# Pressure (physics)

Physical size
Surname pressure
Formula symbol ${\ displaystyle p}$
Size and
unit system
unit dimension
SI Pa  = N / m 2 = kg · m -1 · s -2 M · L −1 · T −2
cgs Ba = dyn / cm 2 = g · cm -1 · s -2 M · L −1 · T −2
Here the force F is perpendicular to the surface A.

In physics , the pressure is the result of a force F acting perpendicularly on a surface A , see picture. The pressure on a flat surface can be mathematically expressed as a quotient

${\ displaystyle p = {\ frac {F} {A}}}$

write. The pressure on a body is positive when the force is directed towards it, a negative pressure corresponds to a pull. According to Pascal's principle (by Blaise Pascal ), pressure spreads in static liquids and gases ( fluids ) on all sides and acts in the volume in all directions but always perpendicular to walls. The usual symbol p is based on the Latin or English word for pressure ( Latin pressio , English pressure ).

Pressure is an intensive , scalar physical quantity that plays an important role in fluid mechanics and thermodynamics . The ratio of force to area is more precisely the mechanical pressure , which is a normal stress that acts equally in all spatial directions (a special case of mechanical stress ). The thermodynamic pressure is a quantity of state that is defined for a gas with an equation of state and can deviate from the mechanical pressure in imbalance .

The Pauli principle of quantum physics leads to a degenerate pressure in fermions , which for example protects a white dwarf star from further collapse. According to the general theory of relativity , pressure also contributes to the effect of gravity .

The spatial dependence of the pressure is often also referred to as a pressure field , since pressure is a scalar field .

## history

Hydrostatic paradox according to Stevin: The little water in the ABCD area of ​​the vessel presses just as strongly against the wall CD as the much water in CDEF.

In ancient times , Archimedes , Ktesibios , Philon of Byzantium , Heron of Alexandria and Sextus Iulius Frontinus were already familiar with the effects of the pressure of water and air. In the Middle Ages is Alhazen to mention that a correct idea of the air pressure went before the Renaissance , the Dutch merchant Simon Stevin (1548-1620), the first principles of hydrostatic and hydrostatic paradox formulated, see picture.

Fundamental research began in the 17th century at the court of Grand Duke Cosimo II. De 'Medici . There the well master was astonished to find that he could not lift water higher than 32 feet (10.26 m) using a suction pump. A vacuum formed above the water column - as in the pipe in the picture in the area AC - which prevents further ascent. This phenomenon was communicated to the teacher and court mathematician Cosimos II, Galileo Galilei , who then dealt with it in his Discorsi (pp. 16-17). Vincenzo Viviani , a colleague of Galileo, was the first to conclude in 1643 that it is the air pressure that pushes the water up in the suction pipe (in the picture at B). Evangelista Torricelli , Galileo's assistant and successor, made experiments with a tube filled with mercury as shown in the picture and explained from the different densities of water and mercury why the former rises 13½ times higher than the latter with 760 mm . This is how Torricelli invented the mercury barometer .

The news of the “Italian experiment” came to Blaise Pascal in 1644 through Marin Mersenne and the physicist Pierre Petit . He repeated Torricelli's experiments and concluded that the pressure in a liquid or a gas is proportional to the depth. Accordingly, if the mercury column is carried by the air pressure, its height on a mountain must be less than in the valley. Petit and Pascal's brother-in-law Florin Périer carried out the relevant measurements on September 19, 1648 in Clermont-Ferrand and on the summit of the 1465 m high Puy de Dôme and received the expected results. Pascal published his results as early as October as a report on the great experiment on the balance of liquids (Pascal: Récit de la grande expérience de l'équilibre des liqueurs ). In the treatise on the equilibrium of liquids and the weight of the mass of air of 1653, Pascal formulated among other things

• the Pascal's principle , according to which the pressure in resting fluids spreads all sides and volume in all directions but always acts perpendicular to walls,
• the Pascal's law for the hydrostatic pressure, which increases linearly with depth, see below, and
• the functional principle of a new machine to multiply forces (Pascal: machine nouvelle pour multiplier les forces ), i.e. the hydraulic press .
Engraving from Guericke's hemisphere experiment

Otto von Guericke performed his famous experiment with the Magdeburg hemispheres in front of the Reichstag in Regensburg in 1654 , see picture.

