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Acta Mech 200, 6978 (2008) DOI 10.1007/s00707-007-0540-y Printed in The Netherlands
I. V. Andrianov1, V. V. Danishevskyy2, D. Weichert1
1 Department of General Mechanics, RWTH Aachen, Aachen, Germany
2 Prydniprovska State Academy of Civil Engineering and Architecture, Dnipropetrovsk, Ukraine Received 9 July 2007; Accepted 15 November 2007; Published online 28 May 2008 Springer-Verlag 2008
Summary. An asymptotic approach is proposed to describe the load-transfer from a single bre to an anisotropic half-space through weak and stiff interfaces. To start with we simplied the input boundary value problem using ratios of the elastic constants as small parameters. The simplied boundary value problem is solved using integral transforms. Inverse transforms are approximately expressed through elementary and special functions. The obtained results can be used for the investigation of the fracture of composites. In Civil Engineering, the proposed solution can describe the behaviour of piles or piers embedded in soil media that exhibit a linear elastic response in the working-load range.
1 Introduction
In the theory of composite materials much attention has been paid to the problem related to the diffusion of load from an elastic bre to its surrounding isotropic matrix [1][6].
Problems involving anisotropy of material are in general more difcult to solve than the isotropic ones. The problem of an anisotropic composite material can be examined on the assumption that the matrix is only slightly anisotropic [7][9]. On the other hand, as it has been shown by Kosmodamianskii for an anisotropic plane with two identical elliptic holes [10], [11], strong anisotropy leads to the possibility of important simplications of the governing boundary value problems.
Independently Manevitch et al. [12], [13] and Everstine and Pipkin [14], [15] (see also [16][19], [20, Chap. 6.1]), beginning with elasticity theory and treating the extensibility of the material in a preferred direction as a small parameter, used the singular perturbation method to obtain approximate boundary value problems. The governing plane problem is reduced to that of solving two Laplace equations and, if higher-order approximations are wanted, a number of Poissons equations. The comparison of approximate solutions with the exact anisotropic elastic solutions showed satisfactory agreement [12][19].
In the paper [15] a cantilever beam with end load was analysed. Spencer [16] studied the problem of a crack parallel to the bres...