Geodesic
Singularities at the tip of a~crack on the interface of piezoelectric bodies
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 39, Tome 362 (2008), pp. 241-271

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Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezoelectric media under various boundary conditions on the crack surfaces. The singularity exponents form the spectrum of a certain polynomial pencil, and although explicit formulas are not available, this spectrum is described completely though. The mathematical results apply to problems in the mechanics of fracture. In this way the Griffith formulae are obtained for increments of energy functionals due to the growth of the crack and the notion of the energy release matrix is introduced. Normalization conditions for bases of singular solutions are proposed to adapt them to the energy, stress, and deformation fracture criteria. Connections between these bases are determined and additional properties of the deformation basis related to the notion of electric surface enthalpy are established. Bibl. – 44 titles. 
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     author = {S. A. Nazarov and M. Specovius-Neugebauer},
     title = {Singularities at the tip of a~crack on the interface of piezoelectric bodies},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {241--271},
     publisher = {mathdoc},
     volume = {362},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_362_a8/}
}
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S. A. Nazarov; M. Specovius-Neugebauer. Singularities at the tip of a~crack on the interface of piezoelectric bodies. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 39, Tome 362 (2008), pp. 241-271. http://geodesic.mathdoc.fr/item/ZNSL_2008_362_a8/