Scrolled Paper
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Hi,
I hope this is the right place for my question.
I've attached a photo of scrolled paper. Lets imagine one takes this scrolled paper and then opens it on a table.
Back in the last century, many people had the experience of handling photographic film, which has similar behavior as the scroll of paper you describe. Some may have handled a roll of photo paper which also scrolls up after being pulled off the roll and laid flat. The scroll up forces are caused by the coating on one side of the backing media (film or paper) having a coating stress. The coating is applied as a liquid, and as it dries, it shrinks, generating the coating stress. The stress is often the same in any direction on the surface, the scroll forms along the long direction of the strip of paper because that is the lower energy state of the system, but you can sometimes hold a short length of it flat and get it to buckle into curling across the width. One property you need to know is the coating stress.
The following is from this paper: http://rosakis.caltech.edu/downloads/pubs/2007/160%20On%20the%20Stoney%20formula%20for%20a%20thin%20film%20substrate.pdf
You have to measure the radius of curvature of a strip of paper. This is not simple, since you can't lay it flat on the table to measure as gravity would change the measurement. You need to have the strip with the narrow dimension vertical, but you can't rest the long bottom edge on the table, as the friction with the table will change the measurement. The trick is to support the center of the long edge of the strip on a short pad to keep most of the long edge from dragging on the table and have a pair of vertical rods supported in the pad with a gap between them of the thickness of the paper. Now the paper is supported at its center, and is free to curl without gravity affecting the curvature. Obviously you can't have too long of a strip of paper such that the circle closes on itself, and the strip can't be too short or there is not enough of a circle to measure the diameter and calculate the radius of curvature.
The coating stress is created by solving a Static Structural analysis that has special values of Isotropic Secant Coefficient of Thermal Expansion (CTE) and a Thermal Condition load that raises the temperature by 1 C. The substrate CTE is set to 0 so the temperature change has no effect on that. The coating CTE is set by an equation that results in the coating stress being developed by a temperature increase of 1 C.
CTEcoating = CoatingStress(1-Nu)/Ec
Where Nu is the Poisson's Ratio for the coating and Ec is the Young's Modulus of the coating. If you don't know Nu, you can set that to 0 to get the largest effect. The reference for this method is Integrated Optomechanical Analysis (2nd Ed), Keith B. Doyle, Victor L. Genberg, Gregory J. Michaels on page 140. Below is a reference on plasma coatings. https://www.researchgate.net/publication/257809497_Several_fundamental_researches_on_structural_integrity_of_plasma-sprayed_coating-based_systems/download
Just for a simple example, let's use a coating stress of 10 MPa in an aluminum coating. Using the equation above, the CTE value is -9.44E-5 /C
Let's apply that to a quarter model of a very thin sheet of glass (to make a mirror!) A rectangle of 0.5 mm thick glass is 80 mm long x 40 mm wide, but the model will be 40 mm x 20 mm long. The two cut faces that create the symmetry will each have a displacement support of 0 in the direction normal to the face. One vertex has a displacement support normal to the sheet of glass. Note that under the Mesh branch, a Method of Sweep has been used to get 4 elements through the thickness of the glass. The Element Order was set to Linear. Under the Structural branch, note that the Environment Temperature is 22 C.
Add a Thermal Condition of 23 C, which represents a 1 C temperature increase.
Under the Geometry branch, Insert a Surface Coating. Pick the top surface of the solid body and fill out the Thickness of the coating, select the Material and set the Stiffness Behavior to Membrane Only.
Now Solve the model and plot the deformation in the Y direction.
The 1 micron thick aluminum coating with a stress of 10 MPa has caused the pre-coated, perfectly flat 0.5 mm thick rectangle of glass to deform so the corner lifted 2.67 microns away from the flat plane.
You may think that is not exactly "scrolling up", but it's an example of a mirror, where 2.67 microns of deformation could be a big deal. If this was too much, then increasing the glass thickness to 1 mm will greatly reduce the deformation.
