Viscoelastic friction in sliding a non-cylindrical asperity

We investigate the 2D contact problem of sliding a non-cylindrical punch on a viscoelastic halfplane, assuming a power law shape <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mfenced><mml:mi>x</mml:mi></mml:mfenced> <mml:mi>k</mml:mi></mml:msup> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> . We find with a full boundary element numerical solution that the Persson analytical solution for friction, which works well for the cylindrical punch case assuming the pressure remains identical in form to the elastic case, in this case leads to significant qualitative errors. However, we find that the friction coefficient follows a much simpler trend; namely, we can use as a first approximation the solution for the cylinder, provided we normalize friction coefficient with the modulus and mean pressure at zero speed, despite that we show the complex behaviour of the pressure distribution in the viscoelastic regime. We are unable to numerically solve satisfactorily the ill-defined limit of sharp flat punch, for which Persson's solution predicts finite friction even at zero speed.