Bulk Modulus and Shear Modulus

Drop a piston into a cylinder of water and push hard. The water level hardly drops — it takes about 2000 atmospheres of pressure to compress water by even 10%. Drop the same piston into a cylinder of air, and the air squeezes easily: double the pressure and the volume halves. Both fluids are "pushable," but water is a million times stiffer than air against uniform compression. That stiffness is its bulk modulus.

Now take a slab of rubber and push the top face sideways while holding the bottom still. The rubber tilts — the top face moves a centimetre or two, and it springs back when you let go. Do the same to a glass of water: the top water moves, but it just keeps moving, forever, with no restoring force. Glass won't tilt at all (it resists and then shatters). The rubber has a modest shear modulus (it resists being sheared but gives a bit). Glass has a large one (it resists fiercely). Water has zero shear modulus — no resistance at all. That zero is the defining difference between a solid and a fluid.

This article tells the story of two elastic constants — the bulk modulus and the shear modulus — and the ratio that ties longitudinal and lateral deformation together, Poisson's ratio. Along the way, we'll see why water at the bottom of the Mariana Trench is only about 5% denser than water at the surface, why diamond is the "hardest" material in two different senses, and why a cube of toothpaste, under a knife, splits along the shear plane instead of flowing like water.

Uniform compression — the bulk modulus

Imagine you have a cube of material — say, a 1 cm cube of steel — and you submerge it deep under water. At depth h below the surface, the water pressure is P = P_\text{atm} + \rho g h, pressing inward on every face of the cube. The pressure is isotropic: every face feels the same compressive stress, from every direction at once.

Under isotropic pressure, the cube shrinks — not in one direction, but uniformly in all three. The length along every side decreases by roughly the same fractional amount, so the volume decreases. How much the volume decreases for a given pressure increment is what the bulk modulus quantifies.

The definition

Start with the cube at volume V_0 and pressure P_0. Increase the pressure by a small \Delta P. The volume decreases by |\Delta V| (since compression means \Delta V < 0). The fractional volume change is the volumetric strain:

Like all strains, \epsilon_V is dimensionless. It is negative for compression.

The applied volumetric stress is the pressure increment \Delta P. Now the linear elasticity assumption — stress proportional to strain — tells you that:

where the minus sign is there because increasing the pressure (\Delta P > 0) produces a volume decrease (\Delta V < 0), so we need the minus sign to make B a positive number for physically sensible materials.

Solving for B:

For an ideal gas at constant temperature, PV = nRT, so dP = -P\,dV/V, which gives B_\text{isothermal} = P. For adiabatic compression (no heat flow), B_\text{adiabatic} = \gamma P, where \gamma is the ratio of specific heats. So the bulk modulus of a gas depends on how you compress it — slowly (isothermal) or quickly (adiabatic). For solids and liquids, the distinction is small because thermal effects of compression are negligible.

Representative bulk moduli

Notice the enormous range: B varies from 10^{-4} GPa (air) to 440 GPa (diamond) — a factor of four million between the most and least compressible materials on this table. Solids are thousands of times stiffer than gases.

Compressibility — how easily a material is squeezed

The compressibility is the reciprocal of the bulk modulus:

It measures how much fractional volume change you get per unit pressure increase. High-compressibility materials (gases) shrink a lot under modest pressure. Low-compressibility materials (diamond, tungsten) hardly budge even under enormous pressure.

Water under the Mariana Trench. The Trench bottom is about 11 km below sea level, where the pressure is roughly P \approx \rho g h = 1030 \times 9.8 \times 11000 \approx 1.11 \times 10^8 Pa \approx 1100 atm.

Fractional volume change at the bottom:

The water at the Trench bottom is compressed by only about 5% — so despite the crushing pressure, water is essentially incompressible compared with the numbers you'd naively expect. This small compressibility is why the speed of sound in water (about 1500 m/s) is much higher than in air (340 m/s): the restoring forces against compression are much stronger.

Air in a bicycle tire. A car tire is pumped to about 2 atm gauge (3 atm absolute). The air inside has been compressed by a factor 3/1 = 3, so the volume is roughly one-third of the free-atmosphere volume of the same gas. This is huge compared to a solid. For a gas, compressibility is a first-order effect; for a solid or liquid, it is a fractional-percent correction.

The shear modulus — resistance to sliding

Now switch to a different kind of deformation. Take a block of material — a rubber eraser will do — and apply a force tangentially to its top face, while holding the bottom face fixed. The block deforms into a parallelogram. The top face slides sideways by a distance x while the bottom stays put, and the block's height h tilts by an angle \theta (the shearing strain) where \theta \approx x/h for small angles.

