Solvability and regularity results to boundary‐transmission problems for metallic and piezoelectric elastic materials

Buchukuri, Tengiz; Chkadua, O.; Natroshvili, David; Sändig, Anna‐Margarete

doi:10.1002/mana.200610790

Cited by 11 publications

(13 citation statements)

References 25 publications

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“…This shows that the Hölder smoothness exponents depend on the material parameters. Moreover, for these particular cases, from Table I it follows that Note that by a standard limiting procedure, the above Green's formulae (A1), (A2), (A3) and (A4) can be generalized to Lipschitz domains and to vector-functions u (m) 4 , then (A1) and (A3) hold true as well (see [13,29,30] [12]). Furthermore, the matrices q , q = 1, 2, are strongly elliptic, i.e.…”

Section: Theorem 43mentioning

confidence: 89%

“…For regular densities the proof for the potentials V (m) and W (m) can be found in [14], in the isotropic case, and in [21,24,25], in the anisotropic case, while for the potentials V and W the proof is given in [26,27] (see also [12]). Note that the main ideas for generalization to the scale of Bessel potential spaces and Besov spaces are based on the duality and interpolation technique that is described in [28], in view of the theory of pseudodifferential operators on smooth manifolds without boundary (see also [29,30]).…”

Section: Proofmentioning

confidence: 98%

“…The matrix −M (m) (x, 1 , 2 ) is positive definite, while −M(x, 1 , 2 ) is strongly elliptic (for details see [12,21,24]); that is, there exist positive constants c (m) and c, depending on the material parameters, such that…”

Section: (M)mentioning

confidence: 99%

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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: Theorem 43mentioning

confidence: 89%

Section: Proofmentioning

confidence: 98%

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Interface crack problems for metallic–piezoelectric composite structures

Natroshvili

Stratis

Zazashvili

2009

Math Methods in App Sciences

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“…The dimensionless constants c and e from (71) are always negative and h is positive. It follows from accessible data for the piezoelectric materials [21,22] that from (74) can be positive, negative and equal to zero. Consider below these cases and simple examples to demonstrate that all these cases take place.…”

Section: Discussionmentioning

confidence: 99%

“…A method of integral equations for doubly periodic problems including anisotropic piezoelectric composites was developed in [19][20][21]. A rigorous mathematical study of the piezoelectric boundary value problems based on the theory of singular integral equations was presented in [22,23].…”

Section: Introductionmentioning

confidence: 99%

Effective anti-plane properties of piezoelectric fibrous composites

Rylko

2013

Acta Mech

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Anti-plane shear of piezoelectric fibrous composites is theoretically investigated. The geometry of composites is described by the 2-dimensional geometry in a section perpendicular to the unidirectional fibers. The previous constructive results obtained for scalar conductivity problems are extended to piezoelectric antiplane problems. First, the piezoelectric problem is written in the form of the vector-matrix R-linear problem in a class of double periodic functions. In particular, application of the zeroth-order solution to the R-linear problem yields a vector-matrix extension of the famous Clausius-Mossotti approximation. The vector-matrix problem is decomposed into two scalar R-linear problems. This reduction allows us to directly apply all the known exact and approximate analytical results for scalar problems to establish high-order formulae for the effective piezoelectric constants. Special attention is paid to non-overlapping disks embedded in a two-dimensional background.

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Solvability, asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

Chkadua

2017

Math Methods in App Sciences

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In the paper, we investigate the mixed type transmission problem arising in the model of fluid-solid acoustic interaction when a piezoceramic elastic body ( C ) is embedded in an unbounded fluid domain ( ). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain C , we have a 4 4 dimensional matrix strongly elliptic second-order partial differential equation, while in the unbounded complement domain , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener-Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev-Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions.The present paper is devoted to the investigation of a mixed type .M ! / interaction boundary-transmission problem associated with the previously described mathematical model with a piezoelectric solid component. The Dirichlet type .D ! / and Neumann type .N ! / problems are studied in [7].Similar interaction problems for the classical model of elasticity are studied in the references [8][9][10][11][12][13][14][15][16][17][18][19][20]. In this case, one has a threedimensional elastic field, the displacement vector with three components in the bounded domain C and a scalar pressure field in the unbounded domain . In our case, in the domain C , we have an additional electric field that essentially complicates investigation of the transmission problems in question. In particular, except transmission conditions, electric potential is given on one part of the boundary of C (the Dirichlet type condition), while on the other part, normal component of electric displacement is given (the Neumann type condition).In the paper [21], uniqueness and existence theorems for the mixed type interaction problem of acoustic waves and piezoelectric structures are stated without proof.We investigate the aforementioned problem with the use of the potential method and the theory of pseudodifferential equations on manifolds with boundary and prove existence and uniqueness theorems in Sobolev-Slobodetskii spaces.The paper is organized as follows. In Section 2, we formulate mixed type .M ! / boundary-transmission problem for steady state oscillation equation in appropriate function spaces. In Section 3, we define Jones modes and Jones eigenfrequencies and prove the uniqueness theorem for the problem .M ! /. In Section 4, we describe properties of potentials related to the Helmholtz equation and the steady-state oscillation equations of piezoelectricity. In Section 5, mixed interaction boundary-transmission problem .M / is formulated for the pseudo-oscillation equations and the unique...

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Solvability and regularity results to boundary‐transmission problems for metallic and piezoelectric elastic materials

Cited by 11 publications

References 25 publications

Interface crack problems for metallic–piezoelectric composite structures

Interface crack problems for metallic–piezoelectric composite structures

Effective anti-plane properties of piezoelectric fibrous composites

Solvability, asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

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