Dynamic Propagation Characteristics of a Mode-III Interfacial Crack in Piezoelectric Bimaterials

This article presents the dynamic behavior of a semi-inﬁnite interfacial crack in piezoelectric bimaterials under impact loading. With the help of the transform methods (the Laplace transforms and Fourier transforms), the problem is studied with the Wiener–Hopf technique. This strict proof guarantees the feasibility of this approach. The dynamic stress intensity factor and dynamic electric displacement intensity factor of the interfacial crack propagation characteristics are expressed. Finally, several classic numerical examples are mentioned and discussed to demonstrate that the theoretical deduction is highly accurate for interfacial crack analysis of piezoelectric bimaterials. The results show that the crack propagation is aﬀected by the electromechanical coupling coeﬃcient. In addition, if the velocity of the dynamic crack propagation reaches the generalized Raleigh wave speed, the dynamic stress intensity factor will disappear. Furthermore, for a given time, the ratio of the dynamic stress intensity factor to load increases with the electromechanical coupling coeﬃcient decreasing. Numerical examples are presented to highlight the result.


Introduction
1.1. Background. Curie brothers have presented the piezoelectric effect. Due to their coupling behaviors, piezoelectric materials have been widely used in modern smart devices and structures such as actuators, sensors, and ultrasonic generators. At present, piezoelectric materials widely used in engineering practice are mainly piezoelectric ceramics and piezoelectric composites made of piezoelectric ceramics and polymers. Piezoelectric ceramics are brittle materials and have the disadvantage of low fracture toughness. However, defects often cause the structural failure of piezoelectric materials. Cracks as their main disadvantages in piezoelectric media have been widely investigated [1][2][3][4][5][6][7][8][9]. One of the challenging problems is the fracture behavior of interfacial crack in piezoelectric bimaterials. A few fundamental problems of the crack in homogeneous and inhomogeneous piezoelectric bimaterials have been discussed [10][11][12][13]. e fracture analyses of these piezoelectric materials have provided much knowledge for improving the performance of piezoelectric devices. However, most results have focused on static propagation.
Dynamic fracture can be divided into three types according to the time dependence of load and crack state: the mechanical-electric load is time-independent and the crack propagates rapidly [14]; not only is the electromechanical load time-dependent but also the crack propagates rapidly [15]; the crack is stable and the electromechanical load varies with time [16].
A great deal of research has been done on the first and the second kind of fracture conditions [17,18]. For the third type of dynamic fracture, the dynamic propagation of electroacoustic wave (elastic wave) [19] and impact is included. e crack propagation becomes dynamic behavior under the action of explosion or impact load. e propagation behavior of the crack was analyzed under a concentrated point load [20,21]. e solutions are derived in an explicit way. When the "electrode" Bleustein-Gulyaev wave velocity is above the crack propagation speed, the stress displacement and electric displacement are exhibited r − 1/2 , which is similar to that of static fracture behavior. Narita [22] researched the crack in a piezoelectric layer. e interface was normal to the interface. e results show it is feasible and practical for the design of piezoelectric devices. e interfacial crack propagation in piezoelectric bimaterials is the most complicated problem in the dynamic fracture of piezoelectric media. Ray investigated the dynamic crack propagation behaviors in piezoelectric layers [23].
e transient response of a Griffith crack between two piezoelectric layers under finite widths was investigated [24]. e problem was formulated using integral transforms, and the path-independent integral G was evaluated at the crack tip to obtain the dynamic energy release rate. e dynamic fracture toughness of a mode-III interfacial crack in two piezoelectric half-spaces was analyzed [25,26]. Shen et al. provided a unified method to analyze a interfacial crack propagating in piezoelectric bimaterials [27]. e problem of an antiplane Griffith crack moving along with the interface of dissimilar piezoelectric materials was performed using the integral transform technique [28]. A semi-infinite crack in piezoelectric and piezomagnetic bimaterials was researched through the Wiener-Hopf techniques together with the integral transform method [29]. Nourazar and Ayatollahi analyzed the multiple moving cracks along with the interface of two piezoelectric layers under electrically impermeable or permeable loading, respectively [30].
At the same time, numerical simulations were used for studying the crack. Fracture in piezoelectric materials was analyzed by using a cell-based smoothed X-FEM [31]. Numerical methods were required to deal with the dynamic fracture behavior under loading conditions and general geometries. e dynamic problems of piezoelectric materials were researched by using the X-FEM [32], and it is effective in solving the dynamic fracture problems of isotropic materials. e SGBEM was raised about the dynamic crack in the piezoelectric composite materials [33].
However, piezoelectric bimaterials are widely used in modern life. e crack propagation caused by impact load is often hard to observe, and the crack runs through the interface of bimaterials, which is a complex transient process.
e fracture mechanism and process of the material cannot be observed. erefore, the macroscopic process of crack propagation is carried out to research questions about the materials.
e interfacial fracture of piezoelectric bimaterials is caused by its mismatching characteristics under working loading conditions. ere are few kinds of research on the mechanical properties of piezoelectric bimaterials and a variation of the crack as time changes. erefore, the analysis of dynamic crack along with the interface in piezoelectric bimaterials is essential to the practical engineering design.

