# What equation predicts at what point a stretched object comes apart?

I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea.

Is there a general equation for an object's threshold and pulling it apart?

• IMO you'd need a stress-strain graph for taffy. A quick internet search reveals nothing. Would be something nice to investigate, worthy of an Ig Nobel prize. – Manishearth Feb 25 '12 at 11:49
• look into elasticity and plasticity en.wikipedia.org/wiki/Elasticity_%28physics%29 , en.wikipedia.org/wiki/Plasticity_%28physics%29 – anna v Feb 25 '12 at 11:54
• @annav Elasticity won't help; taffy is nearly perfectly plastic. Plasticity has no fixed equation; only stress-strain curves. – Manishearth Feb 25 '12 at 12:46
• '...A bit tougher than a simple equation like gravity', classic! – Rumplestillskin Jan 8 '17 at 5:35

So the general rule is that materials scientists usually take some complex thing like steel and plastic and plot a stress-strain curve. The thing about "stresses" is that if you have two beams right next to each other you expect that they can handle twice the load before breaking, so you divide force by cross-sectional area to get some number which doesn't depend on this aspect. The thing about "strains" is that if you have a mass that's hanging from a beam and it's stretching that beam by an amount $s,$ and you hang this from another identical beam (and the beams are relatively massless compared to the hanging-mass) then you expect the other beam to also stretch by an amount $s$, so that both beams stretch together by $2s$. Therefore we divide the stretch by the resting length to get some number which doesn't depend on this aspect.

Once we normalize these two appropriately we get some curve of strain as a function of stress. This doesn't tell you everything, but in general there are a bunch of important stresses in describing how things deviate from this curve.

First off there is the plastic deformation stress. Beneath this we say that the deformation is "elastic", which is a fancy word that means that if you let go it will mostly come back to where it started. (Try this with a paperclip, try to bend the steel so slightly that it snaps right back to where it started.) Above this stress the deformation is "plastic", a fancy word that means that it doesn't come back to where it started. So the stress-strain curve only applies for "stretching something out" and when it returns back to being un-stretched it might not come back to where it was. You can see this for example in plastic shopping bags after they're used to carry a heavy load; the plastic handle is just thinner and stretched out longer, and it doesn't come back to where it used to be.

Above that we might have a (stress, strain) point on the curve where an inversion happens: you ask your stretching machine to stretch the object out 1mm longer, and it reports that the force it's using is the same. You ask your stretching machine to stretch out the object 1mm longer than that and it reports that the force has actually decreased. So this is a point of runaway plastic deformation, right, if you pass the point of inversion then unless you rapidly lower the load (like the stretching machine does automatically) the thing will just totally stretch out until it breaks.

On the other hand other materials have cusps where the stress gets higher and higher as the strain gets to a fixed point, and then after that a break happens. The stress at which the break happens is usually called the ultimate tensile stress, with "ultimate" meaning "after that it done broke" and "tensile" meaning "in tension", or in other words when you're stretching it out. You can also define an ultimate compressive strength for compression; if you haven't seen hydraulic press videos on the internet this is your cue to do so because they're just really fun.

In addition as @rdt2 said, as your physics gets more realistic you're going to want to model the fact that objects might not be able to instantaneously respond to a force. One example of this is a "dashpot" which responds by $dL/dt = \alpha F,$ it doesn't change its length much if you smack it with a hammer but moves if you just put any sustained force on it, no matter how gentle. By combining it with a spring in parallel, if that spring has an equilibrium length, you get a reasonably nice simulation of velocity-dependent elastic deformation. You can then attach a spring to that ensemble in series if you want a little bit of "A/C" response (response to short vibrations) and make the various pieces break to give some sort of plastic deformation. This might be the simplest way to program something which "feels" similar to how things "really are".

Yeah, that's the other thing, is that real materials usually differ between tension and compression. (And bending is a combination of putting one edge under tension and another under compression.) To see why this is, usually I tell inquisitive students to think about concrete. Concrete usually consists of a bunch of rocks that are glued together by a "cement", but you can just think of it as "glue". Now under compression, you're trying to squeeze rocks together -- really difficult! -- but under tension, you're trying to pull glue apart -- really easy!. So that's why you get things like steel rebar inserted into that stuff: it's because it's being "buffed up" for a tensile or bending load, which can be partially absorbed by these long steel beams if they're inside the composite. Springs and dashpots are going to look the same under tension and compression unless you manually program in a difference.

Like most materials, toffee (UK English) is elastic under low loads and plastic under greater loads. However, its response is highly dependent on both temperature (obviously?) and strain rate. If you load it quickly, it will fracture but, if you load it slowly, it will deform plastically. There's then a different stress-strain curve for each value of strain rate, which makes it difficult to analyse (although there are finite element studies of toffee in the literature). In practical terms, if you want to break a slab of toffee, hit it with a (toffee) hammer or bash it against your elbow.