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

5. Conclusions The paper offers a novel method to estimate the minimum condition for the crack to advance in dielectric material obeying the direct flexoelectric laws by matching the inner and outer asymptotic solution of the fracture problem. It is assumed that the inner 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. On the other hand, the outer asymptotic solution suppresses the influence of the strain gradients and the polarization of the material and it coincides with the asymptotic solution from the classical linear elastic fracture mechanics. It makes possible to estimate the amplitude factors of the inner asymptotic solution as functions of the stress intensity factors and and thus avoid demanding numerical calculations using FEM. Moreover, analytical expressions were obtained which allow to easily analyze an influence of various parameters. The classical Griffith postulate regarding a critical energy release rate G c was applied. 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 were estimated. The criterion for the crack to advance was analyze for the conductive crack boundary conditions under mixed mode when pure mechanical Mode I is combined with electrical mode E . Subsequently, it would be desirable to extend the analysis for other cases of boundary conditions. Acknowledgements The authors acknowledge the supports by the Scientific Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic and the Slovak Academy of Sciences VEGA-1/0327/21. References Abdollahi, A., Peco, C., Millán, D., Arroyo, M., Catalan, G., et al, 2015. Fracture toughening and toughness asymmetry induced by flexoelectricity. Phys Rev B Condens Matter Mater Phys , 92(9), p.094101. Baskaran, S., He, X., Chen, Q., Fu, J.Y., 2011. Experimental studies on the direct flexoelectric effect in α -phase polyvinylidene fluoride films. Appl Phys Lett , 98(24), p.242901. Cady, W.Guyton., 2018. Piezoelectricity : an Introduction to the Theory and Applications of Electromechanical Phenomena in Cr ystals. , p.451. Catalan, G., Lubk, A., Vlooswijk, A.H.G., Snoeck, E., Magen, C., et al, 2011. Flexoelectric rotation of polarization in ferroelectric thin films. Nature Materials 2011 10:12 , 10(12), p.963 – 967. Dumitricǎ, T., Landis, C.M., Yakobson, B.I., 2002. Curvature -induced polarization in carbon nanoshells. Chem Phys Lett , 360(1 – 2), p.182 – 188. Gourgiotis, P.A., Georgiadis, H.G., 2009. Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity. J Mech Phys Solids , 57(11), p.1898 – 1920. Hong, J., Catalan, G., Scott, J.F., Artacho, E., 2010. The flexoelectricity of barium and strontium titanates from first principles. Journal of Physics: Condensed Matter , 22(11), p.112201. Hong, J., Vanderbilt, D., 2011. First-principles theory of frozen-ion flexoelectricity. Phys Rev B Condens Matter Mater Phys , 84(18), p.180101. Joseph, R.P., Zhang, C., Wang, B.L., Samali, B., 2018. Fracture analysis of flexoelectric double cantilever beams based on the strain gradient theory. Compos Struct , 202, p.1322 – 1329. Kalinin, S. V., Meunier, V., 2008. Electronic flexoelectricity in low-dimensional systems. Phys Rev B Condens Matter Mater Phys , 77(3), p.033403. Kotoul, M., Ševeček, O., Profant, T., 2010. Analysis of multiple cracks in thin coating on orthotropic substrate under mechan ical and residual stresses. Eng Fract Mech , 77(2), p.229 – 248. Kotoul, M., Skalka, P., Profant, T., Friák, M., Ř ehák, P., et al, 2019. Ab initio aided strain gradient elasticity theory in prediction of nanocomponent fracture. Mechanics of Materials , 136. Kotoul, M., Skalka, P., Profant, T., Řehák, P., Šesták, P., et al, 2020. A novel multiscale approach to brittle f racture of nano/micro-sized components. Fatigue Fract Eng Mater Struct , 43(8). Leguillon, D., Lacroix, C., Martin, E., 2001. Crack deflection by an interface - Asymptotics of the residual thermal stresses. Int J Solids Struct , 38(42 – 43), p.7423 – 7445. Leguillon, D., Lacroix, C., Martin, E., 2000. Interface debonding ahead of a primary crack. J Mech Phys Solids , 48(10), p.2137 – 2161. Majdoub, M.S., Sharma, P., Çaǧin, T., 2008a. Dramatic enhancement in energy harvesting for a narrow range of dimensions in piezoelectric nanostructures. Phys Rev B Condens Matter Mater Phys , 78(12), p.121407.