Fracture Analysis of Griffith Interface Crack in Fine-Grained Piezoelectric Coating/Substrate under Thermal Loading

Coating often plays a role in monitoring and protecting substrates in engineering applications. Interface cracks between the coating and the substrate can lead to crack growth under the action of external loading and will cause device failure. In this paper, the behavior of a fine-grained piezoelectric coating/substrate with a Griffith interface crack under steady-state thermal loading is studied. The temperature field, displacement field, and electric field of the coupling of thermal and electromechanical problems are constructed via integral transformation and the principle of superposition. Thus, problems are transformed into a system of singular integral equations, and the expressions of thermal intensity factor, thermal stress intensity factor, and electric displacement intensity factor are obtained. We used a numerical calculation and a system of singular equations to obtain the relationship of strength factor with material parameters, coating thickness, and crack size.


Introduction
Piezoelectric materials have played an important role in the production of intelligent structures. Due to strength demand, we often bond another piezoelectric material on the surface of the device to protect it or monitor the device. However, piezoelectric composites are prone to cracks at the interface during the process of fabricating defects and load conditions. This can even lead to structural failure. Therefore, problems of interface fracture of the piezoelectric composite under all kinds of loads have attracted wide concern [1][2][3][4][5][6][7][8][9]. Qin and Mai [10] analyzed the interface crack problem of a piezoelectric bimaterial subjected to combined thermal, mechanical, and electrical loads via Stroh's formalism and the singular integral equation method. However, the influence of material size on the thermal intensity factor has not been considered in the numerical analysis. Ueda [11,12] investigated the problem of a parallel crack in a piezoelectric strip under thermoelectric loading and the FGM material strip containing an embedded crack or an edge crack perpendicular to its boundaries. Wang and Noda [13] studied the problems of the piezoelectric material strip with a Griffith interface crack under thermal loading and discussed the effects of polarization direction, crack size, and location on the thermal strength factor. Kang et al. [14] considered the influences of piezoelectric laminated composite shells with interlaminar stresses under electrical, thermal, and mechanical loads. They concluded that the interlaminar shear stress of the laminated shell can be reduced by selecting the appropriate electric field value. Cook and Vel [15] examined a comprehensive multiscale analysis of laminated plates with integrated piezoelectric fiber composite actuators. They used the coupling method of microscale field variables and macroscale field variables to solve the three-dimensional macroscopic equilibrium equation of the piezoelectric laminated plate under arbitrary boundary conditions solved. Li et al. [16] presented a semianalytical technique to solve fracture behaviors of piezoelectric composites under thermal loading. This method could analytically represent the resulting stress and electric displacement distribution along the radial direction obtained. Kuuttia and Virkkunen [17] justified the surface crack behavior under periodic thermal loads. The finite element method and weight function method could simulate the crack behavior over the entire load cycle. Mojahedin et al. [18] considered the mechanical behavior of a solid circular plate which is made of saturated and unsaturated porous material with piezoelectric actuators under thermal loading. They discussed the effect of porosity on the thermal and mechanical stability of the material. Herrmann and Loboda [19] performed fracture analysis on an electrically impermeable interface crack with contact zones in thermopiezoelectrical bimaterials. Using the admissible directions of the heat and the electrical fluxes, the dependencies of the electrical intensity factors on the intensities of the thermal and electrical fluxes were discussed.
Many researchers have studied the interface fracture problems of piezoelectric materials under loading [20][21][22][23][24]. However, all of polycrystalline materials studied above are composed of multidomain large grains, and such sizes cannot meet the requirements of the current intelligent devices, but fine-grained piezoelectric materials are gaining increasing interest. However, there are few studies on the mechanical properties of fine-grained piezoelectric materials.
As an important fracture parameter that can measure structural safety, the intensity factor plays a crucial role in checking the safety of structures. In this article, a Griffith interface crack in fine-grained piezoelectric coating/substrate under steadystate thermal loading is established. An integral transformation method and superposition principle of the solution of the equation are used to transform the thermal load problem into a singular integral equation to obtain the intensity factor conveniently. Furthermore, the interaction between the intensity factor and material parameters is investigated. The results indicate that the larger elastic modulus and thinner coating thickness improve the safety of the coating/substrate structure. Figure 1 shows that a fine-grained ceramic powder coating and the substrate are bond together by plasma spraying, while the crack with a length of 2l is along the interface between the coating and the substrate. A fine-grained piezoelectric coating/substrate structure was obtained by the polarization treatment of the coating. The coating and substrate are polarized along the Y-axis, and both are transversely isotropic. The thicknesses of the coating and substrate are h 1 and h 2 , respectively. We assumed that the crack faces remain thermally and electrically insulated, and T a and T b are the environment temperatures.