New insights came from, among others

## definition

Pressure is the result of a force acting on a surface. The size of the pressure on the reference surface A results exclusively from the force component perpendicular to the surface . Mathematically for a flat surface A : ${\ displaystyle F_ {n}}$

${\ displaystyle p = {\ frac {F_ {n}} {A}}}$

For curved surfaces or location-dependent pressure, a sufficiently small surface element d A must be considered:

${\ displaystyle p = \ lim _ {\ mathrm {d} A \ to 0} {\ frac {\ mathrm {d} F_ {n}} {\ mathrm {d} A}}}$

With:

 ${\ displaystyle p}$ - Pressure ${\ displaystyle \ mathrm {d} F_ {n}}$ - normal force and ${\ displaystyle \ mathrm {d} A}$ - Area on which the force acts.

In vector terms , the pressure is the constant of proportionality between the vector surface element and the normal force that acts on this element: ${\ displaystyle \ mathrm {d} {\ vec {A}}}$${\ displaystyle \ mathrm {d} {\ vec {F}} _ {n},}$

${\ displaystyle \ mathrm {d} {\ vec {F}} _ {n} = - p \, \ mathrm {d} {\ vec {A}} = - p \, {\ hat {n}} \, \ mathrm {d} A}$.

The normal unit vector on the surface is parallel to the force and points outwards away from the body. The minus sign creates positive pressure when the force is directed towards the body . A compressive force acts anti-parallel to this outwardly directed normal vector, i.e. towards the body (a force acting outward in the direction of the normal vector is a tensile force .) ${\ displaystyle {\ hat {n}}}$

It is sometimes said that pressure works in a certain direction. In terms of physics, it would be more correct to speak of the compressive force that can push in one direction . In physics, however, pressure, as a scalar quantity, is directionless or “acting in all directions”.

For incompressible and for compressible fluids different components contribute to the total pressure. In the case of free-flowing fluids, incompressibility can be assumed to a good approximation at speeds well below the wave propagation speed, especially in liquids. Static gases, on the other hand, are compressible.

## Definition in technical mechanics and continuum mechanics

In the strength theory of technical mechanics and continuum mechanics , pressure is a mechanical tension that acts in all spatial directions . The mechanical normal stress is the force component perpendicular to the surface with normal on which it acts: ${\ displaystyle \ sigma _ {n}}$ ${\ displaystyle F_ {n}}$${\ displaystyle A}$${\ displaystyle n}$

${\ displaystyle \ sigma _ {n} = \ lim _ {\ Delta A \ to 0} {\ frac {\ Delta F_ {n}} {\ Delta A}}}$

The pressure is defined as a normal stress acting in all spatial directions.

In continuum mechanics, the rule of signs applies that tensile forces cause positive tension and stresses caused by compressive forces have a negative sign. At the same time, the convention applies that positive pressure has a compressive effect: thus positive pressure creates negative tension.

The stress state in a body is summarized into a mathematical object by the stress tensor σ . The mechanical pressure is defined as the negative third of the trace of the stress tensor:

${\ displaystyle p _ {\ rm {mech}}: = - {\ frac {1} {3}} (\ sigma _ {x} + \ sigma _ {y} + \ sigma _ {z}) = - {\ frac {1} {3}} \ operatorname {Sp} {\ boldsymbol {\ sigma}}}$.

