Some Mechanical Properties of Matter

Steel holds a bridge up, water flows, air fills a room. All three are made of atoms; what differs is how strongly and how closely those atoms hold on to each other.

Layers of structure#

Nuclear forces between protons and neutrons (together, nucleons) shape the nucleus. Electromagnetic forces between electrons and nuclei shape the atom. Forces between atoms build molecules, and forces between molecules give a material the properties we see and touch.

The force between two atoms#

Plot the potential energy of a pair of atoms against their separation r, taking U = 0 when they are far apart. As they approach, U falls below zero: the force is attractive. At a particular separation r₀, U is lowest: the force is zero and the atoms can rest in equilibrium. Closer still, U rises steeply: the force becomes strongly repulsive.

A molecule forms when its atoms settle into the arrangement of lowest total potential energy. Forces between molecules follow the same pattern: weak attraction far away, growing as they approach, zero at r₀, repulsion closer in.

Bonds#

Atoms bind mainly through electrostatic attraction between electrons and nuclei. Two kinds of bond are especially common:

Solids, liquids and gases#

Two molecules at separation r₀ are in equilibrium. Pull them slightly apart and attraction pulls them back; push them together and repulsion pushes them apart. So a displaced molecule oscillates about r₀.

• Solid: molecules sit very nearly at their equilibrium separations and vibrate with tiny amplitude about fixed positions, so the solid keeps its shape.
• Liquid: the average separation is a little larger, the attraction weaker, and molecules move freely through the whole volume.
• Gas: molecules are far apart and the forces between them are very weak.

Kinds of solid#

In many solids the equilibrium positions form a regular three-dimensional pattern, a crystal; the positions are lattice points. Crystalline solids are classified by what holds them together:

Molecular solids

Covalent molecules held together by weaker forces between molecules: dipole–dipole bonds between polar molecules (like water, whose positive and negative charge centres don’t coincide) and van der Waals bonds between nonpolar ones (H₂, O₂, Cl₂). Usually soft, with low melting points; poor conductors.

Ionic solids

Positive and negative ions on the lattice points, bound by strong electrostatic forces. Hard, with fairly high melting points; poor conductors.

Covalent solids

Atoms joined to their neighbours by shared electrons in a network extending through the whole crystal. In diamond each carbon atom bonds to four others. Very hard, high melting point, poor conductors.

Metallic solids

Positive ions on the lattice points, with the detached electrons roaming the whole solid like a gas. Those free electrons make metals excellent conductors.

Amorphous (glassy) solids#

Some solids have only local order: small groups of 4–5 molecules bond together, but these groups are arranged randomly, like the molecules of a liquid. The forces are much stronger than in a liquid, so the material doesn’t flow. Glass (strong Si–O–Si bonds that don’t extend far) is the classic example.

Our “rigid bodies” were an idealisation. Pull on any real rod and it stretches a little; twist it and it twists. How much, and how it springs back, depends on the material.

Elasticity#

When forces deform a body, internal forces appear that try to restore its natural shape. The property of resisting deformation and restoring shape is elasticity.

• A body that regains its shape completely when the forces are removed is perfectly elastic.
• One that stays deformed and doesn’t recover at all is perfectly plastic (inelastic).
• Most bodies are partially elastic: they recover part of the way.

Stress#

Let a body be acted on by forces that add to zero, so it stays put but deforms. Imagine a small cross-section ΔS inside it. The parts on the two sides push or pull on each other with forces F and −F. Split F into a component Fₙ normal to ΔS and a component Fₜ along it:

Longitudinal stress is tensile when the two parts pull on each other (a stretched wire; Fₙ is the tension) and compressive when they push (a rod squeezed at both ends). Unless stated otherwise, the cross-section is taken perpendicular to the length.

Volume stress

A body immersed in a fluid at pressure P feels a force PΔS on every surface element, always perpendicular to the surface. This force per unit area is the volume stress, and it simply equals the pressure (14.3).