The above was a linear analysis, since the deformation was a small fraction of an element thickness. If you try this with a much larger coating stress to get a lot more curl, you will want to make it a nonlinear analysis. Under Analysis Settings, you will want Large Deflection turned On. My solution was accurate with that turned Off.
Making the coating stress much larger and making the length 200 mm long, a deformation that starts to look like it will scroll up becomes evident. The plot below is to True Scale. I mention that because on the Results tab is a display magnification scale factor. When small deformations are in the results, a display magnification factor of arbitrary size can be applied so that a deformation in microns become visible.
I couldn't easily get much more curl out of the solid element model shown above.
Back in SpaceClaim, I used a rectangular surface, 200 mm long, 20 mm wide.
In Mechanical, I defined a Layered Section and in the Worksheet, entered the material and thickness for the substrate and then the material and thickness for the coating. I switched over to using Symmetry Regions for the two cut edges.
Now I can get 204 degrees of curl on the end. That means if I reflect the model about the symmetry plane to recover the full 400 mm long strip, it would have turned into a 109.75 mm diameter circle with 48 degrees of overlap of the two ends.
Now that's what I call a scroll!
Thank you very much for your answers!!!
First, I'm not sure why CTE is needed here for the simulation. Why the coating stress has any relation to the temperature? The scrolling caused by the stress in any temperature, no?
What if I want to do these simulations but not on a straight plane/table, but into a "bowl".
Like in this picture, where the black bold line is the paper, which tends to get the scrolling.
I want to clarify my last questions, the film will scroll "into" the bowl, not in the directions of the red lines.... Is it possible to do so with ANSYS?
And if on the top of film there's a gas (with a known pressure), where this value should be enter?
Formulas for layered materials are not simple. Formulas for shapes with curvature are not simple. That is the beauty of the Finite Element Method, you can use simple equations on each element, but then combine all the elements into a giant matrix and solve the simultaneous system of equations to get an answer to a complex structure with complex material properties.
For a specific modulus of film, coating, and a specific value of coating stress, there is a natural level of curl that develops with no other forces acting on the system. In the example below, that natural curl causes a 400 mm long x 20 mm wide flat strip to lift off the flat table by 73.8 mm.
Search to find the Pressure that would keep the film from lifting off the table. 10 Pa is not enough as shown below.
20 Pa is almost enough. Here you see the effect you were describing of the edge seeming to curl more than the center.
There is no table surface in this model for the film to push against, but if I added that to the model, I could increase the pressure so that the center of the film touched the table, but some of the strip near the edge would still have lifted off the table. If I increased the pressure more, I could ensure that none of the strip was lifting off the table.
Note that if the film was pressed flat against a smooth, non-porous table, so there was no air behind the film, air pressure of 101325 Pa would be pressing it against the table. That is what I meant by needing a very strong vacuum so the film curl could lift off the table. The vacuum pressure would have to be 20 Pa for the lift off to happen.
If you want to simulate just the film without any bowl, use the surface created in the example I provided for the bowl and assign film material properties to it.
Use a Static Structural model to simulate the change in pressure. If you don't have any material creep properties in your model, then time is not real, it is just a way to meter out changes in loads. Therefore, you can ramp the pressure up to one value in 1 second, and change it to another value that ends at time = 2 seconds.
But it sounds like you expect something to change as it sits there, which implies you want to model creep. For that you need creep properties to be added to the material in Engineering Data. Once you do that, time becomes real in Static Structural and to simulate 5 days, you set an end time of 5*24*60*60 = 432,000 seconds.
There are many creep models that you can read about in ANSYS Help under the Mechanical APDL, Material Reference manual.
The archive I provided has a bowl-shaped surface shown below. Did you look at how that was made?
Can you put the SpaceClaim file of your bowl in an archive such as .7z or .rar and attach that to your reply? Was that bowl geometry created in SpaceClaim or imported from another CAD system?
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