# Why doesn't paper tear into 3 pieces?

I realized a strange phenomenon that we often overlook. Why does a paper tear into two pieces when you pull them apart on the 2 sides? Why not 3? I hope someone can answer my query. Thanks in advance! The following post by another user tells us why spaghetti splits into 3. I would like to know why the same does not happen to paper

• Is there a reason why you think it should be 3? Maybe I'm missing something obvious...
– JMac
Dec 29 '17 at 14:17
• @JMac physics.stackexchange.com/questions/71661/…, spaghetti splits into 3. why not paper? Jan 1 '18 at 9:52
• Well, spaghetti specifically breaks because of its brittleness and the wave that the fracture sends out. Paper is structurally very different. It also would have made a bit more sense if you referenced the spaghetti thing. It's not exactly normal behaviour in general for materials.
– JMac
Jan 1 '18 at 16:59

One can imagine the paper as a random network of fibers meeting at contact points. Some fibers are straight, others are slightly bent. The bent ones have extra slack if they are subject to tension, while the straight ones do not. Some fibers are thinner or have flaws, so they will break at a lower tension than the thicker ones. The contact points also have some random mechanical strength. In actual paper there is a grain producing a directionality of properties, but let's ignore it for the moment.

When you subject the paper to tension sooner or later some fibers or crossings will break. That relieves the tension on some of the nearby fibers. Typically a rip or a crack in a material relieves tension at right angles to the rip, but concentrates it near the tip: this is why once the break starts it will expand fast and at right angles to the tension. The stronger the tension the faster it will move, and the local randomness will be less effective since the increase in tension force will be much greater than the breaking stress of the fibers.

In the paper case that means that once the rip starts it will reduce the probability of rips parallel to itself, and it will also move fast so there is little time for other random breaks to start. Even if there are two initial breaks, one will be a bit weaker and move first, getting more concentrated force speeding up its growth.

Note that standard paper tearing involves inducing a shear force on the sheet rather than tension force. This is far easier since the shear strength is lower. The rip also will start at the edge and move inward, with little shear in front and behind: this will naturally keep the rip the only one. Given the grain, one can predict the curve the tear will follow to a high degree.

Sometimes tears branch, apparently because the direction of force changes a bit and one of the pieces finds itself under tension parallel to the tear. This will not happen for a perfect sideways tension, but in more complex situations it happens.

There are of course some tricks to ripping a paper into three. One interesting one is to increase the inertia of the middle piece by taping a coin to it (and have initial tears on the sides): now there is a chance to quickly rip it apart.

I suppose it might tear into three pieces if it was made with exact horizontal symmetry, with symmetrical weak points on each side for it to tear on, and equal force exerted on each side, but if any of those things is missing as it would be, one side is stronger then the other and withstands the force lighter. At least this is my understanding from when I asked my old teacher years ago.