1 Introduction

Piezoelectric materials, due to their intrinsic property of converting electrical energy into mechanical energy and vice-versa, have been widely used in modern smart structures and devices, e.g., actuators, transducers, and sensors. However, during the manufacturing or fabrication processes for such devices made of piezoelectric materials, defects, e.g., inclusions, holes, and cracks, inevitably occur, which will reduce the life time of these devices and even lead to premature failure and fracture in these devices under the influence of external loadings. In order to increase the life time and performance of these electromechanical devices and smart structures, a detailed study on the fracture mechanics of piezoelectric materials is required in the presence of multiple cracks/damages. One such fracture problem is the study of strip saturated collinear cracks in piezoelectric materials. The strip yield collinear cracks in elastic perfectly plastic materials have widely been analyzed in the past with analytical and numerical methods^[1-7].

Theocaris^[1] applied Dugdale's model^[8] to study the two collinear and unequal cracks within a homogeneous isotropic elastic-perfectly plastic infinite plate. Applying the complex variable and non-homogenous Riemann-Hilbert approach, Collins and Cartwright^[2] corrected the previous published solution, and presented a study on two equal-length and collinear strip yield cracks. A particular case of two collinear equal cracks with coalesced plastic zones was also studied as a special case. Nishimura^[3] studied the plastic zone sizes and crack tip opening displacements of two unequal collinear cracks in an infinite sheet subjected to remote stress. The analysis was done with the Fredholm integral equation method in conjunction with the strip yield model. Chang and Kotousov^[4] used the distributed dislocation and Dugdale strip yield model to analyze the non-linear interaction between two equal-length collinear cracks subjected to the remote tensile stress. Two alternative approaches, i.e., the Foppl integral equation and the Gauss-Chebychev quadrature method, were used to obtain the analytical and numerical solutions. Hasan and Akhtar^[5] used Muskhelisvili's complex variable method to investigate the problem of three collinearly equal straight cracks based on Dugdale's model for the load bearing capacity, the yield zone lengths, and the crack tip opening displacements. Bhargava and Hasan^[6] obtained the crack tip opening displacements and lengths of the developed strip yield zones for four collinearly straight cracks considered in an unbounded elastic-plastic domain. The strip yield zones developed at each interior tip of each set for the exterior and interior cracks were assumed to be coalesced. Hasan^[7] derived an analytical solution for three unequal collinearly straight cracks with coalesced yield zones by using the complex variable method.

In analogous to Dugdale's model, Gao et al.^[9] proposed the polarization saturated model or strip saturated model in two-dimensional (2D) piezoelectric media. In this model, the electric yielding zones developed near the crack tips were assumed to be in a strip arrested form by the normally saturated electric displacement loading. The studies based on this nonlinear fracture model were found to be in agreement with the experimental results^[10], which were not explained by the linear theory of piezoelectricity. Fan et al.^[11] presented the numerical results of the polarization saturation (PS) and dielectric breakdown models in 2D piezoelectric media with the nonlinear hybrid extended displacement discontinuity-fundamental solution method. Fan et al.^[12] extended the study of the PS model to semipermeable piezoelectric media, and derived an analytical solution for the problem. The extension of the PS model to three-dimensional (3D) semipermeable cracked piezoelectric media was also done by Fan et al.^[13]. To analyze the fracture parameters for multiple cracks, Bhargava and Jangid^[14] extended the strip saturated model to semipermeable collinear cracks in 2D piezoelectric media with the inner zones coalesced. With the complex variable approach, they derived closed form expressions for the crack opening displacement (COD), the crack opening potential (COP), the local intensity factors (LISFs), and the saturated zone lengths. Bhargava and Jangid^[15] studied the problem of two equal collinear cracks in an infinite piezoelectric domain with the strip electro-mechanical yielding model. Bhargava et al.^[16] extended the strip saturated model for two collinear semipermeable cracks to Mode-Ⅲ cases.

