Mode I solution for micron-sized crack

The feature size of micro-electronic, optoelectronic and biomedical devices is in the sub-micron scale and is pushing toward the nanometer scale. Defects in these small structures are correspondingly smaller such that small crack behavior is becoming an important design consideration for reliability and performance of such devices. Analyses of small cracks are complicated by the rapid variation in the deformation in the crack tip zone. Strain gradients in the near tip zone, which can be ignored in large cracks, will influence the small crack behavior when the crack tip zone is in the order of the crack size. In this paper, we consider two-dimensional crack deformations with strain gradient effect and establish the dual formulations in terms of potentials. The formulations reduce to the conventional linear elastic fracture theory, when the material length scale parameters for the higher order deformation measures are zero. One of the formulations is in terms of two complex stress functions and two pseudo potentials. The complex stress functions are harmonic and the governing equations for the pseudo-potentials are two uncoupled second order partial differential equations. The solutions for these equations are coupled through the boundary conditions. urbation method is used to construct the solution for mode I cracks under a K-field when only the effect of the rotational gradient is included. The perturbation solution has 1 / r induced singularity for the stresses. The induced deformation decays exponentially away from the crack tip. The induced stresses become insignificant beyond 3ε, the typical characteristics of a boundary layer type. To a first order approximation, the induced deformation energy normalized by that of the classical solution under constant applied crack opening load is linearly proportional to ε. This implies that the induced energy release rate is linearly proportional to the length scale parameter. The induced energy release rate under a fixed crack opening load is negative indicating that the rotational gradient shields the crack and lowers the total deformation energy release rate for small crack.