Mechanics of nanoparticle adhesion — A continuum approach

16J. Tomastion and remaining strength after tabletting must be influenced by irreversiblecontact deformations, which are shown for a small stress level in a powder bulk inFigs 2 and 3.If the maximum pressure p max = p f in the center of the contact circle reaches themicro-yield strength, the contact starts with irreversible plastic yielding (index f).From Eqs. (2) and (5) the transition radius r K,f and from Eq. (6) the center approachh K,f are calculated as:hπ ⋅r⋅ p= (29)E12 , fKf , *r2 2π ⋅r12 ,⋅ pfKf ,* 2= (30)EFigure 2 demonstrates the dominant irreversible deformation over a wide rangeof contact forces. This transition point Y for plastic yielding is essentially shiftedtowards smaller normal stresses because of particle adhesion influence.Rumpf et al. [62] and Molerus [13, 14] introduced this philosophy in powdermechanics and the JKR theory was the basis of adhesion mechanics [58, 67, 76,85, 86, 90].2.1.3. Perfect plastic and viscoplastic contact displacementActually, assuming perfect contact plasticity, one can neglect the surfacedeformation outside of the contact zone and obtain with the following geometricalrelation of a sphere( ) 22 2 22K 1 1 K1 , 1 K1 , K1 , 1 K1 ,r = r − r − h = ⋅r ⋅h −h ≈d ⋅ h(31)the total particle center approach of the two spheres:2 2 2K K Kr r rhK = hK1 ,+ hK,2= + = (32)d d 2 ⋅ r1 2 12 ,Because of this, a linear force–displacement relation is found for small sphericalparticle contacts. The repulsive force as a resistance against plastic deformationis given as:F = p ⋅ A = π ⋅d ⋅ p ⋅ h(33)Npl , f K 12 , f KThus, the contact stiffness is constant for perfect plastic yielding behavior, butdecreases with smaller particle diameter d 1,2 especially for cohesive fine powdersand nanoparticles:dFk = = ⋅d ⋅ p(34)NπNpl , dh12 , fK