## 1 Introduction
[17] examined the plane strain problem of finite permeable piezoelectric bi-material interface cracks under load. They used complex function theory to transform the problem into a linear one, and given analytic expressions of the interface stress and crack tip strength factor in simple forms. Finally, by means of finite element method, the results of the influence of different polarization angles and different material combinations on fracture parameters were given. [PERSON] et al. [18] explored the interaction between helical displacements and three different transverse isotropic piezoelectric semi-planar ternary composites and with hypotenuse cracks under mechanical loads in far-antiplane and in-plane electrical loads. The effects of crack length, crack angle and load position on electric field, stress field, generalized stress intensity factor, image force and crack propagation force were studied by the graphic method. [PERSON] et al. [19] dealed the interaction between multiple holes and cracks in piezoelectric devices under thermal-electro-mechanical loading environment. They used extended finite element method, interaction integral method and generalized Stroh formula to predict the stress intensity factor. However, with the higher and higher requirements for precision instruments, ordinary piezoelectric materials can no longer meet the production requirements of high-end instruments, resulting in a series of new materials such as fine grain piezoelectric materials (the grain size is 100 nm-10 um), nano piezoelectric materials. Reducing the particle size to the submicron level can improve the machinability of the material and the mechanical strength of the resulting device. In recent years, fine-grained piezoelectric materials have been used for cutting and grinding, and some high frequency transducers, microbrakes, and thin buzzers have been fabricated in which the advantages of fine-grained piezoelectric materials have been proved [20; 21]. At the same time, many scholars have studied the relationship between grain size and mechanical properties and piezoelectric effect of dielectric piezoelectric materials through experiments [22; 23; 24]. However, the mechanical, dielectric and piezoelectric properties of fine-grained piezoelectric materials are uncertain with the grain size and preparation conditions, so the theoretical research on the defects of fine-grained piezoelectric materials mostly stays on the normal grain size, and the theoretical research on the interface defects of fine-grained piezoelectric materials was less. In this paper, the interaction between screw dislocations and interface edge cracks in fine-grained piezoelectric/substrate structures under steady thermal loading was studied. 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.