Abstract
The behavior of a fine-grained piezoelectric coating/substrate with multiple Griffith interface cracks under electromechanical loads is investigated. In this work, double coupled singular integral equations are proposed to solve the fracture problems. Both the singular integral equation and single-valued conditions are simplified into an algebraic equation and solved by numerical calculation. Thereby, the intensity factors of electric displacement and stress obtained are used to obtain the expression of the energy release rate. Furthermore, numerical results of the energy release rate with material parameters are demonstrated. Based on the obtained results, it could be concluded that the energy release rate is closely related to the size of the interface cracks and the mechanical-electrical loading. For a bimaterial structure, the fine-grained piezoelectric structure exhibited better material performance compared to the large one.
1. Introduction
Piezoelectric bimaterials have been widely applied in many fields, such as sensors, actuators, filters, and storage devices. Owing to the reliability of composite materials, double-layer bonded bimaterial composite structures have played an important role in smart structures. However, piezoelectric composites tend to crack, fracture, and debond at the interface during the process of fabricating defects and load conditions [1]. Therefore, the issue of defects of piezoelectric materials under electromechanical load has attracted wide attention [2–7]. Viun et al. [8] handled the interface problems of bi-piezoelectric materials under tensile mechanical loads and electric displacements. The influence of material parameters on finite-sized piezoelectric plates is analyzed by a finite-element method. Li et al. [9] analyzed the mechanical properties of static and dynamic interface cracks of functionally graded coatings/substrates. In this work, the Fourier transform and Laplace transform were used as the fundamental solutions to solve the problem under static and dynamic loads. Soh et al. [10] studied the behavior of bi-piezoelectric ceramic strips with mode III central interface cracks, and the influences of thickness and material coefficients of piezoelectric strips on intensity factors and stress were obtained. Shavlakadze [11] demonstrated the electroelastic contact problem of piezoelectric plates with elastic coatings. By using the method of analytic function theory, the problem was reduced to a singular integrodifferential equation with fixed singularity. Shin and Kim [12] studied the transient response of mode III interface cracks between the piezoelectric layer and the functional orthotropic material layer. The integral equations were solved by the integral method, and it was concluded that the material parameters could prevent the transient fracture of the interface crack. Gherrous and Ferdjani [13] investigated a Griffith interface crack of bi-piezoelectric materials, and the singular integral equations were reduced into algebraic equations by using Chebyshev polynomials. Nan and Wang [14] studied the effect of residual surface stress on the stress and electric field strength of conductive cracks in piezoelectric nanomaterials. The numerical results showed that the residual stress had a significant effect on the piezoelectric nanomaterials, and the piezoelectric nanomaterials had stronger electromechanical coupling compared to the classical fracture mechanics. Nourazar and Ayatollahi [15] examined the effects of multiple interface cracks in orthotropic-strip-bonded bi-piezoelectric strips, and the results reflected the crack size had a great influence on the intensity factor and the hoop stress of the cavities. Tian and Rajapakse [16] studied a penny-shaped interface crack of double piezoelectric materials, and the Hankel integral transformation method was used to establish a mixed boundary-value problem corresponding to an interface crack. Gao et al. [17] investigated a permeable crack problem in infinite magnetoelectroelastic solids with linear coupling between the elastic and electromagnetic fields, and the results showed in the most general cases that the singularity of the electric-magnetic field was always dependent on that of stress. Kozinov et al. [18] considered a periodic set of limited electrically permeable cracks at the interface of two different piezoelectric semi-infinite spaces and analyzed the influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters.
However, the piezoelectric materials with defects studied above are large-grained materials composed of multidomain grains from several microns to tens of microns, and the size could not meet the needs. Reducing the particle size to the submicron level can improve the machinability of the material and the mechanical strength of the device. Kim et al. studied the piezoelectric characteristics of fine-grained piezoelectric materials compared with conventionally prepared large-grained piezoelectric materials under certain conditions [19]. In this latter work, the influence of grain size on density, dielectric constant, phase structure, and piezoelectric properties was investigated [20–23]. However, the mechanical properties of fine-grained piezoelectric materials with defects have not been studied.
Although many researchers have studied the interface fracture problems of piezoelectric bi-materials [1], few studies have been done on the interface fracture of fine-grained piezoelectric coating/substrate smart structures. As an important fracture parameter with which to measure structural safety, the energy release rate plays a crucial role in safety checking of structures. In this paper, a fine-grained piezoelectric coating/substrate mechanical model is established to describe the interaction of multiple Griffith interface cracks. The Fourier integral transformation method is used to transform the problem into a singular integral equation to obtain the energy release rate conveniently. Furthermore, the interaction between the energy release rate and material parameters is investigated. The results of the studies indicate that the appropriate stress and electric displacement loads could prevent the growth of cracks and that a thinner coating thickness is more conducive to the safety of the structure. In addition, fine-grained piezoelectric materials have more safety than large-grained ones.
