Abstract
The indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. The slip boundary between the two regions depends on μ and the Poisson ratio v, and is found uniquely as an eigenvalue of a certain integral equation.
For power law indentors of the form z∝r n, a group property of the integral operator connecting stresses and displacements makes it possible to derive the contact stress distributions from those under a flat punch by a simple quadrature, and shows that the slip radius is the same in all such cases.
An iterative numerical solution using a dual system of Volterra equations is described, and calculated distributions of surface stress presented for the cases of indentation by a flat punch and by a sphere.
Zusammenfassung
Das Eindringen eines axial-symetrischen Stempels in einen elastischen unendlichen Halbraum, unter Einwirkung einer monotonischen senkrechten Kraft, wird dargestellt als ein gemischtes Grenzwert problem, wobei ein Coulombscher Reibungskoeffizient μ im Kontaktvolumen angenommen wird. Innerhalb eines inneren Kreises der Kontakt is haftend während in dem umgebenden Kreisring eine nach innen gerichtete Bewegung der Ebene stattfindet, die mit der Kraft wächst. Die Gleitgrenze zwischen diesen beiden Gebieten hängt von μ und dem Poisson Verhältnis ν ab und ist ein eindeutiger Eigenwert eines Integralgleichung.
Für Stempel, deren Form z∝r n gehorcht, wird gezeigt, dass eine Gruppeneigenschaft des Integral Operators, welche Spannungen und Verschiebungen verknüpft, ermöglicht, die Kontaktspannungsverteilung in der Umgebung eines flachen Stempels durch einfache Quadratur abzuleiten und zu zeigen, dass der Gleitradius in allen Fällen der gleiche ist.
Es wird eine “iterative numerische” Lösung beschrieben, die Volterra-Gleichungen benutzt. Die Berechnungen der Oberflächenspannungsverteilung für das Eindringen eines ebenen Stempels in eine Kugel wird gegeben.
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Spence, D.A. The hertz contact problem with finite friction. J Elasticity 5, 297–319 (1975). https://doi.org/10.1007/BF00126993
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DOI: https://doi.org/10.1007/BF00126993