Here are the normal stresses in -, - and - direction of a Cartesian coordinate system . Because the stress tensor is objective and the trace is a main invariant , this negative mean value of the normal stresses - the mechanical pressure - is invariant in the reference system . In fluids (liquids and gases) the absolute pressure , see below, is always positive. Negative absolute pressure can also occur in solids. If the stress tensor according to ${\ displaystyle \ sigma _ {x, y, z}}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$

${\ displaystyle {\ boldsymbol {\ sigma}} = - p \, \ mathbf {1}}$

contains only compressive stresses, it is called the pressure tensor . Here 1 is the unit tensor .

In a body bounded by a surface, let the normal unit vector on the surface be directed outwards. The stress vector on the surface then results from . In the special case of pressure it is calculated as above: ${\ displaystyle {\ hat {n}}}$${\ displaystyle {\ vec {T}} ^ {({\ hat {n}})} = {\ boldsymbol {\ sigma}} \ cdot {\ hat {n}}}$

${\ displaystyle \ mathrm {d} {\ vec {F}} _ {n} = {\ vec {T}} ^ {({\ hat {n}})} \ mathrm {d} A = -p \, \ mathbf {1} \ cdot {\ hat {n}} \, \ mathrm {d} A = -p \, {\ hat {n}} \, \ mathrm {d} A}$

I.e. the direction of the force is always normal on a surface and is directed towards the body with positive pressure.

Free-flowing fluids are, to a good approximation, incompressible at speeds well below the wave propagation speed . Then the pressure is a “ constrained tension ”, which maintains the incompressibility as a reaction of the fluid to compression attempts. Mathematically, the pressure here is a Lagrange multiplier for the secondary condition “incompressibility”. An example for calculating the pressure in solid mechanics is given in the article on hyperelasticity .

Material models define the stress tensor as a function of the deformation of the body, whereby the term deformation is taken so broadly that it also includes the flow of a liquid or the flow of a gas. The material models used in fluid mechanics for the ideal gas and Newtonian fluid have the form

${\ displaystyle {\ boldsymbol {\ sigma}} = - p _ {\ rm {thermo}} \, \ mathbf {1} + \ mathbf {S}}$

where the proportion S in the Newtonian fluid is caused by viscosity and is omitted in the ideal gas. The pressure p thermo is the thermodynamic pressure that is determined for a gas from an equation of state and is generally a function of density and temperature . The mechanical pressure is then:

${\ displaystyle p _ {\ rm {mech}} = - {\ frac {1} {3}} \ operatorname {Sp} {\ boldsymbol {\ sigma}} = p _ {\ rm {thermo}} - {\ frac { 1} {3}} \ operatorname {Sp} \ mathbf {S}}$

If the fluid has a volume viscosity , the second summand in the imbalance can be different from zero, so that the mechanical and thermodynamic pressure in the fluid then differ from one another. The difference would be a consequence of an increased resistance to compression due to the bulk viscosity and would approach zero as equilibrium is approached.

The stress tensor is defined at every point of the fluid and thus represents a field . A pressure field that also fills the entire body can be derived from this field . The divergence of the stress tensor represents the flow of force in the fluid and therefore a pressure increase slows down fluid elements, see Navier-Stokes equations and Euler equations of fluid mechanics . ${\ displaystyle \ operatorname {div} (-p \ mathbf {1}) = - \ operatorname {grad} p}$

## Pressure of liquids

Static and dynamic pressure component in a lossless flow
Water shoots out of a defective underground hydrant under high pressure

The pressure in flowing liquids consists of a static and a dynamic component. While both parts depend on the density , they differ in that the hydrostatic pressure (at constant density) increases linearly with the height of the fluid column. He is also of the acceleration due to gravity , so the gravity dependent. The dynamic component, on the other hand, grows quadratically with the flow velocity of the fluid and only manifests itself when the flowing fluid is slowed down ( dynamic pressure ).