Strain#

Each kind of stress has a matching measure of deformation, a strain (a pure number):

In shear, the four tangential forces on a block add to zero and so do their torques, so the block stays in equilibrium while it deforms.

Double the load on a wire and, as long as it’s a modest load, it stretches twice as far. Robert Hooke spotted this in 1660. The constants of proportionality tell you how stiff a material is.

Hooke’s law#

Each kind of stress–strain pair gives its own constant, a modulus of elasticity.

Young’s modulus

A rod of length L and cross-section A is stretched by forces F at its ends and lengthens by ΔL:

Shear modulus (modulus of rigidity)

Bulk modulus and compressibility

A volume stress (pressure) P changes the volume by ΔV (a decrease, so ΔV is negative):

Poisson’s ratio

Stretch a rod and it gets longer and also thinner. The fractional change in width is proportional to the fractional change in length:

Hooke’s law is only the opening chapter. Keep pulling a wire and it stops springing back, stretches like toffee, and finally snaps. Rubber does something different again.

A metal wire, pulled further and further#

Stretch a metal wire steadily and plot stress against strain. Several landmarks appear:

Past the elastic limit the strain grows rapidly and the behaviour is plastic: remove the load and the wire shortens along a line roughly parallel to the original straight part, but keeps a permanent extension, the permanent set. Keep loading and the wire eventually breaks at the fracture point (d); the stress there is the breaking stress.

• Materials that deform a lot between the elastic limit and fracture are ductile (copper, mild steel: they can be drawn into wires).
• Materials that break soon after the elastic limit are brittle (cast iron, glass).

Rubber#

Vulcanised rubber behaves quite differently. It stays elastic even when stretched to several times its original length (eight times, say) and still returns completely. In that sense it is “more elastic” than steel. But the stress needed for a given strain is far smaller than in steel, so in the sense of strong restoring forces steel is much more elastic.

Two features stand out:

• There is almost no region where stress is proportional to strain.
• On unloading, the curve doesn’t retrace the loading curve: it lies below it, though the sample still ends at its natural length.

A drawn bowstring stores the energy that later launches the arrow. Any stretched wire does the same, and measuring how much it stretches is how we find Young’s modulus in the lab.

Energy stored in a stretched wire#

In its natural shape a body’s molecular potential energy is at its minimum; call it zero. Deforming it means doing work against the internal forces, which raises its elastic potential energy.

Fix one end of a wire (natural length L, cross-section A) and stretch it slowly, so the applied force always equals the tension. When the extension is x, Hooke’s law gives $\dfrac{F/A}{x/L} = Y$, so $F = \dfrac{AY}{L}x$. Stretching it from 0 to l takes work

Measuring Young’s modulus in the laboratory#

A simple, accurate arrangement uses two wires hanging side by side from the same rigid support:

• A reference wire (2–3 m long) carries a main scale and a fixed heavy weight that keeps it straight. Any sagging of the support or expansion due to temperature affects both wires equally, so it cancels out.
• The experimental wire carries a vernier that slides against the main scale, and a hanger for slotted weights.

1. Measure the wire’s radius at several places with a screw gauge; from the known breaking stress, find the breaking load and stay below half of it.
2. Hang a small initial load (1–2 kg) to remove kinks; note the vernier reading.
3. Add equal loads (0.5 or 1 kg) one at a time, waiting a minute each time, and note the reading. Then remove them in the same steps, reading again.
4. Plot extension against load: a straight line through the origin.

If the slope of the line is $\tan\theta = l/M$ (extension per unit mass),

A steel needle laid gently on water floats. Insects skate across ponds. Raindrops are round. The surface of a liquid behaves like a stretched elastic skin.

Why surfaces are special#

A molecule deep inside water is surrounded by water molecules on every side, and their pulls balance. A molecule at the surface has water below but air above, so the forces on it are lopsided. The “surface” here means a layer about 10–15 molecular diameters thick: molecular forces become negligible beyond that distance, so a molecule deeper than this feels balanced forces.

