Preprint for "Small-Scale Domain Switching Near Sharp Piezoelectric Bi-Material Notches"

Small-Scale Domain Switching near Sharp Piezoelectric Bi-material Notches Miroslav Hrstka1), Michal Kotoul 1,2)*,Tomáš Profant 1) and Marta Kianicová2) 1Institute of Solid Mechanics, Mechatronics and Biomechanics, Brno University of Technology, Technická 2896/2, 616 69 Brno, Czech Republic 2 Faculty of Special Technology, Alexander Dubček University, Studentská 2, 911 50 Trenčín, Slovak Republic Keywords: Small-scale domain switching; Bi-material piezoelectric sharp notch; Expanded LekhnitskiiEshelby-Stroh formalism; Two-state H-integral. Abstract. Assuming small-scale domain switching, the size and shape of the domain switching zone ahead of a sharp generally monoclinic piezoelectric bi-material notch is calculated adopting the energetic switching criterion and micromechanical domain switching model suggested by Hwang et al., (Hwang et al., 1995) for a specified material combination, geometry, and polarization orientation. As the piezoelectric bi-material the combination of piezoelectric ceramics PZT-5H and BaTiO3 is considered. The asymptotic in-plane field of a bi-material sharp notch is analysed using the expanded Lekhnitskii-Eshelby-Stroh formalism, (Ting, 1996). Next, the boundary value problem with the prescribed spontaneous strain and polarization within the switching domain is solved and their influence on the in-plane intensity of singularity at the tip of interface crack is computed. The effects of the initial poling direction on the resulting variation of the energy release rates are discussed. INTRODUCTION Smart structures, such as sensors and actuators, are made of ferroelectric ceramics which are inherently brittle and susceptible to cracking; thus, the important issue of their reliability is raised. These devices usually exhibit a multilayered architecture very often with dissimilar material-bonded joints. Due to a mismatch of material properties across interfaces, a stress singularity at the edge of their interface develops. There is numerous experimental evidence that large singular stresses at vertices of joints may be a cause of fractures and failures in joints. For example, actuators are often found to crack around the edges of internal electrodes. The singularity behaviour of dissimilar piezoelectric material bonded joints was examined in a number of papers, see e.g.(X. L. Xu & Rajapakse, 2000) (Chue & Chen, 2002) (Ou & Wu, 2003)(Chen, 2006) (Ou & Chen, 2004) (Hwu, 2014) (Hirai et al., 2012) (Abe et al., 2017) (Hrstka et al., 2019) (Luangarpa & Koguchi, 2018) and references therein. To understand the singularity behaviour of the bonded joints, two important parameters are needed to explore; (1) the order of stress singularity and (2) the intensity of singularity. The conservative integral based on Betti’s reciprocal principle (Sinclair et al., 1984) (Hwu, 2010) (Belov & Kirchner, 1996) (Stern et al., 1976) (Banks-Sills & Sherer, 2002) (Profant et al., 2013) has been proved to be a very efficient too for determining the intensities of singularity at a vertex of the interface in piezoelectric bi-material bonded joints (Hwu, 2014) (Hirai et al., 2012) (Abe et al., 2017) (Hrstka et al., 2019) (Luangarpa & Koguchi, 2018) (Sinclair et al., 1984) (Hwu, 2010) (Belov & Kirchner, 1996). However, this sort of analysis is purely linear and does not consider any nonlinear phenomena dominating * _Corresponding author, email address: [email protected] This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed in the vicinity of the vertex. It is well known that ferroelectric ceramics exhibit strong nonlinear and hysteresis behaviour at a large field strength with domain switching being the source for the hysteresis loop. The domain switching as a possible source of toughening under electrical/mechanical loading in ferroelectrics was analysed by a number of researchers (Park & Sun, 1995) (Hwang et al., 1995) (Zhu & Yang, 1997) (Yang & Zhu, 1998) (Ricoeur & Kuna, 2003) (Z. K. Zhang et al., 2006) (Sheng & Landis, 2007) (Kuna, 2010) (Yang