Interaction between Screw Dislocation and Interfacial Crack in Fine-Grained Piezoelectric Coatings under Steady-State Thermal Loading

: The mechanical behavior of ﬁne-grained piezoelectric/substrate structure with screw dislocation and interface edge crack under the coupling action of heat, force and electricity are studied. Using the mapping function method, ﬁrstly, the ﬁnite area plane is transformed into the right semi-inﬁnite plane, then the expression of the temperature ﬁeld is given with the help of the complex function, and then the temperature ﬁeld of the problem is achieved. By constructing the general solution of the governing equation with temperature function, the analytical expression of the image force is derived. Finally, the effects of material parameters, temperature gradient, coating thickness and crack size on image force are analyzed by numerical examples. The results show that the temperature gradient has a very signiﬁcant effect on the image force, and thicker coating is conducive to the stability of dislocation and interface crack.


Introduction
Piezoelectric materials have been extensively used in aviation, aerospace, high-tech precision instruments, intelligent structures, structural health monitoring and other fields due to their unique properties. However, subject to the level of the manufacturing process and the piezoelectric composite structure in the process of serving local damage accumulation or format sample by work environment of the influence of the static, dynamic and random load also can produce all kinds of injury, when these defects and damage further expansion will lead to the overall structural failure, even resulting in economic losses and casualties. Therefore, it was particularly important to study the mechanical behavior of cracks and defects at the interface of piezoelectric composites, which attract a lot of attention [1][2][3][4][5][6]. Gao et al. [7,8] systematically studied the plane problem of piezoelectric material interface defects by using complex function method and Stroh theory. They obtained the piezoelectric materials containing an elliptic hole series approximate solution in the form of plane problems, and based on this as the basic solution of the boundary element method was used to solve more complex problems, at the same time as a special case degenerate elliptic holes crack were derived when the crack in the transverse isotropic piezoelectric body under the action of arbitrary concentrated force and concentrated charge of complex function. Xiao et al. [9] examined the interaction of screw dislocation, circular crack and inclusion in infinite single piezoelectric material under uniform heat flow, but did not discuss the case of piezoelectric composite material with finite size. Ueda [10] discussed the fracture of functionally gradient piezoelectric composites under multi-field coupling by using the Fourier integral transformation method. They calculated that increasing the material gradient helped to reduce the stress intensity factor under pure mechanical loads, and the crack length and material heterogeneity had a great effect on the temperature stress By using the mapping function method, the problem of composite structure with finite thickness was transformed to the right half infinite plane structure. At the same time, the expression of the temperature field was given by using complex function method and Riemann-Schwartz analytic continuation theorem, and the temperature field was obtained. By using the theory of solution of linear equations, the solution of the governing equation with temperature function was regarded as the sum of one particular solution and the general solution of the corresponding governing equation in homogeneous form. According to the construction process of temperature field and the theoretical method adopted, the solution of the governing equation containing temperature function was constructed, and the theoretical expression of the image force was derived from it. Furthermore, the influence of various material parameters on image force were analyzed by numerical calculation.

Problem Formulation
As shown in Figure 1, the fine-grained ceramic powder is prepared by high-efficiency mechanical ball milling, and then bonded with the substrate by plasma spraying technology to form the coating/substrate structure, while an open crack with a length of 2l is along the interface between the coating and substrate. A fine-grained piezoelectric coating/substrate structure was obtained by polarization treatment of coating. The coating is polarized along the z -axis. Assumed that coating and substrate are transversely isotropic. During the heating process of coating preparation, crystal deflection or bending may be caused, resulting in dislocation. Therefore, we assume that the screw dislocation  The control and constitutive equation of elastic field are [9] 0 σ σ The control and constitutive equation of elastic field are [9] ∂σ xz ∂x In Equations (1)-(5) the W, Φ are anti plane displacement and the electric potential, 44 are elastic modulus, e (n) 15 and ε (n) 11 are piezoelectric and dielectric constants, respectively. T n is the temperature change, σ xz , σ yz are stress components, D x and D y are piezoelectric and dielectric constants, respectively. p (n) 1 is pyroelectric constant. the superscript n (n = 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: y are coefficients of thermal conductivity.
Substituting Equations (2)-(5) into Equation (1), we obtained e (n) where ∇ 2 is the two-dimensional Laplacian operator. For the homogeneous form of Equations (7) and (8), let The homogeneous form of Equations (7) and (8) can be rewritten as Introduce the following variables The Equation (9) can be written as follows where without loss of generality, the boundary conditions can be written as The thermal boundary conditions are specified as Considering that the temperature field does not depend on the stress field and displacement field, it can therefore, be calculated separately, and then the expressions of thermal stress and displacement field can be deduced in sequence.

