In simple terms, what is a volume embossing module?

It measures how difficult it is to compress a body under uniform pressure from all sides into a smaller volume. The higher the bulk modulus, the less the change in volume for a given pressure. Thus steel has a value of 160 GPa and water has a value of 2.2 GPa. This value itself gives the theoretical pressure that would be required to halve the volume of a material, assuming it behaves linearly. So if you apply 2.2 GPa to water, its volume will decrease by about 0.045 percent per MPa increase in pressure.

Can the volume embedding module become negative?

This is not a stable material. A negative value means that the material expands when compressed and becomes a source of free energy. The minus sign in the formula depends on the fact that the volume change itself is negative, but both factors cancel each other out so that the result is positive. If this calculator shows a negative value, there is a sign error in one of the two changes. This is usually caused by swapping the initial and final values.

How to calculate the bulk modulus from young's modulus?

The formula used is K = E / (3(1 - 2v)), where v is the Poisson's ratio. For a steel material with E = 200 GPa and v = 0.30, we have: K = 200 / (the value of 3 multiplied by 0.4) = 167 GPa. This relationship only applies to isotropic materials. As the Poisson's ratio approaches 0.5, the denominator approaches zero, which makes it very sensitive and allows the bulk modulus to go to infinity. This limit means incompressibility.

What is the volume compressibility of air? And why are two values shown?

For an ideal gas, the isothermal bulk modulus is equal to the pressure itself, so at sea level it would be 101 kPa. If the compression happens too quickly and there isn't time for heat to escape, then the product of the adiabatic value gamma and the pressure must be used. Since gamma for air is 1.4, this gives a result of about 142 kPa. Sound waves compress air adiabatically, so using the isothermal value in calculating the speed of sound will give incorrect results. For that reason, Newton's first estimate was actually 16 percent lower than the actual value.

Why can small amounts of air in a hydraulic system cause failure?

Since the compressibility of air is about twenty thousand times that of oil, even small amounts of air determine the overall expansion rate. The corrections in the manuals are a function of the square of absolute pressure. So an oil with 2% air at 300,000 psi will lose about one quarter of its stiffness at 500 psi while it will only loose a few percent at 2000 psi. This is why the system feels soft at the beginning of the stroke and gets stiffer as pressure builds up.

Bulk Modulus Calculator

Calculation type

What do you want to work out?

Pick the job. The first two need a pressure and a volume, the third needs two elastic constants and no volume at all, and the fourth derates a catalogue figure for the pressure, temperature and air your system actually runs at.

I have before and after readings instead

Swaps the two change fields for four straight readings: the pressure and volume you measured before loading, and the pressure and volume after. The calculator works out both changes for you and keeps the signs right.

Estimate the speed of sound in this material

Pressure waves travel faster through stiffer material. Give a density and the calculator returns the sound speed a bulk modulus implies.

Pressure and volume

Pressure change (delta P)

Initial volume (V0)

Volume decrease (delta V)

Answers

Bulk modulus (GPa)

Bulk modulus (psi)

Bulk strain (%)

Bulk modulus (GPa)

Bulk modulus (psi)
Compressibility (1/GPa)
Bulk strain (%)

Volume change: l
Volume under pressure: l

At 1.6667 GPa this is in the liquid and elastomer range: water is 2.2 GPa, hydraulic oil 1.5 to 2.0 GPa and rubber around 2 GPa.

Stiff enough that engineers call it incompressible, which is a convenient lie rather than a fact.

A volume change of 1.2 percent is the scale you see in liquids inside high-pressure systems. In a hydraulic circuit that is real lost motion: the pump has to deliver it before anything moves.

Charts

The bulk modulus indicates how difficult it is to compress an object. When a material is uniformly compressed from all sides, its volume decreases slightly. The bulk modulus is the ratio of the applied pressure to the fractional volumetric change.

This calculator supports four tasks related to this concept: measuring the modulus of elasticity by a compression test, predicting how much a known material will compress, converting between the five elastic constants for an isotropic solid, or adjusting values given in a hydraulic oil catalog based on pressure, temperature and air content in a real system.

Formula for volume compression modulus:

The definition of the bulk modulus is the ratio of stress to strain just like with Young's Modulus except that in the numerator there is pressure and in the denominator there is a percentage change in volume.

B = -\frac{\Delta P}{\Delta V / V_{0}} \qquad B = -\frac{\Delta P \cdot V_{0}}{\Delta V}

B is the bulk modulus, V0 is the volume before applying a pressure, ΔP is the applied pressure and ΔV is the resulting change in volume. The value of ΔV divided by V0 has a special name: it's called the volumetric strain, which is dimensionless.