The sum of dynamic and static pressure, the total pressure , is constant in a viscosity-free , horizontal flow, see picture. The constancy of the total pressure is a consequence of the energy conservation of the fluid elements along a stream filament in the flow from which Daniel Bernoulli derived the Bernoulli energy equation named after him .

In a real system, the pressure losses in the flow course must also be taken into account, for example due to the friction of the fluid on the wall of the pipeline.

### Hydrostatic pressure

A fluid resting in a gravitational field exerts an all-round hydrostatic pressure on every body immersed in it according to Pascal's principle , which increases with depth according to Pascal's law. Examples of hydrostatic pressure are water pressure and air pressure .

In the stationary liquid there are only normal stresses that act equally in all directions, precisely the hydrostatic pressure. In the shear -free hydrostatic stress state , Mohr's circle of stress degenerates to a point.

The hydrostatic pressure at the bottom of a standing column of liquid height and density under the effect of gravitational acceleration is determined by the Pascal's law to ${\ displaystyle p (h)}$${\ displaystyle h}$${\ displaystyle \ rho}$ ${\ displaystyle g}$

${\ displaystyle p (h): = \ rho gh + p_ {s}}$

Here is a pressure component that is applied by the environment at the upper end of the liquid column. In a flowing fluid, the pressure can vary from place to place. ${\ displaystyle p_ {s}}$${\ displaystyle p_ {s}}$

### Hydrodynamic pressure

The hydrodynamic pressure corresponds to the dynamic pressure . It results from the kinetic energy of the flowing fluid elements in a flow. The hydrodynamic pressure increases quadratically with the flow velocity of the fluid elements: ${\ displaystyle p_ {d}}$ ${\ displaystyle v}$

${\ displaystyle p _ {\ mathrm {d}}: = {\ frac {1} {2}} \ rho v ^ {2}}$

Therein is the density of the flowing fluid. ${\ displaystyle \ rho}$

The hydrodynamic pressure cannot be measured directly, but with loss-free, horizontal and steady flow it can be determined by measuring the difference between total pressure and static pressure (see Prandtl probe ). The speed of the fluid can then be determined from the hydrodynamic pressure.

### Total pressure

At a constant temperature in the fluid, the total pressure is the sum of the pressure components mentioned: ${\ displaystyle p_ {t}}$

${\ displaystyle p_ {t} = p (h) + p_ {d} = p_ {s} + \ rho gh + {\ frac {\ rho} {2}} v ^ {2}}$

According to Bernoulli's pressure equation , the total pressure along a flow filament in a viscosity-free fluid is constant at constant temperature. When changing from a larger to a smaller cross-section, as in the picture above, the flow velocity (and thus also the hydrodynamic pressure) must increase according to the law of continuity . This can only happen if the static pressure is lower in the smaller cross-sections and vice versa. The static pressure component is the pressure felt by a fluid element floating with the flow. ${\ displaystyle p_ {s} + \ rho gh}$

Pressure losses due to a loss of momentum at the flow edges can be taken into account with pressure loss coefficients in Bernoulli's extended pressure equation for viscous liquids.

In an ideal gas, there would also be a proportion resulting from the thermal expansion:

${\ displaystyle p_ {t} = p_ {s} + \ rho gh + {\ frac {\ rho} {2}} v ^ {2} + \ rho c_ {v} T}$.

Here is the absolute temperature and the specific heat capacity at constant volume of the gas. This is the extended Bernoulli pressure equation for ideal gases. ${\ displaystyle T}$${\ displaystyle c_ {v}}$

## Pressure of gases

Gas particles that are enclosed in a vessel exert pressure on the vessel walls.

