Stress intensity factor of mode-III cracks in thin sheets

The stress field at the tip of a crack of a thin plate of elastic material that is broken due to a mode III shear tearing has a universal form with a nonuniversal amplitude, known as the stress intensity factor, which depends on the crack length and the boundary conditions. We present in this paper exact analytic results for this stress intensity factor, thus enriching the small number of exact results that can be obtained within linear elastic fracture mechanics.

Figure 1

The mode III configuration: a semi-infinite strip of width $2a$ and a cut (C-D) of length $b$ in the middle of the strip. The upper left side of the strip (B-C) is subjected to an antiplane positive load, and the lower left side to an identical but negative load.

Figure 2

Mapping of the upper strip in Fig. 1 onto the upper half of the $\zeta$ plane. The inverse function is $\zeta =\cosh (\frac{\pi (z+b)}{a})$, and the crack tip is at ${\xi }_D=\cosh (\pi b/a)$.