The shear stress is \tau = F/A, the tangential force divided by the sheared area. The shear modulus (also called the rigidity modulus or modulus of rigidity) is the ratio of shear stress to shear strain:

Representative shear moduli

The list reads almost like the bulk modulus table — same order, same order of magnitudes — but at the bottom there is a new entry: zero. All fluids, gases and liquids, have exactly zero shear modulus in the equilibrium sense.

Why fluids have zero shear modulus

This is the single most important conceptual distinction between a solid and a fluid, and it deserves slowing down on.

Imagine a solid block of rubber. Push the top face sideways with force F per area A (shear stress \tau = F/A). The rubber deforms into a parallelogram and then stops deforming at some finite angle \theta. It has reached an elastic equilibrium: the restoring force from the deformed rubber exactly balances your applied force. Remove the force, and the rubber springs back.

Now imagine a block of water with the same setup. Push the top surface sideways. The water does not reach an equilibrium parallelogram. Instead it flows. The water layers slide past each other continuously, as long as you keep pushing, with no bound on how far they will eventually displace. The water has no "equilibrium" configuration that is different from the undeformed one — any attempt to shear it simply results in motion.

Algebraically: if you set G = 0 in the relation \tau = G\gamma, then any non-zero \tau forces \gamma to grow without bound — which physically translates into flow. That is exactly what fluids do.

What fluids do have: viscosity, which is the resistance to the rate of shear (how fast you are shearing), not to the amount of shear. Water resists fast shearing (try pushing your hand quickly through it) more than slow shearing (a boat on a calm lake hardly feels the water). But at any finite shearing rate, no matter how slow, the fluid eventually flows — because the static shear modulus is zero.

This distinction is the physical definition of a fluid: a fluid is any substance with zero static shear modulus. Gases, liquids, plasmas all qualify. Solids — everything that can sit in a cup without reshaping itself — have G > 0.

Earthquakes and what fluids transmit

The zero shear modulus of fluids has a surprising consequence for wave propagation. Elastic waves in a solid come in two kinds:

• Longitudinal (P-waves) — compression waves, where the material alternately compresses and expands along the direction of motion. These exist in any medium that has bulk modulus, so they travel through solids and fluids.
• Transverse (S-waves) — shear waves, where the material oscillates perpendicular to the wave's motion. These require shear modulus, so they travel only in solids.

When an earthquake produces both kinds of waves, the transverse S-waves pass through the Earth's mantle (solid) but stop abruptly at the outer core. The outer core is molten iron, a fluid — and fluids cannot transmit S-waves. This "S-wave shadow zone" is how seismologists first proved that the Earth's outer core is liquid. The P-waves, by contrast, pass straight through the core (refracted, but not stopped), because liquids happily transmit pressure waves.

The same physics explains why you can hear under water (P-waves in water) but you cannot feel the kind of side-to-side shaking of the water itself that you can feel in a solid wall.

Poisson's ratio — sideways shrink when stretched lengthwise

When you pull a rubber band, it gets longer — obvious. But if you watch carefully, you'll notice it also gets thinner. The lateral dimensions shrink as the longitudinal dimension grows. This is universal: under tensile stress, most materials contract sideways by some fraction of their longitudinal stretch.

The ratio of the lateral contraction to the longitudinal extension is Poisson's ratio, denoted \nu (Greek nu):

Typical Poisson's ratios

Why cork is used for wine bottles: its near-zero Poisson's ratio means when you compress it into the bottle neck, it doesn't fatten sideways and get stuck. It just compresses straight through — which is exactly what you want for a seal that can be pushed in and eventually pulled out.

Why rubber behaves like a fluid under shear but like a solid under tension: rubber has \nu \approx 0.49, very nearly the incompressible limit of 0.5. That is why when you squeeze a rubber ball, the ball's volume barely changes — it just bulges out somewhere else. Rubber has a huge bulk modulus (it is nearly incompressible) but a very small shear modulus (easy to deform in shape).

The four elastic constants and their relations

For an isotropic linear elastic solid, there are four elastic constants — Young's modulus Y, bulk modulus B, shear modulus G, Poisson's ratio \nu — but only two are independent. The other two are determined. The relations are:

Give me any two, and I can compute the other two. For example, for steel with Y = 200 GPa and \nu = 0.28:

— which match the values in the tables above.

