Stresses Formed Near a Crack in Half Space in Contact with Liquid Under Harmonic Loading
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STRESSES FORMED NEAR A CRACK IN HALF SPACE IN CONTACT WITH LIQUID UNDER HARMONIC LOADING V. Z. Stankevych
UDC 539.3
We study a three-dimensional bimaterial object formed by an elastic half space with cracks and a liquid. The crack surfaces are subjected to the action of harmonic loads. The problem is reduced to the solution of a system of two-dimensional boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. We analyze the case of a circular crack whose lips are subjected to the action of stationary loads. The inertial effects in the vicinity of the crack contour are investigated.
Numerous works are devoted to the investigation of the stress-strain states of composite bodies with cracks under the action of dynamic loads. Thus, singular integral equations of the corresponding two-dimensional [1–3] and axially symmetric [4, 5] problems were obtained by the method of functions of complex variable. The threedimensional dynamic problems for bimaterial bodies with plane cracks were reduced to the solution of boundary integral equations by the method of potential [6–8]. The stress concentration in the vicinity of the crack contours was found depending on the type of dynamic loading, location of cracks in the body, and the ratio of elastic constants of the components. It is of interest to analyze the case where the role of a component of an object of this sort is played by a liquid. Statement of the Problem Consider a bimaterial body formed by an elastic isotropic half space with density ρ, Poisson’s ratio μ, and shear modulus G and an ideal incompressible liquid with density ρ p . The lower (solid) half space contains M plane cracks located in the planes x1m Om x 2m of local Cartesian coordinates Om x 1m x 2m x 3m (Fig.1). The opposite surfaces Sm± ( m = 1, M ) of the cracks are subjected the the action of self-balanced forces harmonic as functions of time t : N +jm ( x m , t ) = − N −jm ( x m , t ) = N jm ( x m ) exp (− iωt ) ,
j = 1, 3 ,
m = 1, M ,
xm ∈ Sm ,
(1)
where ω is the circular frequency of the applied loads, N j m ( j = 1, 3 ) are the projections of their amplitude values in a coordinate system Om x1m x2m x3m , and xm is a point with coordinates x1m , x2m , and x3m in the same coordinate system. As a consequence of the harmonic law of loading (1), all characteristics of the wave field in the bimaterial body are characterized by the harmonic time dependence. The problem of determination of the stress-strain state of the body is reduced to the solution of the following differential equations for the amplitude values of the vectors of displacements up in the liquid and u in the lower half space: Lviv Department of the Dnipropetrovs’k National University of Railway Transport, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 3, pp. 96–100, May–June, 2005. Original article submitted December 13, 2004. 388
1068–820X/05/4103–0388
© 2005
Springer Science+Business Media, Inc.
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