The gas pressure is the sum of all forces acting on a surface by a gas or gas mixture . If a gas particle hits a wall, both exchange an impulse . The higher the internal energy of the gas, the faster the particles are and the greater the pressure. The momentum transfer depends on the kinetic energy of the gas particle. The momentum transfer also depends on the direction in which the particle hits the wall. For many particles these momentum transfers add up to a total force. This depends on the number of particles that hit the wall per unit of time and their mean momentum. In a gas mixture, the gas pressure arises from the partial pressures of the components of the mixture. Evaporating liquids generate a vapor pressure that can build up to saturation vapor pressure . The air pressure is an example of a gas pressure.

The kinetic gas theory delivers the equation of state from the mechanical and statistical considerations mentioned

${\ displaystyle p: = - {\ frac {\ partial U (S, V, n)} {\ partial V}} \,}$

with which the pressure is defined as an intensive quantity in thermodynamics (see also fundamental equation ). In a second step it is shown that this pressure actually equals the quotient of force and area.

In the special case of an ideal gas , the thermal equation of state applies :

${\ displaystyle p = {\ frac {n \, R \, T} {V}}}$

Based on the kinetic theory of gases it follows

${\ displaystyle p = {\ frac {n \, M \, {\ overline {v ^ {2}}}} {3V}} \,}$

The individual symbols stand for the following quantities :

${\ displaystyle V}$- volume
${\ displaystyle n}$- Amount of substance
${\ displaystyle R}$- Universal gas constant
${\ displaystyle T}$- temperature
${\ displaystyle M}$- molar mass
${\ displaystyle {\ overline {v ^ {2}}}}$ - the mean square of the speed

The averaged momentum transfer is contained in the product of the gas constant and temperature of the equation of state. The gas pressure provides the material model for the ideal gas via the equation of state :

${\ displaystyle {\ boldsymbol {\ sigma}} = - p (\ rho, T) \ mathbf {1} = - {\ frac {nRT} {V}} \ mathbf {1} = -R_ {s} T \ rho \ mathbf {1}.}$

There is - the specific gas constant - a material parameter of the gas. The flow of an ideal gas obeys Euler's equations of fluid mechanics , in which Bernoulli's energy equation applies. ${\ displaystyle R_ {s}}$

## Definition in statistical physics and thermodynamics

In statistical physics , the pressure is generally given by the following expected value :

${\ displaystyle p: = - \ left \ langle {\ frac {\ partial {\ hat {H}}} {\ partial V}} \ right \ rangle}$

this is the Hamiltonian of the system, the volume, an ensemble average over the respective statistical ensemble . ${\ displaystyle {\ hat {H}}}$${\ displaystyle V}$${\ displaystyle \ langle \ ldots \ rangle}$

This definition leads in the microcanonical ensemble to

${\ displaystyle p = - {\ frac {\ partial U} {\ partial V}}}$

( is the inner energy ), in the canonical ensemble to ${\ displaystyle U}$

${\ displaystyle p = - {\ frac {\ partial F} {\ partial V}}}$

( The free energy ) and in the grand canonical ensemble to ${\ displaystyle F}$

${\ displaystyle p = - {\ frac {\ partial \ Omega} {\ partial V}}}$

( is the grand canonical potential). ${\ displaystyle \ Omega}$

According to Stokes' hypothesis from 1845, mechanical pressure is equal to thermodynamic pressure. However, this only applies with restrictions, see above.

## Absolute / relative pressure

The absolute pressure ( English absolute pressure ) refers to the perfect vacuum . In this absolutely particle-free space, the zero point of the absolute pressure is defined. An example of a value that is often given as “absolute” is the air pressure . ${\ displaystyle p_ {abs}}$

A relative pressure relationship between two volumes is called relative pressure. The ambient pressure is often used as a reference variable, but other reference variables can also be used depending on the context. Examples of a pressure that is often given as “relative” are the inflation pressure of a tire and the blood pressure .

To clarify: If you inflate a tire with a relative pressure of 2 bar at an air pressure of 1 bar, the absolute pressure in the tire is 3 bar. Similarly, the air pressure must be added to the blood pressure in order to obtain the absolute blood pressure.