As volumetric strain is only a ratio, the bulk modulus is given in the same units as pressure. In SI this is Pascals but because of very large numbers for solids Gigapascals are usually used while in hydraulics psi is still common.

Why is a minus sign used?

Since the volume becomes smaller as it is compressed harder, delta P is positive and delta V is negative. The ratio of both values results in a negative number, which means that the negative bulk modulus indicates that the material expands when being squeezed together.

This does not happen with stable materials. The minus sign is just a symbolic treatment to turn the ratio into a positive number and has no physical meaning.

This calculator will enter the volume decrements as positive numbers rather than signed changes, thus completely avoiding this error. If you want to use the original measurements, activate the before and after change values and enter all four measurements. The tool will then determine the sign.

Example calculation:

A hydraulic cylinder contains 0.5 liters of motor oil. When the system is pressurized to 20 MPa, the volume of motor oil reduces by 0.006 liters.

B = \frac{20 \times 10^{6}\ \text{Pa} \times 0.0005\ \text{m}^3}{0.000006\ \text{m}^3} = 1.67 \times 10^{9}\ \text{Pa} = 1.67\ \text{GPa}

This result shows that the hydraulic oil is in the normal range of 1.5 to 2.0 GPa and a reasonable check. If the result for the oil is 50 GPa then this indicates an issue with one of the components.

Conversely, you can use the same formula to answer design questions. Consider a fluid with a bulk modulus of 5 GPa, volume of 0.001155 cubic meters and pressure of 21 MPa. The volume loss is the result of multiplying 21e6 by 0.001155, divided by 5e9, which results in a value of 4.851 cubic millimeters.

What the figures mean for volume expansion

It is difficult to judge by volume change alone. If they are given as a ratio of the original volume this can be easily understood.

Bulk strain	What it means in practice
0.01% or less	Barely any compression. Metals under ordinary pressure live here.
0.1%	Small but measurable. Stiff solids under serious pressure.
1%	Noticeable. Liquids in high pressure hydraulic systems, and real lost motion in a cylinder.
5% or more	Substantial. Gases, foams, or a measurement with a unit error in it.
Negative volume change under rising pressure	Not physical. Check which reading is the initial one.

Bulk modulus of typical materials

These are the values used in the material drop down menu and also serve as a reference to check the validity of the measurement results.

Material	Bulk modulus (GPa)	Compressibility (1/GPa)	Volume lost at 100 MPa
Diamond	443	0.00226	0.023%
Tungsten carbide	319	0.00313	0.031%
Steel	160	0.00625	0.063%
Copper	140	0.00714	0.071%
Titanium	110	0.00909	0.091%
Aluminium	76	0.0132	0.13%
Glass	35 to 55	0.0182 to 0.0286	0.18% to 0.29%
Concrete	10 to 20	0.05 to 0.1	0.5% to 1.0%
Lithium	11	0.0909	0.91%
Water	2.2	0.4545	4.5%
Rubber	1.5 to 2.5	0.4 to 0.67	4% to 6.7%
Hydraulic oil	1.5 to 2.0	0.5 to 0.67	5% to 6.7%
Air at sea level, isothermal	0.000101	9901	compresses completely

Rubber is often considered to be a surprising material. Although it feels soft because it stretches under small forces, it has a bulk modulus of elasticity that is very close to water's.

Stretching and uniform compression are different deformations. When rubber is stretched, the entangled molecules can easily be rearranged. However, to squeeze it together in all directions requires bringing the molecules closer together with a resistance similar to that of a liquid.

There is a widespread statement that an important point should be noted. It is often said that the compressibility of water is about one hundred thousand times the compressibility of steel. The exact ratio results from the quotient of 160 and 2.2, which is approximately 73 times. The number one hundred thousand refers to the comparison between air and steel.

Volumetric elasticity modulus, Young's modulus, shear modulus.

An isotropic solid has only two independent elastic constants. Any two of the five can be chosen to determine exactly the remaining three. This is the role of the elasticity constant mode.

G = \frac{E}{2\,(1 + \nu)} \qquad K = \frac{E}{3\,(1 - 2\nu)}

\lambda = K - \frac{2G}{3} \qquad M = K + \frac{4G}{3}

E stands for Young's modulus and represents the resistance to stretching in a single axis. G stands for the shear modulus and represents the resistance to deformation by shearing. K stands for the bulk modulus of elasticity, λ is the first Lame constant and M is the P-wave modulus of elasticity which is actually used in propagation of pressure waves.

