Preprint for "Influence of initial misfit strains on small scale domain switching ahead of interface crack between piezoelectric layer and dielectric isotropic substrate"

Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2025) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi Fracture, Damage and Structural Health Monitoring Influence of initial misfit strains on small scale domain switching ahead of interface crack between piezoelectric layer and dielectric isotropic substrate Miroslav Hrstkaa, Michal Kotoula,b * , Tomáš Profanta, Ferri Aliabadic aInstitute of Solid Mechanics, Mechatronics and Biomechanics, Faculty of Mechanical Engineering, BUT, Technická 2896/2, Brno, 616 69, Czech Republic bFaculty of Special Technology, Alexander Dubček University of Trenčín, Studentska 2, 911 50 Trenčín, Slovak Republic cDepartment of Aeronautics, Imperial College London, Exhibition Rd, SW7 2AZ South Kensington, London, United Kingdom Abstract This study derives the effect of discontinuous initial strain distributions on the small-scale domain switching ahead of the bimaterial notch formed between a piezoelectric layer and dielectric isotropic substrate. As a piezoelectric layer the piezoelectric ceramics PZT-5H grown on the elastic substrate formed by amorphous silicon dioxide (SiO2) is considered. The energetic switching principle and micromechanical domain switching framework proposed by Hwang et al. (1995) is applied. The initial thermal misfit constant strain is included in the constitutive relations. The analysis of the asymptotic in-plane field of the bi-material notch is conducted utilizing the extended Lekhnitskii-Eshelby-Stroh formalism. The asymptotic in-plane field is used to predict the domain switching zone applying the energy-based criterion. The influence of the thermal misfit strain on the size and shape of the switching zone in the piezoelectric layer is computed for various initial poling directions. Keywords: Small-scale domain switching; Piezoelectric interface crack; Expanded LES formalism; Two-state H-integral. 1. Introduction This work follows up our recent papers Hrstka et al. (2025) which investigates small-scale domain switching ahead of the interface crack in the piezoelectric bi-material comprising piezoelectric ceramics PZT-5H and BaTiO3 and its impact on the in-plane intensity of singularity at the tip of interface crack is computed. However, no initial thermal misfit strains have been considered. Initial strain in piezoelectric composites is usually formed during the * Corresponding author: E-mail address: [email protected] PREPRINT 2 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 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 (SiO2) substrate is considered. The analysis of the asymptotic in-plane field of interface crack is conducted employing the extended Lekhnitskii-EshelbyStroh 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 bimaterial 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. eijk = 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 ( ) 00 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. PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 3 The piezoelectric layer of PZT-5H exhibits the tetragonal symmetry. However, because the crystallographic reference frame can be arbitrary rotated about x3 axis with respect to global coordinate system in Fig.1, the elasticity and piezoelectricity matrices possess the structure of monoclinic materials in the global coordinate system, hence the problem is formulated for the monoclinic materials with the symmetry plane at x3 = 0. Fig. 1. Geometry of a bi-material notch characterized by two regions I and II. Notch faces are defined by angles ω1 and ω2. Material interface is always considered at θ = 0. Angle α1 denotes poling direction of the materials I The constitutive laws for a linear elastic piezoelectric material which include homogeneous initial strains due to uniform temperature change ΔT can be written in the matrix form as ,, TT ED TT       −−         ==             −− −−                 σ ε α ε α σ C e S g D E E D eω g β (1) where σ is the stress tensor, ε is the strain tensor (both written in the Voigt notation in the vector form), α is the thermal expansion tensor, E and D are the vectors of electric intensity and electric flux density, CE is the elastic stiffness tensor at constant electric field, e is the stress/charge piezoelectric tensor, and ωε is the dielectric permittivity tensor at constant strain, SD is the elastic compliance tensor at constant electric induction, g is the strain/voltage piezoelectric tensor, and βσ is the dielectric non-permittivity tensor at constant stress (Hwu and Ikeda (2008), Hwu and Kuo (2009), 2010)). The material matrices are related through an inversion as .          =         −−         TT ED C e S g I eω g β (2) The elasticity and piezoelectricity matrices for a monoclinic material poled in the x1x2-plane have the following structure: 11 12 13 16 111 12 22 23 26 222 13 23 33 36 3 33 423 44 45 51 45 55 6 16 26 36 66 00 00 00 , 0 0 0 0 0 0 0 0 00                        = = =                   E E E E E E E E E E E E EEE EE E E E E C C C C C C C C C C C C CC CC C C C C Cσ 111 222 333 423 5 3 13 6 12 12 , 2 2 2                                                 ==                                                     ε (3) PREPRINT 4 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 11 12 13 16 21 22 23 26 34 35 00 00 0 0 0 0     =    e e e e e e e e ee e , 11 12 12 22 33 0 0 00           =     ε ω , 1 2 3     =     E E E E , 1 2 3 D D D   =   D The structure of the matrices SD, g and  β is similar to the structure of their inverse counterparts and is not stated here for the sake of brevity. The directional properties of the matrices depend on the poling axis, which can attain two limit configurations, either it coincides with x1-axis or with x2-axis. Between these states their structure corresponds to the above mentioned monoclinic one. The material constants in the general poling direction are obtained by using transformation relations ( ) 1 1 1 1 , , T DD  − − − − = = = * * * S K S K g Ωg K β Ωβ Ω . (4) where D * S , * g and * σ β are material coefficient matrices with the poling direction aligned with certain principal direction (e.g. in which the properties were measured). The form of the transformation matrices K and Ω are given in Appendix A. External loads are assumed to be parallel to the plane defined by x3 = 0 and plane strain deformations characterized by linear piezoelectricity are considered. These assumptions allow to decouple the in-plane and anti-plane relations and to solve the in-plane and anti-plane problem separately. In the following only the in-plane problem is considered. The g-type constitutive relations for the in-plane problem can be expressed as (Hrstka (2019), Hrstka et al. (2019), Hwu and Ikeda (2008)) '' '' σ ˆ , ˆ ˆ ˆ T ps D T     −  =     −  −   Sg σ ε D Egβ  (5) where the components of the effective plane strain CTE ps  for isotropic thermal expansion α (see Table1) are 13 23 36 33 33 33 1 , 1 , T D D D ps ID D D S S S αS S S   = − − −     (6) and 1 11 1 11 1 11 2 22 2 22 22 6 12 6 12 2 , , , 2 x ED ED x                 −                         = = = = = = =                              −            σ ε E D (7) are the stress vector, the strain vector, the electric field vector, the electric potential, and the electric displacement vector, respectively, ''' 11 12 16 ' ' ' '' 11 12 16 11 12 ' ' ' ' ' ' 12 22 26 ' ' ' '' 21 22 26 12 22 ' ' ' 16 26 66 ˆˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ , , ˆ , ˆ DDD D D D D D D D SSS g g g S S S g g g S S S          = = =          Sgβ (8) PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 5 where ' ' ' ˆˆ ˆ, , D  Sgβ are the compliance matrix at constant induction, the piezoelectric strain/voltage matrix, and the dielectric impermeability matrix at constant stress, respectively, evaluated under the assumption of the generalized plane strain a short circuit 30  = and E3 = 0 as 3 3 3 3 3 3 ' ' ' ' ' 