Averaging of the physicomechanical characteristics of piezocomposites (plane deformation)
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AVERAGING OF THE PHYSICOMECHANICAL CHARACTERISTICS OF PIEZOCOMPOSITES (PLANE DEFORMATION) L. A. Fil’shtyns’kyi and Yu. V. Shramko
UDC 539.3
The mechanical, electric, and piezoelectric characteristics of fibrous piezocomposites with doubly periodic system of fibers are averaged within the framework of a model of regular structures under the conditions of plane deformation. It is shown that, for the general case of stacking of fibers with arbitrary cross section, the problem is reduced to the determination of certain functionals specified by a system of linear algebraic and singular integral equations of the corresponding boundary-value problem of electroelasticity for the analyzed structure.
In the problems of finding physical fields in structural elements made of composite materials, it is customary to replace the composite material with a certain homogeneous material equivalent, in a certain sense, to the material with microstructure. In analyzing fibrous composites with piezoceramic matrix, it is necessary to average their electroelastic characteristics. In the present work, on the basis of the ideas advanced in [1], by using the method of boundary singular integral equations, we propose a procedure of determination of macroscopic parameters of fibrous piezoelectric composites with doubly periodic stacking of fibers under the conditions of plane deformation. Statement of the Problem We consider an unbounded piezoceramic medium (matrix) reinforced along the Ox2 -axis by a doubly periodic system of identical fibers. It is assumed that the cross section of a fiber is a simply connected domain D bounded by an arbitrary sufficiently smooth closed contour L and the properties of the composite are determined by the structure of an elementary cell Π0 (Fig. 1) constructed with periods ω1 and ω 2 ( Im ω1 = 0 and ω1− 1 Im ω2 > 0). If we know the mean values of the stress tensor 〈 σ11 〉, 〈 σ13 〉, and 〈 σ33 〉 and the components 〈 E1 〉 and 〈 E3 〉 of the vector of electric-field intensity, then we can pose the problem of plane deformation of a regularly reinforced piezoceramic medium. This problem reduces to the evaluation of the mechanical and electric fields in the structure described above. We assume that the fibers are in perfect mechanical and electric contact with the matrix, i.e., the fibers are continuously fastened to the matrix without tension over the entire contact surface and the congruent fibers have identical physicomechanical characteristics. For a homogenous piezoelectric medium, we have the following balance equations [2, 3]: ∂1 σ11 + ∂3 σ13 = 0,
∂3 σ13 + ∂3 σ33 = 0,
∂i =
∂ ∂xi
(i = 1, 3),
(1)
the Cauchy relations: eik = 0.5 ( ∂i uk + ∂k ui )
( i, k = 1, 3 ),
(2)
Sumy State University, Sumy. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 2, pp. 67–72, March–April, 2004. Original article submitted September 7, 2003. 232
1068–820X/04/4002–0232
© 2004
Springer Science+Business Media, Inc.
AVERAGING OF THE P HYSICOMECHANICAL CHARACTERISTICS OF PIEZOCOMPOSITES (PLANE DEFORMATIO
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