We consider the free-surface shape near, order O(e-05 L), the upper corner of the punch (width = L) as predicted by both the asymptotic and the computational results obtained from the solution of the bonded punch problem. The mathematical formulation of this problem is stated elsewhere in this site.
In Figure 1, we show a sequence of frames extracted from the MPEG movie free (2.25 Mb) (for a short version, see the movie short free (1.03 Mb) ) which illustrates the self-intersecting material behavior in this neighborhood of the corner for increasing values of the punch displacement U.
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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In Figure 1.a (9.9 Kb), we identify the contour of the punch in the vicinity of the upper corner in the undistorted state. On the left side we show both the punch surface corresponding to the thick white line and the free-surface corresponding to the yellow line. The scale is represented by the yellow line of length 2 e-05 L. On the right side we show a color map used to represent the values of the Jacobian determinant of the deformation gradient F (det F) between -2 and +2. Values of det F greater than +2 are represented by the red color, while values of det F smaller than -2 are represented by the blue color. In particular, the free-surface is yellow in the undistorted state since det F = 1. We also show a plot of the total force P under the punch surface versus the horizontal displacement U prescribed at infinity.
In Figure 1.b (12.2 Kb), the behavior of the free-surface shape is indicated by two lines; a thick line obtained from the asymptotic solution and a thin line obtained from numerical computations using the Finite Element Method. Approximation errors in both the asymptotic and the computational procedures account for the difference between the two solutions which, according to the scale, is small. In both cases, material is drawn under the punch and, as we approach the corner, det F becomes first negative and then positive again. The transition region between negative and positive values is very thin and is not perceptible in this image.
In Figure 1.c (13.7 Kb), the free-surface (predicted by the numerical computations) becomes cusp-like for a moderately large load. Now it is possible to see a smooth transition region between positive and negative values of det F. Notice also the good agreement between the asymptotic and computational results with respect to both the deformed shape of the free-surface and the values of det F.
In Figure 1.d (17.2 Kb), the cusp becomes looped and the free-surface intersects itself. The good agreement between the asymptotic and computational results is more visible.
In Figure 1.e (16.9 Kb), we change the scale from 2 e-05 L to 8 e-05 L to show the intersection of the free-surface with the punch surface. The good agreement between the asymptotic and computational results is observed once again.
Figure 1.f (14.6 Kb) is the last frame of the MPEG movie free (2.25 Mb) (for a short version, see the movie short free (1.03 Mb) ) and shows that the intersection of the free-surface with the punch surface persists.
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