Known pair	What the calculator returns
E and Poisson's ratio	G = E / 2(1+v), K = E / 3(1-2v)
E and G	v = E/2G - 1, K = EG / 3(3G - E)
E and K	v = (3K - E) / 6K, G = 3KE / (9K - E)
Poisson's ratio and G	E = 2G(1+v), K = 2G(1+v) / 3(1-2v)
Poisson's ratio and K	E = 3K(1-2v), G = 3K(1-2v) / 2(1+v)
G and K	E = 9KG / (3K+G), v = (3K - 2G) / 2(3K+G)

Let's see what happens when the Poisson ratio approaches 0.5. When this term one minus two nu goes to zero, then k approaches infinity. This indicates that the material becomes incompressible.

This explains why rubber with a Poisson's ratio of about 0.4999 has both a large bulk modulus and small Young's modulus. The second law of thermodynamics places strict limits here, since E, G, and K must all be positive, the Poisson's ratio must lie between -1 and 0.5.

The speed of sound and a common mistake.

Pressure waves travel faster through stiffer materials. For liquids this relationship is simple as they cannot resist shear.

v = \sqrt{\frac{B}{\rho}}

For water with a bulk modulus of 2.2 GPa and a density of 1000 kg/m3 the result is 1483 meters per second, which agrees with experimental values.

If we use the same formula for steel with B=160 GPa and density of 7850 kg/m3, we get 4514 meters per second. The actual value is closer to 5900.

The difference comes from the shear. Since solids resist both deformation and compression, the propagation speed of pressure waves depends not only on K but also on K + 4G/3. For steel this is approximately 266 GPa (calculated as 160 plus four times 79.3, divided by three). This gives a velocity of 5818 meters per second, which matches the actual conditions. This calculator displays both values simultaneously and shows the corresponding results.

Hydraulic fluids: catalog values are only a starting point.

Hydraulic handbooks usually list the volumetric properties tangentially at standardized conditions such as 100F, 2500psi and no air bubbles but your system may not meet those conditions.

The Lee Company's Handbook of Hydraulic Engineering presents this as a product of three factors that must be applied to the reference values in order to make a correction.

B = E_{P} \cdot E_{T} \cdot E_{A} \cdot B_{ref}

The pressure makes the liquid more solid because the molecules are already close together. Heat makes the liquid softer. Bubbles make it less stiff.

For the air component there is a closed formula and the absolute pressure must be used.

E_{A} = \frac{1}{1 + \dfrac{0.147 \, B_{ref} \, a}{P_{1} P_{2}}}

Consider this as a warning about underpressure. The amount of correction depends on the square of absolute pressure. So an oil with 2% air content at 300,000 psi will lose one quarter of its stiffness at 500 psi while it only loses 2% of its stiffness if the pressure is 2,000 psi.

The worst feeling when operating the brake pedal is just then because the air bubbles are not completely compressed yet.

Fluid	Reference bulk modulus (psi)	Thermal expansion (per F)
Gasoline	150000	0.00072
JP-4	200000	0.00057
MIL-H-5606	260000	0.00046
MIL-H-83282	300000	0.00046
MIL-H-6083	260000	0.00044
Skydrol 500B-4	340000	0.00047
Silicone, 100 cSt	150000	0.00054
Water	310000	0.00021

pressure due to enclosed volume.

This calculation prevents damage to the installation. A trapped liquid between two closed valves cannot expand even if it is heated. Instead of expanding outward, it creates a thrust force.

\Delta P = B \cdot \gamma \cdot \Delta T

The volumetric modulus of elasticity per MIL-H-83282 is 300000 psi and the coefficient of cubic expansion is 0.00046 per degree Fahrenheit. If the entrapped liquid is heated by 50 degrees Fahrenheit, a pressure of 6900 psi will be generated.

Notice what is missing in this formula. Since the volume itself does not explicitly appear in the formula, the pressure will be the same whether it's a closed pipeline with a diameter of 6 inches or an entire storage tank. By multiplying the fraction that results from free expansion (V) and changes in gamma and delta T, the pressure can be maintained at its original level.

What actually determines the volume imprint module:

Hydraulic rigidity and responsiveness.

For every cubic centimeter of fluid lost there is a volume delivered by the pump that has not been received by the hydraulic cylinder. In a brake system with 10 MPa pressure and 50 ml of fluid, at an elastomer bulk modulus of 2 GPa, the loss would be about 0.25 ml. This is negligible. If air gets in, the effective bulk modulus drops dramatically, and the lost volume corresponds to pedal travel.

Deep Sea:

At a depth of 4000 m the pressure exceeds 40 MPa. In a titanium housing with a bulk modulus of 110 GPa, the volume is reduced by 0.036 percent, which no seal can notice. In a plastic housing with a bulk modulus of 3 GPa, the volume is reduced by 1.3 percent, and seals are clearly affected.

Seismology and ultrasound.

The P-wave velocity in rocks depends on the bulk modulus and density. So by measuring the travel time of waves, it is possible to probe rock at depths of several thousand meters. The same physical principle can be applied in reverse in a laboratory. By measuring the speed with which ultrasound waves propagate through a small sample, it is possible to determine the elastic constants of the sample without destroying it.