Temperature Field
According to reference [25], the finite interface region problem can be transformed into the right semi-infinite region problem by the following two mapping functions.
From Equation (6), without losing generality, assuming that the thermal conductivity along X-axis and Y-axis is the same, that is, k x = k y = k, it can be obtained According to the theory of complex variable function, the solution of Equation (26) can be expressed in the following form T = Img (ξ). (27) where g (ξ) is the temperature analytic function and g(ξ) = [g 1 (ξ), g 2 (ξ)] T , T = [T 1 , T 2 ] T , ξ = x + iy, Re and Im represent the real and imaginary parts of the complex potential function respectively. Substitute Equation (27) into Equation (6), we obtain q (n) where q (n) y ] T . According to reference [26], the function g 2 (ξ) in the y ≤ 0 plane can be written in the following form At the same time, according to Riemann-Schwarz analytical continuation theorem, the function g 2 (ξ) in the whole plane has the following form Similarly, let the function g 1 (ξ) have the following form in the y ≥ 0 plane In the whole plane, the function g 1 (ξ) is extended to Here, h 0 represents the uniform heat flow components in the X and Y directions, m is the undetermined complex number, and g 1 * (ξ) is a holomorphic function.

Thermal Stress and Electric Displacement Field
According to the theory of solution of the equations, the solution of the in-homogeneous Equations (7) and (8) can be expressed as the sum of the general solution of the corresponding homogeneous equations and a particular solution of the in-homogeneous equation. Therefore, in the following discussion, we will construct the particular solution of Equations (7) and (8) and the corresponding general solution of homogeneous form.
Similar to the solution process of temperature field, the solution of homogeneous Equation (9) can be expressed as Therefore, the stress and potential shift can be expressed as where G(z) = g (z).

]
T . By solving the Equations (52) and (53), we can obtain are holomorphic functions. According to the above discussion, the following functions may be taken as the special solutions of Equations (7) and (8) where Here, f 1 * (z) is a holomorphic function. For the non-homogeneous Equations (7) and (8), the relationship between stress and potential shift is also obtained According to Equations (51), (57), the solutions of Equations (7) and (8) can finally be expressed as Substituting Equation (46) into Equation (57) and find the derivative, then (N 1 −a) )] + f 1 * (z) where Similarly, let the function f 2 (z) have the following form where (60) and (61), we obtain )] + f 1 * (z) Substituting Equation (62) into boundary continuity conditions (14)- (17), we can obtain (63) where According to the solution method of Equation (37), we obtain Substituting Equations (57), (62) and (63) into Equations (2)-(5) can obtain the thermal stress and potential shift field of the problem.

Image Force
According to the above discussion, the general solution of Equations (6) and (7) can be written as: Obviously, it satisfies the boundary continuity conditions (12)-(23). According to Equations (51), (58) and (66), we obtain: According to reference [2], the image force acting on the dislocation can be expressed as: where Substituting Equations (44), (48), (64) into Equations (67)-(69), the image force expression can be obtained: Take the real part and imaginary part of Equation (70) respectively to obtain the image force along the x-axis and y-axis.

Numerical Example
We select cadmium selenide piezoelectric ceramics as the base material and finegrained cadmium selenide piezoelectric ceramics as the coating material. The material parameters are: Substitute z 0 = re iθ into Equation (60) to discuss the variation of image force with angle θ.

Effect of Temperature Gradient on Image Force
. The Figures shows that the image force F x first decreases and then increases with the increase of θ, while the image force first increases and then decreases with the increase of θ. At the same time, it is observed that the peak values of image force F x and F y change greatly for different temperature gradient values when θ ≥ 1.4, indicating that the influence of temperature gradient on image force is significant, but the image force changes little when θ ≤ 1.4. On the other hand, it is found that the peak value of image force F x and F y increases with the increase of temperature gradient.      ( + ) = z F c b h h . It can be seen from the figures that the influence of elastic modulus on image force is similar to that of temperature gradient, that is, image force x F first decreases and then increases with the increase of θ , while image force y F first increases and then decreases with the increase of θ , however, the peak value of the image force corresponding to the elastic modulus is much larger than that corresponding to the temperature gradient, indicating that the influence of the elastic modulus of the material on the image force is more significant than that of the temperature gradient.