33 33 33 3 3 3 3 3 3 33 33 33 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ , , , , , , , ˆ ˆ ˆ ˆ i j i j i j D D D ij ij ji ij ij ij ij ji D D D i j i j i j D D D ij ij ji ij ij ij ij ji D D D g g g S S S g g S S g S g g S S S g g S S S                         = + = = − = − = +==−==−= (9) for ,3ij , where 1 1 1 1 1 1 1 , , TT D E E E E E     − − − − − − − = − = + =S C C e ω eC ω eC e ω g ω eC . The inverse form of the constitutive relations in Eq. (5) reads '' '' ˆˆ ˆˆ , Tps ET     −  =     −  −      Ce σε DE eω   (10) where the in-plane elasticity and piezoelectricity matrices under the assumption of the generalized plane strain and short circuit have the following structure: ' ' ' 11 12 16 ' ' ' '' 11 12 16 11 12 ' ' ' ' ' ' 12 22 26 ' ' ' '' 21 22 26 21 22 ' ' ' 16 26 66 , ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆ , , ˆ ˆ ˆ               = = =            E E E E E E E E E E C C C e e e C C C e e e C C C Ceω (11) with 3 3 3 3 3 3 '' 33 33 33 3 3 3 3 3 3 'ε ε ε ε 'ε ε 'ε 33 33 33 ' ' ' ' , , , , , , ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ , ˆ , ˆˆ.,                      = − = − = − = − = + = − = = = = = E i j i j i j ij ij ij ij ij ij E i j i j i j ij ij ij ij ij ij ji E E E E ij ij ij ij ij ij e e C e e e e e e e C ee C C C C e e e (12) The thermoelastic constitutive law for the isotropic substrate in plane strain is ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) 11 22 66 10 1 1 2 1 1 2 1 10 1 , 1 1 2 1 1 2 00 1 II II T EE E E E T                       −  + − + −   +  −  =+      + − + −         − −   +  (13) where PREPRINT 6 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 ( ) 12 12 44 44 12 44 12 44 32 , . 2 E E E E EE EE C C C EC CC CC  + == + + The in-plane problem of the stress singularity at the sharp notch composed of the monoclinic piezoelectric material and the isotropic non-piezoelectric material requires some care. The monoclinic piezoelectric material is described in terms of the LES formalism while the isotropic non-piezoelectric material needs to modify the LES formalism by employing the Muskhelishvili complex potentials. We start with the description of the asymptotic field in the monoclinic piezoelectric material referred to as the Material I. The following vectors are introduced both in the case of anisotropic piezoelectric material and isotropic non-piezoelectric one 11 22 ,, D uT uT T          ==             uT (14) where u1, u2 are the displacement components,  is the electric potential, T1, T2 and TD are the components of the generalized stress function vector of the mechanical and electrical quantities. The generalized stress function components T1, T2 and TD are related to the stresses and electric displacements by i1 ,2 i2 ,1 1 ,2 2 ,1 , , 1,2, , . i i D D T T D T D Ti  === − = = − (15) The complex potential for expression of the stress singularity has the form 1 2 3 1 2 diag , , , , 1,2,3, ii z z z z x x i       = = + =  Z (16) where i  are the material eigenvalues and they are obtained by solving the characteristic equation Eq. (A5) (see Appendix A). The coupled electromechanical field near the notch vertex is sought in the form of the ansatz for unknown singularity exponents and corresponding eigenvectors as ( ) ( ) ( ) ( ) ( ) ( ) δδ δδ ,, , , r r       =+ =+ u AZ v AZ w T LZ v LZ w (17) where the complex function  Z and matrices A and L are defined in Appendix A, the bar above the symbols denotes complex conjugate quantities. v and w are eigenvectors pertinent to the singularity exponent eigenvalue problem of the considered sharp notch in Fig. 1. The unknown singularity exponents and eigenvectors are determined through the satisfaction of the boundary conditions at the notch tip. The material eigenvectors matrices A, L in Eq. (17), see also Eq. (A7), are non-degenerate when the poling direction is perpendicular to the x3 axis and semi-degenerate or degenerate otherwise. Only non-degenerate cases are considered thereinafter. In case of the asymptotic field in the isotropic non-piezoelectric material, the