Two different approaches exist for simulating the evolution of localized damage, i.e. continuous and discontinuous ones. Representing a hyperbolic-to-elliptic transition with a parabolic equation, damage diffusion laws have recently been proposed to develop a robust numerical procedure without invoking higher order terms in the single continuous governing differential equation. In this paper, the relationship among rate-dependency, strain gradient and damage diffusion is explored to demonstrate that all these continuous approaches are of higher orders in space and/or time from a mathematical viewpoint although a single higher order equation could be decomposed into a set of 2nd order equations governing different problem domains. It appears that a combined rate-dependent damage and decohesion approach could be sound in physics and efficient in computation if a discontinuous bifurcation analysis is performed to bridge the gap between the continuous and discontinuous approaches. Sample problems are considered to illustrate the potential of the proposed approach in simulating the evolution of impact failure.
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