Problem Formulation
The constitutive equations for the elastic field are [12] In Equations (1), (2), (3), (4) and (5), the w and u are the displacement components; ϕ is the electric potential; T is the temperature change; σ xx , σ yy , and σ xy are stress components; D x and D y are electric displacement components;c is the pyroelectric constant. The superscript m(m = 1, 2) stands for the fine-grained piezoelectric coating and piezoelectric substrate, respectively.
Assume that the temperature satisfies the Fourier heat conduction equation, as follows: where k m = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðmÞ x /k ðmÞ y q , k ðmÞ x , and k ðmÞ y are coefficients of thermal conductivity. m = 1, 2.
The equations of equilibrium are [12] c m ð Þ 11 Without loss of generality, the boundary conditions can be written as Substrate Piezoelectric coating Figure 1: Mechanical structure of fine-grained piezoelectric coating/substrate with interface crack. 2 Advances in Mathematical Physics The thermal boundary conditions are specified as where T a ðxÞ and T b ðxÞ are environment temperatures, Q 0 ðxÞ is the thermal flow, and k ð1Þ y and k ð2Þ y are coefficients of thermal conductivity.
Since the temperature field does not depend on the stress and displacement field, it can therefore be calculated separately, and then, the thermal stress and potential displacement field can be solved.

Solution to the Problem
3.1. Temperature Field. By using the Fourier integral transform, the solution of temperature field can be expressed as where k m = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ðmÞ x /k ðmÞ y q , A 1m ðsÞ, and A 2m ðsÞ are the unknown functions ðm = 1, 2Þ.
To solve the problem easily, the density function is defined as follows: According to Equation (11a), the dislocation density function should make the following single-valued conditions valid: Substituting formula (12) into Equations (11a), (11b), (11c) and (11d), we obtain Equations (16), (17) and (18) can be written in the form of a matrix as follows: , Solving the linear Equation (19), we get The expressions of F q and Δ, q = 1, 2, :::12, are given in Appendix A. Obviously, the unknown functions A 11 ðsÞ, A 21 ð sÞ, A 12 ðsÞ, and A 22 ðsÞ depend on I 0 . Once I 0 is determined, the temperature field could be obtained.

