J.E. Gordon, a leading engineer at the Royal Aircraft Establishment at Farnborough and holder of the British Silver Medal of the Royal Aeronautical Society, wrote two brilliant books on engineering: “The New Science of Strong Materials” and “Structures – Or Why Things Don’t Fall Down”. Elon Musk has recommended the latter of the two books, and I can only encourage you to read both. In my eyes, the role of a good non-fiction writer is to explain the intricacies of a non-trivial topic that we can see all around us but nevertheless rarely fully appreciate. Something interesting hidden in plain sight, if you will.

With this in mind, let’s discuss an underappreciated topic from the world of materials science.

First of all, what do we mean by a material’s stiffness and strength?

To be able to compare the load and deformation acting on components of different sizes, engineers prefer to use the quantities of stress and strain over load and deformation. Imagine a solid rod of a certain diameter and length which is being pulled apart in tension. Naturally, two rods of the same material but of different diameters and lengths will deform by different amounts. However, if both rods are stressed by the same amount, then they will experience the same amount of strain. In our simple one-dimensional rod example, the stress $\sigma$ is given by

$\sigma = \frac{P}{A}$

where $P$ is the tensile force and $A = \pi d^2 / 4$ is the cross-sectional area for a diameter $d$, i.e. force normalised by cross-sectional area.

The engineering strain $\epsilon$ is given by

$\epsilon = \frac{\Delta L}{L}$

where $\Delta L$ is the change in length (deformation) of the rod and $L$ is its original length, i.e. the deformation normalised by original length.

For an elastic material deforming linearly (i.e. no plastic deformation), the ratio of stress to strain is constant, and for our simple one-dimensional example the constant of proportionality is equal to the stiffness of the material.

$E = \frac{\sigma}{\epsilon}$  (Hooke’s Law).

This stiffness $E$ is known as the Young’s modulus of the material.

These two definitions of stress and strain illustrate a simple point. By dividing force by cross-sectional area and change in length (deformation) by original length, the role of geometry is eliminated entirely. This means we can deal purely in terms of material properties, i.e.  Young’s modulus (stiffness), stress to failure (strength), etc., and can therefore compare the degree of loading (stress) and deformation (strain) in components of different sizes, shapes, dimensions, etc.

We can all appreciate that metals are incredibly strong and stiff. But why are some materials stronger and stiffer than others? Why don’t all materials have the same strength and stiffness? Aren’t all materials just an assemblage of molecules and atoms whose molecular bonds stretch and eventually separate upon fracture? If this is so, why don’t all materials break at the same value of stress and strain?

The stiffness and strength of a material does indeed depend on the relative stiffness and strength of the underlying chemical bonds, and these do vary from material to material. But this difference is not sufficient to explain the large variations in strength that we observe for materials such as steel and glass – that is, why does glass break so easily and steel does not?

In the 1920s, a British engineer called A.A. Griffith explained for the first time why different materials have such vastly different strengths. To calculate the theoretical maximum strength of a material, we need to use the concept of strain energy. When we stretch a rod by 1 mm using a force of 1,000 N, the 1 J of energy we exerted (0.001 m times 1,000 N) is stored within the material as strain energy because individual atomic bonds are essentially stretched like mechanical springs. Written in terms of stresses and strains, the strain energy stored within a unit volume of material is simply half the product of stress and strain:

$\text{Strain Energy per unit volume} = \frac{1}{2} \sigma \times \epsilon$

Griffith’s brilliant insight was to equate the strain energy stored in the material just before fracture to the surface energy of the two new surfaces created upon fracture.

Surface energy??

It is probably not immediately obvious why a surface would possess energy. But from watching insects walk over water we can observe that liquids must possess some form of surface tension that stops the insect from breaking through the surface. When the surface of a liquid is extended, say by inflating a soap bubble, work is done against this surface tension and energy is stored within the surface. Similarly, when an insect is perched on the surface of a pond, its legs form small dimples on the surface of the water and this deformation causes an increase in the surface energy. In fact, we can calculate how far the insect sinks into the surface by equating the increase in surface energy to the decrease in gravitational potential energy as the insect sinks. Furthermore, liquids tend to minimise their surface energy under the geometrical and thermodynamic constraints placed upon them, and this is precisely why raindrops are spherical and not cubic.