Outline.
e presentation of the study is structured as follows. In Section 2, the mixed initial boundary conditions and constitutive equations are presented in detail. In Section 3, the solution method of dynamic crack propagation in the transformed domain is presented. e dynamic stress intensity factor and electric displacement intensity factor of the crack propagation behavior in the interface are given in Section 4. Section 5 presents the numerical results. In Section 6, we give some immediate conclusions.

Governing Equations
According to [34] and the variational principle, the energy conservation laws can be written as and v are, respectively, the kinetic energy, internal energy (including electrical and mechanical energies), a surface in the volume domain V, outward normal vector, the stress tensor, density, electric field, electric displacement, and velocity vector. e basic equations for linear piezoelectric medium can be written as follows.
Kinematic equations: where w is the mechanical displacement. Electrostatic charge conservation: Gradient equations: where w is the mechanical displacement, ∈ ij is the strain tensor, and φ is the electric potential. From equations (2) and (3), we will obtain the local form of the energy conservation laws: We introduce electric enthalpy density H(∈ ij , E i ), e electric enthalpy density H(∈ ij , E i ) is expanded in Taylor series near the natural state (τ ij � 0, E i � 0): where c ijkl are the elastic stiffness constants, e kij are the piezoelectric constants, and ε ij are the dielectric constants.

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By substituting equation (7) to (8), the change of H can be written as follows: It is noted that According to equations (10) and (12), the nontrivial constitutive equations can be obtained: Using the symmetry of strain tensor and stress tensor, the symmetry relation of material constants can be obtained: From equations (3)-(4) and (13)-(15), we have the following basic governing equations of piezoelectric theory: For the transversely isotropic piezoelectric medium, the constitutive equations [35] are τ xx � c 11 ∈ xx + c 12 ∈ yy + c 13 ∈ zz − e 31 E z , τ yy � c 12 ∈ xx + c 11 ∈ yy + c 13 ∈ zz − e 31 E z , τ zz � c 13 ∈ xx + c 13 ∈ yy + c 33 ∈ zz − e 33 E z , If only the in-plane electric fields and the out-of-plane displacement are considered, the dynamic antiplane governing equations for the transversely isotropic piezoelectric material can be described by In Figure 1, the piezoelectric material one and piezoelectric material two occupy the domains Ω 1 and Ω 2 , respectively. A semi-infinite crack along with the interface in the transversely isotropic piezoelectric bimaterials is considered. It is located at Y � 0, X < 0. e crack at any time t < 0 is in a static equilibrium state in Figure 1. e crack tip begins to move when t � 0. e position is x � vt when t > 0.
If only the in-plane electric fields and the out-of-plane displacement are considered, the dynamic antiplane governing equations for the transversely isotropic piezoelectric bimaterials can be described by e boundary conditions of mixed mechanical are where H(t) is the Heaviside step function of the time t. e mechanical displacements and electric potential will disappear when y ⟶ ∞: e static initial conditions are

