PSI - Issue 80

Miroslav Hrstka et al. / Procedia Structural Integrity 80 (2026) 471–492 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000 – 000

472

2

manufacturing process, especially in piezoelectric layer/film and elastic substrate structures. These structures inevitably produce cracks and other defects inside the coating or at the interface junction during the manufacturing process, which reduces the reliability of the device and shorten the service life of the device under the action of loads. Here, the piezoelectric layer of PZT-5H grown on amorphous silicon dioxide (SiO 2 ) substrate is considered. The analysis of the asymptotic in-plane field of interface crack is conducted employing the extended Lekhnitskii-Eshelby Stroh formalism. The initial substrate-induced constant elastic misfit strains are included in the analysis. The asymptotic in-plane field is employed to predict the domain switching zone using the energy-based criterion proposed by Hwang et al. (1995). To the best of our knowledge there is no literature where the effect of the domain switching including the effect of the thermal misfit strain parallel to the interface on the change in singularity intensity at the bi material notch tip has been studied. This particular configuration is common in sensors and actuators within intelligent structures, and it will be the subject of discussion in the present study. 2. Problem formulation In our previous study of small scale switching ahead of bi-material notches we considered bi-material configuration formed by two piezoelectric ceramics and the analysis of the asymptotic in-plane field around a bi-material sharp notch was conducted applying the extended Lekhnitskii-Eshelby-Stroh formalism, see e.g. Ting (1996), while no initial thermal misfit strains were taken into account. In structures which employ piezoelectric elements, piezoelectric materials are coupled to electrodes, which conduct the electric charge, or to insulators, e.g. an underlay or insulating pads between piezoelectric and electrodes, or simply to the body of a construction. Solving problems of bi-materials consisting of combinations of piezoelectric and non-piezoelectric solids requires specific changes in the expanded LES formalism used for modelling of bi-material notches. The first step in the modification of the expanded LES formalism for piezoelectric materials to pure elastic non-piezoelectric materials is to set the piezoelectric constants to zero, i.e. e ijk = 0 for any i , j , k . The elastic and electric fields are then decoupled and both direct and converse piezoelectric effects vanish. The second step is to modify the problem according to the case of an insulator or conductor. Both cases are different from the physical point of view. In the framework of the Lekhnitskii and Stroh formalism, Hwu and Kuo (2010) proposed a method which fulfil the condition of the interface impermeability by reducing the permittivity to a sufficiently small value when modelling an insulator or increasing to a very large value when considering a conductor. Its purpose was to be in agreement with authors in Xu and Rajapakse (2000), Weng and Chue (2004), Chen et al. (2006), who modelled the insulator/piezoelectric bi-material by prescribing electric displacement ( ) 0 0 insulator D  = along the interface. However, the condition for an insulator/piezoelectric interface is not physically exact. The assumption of zero electric displacement expresses an impermeable interface condition, i.e. the surfaces are free of charge. This effect is not violated, if one material has significantly higher permittivity than the second one, e.g. a piezoelectric ceramic in a contact with air (Qin (2013)), which is actually prescribed on the notch face. But this cannot be applicable to an insulator/piezoelectric interface, because relative permittivity of insulators attains a wide range on values. Then, an insulator/piezoelectric bi-material notch can be modelled in the same way as a piezoelectric bi-material, but by prescribing zero piezoelectric constants and given permittivity, if known. Further, it should be noted that the LES formalism is primarily derived for anisotropic materials. In case of an isotropic material the key matrices of LES formalism A and L (see Appendix A) are degenerate or non-semisimple and cannot be no longer inverted. Since most insulators have isotropic properties, the Muskhelishvili complex potential method needs to be implemented in the framework of LES formalism to describe the elastic field of the isotropic material. The last problem relates to the combined (thermal +electro/mechanical) loading. The thermal loading due to cooling down from the processing temperature to room temperature induces residual stresses in the specimen. In case of the electro/mechanical loading, the results obtained from 2D plane strain calculations entirely correspond to those obtained from the full 3D analysis in the centre of the specimen. However, in case of the thermal loading of laminate, certain disproportions are found between the results obtained with 2D and 3D model, respectively. The reason is that the plane strain conditions cannot be fulfilled in case of the thermal loading (because of the material extension in all directions) and the biaxial thermal residual stress  11  0,  33  0,  22 = 0 is not reproduced. To overcome this problem while still taking an advantage of 2D calculations also for the thermal problems, certain modification of the material coefficients of thermal expansion (CTEs), in all directions, has to be performed.