2. Problem Formulation
As shown in Figure 1, the fine-grained ceramic powder was sprayed uniformly to the surface of the piezoelectric substrate by plasma spraying technology to form a coating. A fine-grain piezoelectric coating/substrate structure was obtained by polarization treatment of coating. The thickness of the coating is and of the substrate is . The rectangular coordinate system was built with the -axis along the structure interface and the -axis along the structure thickness direction. The multiple Griffith interface cracks are 2-dimensional and all within the same orientation and occupying the intervals , respectively. The fine-grained piezoelectric coating and piezoelectric substrate are polarized along the -axis, and they are transversely isotropic.
Assume that the problem of the fine-grained piezoelectric coating/substrate interface cracks is considered as only the out-of-plane displacement and the in-plane electric fields such that
In this case, the constitutive equations become [4]where and are the stress components, and are the electric displacement components, is the elastic modulus, and are the piezoelectric and dielectric constants, and the superscript stands for the fine-grained piezoelectric coating and piezoelectric substrate, respectively.
According to Park, the governing equations can be written as follows:where is the two-dimensional Laplacian operator.
Assume that there is no loading on the surface of the coating/substrate and the antiplane shear stress and in-plane electric displacement are loaded on the surface of the Griffith interface cracks. Therefore, the mixed boundary conditions of the problems can be written as follows:
3. Solution to the Problem
By Fourier integral transform of equations (6) and (7), the loading displacement and electric field are obtained, respectively:where , , , and are the unknown functions .
Substitution formulas (16) and (17) into equations (4) and (5), the stress and the electric displacement are obtained, respectively:
In order to solve the problem easily, we utilize continuously distributed dislocations to simulate collinear cracks. For this, we introduce the density function as follows:
According to equations (12) and (13), the dislocation density function should make the following single-valued conditions valid:
Using the boundary conditions (8) and (9), we obtain
Inserting equations (16) and (17) into equations (20) and (21) and inserting equations (18) and (19) into the boundary conditions (8)–(11), (14), and (15), we obtain
Equations (25)–(32) can be written in the form of matrix as follows:where
Solving the linear equation (32), we obtain
The expressions of and , are given in Appendix.
Obviously, the unknown functions , and , depend on and and, consequently, on the density functions and . Once and are determined, the stress expressions and electric displacement expressions could be obtained.
For this, by substituting equations (16) and (17) into equations (4) and (5), we obtain
Next, we need to transform equations (35) and (36) into singular integral equations of the first kind with Cauchy kernel. In another form, the integral equations (35) and (36) will bewhere
Equations (37) and (38) can be rewritten in the following forms:
By substituting the expressions of and into equations (40) and (41), we obtain
Through changing the order of the integration, we obtain
On the cracks’ surface,where
Ultimately, equations (4) and (5) are reduced to coupling the first-kind Cauchy singular integral equations.
In equations (45) and (46), and are unknown functions with the following conditions:
By the principle of superposition and mixed boundary conditions, equations (45) and (46) can be formulated as
4. Resolution of Singular Integral Equations
4.1. Normalization of Singular Integral Equations
For the convenience of numerical calculations, we define the normalized quantities as follows:
Using equations (51) and (52), the integral equations (49) and (50) can be expressed as
The dislocation density functions and in equations (54) and (55) have the squareroot-type singularity. So we can express , where is a continuous function defined in the interval [−1, 1]. In the same way, may be expressed as
Using the Chebyshev collocation method, one can transform equations (54), (55), and (48) into the system of algebraic equations as follows:where
is the node number of the quadrature formula and and are the zero points of the first and second kinds of Chebyshev polynomials. Equations (58)–(60) are solved numerically to get the solutions of and , which can permit to get the stress and electric displacement.
4.2. Fracture Parameters
The intensity factors are defined by [4]
According to equations (45) and (46), the singular parts of and are
Substituting equations (56) and (57) into equations (66)–(69), we obtain
Substituting equations (70)–(73) into equations (62)–(65), we obtain
The energy release rate of piezoelectric materials has the following form [24]:where
5. Numerical Process and Results
This section is divided into two parts. Firstly, some numerical applications to verify the obtained solutions are given, in the case when we have a single interface crack. Then, the influence of material constants on the energy release rate is studied when we have multiple interface cracks.
Initially, we assume that the piezoelectric substrate is the PZT-5H ceramic. Its parameters are given as [2]where superscript 2 represents the substrate material.
5.1. A Single Interface Crack
To validate the solution, we chose the case that material 1 is a fine-grained coating made of PZT-5H ceramics by plasma spraying technology and material 2 is normal PZT-5H ceramics (where 1 and 2 represent the fine-grained coating and substrate material, respectively), which means that material 1 and material 2 have approximately the same piezoelectric and dielectric constants under suitable conditions, and the elastic modulus of material 1 is larger than that of material 2. So , , and the single interface crack of loaded by an antiplane stress of and a variable electric displacement are considered. The curves of the energy release rate as a function of under different values of are plotted (Figure 2). Then, the same interface crack is loaded by an electric displacement and a variable stress . For different values of , the curves as a function of are plotted (Figure 3).