When a liquid freezes into a solid, the underlying molecular structure changes, but the overall surface energy remains largely the same. Because the molecular bonds in solids are so much stronger than those in liquids, we can’t actually see the effect of surface tension in solids (an insect landing on a block of ice will not visibly dimple the external surface). Nevertheless, the physical concept of surface energy is still valid for solids.

So, back to our fracture problem. What we want to calculate is the stress which will separate two adjacent rows of molecules within a material. If the rows of molecules are initially $d$ metres apart then a stress $\sigma$ causing a strain $\epsilon$ will lead to the following strain energy per square metre

$\text{Strain Energy per unit area} = \frac{1}{2} \sigma \times \epsilon \times d$

From Hooke’s law we know that

$\epsilon = \frac{\sigma}{E}$

and therefore replacing $\epsilon$ in the first equation we have

$\text{Strain Energy per unit area} = \frac{d\sigma^2}{2E}$

Now, if the surface energy per square metre of the solid is equal to $G$, then the separation of the two rows of molecules will lead to an increase in surface energy of $2G$ (two new surfaces are created). By assuming that all of the strain energy is converted to surface energy:

$\frac{d\sigma^2}{2E} = 2G \Rightarrow \sigma = 2 \sqrt{\frac{G E}{d}}$

There is typically a considerable amount of plastic deformation in the material before the atomic bonds rupture. This means that the Young’s modulus decreases once the plastic regime is reached and the strain energy is roughly half of the ideal elastic case. Hence, we can simply drop the 2 in front of the square root above to get a simple, yet approximate, expression for the strength of a material

$\sigma = \sqrt{\frac{G E}{d}}$

As the values of $E$ and $G$ vary from material to material, the theoretical strengths will be different as well. The surface tension of a material is roughly proportional to the Young’s modulus because the same chemical bonds give rise to both these properties. In fact, the relationship between surface energy and Young’s modulus can be approximated as

$G \approx \frac{Ed}{20}$

such that the strength of a material is approximately proportional to the Young’s modulus by the following relation

$\sigma \approx \sqrt{\frac{E^2}{20}} \approx \frac{E}{5}$

Given, the relationship between stress and strain we can conclude that the theoretical failure strain of most materials ought to be, approximately,

$\epsilon = \frac{\sigma}{E} \approx \frac{1}{5}$

or 20% for basically all materials.

In everyday practise, most materials have failure strengths far beneath the theoretical maximum and also vary widely in their failure strains. To explain why, Griffith conducted some simple experiments on glass.  After calculating the Young’s modulus $E$ from a simple tensile test and assuming a molecular spacing of $d = 3$ Angstroms, Griffith arrived at a theoretical strength for glass of 14,000 MPa. Griffith then tested a number of 1 mm diameter glass rods in tension  and found the strength to be on average around 170 MPa, i.e. $1/100$th of the theoretical value.

The pultrusion process used to create the glass rods allowed Griffith to pull thinner and thinner rods, and as the diameter decreased, the failure stress of the rods started to increase – slowly at first, but then very rapidly. Glass fibres of 2.5 $\mu$m in diameter showed strengths of 6,000 MPa when newly drawn, but dropped to about half that after a few hours. Griffith was not able to manufacture smaller rods so he fitted a curve to his experimental data and extrapolated to much smaller diameters. And lo and behold, the exponential curve converged to a failure strength of 11,000 MPa – much closer to the 14,000 MPa predicted by his theory.

Variation of tensile strength with fibre diameter. From W.H. Otto (1955). Relationship of Tensile Strength of Glass Fibers to Diameter. Journal of the American Ceramic Society 38(3): 122-124.

Griffith’s next goal was to explain why the strength of thicker glass rods fell so far below the theoretical value. Griffith surmised that as the volume of a specimen increases, some form of weakening mechanisms must be active because the underlying chemical structure of the material remains the same. This weakening mechanism must somehow lead to an increase in the actual stress around a future failure site and act as a stress concentration. Luckily, the idea of stress concentrations had previously been introduced in the naval industry, where the weakening effects of hatchways and other openings in the hull had to be accounted for. Griffith decided that he would apply the same concept at a much smaller scale and consider the effects of molecular “openings” in a series of chemical bonds.

The idea of a stress concentration is quite simple. Any hole or sharp notch in a material causes an increase in the local stress around the feature. Rather counter-intuitively, the increase in local stress is solely a function of the shape of the notch and not of its size. A tiny hole will weaken the material just as much as a large one will. This means a shallow cut in a branch will lower the load-carrying capacity just as well as a deep one – it is the sharpness of the cut that increases the stress.

