig. 10(a) shows that the results for the spherical cap crack obtained from the impermeable, permeable, and semi-permeable models are not identical and that the results for the semi-permeable crack depend on the medium inside the crack gap, whereas the results for the impermeable and permeable cases are completely different. Also, one can see that K_{\mathrm{I}} for the semi-permeable crack varies from the impermeable solution to the permeable solution (which clearly represents the upper bound solution) as the value of the permittivity of the medium increases. This implies that the non-planar feature of the crack surface has the significant influence on the mode-I stress intensity factor of the three crack models. In addition, the numerical results also indicate that the lower bound of K_{\mathrm{I}} for the special cap crack switches from the energetically consistent solution to the impermeable solution as the relative permittivity increases.

Figures 10(b) and 10(c) indicate that the mode-II stress intensity factor K_{\mathrm{I}\mathrm{I}} and the electric intensity factor K_{\mathrm{IV}} of the spherical cap crack under four crack-face conditions are significantly different according to the electrical boundary condition adopted on the crack surface. Moreover, it can be observed from obtained results that K_{\mathrm{II}} and K_{\mathrm{IV}} are strongly dependent on the permittivity of a medium inside the crack gap for both the semi-permeable and energetically consistent models. Such the results for both crack-face conditions approach the permeable solution as the permittivity of the medium increases. In addition, it can be seen that the impermeable solution represents the upper bound for both K_{\mathrm{II}} and K_{\mathrm{IV}} while the energetically consistent solution is the lower bound for the mode-II stress intensity factor K_{\mathrm{II}} and the permeable solution serves as the lower bound for the electric intensity factor K_{\mathrm{IV}}. Finally, it is important to note that the distribution of K_{\mathrm{IV}} for the spherical cap crack under four crack-face conditions is similar to that of the penny-shaped crack.

Cylindrical crack

The normalized stress intensity factors K_{\mathrm{I}}, K_{\mathrm{II}}, K_{\mathrm{III}}, and the electric intensity factor K_{\mathrm{IV}} under four types of crack-face conditions are reported in Figs. 11(a)–11(d), respectively. Results in Fig. 11(a) indicate that the mode-I stress intensity factor K_{\mathrm{I}} at the top point is greater than the middle and bottom points of the cylindrical crack. It can be also observed K_{\mathrm{I}} for the semi-permeable and energetically consistent models depends strongly on the permittivity of a medium inside the crack gap. In addition, the energetically consistent solution is significantly different from those of the other three crack-face models and it serves as the lower bound for the entire range of the relative permittivity \alpha. This similar behavior is observed for the top, middle, and bottom points of the cylindrical crack. It is interesting to remark that the upper bounds of K_{\mathrm{I}} at the three points of the cylindrical crack are not identical. More specifically, results at the top point for the impermeable, permeable, and semi-permeable crack-face conditions are nearly identical and serve as the upper bound, whereas the permeable solutions represent the upper bounds for both the middle and bottom points of the cylindrical crack.

Finally, it can be concluded that the lower and upper bounds of K_{\mathrm{I}} for the penny-shaped crack, spherical cap crack, and cylindrical crack under four crack-face conditions are different. For the penny-shaped crack, the three crack-face models (i.e., impermeable, permeable, and semi-permeable models), and the energetically consistent crack-face model yield the upper and lower bound solutions, respectively, whereas, for the spherical cap crack, the permeable solution serves as the upper bound while the lower bound switches from the energetically consistent solution to the impermeable solution as the permittivity of the medium inside the crack gap increases. However, for the cylindrical crack, the energetically consistent solution serves as the lower bound for all three points. One can also see that the semi-permeable solution is independent of the permittivity of the medium inside the crack gap for the penny-shaped crack but strongly depends on such quantity for both the spherical cap crack and the cylindrical crack.

Results in Fig. 11(b) indicate that the mode-II stress intensity factor K_{\mathrm{II}} at the bottom point is greater than those at the middle and top points of the cylindrical crack. Similar to the mode-I stress intensity factor, K_{\mathrm{II}} for the semi-permeable and energetically consistent condition is also strongly dependent on the permittivity of medium inside the crack gap. Results for both models vary from the impermeable solution to the permeable solution as the permittivity of the medium inside the crack gap increases. This finding is true for all three points of the cylindrical crack. In addition, one can observe that the impermeable solutions represent the upper bound for all three points of the cylindrical crack whereas the permeable solution serves as the lower bound for the top point and the energetically consistent solution represents the lower bound for the middle and bottom points for the cylindrical crack. This implies that the distribution of K_{\mathrm{II}} for the middle and bottom points of the cylindrical crack is similar to that of the special cap crack. As a result, one can see that the upper and lower bounds of the middle and bottom points of the cylindrical crack are the same as those for the spherical cap crack.

From results of the non-zero mode-III stress intensity factor K_{\mathrm{III}} at the middle point of the cylindrical crack shown in Fig. 11(c), the semi-permeable and energetically consistent solutions, again, depend strongly on the permittivity of the medium inside the crack gap and both results varies from the impermeable solution to the permeable solution as the permittivity of the medium increases. Clearly, the impermeable solution serves as the upper bound whereas the permeable solution represents the lower bound for the entire range of the relative permittivity \alpha.

