Crack-Type Boundary Value Problems of Electro-Elasticity

Buchukuri, Tengiz; Chkadua, O.; Duduchava, Roland

doi:10.1007/978-3-0348-7926-2_26

Cited by 5 publications

(2 citation statements)

References 6 publications

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“…Recall that A (m,0) (*) and A (0) (*) are the principal homogeneous parts of the differential operators A (m) (*, ) and A(*, ), respectively, see (8) and (18). The principal singular parts of the matrices (m) (·, ) and (·, ) can be represented as [13] (m,0) (…”

Section: (M)mentioning

confidence: 99%

Interface crack problems for metallic–piezoelectric composite structures

Natroshvili

Stratis

Zazashvili

2009

Math Methods in App Sciences

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We investigate three-dimensional interface crack problems (ICP) for metallic-piezoelectric composite bodies. We give a mathematical formulation of the physical problem when the metallic and piezoelectric bodies are bonded along some proper part of their boundaries where an interface crack occurs. By potential methods the ICP is reduced to an equivalent strongly elliptic system of pseudodifferential equations on manifolds with boundary. We study the solvability of this system in different function spaces and prove uniqueness and existence theorems for the original ICP. We analyse the regularity properties of the corresponding electric and mechanical fields near the crack edges and near the curves where the boundary conditions change. In particular, we characterize the stress singularity exponents and show that they can be explicitly calculated with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators. We present some numerical calculations that demonstrate that the stress singularity exponents essentially depend on the material parameters.

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Section: (M)mentioning

confidence: 99%

Interface crack problems for metallic–piezoelectric composite structures

Natroshvili

Stratis

Zazashvili

2009

Math Methods in App Sciences

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“…From Theorems 4.3 and 4.4 we conclude that for arbitrary p, q and s satisfying the conditions (4.52), the operatorsN 0 : [ H s p (S N )] 5 × [H s p (∂Ω)] 5 × [ H s p (S)] 3 → [H s−1 p (S N )] 5 × [H s p (∂Ω)] 5 × [H s−1 p (S)] 3 , N 0 : [ B s p,q (S N )] 5 × [B s p,q (∂Ω)] 5 × [ B s p,q (S)] 3 → [B s−1 p,q (S N )] 5 × [B s p,q (∂Ω)] 5 × [B s−1 p,q (S)]3 , are invertible. Therefore the operators (4.53) are Fredholm with zero index.…”

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confidence: 99%

Mixed Boundary Value Problems of Thermopiezoelectricity for Solids with Interior Cracks

Buchukuri¹,

Chkadua²,

Natroshvili

2009

Integr. equ. oper. theory

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We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well. (2000). Primary 35J55; Secondary 74F05, 74F15, 74B05. Mathematics Subject Classification

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