et al., 2001) (Fulton & Gao, 2001a) (Tan et al., 2014) (Fulton & Gao, 2001b) (Rajapakse & Zeng, 2001) (Zeng & Rajapakse, 2001). It was found that domain switching plays an important role in the toughness variation. Due to the concentrated stress and electric fields near the crack tip a domain reorientation takes place. The switched domains induce incompatible strain under the constraint of surrounding un-switched material which leads to the stress redistribution near the crack tip. The apparent toughness of ferroelectrics thus varies due to domain switching. Assuming that these non-elastic processes are restricted to regions around the crack tip, which are small compared to relevant crack lengths, the concepts of linear elastic fracture mechanics can be applied. As a results, several simple domain switching models have been proposed in literature to estimate crack tip process zones in ferroelectrics, which can 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. Angles α1 and α2 denote poling direction of the materials I and II, respectively successfully predict possible toughening or softening effects (Hwang et al., 1995) (Ricoeur & Kuna, 2003) (Rajapakse & Zeng, 2001) (Wang & Zhang, 2007). It should be noted that a large scale domain switching was modelled using the phase field model, which is not restricted to small-scale domain switching and the effects on fracture of ferroelectric materials were analysed in (Schrade et al., 2007) (Wang & Kamlah, 2009) (B. X. Xu et al., 2010) (Sluka et al., 2012) (Abdollahi & Arias, 2012) (Su et al., 2015) (Van Lich et al., 2015) (Yu et al., 2016) (Q. Zhang & Su, 2018). However, it should be noted that all these estimates were performed for crack-like flaws in homogenous materials. There are no attempts to analyse the domain switching and the resulting variation of the intensity of singularity at a vertex of the interface in piezoelectric bi-material bonded joints or at the tip of interface crack. Such a configuration is however typical for sensors and actuators in smart structures and it will be addressed in this paper. PROBLEM FORMULATION AND SOLUTION METHODS General geometry of the considered bi-material notch is illustrated in Fig. 1. Each wedge occupies the region 0 < θ < ω1 or ω2 < θ < 0. With respect to the material symmetry, the monoclinic materials with the symmetry axis parallel to x3 = 0 are considered. Their elasticity and piezoelectricity matrices have the following structure: This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed 𝑪 𝐸 = [ 𝐶 𝐸 11 𝐶 𝐸 12 𝐶 𝐸 13 0 0 𝐶 𝐸 16 𝐶 𝐸 12 𝐶 𝐸 22 𝐶 𝐸 23 0 0 𝐶 𝐸 26 𝐶 𝐸 13 𝐶 𝐸 23 𝐶 𝐸 33 0 0 𝐶 𝐸 36 0 0 0 𝐶 𝐸 44 𝐶 𝐸 45 0 0 0 0 𝐶 𝐸 45 𝐶 𝐸 55 0 𝐶 𝐸 16 𝐶 𝐸 26 𝐶 𝐸 36 0 0 𝐶 𝐸 66 ] , (1) 𝒆 = [ 𝑒 11 𝑒 12 𝑒 13 0 0 𝑒 16 𝑒 21 𝑒 22 𝑒 23 0 0 𝑒 26 0 0 0 𝑒 34 𝑒 35 0 ] , 𝝎 𝜺 = [ 𝜔 𝜀 11 𝜔 𝜀 12 0 𝜔 𝜀 12 𝜔 𝜀 22 0 0 0 𝜔 𝜀 33 ] , (2) where 𝑪 𝐸 is the elastic stiffness tensor at constant electric field, 𝒆 is the piezoelectric tensor, and 𝝎 𝜺 is the dielectric permittivity tensor at constant strain. 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. 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 constitutive relations can be expressed as { 𝜺 ― 𝑬 } = [ 𝑺 ′ 𝐷 𝒈 ′ 𝑇 𝒈 ′ ― 𝜷 ′ σ ] { 𝝈 𝑫 } , (3) where 𝝈 = { 𝜎 1 𝜎 2 𝜎 6 } = { 𝜎 11 𝜎 22 𝜎 12 } , 𝜺 = { 𝜀 1 𝜀 2 𝜀 6 } = { 𝜀 11 𝜀 22 2𝜀 12 } , 𝑬 = { 𝐸 1 𝐸 2 } = { ― ∂ 𝜙 ∂ 𝑥 1 ― ∂ 𝜙 ∂ 𝑥 2 } , 𝑫 = { 𝐷 1 𝐷 2 } (4) are the stress vector, the strain vector, the electric field vector, the electric potential, and the electric displacement vector, respectively, 𝑺 ′ 𝐷 = [ 𝑆 ′ 𝐷 11 𝑆 ′ 𝐷 12 𝑆 ′ 𝐷 16 𝑆 ′ 𝐷 12 𝑆 ′ 𝐷 22 𝑆 ′ 𝐷 26 𝑆 ′ 𝐷 16 𝑆 ′ 𝐷 26 𝑆 ′ 𝐷 66 ] , 𝒈 ′ = [ 𝑔 ′ 11 𝑔 ′ 12 𝑔 ′ 16 𝑔 ′ 21 𝑔 ′ 22 𝑔 ′ 26 ] , 𝜷 ′ 𝜎 = [ 𝛽 ′ 𝜎 11 𝛽 ′ 𝜎 12 𝛽 ′ 𝜎 12 𝛽 ′ 𝜎 22 ] (5) where 𝑺 ′ 𝐷 , 𝒈 ′ , 𝜷 ′ 𝜎 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 𝜀 3 =0 and E3 = 0 as 𝑆 ′ 𝐷 𝑖𝑗 = 𝑆 𝐷 𝑖𝑗 + 𝑔 3 𝑖 𝑔 3 𝑗 𝛽 𝜎 33 = 𝑆 ′ 𝐷 𝑗𝑖 , 𝑔 ′ 𝑖𝑗 = 𝑔 𝑖𝑗 ― 𝛽 𝜎 3 𝑖 𝑔 3 𝑗 𝛽 𝜎 33 , 𝛽 ′ 𝜎 𝑖𝑗 = 𝛽 𝜎 𝑖𝑗 ― 𝛽 𝜎 3 𝑖 𝛽 𝜎 3 𝑗 𝛽 𝜎 33 = 𝛽 ′ 𝜎 𝑗𝑖 , (6) 𝑆 𝐷 𝑖𝑗 = 𝑆 𝐷 𝑖𝑗 𝑆 𝐷 3 𝑖 𝑆 𝐷 3 𝑗 𝑆 𝐷 33 = 𝑆 𝐷 𝑗𝑖 , 𝑔 𝑖𝑗 = 𝑔 𝑖𝑗 𝑔 𝑖 3 𝑆 𝐷 3 𝑗 𝑆 𝐷 33 , , 𝛽 𝜎 𝑖𝑗 = 𝛽 𝜎 𝑖𝑗 𝑔 𝑖 3 𝑔 𝑗 