Solution to the Problem
Following [20,21], a function is introduced as e equations are converted into the canonical form by substituting equations (38) To obtain the solution of equations (22)- (25), the following are the Fourier transform concerning x and Laplace transform concerning the time t: where ξ � ξ 1 + iξ 2 is complex, and B r denotes the Bromwich path of integration. From equations (41) and (42), the governing equations (27), (28), and (38) in the transformed domain can be obtained as follows:  Advances in Materials Science and Engineering By substituting equations (43) and (45) to equations (39) and (40), respectively, using zero initial conditions in equations (36) and (37), the transformed governing equations become where Applying the conditions given in equations (35), (36), and (37) at infinity, the solutions of equation (46) are where c k � ����� f k c k 2 denotes the shear bulk wave velocity.
We have the following form with the inverse Fourier transform to equation (50): From equations (52a) and (52b), we have the following form: By using equations (52a), (52b) and (53), the transformed results of the stress and electric displacement are Advances in Materials Science and Engineering To further improve the speed for solving problems, the boundary conditions equations (29)-(34) expressed in the Laplace transform domain are By substituting equations (59) and (60) into equations (52a) and (53), respectively, we have 1 2π Substitution equations (57) and (58) into equations (55) and (56), we will have From equation (64), we have 1 2π en, rewriting a k , we have in which When v � 0, g k and h k become It can be verified that this result is consistent with reference [36] in the case v � 0. Now, let us calculate the case v ≠ 0. Given the four boundary conditions equations (59)-(62), the displacement and stress boundary conditions must extend to the entire range of the x-axis.
e inverse Fourier transform is applied to equations (76)-(77): 6 Advances in Materials Science and Engineering Equation (79) can be rewritten as where where where c min and c max are the minimum and maximum of the wave speeds c k and d k (k � 1, 2), respectively. Since M k (ζ) (Maerfeld-Tournois wave function) depends on the MT wave, M k (ζ) are defined and coefficiented as follows: Advances in Materials Science and Engineering e product decomposition of equation (80) is given by [20,21,37] where Moreover, equation (85) is expressed as follows: On the basis of equation (41), the Laplace transform of τ yz (x, y, t) is where p is a positive real variable in the case. From the integrable energy density and continuity of displacement, the displacement function w * − (x, 0, p) and stress function τ * x + (x, 0, p) satisfy the following conditions: Using the Able theorem [38], we can obtain It follows that as |ζ| ⟶ ∞, According the extended Liouville's theorem [37], the polynomial must be less than the maximum value . It can only be equal to zero in the case that According to the procedure by Li and Mataga [20,21], the functions B k (ζ) can be obtained:

Intensity Factors
Substituting equations (93) and (94) into equations (55) and (56), respectively, and applying the inverse Fourier transform, D * y (x, p) and τ * yz (x, p) can be obtained. According to Abel's theorem [37], we can obtain the stress intensity factor and electric displacement intensity factor as follows: Using the inverse Laplace transform to equations (95)-(97) leads to According to the procedure in [38] for the problem of the static semi-infinite crack propagation, we introduce a normalization. e stress intensity factor for the static semi-infinite crack is e electric displacement intensity factor for a stationary semi-infinite crack is As a result, the dynamic stress intensity factor and the dynamic electric displacement intensity factors can be represented as are the nondimensional dynamic intensity factors.
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Numerical Examples
e material properties and the mass density [20,36], elastic stiffness constant, piezoelectric constant, dielectric constant, electromechanical coupling coefficient, and shear wave speed are given in Table 1. Incidentally, the values c and d for PZT65/35 and ZnO, respectively, in [20] are incorrect, and the correct values are given in Table 1.
ere are three parts in this section. First, when the research is degenerated into the simple case, some numerical applications are provided to verify the conclusions obtained. Second, some analyses and comparisons are made between the results obtained by us and the existing references [25,26] (the methods used are different from those we used). ird, the influence of material constants, velocity of the crack propagation, and time of impact are discussed. e researches have led to some new conclusions.