Effect of Material Elastic Modulus on Image Force
. It can be seen from the figures that the influence of elastic modulus on image force is similar to that of temperature gradient, that is, image force F x first decreases and then increases with the increase of θ, while image force F y first increases and then decreases with the increase of θ, however, the peak value of the image force corresponding to the elastic modulus is much larger than that corresponding to the temperature gradient, indicating that the influence of the elastic modulus of the material on the image force is more significant than that of the temperature gradient.    ( + ) = z F c b h h . It can be seen from the figures that the influence of elastic modulus on image force is similar to that of temperature gradient, that is, image force x F first decreases and then increases with the increase of θ , while image force y F first increases and then decreases with the increase of θ , however, the peak value of the image force corresponding to the elastic modulus is much larger than that corresponding to the temperature gradient, indicating that the influence of the elastic modulus of the material on the image force is more significant than that of the temperature gradient.

Effect of Coating Thickness on Image Force
(2) 44 great influence on the dislocation. It is also found that the peak value of image force decreases when the coating thickness is large.

Effect of Coating Thickness on Image Force
. It can be seen from the figures that image force F x increases with the increase of θ, and image force F y first increases and then decreases with the increase of θ, but for h 1 = 2 mm, image force F y decreases with the increase of θ. For the coating thickness h 1 , when different values are taken, the image forces F x and F y show great changes, indicating that the coating thickness also has a great influence on the dislocation. It is also found that the peak value of image force decreases when the coating thickness is large.

Effect of Coating Thickness on Image Force
(2) 44 great influence on the dislocation. It is also found that the peak value of image force decreases when the coating thickness is large.
. It can be seen from the figures that image force F x increases with the increase of θ and then tends to be stable, and image force F y first increases and then decreases with the increase of θ. At the same time, when θ ≥ 1.9, the crack length has little effect on the image force F x . Compared with other parameters, the effect of crack length on the image force is not so significant.  Figures 8 and 9 shows the variation law of image force with angle under the same   crack length, and the other parameters are: , ,

Discussion and Conclusions
In this paper, the interaction between screw dislocation and interfacial crack in finegrained piezoelectric/substrate under steady-state thermal load is studied. The effects of material parameters, temperature gradient, coating thickness and crack size on image force are discussed through numerical examples. The results show that: (1) the image force increase with the increase of angle , when 1.4 θ ≥ the growth rate of image force is obviously accelerated, and the temperature gradient has a very significant effect on the image force ; (2) the peak value of image force corresponding to different elastic modulus ratio is larger than that corresponding to temperature gradient, indicating that influence of elastic modulus on image force is more significant than that of temperature gradient; (3) both coating thickness and crack size have an impact on the image force. The impact of crack size on the image force will gradually decrease with the increase of angle . At the same time, the image force peak corresponding to a smaller coating thickness is less. Relatively speaking, a thicker coating helps to the stability of dislocations and interface cracks.

Discussion and Conclusions
In this paper, the interaction between screw dislocation and interfacial crack in finegrained piezoelectric/substrate under steady-state thermal load is studied. The effects of material parameters, temperature gradient, coating thickness and crack size on image force are discussed through numerical examples. The results show that: (1) the image force increase with the increase of angle θ, when θ ≥ 1.4 the growth rate of image force is obviously accelerated, and the temperature gradient has a very significant effect on the image force; (2) the peak value of image force corresponding to different elastic modulus ratio is larger than that corresponding to temperature gradient, indicating that influence of elastic modulus on image force is more significant than that of temperature gradient; (3) both coating thickness and crack size have an impact on the image force. The impact of crack size on the image force will gradually decrease with the increase of angle θ. At the same time, the image force peak corresponding to a smaller coating thickness is less. Relatively speaking, a thicker coating helps to the stability of dislocations and interface cracks.

Conflicts of Interest:
The authors declare that there are no conflict of interest regarding the publication of this paper.