1 Introduction

With the development in the modern electromechanical devices and smart embedded structures, piezoelectric ceramics are extensively used in such devices/structures as an essential part of transducers, actuators, sensors, resonators, and ultrasonic generators. However, these materials are brittle in nature, and due to their low fracture toughness, they are always susceptible to fracture or failure under high mechanical and electrical loadings. Moreover, the presence of defects, such as voids, inclusions, and cracks, induces the failure of such materials and reduces the lifetime and applications of these devices. Therefore, to increase the performance and utility of these materials, a thorough investigation of fracture behaviour of such materials is required.

The study of fracture mechanics for brittle materials is quite old, and to analyze the fracture mechanics problems, various mathematical techniques were applied^[1-4]. Many researchers^[5-12] have extended the concept of linear elastic fracture mechanics to develop the fracture criteria for these materials within the theory of linear piezoelectricity. However, the fracture parameters obtained on the basis of linear piezoelectricity were not found in agreement with the experimental evidences given by Park and Sun^[8]. As per the experimental results, the crack growth depends upon the direction of the applied electric field. It increases if the applied electrical field and poling are in the same direction and decreases if they are in the opposite directions. This discrepancy was resolved by Gao et al.^[13] who considered the role of electric nonlinearity near the crack-tip and defined the local and global energy release rate concept. For the same, they proposed a strip polarization saturation (PS) model in analogy with Dugdale's^[1] elastoplastic model (where the plastic zone was limited to a strip just in front of the crack-tip and the strip was bounded by the yield stress) by considering the piezoelectric materials as mechanically brittle and electrically ductile. In their proposed model, an electric yielding zone near the crack-tip was approximated to a line segment along with a constant normal electric displacement equal to its saturated value imposed on the strip. Thereafter, various researchers^[14-24] applied the PS model to study the various types of crack (s) problems in two-dimensional (2D) piezoelectric media.

Ru^[14] derived the closed form solution of intensity factors using a complex variable technique for a generalized case of PS model in 2D piezoelectric media. He generalized the impermeable PS model by considering the electric displacement discontinuity across the saturated strip. On similar basis of the PS model, Zhang et al.^[16] proposed and studied the strip dielectric break-down model in 2D piezoelectric materials. Fan et al.^[18] presented detailed numerical studies for the PS and dielectric breakdown (DB) models in 2D finite piezoelectric media using the nonlinear hybrid extended displacement discontinuity method. Fan et al.^[20] applied the extended Stroh formalism and distributed the dislocation method to obtain the analytical solution for a semipermeable crack in 2D piezoelectric media based on the PS model. Bhargava and Jangid^[23-24] applied the complex variable and Stroh formalism technique to analyze the strip-saturation model solution for two collinear cracks in 2D piezoelectric media considering impermeable and semi-permeable conditions. Bhargava and Hasan^[25], Hasan and Akhtar^[26], and Hasan^[27] proposed a modified Dugdale model for multiple collinear cracks by quadratically varying normal cohesive stress distribution over the rims of yield zones. Explicit expressions for yield zones and crack opening displacements (CODs) were also established using the complex variable technique.

In view of the above literature on the PS model and modified Dugdale's model, the authors propose a modified PS model (a linearly varying electric displacement based nonlinear fracture model) in 2D piezoelectric media to consider the effect of electric nonlinearity at the crack-tip. Here, in the proposed model, the unknown saturated zone in front of the crack-tip is approximated as a strip ahead of the crack-tip, and the unknown saturated condition on the strip is replaced by linearly varying normal saturated electric displacement value. Further, this problem of nonlinear fracture model in 2D piezoelectric media is mathematically modeled into a non-homogenous Riemann-Hilbert problem by using the complex variable technique and extended Stroh formalism. To analyze the effect of the modified PS model on the fracture parameters, the closed form expressions for the saturated zone length, the crack COD, the crack opening potential (COP), and the local stress intensity factor (SIF) are obtained. The results of these parameters are plotted and discussed under the arbitrary poling, semipermeable crack face, and different sets of loading conditions.

2 Modified PS model

This model is proposed considering the following points observed from the literature survey carried on the PS model:

(ⅰ) A significant role of electric nonlinearity in defining the new fracture parameters (local energy release rate and local intensity factor) for 2D cracked piezoelectric materials which have shown the behaviour of these parameters in agreement with the experimental results.

(ⅱ) Piezoelectric materials may be considered as mechanical brittle and electrically ductile, and hence the electric displacement saturation zone develops near the crack-tip under the high electric field.