Study of Degenerate into the Simple Case. When bimaterials degenerate into a single material, it follows that
. It is noted that the conclusions agree with [20,21]. We choose the single material PZT-4 and PZT-5H, respectively, for research. e corresponding analysis diagram is shown in Figure 2. ey show the variations f(v) and g(v) with the normalized speed v/d of the dynamic crack propagation. As shown in Figure 2(a), f(v) decreases with increasing velocity v of the dynamic crack propagation. e variations f(v) vanish when v reaches Bleustein-Gulyaev d. Furthermore, the nondimensional function of the stress f(v) for an antiplane crack propagating in piezoelectric solids as considered is plotted for comparison [20,21]. eir changes are consistent. However, as shown in Figure 2(a), when the speed of the dynamic crack propagation reaches Bleustein-Gulyaev d, the variations g(v) do not have to go to zero. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. e variations g(v) increase again as the speed v of the dynamic crack propagation reaches Bleustein-Gulyaev d. Furthermore, the variations g(v) for the crack propagating in piezoelectric solids [20,21] as considered are plotted for comparison. eir changes are consistent.

Analysis and Comparison with the Existing References.
Some analyses and comparisons are made between the results obtained by us and the existing references [25,26] (the methods used are different from those we used). We choose PZT-4&PZT65/35 and PZT-2&ZnO, respectively, for research. e corresponding analysis diagram is shown in Figure 3. In Figures 3(a) and 3(c), it represents f(v), g 1 (v), and g 2 (v) of PZT-4&PZT65/35 versus the normalized speed v/d and v/c of the dynamic crack for two cases. In Figures 3(b) and 3(d), it represents f(v), g 1 (v), and g 2 (v) of PZT-2&ZnO versus the normalized speed v/d and v/c of the dynamic crack for two cases.
It is obvious that the nondimension functions f(v) and g 1 (v) decrease with the velocity v of the crack propagation increasing, as shown in Figure 3.
e nondimension functions f(v) and g 1 (v) reach zero, with the velocity v of the dynamic crack propagation reaching the slower velocity d. It is consistent with [25,26]. However, the nondimension function g 2 (v) does not necessarily decrease to zero and is relatively complex, with the dynamic crack speed reaching the Bleustein-Gulyaev wave speed d. It has an initial decrease as the speed v of the dynamic crack propagation increases from zero. e variations increase again as the speed v of the dynamic crack propagation reaches Bleustein-Gulyaev d. e whole process of changes are consistent with [25,26].  Figure 5, the change of each function g(v) remains almost constant as k increases. e change of each g(v) drops sharply as k approaches unity. e Bleustein-Gulyaev wave speed d and the shear wave speed c both increase as k increases. e change of g(v) decreases as c increases, and the change of g(v) increases as c increases. However, the effect is not an obvious effect when k is at low values. e variation of g(v) is complex, and the nondimension functions g(v) attains a finite nonzero value with v approaches c or d. e change of g(v) is affected by the Bleustein-Gulyaev wave term 1 − k 2 k . Since 1 − k 2 k go to zero with v approaching the velocity d of Bleustein-Gulyaev wave, the functions g(v) do not have to go to zero at large v � d.