Common mistakes:

A common mistake is to have the wrong sign for the volume change. When a material is compressed, the recorded change will be negative. Entering it as a positive number will give a negative bulk modulus. This calculator does not directly output this result but points out this problem.

Another mistake is to confuse MPa and cubic meters and forget the conversion. Use these two units as the tool will automatically convert them so you don't have to do any manual conversions.

Treating gases as solids: While the bulk modulus of liquids and solids is generally constant, this is not true for gases. The bulk modulus of a gas under isothermal conditions is its own pressure, while under adiabatic conditions it is the product of the adiabatic exponent gamma and the pressure. Thus, it changes very rapidly when compressed.

Tangential elasticity as a secant elasticity: Over a large range of pressure changes the two values are not identical and handbooks usually give the tangential values for particular pressures.

What this calculator cannot do:

This tool uses a constant bulk modulus over the entire pressure range. This is a good approximation for solids and liquids at moderate pressures but it is not suitable for gases or pressure ranges of several hundred MPa since the bulk modulus itself increases.

The pressure and temperature corrections in the hydraulic mode are based on a linear reconstruction of known curves using reference conditions as a base point. The coefficients used therein are 1.00. Within the range of about 0 to 3,000 psi or 40 to 200 F, the error required to reproduce the values given in the manual is less than one half percent. Outside this range, the values are limited. If you use these data for a design, please consult published curves.

The conversion of the modulus of elasticity assumes that the material is an isotropic, linear elastic body. Composite materials, sheets, wood and single crystals do not meet these conditions.

Frequently asked questions

In simple terms, what is a volume embossing module?: It measures how difficult it is to compress a body under uniform pressure from all sides into a smaller volume. The higher the bulk modulus, the less the change in volume for a given pressure. Thus steel has a value of 160 GPa and water has a value of 2.2 GPa. This value itself gives the theoretical pressure that would be required to halve the volume of a material, assuming it behaves linearly. So if you apply 2.2 GPa to water, its volume will decrease by about 0.045 percent per MPa increase in pressure.
Can the volume embedding module become negative?: This is not a stable material. A negative value means that the material expands when compressed and becomes a source of free energy. The minus sign in the formula depends on the fact that the volume change itself is negative, but both factors cancel each other out so that the result is positive. If this calculator shows a negative value, there is a sign error in one of the two changes. This is usually caused by swapping the initial and final values.
How to calculate the bulk modulus from young's modulus?: The formula used is K = E / (3(1 - 2v)), where v is the Poisson's ratio. For a steel material with E = 200 GPa and v = 0.30, we have: K = 200 / (the value of 3 multiplied by 0.4) = 167 GPa. This relationship only applies to isotropic materials. As the Poisson's ratio approaches 0.5, the denominator approaches zero, which makes it very sensitive and allows the bulk modulus to go to infinity. This limit means incompressibility.
What is the volume compressibility of air? And why are two values shown?: For an ideal gas, the isothermal bulk modulus is equal to the pressure itself, so at sea level it would be 101 kPa. If the compression happens too quickly and there isn't time for heat to escape, then the product of the adiabatic value gamma and the pressure must be used. Since gamma for air is 1.4, this gives a result of about 142 kPa. Sound waves compress air adiabatically, so using the isothermal value in calculating the speed of sound will give incorrect results. For that reason, Newton's first estimate was actually 16 percent lower than the actual value.
Why can small amounts of air in a hydraulic system cause failure?: Since the compressibility of air is about twenty thousand times that of oil, even small amounts of air determine the overall expansion rate. The corrections in the manuals are a function of the square of absolute pressure. So an oil with 2% air at 300,000 psi will lose about one quarter of its stiffness at 500 psi while it will only loose a few percent at 2000 psi. This is why the system feels soft at the beginning of the stroke and gets stiffer as pressure builds up.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Bulk modulus
Definition, the isothermal and adiabatic distinction, and typical values for solids, liquids and gases.
Wikipedia: Compressibility
The reciprocal of bulk modulus and how it is defined thermodynamically.
Wikipedia: Elastic modulus
The conversion table between the five isotropic elastic constants used by the elastic constants mode.
Wikipedia: Poisson's ratio
Why Poisson's ratio has to lie between -1 and 0.5, and what happens to the bulk modulus at the upper limit.
Wikipedia: Speed of sound
The fluid form using bulk modulus and the solid form using the P-wave modulus.
The Lee Company: Technical Hydraulic Handbook, Bulk Modulus
Reference bulk modulus and thermal expansion data for hydraulic fluids, the pressure, temperature and entrained-air corrections, and the trapped-volume pressure rise.