3 𝑆 𝐷 33 = 𝛽 𝜎 𝑗𝑖 (7) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed for 𝑖,𝑗 ≠ 3 , where 𝑺 𝐷 = 𝑪 ― 1 𝐸 ― 𝑪 ― 1 𝐸 𝒆 𝑇 𝝎 ― 1 𝜎 𝒆 𝑪 ― 1 𝐸 , 𝝎 𝜎 = 𝒆 𝑪 ― 1 𝐸 𝒆 𝑇 + 𝝎 𝜺 , 𝒈 = 𝝎 ― 1 𝜎 𝒆 𝑪 ― 1 𝐸 . The inverse form of the constitutive relations in Eq. (3) reads { 𝝈 𝑫 } = [ 𝑪 ′ 𝐸 𝒆 ′ 𝑇 𝒆 ′ ― 𝝎 ′ 𝜀 ] { 𝜺 ― 𝑬 } , (8) where the in-plane elasticity and piezoelectricity matrices under the assumption of the generalized plane strain and short circuit have the following structure: 𝑪 ′ 𝐸 = [ 𝐶 ′ 𝐸 11 𝐶 ′ 𝐸 12 𝐶 ′ 𝐸 16 𝐶 ′ 𝐸 12 𝐶 ′ 𝐸 22 𝐶 ′ 𝐸 26 𝐶 ′ 𝐸 16 𝐶 ′ 𝐸 26 𝐶 ′ 𝐸 66 ] , 𝒆 ′ = [ 𝑒 ′ 11 𝑒 ′ 12 𝑒 ′ 16 𝑒 ′ 21 𝑒 ′ 22 𝑒 ′ 26 ] , 𝝎 ′ 𝜀 = [ 𝜔 ′ 𝜀 11 𝜔 ′ 𝜀 12 𝜔 ′ 𝜀 21 𝜔 ′ 𝜀 22 ] (9) with 𝑒 ′ 𝑖𝑗 = 𝑒 𝑖𝑗 ― 𝜔 𝜀 3 𝑖 𝑒 3 𝑗 𝜔 𝜀 33 , 𝑒 𝑖𝑗 = 𝑒 𝑖𝑗 ― 𝑒 3 𝑖 𝐶 𝐸 3 𝑗 𝐶 𝐸 33 , , 𝑒 ′ 𝑖𝑗 = 𝑒 𝑖𝑗 ― 𝜔 𝜀 3 𝑖 𝑒 3 𝑗 𝜔 𝜀 33 , 𝜔 ′ ε 𝑖𝑗 = 𝜔 ε 𝑖𝑗 ― 𝜔 𝜀 3 𝑖 𝜔 𝜀 3 𝑗 𝜔 𝜀 33 , 𝜔 ε 𝑖𝑗 = 𝜔 ε 𝑖𝑗 + 𝑒 3 𝑖 𝑒 3 𝑗 𝐶 𝐸 33 , , 𝜔 ′ ε 𝑖𝑗 = 𝜔 ε 𝑖𝑗 ― 𝜔 𝜀 3 𝑖 𝜔 𝜀 3 𝑗 𝜔 𝜀 33 = 𝜔 ′ ε 𝑗𝑖 𝐶 ′ 𝐸 𝑖𝑗 = 𝐶 ′ 𝐸 𝑖𝑗 = 𝐶 𝐸 𝑖𝑗 , 𝑒 ′ 𝑖𝑗 = 𝑒 ′ 𝑖𝑗 = 𝑒 𝑖𝑗 , . (10) Characteristic matrices of the material 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. Note that the in-plane problem of the stress singularity at the sharp notch composed of the two monoclinic piezoelectric materials is characterized by two generally complex exponents δ1 -1, δ2 -1 and one real exponent δ3 -1. The resulting displacements and stresses are obtained as the superposition of these particular singular contributions weighted by the generally complex amplitudes. The amplitudes are introduced in the similar manner as for a crack and referred to as the generalized stress intensity factors (GSIFs). The following vectors are introduced 𝒖 = { 𝑢 1 𝑢 2 𝜙 } , 𝑻 = { 𝑇 1 𝑇 2 𝑇 𝐷 } , (11) where u1, u2 are the displacement components, φ is the electric potential, T1, T2 and TD are the components of the resulting tractions and the electric charge q along the semi-infinite line passing through the origin of the coordinate system x1x2. The coupled electromechanical field near the notch vertex can be approximated as 𝒖 ( 𝑟 , 𝜃 ) = 𝐻 1 𝑟 𝛿 1 𝜼 1 ( 𝜃 ) + 𝐻 2 𝑟 𝛿 2 𝜼 2 ( 𝜃 ) + 𝐻 3 𝑟 𝛿 3 𝜼 3 ( 𝜃 ) , 𝑻 ( 𝑟 , 𝜃 ) = 𝐻 1 𝑟 𝛿 1 𝝀 1 ( 𝜃 ) + 𝐻 2 𝑟 𝛿 2 𝝀 2 ( 𝜃 ) + 𝐻 3 𝑟 𝛿 3 𝝀 3 ( 𝜃 ) , (12) where Hi are generalized stress intensity factors, r and θ are polar coordinates, see Fig. 1, and 𝜼 𝑖 ( 𝜃 ) = 𝑨 𝒁 𝛿 𝑖 ( 𝜃 ) 𝒗 𝑖 + 𝑨 𝒁 𝛿 𝑖 ( 𝜃 ) 𝒘 𝑖 , (13) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed 𝝀 𝑖 ( 𝜃 ) = 𝑳 𝒁 𝛿 𝑖 ( 𝜃 ) 𝒗 𝑖 + 𝑳 𝒁 𝛿 𝑖 ( 𝜃 ) 𝒘 𝑖 . where the complex function 𝒁 𝛿 𝑖 and matrices 𝑨 and 𝑳 are defined in Appendix A, the bar above the symbols denotes complex conjugate quantities. 𝒗 𝑖 and 𝒘 𝑖 are eigenvectors pertinent to the singularity exponent eigenvalue problem of the considered sharp notch in Fig. 1. Considering traction and charge free notch faces the following boundary conditions are imposed: 𝑻 I ( 𝜔 1 ) = 0, 𝑻 II ( 𝜔 2 ) = 0. (14) The displacement and traction continuity conditions are prescribed along the interface 𝜃 = 0 as 𝒖 I ( 0 ) = 𝒖 II ( 0 ) , 𝑻 I ( 0 ) = 𝑻 II ( 0 ) . (15) By substituting vector functions 𝝀 𝑖 ( 𝜃 ) and 𝜼 𝑖 ( 𝜃 ) from Eq. (13) into Eqs. (14) and (15), one gets the singularity exponent eigenvalue problem formed by twelve homogeneous algebraic equations, which can be written in the matrix form as [ 𝑿 I 1 𝑿 I 1 𝟎 𝟎 𝟎 𝟎 𝑿 II 2 𝑿 II 2 𝑩 I 0 𝑩 I 0 ― 𝑩 II 0 𝑩 II 0 𝑰 𝑰 ― 𝑰 ― 𝑰 ] . { 𝑳 I 𝒗 I 𝑳 I 𝒘 I 𝑳 II 𝒗 II 𝑳 II 𝒘 II } = 𝟎, (16) where 𝑿 𝑗 = 𝑳 𝒁 𝛿 𝑗 ( 𝜔 𝑗 ) ( 𝑳 ) ― 1 , 𝑿 𝑗 = 𝑳 𝒁 𝛿 𝑗 ( 𝜔 𝑗 ) ( 𝑳 ) ― 1 , ( 𝑗 = 1,2 ) 𝑩 0 = i 𝑨 𝑳 ― 1 , 𝑩 0 = ― i 𝑨 𝑳 ― 1 . (17) and 0 denotes 3 x3 zero matrix on the left-hand side and 12 x 1 zero vector on the right-hand side of Eq. (16). After some algebraic manipulations one receives the characteristic equation for the singularity exponent values in the form det [ 𝑲 ( 𝛿 ) ] = 0, (18) where 𝑲 = 𝑩 I 0 + 𝑩 I 0 𝒀 I 1 ― ( 𝑩 II 0 + 𝑩 II 0 𝒀 II 2 ) ( 𝑰 ― 𝒀 II 2 ) ― 1 ( 𝑰 ― 𝒀 I 1 ) (19) and 𝒀 𝑗 = 𝑿 ― 1 𝑗 𝑿 𝑗 , ( 𝑗 = 1,2 ) . For details see (Hrstka et al., 2019). Let us introduce the following functions: 𝑑 𝝀 𝑖 ( 𝜃 ) 𝑑 𝑥 1 = 𝑳 𝛿 𝑖 𝒁 𝛿 𝑖 ― 1 ( 𝜃 ) 𝒗 𝑖 + 𝑳 𝛿 𝑖 𝒁 𝛿 𝑖 ― 1 ( 𝜃 ) 𝒘 𝑖 , 𝑖 = 1,2,3 𝑑 𝝀 𝑖 ( 𝜃 ) 𝑑 𝑥 2 = 