We consider the behavior of the free-surface shape at small scale distance, order O(e-10 L), from the upper corner of the punch (width = L) as predicted by the asymptotic solution of the bonded punch problem. The mathematical formulation of this problem is stated elsewhere in this site.
In Figure 3, we show a sequence of frames extracted from the MPEG movie zoom in (2.53 Mb) (for a short version, see the movie short zoom in (1.09 Mb) ) which illustrates the material behavior of the free-surface in this neighborhood of the corner for increasing values of the punch displacement U. This movie contains more frames and shows the deformation of the free-surface for higher values of the punch displacement than both movies free (2.25 Mb) and grid (2.25 Mb), (for short versions, see, respectively, short free (1.03 Mb) and short grid (1.14 Mb) )
Click on each individual frame to see an enlarged version of the image, or, click anywhere else on the figure to see an enlarged version of the whole image.
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Figure 3.a (10.1 Kb) shows the undistorted state and is similar to Figure 1.a (9.9 Kb), except for the length scale 2 e-10 L. See the description of Figure 1.a for more details.
In Figure 3.b (11.0 Kb), we show the behavior of the free-surface shape for a small punch displacement. Observe that det F alternates between positive large and negative small values as the corner is approached.
Figure 3.c (11.3 Kb) shows the continuation, close to the corner, of the cusp-like shape of the free-surface as shown at larger scale in Figure 1.c (34.9 Kb). Compare these two figures and observe the tendency of the free-surface of having a spiral-like shape as the corner is approached.
Figure 3.d (10.8 Kb) is the last frame of the MPEG movie zoom in (2.53 Mb) (for a short version, see the movie short zoom in (1.09 Mb) ) and indicates further that the free-surface is spiral-like under further magnification at the corner.
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