I haven't applied a force to be honest, but a vertical displacement (scoped to the upper flat edge of the semisphere) to which correspond a certain surface force (i.e. pressure). You're right, the pic I provided is a little bit misleading. Anyway, ok, the important thing is that I have to consider just the vertical component of the total reaction force vector.
Could I kindly ask you another thing? I'm having serious troubles to find the results that the theoretical mathematical model predicts (the Hertz contact model). I've already put myself in the physical condition to be sure that the theoretical model is valid. When I indent the sphere into the plane and plot [vertical component of the surface reaction force VS indentation] I should obtain a power law like F= k(E)*indentation^(1.5) where k(E) is a prefactor containing the Young's Modulus. What I've obtained so far is an exponent oscillating between 1.45 and 1.47 (depending on the mesh and other nonlinear parameters). The first thing I've thought was a problem of the mesh or nonlinear analysis, but after have tried everything the exponent still oscillates between 1.45 and 1.47, so I'm lost.
The theoretical model is: F = 4/3*E/(1-v^2)*sqrt(R)*x^(1.5).
where F = normal force applied to the sphere which indents the plane, R = radius of the sphere, v = Poisson ratio, x = indentation (that is, the surface plane deformation along the axis).
I've put Large Deflections on On, the solver preference on Mechanical Nonlinear, the contact is frictionless and the contact status is closed. I also used Nonlinear Adaptive Region to take into account any bad mesh deformation. The material of the sphere is Structural Steel; for the plane I've created a simple elastic material with density 0.97 kg/m^3, E=0.5 MPa, v=0.49 .
The vertical component of the force data are reasonable, It means that applying the aforementioned equation to compute F for a given indentation X returns values very similar to those of the simulation. But, there is a but, the fit still gives me a coefficient of x different from the theoretical 1.5 :
Have you any idea? Thank you.
Best Regards