Abstract
The flexoelectric effect is a significant electromechanical coupling phenomenon between strain gradients and electric polarization. Since the design of materials with high flexoelectricity should be accompanied with stress concentration/intensity, the strength and fracture analysis of flexoelectric materials with large strain gradients is desired. The famous J-integral can be used to characterize the singularity at crack tips and predict the fracture behavior of flexoelectric solids. However, the definition of J-integral in flexoelectric solids is lacked or incomplete in the open literature. In this study, an explicit expression of J-integral associated with material configurational forces is derived from the gradient operation of electric enthalpy density function for centrosymmetric flexoelectric solids, where the electric enthalpy density depends not only on the strain and strain gradient, but also on the polarization and polarization gradient. The path-independence of J-integral in flexoelectric solids is also examined through the Gauss–Green’s theorem. Then the derived J-integral is applied to study a cylindrical cavity and a mode III crack problem in flexoelectric solids. The results indicate that, in flexoelectric solids, there is a conservation law of the J-integral. That is, the J-integral defined in a global coordinate system vanishes when the integration contour chosen to calculate the J-integral encloses whole cavity. The present complete expression of J-integral in flexoelectric solids is addressed from the self-consistent theory of flexoelectricity. It corrects the inaccurate definition of J-integral in the previous literature. The J-integral obtained in this paper will provide a useful way to study fracture problems in flexoelectric solids.
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References
Abdollahi A, Millán D, Peco C, Arroyo M, Arias I (2015) Revisiting pyramid compression to quantify flexoelectricity: a three-dimensional simulation study. Phys Rev B 91:104103
Abdollahi A, Peco C, Millán D, Arroyo M, Arias I (2014) Computational evaluation of the flexoelectric effect in dielectric solids. J Appl Phys 116:093502
Abdollahi A, Peco C, Millán D, Arroyo M, Catalan G, Arias I (2015) Fracture toughening and toughness asymmetry induced by flexoelectricity. Phys Rev B 92:094101
Ahmadpoor F, Sharma P (2015) Flexoelectricity in two-dimensional crystalline and biological membranes. Nanoscale 7:16555–16570
Aravas N (2011) Plane-strain problems for a class of gradient elasticity models—a stress function approach. J Elast 104:45–70
Aravas N, Giannakopoulos AE (2009) Plane asymptotic crack-tip solutions in gradient elasticity. Int J Solids Struct 46:4478–4503
Askar A, Lee PCY (1974) Lattice-dynamics approach to the theory of diatomic elastic dielectrics. Phys Rev B 9:5291–5299
Askar A, Lee PCY, Cakmak AS (1970) Lattice-dynamics approach to the theory of elastic dielectrics with polarization gradient. Phys Rev B 1:3525–3537
Askar A, Lee PCY, Cakmak AS (1971) The effect of surface curvature and discontinuity on the surface energy density and other induced fields in elastic dielectrics with polarization gradient. Int J Solids Struct 7:523–537
Baskaran S, Ramachandran N, He X, Thiruvannamalai S, Lee HJ, Heo H, Chen Q, Fu JY (2011) Giant flexoelectricity in polyvinylidene fluoride films. Phys Lett A 375:2082–2084
Catalan G, Lubk A, Vlooswijk AHG, Snoeck E, Magen C, Janssens A, Rispens G, Rijnders G, Blank DHA, Noheda B (2011) Flexoelectric rotation of polarization in ferroelectric thin films. Nat Mater 10:963–967
Chen YH (2002) Advance in conservation laws an energy release rates: theoretical treatments and applications. Kluwer Academic Publishers, Netherlands
Chen YH, Lu TJ (2001) Conservation laws of the \(J_{k}\)-vector for microcrack damage in piezoelectric materials. Int J Solids Struct 38:3233–3249
Cherepanov GP (1967) Cracks propagation in continuous media. J Appl Math Mech 31:503–512