We can visualise quite easily what must happen at a molecular scale when we introduce a notch in a series of molecules. A single strand of molecules must reach the maximum theoretical strength. Similarly, placing a number of such strands side by side should not effect the strength. However, if we cut a number of adjacent strands at a specific location perpendicular to the loading direction, then the flow of stress from molecule to molecule has been interrupted and the load in the material has to be redistributed to somewhere else. Naturally, the extra load simply goes around the notch and will therefore have to pass through the first intact bond. As a result, this bond will fail much earlier than any of the other bonds as the stress is concentrated in this single bond. As this overloaded bond breaks, the situation becomes slightly worse because the next bond down the line has to carry the extra load of all the broken bonds.

Stress concentration at a notch

The stress concentration factor of a notch of half-length $a$ and radius of curvature at the crack tip $R$ is given by

$1 + 2 \sqrt{\frac{a}{R}}$

If we now consider a crack about 2 $\mu$m long and 1 Angstrom tip radius, this produces a stress concentration factor of

$1 + 2 \sqrt{\frac{1 \times 10^{-6}}{1 \times 10^{-10}}} = 201$

and therefore this would lower the theoretical strength of glass from 14,000 MPa to around 70 MPa, which is very close to the average strength of typical domestic glass.

As a result, Griffith made the conjecture that glass and all other materials are full of tiny little cracks that are too small to be seen but nevertheless significantly reduce the theoretical maximum strength. Griffith did not give an explanation for why these cracks appeared in the first place or why they were rarer for thinner glass rods. As it turns out, Griffith was correct about the mechanism of stress concentrationa, but wrong about their origins.

It took quite some time until a more satisfactory explanation was provided, dispelling the notion that the reduction in strength could be attributed to inherent defects within the material. After WWII, experiments showed that even thick glass rods could approach the theoretical upper limit of strength when carefully manufactured. It was also noticed that stronger fibres would weaken over time, probably as a result of handling, and that weakened fibres could consequently be strengthened again by chemically removing the top surface. By depositing sodium vapour on the external surface of glass, the density of cracks could be visualised and was found to be inversely proportional to the strength of the glass – the more cracks, the lower the strength, and vice versa.

These cracks are a simple result of scratching when the exterior surface comes in contact with other objects. Larger pieces of glass are more likely to develop surface cracks due to the larger surface area. Furthermore, thin glass fibres are much more likely to bend when in contact with other objects, and are therefore less likely to scratch. This means that there is nothing special about thin fibres of glass – if the surface of a thick fibre can be kept just as smooth as that of a thin fibre then it will be just as strong.

This means that an airplane cast from one piece of 100% pristine glass could theoretically sustain all flight loads, such an idea ludicrous in reality, because the likelihood of inducing surface cracks during service is basically 100%.

At this point you might be asking, what is different about metals – why are they used on aircraft instead?

The difference boils down to differences between the atomic structure of glasses and metals. When liquids freeze they typically crystallise into a densely packed array and form a solid that is denser than the liquid. Glasses on the other hand do not arrange themselves into a nicely packed crystalline structure but rather cool into a purely solidified liquid. Glasses can crystallise under some circumstances under a process known as devitrification, but the glass is often weakened as a result. When a solid crystallises, it can deform via a new process in which it starts to flow in shear just like Plasticine or moulding clay does when it is formed.

There is no clear demarcation line between a brittle (think glass) and ductile (think metal) material. The general rule of thumb is that a brittle material does not visibly deform before failure and failure is caused by a single crack that runs smoothly through the entire material. This is why it’s often possible to glue a broken vase back together.

In ductile materials, there is permanent plastic deformation before ultimate failure and so these materials behave more like moulding clay. Before a ductile material, like mild steel, finally snaps in two, there is considerable plastic deformation which can be imagined along the lines of flowing honey or treacle. This plastic flowing is caused by individual layers of atoms sliding over each other, rather than coming apart directly. As this shearing of atomic bonds takes place, the material is not significantly weakened because the atomic bonds have the ability to re-order, and the material may even be strengthened by a process known as cold working (atomic bonds align with the direction of the applied load). The amount of shearing before final failure depends largely on the type of metal alloy and always increases as a metal is heated; hence a blacksmith heats metal before shaping it.

Generally, these two fracture mechanism, brittle cracking and plastic flowing, are always competing in a solid. The material will break in whatever mechanism is weakest; yield before cracking if it is ductile or crack directly if it is brittle.