5.1.1. Studies of the Effect of Stress and Electric Displacement on Energy Release Rate
Effects of electric displacement and stress on the energy release rate for different are shown in Figures 2 and 3, respectively.
5.2. Multiple Interface Cracks
The interaction between multiple interface cracks is investigated. The distance between double interface crack centers is chosen as (where , , and and other material parameters are the same as in the previous selection). The influences of electric displacement on the normalized energy release rate are shown in Figures 4 and 5 and the change rules of the normalized energy release rate and the value of are shown in Figures 6–8 for the interaction between three identical and different Griffith cracks. The rules of the normalized energy release rate versus the value of are shown in Figures 9 and 10 (where , , , , , and and other material parameters are the same as in the previous selection).
5.2.1. Studies of the Effect of Stress and Electric Displacement on Energy Release Rate for Double Griffith Interface Cracks
Effects of electric displacement and stress on the energy release rate for double Griffith interface cracks are shown in Figures 4 and 5.
5.2.2. Studies of the Influence of on Energy Release Rate for Double Interface Cracks
Effects of on the energy release rate for double Griffith interface cracks are shown in Figures 6–8.
5.2.3. Studies of the Influence of on Energy Release Rate for Three Interface Cracks
Effects of on the energy release rate for three interface cracks are shown in Figures 9 and 10.
6. Results and Discussion
Figure 2 displays the variation of the normalized energy release rate against the applied electric displacement . It can be clearly seen that by varying , the curves are parabolas, and as decreases from zero, always decreases. However, when increases from zero, firstly increases and then begins to decrease. The results show that negative electric displacement loading always prevents crack propagation and positive electric displacement loading may prompt or prevent crack propagation, which are consistent with the conclusions of Pak et al. [4, 10, 13]. However, the peak energy release rate of the fine-grained piezoelectric coating/substrate structure is higher than that of the large-grained piezoelectric coating/substrate structure.
Figure 3 indicates the effects of applied stress on the normalized energy release rate . It can be easily seen that as stress increases from zero, always increases. When the stress decreases from zero, the first decreases and then increases, and it indicates the initial closure of a crack followed by a type of crack propagation due to compressive loading. The results indicate that stress loading may prompt or retard crack propagation.
The diversification of the normalized energy release rate versus the applied electric displacement for double identical and different interface cracks is shown in Figures 4 and 5. It is found that the variation of the energy release rate against the applied electric displacement is similar to that shown in Figure 2. However, the normalized energy release rate of the double interface crack has a larger peak compared to that shown in Figure 2. Moreover, the peak values of the energy release rate near the crack tips , are larger than those near the crack tips , . Because of symmetry when the length of double interface cracks is equal, the variations of the energy release rate near the crack tips , and , are the same.
Figures 6–8 present the effects of the values of (where stands for the crack length) on the normalized energy release rate for double identical and different interface cracks. We found that the energy release rate increased with the increase of . The change rate of the inner crack tips is higher with the variation of than that of the outer crack tips. The lines in Figure 8 intersect with the variation of in this case; if the double cracks have the same length, they are consistent with previous conclusions.
Figures 9 and 10 illustrate the change rule of the normalized energy release rate and for three identical and different interface cracks. The results reflect that the values of the energy release rate susceptible to near the inner crack tips are larger than those that approach the outer crack tips, regardless of the number of cracks. At the same time, we noticed that when the value of thickness of coating was small, the value of the energy release rate was large, which indicated that the structure was more safe when the coating was thin. This reflects the superiority of the fine-grained piezoelectric material structure because it can make the structure thinner.
7. Conclusions
In this paper, multiple Griffith cracks at the interface of a fine-grained piezoelectric coating/substrate are studied. The variation of the energy release rate with applied stress, electric displacement, crack length, and coating thickness is demonstrated. Based on the change rule of the energy release rate and various parameters, the following conclusions are drawn:(1)The interaction between cracks, applied electromechanical loads, and coating thickness has a great influence on the energy release rate for Griffith interface cracks. Moreover, the negative electric displacement loading always prevents crack propagation, and positive electric displacement loading may prompt or delay crack propagation. Furthermore, the appropriate stress loading can restrain the crack growth. Therefore, in the practical application of engineering, we can choose the appropriate load to ensure the safety of the structure.(2)For a bimaterial structure, compared with previous works [10, 13], we found that the peak of the energy release rate at the crack tips of a fine-grained piezoelectric coating/substrate interface is larger than that at the crack tips of a large-grained piezoelectric material/substrate structure interface, and thus, the fine-grained piezoelectric coating/substrate structure has safer material properties. Therefore, in order to better resist fractures, fine-grained piezoelectric materials are a better choice in engineering.
Appendix
The expressions of and , are as follows:
Nomenclature
| : | Left and right tips of arbitrary interface cracks |
| : | Unknown functions satisfying the loading displacement |