Abstract
We present uniform asymptotic solutions (UAS) for displacement discontinuities (DD) that lie within the middle layer of a three layer elastic medium. The DDs are assumed to be normal to the two parallel interfaces between the leastic media, and solutions will be presented for both 2D and 3D elastic media. Using the Fourier transform (FT) method we construct the leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three layer medium. Although a closed form solution will require an infinite series solution, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We present an explicit UAS for elements in which the DD fields are assumed to be piecewise constant throughout a line segment in 2D and a rectangular element in 3D. We demonstrate the usefulness of this UAS by providing a number of examples in which the UAS is used to solve problems in which cracks just touch or cross an interface. The accuracy and efficiency of the algorithm is demonstrated and compared with other numerical methods such as the finite element method and the boudary integral method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ABAQUS, HKS, Rhode Island, 1998.
Bender, C.M. and Orszag, S.A. (1978). Advanced Mathematical Methods for Scientists and Engineers McGraw-Hill, New York.
Buffler, H. (1961). Der Spannungszustand in einem geschichteten Korper bei axialsymmetrischer Belastung. Ingenieur-archiv. 30, no 6, 417–430.
Buffler, H. (1962). Die Bestimmung des Spannungs und Verschiebungszustandes eines geschichteten Korpes met Hilfe von Ubertragungsmatrizen. Ingenieur-archiv. 31, no 1, 229–240.
Buffler, H. (1971). Theory of elasticity of a multilayered medium. Journal of Elasticity 1, no 2.
Crouch, S.L. (1998). Personal communication.
Crouch, S.L. and Starfield, A.M. (1990). Boundary Element Methods in Solid Mechanics, Unwin Hyman, London.
Erdogan, F. and Biricikoglu (1973). Two bonded half planes with a crack going through the interface. International Journal of Engineering Science 11, 745–766.
Gilbert, F. and Backus, G.E. (1966). Propagator matrices in elastic wave and vibration problems. Geophysics 31, 326–332.
Kennet, B.L.N. (1983). Seismic Wave Propgation in Stratified Media, Cambridge University Press, Cambridge.
Kuo, C.H. and Keer, L.M. (1995). Three-dimensional analysis of cracking in a multilayered composite, ASME Journal of Applied Mechanics 62, 273–281.
Lin, W. and Keer, L.M. (1989). Analysis of a vertical crack in a multilayered medium. ASME Journal of Applied Mechanics 56, 63–69.
Lin, W. and Keer, L.M. (1989). Three-dimensional analysis of cracks in layered transversely isotropic media. Proceedings Royal Society London A, 424, 307–322.
Linkov, A. and Filippov, N. (1991). Difference equations approach to the analysis of layered systems. Meccanica 26, 195–209.
Napier, J.A.L. (1998). Personal communication.
Olver, F.W.J. (1974). Asymptotics and Special Functions, Academic Press Inc., New York.
Pan, E. (1997). Static Green's functions in multilayered half spaces, AppliedMathematical Modelling 21, 509–521.
Peirce, A.P. and Siebrits, E. (2001). The scaled flexibility matrix method for the efficient solution of boundary value problems in 2D and 3D layered elastic media. Computer Methods in Applied Mechanics and Engineering (to appear).
Ryder, J.A. and Napier, J.A.L. (1985). Error analysis and design of a large-scale tabular mining stress analyzer, Proceedings 5th International Conference of Numerical Methods in Geomechanics, Nagoya, 1549–1555.
Selcuk, S. (1998). Personal communication via Crouch, S.L.
Singh, S.J. (1970). Static deformation of a multilayered half-space by internal sources. Journal of Geophysical Research 75, 3257–3263.
Singh, S.J. (1986). Static deformation of a transversely isotropic multilayered half-space by surface loads. Physics of the Earth and Planetary Interiors 42, 263–273.
Sneddon, I.N. (1995). Fourier Transforms, Dover, New York.
Thompson, W.T. (1950). Transmission of elastic waves through a stratified medium, Journal of Applied Physics 21, 89–93.
Wallace, P.R. (1984). Mathematical Analysis of Physical Problems, Dover, New York.
Wardle, L.J. (1980). Stress analysis of multilayered anisotropic elastic systems subject to rectangular loads, CSIRO Institute of Earth Resources, Division of Applied Geomechanics, Technical Paper 33, Australia.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Peirce, A.P., Siebrits, E. Uniform asymptotic approximations for accurate modeling of cracks in layered elastic media*. International Journal of Fracture 110, 205–239 (2001). https://doi.org/10.1023/A:1010861821959
Issue Date:
DOI: https://doi.org/10.1023/A:1010861821959