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Uniaxial compression of heterogeneous/cellular material (bone)

Frank Richter's picture


Dear colleagues,

I kindly ask for advice on the following problem:

I simulate the uniaxial compression of a heterogeneous cylinder representing bone. Thus, it is not the visible solid exterior rim of the bone, but represents a porous structure within the bone, so-called trabecular bone, see the figure underneath.

The simulation is done in ABAQUS. The sample contains 164175 elements of type C3D4. The heterogeneous, porous structure makes it impossible to use axisymmmetric elements. The constitutive behavior is ELASTIC.

The loading is imposed by displacing (linearly in time) a fictive node above the sample, linked to the nodes on top of the sample by
BEAM, (node set on top of sample), (fictive node)
The displacement is vertical = parallel to the cylinder axis (3-direction). We track the reaction force in this fictive node.


1) ABAQUS Standard: the forces in all three spatial directions start at zero and are linear in time. My conjecture is that forces in the lateral directions (1,2) arise because the fictive node is not centered (not centered in space, but slightly shifted from radius=zero; and the different number of nodes on the top face to both sides (+x, -x; +y, -y) in the two directions might also give rise to these forces because of the MPC link).

Question: is a force linear in time to be expected in a heterogeneous structure ? I'd expect the structure to crumble continually, causing peaks in the signal.

Watching the sample the heterogeneity is distinctly visible, there is significant porosity. The simulation lists 12 time steps, thus the line is not simply generated by linking data for the instants time zero and time = end time.

My boss told me: the linear rise of the forces is due to omitting the definition of self-contact between the elements. Is this the correct explanation ? If so, the significance of the simulation must be questioned. And what then gives rise to the force output, if not displacing the elements within ??

2) ABAQUS Explicit: the modifications to case 1 consist in replacing the displacement boundary conditions with velocity boundary conditions (constant velocity) and the additional command

The reaction forces do NOT start at zero. The one for the vertical direction starts at -2300 N, the others at +25 N and +175 N. The forces are not linear in time.

1) why is that and how can it be suppressed ?
2) if I divide the speed by 1000 while keeping the total time of one second, one can expect the force to represent the initial portion (1/1000) of the trace from the simulation with the initial speed. Instead, the simulation generates approximately the same force history with values divided by 1000. How come ?
3) Is this a purely numerical effect ? How to set up the simulation in ABAQUS Explicit ? I am not at all familiar with the concept of variable mass scaling. This command has parameters to it. Can anyone make a guesstimate for to set for DT, frequency and type ? Do they impact the simulation significantly ?

Does anyone know a good reference for the simulation of uniaxial compression of heterogeneous materials ? Perhaps a thesis posted online ? I hope to find therein a detailed discussion with excerpts from the simulation code.

Thank you in advance for sharing your expertise.



Image icon trabecular.png241.39 KB


jason Zhu's picture





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Dear Frank,


Really a interesting analysis. I thus would like to express
my experience on FEA.

1)I think it is possible for this structure to behave

If you want to get the peak force, I recommend you using
RIKS method for this geometry.


2) I notice a velocity boundary condition was used in your
model and MASS SCALING was specified.

In my opinion, displacement boundary condition may be a
better choice.

Additionally, caution must be taken notes of when the MASS
SCALING was used. Did you output the ALLAE compared to ALLIE in the history

Additionally, I think the force +25N and +175 N is quite
small compared with -2300N


PS: An attachment of an inp file may be helpful to discuss further. 




Jason Zhu Best Regards

Frank Richter's picture

Dear Jason,

thank you for your advice.

I refrained from pasting a 27 MB input file with approx. 656000 text lines in my blog :-)

Abaqus Explicit demands displacement BC be converted to velocity BC; otherwise you get a warning message in the sta file stating that displacement BCs will be interpreted as displacement jumps.

How should ALLAE and ALLIE be related to each other if confidence in the model is desired ?

The values provided for the three forces were the initial forces at time zero. They are nonlinear in time. I can make myself believe that the origin of the lateral forces is the one I suspect and go on living with that, but the deviation from zero is the thing that drives me up the wall.

I started my first Static,Riks - simulation. It threw out some numbers. The force in the vertical direction is still linear, now with arc length.


------------------------------------------ Ruhr-University Bochum Germany

jason Zhu's picture

Dear Frank,

1)It is a very big inp file. I am afraind my computer may not be abel to  open it.

2) Actually, Abaqus Explicit will give a stating of as displacement jumps anytime if the displacement BCs is applied.  This message, however,does not mean validity of  displacement BCs.

3) After running a model in Abaqus Explicit, try to output ALLAE and ALLIE. If the ratio of ALLAE/ALLIE is not less than 10%(or more rigorous 5%), this result will not be regarded as robustness. You may search "ALLAE" in abaqus DOCUMENTATION for reference.


Hope this help. 



Jason Zhu
Best Regards

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