Reaction Forces

[VCB]

Hello,

I was wondering how reaction forces are calculated from prescribed displacements. In PreView I made a 100 mm tall hollow cylinder that had a mid-radius of 15 mm and a shell thickness of 3 mm. One end of the cylinder was completely fixed in all directions, while the other end had a long-axis displacement of 5 mm, resulting in a 5% longitudinal strain. The cylinder was modeled as an isotropic elastic material, with a density of 0.00105 g/mm^3, a Young?s modulus of 1,000,000 Pa, and a Poisson?s ratio of 0.49. A general Structural Mechanics step was used. The reaction force from the displaced ring of nodes was ~16 N. If I calculate the force from E*strain*final wall area the answer is closer to 13 N, but I assume the difference is because the strain is not homogenous throughout the cylinder since I fixed the one end of the cylinder in all 3 directions?

My main question is about what happens to the reaction forces when I apply a prescribed displacement to multiple rows of nodes. For example, when I made a new cylinder that was 105 mm tall (with the same additional dimensions and material properties as before), I selected the top two rings of nodes and displaced them by 5 mm. There was still a 5% longitudinal strain in the cylinder below the displaced nodes, but the furthest displaced ring of nodes did not shrink in circumference or shell thickness. The total reaction force from the two rings of nodes was 46 N. When I measured the reaction force for the elements between the two rings of nodes the sum was 23 N, and when I measured the reaction force for all elements with non-zero values, the sum was 39 N.

Similarly, when I displaced 6 rings of nodes in a 125 mm tall cylinder, the node reaction forces summed to 68 N, the elements between the displaced nodes summed to 50 N, and the total reaction force for all non-zero elements summed to 68 N.

The reaction forces were the same whether I pulled on two rings of nodes, or one ring of elements. I?ve attached a photo of the reaction force distributions for the 100 mm, 105 mm, and 125 mm cylinders. Should I only displace one ring of nodes in the cylinder to get accurate reaction force values, or is it possible to displace multiple rings and if so how should I measure the total reaction force?

Thank you!

[VCB]

Hello,

I was wondering if there was any update for this question.

Thank you!

[maas]

Hi,

I tried what you did and I am not finding any differences in total reaction forces, regardless of how many rings I displace. I've attached my input files for displacing one and two rings of nodes. In either case, the total reaction force is 17.54. I calculated the total reaction force by selecting all the prescribed nodes in PostView and summing the values with the integration tool (for selected nodes, the integration tool just calculates the sum). Please take a look at these files and hopefully this will help you resolve the issue. If not, please attach your input files so that we can try reproducing the issue on our end. Let us know what you find!

Cheers,

Steve

[VCB]

Dear Steve,

Thank you for getting back to me. I have a few related follow-up questions.

1. After looking through your .feb files, I realized the difference between our two models is that I did not assign any X or Y values to the displaced end of my cylinder. Is it not possible to accurately solve a FEA model with a 3-direction BC on one end and only a 1-direction BC on the other end?

2. The reaction force value changes depending on if I constrain the X and Y directions using a Zero Displacement or Prescribed Displacement BC. The FeBio and PostView files you attached constrain the displaced ring using a Prescribed Displacement of 0 in the X and Y direction and had a total reaction force of 19.47. If I assigned a Zero Displacement BC in the X and Y direction instead, the reaction force was 17.54. (This was true for both the one and two ring versions).

3. Do you have any suggestion for material choice? I am trying to model the human aorta with an average stiffness of 1 MPa and long-axis strains around 8%. The two materials that I have been using are Isotropic Elastic and Neo-Hookean. Would you suggest one over the other? Also, I am assuming the material is incompressible (Poisson’s ratio = 0.49) so one of the Nearly-Incompressible materials might be better, such as the Mooney-Rivlin material. However, I’m not sure how I can empirically calculate the material constants.

4. Finally, I was hoping you could clarify the reaction force calculation. If I calculate reaction forces using Young’s modulus * strain * final wall area I get values that are about 17% smaller than what FEBio outputs. The difference is probably due to the use of stiffness matrices in FEA models, but an explanation for why calculating force from Young’s modulus is incorrect would be very much appreciated.

Thank you!

[maas]

Regarding your questions,

1. Perhaps I don't understand the question, but you can apply any boundary condition you want and FEBio will give you an answer. The question is do the boundary conditions reflect what you had intended.

2. FEBio only calculates reaction forces for prescribed degrees of freedom, not for fixed ones. The z-component of the force should be the same either way, but of course the total reaction force will be different. Now, the only way I get the 19.47 is if I sum the "Total reaction force" for all nodes, but that's not the same as the net reaction force. You have to sum component by component in which case the x,y components will sum up to zero and the z-component sums up to 17.54.

3. I'm not the best person to ask about what constitutive models to use, but I suspect you would need at least a transversely isotropic material (e.g. the trans-iso Mooney-Rivlin). Perhaps someone else can contribute. I also recommend checking the literature. I know there are a lot of studies out there doing FE analysis on arteries.

4. The calculation using Young's modulus would only be valid for uni-axial extension and a Poisson's ratio of zero. The deformation is not entirely uni-axial (the circumference of the center of the cylinder shrinks), and your Poisson ratio is 0.49. Also, the isotropic elastic and neo-Hookean have a non-linear stress-strain curve which could further explain the difference.

Hope this helps. Let me know if you have more questions.

Cheers,

Steve