With reference to Dugdale's model, many scholars modified the hypothesis and proposed other non-linear fracture models^[17-21]. Bhargava and Hasan^[19-20] studied two collinear unequal cracks with coalesced zones based on the modified Dugdale's model, considering the normal cohesive quadratically varying stress distribution over the rims of the plastic zones. Mukhtar and Ali^[21] obtained the analytical solution for two unequal collinear cracks with coalesced zones with the generalized Dugdale's model, considering the normal cohesive linearly varying stress distribution over the rims of the plastic zones. Hasan^[22] used the modified Dugdale's model to study two pairs of collinear cracks with coalesced yield zones.

In piezoelectric materials, Ru^[23] obtained the analytic solution for a center crack problem under the generalized strip saturated electric displacement condition. Bhargava and Setia^[24-25] extended the concept of constant saturated loading to linear and quadratic variable saturated loadings in studying the 2D impermeable center cracked piezoelectric strip with the Fourier transform technique. Singh et al.^[26] used the complex variable technique to study the linearly varying modified PS model in 2D semipermeable center cracked piezoelectric media. Recently, Sharma and Singh^[27] presented numerical studies of polynomially varying polarization saturation models in center cracked 2D piezoelectric media with the distributed dislocation method.

Considering the importance of studies based on the modified nonlinear fracture models in piezoelectric materials, the unknowingness of the saturated condition in nonlinear fracture models, and the possibility of interpolated electric displacement variations based on near tip behaviors, we propose and investigate the polynomially varying modified strip saturated models for two equal collinear cracks with coalesced zones in 2D arbitrary polarized piezoelectric media.

The mathematical formulation and analytical solution obtained with the complex variable approach are presented in Section 4. The closed form solutions for the outer saturated zone lengths, the COP, the COD, and the LSIF are expressed in Section 5. The numerical studies on PZT-4 media are demonstrated in Section 6, whereas the analytical and numerical studies on the modified strip saturated models considering two equal collinear cracks with coalesced saturated zones are concluded in the last section.

2 Fundamental equations for 2D piezoelectric media

In this section, the generalized governing equations based on the 3D formulations of linear piezoelectric media as described in Refs. [28]-[32] are presented.

The constitutive equations are

(1a)

(1b)

where σ_ij, γ_kl, D_k, and E_l denote the stress, the strain, the electric displacement, and the electric field, respectively. c_ijkl, e_kij, and κ_kl stand for the elastic constant, the piezoelectric constant, and the dielectric constant, respectively, which are defined for the transversely isotropic piezoelectric material with the poling direction along the y-axis.

In case of arbitrary polarization, i.e., when the poling direction makes an arbitrary angle α with the positive y-axis as in Fig. 1, the material constants are changed, and hence the aforementioned constitutive equations converted to

(2a)

(2b)

where the transformed material constants (c^Θ, e^Θ, κ^Θ) are obtained with the help of the transformed matrices, and the generalized material constants matrices (c, e, κ) have the following relations:

(3)

Here, the rotation matrices are defined as follows:

The gradient equations are

(4)

In the absence of body forces and free charges, the equilibrium equations are

(5)

(6)

where u_i (i, j=1, 2, 3) and ϕ denote the mechanical displacement components and the electric potential, respectively, and the subscript denotes the partial differentiation with respect to one of the coordinates x, y, and z.

2.1 Crack face boundary conditions

The earlier research works on the fracture in piezoelectric materials are mainly based on the impermeable crack face conditions, in which the permittivity of the medium within the crack faces is not considered. It is mainly based on the assumption that the permittivity of the piezoelectric medium is very large in comparison with that of the air/medium within the crack faces. Also, these conditions are analogous to the mechanical traction free conditions defined in elastic-plastic materials. However, due to the presence of electric displacement singularity near the crack tips, Hao and Shen^[33] considered the role of permittivity of the medium within the crack faces, and proposed new crack face conditions, i.e., the semipermeable crack face conditions. These conditions later on are also called as realistic or actual crack face conditions, and have been widely used^{[12-16, 26-27]}. The mathematical expressions are as follows:

(7a)

(7b)

where the indices “+” and “-” represent the upper and lower crack surfaces, respectively. ΔΦ (x₁) is the electrical potential jump, and Δ u₂(x₁) is the crack opening displacement. k_c is the permittivity of the medium between the crack faces.