Deng F, Deng Q, Yu W, Shen S (2017) Mixed finite elements for flexoelectric solids. ASME J Appl Mech 84:081004
Deng Q, Liu L, Sharma P (2014) Flexoelectricity in soft materials and biological membranes. J Mech Phys Solids 62:209–227
Eshelby JD (1951) The force on an elastic singularity. Philos Trans R Soc A 244:87–112
Kogan SM (1964) Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals. Sov Phys Solid State 5:2069–2070
Lee D, Yoon A, Jang SY, Yoon JG, Chung JS, Kim M, Scott JF, Noh TW (2011) Giant flexoelectric effect in ferroelectric epitaxial thin films. Phys Rev Lett 107:057602
Li Q, Kuna M (2012) Inhomogeneity and material configurational forces in three dimensional ferroelectric polycrystals. Eur J Mech A Solid 31:77–89
Ma W, Cross LE (2002) Flexoelectric polarization of barium strontium titanate in the paraelectric state. Appl Phys Lett 81:3440–3442
Majdoub MS, Sharma P, Cagin T (2008) Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys Rev B 77:125424
Mao S, Purohit PK (2014) Insights into flexoelectric solids from strain-gradient elasticity. ASME J Appl Mech 81:081004
Mao S, Purohit PK (2015) Defects in flexoelectric solids. J Mech Phys Solids 84:95–115
Mao S, Purohit PK, Aravas N (2016) Mixed finite-element formulations in piezoelectricity and flexoelectricity. Proc R Soc A Math Phys 472:20150879
Maranganti R, Sharma ND, Sharma P (2006) Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Phys Rev B 74:014110
Mashkevich VS, Tolpygo KB (1957) Electrical, optical and elastic properties of diamond type crystals. Sov Phys JETP 5:435–439
Mindlin RD (1968) Polarization gradient in elastic dielectrics. Int J Solids Struct 4:637–642
Mohammadi P, Liu LP, Sharma P (2014) A theory of flexoelectric membranes and effective properties of heterogeneous membranes. ASME J Appl Mech 81:011007
Pak YE (1990) Crack extension force in a piezoelectric material. ASME J Appl Mech 57:647–653
Pan S, Li Q, Liu Q (2017) Ferroelectric creep associated with domain switching emission in the cracked ferroelectrics. Comput Mater Sci 140:244–252
Rice JR (1968) A path independent integral and approximate analysis of strain concentration by notches and cracks. ASME J Appl Mech 35:379–386
Sahin E, Dost S (1988) A strain-gradients theory of elastic dielectrics with spatial-dispersion. Int J Eng Sci 26:1231–1245
Sharma ND, Maranganti R, Sharma P (2007) On the possibility of piezoelectric nanocomposites without using piezoelectric materials. J Mech Phys Solids 55:2328–2350
Shen S, Hu S (2010) A theory of flexoelectricity with surface effect for elastic dielectrics. J Mech Phys Solids 58:665–677
Tagantsev AK (1986) Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys Rev B 34:5883–5889
Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414
Trabi CL, Brown CV, Smith AAT, Mottram NJ (2008) Interferometric method for determining the sum of the flexoelectric coefficients \((e_1+e_3)\) in an ionic nematic material. Appl Phys Lett 92:223509
Yu PF, Chen JY, Wang HL, Liang X, Shen SP (2018) Path-independent integrals in electrochemomechanical systems with flexoelectricity. Int J Solids Struct 147:20–28
Zhang L, Huang Y, Chen JY, Hwang KC (1998) The mode III full-field solution in elastic materials with strain gradient effects. Int J Fract 92:325–348
Zhang X, Sharma P (2005a) Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems. Int J Solids Struct 42:3833–3851
Zhang X, Sharma P (2005b) Size dependency of strain in arbitrary shaped anisotropic embedded quantum dots due to nonlocal dispersive effects. Phys Rev B 72:195345
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11772245, 11472205, 11672222), the 111 Project (B18040), and the Fundamental Research Funds for the Central Universities in China.
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Tian, X., Li, Q. & Deng, Q. The J-integral in flexoelectric solids. Int J Fract 215, 67–76 (2019). https://doi.org/10.1007/s10704-018-0331-6
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DOI: https://doi.org/10.1007/s10704-018-0331-6