(ⅲ) It is cumbersome to determine the exact zone behaviour (the shape of saturation zone and saturation condition of electric displacement on the zone) near the crack-tip, and it is incorporated into the linear theory of piezoelectricity.

Therefore, considering the aforementioned points, here too similar to the PS model, the unknown saturated zone is approximated with a line segment in front of the crack-tip. However, the unknown saturated condition defined over the strip is modified from a constant normal saturated value to a linearly varying normal electric displacement saturated value, i.e., ( is the saturated zone length). The modified saturated condition defined here is bounded over the strip and is always less than or equal to the electric displacement saturated value, i.e., D_s satisfying the condition as explained by Ru^[14].

The basic objective of this model is to study the effect of linearly varying saturation condition (imposed on the strip) on the strip zone size, the COD, the COP, and the applied electrical loadings under the influence of arbitrary poling direction and semipermeable crack face condition. In addition to this, the authors are interested in investigating the comparative study of the modified PS model and PS model.

3 Basic equations for piezoelectric media

The fundamental equations and the boundary conditions for linear piezoelectric media are defined as below.

3.1 Constitutive equations

(1)

(2)

3.2 Gradient equations

(3)

3.3 Equilibrium equations

(ⅰ) The stress in absence of body forces,

(4)

(ⅱ) Maxwell equations in absence of charge,

(5)

where σ_ij, γ_ij, D_i, E_i, u_i(i, j = 1, 2, 3), and ψ denote the stress, the strain, the electric displacement, the electric field, the mechanical displacement component, and the electric potential, respectively. c_ijkl, e_kij, and k_kl stand for the elastic constants, the piezoelectric constants, and dielectric constants, respectively.

3.4 Crack face boundary conditions

There are mainly three crack face boundary conditions defined in the literature, i.e., impermeable, permeable, and semipermeable. In 1994, Hao and Shen^[28] proposed the semipermeable crack face boundary conditions by considering the permittivity of the air as a medium inside the crack. Mathematically, these conditions are expressed as

(6)

where the indices "+" and "-" represent the upper and lower crack surfaces, respectively, is the electrical potential jump, and is the COD. κ_c is the permittivity of medium between the crack faces.

The semipermeable electric boundary conditions can be degenerated to the impermeable one when κ_c=0, and to the permeable one when the jump in electric potential vanishes.

4 Solution for 2D piezoelectric domain using Stroh formalism

For a 2D problem, all field variables are taken to depend on independent variables x and y. Introduce the generalized displacement vector u as

(7)

Equations (1) -(5) satisfy Eq. (7) for an arbitrary analytic function f(x+py) if

(8)

which has a non-trivial solution only if

(9)

Considering the generalized case of poling direction (a direction of poling which makes an angle φ with the axis perpendicular to crack), the material matrices W, R, and Q are given by

(10)

respectively, where the transformed material constants are obtained from the generalized material constants matrices (C, e, ε) using the following relations:

(11)

Here, the rotation matrices are

Now, if the eight roots of Eq. (9) are denoted by p_α and p_α, ζ is obtained from the following standard eigen-equation:

(12)

where

According to the Stroh formulation, the general solution satisfying Eqs. (1) -(5) may be written as

(13)

(14)

where

The column vector of matrix is related to the column vector of matrix M=(m₁, m₂, m₃, m₄) as

(15)

and Φ is the generalized stress function such that

(16)

5 Mathematical formulation of modified PS model in 2D piezoelectric media

An infinite transversely isotropic piezoelectric 2D domain is considered for the analysis in the xOy-plane. A center crack is taken along the x-axis occupying the interval [-a, a]. The saturated zone which develops near the crack-tip due to the high electric field is represented as a line segment along the x-axis satisfying a < |x| < c₁. The semipermeable crack face condition is taken for the analysis. The remote boundary of the plate is prescribed under a uniform tensile loading σ₂₂^∞ and an in-plane electric displacement D₂^∞. The entire configuration is schematically presented in Fig. 1. The physical boundary conditions stated above may be written as

(ⅰ) and

(ⅱ) and

(ⅲ) and

(ⅳ) and

The continuity of defined on the whole real axis relates the complex function as

(17)

According to Muskhelishvili^[2], Eq. (17) may be written as

(18)

Let

(19)

where F(z) is the vector complex function of the inhomogeneous field.