One can reduce these conditions to an impermeable one by considering κ_c=0 and to a permeable one by considering a high value of κ_c.

3 Stroh formalism based solution for 2D arbitrary polarized piezoelectric media

Adopting the procedure of Eshelby et al.^[34] for deriving the generalized solution for anisotropic elasticity, the Stroh formalism based solution for 2D arbitrary polarized piezoelectric media is presented here.

In order to find the generalized 2D deformations, the generalized displacement vector u=(u₁, u₂, u₃, ϕ) depending on two independent variables x and y are considered as follows:

(8)

where f(z) is an analytic function, q is an unknown complex number, and s is an unknown column vector to be determined from the field equations.

Since u is the generalized solution of the problem, it must satisfy the field equations (2)-(6). Therefore, Eqs. (5) and (6) become

(9)

(10)

Now, based on the generalized solution, the displacement components and the potential can be written as follows:

Substituting u_k and u₄ into Eqs. (9) and (10) yields

(11)

(12)

where δ_mn is the Kroneckor delta function, which is equal to 1 if m=n and 0 if m≠ n.

Further, these equations can be expressed as follows:

(13)

(14)

In the matrix form, Eqs. (13) and (14) can be written as follows:

(15)

This forms an eigenvalue problem with the eigenvector (s_k s₄) and the eigenvalue q. Now, in order to find the nontrivial solution of (s_k s₄)^T, we let

(16)

or

(17)

where

After solving the determinant, Eq. (16) forms a polynomial equation in q of degree 8, the roots of which are four distinct complex conjugate pairs as the real root violates the condition of the positive definite strain energy.

Now, if q_n represents the eigenvalue of Eq. (15) with a positive imaginary part and s⁽ⁿ⁾ is its corresponding eigenvector, the generalized solution obtained by superimposing all the independent solutions of the form of Eq. (8) is expressed as follows:

(18)

where f_n (n=1, 2, ..., 8) are arbitrary analytic functions obtained from the boundary conditions of the problem, and

In the matrix form, it can be written as

(19)

where H=(s⁽¹⁾ s⁽²⁾ s⁽³⁾ s⁽⁴⁾) is a square matrix with four independent eigenvectors associated with the eigenvalues q_n (n=1, 2, ..., 8) with positive imaginary pars, and

Similarly, one can obtain the stress and electric displacement components from Eq. (2) and the generalized solution (see Eq. (8)) as follows:

(20)

In particular,

(21)

where

From Eq. (20), it is possible to define a generalized complex variable function for the field components as follows:

(22)

where f_n (n=1, 2, ..., 8) are arbitrary functions, z_n=x+q_n y, and M=(L₁ L₂ L₃ L₄) is a square matrix associated with the eigenvectors L_n.

Hence, the stress and electric displacement components can be obtained by differentiating the generalized stress function with respect to x or y as follows:

(23)

(24)

where

4 Mathematical formulation and analytical solution of modified strip saturated models in piezoelectric media

An infinite 2D piezoelectric specimen occupying the xOy-plane with the arbitrary poling direction, i.e., the direction which makes an angle α w.r.t. the y-axis, is considered. It is cut with two equal collinear cracks symmetrically situated about the origin of the plane and occupying the intervals [-d, -c] and [c, d] along the x-axis. The specimen is subjected to the semipermeable crack face conditions and under the effect of the remote uniform tensile and in-plane electric displacement loadings, i.e., σ₂₂= σ₂₂^∞ and D₂ = D₂^∞. The applied remote loading opens the cracks in a self-similar fashion, and forms a strip saturated zone ahead each tip of the two cracks, i.e., 0 ≤ |x|≤ c, and d ≤ |x|≤ b. The same results are arrested by the polynomially varying saturated electric displacement value. The entire configuration of the problem is schematically presented in Fig. 1. Further, to analyze the behaviors of the fracture parameters for these modified models, the mathematical modeling and analytical solutions are presented for each of the models in this section.

