Smoothing defected welds and hairline cracks

Let a smooth curve (hairline crack) split a planar domain into two pieces. We consider a homeomorphism of the domain (hyperelastic deformation), which is a diffeomorphism on each side of the curve. However, the one-sided derivatives along the curve need not match. The goal is to smooth such a deformation near the curve (welding area) to become a diffeomorphism of the entire domain. Apart from interpretations and relevance to engineering practice (strengthening deformed welds), there is a mathematical interest in this subject, beginning with the work of J. Munkres [J. Munkres, Ann. of Math. (2), 72 (1960), pp. 521-554] on smoothing piecewise-differentiable homeomorphisms, and, most recently, approximation of Sobolev homeomorphisms [T. Iwaniec, L. V. Kovalev, and J. Onninen, Arch. Ration. Mech. Anal., 201 (2011), pp. 1047-1067] and monotone Sobolev mappings with diffeomorphisms [T. Iwaniec and J. Onninen, Arch. Ration. Mech. Anal., to appear].