More of the same ‘out loud’ thinking on the issue of warping in this post to follow on from the last post on warping fundamentals. In particular, thinking about how the speed of a print, the size of an object and they type of plastic all effect warping. Also some other obvious points about why a heated print bed helps reduce warping.
The effects of printing speed, object size and plastic types on warping.
By printing small objects quickly it seems you can limit the amount of warping that takes place. This may be due to the limited time that the outside of the object has to cool and so results in the whole object being of a more even temperature over the short duration of the print.
Top of image: A large object or an object that is printed at a slow speed. Bottom of image: A quickly printed or smaller object. The left hand side is a cut away of the solid object on the right.
Again, the image above also applies if the top object was larger than the bottom and both were printed at the same speed. The larger object will take longer to print and so will have more time for the sides of the object to cool and possibly resulting in less warping.
The image above also applies if different plastics are used, with the top object being a high temperature Tg plastic (Eg: HDPE) while the lower image being a low temperature Tg plastic (Eg:PLA). The higher temperatures relative to ambient (often 25ºC) required to print HDPE will result in the outside edges falling in temperature considerably faster with respect to the centre than compared to PLA.
Two fictional temperature profiles through the centre of a 50mm wide object during printing. Zero mm is one side of the object, 25mm is its centre and 50mm is the other side of the object.
Although the sides of the HDPE object are hotter than the core of a PLA object, they are still relatively a lot cooler than the core of the HDPE object. It is this steeper temperature gradient that leads to warping. Differences in the expansion coefficient between the different types of plastic may also play a part.
Heated beds and warping
Heated beds are the obvious solution to the problem of cool down on big objects or high Tg plastics. The wide spread adoption of heated beds is a tribute to their effectiveness.
A heated bed (in red) is on the right and a room temperature bed on the left. The direction of the arrows and their size represent internal stresses caused by contraction of the hot core after the exterior has already become ridged.
Unfortunately it seems even a heated print bed has its limits. As an object is printed the warming effects of the heated bed will diminish with height. I imagine this then leads to the same warping effect in the top section of the object that is present in objects printed with out a heated bed. These internal stresses that build in the top layer would weaken the object even if it is not enough to cause the lower, stress free layers, to curl up at the edges.
This problem could be solved by streams of hot air blowing from above, a heated build chamber or even an infrared globe above the print bed. However, with out the dissolvable support material used in commercial printers (work in progress for reprap’s) I imagine this will also have its limits as slight overhangs or teardrop through holes begin to slump on larger objects. A simple solution to slumping is to strap on a lot of fans, but then you would be back at square one with the warping problem*…
It may also be possible to greatly reduce warping by having a slow controlled cool down such as used when casting large objects. So instead of the heated bed switching from ‘full on’ to ‘full off’ there could be a gradual decline of 2 or 3 degrees per minutes until room temperature is reached. This might aid in reducing warping but has not been tested as far as I’m aware.
It should also be noted that the heated print bed seems to also allow for greater adhesion between the print bed surface and the first layer of plastic. From what I can gather this is due to an increase in intermolecular contact brought about by the higher temperatures (lower molten plastic surface tension) and the longer time frame were the plastic is molten at the surface and so can spread (wet) more. More info.
Anyway, I hope those new to the reprap community find this helpful. If I missed a few things or something doesnt seem quite right please, by all means, let me know.
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* Just as a side note: What would be really nice is if there was a way to analyse a 3d object and determine where slumping is most likely to occur. Then throughout the print a fine jet of room temperature air aimed at the print nozzle could be turned on or off as the print head prints the layers above these ‘high slump risk zones’. You might be able to get away with having an elevated build chamber temperature and reduce slump at the same time. Just a thought.
Capolight Electronics Projects. 
I really think this kind of theoretical analysis is important to solve the warping problem. I wrote my own short analysis in this thread: http://forums.reprap.org/read.php?1,55300 but I might do a more detailed write up as well. Writing that post made me consider in more detail the causes of warping. My analysis differs in important ways from yours, though, which leads me to different conclusions. So I feel it’s important to comment here.