structural and electrical constitutive equations are decoupled due to the assumption of zero piezoelectric coefficient ' ˆ g , see Eq. (5). Thus, the modification of the formalism unifies the relations for pure isotropic elasticity with equations describing the electrostatic field. By substituting the material parameters into the in-plane characteristic equation (A5), triple complex conjugate roots 𝜇1,2,3 = 𝑖 are obtained. The complex potentials of an isotropic media (marked with a star) have the form ( ) ( ) ( ) ( ) *d, d z z z z z z = + − f f f Q (18) PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 7 where 000 1 0 0 000   =   Q The complex potentials f(z) are defined as ( ) ( ) ( ) ( ) ( ) , cos sin , z z z z r i z        = = +    f (19) in which ( ) ( ) ( ) 1 2 3 , , .z z v z z v z z v       = = = (20) The displacements and stress have the form ( ) ( ) * * * * * * * * * * , , z z     =+ =+ u A Z v A Z w T L Z v L Z w (21) where the matrices A* and L* are expressed by 33 ** 11 0 44 0 1 1 1 0 , 1 1 0 4 4 2 0 0 2 00 ii Gi Gi ii Gi Gi a    −  −     ==     −     AL (22) with ' 2233 3 ˆ a   = , 34  =− for plane strain and 44 E GC= and ( ) 12 12 44 2 E EE C CC  =+ . The complex functions *  Z are ( ) ( ) 1 *1 e 0 0 00 0 2 e sin e 0 . 0 0 0 0 e i ii i r z z z z z ir r zr                 − −      = − = −        Z (23) Note that the upper left 2×2 matrix describes pure isotropic elasticity, while the third diagonal element describes the electric field. Complex conjugation of the function (23) leads to PREPRINT 8 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 ( ) 1 * e 0 0 2 e sin e 0 0 0 e i ii i r ir r r          − −− − −    =   Z (24) Considering traction and charge free notch faces the following boundary conditions are imposed by: ( ) ( ) I II 12 0, 0.  ==TT (25) The displacement and traction continuity conditions are prescribed along the interface 0  = as ( ) ( ) ( ) ( ) 0 0 , 0 0 , I II I II u u T T== (26) The eigenvalue problem for a bi-material notch composed of a piezoelectric material and an insulator is defined in terms of the equations (17) and (21). A bi-material notch with the geometry in Fig. 1 is considered, where material I is the piezoelectric one and material II non-piezoelectric one defined by elastic constants E ij C and permittivities ij   . Let us define the following identities: ** ** * , , , , , . I I II II I I II II I II     = = = = = = = = == A A, L L, A A L L u u T T, u u T T Z Z Z,Z (27) The eigenvalue problem is introduced by the boundary conditions (25) and (26) and can be written in the matrix form as II II 11 II II II 22 II II I I II II 0 0 0 0 II II 00 00 . 0, Lv XX Lw XX Lv B B B B I I I I Lw            =    −      −−    (28) where ( ) ( ) ( ) ( ) ( ) 1 1 11 00 , , 1,2 i , i   − − −− = = = = = − j j j j j j jX LZ L X LZ L B AL B AL (29) and 0 denotes 3×3 zero matrix on the left-hand side and 12×1 zero vector on the right-hand side of Eq.(28). After some algebraic manipulations one receives the characteristic equation for the singularity exponent values. Definitions for the asymptotic stresses and electric displacements of the piezoelectric monoclinic material and of the isotropic non-piezoelectric material, respectively, have the form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 12 2 2 2 3 12 1 1 1 1 11 1 1 1, 2 2, 3 3, 1 11 2 1 1, 2 2, 3 3, , , ,,             − −− − −− = − − − = + + x x x x x x r H r H r H r r H r H r H r σ λ λ λ σ λ λ λ (30) PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 9 for piezoelectric material I, and ( ) ( ) ( ) ( ) ( ) ( ) 3 12 2 2 2 3 12 1 1 1 1 11 *1 * * * 1 1, 2 2, 3 3, 1 11 *2 * * * 1 1, 2 2, 3 3, , ,           − −− − −− = + + = + + x x x x x x H r H r H r H r H r H r σ λ λ λ σ λ λ λ (31) for isotropic non-piezoelectric material II, where Hi are generalized stress intensity factors and 11 21 12 12 22 12 , .                   ==                     DD σσ (32) The derivatives of the shape functions (subscripts , 𝑥 1 and , 𝑥 2 denote differentiation with respect to x1 and x2) in Eqs.