Growth or healing of crack surfaces

The factor 2 in front of the second integral accounts for the fact that two surfaces, each with surface energy Us, are created from the surface S in the reference configuration. Continuum stress and deformation fields near a geometrical discontinuity such as a crack edge are potentially singular. In such a case, the differentiability requirements on these fields for interpretation of the continuum equilibrium equation or the divergence theorem may not be satisfied consequently, a special interpretation is required (Freund 1990). 

The local field near the crack edge has an autonomous dependence on local coordinates (Irwin 1957), a feature that has many consequences. An important consequence for present purposes is that the particle velocity Ui which appears in (8.27) can be replaced by a spatial gradient in a direction normal to the crack edge C according to 

---