Using the boundary condition (ⅰ) in Eq. (14), we have

(20)

and the jump displacement vector Δ_{u, 1} may be written as

(21)

where in which

Now, consider

(22)

Equation (20) can be represented in its component as

(23)

(24)

From Eqs. (23) and (24), we have

(25)

where

The solution to Eq. (25) together with the single-valuedness condition of the mechanical displacement is

(26)

Now, from Eqs. (23) and (24),

(27)

Similarly, using the boundary condition (ⅲ), one can write

(28)

The solution to Eq. (28) together with the single-valuedness condition of the electrical displacement is obtained as

(29)

5.1 Derivation of fracture parameters

In this section, the expressions are determined for the length of saturation zone, the SIF, the CODs, and the COP.

5.1.1 Saturation zone

The electric displacement is determined using Eq. (14) as

(30)

Taking the fourth component of the aforementioned equation, we have

(31)

Substiuting the values of Σ_k(x) and Σ₄(x) from Eqs. (26) and (29), we have

(32)

The saturation zone lengths are now obtained by extending the Dugdale hypothesis to the electric displacement to remain finite at the point c₁ of the domain. This leads to the following relation in the saturation zone length to half crack length:

(33)

5.1.2 SIF

The field components at any point (|x| < a) on the x-axis of the domain are given as

(34)

From Eq. (34), one can obtain σ₂₁(x) as

(35)

The local SIF K_I(a) is determined at the tip x=a as

(36)

Since the local SIF is evaluated at the actual (mechanical) crack-tip, it is independent of the normal electric displacement saturation condition imposed on the strip zone. Hence, the analytic expression for the local SIF obtained in Eq. (36) is the same as that in Fan et al.^[20].

5.1.3 COD

To find the relative COD at any point of the crack face, i.e., Δu₂, put the value of Σ_k(x) from Eq. (26) to Eq. (21) and integrate it. Then, we have

(37)

where

5.1.4 COP

Similar to the COD, the relative COP at any point of the crack face, i.e., Δφ(x), is obtained by using Eq. (29) into Eq. (21) as

(38)

Using Eqs. (6), (37), and (38), the value of D₂^c(x) is obtained as a function of x. However, since the first-order approximation D₂^c(x) is assumed to be constant, similar to Fan et al.^[20], we also consider D₂^c as a constant in this paper, and for its calculation, the permittivity of air is κ_c=8.85× 10^-12F·m^-1.

5.1.5 Critical applied electric displacement loading

To understand the role of saturation condition in defining the maximum value of applied electric loading, the following mathematical analysis is done using Eq. (33).

From Eq. (33), one can interpret that the electric zone length increases with the applied electric displacement loading. Therefore, to obtain the maximum possible electrical load applied in the case of the proposed model, we consider in Eq. (33) and this implies

(39)

Applying the similar approach on the analytical expression obtained for the saturation zone length in the case of the PS model as in Fan et al.^[20], it is found that D₂^∞=D_s. It concludes that the linearly varying saturated condition (defined in the proposed model) imposed on the strip significantly reduces the possible value of applied electrical loading than the constant saturation condition (as in the PS model).

6 Numerical studies

In this section, numerical studies are presented for the modified PS model and PS model subject to arbitrary poling direction, semipermeable crack face condition, and electro-mechanical loadings. The effects of these parameters on the saturation zone length and local intensity factor are demonstrated by considering the PZT-4 material with the material constants given in Table 1. Further, the half-crack length a=1, σ₂₂^∞=10 MPa, and the saturated electric displacement, D_s=0.2 C/m² are considered for these studies.

Figure 2 shows the variation in the electric displacement D₂^c in the crack cavity versus the applied electromechanical loading (i.e., a constant stress and variable electric displacement loading) subject to the semipermeable crack face condition. It is found that D₂^c increases with the increase in the electric displacement loading, and also for a particular applied loading, it increases with an increase in the polarization angle. This implies that D₂^c increases as the poling direction moves towards the crack-axis.

Figure 3 presents the variation in the normalized saturation zone length (the zone length over half crack length) versus the electric displacement loading considering both the modified PS model and the PS model. These results are obtained at different poling directions, i.e., φ=0°, 30°, 50°, and 75° for impermeable and semipermeable crack face conditions. The behaviours of electric saturated zone with respect to the electric loading are the same for both the models irrespective of crack face conditions and poling direction. It shows that the electric saturated zone length increases with the electric displacement loading, but the size of electric saturated zone is significantly less in the case of semipermeable crack face condition than impermeable condition, and it further decreases with the increase in the polarization angle. However, in the case of impermeable crack face condition, no effect has been observed in the electric zone length with respect to the variation in the poling direction. The aforementioned behaviour of the saturated zone length versus the polarization angle can also be confirmed from the results shown in Fig. 4. The obtained behaviour of saturated zone length with respect to electric displacement loading and crack face conditions is found in agreement with Fan et al.^[20].