𝑳 𝛿 𝑖 𝒁 𝛿 𝑖 ― 1 ( 𝜃 ) 𝝁 𝒗 𝑖 + 𝑳 𝛿 𝑖 𝒁 𝛿 𝑖 ― 1 ( 𝜃 ) 𝝁 𝒘 𝑖 , 𝑖 = 1,2,3. (20) Using Eq. (20), the asymptotic stresses and electrical displacements can be written as This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed 𝝈 1 ( 𝑟 , 𝜃 ) = ― 𝐻 1 𝑟 𝛿 1 ― 1 𝑑 𝝀 1 ( 𝜃 ) 𝑑 𝑥 2 ― 𝐻 2 𝑟 𝛿 2 ― 1 𝑑 𝝀 2 ( 𝜃 ) 𝑑 𝑥 2 ― 𝐻 3 𝑟 𝛿 3 ― 1 𝑑 𝝀 3 ( 𝜃 ) 𝑑 𝑥 2 , 𝝈 2 ( 𝑟 , 𝜃 ) = = 𝐻 1 𝑟 𝛿 1 ― 1 𝑑 𝝀 1 ( 𝜃 ) 𝑑 𝑥 1 + 𝐻 2 𝑟 𝛿 2 ― 1 𝑑 𝝀 2 ( 𝜃 ) 𝑑 𝑥 1 + 𝐻 3 𝑟 𝛿 3 ― 1 𝑑 𝝀 3 ( 𝜃 ) 𝑑 𝑥 1 , (21 ) where 𝝈 1 = { 𝜎 11 𝜎 12 𝐷 1 } , 𝝈 2 = { 𝜎 21 𝜎 22 𝐷 2 } , 𝝁 = [ 𝜇 1 0 0 0 𝜇 2 0 0 0 𝜇 3 ] , 𝝁 = [ 𝜇 1 0 0 0 𝜇 2 0 0 0 𝜇 3 ] . To determine generalized stress intensity factors in Eq. (12), a conservative line integral for the bimaterial notch developed from Betti’s reciprocal principle mentioned in the Introduction is applied. Contrary to the J-integral, the path-independence of the line integral developed from Betti’s reciprocal principle is also preserved for multi-material stress concentrators if the body forces and charges are neglected. This line integral referred to as the H-integral can be written for a bimaterial notch characterized by angles ω1 and ω2, see Fig. 1, as 𝐻 ( 𝒖 , 𝒖 𝑖 ) = 𝜔 1 𝜔 2 ( 𝒖 𝑇 𝒕 𝑖 ― 𝒖 𝑇 𝑖 𝒕 ) 𝑟 d 𝜃. (22) The vectors 𝒖 𝑇 𝑖 and 𝒕 𝑖 are the auxiliary solutions to the displacements, tractions, electric potential and the charge and correspond to the exponent 𝛿 𝑖 = ― 𝛿 𝑖 . The auxiliary solutions are defined as 𝒖 𝑖 ( 𝑟 , 𝜃 ) = 𝑟 ― 𝛿 𝑖 𝜼 𝑖 ( 𝜃 ) , 𝒕 𝑖 ( 𝑟 , 𝜃 ) = ― 1 𝑟 ∂ 𝑻 𝑖 ( 𝑟 , 𝜃 ) ∂𝜃 = ― 𝑟 ― 𝛿 𝑖 ― 1 𝝀 ′ 𝑖 ( 𝜃 ) , ( 𝑖 = 1,2,3) (23) where 𝜼 𝑖 ( 𝜃 ) = 𝑨 𝒁 ― δ 𝑖 ( 𝜃 ) 𝒗 𝑖 + 𝑨 𝒁 ― δ 𝑖 ( 𝜃 ) 𝒘 𝑖 , 𝝀 ′ 𝑖 ( 𝜃 ) = 𝑳 ( 𝒁 ― 𝛿 𝑖 ( 𝜃 ) ) ′ 𝒗 𝑖 + 𝑳 ( 𝒁 ― 𝛿 𝑖 ( 𝜃 ) ) ′ 𝒘 𝑖 , where (.)’ denotes the differentiation with respect to 𝜃 . Observe that the vectors u and t in Eq.(22) represent either the regular asymptotic or a full field solution obtained numerically e.g. by FEM. In the first case, the vector u is given by Eq.(12)1 and the vector t is given by the derivative of Eq.(12)2 with respect to 𝜃 similarly as 𝒕 𝑖 in Eq. (23)2. The full-field solution reduces to the asymptotic solution if the integration contour shrinks to the notch tip. Since the regular and corresponding auxiliary solutions are orthogonal with respect to the “scalar product“ defined by the integral (22), i.e. 𝐻 ( 𝑟 δ 𝑗 𝜼 𝑗 ( θ ) , 𝑟 ― δ 𝑖 𝜼 𝑖 ( θ ) ) = { const ≠ 0 for 𝑖 = 𝑗 , 0 for 𝑖 ≠ 𝑗 , (24) an important result for the GSIFs evaluation follows as This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed 𝐻 𝑖 = 𝐻 ( 𝒖 𝐹𝐸𝑀 , 𝑟 ― δ 𝑖 𝑐 𝜼 𝑖 ( θ ) ) 𝐻 ( 𝑟 δ 𝑖 𝜼 𝑖 ( θ ) , 𝑟 ― δ 𝑖 𝜼 𝑖 ( θ ) ) ( 𝑖 = 1,2,3 ) , (25) where 𝐻 ( 𝒖 𝐹𝐸𝑀 , 𝑟 ― δ 𝑖 𝑐 𝜼 𝑖 ( θ ) ) = = ω 1 ω 2 ( ( 𝒖 FEM ) 𝑇 𝑟 ― δ 𝑖 ― 1 𝑐 𝝀 𝒊 ′ ( θ ) + 𝑟 ― δ 𝑖 𝑐 𝜼 𝑖 𝑇 ( θ ) 𝒕 FEM ) 𝑟 𝑐 d θ (26) with 𝒖 𝐹𝐸𝑀 and 𝒕 𝐹𝐸𝑀 standing for the FEM approximation to the vectors u and t, respectively, and with rc denoting the radius of the circular path remote from the notch singularity. Elements of vector 𝒕 𝐹𝐸𝑀 along the integrating contour have to be computed from the stresses using the Cauchy formula ti = σijnj, in the matrix form written as 𝒕 𝐹𝐸𝑀 = 𝝈 𝐹𝐸𝑀 𝒏 , where 𝝈 𝐹𝐸𝑀 is the two-dimensional generalized stress tensor and 𝒏 is the outer normal to the domain enclosed by the circular integrating path of the radius rc defined as 𝝈 𝐹𝐸𝑀 = [ 𝜎 𝐹𝐸𝑀 11 𝜎 𝐹𝐸𝑀 12 𝜎 𝐹𝐸𝑀 21 𝜎 𝐹𝐸𝑀 22 𝐷 𝐹𝐸𝑀 1 𝐷 𝐹𝐸𝑀 2 ] , 𝒏 = { cos ( 𝜃 ) sin ( 𝜃 ) } . (27) For to predict the domain switching zone, the energy-based criterion proposed in (Hwang et al., 1995) is applied 𝜎 𝑖𝑗 ∆ ε 𝑖𝑗 + 𝐸 𝑖 ∆ 𝑃 𝑖 ≥ 2 𝑃 𝑠 𝐸 𝑐 , (28) where ∆ ε 𝑖𝑗 and ∆ 𝑃 𝑖 are the changes in the spontaneous strain and the spontaneous polarization during switching, respectively. 