The physical boundary conditions of the problem can be mathematically written as follows:

Now, on the basis of g(x), different modified strip saturated models can be defined. Sine the objective of the study is to analyze the behaviors of the fracture parameters under polynomial varying conditions, we propose the following cases:(Ⅰ) the strip saturated model, i.e., g(x)=1;(Ⅱ) linearly varying strip saturated model, i.e.,

(Ⅲ) quadratically varying strip saturated model, i.e.,

(Ⅳ) cubically varying strip saturated model, i.e.,

In the above equations, .

The continuity of Φ, ₁ (x) on the whole real axis in the xOy-plane is as follows:

(25)

Following Musklishvili^[35], we can write the solution of Eq. (25) as follows:

(26)

Let

(27)

where F(z) is the complex function vector of the inhomogeneous field. Using the boundary condition in Eq. (23), we have

(28)

Then, we can write the jump displacement vector Δ u_{, 1} as follows:

(29)

where

Equation (28) can be represented in its components as follows:

(30)

(31)

From Eqs. (30) and (31), we have

(32)

where

The solution of Eq. (32) together with the single-aluedness condition of the mechanical displacement is represented as follows:

(33)

where

Now, from Eq. (31), we have

(34)

Using the boundary condition in Eq. (34), we have

(35)

where |x| < c, and d < |x| < b.

The solution of Eq. (35) depends upon the definition of the function g(x) and the single-aluedness condition of the electrical displacement. Therefore, all the aforementioned four cases are taken here for study one by one, and are represented as follows:

Case Ⅰ  Strip-saturation model

(36)

where

Case Ⅱ  Linearly varying strip saturated model

(37)

where

Case Ⅲ  Quadratically varying strip saturated model

(38)

where

Case Ⅳ  Cubically varying strip saturated model

(39)

where

in which

5 Derivation of the fracture parameters

In this section, the explicit expressions are determined for the length of the outer saturation zone, the LSIF, the COD, and the COP.

5.1 Saturation zone

The electric displacement is determined with Eq. (23) as follows:

(40)

Taking the fourth component of the above equation, we have

(41)

Substituting the values of Υ_σ (x)and Υ₄^Ⅰ (x) into Eqs. (33) and (37), we have

(42)

for the strip saturated model.

Substituting the values of Υ_σ (x) and Υ₄^Ⅱ (x) into Eqs. (33) and (37), we have

(43)

for the linearly varying strip saturated model.

Substituting the values of Υ_σ (x) and Υ₄^Ⅲ (x) into Eqs. (33) and (38), we have

(44)

for the quadratically varying strip saturated model.

Substituting the values of Υ_σ (x) and Υ₄^Ⅳ (x) into Eqs. (33) and (39), we have

(45)

where

for the cubically varying strip saturated model.

The lengths of the outer saturation zone are now obtained by extending the Dugdale hypothesis to the electric displacement to remain finite at Point b of the domain. This leads to the following relations in the outer saturation zone length:

(ⅰ) For the strip saturated model,

(46)

(ⅱ) For the linearly varying strip saturated model,

(47)

(ⅲ) For the quadratically varying strip saturated model,

(48)

(ⅳ) For the cubically varying strip saturated model,

(49)

5.2 LSIFs

The field components at any point (|x|>d, |x| < c) on the x-axis of the domain are given as follows:

(50)

From Eq. (50), we have

(51)

The LSIFs K_I(d) and K_I(c) are determined at the tips x =d, x=c as follows:

(52)

(53)

Since the LSIFs are evaluated at the actual (mechanical) crack tips, they are independent of the normal electric displacement saturation condition imposed on the saturated zone. Hence, the analytic expressions for the LSIFs are obtained only for the strip saturated model.