Results reported in Fig. 11(d) indicate that the electric intensity factor K_{\mathrm{IV}} at the top point is greater than those at the middle and bottom points of the cylindrical crack. Moreover, it was found that the behavior of K_{\mathrm{IV}} at the three points of the cylindrical crack is similar to that of the penny-shaped crack and the spherical cap crack. More specifically, K_{\mathrm{IV}} generated from the semi-permeable and energetically consistent models strongly depends on the relative permittivity \alpha of the medium inside the crack gap and both results vary from the impermeable solution that is the upper bound to the permeable solution that is the lower bound as \alpha increases.

Conclusions

In the present study, we have fully investigated the influence of half subtended angle on the intensity factors of three crack configurations including the penny-shaped crack, the spherical cap crack, and the cylindrical crack under impermeable, permeable, semi-permeable, and energetically consistent crack-face conditions. Obtained results have indicated that the mode-I stress intensity factor K_{\mathrm{I}} and the electric intensity factor K_{\mathrm{IV}} at the top and bottom points of the cylindrical crack have the similar behavior to that of the penny-shaped crack and the spherical cap crack, respectively, whereas \{K_{\mathrm{I}},K_{\mathrm{IV}}\} at the middle point of the cylindrical crack have different trend when compared with those for the penny-shaped crack and the spherical cap crack. However, for the relatively small half-subtended angle (e.g., \theta =\mathrm{5}°,\mathrm{10}°), the characteristics of \{K_{\mathrm{I}},K_{\mathrm{IV}}\} for the spherical cap crack and the cylindrical crack are similar to those for the penny-shaped crack.

For the penny-shaped crack subjected to a uniform loading, the mode-II and mode-III stress intensity factors vanish identically for all four types of crack-face conditions considered due to the symmetry. In contrast, due to the non-flat feature of the crack surface, the mode-II stress intensity factor of the spherical cap crack and the cylindrical crack and the mode-III stress intensity factor of the cylindrical crack do not vanish along the crack front. It was also found that when the half-subtended angle of the spherical cap crack and the cylindrical crack are sufficiently large, the non-zero mode-II and mode-III stress intensity factors under the four crack-face conditions are different according to the electrical boundary condition adopted on the crack surface. As a result, the upper and lower bounds of the mode-II and mode-III stress intensity factors could be seen. In particular, the impermeable solution represents the upper bound of the mode-II stress intensity factor of the spherical cap crack and the three representative points for the cylindrical crack, and as well as the upper bound of the mode-III stress intensity factor at the middle point of the cylindrical crack. However, the energetically consistent solution represents the lower bound of the mode-II stress intensity factor of the spherical cap crack and the middle and bottom points of the cylindrical crack, whereas the permeable solution serves as the lower bound of the mode-II and mode-III stress intensity factors at the top and middle points of the cylindrical crack, respectively.

Results from the investigation of the influence of the electric field on the intensity factors for the three representative crack configurations under four different crack-face conditions have indicated that the non-zero intensity factors \{K_{\mathrm{I}},K_{\mathrm{II}},K_{\mathrm{III}},K_{\mathrm{IV}}\} under the permeable and energetically consistent crack-face conditions for both planar (i.e., penny-shaped crack) and non-planar cracks (i.e., spherical cap and cylindrical cracks) were independent and dependent of the electric field, respectively. In addition, it was found that the influence of the electric field on the electric intensity factor K_{\mathrm{I}\mathrm{V}} for the penny-shaped crack was similar to that for the spherical cap crack and cylindrical crack under the four crack-face conditions. In particular, K_{\mathrm{IV}} obtained from the impermeable, semi-permeable, and energetically consistent models depends strongly on the electric field, whereas K_{\mathrm{IV}} of the permeable crack is independent of the electric filed. On the contrary, the behavior of the mode-I stress intensity factor K_{\mathrm{I}} under the impermeable and semi-permeable conditions for the planar crack (i.e., penny-shaped) and the non-planar cracks (i.e., spherical cap crack, and the middle and bottom points of the cylindrical crack) are different. In particular, K_{\mathrm{I}} of the impermeable and semi-permeable spherical cap crack depends strongly on the electric field whereas such results for the penny-shaped crack are independent of the electric field. This implies that the non-flat crack surface renders the dependence on the electric field for K_{\mathrm{I}} for both the impermeable and semi-permeable crack-face conditions.

Finally, it can be concluded that all the intensity factors \{K_{\mathrm{I}},K_{\mathrm{II}},K_{\mathrm{III}},K_{\mathrm{IV}}\} for the three representative crack configurations under the energetically consistent crack-face condition strongly depend on the permittivity of the medium inside the crack gap. The same behavior can also be observed for the semi-permeable crack-face condition, except for the mode-I stress intensity factor K_{\mathrm{I}} of the penny-shaped crack that is found independent of the medium inside the crack gap.

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