(30) and (31) are given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 11 , 11 , 1 *1 * * * * , 1 *1 * * * , , 1,2,3 , 1,2,3, , 1,2,3, , 1,2,3, ii ii i i i i i i x i i i i i x i i i i i x i i i i i x i i i i i i i i * λ L Z v L Z w λ L Z μv L Z μw λ B Z v B Z w λ L Z v L Z w                           −− −− − − − − = + = = + = = + = = + = (33) where 1 1 22 33 00 00 0 0 , 0 0 00 00         ==        μμ , * 3 1 0 10 20 0 2 ii i −   =−    B and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 1 2 3 i i i 1 2 3 i i i 1 2 3 diag e , e , e , diag e , e , e , i i i i i i i i i i i i i i R R R R R R                                −  −  −   =   =  Z Z (34) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 21 *1 1 1 121 * 1 e 0 0 2 1 e sin e 0 , 0 0 e e 0 0 2 1 e sin e 0 . 0 0 e i ii i i ii i i i                    − −− − − −− −− − − − −−    = − −       =−    Z Z (35) PREPRINT 16 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 Fig. 5. The stress components, displacements, electric displacement components and electric potential of a PZT-5H/SiO2 interface crack on the circular path r = 0.001mm, ω1 = 180°, ω2 = –180°, poling direction α1 = 90°, loaded by σ2appl = 20 MPa and ΔT = –160 °C. PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 17 Fig. 6. The stress components, displacements, electric displacement components and electric potential of a PZT-5H/SiO2 interface crack on the circular path r = 0.001mm, ω1 = 180°, ω2 = –180°, poling direction α1 = 0°, loaded by σ2appl = 20 MPa and ΔT = –160 °C. Table 2. The conventional stress intensity factors KI, KII, KIV calculated for both initial poling directions Stress intensity factor α1 = 90° ΔT = 0 °C α1 = 90° ΔT = –160 °C α1 = 0° ΔT = 0 °C α1 = 0° ΔT = –160 °C KI [MPa m1/2] 3.44 4.34 3.49 4.32 KII [MPa m1/2] -0.27 1.18 -0.73 0.83 KIV [μC m1/2] 0.40 1.18 -1.25 -0.55 Notice, that for both considered initial poling directions α1 =0°, 90° is the change of spontaneous strain Δε12 = 0. The switching zone predicted from the criterion (42) by employing the linear asymptotic field in Eq. (17) with evaluated GSIFs Hi is shown for two selected initial poling directions in Fig. 7 and Fig. 8. For comparison, the switching zone is displayed for both the temperature change ΔT = –160 °C and ΔT = 0 °C. The orientation of the switching angle i.e. 90°or –90°, depends on the criterion (42), priority is given to the orientation for which the criterion (42) is met first. Most surprising is the extraordinary increase in the size of the switching zone by more than one order of magnitude for the initial poling direction 0° when thermal misfit strains are included, cf. Fig. 7. Fig. 7. Domain switching zone for an interface crack loaded by σ2appl = 20 MPa for the initial poling direction 0° with respect to the interface. For comparison, the switching zone is displayed both for temperature change ΔT = –160 °C and ΔT = 0 °C; on the right-hand side the detailed view is shown PREPRINT 18 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 Fig.8. Domain switching zone for an interface crack loaded by σ2appl = 20 MPa for the initial poling direction 90° with respect to the interface. For comparison, the switching zone is displayed both for temperature change ΔT = –160 °C and ΔT = 0 °C; on the right-hand side the detailed view is shown. To understand this peculiar behavior, it is necessary to analyze individual terms in the criterion (42). First of all, notice that for α1 =0° the change in spontaneous strain components are Δε11 < 0, Δε22 > 0 and the change in spontaneous polarization components ΔP1 < 0, ΔP2 >0 or < 0, the latter depending on the orientation of the switching angle. When the thermal misfit strains are included, the stress component σ11 >0 for 0 < θ < π/2, thus making the contribution to the switching criterion negative in this range, and σ11 < 0 for π/2 < θ < π, thus making the contribution to the switching Fig. 9. Comparison of stress components σ11, σ22 in case of the initial poling direction 0° for thermal misfit strains included (ΔT = –160 °C ) and thermal misfit strains not included (ΔT = 0 °C ) Fig. 10. Comparison of electric field components in case of the initial poling direction 0° for thermal misfit strains included (ΔT = –160 °C ) and thermal misfit strains not included (ΔT = 0 °C ) PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 19 criterion positive in this range, cf. Fig. 9. σ22 is greater than 0 in the whole range thus making the contribution to the switching criterion positive, cf. Fig. 9. When the thermal misfit strains are not included, the stress component σ11 >0 in the whole interval of θ and its contribution to the switching criterion is always negative. σ22 is also greater than 0, but about of 15% smaller. The electric field plays a significant role in the switching criterion. The E1 component does not differ much whether or not misfit strains are considered, however the E2 component differs significantly between the two cases, see Fig. 10. While with the thermal misfit strains included is the E2 component negative in the whole range, thus providing a positive contribution to the switching criterion, in the case when the thermal misfit strains are not included is the E2 component negative only in the range 0 < θ < π/2 and having the absolute value more as 3 times lower. The electrical part of the switching criterion is thus the main source of the large increase in the switching zone for the initial poling direction 0° under conditions of thermal misfit strains. However, it should be emphasized that despite the large increase in the switching zone in the presence of thermal misfit strains for an initial polarization of 0°, the size of the switching zone is still an order of magnitude smaller than the size of the switching zone for an initial polarization of 90°. The explanation of the increase in the size of the switching zone for polarization 90° due to thermal misfit strain can be done in a similar vein as for the initial polarization 0°. However, since the differences between the mechanical and electrical quantities in this case are not as significant as in the previous case, see Fig. 11 and 12, the increase in the switching zone due to thermal misfit strain is less pronounced. Fig. 11. Comparison of stress components σ11, σ22 in case of the initial poling direction 90° for thermal misfit strains included (ΔT = –160 °C ) and thermal misfit strains not included (ΔT = 0 °C ) Fig. 12. Comparison of electric field components in case of the initial poling direction 90° for thermal misfit strains included (ΔT = –160 °C ) and thermal misfit strains not included (ΔT = 0 °C ) PREPRINT 20 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 4. Concluding remarks The presented work studied effect of initial misfit strains emerging from temperature loading of a bi-material notch composed of piezoelectric layer PZT-5H and non-piezoelectric substrate SiO2. The work followed up author's previous research where robustness and settings of the algorithm were tested. The radius of the integration path encircling the crack tip was set to r=1 mm, so as the radius of the circle of the domain integral containing temperature strains. As the gradient of complex potential of the auxiliary strains is large in the close vicinity of the crack tip, the integration over finite elements was realized by incorporating 7-point Gauss quadrature to achieve acceptable accuracy. The mesh sensitivity study was not performed as these aspects has been tested in the previous author's works. It has been shown that the temperature change generating initial thermal strains has significant impact to the domain switching zone shapes. The effect is stronger for the piezoelectric material poled in the x1 axis, even though the domain switching zone was smaller than for poling in x2 direction. Since the formation of the switching zone significantly affects the shielding or anti-shielding of the crack tip, leading to an increase or decrease in the apparent fracture toughness, it is obvious that the thermal misfit strain arising