PSI - Issue 42

Jan Sladek et al. / Procedia Structural Integrity 42 (2022) 1584–1590 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction The delamination is observed frequently in layered structures due to induced high stresses on the interface at a thermal load. Then, for the prediction of failure, it is needed to analyse stress and strain fields near the tip of interface cracks. In literature one can find a lot of studies on interface cracks analysed by the classical theory of elasticity, where the microstructure of the materials is ignored (Askes and Gitman 2009). The microstructure has to be considered if the characteristic length of the material structure is comparable with the size of the structure (Buhlmann et al. 2002; Catalan et al. 2011). The gradient theory has been successfully applied to modelling micro/nano-sized structrures and cracks in homogeneous materials (Altan S, Aifantis 1992; Aifantis 2003; Aravas and Giannakopoulos 2009; Sladek et al. 2017; Hu SL and Shen SP 2009). Results for interface cracks between two dissimilar materials analysed by the gradient theory are very seldom in literature (Itou 1991; Piccoroaz et al. 2012; Kotoul and Profant 2018). In the direct flexoelectricity the electric polarization is induced by strain gradients. Due to the large strain gradients the polarization is induced even in all dielectric materials (Yudin and Tagantsev 2013). The biggest influence of flexoelectricity is observed at the crack tip vicinity since the strain gradients are the largest there (Huang et al. 1999; Georgiadis 2003; Gourgiotis and Georgiadis 2009). Recent experimental observations and analytical studies of the flexoelectric effect near the crack tip have been reported by Wang et al. (2020) and Tian et al. (2022) in homogeneous materials. Numerical investigation of the flexoelectric effect near interface crack tips between two dissimilar materials is still missing. Therefore, we solve this problem in this paper. For this purpose it is needed to have a reliable computational tool. The finite element method (FEM) seems to be a convenient tool to solve general boundary-value problems in the gradient theory of elasticity with flexoelectric effect. The standard C 0 continuous finite element method cannot be applied to solve problems in the gradient theory due to higher order derivatives in the governing equation (Sladek et al. 2017, 2019; Tian et al. 2021). In this paper the mixed FEM is developed for an interface crack between two dissimilar dielectrics, where the C 0 continuous approximation is employed independently for displacement and displacement gradients. The constraints between them are satisfied by collocation inside elements (Tian et al. 2021). Numerical results illustrate the influence of the flexoelectric coefficient and ratio of elastic coefficients on the crack opening displacement and induced electric potential in layered cracked structures. 2. Boundary value problems for direct flexoelectricity In the direct flexoelectricity electro-mechanical fields are coupled by strain gradients. The large strain gradients can break the inversion symmetry in centrosymmetric crystals. Then, the polarization is observed also in dielectrics. The constitutive equations for Cauchy stresses ij  , higher-order stresses ijk  and electric displacement i D in dielectric materials (non piezoelectric) can be written as (Hu and Shen, 2009) ij ijkl kl c   = , (1) where a ij and c ijkl are the permittivity and elastic stiffness tensors, respectively. The direct flexoelectric coefficients are denoted by f ijkl and the higher-order elastic coefficients by g ijklmn . Strains ij  , electric intensity vector j E and strain-gradients  ijk can be expressed through displacements i u and electric potential  , respectively: ( ) , , , / 2, ij i j j i j j u u E   = + = − , (2) ( ) , , , / 2 ijk ij k i jk j ik u u   = = + . (3) In the simplified gradient elasticity, the higher-order elastic parameters jklmni g are reduced to elastic stiffness coefficients klmn c and one additional internal length material parameter l (Gitman et al., 2010) as 2 jklmni li jkmn g l c  = . Similarly two independent parameters 1 f and 2 f are used for expression of the direct flexoelectric coefficients f ijkl , (Deng et al. 2017): jkl ijkl i jklmni mni f E g   = − + i ij j ijkl jkl D a E f  = + ,