Thermal Expansion and Thermal Stress

From more jiggle to more room

Guide one left the lattice buzzing: heating a solid pours energy into its atoms, and they vibrate harder about their resting places — the vibration we called phonons, the store we called heat capacity. This guide asks the obvious next question: what does all that extra jiggling actually do to the size of the object? The answer is that almost everything gets bigger when it warms and smaller when it cools — thermal expansion — and it is no small curiosity. A steel bridge, a jet engine, and the solder under a computer chip all live or die by it. Let us first pin down how we measure it, then uncover why it happens, and finally meet the trouble it causes when a material is not free to grow.

The measure is the coefficient of thermal expansion — call it alpha. It answers a simple question: for each one-degree rise, by what fraction does a length grow? Formally the linear expansion coefficient alpha_l is the fractional change in length per degree, and the length change is delta L = alpha_l x L0 x delta T, where L0 is the starting length and delta T the temperature change. The numbers are tiny — alpha for most solids sits at a few times 10^-6 per degree C — so we usually quote it in 'parts per million per degree'. A steel beam with alpha near 12 x 10^-6 that is 10 metres long and warms by 30 degrees C grows by 12 x 10^-6 x 10 m x 30 = 0.0036 m, about 3.6 mm. Small — but a bridge that cannot accommodate 3.6 mm will buckle.

The size of alpha sorts the material classes almost by itself, and it tracks bond strength. Ceramics expand least (fused silica about 0.5, alumina about 8 x 10^-6) because their stiff ionic-covalent bonds resist being stretched. Metals sit in the middle — tungsten about 4.5, iron and steel about 12, aluminium about 23, all in units of 10^-6 per degree C. Polymers expand the most, often 50 to 200 x 10^-6, because much of a polymer is held by feeble van der Waals forces between chains that uncoil and spread easily with heat. Notice the pattern already forming: the stronger and more directional the bonding, the smaller the expansion. That is no coincidence, and the next section is the reason why.

The lopsided bond well

Way back in the bonding rung we drew the bonding-energy curve: energy plotted against the spacing between two atoms, dipping to a low point at the equilibrium spacing r0 — the comfortable distance the atoms rest at when cold. Here is the crucial detail we glossed over then. That well is not symmetric. Squeeze the atoms closer than r0 and the energy shoots up steeply — the inner wall is stiff because electron clouds refuse to overlap. Pull them apart and the energy rises far more gently — the outer wall is a lazy slope. The well is lopsided: steep on the near side, shallow on the far side.

Now warm the pair up. At a given temperature the atoms rattle back and forth, sweeping across the well at some energy level, spending as much time against the far wall as against the near wall. But because the far wall is shallow, the atom can wander much further out before it is turned back than it can push in — so the midpoint of its swing sits a little to the right of r0. The hotter it gets, the higher the swing, and the further that midpoint drifts outward. The average spacing grows even though the resting spacing r0 has not moved. Multiply that atom-scale drift over billions of bonds and you get a beam that is measurably longer. That, at heart, is thermal expansion.

BONDING-ENERGY WELL  (energy E vs atom spacing r) -- why solids expand

  E
   \                                        ____ gentle OUTER wall
    \                                 __.--'
     \                          __.--'      stretching is "cheap"
      \                   __.--'
       \___         __.--'   <-- at temperature T the atom rattles
  steep \   \      /   |          between the walls at energy E(T)
  INNER  \   \    /    |
  wall    \   \  /     |     because the well is LOPSIDED, the swing's
 (squeeze  \   \/      |     midpoint sits to the RIGHT of r0
  is "dear") \  r0     |
                    r_avg(T) > r0   ->   the solid has EXPANDED

  deeper, more symmetric well (strong bond, high melting point)
      ->  midpoint drifts less  ->  smaller alpha
  a PERFECT symmetric parabola  ->  r_avg = r0 always  ->  ZERO expansion

Thermal expansion is the sideways drift of the vibration's midpoint. It exists only because the bond well is lopsided (anharmonic) — a symmetric well would not expand at all.

This picture carries an honest and surprising corollary: if the bond well were a perfect symmetric parabola, the near and far excursions would match exactly, the midpoint would never move, and the solid would not expand at all. Thermal expansion is entirely an effect of the well's lopsidedness — its anharmonicity. It also explains our pattern: a deep, strong bond well (which also means a high melting point) has steeper, more symmetric walls near the bottom, so the midpoint drifts less and alpha is small. As a loose but genuinely useful rule of thumb, high-melting materials expand little and low-melting ones expand a lot — tungsten melts near 3400 degrees C and barely expands, while a soft polymer melts low and expands freely. Like most such rules it has exceptions, but the bonding logic behind it is real.

Thermal stress: the price of being held

Here is the misconception to kill early: expansion by itself does no harm. A rod lying free on a table can grow as long as it likes with zero internal stress — it just gets longer. The damage begins the moment something stops it from moving. Clamp that rod rigidly between two immovable walls and then heat it. It wants to lengthen by a strain of alpha x delta T, but the walls forbid it, which is mechanically identical to squashing an already-expanded rod back to its original length. By Hooke's law that forced compression sets up a real internal stress — a thermal stress.

