## Friday, December 21, 2018

### Structures: Or Why Things Don't Fall Down

 Structures: Or Why Things Don't Fall Down, by James Gordon.
When I was in graduate school, I read a fascinating book by James Gordon titled Structures: Or Why Things Don’t Fall Down. It showed me to how engineers think about mechanics. Recently, I reread Structures and read for the first time its sequel The New Science of Strong Materials: Or Why You Don’t Fall Through the Floor. I enjoyed both books thoroughly.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss two mechanical properties of a material: stiffness and strength. Stiffness describes how much a material lengthens when pulled (that is, strains when stressed), and is quantified by its Young’s modulus. Strength measures how much stress a material can withstand before failing. Gordon summarizes these ideas succinctly.
A biscuit is stiff but weak, steel is stiff and strong, nylon is flexible and strong, raspberry jelly is flexible and weak. The two properties together describe a solid about as well as you can reasonably expect two figures to do.
Just two figures, however, are not sufficient to characterize a material, especially when it's used to build a structure.
The worst sin in an engineering material is not lack of strength or lack of stiffness, desirable as these properties are, but lack of toughness, that is to say, lack of resistance to the propagation of cracks.
Toughness is opposite to brittleness, and is related to but not identical to ductility. It is quantified by the work of fracture—the energy needed to produce a new surface by propagation of a crack through the material—a concept introduced by Alan Griffith during his research on fracture mechanics.
A strained material contains strain energy which would like to be released just as a raised weight contains potential energy and would like to fall…The relief of strain energy …. [is] proportional to the square of the crack length…On the other side of the account book is the surface energy…needed to form the new surfaces and clearly increases as only the first power of the depth of the crack…When the crack is shallow it is consuming more energy as surface energy than it is releasing as relaxed strain energy and therefore conditions are unfavorable for it to propagate. As the crack gets longer however these conditions are reversed and beyond the ‘critical Griffith length’ lg the crack is producing more energy than it is consuming, so it may start to run away in an explosive manner.
In heterogeneous materials, internal interfaces act as crack stoppers. This makes wood exceptionally tough; Its cellular, fibrous structure prevents a crack from propagating. Toughness is important in biological materials that must undergo large strains without breaking. Wood is not dense (compared to, say, steel), so you get lots of toughness for little weight, which is one reason wood is so popular as a building material. On the other hand, wood isn’t very stiff, and it swells, burns, and rots.

Gordon provides deep insight into the behavior of structures and materials. Consider the stress in the wall of a cylindrical pressure vessel (a long cylinder with spherical end caps). The circumferential stress in the cylinder's wall is given by the Law of Laplace (see IPMB, Chapter 1, Problem 18). The longitudinal stress is equal to the stress in the end caps (the stress in a sphere is two times that in a cylinder, see Problem 19). Thus
the circumferential stress in the wall of a cylindrical pressure vessel is twice the longitudinal stress...One consequence of this must have been observed by everyone who has ever fried a sausage. When the filling inside the sausage swells and the skin bursts, the split is almost always longitudinal.
Then Gordon develops this theme.
 Figure 5 from Structures.
If we make a tube or cylinder from such a material [as rubber] and then inflate it, by means of an internal pressure, so as to involve a circumferential strain of 50 per cent or more, then the inflation or swelling process will become unstable, and the tube will bulge out...into a spherical protrusion which a doctor would describe as an “aneurism”....Since veins and arteries do, in fact, generally operate at strains around 50 per cent, and since, as any doctor will tell you, one of the conditions it is most desirable to avoid in blood-vessels is the production of aneurisms, any sort of rubbery elasticity is quite unsuitable....The only sort of elasticity which is completely stable under fluid pressures at high strains is that which is represented by Figure 5 [showing the stress increasing exponentially with the strain]. With minor variations, this shape of stress-strain curve is very common indeed for animal tissue....Materials with this [exponential] type of stress-strain curve are extremely difficult to tear. One reason is, perhaps, that the strain energy stored under such a curve--and therefore available to propagate fracture...is minimized."
He continues
Perhaps partly for these reasons the molecular structure of animal tissue does not often resemble that of rubber or artificial plastics. Most of these natural materials are highly complex, and in many cases they are of a composite nature, with at least two components; that is to say, they have a continuous phase or matrix which is reinforced by means of strong fibres of filaments of another substance. In a good many animals this continuous phase or matrix contains a material called 'elastin', which has a very low modulus and a [flat] stress-strain curve...The elastin is, however, reinforced by an arrangement of bent and zig-zagged fibres of collagen...a protein, very much the same as tendon, which has a high modulus...Because the reinforcing fibres are so much convoluted, when the material is in its resting or low-strain condition they contribute very little to its resistance to extension, and the initial elastic behavior is pretty well that of elastin. However, as the composite tissue stretches the collagen fibres begin to come taut; thus in the extended state the modulus of the material is that of the collagen, which more or less accounts for Figure 5.
As you probably can tell, Gordon writes wonderfully and explains mechanics so it's understandable to a layman. His writing is a model of clarity.

 Structures: Or Why Things Don't Fall Down.
Structures was my first exposure to continuum mechanics, but certainly not my last. I was a member of the Mechanical Engineering Section when I worked at the National Institutes of Health, so I was surrounded by outstanding mechanical engineers. My friend Peter Basser—himself a mechanical engineer—would lend me his books, and I recall reading classics such as Love’s A Treatise on the Mathematical Theory of Elasticity and Schlichting’s Boundary Layer Theory. I was impressed by Basser’s model of infusion-induced swelling in the brain and Richard Chadwick’s studies of cardiac biomechanics (Richard was another member of our Mechanical Engineering Section). In many ways, NIH provided a liberal education in physics applied to biology and medicine.

Throughout my career, most of my research has focused on bioelectricity and biomagnetism. Recently, however, I have been working on problems in biomechanics. But that is another story.