In the classical bonded punch problem, a finite segment of the boundary of a half-plane is forced to undergo a translation by means of a flat-ended rigid punch that is fully bonded to this boundary segment; the remainder of the boundary, as well as the points infinitly far removed from the boundary, are free of traction. We study the solution of this problem within the context of elasticity theory when the punch is displaced normal and into the half-space. Our interest is in the possibly singular asymptotic behavior of the solution in the neighborhood of the corners of the punch.
In linear elasticity theory the solution of this problem is highly irregular; in fact, the stress and strain fields are singular and the material is predicted to self-intersect in the vicinity of the corner. What is the consequence of this anomalous, physically unacceptable behavior?
We have developed an analytical asymptotic solution which contains this curious behavior and we have verified it by means of global numerical computations based on the Finite Element Method. Here, we display the highly localized self-intersecting deformation behavior. Similar anomalous behavior is predicted in nonlinear elasticity theory for certain strain energy functions.
For illustrative purposes, we fix the flat-ended rigid punch in a vertical position and impose a prescribed horizontal displacement on the half-plane at infinity. The mathematical statement of the problem is available in PostScript format (57.3 Kb).
To compare both the asymptotic and computational results
in the vicinity of the corners we consider the deformation of the
free-surface
in the vicinity of the upper corner of the punch. We then show
the corresponding behavior of internal
material lines
and the free-surface shape in the neighborhood of this corner.
Finally, we zoom in further and show the free-surface shape at
small scale
distance, order O(e-10 L), from the corner of the punch
(L = width of the punch). At smaller scales the pattern is similar
as it follows a logarithmic spiral behavior.
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