PSI - Issue 80

Xinpeng Tian et al. / Procedia Structural Integrity 80 (2026) 451–461 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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materials that exhibit electric polarization due to strain gradients and/or mechanical stresses induced by electric field gradients. Flexoelectricity offers unconventional ways to enhance the electromechanical coupling response of piezoelectric materials and composites compared to the conventional piezoelectricity, which is a coupling between strain and electric fields. Most notably, flexoelectricity is a size-dependent effect that becomes increasingly significant as the length scale decreases. With the growing trend towards miniaturization of electronic devices, there is a great utilization potential of flexoelectric materials. For more detailed insights into flexoelectricity, readers are referred to the reviews (Tagantsev 1986, Marangati et al. 2006, Hu and Shen 2009). Various numerical methods, such as the mesh free method (Abdollahi et al. 2014, Sladek et al. 2018), higher-order finite element method (FEM) (Sladek et al. 2017), collocation mixed FEM (Tian et al. 2021) and isogeometric analyses (Ghasemi et al. 2017) have been developed in literature to analyze the flexoelectric effect around crack tips. The strength of the flexoelectric effect is strongly influenced by strain gradients generated in dielectrics. It is well known that at the crack tip vicinity, it is a high stress concentration. Therefore, one can expect a large flexoelectric effect therein (Xu at al. 2023ab, Sladek et al. 2017, Tian et al. 2023). This phenomenon has been also supported experimentally in works (Wang et al. 2020, Xu et al. 2023b). The assumption of reduction of the amount of the flexoelectric coefficients in constitutive equation for electric displacement leads to certain simplification of the flexoelectricity theory. The fourth order tensor of the direct flexoelectric coefficient is represented only by two independent parameters (Deng et al. 2017). Another simplification in earlier research works on crack problems is that real 3D cracks are replaced by 2D models, where only dominant strain gradient components are considered (Mao and Purohit 2015, Tian et al. 2022). Therefore, 3D models give more real insight on the flexoelectric effect at the crack tip vicinity due to large number of non-zero strain gradients in constitutive equations. It requires effective computational tools to solve general 3D boundary value problems. In literature on crack problems in flexoelectricity, one can find research works devoted mostly to analyses of 2D crack problems. However, only one work (Hu at al. 2025) has been published recently, where the collocation mixed FEM (CMFEM) has been developed for 3D boundary value problems. The C 0 continuous approximation is applied independently to displacements and strains. Then, the kinematic constraints between these quantities are satisfied by the collocation method at several internal points of elements. The J-integral plays important role as the fracture parameter in classical fracture mechanics, where Cauchy stresses have singularity r -1/2 , with r being the distance from the crack tip. The J-integral is path independent and the integration path can be selected sufficiently far from the crack tip, where accuracy of computed fields utilized for evaluation of the J-integral is much higher than ones close to the crack tip. Therefore, J-integral value representing electromechanical energy release rate for coupled fields is very popular (Abendroth et al. 2002). In literature, one can find J-integral for 2D crack problems in flexoelectric solids only (Sladek et al. 2017, Tian et al. 2019). In this paper, we present J-integral expression for 3D crack problem in flexoelectric solids.

Nomenclature j E

electric intensity vector electric displacement

i D

Q

electric flux

electric potential Cauchy stress higher order stress strain gradients permittivity elastic coefficients strains



ij 

jkl 

ij 

ijk 

ij a

klmn c

ijkl f direct flexoelectric coefficients jklmni g higher order elastic coefficients l

internal length elastic material parameter

i t

traction vector

i R i s i u

higher order traction vector normal derivative of displacement

displacements