Mechanics of nanoparticle adhesion — A continuum approach

22J. TomasThus, the elastic–plastic force–displacement models introduced by Schubert etal. [57], Eq. (44), and Thornton [60] Eq. (47)( )F N =π⋅p f ⋅r 1,2 ⋅ h K −hK,f / 3(47)should be supplemented here with a complete attractive force contribution due tocontact flattening described before. Taking into account Eqs. (41), (42) and (44),the particle contact force equilibrium between attraction (-) and elastic plus, simultaneously,plastic repulsion (+) is given by ( r * Krepresents the coordinate ofannular elastic contact area):2 2H0 VdWπK N fπK,plF = 0 =−F − p ⋅ ⋅r − F + p ⋅ ⋅rrKK,pl* * *el K K K+ 2 ⋅π⋅ p ( r ) ⋅r drr3/2(48)222 2 2⋅π⋅ pmax ⋅r rK K,pl FN + F H0+ pVdW ⋅π ⋅ rK = pf ⋅π ⋅ rK,pl+ ⋅ 1− 3 (49)r K At the yield point r K = r K,pl the maximum contact pressure reaches the yieldstrength p el = p f .222 2 2⋅π⋅rK pf FN + F H0+ pVdW ⋅π ⋅ rK = pf ⋅ π ⋅ rK,pl+ ⋅3 (50) pmax Because of plastic yielding, a pressure higher than p f is absolutely not possibleand thus, the fictitious contact pressure p max is eliminated by Eq. (1):2 22 2 2⋅π⋅r r K K,plFN + FH0 + p VdW⋅π ⋅ rK = pf ⋅ π ⋅ rK,pl + ⋅ 1−(51)32 r K Finally, the contact force equilibrium2r2K,pl+ + ⋅ = π ⋅ ⋅ ⋅ 2+ 1⋅N H0 VdW K f K 3 3 2 rKF F p A p rand the total contact area A K are obtained: A= p ⋅A⋅ 2+ 1⋅f K 3 3 AplK(52)