Delaminated Interface Inclusion in a Piecewise Homogeneous Transversely Isotropic Space
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DELAMINATED INTERFACE INCLUSION IN A PIECEWISE HOMOGENEOUS TRANSVERSELY ISOTROPIC SPACE O. F. Kryvyi
UDC 393.375
We obtain the explicit solution of a nonaxisymmetric problem of delaminated circular interface inclusion in a piecewise homogeneous transversely isotropic space under the action of arbitrary loads. We establish the asymptotics of solutions and deduce expressions for the generalized stress intensity factors on the boundary of the inclusion and also their values for some combinations of transversely isotropic materials. Keywords: rigid interface inclusion, transversely isotropic, singular integral equations, asymptotics of stresses.
Numerous researchers studied the problems of interface defects in piecewise homogeneous media. However, they mainly focused their attention either on two-dimensional anisotropic media [1–4] or on piecewise homogeneous isotropic spaces and axisymmetric problems for transversely isotropic spaces [5–9]. At the same time, only numerical solutions are known for piecewise homogeneous anisotropic spaces and, in particular, for piecewise homogeneous transversely isotropic media [10–12]. To reduce the problems of defects to systems of singular integral equations (SIE), a discontinuous solution was constructed in [6, 13] for piecewise homogeneous isotropic and homogeneous transversely isotropic spaces. The solutions of nonaxisymmetric problems for interface cracks and circular rigid inclusions were obtained in [14–16] for different conditions of contact interaction with piecewise homogeneous transversely isotropic space.
In the present work, we study a nonaxisymmetric problem for a delaminated thin interface rigid inclusion in the piecewise homogeneous transversely isotropic space. With the help of a discontinuous solution [17, 18], we reduce the problem to a two-dimensional system of SIE, which is solved in the explicit form. Furthermore, we obtain the distribution of stresses near the inclusion, the expressions for the generalized stress intensity factors (SIF), and a necessary criterion for the determination of strength near the inclusion. Statement of the Problem and the System of SIE Suppose that a perfectly rigid inclusion occupying a domain Ω is located in the plane z = 0 of joint of two different transversely isotropic half spaces. An arbitrary load whose action is assumed to be equal to the action of the resultant force P = (P1, P2 , P3 ) and the resultant moment M = (M1, M 2 , M 3 ) is applied to the inclusion. The location of its faces after deformation is described by the functions
ζ 6± = ζ 06 + ϑ 0± (x, y),
ζ ±k = ζ 0k ,
k = 4, 5;
(x, y) ∈Ω ,
Odessa National Maritime Academy, Odessa, Ukraine; e-mail: [email protected]. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 50, No. 2, pp. 77–84, March–April, 2014. Original article submitted October 26, 2012. 1068-820X/14/5002–0245
© 2014
Springer Science+Business Media New York
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O. F. KRYVYI
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ζ 04 = δ x − φ z y,
ζ 05 = δ y + φ z x,
ζ 06 = δ z + φ y x + φ x y ,
(1)
where ϑ 0± (x, y) is the thic
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