## units

In honor of Blaise Pascal, the SI unit of pressure is named Pascal (with the unit symbol Pa), which corresponds to a force of one Newton (i.e. the weight of about 100 grams ) vertically distributed over an area of ​​one square meter :

${\ displaystyle \ mathrm {1 \, Pa = 1 \, {\ frac {N} {m ^ {2}}} = 1 \, {\ frac {kg} {m \ cdot s ^ {2}}}} }$

In engineering, the unit N / mm² or MPa is used for pressure (as well as for mechanical tension ) :

${\ displaystyle 1 \, {\ frac {\ mathrm {N}} {\ mathrm {mm} ^ {2}}} = 1 \, \ mathrm {MPa}}$

### Conversion between the most common units

Other common units were or are:

The conversion between these units is specified in the table to the nearest five significant digits .

Pa bar at atm Torr psi
1 Pa = 1 1.0000 · 10 −5 1.0197 · 10 −5 9.8692 · 10 −6 7.5006 · 10 −3 1.4504 · 10 −4
1 bar = 1.0000 · 10 5 1 1.0197 9.8692 · 10 −1 7.5006 · 10 2 1.4504 · 10 1
1 at = 9.8067 · 10 4 9.8067 · 10 −1 1 9.6784 · 10 −1 7.3556 · 10 2 1.4223 · 10 1
1 atm = 1.0133 · 10 5 1.0133 1.0332 1 7.6000 · 10 2 1.4696 · 10 1
1 torr = 1.3332 · 10 2 1.3332 · 10 −3 1.3595 · 10 −3 1.3158 · 10 −3 1 1.9337 · 10 −2
1 psi = 6.8948 · 10 3 6.8948 · 10 −2 7.0307 · 10 −2 6.8046 · 10 −2 5.1715 · 10 1 1

### More units

The following non- SI-compliant pressure units can be found in literature:

• 1 meter water column (mWS) = 0.1 at = 9.80665 kPa
• inch of mercury ( English inch of mercury , inHg) = 25.4 Torr = 3386.389 Pa at 0 ° C
• 1 micron (1 µm) mercury column = 1 µm Hg = 1 mTorr (is occasionally used in vacuum technology)
• 1 poundal per square foot (pdl / ft²) = 1.4882 Pa
• 1 inch of water 'inch of water column' (inH 2 O) = 249.089 Pa
• 1 foot of water 'foot of water' (ftH 2 O) = 2989.07 Pa

## Pressure gauges and methods

A pressure measuring device is also called a manometer . In most applications, the relative pressure - i.e. in relation to the atmospheric air pressure - is measured. Absolute pressure measuring instruments use a vacuum as reference pressure (e.g. barometer ). Like the others, differential pressure gauges measure a pressure difference, but between any two systems. Pressure gauges are based on different measuring principles :

• Simple Bourdon tube pressure gauges or Bourdon tube pressure gauges are used to measure the tire pressure on the car or the house water and house gas pressure. These are based on the principle of a rolled-up hose that unrolls under pressure.
• Static pressure gauges usually measure the pressure difference based on the deflection of a mechanical separation by comparing the pressure with a reference pressure such as a vacuum. For example, barometers and ring scales measure by translating the deflection directly into a display, or differential pressure sensors by measuring the force of the deflection.
• Indirect pressure measurement is based on the effects of particle number density
• Measuring devices for pressures in flowing media (fluids) use the conclusions from the Bernoulli equation , such as the pitot tube (pitot tube) or the Venturi nozzle
• Blood pressure monitors measure indirectly by absorbing acoustic events when the previously compressed veins relax
• Pressure transmitters are pressure measuring devices that can be used in industrial environments. For this purpose, the pressure measurement signal obtained is converted into a defined signal.
• Pressure sensitive paint (PSP) make local pressure distributions at interfaces visible.
• A ring balance measures very small pressures using a mechanical process between any two systems.

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