Moreover, even after getting the same behaviour for both the models, a higher electric zone length is obtained in the case of the proposed modified PS model than the PS model for a particular electric loading, and the difference in their values increases with the increase in the electric displacement loading. To further compare these models, a critical electrical displacement loading is obtained for both the models through mathematical analysis (mentioned in Subsection 5.1.5). It is found that the critical value of applied electric displacement loading for the proposed model is just half of the saturated value, whereas it is the same in the case of PS model.

Figure 5 shows that the COD increases with (a-x)/a, and it is the maximum at the centre of the crack, i.e., at x=0 but decreases corresponding to the increase in the polarization angle. The same behaviour is observed for both the crack face conditions, but significantly smaller numerical values of COD are obtained for the semipermeable condition than the impermeable one. Similar to the COD, the COP is also obtained for all x(0≤x≤c₁) and plotted in Fig. 6. It shows that the COP values are negative for all x but increase in magnitude from the electric zone tip to the centre of the crack. This behaviour of COP is independent of the crack face conditions. However, numerical values (in magnitude) obtained under the semipermeable crack face condition are also significantly less than the impermeable one. A significant effect of poling direction is observed on the COP for both the crack face conditions. The COP increases in magnitude with the increase in the polarization angle for the impermeable crack face condition, but a reverse behaviour is observed in the case of semipermeable crack face condition. The behaviours of COD and COP obtained here with respect to and are found in agreement with Fan et al.^[18].

Since the local intensity factors are the same for both the models, the study of normalized local intensity factor (K_I^*) is presented only for the proposed model and shown in Fig. 7. It shows the effects of poling direction and crack face conditions on K_I^* with respect to electric displacement loadings. K_I^* increases with respect to the electric displacement loading irrespective of crack face conditions and poling direction. However, it decreases with respect to the increase in the polarization angle. This behaviour is observed for both the crack face conditions. This is mainly due to the increase in D₂^c with respect to the increase in the polarization angle. Also, the results of K_I^* obtained under the semipermeable crack face condition are significantly less than the results of impermeable crack face condition, and this behaviour is found in agreement with the results of Fan et al.^[20].

7 Conclusions

Considering the aforementioned analytical and numerical studies done on the proposed model, the following points are concluded.

(ⅰ) A complex variable and Riemann Hilbert approach is successfully applied to study the linearly varying modified PS model proposed for impermeable and semipermeable 2D piezoelectric media.

(ⅱ) The closed form expressions are obtained for the saturated zone length, the COD, the COP, and the local SIF subject to the modified PS model defined in arbitrary polarized 2D piezoelectric media.

(ⅲ) The effect of electric displacement loading is observed on the saturated zone length and the local SIF. Both the parameters are increased with the increase in the applied electric displacement loading.

(ⅳ) For the saturation zone length, a significant effect of the poling direction is observed in the case of semipermeable condition, whereas no effect has been observed on it for the impermeable condition.

(ⅴ) The effect of the poling direction is also observed on the COD, the COP, and the local intensity factor. A similar kind of effect of poling direction is observed on the COD for both impermeable and semipermeable conditions, but in the case of COP, a reverse effect is observed with respect to crack face conditions.

(ⅵ) Behaviours of obtained saturated zone length with respect to the electrical loading and poling direction are the same for both models. However, higher values are obtained in the case of the modified PS model than the PS model except for the case with higher polarization angle and semipermeable condition.

(ⅶ) The critical value of the applied electrical load is obtained for both models. It is found that the critical electric displacement load is just half of the saturated electric displacement load in the case of modified PS model, whereas it is equal to the saturated electric displacement load for the PS model. This shows the effect of saturation condition defined in the PS model.

(ⅷ) The variation in the saturated condition shows a significant effect in defining the saturated zone length and critical value of applied electrical loading, whereas no effect is observed on the local intensity factor and the COD.

(ⅸ) Numerical comparative studies conclude that a linearly varying saturation condition (considered in the proposed model) can significantly increase the saturated zone length (for a particular applied electric loading) and decrease the critical value of applied electric loading (i.e., half of its saturated value) than that obtained in the PS model irrespective of crack face conditions and poling direction. The reduction in the value of critical applied electric loading also concludes that the proposed modified PS model may be considered as one of the crack arrest models in 2D semipermeable piezoelectric media under small-scale saturation conditions.