The size of that stress is beautifully simple: sigma = E x alpha x delta T, where E is the material's Young's modulus. Put steel numbers in — E about 200 GPa (that is 200,000 MPa), alpha about 12 x 10^-6 per degree C, and a modest heating of delta T = 100 degrees C. The blocked strain is 12 x 10^-6 x 100 = 0.0012, and the stress is 200,000 MPa x 0.0012 = 240 MPa of compression. That is enormous — right up against the yield strength of ordinary mild steel. A mere 100-degree rise, if fully constrained, can push steel to the edge of permanent deformation. And notice a startling thing about the formula: the length and the cross-section have vanished. A fully clamped rod builds the same thermal stress whether it is a centimetre or a kilometre long.

The sign matters more than anything. Heat a constrained part and it goes into compression, which metals shrug off. Cool a constrained part and it goes into tension — it is being stretched — and tension is where trouble lives. Concrete poured on a hot day and cooled overnight, a weld shrinking against the cold plate around it, a glass dish yanked from the oven onto a cold counter: each develops tensile thermal stress, and brittle materials answer tension by cracking — a brittle fracture. When the temperature change is sudden, the outside cools and tries to shrink while the still-hot inside holds it stretched — a self-inflicted mismatch that is the essence of thermal shock, the subject of guide four. Here we simply flag the mechanism: constraint plus a temperature change equals stress, and cooling is the dangerous direction for anything brittle.

When two materials disagree

There is a second, sneakier way to build thermal stress with no clamp in sight: bond two materials with different alphas and change the temperature. Now each layer is the other's wall. Heat the pair and the high-alpha layer strains to grow more than its partner allows; the interface fills with shear and peeling stresses. This thermal expansion mismatch is one of the most common failure drivers in real engineering — and, depending on whether you fight it or exploit it, it is either a nuisance or a tool.

Exploit it and you get the bimetallic strip, the beating heart of the old-school thermostat. Bond a strip of high-expansion brass (alpha about 19 x 10^-6) to a strip of low-expansion Invar (an iron-nickel alloy near 1.5 x 10^-6). Warm the sandwich and the brass side lengthens far more than the Invar side, but they are glued together and cannot slide — so the whole strip has no choice but to curl, bowing toward the Invar side. That curl trips a switch at a set temperature and cuts the heater; as it cools it straightens and closes again. A bimetallic strip turns an invisible few-parts-per-million difference into a visible mechanical motion — the same trick runs oven thermometers and old turn-signal blinkers.

Far more often you are fighting mismatch, and the front line today is electronics. A silicon chip has a very low alpha near 2.6 x 10^-6; the circuit board beneath it is nearer 15 to 18. Every time the device powers up and heats, then cools, the tiny solder joints holding the chip are worked back and forth by the difference — and repeated cycling fatigues them until a joint cracks and the part goes dark. Engineers fight back by matching expansion coefficients (choosing ceramic substrates close to silicon), by adding a compliant 'underfill' that shares the strain, and by keeping joints small. The same law governs glass-to-metal seals: the metal lead through a light bulb or a sensor must be a special alloy like Kovar whose alpha is tuned to match the glass, or the cooling seal cracks the glass in tension. Match the expansion, or pay for the mismatch.

Designing around it

If constraint is what turns harmless growth into dangerous stress, the first design cure is obvious: stop constraining. Let the part move and the stress never appears. That is what an expansion joint does — the finger-like gaps you feel your car thump over on a bridge, the loops welded into long steam and gas pipelines, the sliding supports under railway rails. Each is a deliberate place for the structure to grow and shrink freely with the seasons. Design an expansion joint correctly and a 100-metre bridge that stretches several centimetres between winter and summer never builds a single MPa of thermal stress. It is the cheapest solution whenever you can afford the gap.

When you cannot let it move — a coating must stay bonded, a seal must stay sealed — you manage the mismatch instead. A thermal-barrier coating on a turbine blade is the vivid case: a thin ceramic top layer of yttria-stabilized zirconia insulates the metal blade from the searing gas. But ceramic and superalloy have different alphas, so on every engine start and stop the coating is strained against the metal. Designers survive it three ways: they pick a zirconia whose alpha (about 10 x 10^-6) is unusually high for a ceramic, deliberately closer to the metal; they insert a metallic 'bond coat' as a compliant buffer; and they grow the ceramic as loosely-packed columns that can open and close like a picket fence instead of a solid slab. A thermal-barrier coating is a whole engineering discipline built around living with one stubborn mismatch.

A third route is to pick a material that barely expands. Cookware glass (Pyrex, a borosilicate with alpha about 3.3 x 10^-6) survives the oven-to-counter jump that would shatter ordinary soda-lime glass; fused silica (about 0.5) and the alloy Invar (about 1.5, thanks to a magnetic quirk that cancels the normal expansion) anchor precision instruments and moulds. Two honest caveats before we close. First, alpha is not a fixed constant: it creeps up with temperature and can jump abruptly when a material changes phase, so a single quoted value is really an average over some stated range. Second, alpha is a single number only for cubic crystals and randomly-oriented polycrystals — in graphite, or an HCP metal, or any low-symmetry crystal, expansion differs along different directions, so 'the' coefficient hides a direction you must respect.