Multiscale continuum modeling of a crack in elastic media with microstructures

Cosserat type continuum theories have been employed by many authors to study cracks in elastic solids with microstructures. Depending on which theory was used, different crack tip stress singularities have been obtained. In this paper, a microstructure continuum theory is used to model a layered elastic medium containing a crack parallel to the layers. The crack problem is solved by means of the Fourier transform. The resulting integrodifferential equations are discretized using the Chebyshev polynomial expansion method for numerical solutions. By using the present theory, the explicit internal length effects upon the crack opening displacement and stress field can be observed. It is found that the stress field near the crack tip is not singular according to the microstructure continuum solution although the level of the opening stress shows an increasing trend until it gets very close to the crack tip. The rising portion of the near tip opening stress is used to project the stress intensity factor which agrees fairly well with that obtained using the FEM to perform stress analyses of the cracked layered medium with the exact geometry. The numerical solutions also indicate that treating the layered medium as an equivalent homogeneous classical elastic solid is not adequate if cracks are present and accurate stress intensity factors in the original layered medium is desired.