PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471 T. Profant et al / Structural Integrity Procedia 00 (2023) 000 – 000

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the state-of-the-art understanding of this field is still in its initial stage and lots of the fundamental problems regarding the flexoelectricity are unresolved. Here, we will use the results of our recent paper concerning the matched asymptotic expansion analysis (Profant et al, 2023) for the study of crack propagation. The principal advantage is that the amplitude factors in the flexoelectric asymptotic solution do not need to be calculated through finite element simulation of a finite crack, which due to gradient effects requires physical fields to be approximated by conforming elements with C 1 continuity. It is assumed that the asymptotics is only valid within a region around the crack tip that is on the order of the flexoelectric length scale or the strain gradient elasticity (SGE) length scale. The classical Griffith postulate regarding a critical energy release rate G c and the critical crack opening displacement (COD) for the crack to advance are to be applied. The aim is to estimate the contributions of direct flexoelectric effects and strain gradient effects for various combinations of flexoelectric material properties to the expected reduction of the energy release rate and/or the critical COD. While in the paper (Profant et al, 2023) only mechanical boundary loads are considered, in this contribution also electrical boundary loads are taken into account and the coupling of mechanical and electrical quantities is considered. 2. Basic equations The simplest form of the linear constitutive laws for dielectric solid are applied in the following and the strain gradient energy term is adopted from (Mao & Purohit, 2014) An isotropic dielectric solid with direct flexoelectricity is considered, whose energy function per unit volume ( ) , , U  ε ε P depends on the infinitesimal strain tensor (1) its gradient ∇ and the polarization vector field P . A symbol ∇ represents the vector differential operator. The corresponding conjugates, i.e. the Cauchy stress σ , the double stress τ and the electric field E must satisfy the governing equilibrium equations ( ) 2 0 0, 0, 0,    − = −   + = + = σ τ P E (2) where the body force per volume is omitted, 0 is the permittivity of the vacuum and is the electric potential. The isotropic flexoelectric solid is characterized by the Lamé coefficients and , the length scale parameter , the two flexoelectric constants 1 and 2 , the dielectric permittivity and the dependence of the polarization vector field on the strain gradients ∇ . The constitutive equations describe all these circumstances ( ) ( ) 1 1 2 2 , 2 , Tr a f Tr f    − = +   =  −  −   σ I ε P E ε ε ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 2 1 2 2 i i j j i i j j i i j j i i i i i j j j j i i j j j i i i i j j i j i i j j j j i i i i j i j j i j i j j j l f f f l f P P f P P f P P =  + + + + + + =      + + + + + + + +       + + + +   τ σ Pe e Pe e eeP eeP ePe ePe σ e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e (3) where ≡ , ≡ or ≡ , ≡ . Hence the basis vectors and represent the basis ( , ) of a Cartesian or the basis ( , ) of a polar coordinate system. The reciprocal susceptibility constant is given by the relation ( 1 2 =  +  ε u u )