Surface tension#

Imagine a line AB drawn on the surface of a liquid. It divides the surface into two parts, and experiment shows they pull on each other with a force proportional to the length of the line. The forces are tangential to the surface and perpendicular to the line, just as two halves of a stretched rubber sheet pull on each other.

Seeing it happen#

The thread loop

Dip a wire ring in soap solution to get a film, and lay a wet loop of thread on it. The loop sits in whatever shape it lands, because the film pulls it equally from inside and outside. Prick the film inside the loop: now only the outside film pulls, radially outward all round, and the loop snaps into a perfect circle.

The sliding wire

Make a soap film on a U-shaped wire frame closed by a light sliding wire. Held horizontal (and with little friction), the wire is pulled toward the closed end of the frame. Held vertical with the sliding wire at the bottom, a small weight hung from it can be balanced by the film’s pull.

Surfaces try to shrink#

Because every line in the surface is under tension, the surface shrinks whenever it can. Both demonstrations show this: the punctured film leaves the largest hole a loop of fixed length can enclose (a circle), so the remaining film is as small as possible; the sliding wire moves to reduce the film’s area. Paintbrush bristles splay freely under water but cling together once lifted out, cutting the area of the liquid surfaces between them. A small drop is nearly spherical, because a sphere has the least surface for a given volume (gravity flattens large drops a little).

Blowing a soap bubble takes effort, and it’s hardest right at the start, when the bubble is tiny. Surface tension explains both why making surface costs energy and why small bubbles push back harder.

Surface energy#

A molecule in the surface feels a net inward pull (fewer neighbours above it). Bringing a molecule from the interior up to the surface means working against that pull, so surface molecules have extra potential energy. This extra energy of the surface layer is the surface energy. Molecules constantly swap between surface and interior, but the energy cost of having surface remains.

Take the U-frame and sliding wire (length l) again. A soap film, though it looks thin, is hundreds of thousands of molecules thick, so it has two surfaces, and the wire is pulled with $F = 2Sl$. Pull the wire out slowly through x: the work done is $W = Fx = S(2lx)$, and new surface of area 2lx is created. With no change in kinetic energy, this work is stored as surface energy:

Excess pressure inside a drop#

Cut a small spherical drop of radius R into two hemispheres along a diameter and consider the forces on one of them (its surface layer):

• Surface tension from the other half, acting all round the rim of length 2πR, parallel to the axis: $F_1 = 2\pi RS$.
• The air outside pushes on every bit of the curved surface. By symmetry only components along the axis survive, and each element’s component equals P₀ times the element’s projection on the flat diametric plane. The projections fill the disc of area πR², so $F_2 = P_0\pi R^2$.
• The liquid inside, at pressure Pᵢ, pushes outward the same way: $F_3 = P_i\pi R^2$.

The same holds for an air bubble inside a liquid: there is one surface, with air on the concave side, so the air inside is at a pressure 2S/R above the surrounding liquid.

Excess pressure in a soap bubble#

A soap bubble has air both inside and outside, and a film with two surfaces. Let P₀ be the outside air pressure, P′ the pressure in the film and Pᵢ the air pressure inside. The outer surface has the film on its concave side: $P' - P_0 = 2S/R$. The inner surface has the inside air on its concave side: $P_i - P' = 2S/R$. Adding:

Dip the corner of a paper towel in water and the water climbs. Stand a thin glass tube in water and the water rises inside it, the higher the narrower the tube. In mercury, the same tube shows the opposite.

The contact angle#

Where a liquid surface meets a solid, it curves. Dip a glass plate in water and the water near it curls upward, as if pulled up; dip it in mercury and the surface curls down.

Liquids that rise along the solid have contact angles below 90°; liquids that are depressed have angles above 90°.

Why the surface bends#

A liquid at rest can’t sustain tangential stress, so the resultant force on any small piece of its surface layer must be perpendicular to the surface. Near the solid, a piece of the surface feels:

• the adhesive force Fₛ from the nearby solid molecules (force between unlike molecules), pointing into the solid;
• the cohesive force F_L from the neighbouring liquid molecules (force between like molecules), pointing into the liquid;
• its weight.

