I like sausages. I like grilled sausages. I like fried sausages. I like beef, pork and chicken sausages. I even like vegetarian sausages. Sometimes when you cook sausages, they burst. This occasionally happens if you try and cook them too fast. It can also happen if the brand of sausage you are cooking is overly generous with their filling.

Sausages split because pressure builds up on the inside as they cook (typically from vapor produced by the heat, but also from the expansion of the foody goodness inside).

As the pressure increases, so does the stress in the casing of the sausage (the skin). When the stress in the casing reaches a critical value, it fails catastrophically and the sausage bursts.

Have you noticed, when sausages burst, they almost always split lengthwise? Why is that? Let's investigate...

Let's model our sausage as a thin-walled cylinder with hemispherical end caps.

What does that mean? It means that we're only concerned with modelling the casing of sausage (the thing that fails). Thin-walled means that the thickness of skin is considered negligible compared to the diameter of the sausage (in this way, we only need to consider the tension in the skin).

According to Pascal's Principle we'll assume the pressure of the 'fluid' filing (sausage meat and vapor) inside is uniform.

Because of cylindrical symmetery we don't need to worry about shear stress in the skin. If we examine a small square on the surface we can consider the forces resolved in two perpendicular directions: Hoop Stress which runs around the sausage like a ring, and Longitudinal Stress which runs axially up the sausage.

Stress is defined as *Force per Area*, and can be calculated by dividing the forces experienced over the cross-section upon which they are applied. We'll use the Greek letter σ to represent stress.

Imagine a cross section cut through the sausage revealing a circular section.

Axial stress passes passing longitudinally through this ring section and must balance out the pressure force experienced internally (equilibrium).

**σ _{e} = Force / Area**

The force pulling the sausage apart, end-to-end, is the internal pressure multiplied by the cross sectional area.

**Force = Pressure x (Cross-sectional area of sausage)**

**Force = Pπr ^{2}**

The area this force is applied over is the thin ring of casing around the edge:

**Area = 2πrt**

Combining these, we can calculate the longitudinal stress:

**σ _{e} = Pπr^{2} / 2πrt**

** σ _{e} = Pr / 2t **

(You can see from the derivation of this stress calculation that the longitudinal stress is proportional to the cross section area of the sausage and is not related to the end caps. Whilst we have drawn our sausage with hemispherical ends, the calculation would result in the same answer if the ends of the sausage were slab sided.)

To calculate the hoop stress, we look at a slice down the center line of the sausage and consider the forces in a small hoop of width *dx*.

As before we calculate the force by the internal pressure multiplied by the area it is applied over:

**Force = P.2r.dx**

The area this force is applied over is the two parts of the ring at the top and bottom of the cylinder:

**Area = 2.t.dx**

Combining these, we can calculate the longitudinal stress:

**σ _{h} = P.2r.dx / 2.t.dx**

**σ _{h} = Pr / t**

Comparing the two results, we can see that the hoop stress in the casing is *twice* that of the of the longitudinal stress. It should, therefore, be no surprise that sausages burst through a hoop failure and create a rupture that runs perpendicular to the line of maximum stress.

** σ _{e} = Pr / 2t**

** σ _{h} = Pr / t **

Whilst our examples have been very simplified calculations, the principles of hoop stress and longitudinal stress apply to many engineering structures: aicraft fuselages, submarine hulls, pressure boilers, roof domes, air-cylinders...

...oh, and have you ever seen a burst water pipe?

*Image: Flickr/Oxfordian*

*This post was originally published on Data Genetics, a site created by Nick Berry was educated as a rocket scientist and aircraft designer, graduating with a Masters Degree in Aeronautical and Astronautical Engineering, and currenty works as a Data Scientist at Facebook. You can follow DataGenetics on Twitter here or on Facebook page here.*

*This post has been republished with permission from Nick Berry.*

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