In this paper, the coupled local-global buckling behavior in laminated composite plates with elliptic delaminations and the associated mechanisms of delamination growth under compressive loads are critically examined. The J-integral technique is used for delamination growth prediction in terms of pointwise energy release rate distribution along the delamination edge. A Multi-plate model, in conjunction with a 3-noded quasi-conforming shell element, is used to model the delaminated plates. The incremental equilibrium equations are set up based on total Lagrangian formulation. The solution strategy incorporates Gauss elimination in a cycle of Newton-Raphson iterations and is augmented with automated arc-length controled load incrementation and equilibrium iterations; and with automated post-buckling path tracing based on a linearised asymptotic solution. The effects of structural parameters such as delamination thickness, size and shape, on the post-buckling behavior and on the delamination growth are critically examined.