PSI - Issue 52

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

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T. Profant et al/ Structural Integrity Procedia 00 (2023) 000 – 000

scale. However, several experimental studies have demonstrated the size-dependent linear electro-mechanical coupling at micro/nanoscale (Baskaran et al, 2011; Catalan et al, 2011). This occurs when the structure dimensions become comparable to the material length scale and the state of stress at a point is not only dependent on the strain but also on the strain gradient. The electromechanical coupling between polarization and strain gradient is termed as flexoelectricity (Mao & Purohit, 2014; Sladek et al, 2017a). Following its discovery several decades ago, studies of flexoelectricity in solids have been scarce due to the seemingly small magnitude of this effect in bulk samples. The development of nanoscale technologies, however, has renewed the interest in flexoelectricity, as the large strain gradients often present at the nanoscale can lead to strong flexoelectric effects. In fact, the concept of flexoelectricity was firstly originated from liquid crystals (Newnham, 2004). The irregularly-shaped polarized molecules existing in liquid crystals can be reoriented by the mechanical strain gradient. Later, similar strain gradient induced polarization phenomenon was also found in crystalline dielectrics, especially in the perovskites. Contrary to the piezoelectric effect, flexoelectricity is not just limited to non-centrosymmetric materials but it may induce electric polarization in the centrosymmetric material by breaking the material’s symmetry (Yan & Jiang, 2013). Therefore, due to flexoelectric effect, non-piezoelectric materials may also be used to produce piezoelectric composites (Sharma et al, 2010) and thus call out new challenges for researchers in the field of nanotechnology. Since flexoelectricity is a gradient effect, thus size-dependent, it cannot be directly incorporated into continuum mechanics, which does not possess an intrinsic length scale. Rather, flexoelectricity needs to be modelled under the framework of strain-gradient elasticity theory (SGET). The first attempts to capture flexoelectricity within this phenomemological framework pertain to Mindlin (Mindlin, 1968) and Toupin (Toupin, 1962). Recently, the phenomenological description of flexoelectricity and its application has been developed in a number of studies. Maranganti et al. (Maranganti et al, 2006) developed fundamental solutions (Green’s functions) for the governing equations of ferroelectricity, (Majdoub et al, 2008b) (Majdoub et al, 2008a) have shown that flexoelectricity may lead to dramatic increase in the energy harvesting capability for piezoelectric nanostructures. Ultimate strain gradients that any material can withstand is that which exist around a crack tip, hence cracks are therefore a natural ground where to search for flexoelectric effects. Abdollahi at al.(Abdollahi et al, 2015) found that, in the presence of flexoelectricity, the resistance to fracture significantly increases in a size-dependent manner for thin BaTiO3 or PbTiO3 films, and moreover that this toughening is asymmetric with respect to the sign of the polarization. Variational formulation for boundary-value problems for flexoelectric solids and its implementation in the FEM was developed e.g. in (Sladek et al, 2017a)(Mao et al, 2016; Sladek et al, 2018; Sladek et al, 2017b) and applied to solution of crack problems. Joseph et al. (Joseph et al, 2018) performed analytical study of fracture in a flexoelectric double cantilever beam using the strain gradient theory. Comparing to the phenomenological Landau Ginzburg-Devonshire theory,(Tagantsev & Yudin, n.d.) the continuum theory of flexoelectricity is complicated by higher order and nonlocal mechanical and electromechanical couplings in the internal energy, multiple definitions of deformation metrics, and subsequently more comprehensive governing equations and boundary conditions. Physical models of the flexoelectricity were first formulated in terms of microscopic theories. The ionic contribution to flexoelectricity was evaluated for several perovskite ferroelectrics and bi-atomic crystals by Maranganti and Sharma (Maranganti & Sharma, 2009) using the framework offered by Tagantsev (Tagantsev, 1986).Ab initio calculations of this contribution were performed by Hong et al (Hong et al, 2010) and Ponomareva et al (Ponomareva et al, 2012) for SrTiO3, BaTiO3, and their solid solution. The first-principles calculations of the purely electronic contribution to flexoelectricity have been done by Hong and Vanderbilt (Hong & Vanderbilt, 2011) for a number of crystals, including classical perovskites. The concept behind these calculations, stemming from the classical work by Martin (Martin, 1972) was formulated in (Resta, 2010). The electronic contribution to flexoelectricity in carbon nanosystems was evaluated in(Dumitric ǎ et al, 2002)(Kalinin & Meunier, 2008) using ab initio calculations. While both first principles and continuum modelling of flexoelectricity have undergone impressive progress in the past few years, there are strengths and limitations to either approach, suggesting that only a combined effort will eventually prove itself effective. Perhaps a first attempt to unify ab-initio formulation of flexoelectricity and strain gradient elasticity in crystalline insulators was reported in (Stengel, 2016) with claiming that first-principles data can been used as the “exact” reference on which the continuum model is built. We used a similar strategy in prediction of brittle fracture of nanocomponents have (Kotoul et al, 2020; Kotoul et al, 2019), which can be naturally extended into flexoelectric fracture problems. Detailed impacts of flexoelectricity on materials and related applications were treated in recent review papers (Shu et al, 2019)(Wang et al, 2019). They state that although the study of flexoelectricity has an impressive achievement,