Linear and Hertz-Mindlin Contact models are the most used models in modelling the granular material. The contact forces between two spheres are calculated based on the equations of motion (Summation of forces=0). However, does these equations account for the inertial forces (F=ma), where F is the inertial force due the acceleration of the sphere (a) with mass=m.
The question has in my opinion two aspects:
Firstly, the contact forces, regardless of the contact model, will of course simply enter the equation of motion, as it was said in the two previous answers (Note, that the contact models don't "account for inertia", they just give forces as functions of displacements and velocities).
However, there is another problem: the Hertz- or Hertz-Mindlin solution (as basically all contact models) are derived from equilibrium conditions of an infinite elastic medium, i.e. dynamic effects like wave propagation etc. are neglected. Thus, if you incorporate these contact solutions in dynamic simulations, this is only possible in the quasi-static limit, i.e.
i) the contact area (or more precise the deformed volume) must be small compared to the macroscopic bodies (spheres)
ii) all macroscopic velocities must be small compared to the smallest speed of wave propagation in the elastic medium
If both those assumptions are fulfilled, you can use static contact models like the Hertzian solution in dynamic simulations.
internal forces F_(Hertz)=k u,
where k is "structural" (material) stiffness given by e.g. Hertz eq. and u is deformation
inertial forces F_(inertia)= m a
F_(Hertz)[t]=F_(inertia)[t]
Off course these models accounts for the internal forces.
For linear contact model, also referred to as spring model, F=kd, k=spring constant, d=displacement where as
For Hertz-Mindlin Contact models , F=Ea^3/R, where E=elastic constant, a= contact radius, R=effective radius of curvature. You can refer to following papers: Conference Paper Simulation of triaxial test with nonlinear contact model by ...
The question has in my opinion two aspects:
Firstly, the contact forces, regardless of the contact model, will of course simply enter the equation of motion, as it was said in the two previous answers (Note, that the contact models don't "account for inertia", they just give forces as functions of displacements and velocities).
However, there is another problem: the Hertz- or Hertz-Mindlin solution (as basically all contact models) are derived from equilibrium conditions of an infinite elastic medium, i.e. dynamic effects like wave propagation etc. are neglected. Thus, if you incorporate these contact solutions in dynamic simulations, this is only possible in the quasi-static limit, i.e.
i) the contact area (or more precise the deformed volume) must be small compared to the macroscopic bodies (spheres)
ii) all macroscopic velocities must be small compared to the smallest speed of wave propagation in the elastic medium
If both those assumptions are fulfilled, you can use static contact models like the Hertzian solution in dynamic simulations.
often Hertz theory and all derivations are used in dynamic models to infer some non-linear stiffness (see for example the classical solutions by Carson, R. M., & Johnson, K. L. (1971). Surface corrugations spontaneously generated in a rolling contact disc machine. Wear, 17(1), 59-72. , and perhaps but much more rarely non-linear damping. But the dynamic problem is typically solved for concentrated bodies. Is this the answer to your question?
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