𝑃 𝑠 is the magnitude of the spontaneous polarization; and 𝐸 𝑐 the coercive electric field. As a first approximation it is assumed that stresses 𝜎 𝑖𝑗 and electrical fields 𝐸 𝑖 remain unchanged during switching. That means that the linear asymptotic field in Eq. (12) dominates at the notch tip and the loading is thus controlled by the GSIFs Hi. This case is referred to as small scale switching when the size of the switching zone is much smaller than other specimen dimensions. Note that the left-hand side of Eq. (28) represents the specific work dissipated during switching. The threshold value on the right-hand side of Eq. (28) represents approximately half of the area of a polarization hysteresis. Due to electric and/or stress loading the spontaneous polarization of a domain near the notch tip can rotate by 180°, +90° or −90°. Considering that a ferroelectric domain forms an angle α with x1 axis the changes in spontaneous strain ∆ ε 𝑖𝑗 and polarization ∆ 𝑃 𝑖 for 90° domain switchings can be expressed in matrix form as: ∆ 𝜺 = { ∆ 𝜀 1 ∆ 𝜀 2 ∆ 𝜀 6 } = { ∆ 𝜀 11 ∆ 𝜀 22 2∆ 𝜀 12 } = 𝛾 𝑠 { ― cos 2 𝛼 cos 2 𝛼 ― 2sin 2 𝛼 } , (29) ∆ 𝑷 = 2 𝑃 𝑠 [ cos ( 𝛼 ± 3 𝜋 4 ) sin ( 𝛼 ± 3 𝜋 4 ) ] , (30) where 𝛾 𝑠 denotes the spontaneous strain associated with 90° domain switching and ― 3 𝜋 4 and 3 𝜋 4 in Eq. (30) correspond to clockwise and counter clockwise 90° switching, respectively. Note that for 180° ∆ ε 𝑖𝑗 = 0 and ∆ 𝑷 = ― 2 𝑃 𝑠 [ cos 𝛼 sin 𝛼 ] 𝑇 . A variation of the local stress/electric fields near the vertex induced by domain switching leads to the additional GSIFs ∆Hi which are evaluated again using Betti’s reciprocal principle. This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed To start with, consider initial generalized stress tensor { 𝝈 𝑟 ( 𝒖 𝑟 ) , 𝑫 𝑟 ( 𝒖 𝑟 ) } 𝑇 , and initial generalized strain tensor { 𝜺 𝑟 ( 𝒖 𝑟 ) , ― 𝑬 𝑟 ( 𝒖 𝑟 ) } 𝑇 in the body. Generalized stress { 𝝈 𝑟 ( 𝒖 𝑟 ) , 𝑫 𝑟 ( 𝒖 𝑟 ) } 𝑇 developed due to changes in spontaneous strain ∆ 𝜺 and polarization ∆ 𝑷 satisfies equilibrium ∂ { 𝝈 𝑟 𝑫 𝑟 } = 0 in Ω, (31) where the matrix differential operator is defined as ∂ = [ ∂ ∂ 𝑥 1 0 ∂ ∂ 𝑥 2 0 0 0 ∂ ∂ 𝑥 2 ∂ ∂ 𝑥 1 0 0 0 0 0 ∂ ∂ 𝑥 1 ∂ ∂ 𝑥 2 ] , and Ω denotes the whole bimaterial body with the body forces and free charges absent. Further, { 𝝈 𝑟 ( 𝒖 𝑟 ) , 𝑫 𝑟 ( 𝒖 𝑟 ) } 𝑇 satisfies the constitutive law in Ω { 𝝈 𝑟 𝑫 𝑟 ― ∆ 𝑷 } = [ 𝑪 ′ 𝐸 𝒆 ′ 𝑇 𝒆 ′ ― 𝝎 ′ 𝜀 ] { 𝜺 𝑟 ― ∆𝜺 ― 𝑬 𝑟 } , (32) or rewritten as { 𝝈 𝑟 𝑫 𝑟 } = [ 𝑪 ′ 𝐸 𝒆 ′ 𝑇 𝒆 ′ ― 𝝎 ′ 𝜀 ] { 𝜺 𝑟 ― 𝑬 𝑟 } ― { 𝑪 ′ 𝐸 ∆𝜺 𝒆 ′ ∆𝜺 ― ∆ 𝑷 } , (33) with 𝑪 ′ 𝐸 , 𝒆 ′ , 𝝎 ′ 𝜀 having the same structure as given in Eq. (9) and ∆𝜺 = ∆ 𝑷 =0 outside the switching zone Ω 𝑆𝑍 . The boundary and continuity conditions are as follows: 𝒕 𝑟 = [ 𝒏 ] { 𝝈 𝑟 𝑫 𝑟 } = [ 𝑛 1 0 𝑛 2 0 0 0 𝑛 2 𝑛 1 0 0 0 0 0 𝑛 1 𝑛 2 ] ∙ { 𝝈 𝑟 𝑫 𝑟 } = 0 , on ∂ Ω (34) where n1, n2 are components of the unit outer normal vector. Across the boundary of the switching zone ∂ Ω 𝑆𝑍 the tractions and the displacements are continuous ⟦ 𝒕 𝑟 ⟧ = ⟦ [ 𝒏 ] { 𝝈 𝑟 𝑫 𝑟 ― ∆ 𝑷 } ⟧ = 0, ⟦ 𝒖 𝑟 ⟧ = ⟦ { 𝑢 1 𝑟 𝑢 2 𝑟 𝜙 𝑟 } ⟧ = 0 on ∂ Ω 𝑆𝑍 , (35) where the double brackets ⟦ . ⟧ denote a jump of the of quantity inside. Finally, the tractions and displacements are required to be continuous along the interface 𝒖 𝐼 𝑟 = { 𝑢 𝐼 1 𝑟 𝑢 𝐼 2𝑟 𝜙 𝐼 𝑟 } = { 𝑢 𝐼𝐼 1 𝑟 𝑢 𝐼𝐼 2 𝑟 𝜙 𝐼𝐼 𝑟 } = 𝒖 𝐼𝐼 𝑟 , 𝒕 𝐼 𝑟 = 𝒕 𝐼𝐼 𝑟 . (36) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed The generalized strain tensor { 𝜺 𝑟 ( 𝒖 𝑟 ) , ― 𝑬 𝑟 ( 𝒖 𝑟 ) } 𝑇 satisfies { 𝜺 𝑟 ― 𝑬 𝑟 } = ∂ 𝑇 𝒖 𝑟 . (37) For arbitrary virtual displacement 𝛿 𝒖 the weak form of the problem given by the equations (31) -(36) reads 𝛺 ( 𝑪 ′ 𝐸 𝜺 𝑟 ― 𝒆 ′ 𝑇 𝑬 𝑟 ) 𝛿 𝜺 d 𝛺 + 𝛺 ( 𝒆 ′ 𝜺 𝑟 + 𝝎 ′ 𝜀 𝑬 𝑟 ) 𝛿 𝑬 d 𝛺 = = 𝛺 𝑆𝑍 [ 𝑪 ′ 𝐸 Δ 𝜺 𝛿 𝜺 + ( 𝒆 ′ Δ 𝜺 + Δ 𝑷 ) 𝛿 𝑬 ] d 𝛺 . (38) Eq. (38) forms a basis of mixed finite element formulation. Considering the initial generalized stress and strain tensor, the reciprocal theorem for two arbitrary admissible fields 𝒖 and 𝒖 (the auxiliary field in our case) reads ∫ 𝐷 = 𝐷 𝐼 + 𝐷 𝐼𝐼 [ ( { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } ― { 𝝈 𝑟 ( 𝒖 𝑟 ) , 𝑫 𝑟 ( 𝒖 𝑟 ) } ) { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } ― { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } ( { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } ― { 𝜺 𝑟 ( 𝒖 𝑟 ) ― 𝑬 𝑟 ( 𝒖 𝑟 ) } ) ] d 𝑆 = 0 , (39 ) or ∫ 𝐷 = 𝐷 𝐼 + 𝐷 𝐼𝐼 [ { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } ∂ 𝑇 𝒖 ― { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } ∂ 𝑇 𝒖 ― { 𝝈 𝑟 ( 𝒖 𝑟 ) , 𝑫 𝑟 ( 𝒖 𝑟 ) } { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } + { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } { 