Hertz-Mindlin Nassauer-Kuna model

The Hertz-Mindlin Nassauer-Kuna model is a modification of the Hertz-Mindlin model.

The effective radius is calculated using the Nassuaer-Kuna formula:
R N K = V π d 2 MathType@MTEF@5@5@+= feaahyart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGobGaam4saaqabaGccqGH9aqpdaWcaaqaaiaadAfaaeaa cqaHapaCcaWGKbWaaWbaaSqabeaacaaIYaaaaaaaaaa@3E25@
Where V is the overlap volume and d is the penetration depth. The effective radius, RNK is used to determine the equivalent radius, R* which is substituted into the Hertz-Mindlin formula as follows:
F n = 4 3 E * R *1/2 d 3/2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCa aaleqabaGaamOBaaaakiabg2da9maalaaabaGaaGinaaqaaiaaioda aaGaamyramaaCaaaleqabaGaaiOkaaaakiaadkfadaahaaWcbeqaai aacQcacaaIXaGaai4laiaaikdaaaGccaWGKbWaaWbaaSqabeaacaaI ZaGaai4laiaaikdaaaaaaa@4354@

All forces are multiplied by the correction factor of 0.62/0.752 specified in Nassauer, B., 2013, and is defined as:

f n = 0.62 0.752 4 3 E * R N K d 3 / 2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGUbaabeaakiabg2da9maabmaabaWaaSaaaeaacaaIWaGa aiOlaiaaiAdacaaIYaaabaGaaGimaiaac6cacaaI3aGaaGynaiaaik daaaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiaaisdaaeaacaaI ZaaaaaGaayjkaiaawMcaaiaadweadaahaaWcbeqaaiaacQcaaaGcda GcaaqaaiaadkfadaWgaaWcbaGaamOtaiaadUeaaeqaaaqabaGccaaM c8UaamizamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaa@4D8A@