Advances in Mathematical Physics
By substituting Equation (12) into Equation (11c), we obtain where Furthermore, Equation (23) can be written as where Through changing the order of the integration on the cracks' surface, we obtain where Equation (26) is a singular integral equation with Cauchy kernel. The density function GðtÞ in Equation (26) has the square root-type singularity. Thus, we can express GðtÞ as follows: where g 0 ðtÞ is a continuous function defined in the interval ½−l, l.
The thermal flow intensity factors are defined by [13].
According to Equation (26), the singular part of k y ð∂T m ðx, 0Þ/∂yÞ is Substituting Equation (28) into Equations (31) and (32), we obtain 3.2. Thermal Stress and Electric Displacement Fields. Similar to the solution of linear equation, the solutions to Equations (7), (8) and (9) of equilibrium for displacements and electric potential consist of a homogeneous solution and a particular solution. Equations (7), (8) and (9) can be written into homogeneous forms as follows: By using the Fourier integral transform, the solutions of Equations (34), (35) and (36) can be expressed as [13] u p x, y ð Þ= 1 2π where 3k , and f k ðsÞ, ðm = 1, 2Þ, are unknown functions to be determined.
In Equations (61), (62) and (63), the unknown function d k ðtÞ, k = 1, 2, 3 with the following conditions: Considering that the solution of the singular integral equation will use Chebychev polynomials which are defined in the interval [−1, 1], we introduce the following normalized quantities: Using Equations (84) and (85), integral Equations (79), (80) and (81) may be expressed as Advances in Mathematical Physics The crack-tip behavior can be characterized by squareroot singularity weighed down by the oscillation. However, because the zone of oscillation is very small [25], the unknown function d k~ð sÞ, k = 1, 2, 3, can be expressed as follows: where g k ðsÞ, k = 1, 2, 3, are continuous functions defined in the interval ½−1, 1.
Using the Lobatto-Chebyshev collocation method, one can transform Equations (79), (80), (81) and (83) into a system of algebraic equations: where χ 0 = χ N = 1 2 , N is the node number of quadrature formula; s R and r q are the zero points of the first and second kinds of Chebyshev polynomials. According to Theocaris and Ioakimidis [26] as well as Lu [27], the convergence rate of the Lobatto-Chebyshev collocation method is first order (i.e., oðhÞ, where h stands for step size).
The intensity factors are defined by [13] Substituting Equation (87)  4.1. Verification. We chose a special case to verify the foregoing derivation. We first assumed that the fine-grained piezoelectric coating and substrate are the same material. The problem in Figure 1 then becomes that of a Griffith crack in a single material plate with thermal loading.
Therefore, in Equation (26), M 0 = −k 1 k 2 k ð1Þ y k ð2Þ y /ðk 1 k ð1Þ y + k 2 k ð2Þ y Þ becomes Thus, the thermal flow intensity factors (29) and (30) can be written as follows: Because the crack is symmetrical about the Y-axis, we get In Figure 2, the effects of ðh 1 + h 2 Þ/l on thermal flow intensity factor for different strip height are displayed (where . It is shown that with an increase in the thickness of the material plate, the size of the thermal strength factor decreases gradually and tends to stabilize. The results show that the thickness of the sheet has a great influence on the thermal strength factor, which is consistent with the conclusion of Wang and Noda [13].

Thermal Intensity Factors of Fine-Grained Piezoelectric
Coating/Substrate Structure . These data indicate that with an increase in the crack length, the thermal stress intensity factor and the electric displacement intensity factor increase, and with the increase of the ratio of the elastic modulus between the fine-grained piezoelectric coating and substrate. The peaks of the thermal stress intensity factor and the electric displacement intensity factor increase correspondingly with the increase in the ratio of the elastic modulus between the fine-grained piezoelectric coating and substrate. Concurrently, when l/h 1 increases from 2, the growth rates of K I and K II are greater than that of K D , and the interface fracture will eventually occur with a continuous increase in l/h 1 .  K D ). This shows that with an increase in crack length, the thermal stress intensity factor and the electric displacement intensity factor increase monotonically. The thermal stress intensity factor and the electric displacement intensity factor have a higher peak value when the coating thickness is small. However, when h 1 /h 2 = 0:01, the influence of l/h 2 on K D is not significant enough. The influence of l/h 2 on K D is obviously significant with an increase of h 1 /h 2 .

Conclusions
The Griffith crack problem of fine-grained piezoelectric coating/substrates under steady-state thermal loading is studied. Based on the principle of integral transformation and superposition of equation solutions, the thermomechanical problem is transformed into a solution of thermal stress and electric displacement intensity factors. We then analyzed the effects of crack geometry, coating thickness, and material modulus of elasticity on the fracture behavior of the crack under thermal load. The results show that a larger elastic modulus and thinner coating thickness improve the safety of the coating/substrate structure. This reflects the advantages of fine-grained structure materials.

Appendix
A.