𝜺 𝑟 ( 𝒖 𝑟 ) ― 𝑬 𝑟 ( 𝒖 𝑟 ) } ] d 𝑆 = 0 , (40 ) where the domain D is any subset of the original domain obtained by excluding the crack tip. The boundary of D is made up of arbitrary contours C1 and C2 circumventing the crack tip and connecting the traction free crack faces. C2 is a remote path whereas C1 is a path very close to the crack tip. Clearly, the integration over the subdomains DI and DII is performed using pertinent generalized stress and strain fields in the respective subdomains. Observe that it holds { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } = { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } 𝑇 [ 𝑪 ′ 𝐸 𝒆 ′ 𝑇 𝒆 ′ ― 𝝎 ′ 𝜀 ] 𝑇 (41) and the last term in Eq. (40) can be written using Eq. (33) as { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } { 𝜺 𝑟 ( 𝒖 𝑟 ) ― 𝑬 𝑟 ( 𝒖 𝑟 ) } = { 𝜺 𝑟 ( 𝒖 𝑟 ) , ― 𝑬 𝑟 ( 𝒖 𝑟 ) } [ 𝑪 ′ 𝐸 𝒆 ′ 𝑇 𝒆 ′ ― 𝝎 ′ 𝜀 ] { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } = { 𝝈 𝑟 ( 𝒖 𝑟 ) , 𝑫 𝑟 ( 𝒖 𝑟 ) } { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } + { 𝑪 ′ 𝐸 ∆𝜺 𝒆 ′ ∆𝜺 ― ∆ 𝑷 } { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } . (42 ) Applying the divergence theorem to the first two terms in Eq. (40) one obtains 𝐶 = 𝐶 1 ∪ 𝐶 2 [ { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } [ 𝒏 ] 𝑇 𝒖 ― { 𝝈 ( 𝒖 ) ,𝑫 ( 𝒖 ) } [ 𝒏 ] 𝑇 𝒖 ] d 𝑠 = ― 𝐷 = 𝐷 𝐼 + 𝐷 𝐼𝐼 { 𝑪 ′ 𝐸 ∆𝜺 𝒆 ′ ∆𝜺 ― ∆ 𝑷 } { 𝜺 ( 𝒖 ) ― 𝑬 ( 𝒖 ) } d 𝑆 . (43 ) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig 5. Domain switching zones for a bi-material notch with the angle 𝜔 1 = 150 loaded by 𝜎 appl 2 = 5 MPa for various orientations of the initial poling direction with respect to the interface; on the left-hand side the detailed view is shown This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed about 90° and the major lobe exhibits switching about -90°. With increasing α the size of the switching zone decreases reaching minimum for 𝛼 ≈ 60 ° . It is interesting to note that the lobe with switching about -90° in Material 2 completely disappears for 𝛼 ≈ 80 ° . Then the size of the switching zone increases again reaching maximum for α =180° and the lobe with switching about 90° disappears. It can be observed in Fig. 5 that domain switching zone for a bi-material notch with the angle 𝜔 1 = 150° develops for various orientations of the initial poling direction in a similar fashion as in the case of the interface crack. Figs. 6-10 show asymptotic displacements, stress components, electric displacement components and electric potential along the circular path with radius r = 1 mm around the interface crack tip for orientations of the initial poling direction 𝛼 = 0 °, 70°, 90°, 120°, 180°. The switching zone markedly influences resulting asymptotic fields as it can be seen when the local GSIFs 𝐻 𝑡𝑖𝑝 𝑖 = 𝐻 𝑖 + 𝛥 𝐻 𝑖 are used to evaluate the asymptotic fields. The results in Figs. 6-10 show that, except of the orientations of the initial poling direction close to limit values 𝛼 = 0 °, 180°, the presence of switching zone for considered material properties of bimaterial sample reduces mechanical asymptotic fields but amplifies electrical asymptotic fields. Remarkable is there an increase of potential difference between crack surfaces, especially for 𝛼 ∈ 90 ° ― 130° accompanied by a large increase of the electric displacement component D2 ahead of the crack tip. Another point is also interesting - whilst before switching is the stress σ11 discontinuous across the interface, after switching it gets closer for 𝛼 ∈ 90 ° ― 130° and the jump is not very significant (the stress is almost continuous). This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig 6. The displacements, stress components, electric displacement components and electric potential along the circular path r = 1 mm for a PZT-5H/BaTiO3 interface crack, 𝛼 = 0°, loading 𝜎 appl 2 = 5 MPa This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig. 7. The displacements, stress components, electric displacement components and electric potential along the circular path r = 1 mm for a PZT-5H/BaTiO3 interface crack, 𝛼 = 70°, loading 𝜎 