e Influence of Material Constants, Velocity of the Crack Propagation, and Time of Impact.
It is obvious that the nondimension functions f(v) and g 1 (v) decrease with the velocity v of the crack propagation increasing, as shown in Figures 6(a) and 6(b). e nondimension functions f(v) and g 1 (v) reach zero, with the velocity v of the dynamic crack propagation reaching the slower velocity d. It is similar to the nondimension function K III (v) of purely elastic solids. However, the nondimension function g 2 (v) does not necessarily decrease to zero and is relatively complex, with the dynamic crack speed reaching the Bleustein-Gulyaev wave speed d. In Figures 6(c) and 6(d), it represents f(v), g 1 (v), and g 2 (v) versus the   1.6   is characteristic has also not been reported in the piezoelectric bimaterials.
In Figures 9(a)-9(c), the effects of the electromechanical coupling coefficient k on K III (t, v)/τ 0 are investigated numerically by calculating K III (t, v)/τ 0 of six different piezoelectric bimaterials, respectively. In this study, we specify M � 0.5, M � 0.6, and M � 0.7, respectively, and the corresponding results of K III (t, v)/τ 0 are shown in Figures 9(a)-9(c). e calculated results K III (t, v)/τ 0 of six different curves are shown. e following characteristics are obvious: (a) the dependence of K III (t, v)/τ 0 on the electromechanical coefficient k is significant; (b) it can also be observed that K III (t, v)/τ 0 are equal to zero at t � 0. e results show that the electromechanical coupling coefficient k of the piezoelectric half-space can be increased from zero to unity. In contrast, the material properties of the elastic half-space are held constant; (c) the values of K III (t, v)/τ 0 decrease with increasing electromechanical coefficient k for a given time and a given v. is characteristic has also not been reported in the piezoelectric bimaterials.
In Figures 10(a) decreasing M; (d) as a consequence, the effects of the electromechanical coefficient k on K III (t, v)/τ 0 in piezoelectric bimaterials are significant.

Conclusions
e dynamic behavior of an interfacial mode-III crack under mechanical impact loading is considered. e dynamic stress intensity factor, the dynamic electric displacement intensity factor, and their nondimension functions are derived with the analytical expressions. According to the solution established in the study, we have some conclusions: (1) the behavior of the dynamic fracture in piezoelectric bimaterials is more to do with the electromechanical coefficient k; (2) the dynamic stress intensity factor always vanishes with the velocity of dynamic crack propagation arriving at the speed of the generalized Raleigh wave; (3) it is verified that the nondimensional stress intensity factors are controlled by the electromechanical coefficient k, that is, the bigger the electromechanical coefficient k, the smaller the nondimensional stress intensity factor; (4) the existence of the Maerfeld-Tournois wave will increase the value of the dynamic intensity factors; (5) the values of K III (t, v)/τ 0 increase with M decreasing; (6) consequently, the effects of the electromechanical coefficient k on K III (t, v)/τ 0 in piezoelectric bimaterials are significant; (7) it is found that for a given strength of the crack, the larger M is, the smaller the time is, under the same loading. In a word, it analyzes the relationship between the dynamic crack change of piezoelectric bimaterials with time, which makes the theoretical analysis more perfect and provides more detailed theoretical support for engineering applications. e stress and electric field concentration caused by cracks will lead to the loss of its design function. e fracture of piezoelectric materials we studied is mainly a piezoelectric solid with cracks. e propagation of cracks is studied to provide theoretical reference for reliability analysis and prolonging life-span. e cracks of piezoelectric materials include mode-I, mode-II, mode-III, and compound. e research idea is consisted of analyzing fracture, selecting the type of crack, establishing the model, defining the boundary conditions, and confirming fracture criterion. In this study, the existing crack is mode-III crack. Dynamic propagation characteristics of the crack under impact loading are presented. It is considered under the in-plane electric field and out-of-plane displacement. e mode-I, mode-II, and composite cracks will be analyzed and discussed in the future research. At the same time, nanoscale piezoelectric bimaterials are widely used in daily life. Our analysis in this section does not apply to nanomaterials. In the next research, we can consider discussing the interface cracks of nanoscale piezoelectric bimaterials in not only analyzing the semi-infinite III cracks but also thinking about deeply the influence of nanomaterial thickness on material design. study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.