5.3 COD

Similar to the LSIFs, the COD is defined as per actual crack length, i.e., for c≤ |x|≤ d. Therefore, it is also independent of the modified saturated conditions defined in this paper. Hence, the relative crack opening displacement at any point of the crack face, i.e., Δ u₂, can be obtained after substituting the value of Υ_σ (x) into Eqs. (29)-(33) and then integrating. Δ u₂ is defined as follows:

(54)

where

5.4 COP

Unlike the COD, the relative crack opening potential depends on the saturation condition. Therefore, we define the following cases after integrating Eqs. (36), (37), (38), and (39), respectively:

(Ⅰ) For the strip saturated model, when c ≤ |x|≤ d,

(55)

when 0 ≤ |x|≤ c and d ≤ |x|≤b,

(56)

where

(Ⅱ) For the linearly varying strip saturated model, when c ≤ |x|≤ d,

(57)

when 0 ≤ |x|≤ c and d ≤ |x|≤ b,

(58)

where

(Ⅲ) For the quadratically varying strip saturated model, when c ≤ |x|≤ d,

(59)

when 0 ≤ |x|≤ c and d ≤ |x|≤ b,

(60)

(Ⅳ) For the cubically varying strip saturated model, when c≤ |x|≤ d,

(61)

when 0 ≤ |x|≤ c and d ≤ |x|≤ b,

(62)

where

6 Numerical results and discussion

In this section, numerical studies for modified strip saturated models considered in two collinear equal-length cracks with coalesced interior zones are presented. A problem of two semipermeable collinear cracks with coalesced zones is considered in a 2D infinite PZT-4 domain. The length of each of the cracks is 4 m. The domain is subjected to the far-field electromechanical loading with the saturated electric displacement value 0.2 C·m^-2 and the mechanical loading 10~MPa. The behaviors of the fracture parameters obtained in Section 5 are demonstrated in this section graphically with various electrical loadings, crack face conditions, inter crack space distances, and poling directions. The details of the material constants taken for the study are tabulated as follows^[26]:

(ⅰ) The elastic constants are c₁₁=126 Gpa, c₁₂ = 55 GPa, c₁₃ = 53 GPa, c₃₃ = 117 GPa, and c₄₄ =35.3 GPa.

(ⅱ) The piezoelectric constants are e₁₅ = 17 C·m^-2, e₃₁= -6.5 C·m^-2, and e₃₃ = 23.3 C·m^-2.

(ⅲ) The permittivities are k₁₁=15.1 nC·V^-1·m^-1 and k₃₃ = 13.1 nC·V^-1·m^-1.

Throughout this study, the normalized zone length is defined as (2 × the value of the outer zone length)/(the value of the coalesced zone length).

In the first analysis, the normalized zone lengths are evaluated for all the modified strip saturated models with respect to D₂^∞/D_s.

Figure 2 shows the results subjected to both the crack face conditions. It is found that the normalized zone length increases with the increase in the electrical loading for each modified strip saturated case. But for a particular applied electrical loading, the value of the normalized zone length increases with the increase in the degree of the polynomial varying saturation condition. This confirms the effect of the saturation condition in evaluating the unknown normalized saturated zone lengths and hence the importance of defining the nonlinear fracture models and other crack arrest models. This is due to the decrease in the increase in the effect of the saturated condition with the increase in the polynomial degree. In each case, it is also observed that the normalized zone length initially increases significantly w.r.t. the electrical loading but later on asymptotically converges to a vertical line at a critical value of the electrical loading. This critical electrical loading decreases with the increase in the polynomial degree. The aforementioned behaviors are found to be independent of the crack face conditions. However, the normalized zone length with reference to the electrical loading under the semipermeable crack face conditions is smaller than that under the impermeable crack face conditions. Similar behaviors are found for each modified strip saturated model. This confirms the established semipermeable results in Refs. [11]-[16], [26], and [27].

Figure 3 represents the behaviors of the normalized electrical loading with different μ. The results are calculated for keeping the outer zone length fixed. It is observed that higher electrical loading is required for higher μ and this is true for all the modified strip saturated models. This is mainly due to the increase in the inter-crack tip distance. The effect of the saturated condition can also be observed, i.e., the value of the electrical loading decreases with the increase in the polynomial degree for a fixed value of μ.

