Longitudinal shear of a body with mutually immobile rigid collinear inclusions
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LONGITUDINAL SHEAR OF A BODY WITH MUTUALLY IMMOBILE RIGID COLLINEAR INCLUSIONS I. P. Shats’kyi and A. M. Kundrat
UDC 539.3: 539.4
We study the problem of antiplane deformation of an elastic body with collinear perfectly rigid inclusions connected to form a single skeleton. The problem is reduced to a system of singular integral equations with an additional condition guaranteeing the absence of mutual displacements of the inclusions. The influence of mutual immobility and arrangement of the inclusions on the distributions of stresses and displacements in the body is analyzed.
As a rule, the analysis of elastic bodies with foreign inclusions is based on the assumption of local equilibrium of each inclusion, whereas the freedom of mutual displacements of the inclusions is restricted only by the material of the matrix [1–6]. This approach is applicable to the problems of materials science but requires corrections for the problems of controlled reinforcement of structures when the elements of reinforcement can form a single skeleton. A significant increase in the stiffness of the structure caused by the fact that the reinforcement works as a single body is accompanied by the elevated stress concentration, which requires separate investigation. In what follows, we consider a simple model problem of this type. Some results obtained in the present work were announced in [7]. Statement and Integral Equations of the Problem We consider an elastic solid body ( x, y, z ) ∈ R 3 subjected to shear along the z-axis. The body is reinforced with two perfectly rigid inclusions in the form of strips whose width is equal to 2l and the distance between their centers to d (Fig. 1). The inclusions are perfectly fastened to the matrix and mutually immobile. The principal state of the body without inclusions under loading is regarded as known. Our aim is to study the perturbations induced by the inclusions. The boundary-value problem of antiplane deformation has the form µ ∆ w = q ( x, y ), µ
∂w = τ1∞ , ∂x ∂w = 0, ∂x
µ
( x, y ) ∈ R2 ,
∂w = τ 2∞ , ∂y
y = 0,
x ∈ Ln ,
(1)
( x, y ) → ∞,
(2)
n = 1, 2,
(3)
Ivano-Frankivs’k Department of the Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, Ivano-Frankivs’k; Ivano-Frankivs’k National Technical University of Oil and Gas, Ivano-Frankivs’k. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 3, pp. 69–73, May–June, 2004. Original article submitted March 11, 2004. 376
1068–820X/04/4003–0376
© 2004
Springer Science+Business Media, Inc.
L ONGITUDINAL SHEAR OF A BODY
WITH
MUTUALLY IMMOBILE RIGID COLLINEAR INCLUSIONS
377
Fig. 1. Schematic diagram of the problem. O2
∫
O1
∂w ( x, 0) dx = 0, ∂x
(4)
2
∑ ∫ [τ yz ](ξ) dξ
= 0,
(5)
k = 1 Lk
where w ( x, y ) is the z-component of the vector of displacements, q is the intensity of bulk forces, τi∞ are stresses acting at infinity, [ τyz] is the discontinuity of stresses on inclusions, µ is the shear modulus, ∆ =
∂2 ∂2 2 + ∂x ∂y 2
is the Laplace operator, and L1
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