The surface settles perpendicular to their resultant. If the resultant points into the solid, the surface is concave upward and the liquid climbs the wall; if it points into the liquid, the surface is convex and the liquid is depressed.

Capillary rise#

Dip a narrow (capillary) tube of radius r in a liquid of density ρ, surface tension S and contact angle θ. The surface inside is nearly spherical. Consider the column of liquid raised to a height h:

• Along the contact circle (length 2πr) the tube pulls the liquid surface with force S per unit length along the liquid surface. The vertical components add up to $2\pi rS\cos\theta$; horizontal ones cancel.
• Air pressure on top and liquid pressure at the bottom (at the level of the outside surface) are both atmospheric, so they cancel.
• The weight of the raised column (ignoring the meniscus) is $\pi r^2h\rho g$.

Pour water and honey side by side. Both are liquids, but honey crawls. The difference is internal friction between layers of the fluid: viscosity.

Internal friction#

When one layer of a fluid slides (or tries to slide) over another, the two layers exert tangential forces on each other that oppose the relative motion. This property is viscosity, and the forces are viscous forces.

A solid surface moving through a fluid drags the layer touching it along. A boat moving slowly on a calm river carries the water in contact with it, while the water at the river bed stays still; layers in between move at intermediate speeds. If the speed changes by dv across a distance dz perpendicular to the flow, $dv/dz$ is the velocity gradient.

The SI unit of η is N·s/m². The CGS unit, dyne·s/cm², is called the poise (after Poiseuille): 1 poise = 0.1 N·s/m². Its dimensions:

Flow through a narrow tube: Poiseuille’s equation#

In steady flow through a tube, the fluid touching the wall is at rest and the centre moves fastest. Poiseuille found how the volume flowing per second depends on the pressure difference P across a tube of length l and radius r. Assume the rate depends on the pressure gradient P/l, on η and on r as a product of powers:

Matching M, L, T: a + b = 0, −2a − b + c = 3, −2a − b = −1, so a = 1, b = −1, c = 4. With the full theory giving k = π/8:

Critical velocity and the Reynolds number#

At low speeds the flow through a tube is steady; above a certain speed it becomes turbulent. The largest speed that still allows steady flow is the critical velocity. Whether flow is steady depends on the fluid’s density ρ, speed v and viscosity η and on the tube diameter D, all combined in a dimensionless number:

Drop a steel ball into a jar of honey and it sinks at a slow, perfectly steady speed. Drop it in air and it keeps speeding up. Viscous drag, growing with speed, makes the difference.

Drag on a sphere: Stokes’ law#

A solid moving through a fluid drags the neighbouring fluid with it, setting up velocity gradients, so viscous forces oppose its motion. For a sphere of radius r moving slowly at speed v through a fluid of viscosity η, assume $F = k\,r^av^b\eta^c$:

M: 1 = c. L: 1 = a + b − c. T: −2 = −b − c. So a = b = c = 1, and with the full theory giving k = 6π:

Terminal velocity#

A sphere dropped into a fluid feels its weight W (down), the buoyant force B (up) and the viscous drag F (up).

• W and B are constant; F grows with speed.
• At first v = 0, so F = 0 and the sphere accelerates under W − B.
• As it speeds up, F grows until F = W − B. The net force is then zero and the sphere falls at a constant terminal velocity.

With the sphere’s density ρ and the fluid’s σ:

Measuring viscosity: Stokes’ method#

A tall tube of the test liquid sits in a water bath at a known temperature, with three equally spaced marks P, Q, R well below the top. Drop a small metal ball in through a narrow guide tube and time it over PQ and over QR. If the two times differ, the ball hasn’t reached terminal velocity before P; try a smaller ball. When the times are equal, $v_t = d/t$. Measure the ball’s radius (screw gauge) and mass, so ρ = m/(4/3 πr³), and then