To demonstrate the study on the arbitrary polarization angle, the normalized zone lengths and the local intensity factors at different polarization angles are obtained (see Figs. 4 and 5) and analyzed. As shown in Fig. 4, the normalized zone length increases with the increase in the electrical loading. The results are the same for the entire modified strip saturated models and the polarization angles. However, a higher electrical loading is required for obtaining the same zone length when the polarization angle is higher, and the critical electrical loading increases with the increase in the polarization angle. This is true for each modified strip saturated model. Moreover, it is found that the strip saturated condition has no significant effect on the normalized zone length at smaller values of electrical loadings, and this lower limit of the electrical loading increases with the increase in the polarization angle. This is mainly due to the fact that increasing the polarization angle means the poling direction moves towards the crack axis and in that the effect of the electrical loading is insignificant on the fracture parameters.

The normalized local intensity factor (K_I^*) is evaluated at the inner and outer tips of the cracks with the increase in the electrical loading for the same polarization angles, i.e., α =0°, 50°, and 80°. The results are evaluated only for the strip saturated model since K_I^* depends only on the actual crack tips.

Figure 5 depicts the results subjected to both impermeable and semipermeable crack face conditions. It shows that K_I^* increases when the electrical loading increases, which is irrespective of the poling direction and the crack face conditions. However, the effects of the poling direction and the crack face conditions are significantly on K_I^*. The values of K_I^* under the semipermeable conditions are less than those under the impermeable conditions, which are in agreement with the established results^[11-12]. No significant effect of the electrical loading is observed at higher polarization angles. The smaller values of K_I^* obtained for the semipermeable crack face condition are due to the consideration of the electric displacement developed at the crack surfaces based on the assumption that the air is considered as the electric conducting medium between the crack surfaces. The behavior of K_I^* at higher polarization angles further confirms the fact that the electrical loading has a less effect on the fracture parameters when the poling direction is along the crack axis.

One can clearly view the effects of the polarization angle on the normalized zone length and the local intensity factors in Fig. 6. It shows the behaviors of the fracture parameters with the increase in the polarization angle evaluated at a particular normalized electrical loading of 0.4. It is found that both the normalized zone length and the LSIF decrease with the increase in the polarization angle for the same reason as stated above. Additionally, the effects of the modified saturated conditions and the crack tip position are observed on the fracture parameters. The LSIFs evaluated at the inner tips are higher than those at the outer tips due to the effects of the inter cracks space distance.

Figure 7 shows the behaviors of the normalized local intensity factor (K_I^*) versus the distance of the center of the crack from the origin (μ). It represents actually the effects of the inter-cracks space distance on the normalized LSIF. As expected, it is found that the LSIFs evaluated at both inner and outer tips of the crack decrease with the increase in the inter-crack space distance. When the distance is large, this effect is insignificant. This is because the mutual crack effect decreases with the increase in μ. Moreover, the values of K_I^* are higher at the inner tips than those at the outer tips, validating the inter-crack space distance effects on the LSIF. The effects of the inter-crack space distance on the LSIF are irrespective of the crack face conditions. However, the values obtained under the semipermeable condition are less than those obtained under the impermeable condition.

7 Conclusions

The following conclusions are made from the analytical and numerical studies presented for the proposed modified strip saturated models in two equal collinear cracks with coalesced inner zones in 2D piezoelectric media:

(Ⅰ) The complex variable approach is successfully used to model and obtain the analytical solution for the modified strip saturated models in two equal collinear cracks with coalesced inner zones proposed in 2D arbitrary polarized semipermeable piezoelectric media.

(Ⅱ) The closed form of expressions are obtained for the outer saturated zone lengths, the COP, the COD, and the LSIFs for each case of the coalesced modified strip saturated models.

(Ⅲ) The effects of the electrical loading, the crack face conditions, the coalesced zone length, and the polarization angle are found on the normalized zone length and the LSIF, whereas the effects of the polynomially varying saturated electric displacement condition are observed only on the normalized saturated zone lengths and the COP.

(Ⅳ) The value of the normalized zone length increases with the increase in the degree of the polynomially varying modified saturated electric displacement conditions subjected to a particular electrical loading.

(Ⅴ) The numerical studies show that a critical value of the applied electrical loading can be obtained for each strip saturated case and it depends upon how the saturated condition is defined. This critical value decreases with the increase in the degree of the polynomially varying modified saturated electric displacement conditions.