Week 5 Sphere pressing on a plate

Objective: Simulating a sphere pressing a flat plate and observing the plastic deformation in the plate.

When the sphere is pressed onto the plate, the plate material will yield under the force applied on it due to the sphere being pressed into it. Once the sphere is retracted, the plate will undergo minute elastic spring back and will stay plastically deformed.

Materials:
Both the sphere and the plate are made of structural steel. Since we do not want to study the sphere, we prefer if the sphere is rigid compared to the plate, i.e. there is minimal deformation in the sphere. If both the sphere and plate have non-linear materials (with bilinear isotropic hardening material models), then the sphere will also plastically deform significantly when it is pressed into the plate. If the sphere material is kept as linear, then in the plastic region (beyond the compressive yield point of structural steel), the solver will assume the stiffness of the sphere to be the linear elastic modulus of structural steel (200 GPa) instead of the tangent stiffness (1.45 GPa), thus there will be minimal plastic deformation in the sphere. The plate material is made of non-linear structural steel.

Symmetry:
In order to reduce computation time, the whole sphere and plate is not modelled. Instead, only a quarter of the hemisphere and quarter of the plate is modelled and then 2 symmetry planes and 1 anti-symmetry plane is introduced.

The two symmetry planes act like mirrors, where the geometry and the boundary/load conditions are mirrored by the solver. The anti-symmetry plane mirrors the geometry but makes the boundary/load condition opposite of what the mirrored condition would be.  

Mesh:
Mesh refinement is applied on the 3 faces of the plate in the vicinity of the sphere (one face in contact and the other two undergoing the majority of the deformation) with the help of face sizing. In order to conduct a grid dependency test, the refined mesh element sizes are varied. The mesh element type for the plate is hexahedral.

Contacts:
Frictional contacts are defined between the plate and the sphere, with the coefficient of friction being 0.74 (between steel surfaces). Since the sphere is going to be applying the pressing force on the plate, the plate is set as the target body and the sphere is set as the contact body. Rest of the contact settings are set to default and program controlled.

Boundary and load conditions:

The hemisphere quarter is given a displacement in the Y direction (global coordinate system) with it being free to deform in the X and Z directions. The displacement load is applied gradually, with there being 8 total loadsteps, within which the sphere is pressed onto the plate a distance of 4mm and then retracted back to its original position. Since, when the hemisphere is mirrored to make a full sphere using the 3^rd symmetry plane, the displacement load on the mirrored hemisphere needs to be in the same direction as the displacement load defined currently, thus the type of the 3^rd plane is set as anti-symmetric.

The plate is provided a fixed support at its base.

Results:
The plastic strain, directional deformation along Y axis and the equivalent VM stress is calculated in the plate for each of the design points.

 Due to the limitations of the student version of the software, 0.45 mm element size mesh is the finest mesh we can have in the plate. We can observe that mesh convergence has not been achieved thus, we will have to take into consideration the results obtained for element size of 0.45 mm.

Plastic strain in plate:

In the above plot, the red curve denotes the maximum plastic strain in the model during the loadsteps. As can be observed, the plastic strain reaches a maxima till t = 5 seconds (end of 5th loadstep), and then becomes constant. In the displacement load given, the sphere starts retracting after pressing the plate when the 5th loadstep ends, thus meaning that after the 5th loadstep, the plate doesn't undergo any more plastic deformation, thus meaning that the plastic strain in the plate will remain constant beyond t = 5 seconds.

If we observe the total strain plot for the plate, we can see it follows a similar trend as the plastic strain, but after t = 5 seconds, it does not remain constant, but reduces a little and then becomes constant. This is because the total strain constitutes both elastic and plastic strain and once the load (pressing force by the sphere in this case) is removed, due to the existence of residual stresses in the plate, there will be some elastic springback in the material, that is it will try to bounce back a little (very minute amount) to its original state once the material is unloaded. This occurs mainly because before the plastic deformation occurs, the plate material goes through elastic deformation which is always restored back completely.

Y deformation in plate:

VM stress in plate:

The maximum stress produced in the plate is 4709.6 MPa at t=5, i.e. when the sphere has pressed into the plate for a 4 mm distance. Once the sphere is retracted, while the stress in the plate reduces as the pressing force is not being applied anymore, but due to the existence of residual stresses in the material, we still observe significantly high values of stresses in the plate. In any case, the compressive stresses being generated in the plate are much higher than the ultimate compressive strength of structural steel, thus we can conclude that if a rigid (compared to the plate) sphere is pressed onto a structural steel plate fby an amount of 4mm, the plate will get crushed/fractured under the pressing load.

Animation