from the cooling from the poling temperature of the piezoceramic layer on the substrate will have a significant indirect impact on the resulting apparent toughness of the investigated bimaterial structure. Therefore, the next step is to solve the boundary value problem with the prescribed spontaneous strain and polarization within the switching domain using FEM and to employ the computed electroelastic field in the Betti’s reciprocal principle with the aim to calculate the new local GSIF tip i H according to Eq. (44). The results of the calculations will be presented at the conference. Data availability Regarding the computational procedures see Hrstka, M. (2025). Data for “Influence of initial misfit strains on small scale domain switching ahead of interface crack between piezoelectric layer and dielectric isotropic substrate” (1.0.0). Zenodo. https://doi.org/10.5281/zenodo.17229890. Acknowledgements The authors acknowledge the supports by the project BAANG – "Building Actions in Smart Aviation with Environmental Gains" funded by the European Union Programme Horizon Europe under grant agreement no. 101079091 and by the project "Mechanical Engineering of Biological and Bio-inspired Systems", funded as project No. CZ.02.01.01/00/22_008/0004634 by Programme Johannes Amos Commenius, call Excellent Research. Appendix A The transformation matrices in Eq. (4) are defined as: 22 22 22 cos sin 0 0 0 2cos sin sin cos 0 0 0 2cos sin 0 0 1 0 0 0 0 0 0 cos sin 0 0 0 0 sin cos 0 cos sin cos sin 0 0 0 cos sin                     −     =  −      −−   K (A1) PREPRINT M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 21 cos sin 0 sin cos 0 0 0 1     =−    Ω The complex function i  Z in Eq.(16) is defined as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 1 2 3 i i i 1 2 3 i i i 1 2 3 diag e , e , e , diag e , e , e , i i i i i i i i i i i i i i R R R R R R                                −  −  −   =   =  Z Z (A2) where ( ) ( ) ( ) ( ) 22 2 ' '' cos sin sin , 1,2,3 , k k k Rk       = + + = (A3) ( ) ( ) '' ' sin arctan for , 1,2,3 cos sin for k kkk          −    = = +  − = −  (A4) The symbols ' k  , '' k  denote real and imaginary part of the material eigenvalue k  , which is the root of the following characteristic equation ( ) ( ) ( ) ( ) 2 2 4 2 3 0,l l m       −=  (A5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ' 2 ' ' 2 55 45 44 ' 4 ' 3 ' ' 2 ' ' 4 11 16 12 66 26 22 ' 3 ' ' 2 ' ' ' 3 11 21 16 12 26 22 ' 2 ' ' 2 11 12 22 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 ˆˆ , 2 ˆ ˆ ˆ , 2 2 , , 2 D D D D D D D D D S S S l S S S S S S m g g g g l gg                       = − + = − + + − + = − + + + − = − + − (A6) and matrices A and L in Eq.(17) are 11 12 13 1 2 21 22 23 31 32 1 2 33 3 33 , 1 1 , 1 a a a a a a a a a       − − −     ==     − − −   AL (A7) where PREPRINT 22 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000–000 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ' ' ' ' ' 1 11 12 16 11 21 2 ' ' ' ' ' 2 12 22 26 12 22 2 ' ' ' ' ' 3 21 22 2 3 6 12 22 2 ' ' ' 13 11 12 163 , 1, 2 / ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆˆ , 1,2 , / , 1,2 ˆ ˆ ˆ ˆ ˆ ˆ D D D k k k k k D D D k k k k k k k k k k k k D D D a S S S g g k a S S S g g k a g g g k a S S S                     = + − + − =  = + − + − =   = + − + − + =   = + − ( ) ( ) 33 3 3 3 3 3 33 3 3 3 3 3 '' 11 21 2 ' ' ' ' ' 23 12 22 26 12 22 2 ' ' ' ' ' 21 22 26 12 22 ˆˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ, ˆ , / / ˆ , D D D gg a S S S g g a g g g               +−  = + − + −    = + − − +  (A8) and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 23 22 23 24 , 1,2 , . 3 . kk k kk kk k kk lm k l lm k ll       = − = = − = (A9) References Chen, M.C; Zhu, J.J.; Sze, K.Y.,2006. Electroelastic singularities in piezoelectric-elastic wedges and junctions. Engineering Fracture Mechanics, 73, pp. 855–868. 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Domain switching induced fracture toughness variation in ferroelectrics. Smart Materials and Structures, 10, pp.203–211. PREPRINT