appl 2 = 5 MPa This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig. 8. The displacements, stress components, electric displacement components and electric potential along the circular path r = 1 mm for a PZT-5H/BaTiO3 interface crack, 𝛼 = 90°, loading 𝜎 appl 2 = 5 MPa This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig. 9. The displacements, stress components, electric displacement components and electric potential along the circular path r = 1 mm for a PZT-5H/BaTiO3 interface crack, 𝛼 = 120°, loading 𝜎 appl 2 = 5 MPa This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig. 10. The displacements, stress components, electric displacement components and electric potential along the circular path r = 1 mm for a PZT-5H/BaTiO3 interface crack, 𝛼 = 180°, loading 𝜎 appl 2 = 5 MPa Fig. 11 shows the conventional stress intensity factors calculated using Eq. (50) as functions of the initial poling direction before switching KII, KI, KIV, and after switching 𝐾 𝑡𝑖𝑝 𝐼𝐼 , 𝐾 𝑡𝑖𝑝 𝐼 , 𝐾 𝑡𝑖𝑝 𝐼𝑉 . It is seen that while before switching the stress intensity factors depend only weakly on the initial poling direction, after switching they strongly vary with the initial poling direction. Mechanical SIFs 𝐾 𝑡𝑖𝑝 𝐼𝐼 , 𝐾 𝑡𝑖𝑝 𝐼 are significantly reduced for the initial poling direction values 𝛼 ∈ 〈 40 °,120° ― 140° 〉 , while the electrical SIF 𝐾 𝑡𝑖𝑝 𝐼𝑉 is significantly amplified within this interval of 𝛼 , which is in accord with the asymptotic fields shown in Figs. 6-10. The ratios 𝐾 𝑡𝑖𝑝 𝐼𝐼 𝐾 𝐼𝐼 , 𝐾 𝑡𝑖𝑝 𝐼 𝐾 𝐼 , 𝐾 𝑡𝑖𝑝 𝐼𝑉 𝐾 𝐼𝑉 are plotted in Fig. 12, which better represent the influence of switching Fig. 11. Stress intensity factors as functions of the orientation of the initial poling direction calculated from Eq. (50) (a) (b) (c) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed Fig. 12. Ratios 𝐾 𝑡𝑖𝑝 𝐼𝐼 𝐾 𝐼𝐼 , 𝐾 𝑡𝑖𝑝 𝐼 𝐾 𝐼 , 𝐾 𝑡𝑖𝑝 𝐼𝑉 𝐾 𝐼𝑉 plotted as functions of the initial poling direction domain independently on the applied load under assumption of small scale switching. It is known, see e.g. (Hwu & Ikeda, 2008), that in case of interface crack the mechanical load alone can induce nonzero value of the electrical intensity factor 𝐾 𝐼𝑉 for in-plane poling. The presented results show that this effect can be distinctly intensified due to poling switching. As already mentioned in the previous section, it is more convenient to compare the energy release rate values to assess the overall influence of the domain switching zone in case of interface crack between two dissimilar piezoelectric materials. The energy release rate before switching, G, and after switching, Gtip, was calculated using Eq. (54) as function of the orientation of the initial poling direction. The total energy release rate, the mechanical energy release rate (labelled in Fig.12 as “without 𝐾 2 𝐼𝑉 ”) and the pure electrical energy release rate were plotted both before and after switching. It is well known that linear piezoelectricity always predicts that the electrical contribution to the energy release rate is negative. Here, the pure electrical energy release rate before switching is negligible in comparison to mechanical one, as it can be seen in Fig. 13, because only mechanical load of the interface crack is considered and the induced electrical intensity factor 𝐾 𝐼𝑉 is low, see red line in Fig.11c. Nevertheless, the switching model predicts that the electrical contribution to the energy release rate after switching becomes dominant and makes the total energy release rate negative for the initial poling direction values 𝛼 ∈ 〈 40 °,150° 〉 , see Fig. 13. Fig. 14 then shows the ratios 𝐺 𝑡𝑖𝑝 𝐺 of the total energy release rate and the mechanical energy release rate only plotted as functions of the initial poling direction. The strong negative electrical contribution to the energy release rate due to switching can be explained using the crack closure integral expression 𝐺 = lim 𝛥𝑎→0 1 2𝑎 𝛥𝑎 0 [ 𝛥 𝑢 1 ( 𝛥𝑎 ― 𝑠 ) 𝜎 12 ( 𝑠 ) + 𝛥𝑢 2 ( 𝛥𝑎 ― 𝑠 ) 𝜎 22 ( 𝑠 ) + 𝐷 2 ( 𝑠 ) 𝛥 𝜙 ( 𝛥𝑎 ― 𝑠 ) ] d 𝑠 (55) with 𝛥 𝑢 𝑖 ,𝛥 𝜙 standing for the jumps of displacements and of the electric potential