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A related problem is that of the wetting process of a disorder substrate. See, for example, PRBMDO 0163-1829 10.1103/PhysRevB.65.064517
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See S. Ramanathan, Ph.D. thesis, Harvard University, 1997 (unpublished) and take the zero-velocity and -frequency limits in Eqs. (3.48), (A.26), and (A.35). Note, however, that a rotation of the "stress intensity factor," from the propagation direction to the normal direction, is missing here and appears properly in. Moreover, a proper rotation of the crack-front velocity, back from the normal direction to the propagation direction [resulting in a Kardar-Paris-Zhang- (KPZ-) like term 1+ h′2], is included in.
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Y. Zhou, Ph.D. thesis, Johns Hopkins University, 1999 (unpublished).
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