Why the limit \nu = 0.5 is "incompressible"

Take a cube of side L_0 and stretch it to L_0 + \Delta L along the x-axis. The lateral dimensions shrink by a factor (1 - \nu\,\Delta L/L_0). The new volume is approximately:

(keeping only first-order terms in the small strain).

So:

When \nu = 0.5, the factor (1 - 2\nu) = 0 — the volume does not change under uniaxial stretch. That is the signature of incompressibility. When \nu < 0.5, stretching a solid uniaxially increases its volume. When \nu > 0.5, stretching would decrease the volume, which is thermodynamically unstable (it means the material gains energy from being stretched, which leads to spontaneous contraction) — hence the upper limit.

Rubber (\nu \approx 0.49) is essentially incompressible: stretch it and it barely changes volume, just reshapes. Cork (\nu \approx 0) is the opposite extreme: stretch it lengthwise and its cross-section doesn't care; the volume increases by the full longitudinal strain.

Explore — compare materials side by side

Drag the material selector below to see how water, steel, and rubber respond under 1 GPa of isotropic pressure (a huge pressure, chosen for visualisation). The output shows the fractional volume change, the bulk compression, and where each material would fall on a log scale of bulk moduli.

Worked examples

Common confusions

• "Bulk modulus has weird sign convention." The minus sign in B = -V\,\Delta P/\Delta V exists because increasing pressure causes a decrease in volume. Without the minus sign, B would come out negative, which would be awkward because "stiffness" is intuitively a positive quantity. The sign is a bookkeeping choice.

• "Compressibility and bulk modulus are different concepts." They're the same concept inverted: \kappa = 1/B. A material is highly compressible if its bulk modulus is small, and vice versa. Some engineering contexts favour one notation; physicists favour the other.

• "Fluids have small shear modulus." No — fluids have exactly zero static shear modulus. This is not "a small number that we round to zero"; it is a true zero, and it is what makes a fluid a fluid. Even the tiniest sustained shear force causes a fluid to flow indefinitely. What fluids have is a non-zero viscosity, which resists the rate of deformation (not the deformation itself).

• "Poisson's ratio can be negative." In ordinary materials, no — thermodynamic stability demands -1 < \nu < 0.5, and for solids specifically 0 \leq \nu \leq 0.5. However, exotic structured materials called auxetics (specially engineered foams, re-entrant honeycombs) can have negative Poisson's ratios, meaning they get fatter when stretched. These are research curiosities, not everyday materials, but they are real and physically allowed by the bounds.

• "Bulk modulus equals Young's modulus." They're related but different. Y measures resistance to uniaxial stretching (you pull on one face and measure how much it stretches). B measures resistance to uniform compression (you squeeze from all sides and measure volume change). The relation B = Y/[3(1-2\nu)] shows that for \nu \to 0.5, B \to \infty even if Y is finite — an incompressible material has infinite bulk modulus.

• "If water has nonzero bulk modulus, how is it compressible at all?" Zero bulk modulus would mean infinite compression under any pressure (infinitely soft). Large-but-finite bulk modulus means small-but-nonzero compression. Water has B \approx 2.2 GPa, large in absolute terms but four orders of magnitude smaller than steel — so water is much more compressible than steel, even though both have finite positive bulk moduli.

• "Shear modulus is only for solids." Right — for the static shear modulus. But for a fluid subjected to oscillating shear (at a frequency high enough that the fluid doesn't have time to flow), you can still talk about an effective shear response — this is the basis for viscoelasticity and explains why some fluids (like silly putty) can behave solid-like under fast deformation and fluid-like under slow deformation.

• "Poisson's ratio is just a curiosity." It is not. The factor (1 - 2\nu) appears in the bulk modulus formula; \nu directly controls how lateral and longitudinal strains are coupled; and in seismology, the ratio of P-wave to S-wave speeds depends on \nu. Incompressible tissues (biological soft tissue is nearly \nu = 0.5) are modelled completely differently from compressible ones.

Where this leads next

• Stress and Strain — the foundation: internal force per area, fractional deformation, and the atomic picture that makes them linearly related.
• Hooke's Law and Young's Modulus — the uniaxial-tension version of what this article did for bulk and shear deformations.
• Elastic Potential Energy and Stress-Strain Curves — how the moduli determine the energy stored per unit volume, and the full picture beyond the linear regime.
• Pressure in Fluids — the hydrostatic pressure that gave us the "pressure at depth" number used in the brass-cube example.
• Spring Force and Hooke's Law — the single-spring analogue of the atomic-bond picture behind elastic moduli.