over the crack faces, and results shown in Figs. 7-9. In these figures one can observe that poling switching induces a remarkable negative increase of potential difference over the crack faces accompanied by a large increase of the electric This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed displacement component D2 ahead of the crack tip. As a results, the last term in in the integrand of the crack closure integral obtains large negative values for the initial poling direction values 𝛼 ∈ 〈 40 °,150° 〉 . Fig. 13. Energy release rate before switching, G, and after switching, Gtip, as function of the orientation of the initial poling direction calculated from Eq. (54) Fig. 14. Ratios 𝐺 𝑡𝑖𝑝 𝐺 plotted as functions of the initial poling direction This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed CONCLUSIONS For the first time a theoretical model was proposed to analyse small-scale domain switching near an interface crack in PZT-5H/BaTiO3 bi-material and its impact upon the energy release rate under pure mechanical loading. The simple energy-based criterion proposed in (Hwang et al., 1995) was applied for prediction of the domain switching zone ahead of the interface crack and the bi-material notch with the angle 𝜔 1 = 150. As boundary conditions only traction and charge free faces were considered. It was shown that the orientation of the initial poling has a marking influence on the size and the shape of the switching zone. The effects of the switching zone upon the local electromechanical field was calculated only for the special case of an interface crack under pure mechanical loading. Consequently, the energy release rates before switching, G, and after switching, Gtip, were calculated as functions of the orientation of the initial poling direction. A remarkable influence of the initial poling direction was found making the total energy release rate negative for the initial poling direction values 𝛼 ∈ 〈 40 °,150° 〉 due to dominant electrical contribution to the energy release rate after switching. To conclude, a robust computational procedure based upon the expanded Lekhnitskii-Eshelby-Stroh formalism was developed to evaluate the influence of domain switching zone on the in-plane asymptotic field of the bi-material sharp notch and the interface crack in a generally monoclinic piezoelectric bimaterial. Betti’s reciprocal principle was applied to calculate the general stress intensity factors and also to capture the influence the domain switching zone, thus avoiding a derivation of weight functions. As a next step, formulation of fracture criteria and experimental verification would be desirable. Also application of the semi-permeable crack boundary conditions and so-called energetically consistent boundary conditions is required to analyse the role of electric crack face boundary conditions on domain switching in case of interface cracks and bi-material notches. APPENDIX A The complex function 𝒁 𝛿 𝑖 in Eq. (13) is defined as follows: 𝒁 𝛿 𝑖 ( 𝜃 ) = diag [ 𝑅 𝛿 𝑖 1 ( 𝜃 ) e i 𝛿 𝑖 𝛹 1 ( 𝜃 ) , 𝑅 𝛿 𝑖 2 ( 𝜃 ) e i 𝛿 𝑖 𝛹 2 ( 𝜃 ) , 𝑅 𝛿 𝑖 3 ( 𝜃 ) e i 𝛿 𝑖 𝛹 3 ( 𝜃 ) ] , 𝒁 𝛿 𝑖 ( 𝜃 ) = diag [ 𝑅 𝛿 𝑖 1 ( 𝜃 ) e ― i 𝛿 𝑖 𝛹 1 ( 𝜃 ) , 𝑅 𝛿 𝑖 2 ( 𝜃 ) e ― i 𝛿 𝑖 𝛹 2 ( 𝜃 ) , 𝑅 𝛿 𝑖 3 ( 𝜃 ) e ― i 𝛿 𝑖 𝛹 3 ( 𝜃 ) ] , (A1) where 𝑅 2 𝑘 ( 𝜃 ) = ( cos 𝜃 + 𝜇 ′ 𝑘 sin 𝜃 ) 2 + ( 𝜇 ′ ′ 𝑘 sin 𝜃 ) 2 , ( 𝑘 = 1,2,3 ) , (A2) 𝛹 𝑘 ( 𝜃 ) = { arctan ( 𝜇 ′ ′ 𝑘 sin 𝜃 cos 𝜃 + 𝜇 ′ 𝑘 sin 𝜃 ) for 𝜃 > ― 𝜋 ― 𝜋 for 𝜃 = ― 𝜋 , ( 𝑘 = 1,2,3 ) . (A3) The symbols 𝜇 ′ 𝑘 , 𝜇 ′ ′ 𝑘 denote real and imaginary part of the material eigenvalue_ 𝜇 𝑘 , which is the root of the following characteristic equation 𝑙 2 ( 𝜇 ) [ 𝑙 4 ( 𝜇 ) 𝜌 2 ( 𝜇 ) ― 𝑚 2 3 ( 𝜇 ) ] = 0, (A4) 𝑙 2 ( 𝜇 ) = 𝑆 ′ 𝐷 55 𝜇 2 ― 2 𝑆 ′ 𝐷 45 𝜇 + 𝑆 ′ 𝐷 44 , 𝑙 4 ( 𝜇 ) = 𝑆 ′ 𝐷 11 𝜇 4 ― 2 𝑆 ′ 𝐷 16 𝜇 3 + ( 2 𝑆 ′ 𝐷 12 + 𝑆 ′ 𝐷 66 ) 𝜇 2 ― 2 𝑆 ′ 𝐷 26 𝜇 + 𝑆 ′ 𝐷 22 , 𝑚 3 ( 𝜇 ) = 𝑔 ′ 11 𝜇 3 ― ( 𝑔 ′ 21 + 𝑔 ′ 16 ) 𝜇 2 + ( 𝑔 ′ 12 + 𝑔 ′ 26 ) 𝜇 ― 𝑔 ′ 22 , 𝜌 2 ( 𝜇 ) = ― 𝛽 ′ 𝜎 11 𝜇 2 + 2 𝛽 ′ 𝜎 12 𝜇 ― 𝛽 ′ 𝜎 22 . (A5) and matrices 𝑨 and 𝑳 in Eq. (13) are This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=4707190 Preprint not peer reviewed