It’s clear to both of us that the key factor when it comes to warping is the thermal contraction of layers. For a qualitative analysis, we can imagine that any short segment of filament has two lengths: The length when hot (Lh), and the length when cold (Lc), where Lh > Lc for any particular filament.
Any time you print a hot filament on a cold one, you have four lengths to consider: The length of the hot new filament when freshly printed (L1h), the length of the new filament after it has cooled (L1c), the length of the old filament when it was freshly printed (L2h), and the length of the old filament when it has cooled (L2c).
Due to thermal contraction, we know that:
L1h > L1c
L2h > L2c
But we also know that the new, hot filament is printed on the old, cold filament. This forces their lengths to be equal:
L1h = L2c
That’s where the trouble happens. When the object cools, L1c < L2c, so the object warps. Uneven cooling does not lead to warping; it doesn't matter which part of the printed part cools first. What matters is that hot filament is deposited onto cold plastic. A slow cooldown shouldn't be any different than a rapid quench.
If you take an un-warped object at a uniform temperature, and impose a temperature gradient on it (at any speed), it will warp. But when you return it back to uniform temperature (at any speed), it will return to its original shape. The trouble is that a RepRapped part is an un-warped object at a nonuniform temperature. When it is brought to a uniform temperature, it warps. It shouldn't matter how quickly the transition happens. The exception is if the temperature change is high enough to cause a phase change, in which case the internal structure will be different before and after the process.
Now to consider the effect of print speed and object size on warping. The difference between Lh and Lc is proportional to the temperature change (Lh = Lc + a dT). For an infinitely fast or small print, all the filament will be deposited instantaneously with no time to cool between layers, so dT = 0, and no warping occurs. For an infinitely large or slow print, the old filament has always cooled to the ambient temperature by the time the new filament is laid down, so dT = max.
For a print that's neither infinitely fast nor infinitely slow, there's two important factors to consider (that I can think of). There's cooling from the surface of the part (convection and radiation) and thermal diffusion inside the part (conduction). Heat is added from the surface, in the form of freshly deposited filament, and then conducts through the body of the part toward the cold surface.
It is not the steepness of the temperature gradient that leads to warping. What this non-uniform temperature means is that at the edges, hot filament is printed on cold plastic, whereas at the center, hot filament is printed on warm plastic. That means that the warping stresses will be highest at the edges, and less at the center, because delta T is less at the center. However, even if there were no temperature gradient in the object (ie, if it were uniformly cold, as in the case of the slow print), warping would happen because the fresh filament is hot. It's just that in this case, the warping stresses will be even throughout the part.
Qualitatively, there's not a big difference between these two cases; one warps evenly (so it would probably form a parabolic shape at the bottom) and one warps unevenly (so it might be more of a quartic curve, with the steepest bend at the edges). But they'd both warp.
An object would have to be very tall before the effect of the heated bed could really wear off, I imagine. Because the heated bed is generally wider than the object, there should be warm convection currents rising and warming the sides of the object, reducing convective cooling. If it became a problem, then up to a certain point (the point where the base of the part would undergo a phase change), the temperature of the heated bed can be increased as the object gets taller, so as to maintain a constant temperature at the upper surface. That would probably take some sophisticated software, though.
Jacob
Warping at the top is not generally a problem. The parts warp a fraction of a mm on each layer. It is only when the bottom is pulled up by dozens of layers above it that the accumulated warping is significant. When a new layer is put on top of a warped layer, it will be thicker in the low areas and thinner in the high areas, thus removing the error. Each layer removes a tiny bit of error and so it never accumulates to anything significant. In essence, the plastic gets pushed around a bit, pushing out the perimeter or filling in holes if the infill is less than 100% rather than copy the existing error up to the next layer.
That’s a good point! It explains why the top of an object stays flat even though the bottom warps.
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Nice areticle there. I am just coming to grips with this problem on a high precision object I m printing. the warping. is causing me grief. Exactly as described in your diagrams. the corners nearest the print bed bend up from the bed so the bottom is not flat.. Your article has me thinking… I have a fan that is keeping the top end of the print head cool. Much of this air is hitting the object too. This could be making the temperature differential worse? I wonder if adding a shield below the print head to stop are from hitting the plastic surface would improve the situation.
Bill D