Coefficient of friction: what it is, formula, table and test (ASTM D1894)

The coefficient of friction (μ) is the dimensionless number that expresses how much of the normal force is needed to make two surfaces slide over each other: μ = F_at / N. It is a property of the pair of surfaces, not of a single material, and it has two values: the static coefficient (μ_s), which measures the resistance to starting motion, and the kinetic coefficient (μ_k), which measures the resistance to keeping it going. Below we go from the physics (formula, inclined plane, table and worked examples) to film testing under ASTM D1894 and ISO 8295.

In short:

• Formula: μ = F_at / N. On a horizontal plane, N = m·g; on an inclined plane, N = m·g·cos θ.
• Quick reading: μ = 0.4 means a tangential force equal to 40% of the normal force is needed to overcome friction.
• Measurement: a sled of known mass is pulled over the material at constant speed; the force peak gives μ_s and the average during sliding gives μ_k.
• Film standards: ASTM D1894 (150 mm/min) and ISO 8295 (100 mm/min). Results from the two are not interchangeable.

The photo shows the Biopdi coefficient of friction fixture mounted on the universal testing machine. One of the test specimens lies on the horizontal table; the sled, covered with the other, slides over it; and a cable, redirected by the pulley, connects the sled to the load cell, which measures the pulling force as the crosshead moves.

What the coefficient of friction is and its formula

Friction is the force that opposes sliding between two surfaces in contact. It acts tangent to the interface, against the motion or the tendency to move, and arises from the asperities and adhesion between the faces. ISO 20808 defines the coefficient of friction as the dimensionless ratio between the friction force, F_f, and the applied normal force, F_p. ISO 8295, for plastic films, applies the same ratio to both regimes: μ_s = F_s / F_p for static and μ_d = F_d / F_p for dynamic, which is the same as kinetic.

Three consequences follow from the definition:

• It is dimensionless. It is the ratio of two forces (N/N), so it has no unit.
• It belongs to the pair, not the material. ASTM D3108 points out that it measures the interaction between two bodies: steel against steel, against PTFE or against rubber gives very different values. The condition of the surfaces (clean, lubricated, wet), the temperature and, for polymers, the speed and contact time also matter.
• It can exceed 1. Clean, identical metals with strong adhesion between the faces exceed 1: dry aluminum on aluminum reaches 1.05 to 1.35 static.

It is one of the surface mechanical properties most used in design and quality control. The table summarizes the formulas used throughout the text:

Quantity	Formula	Note
Coefficient of friction	μ = Fat / N	dimensionless
Maximum static friction	Fat,e ≤ μs·N	equality only at the onset of sliding
Kinetic friction	Fat,c = μk·N	during sliding
Normal force on a horizontal plane	N = m·g	g = 9.80665 m/s² (standard value)
Normal force on an inclined plane	N = m·g·cos θ	θ is the angle of the plane
Inclined plane at the onset of sliding	μs = tan θc	θc: angle at which sliding begins
Sled test, static	μs = Fs / (mtrenó·g)	Fs: initial force peak
Sled test, kinetic	μk = F̄k / (mtrenó·g)	F̄k: average force during uniform sliding

Static friction and kinetic friction

While the force applied to a body at rest is small, static friction grows with it and cancels it: if a box pushed with 20 N does not move, friction equals 20 N. This adjustment has a ceiling, μ_s·N. Beyond it, the body slides and the resistance becomes kinetic friction, μ_k·N. As TAPPI T 549 puts it, static friction resists the start of motion and kinetic friction resists its continuation.

In general μ_s is greater than μ_k: clean, dry steel on steel has μ_s of 0.5 to 0.8 and μ_k of 0.42. The rule is not a law: in brake materials, μ_s at room temperature can be 40 to 50% of the stated average μ_k (Engineering ToolBox). In a force record, static friction appears as the peak that precedes sliding and kinetic friction as the plateau that follows, as the curve in the test section shows.

Amontons-Coulomb laws and where they fail

Leonardo da Vinci studied friction around 1493, without publishing. Amontons published the laws in 1699, Euler distinguished static from kinetic friction in 1750, and Coulomb consolidated them in 1785 (Popova and Popov, Friction, 2015). The classical statement says that:

1. the friction force is proportional to the normal force;
2. it does not depend on the apparent contact area;
3. kinetic friction is practically independent of sliding speed.

These are approximations that work for rigid solids at moderate pressures. At extreme pressure, friction rises faster than the load and the surfaces can seize; speed independence only holds at low speeds; with a lubricant, friction varies with speed and temperature. Polymers and elastomers depart further from the rule: ISO 8295 warns that films generally do not behave ideally, which is why it fixes all test parameters, and ASTM D1894 notes that the static peak depends strongly on the loading rate and on the contact time before motion.

How to calculate the friction force and the coefficient

The calculation starts from the normal force, which presses one surface against the other: F_at = μ·N or, when measuring the friction force, μ = F_at / N. What changes from one problem to another is how to find N.

On a horizontal plane

For a block resting on a horizontal plane, with no other vertical forces, the normal force equals the weight: N = m·g. A 10 kg block has N = 10 × 9.81 = 98.1 N. With μ_k = 0.4, the friction force during sliding is F_at = 0.4 × 98.1 = 39.2 N.

This same number answers a frequent question: what does a coefficient of friction of 0.4 mean? It means a tangential force equal to 40% of the normal force is needed to overcome friction. On a horizontal plane, that is 40% of the weight: about 39 N for a 10 kg object. The same μ corresponds to a 21.8° ramp, on which the object would start sliding by itself, because tan 21.8° = 0.4.

On an inclined plane

On an inclined plane, the weight splits into two components: the one parallel to the ramp, m·g·sin θ, tends to make the block slide down; the perpendicular one, m·g·cos θ, is balanced by the normal force, so N = m·g·cos θ (taking N as the full weight is a common mistake). By gradually increasing the angle, you reach the critical angle θ_c, at which the block starts to slide and the downhill component equals the maximum static friction:

m·g·sin θ_c = μ_s·m·g·cos θ_c, therefore μ_s = tan θ_c.

The mass cancels out: the sliding angle depends only on the pair of surfaces. The diagram below brings both cases together. On the left, the block on the horizontal plane, with the weight P, the normal force N, the applied force F and the friction F_at at the interface, opposite to F. On the right, the block on the ramp, with the weight resolved into its two components and friction pointing up the ramp, against the tendency to slide down.

This is the principle of the inclined plane method, which TAPPI T 815 uses to measure the slide angle of packaging materials. The method gives only the static coefficient, because what is read is the angle at which motion begins. ASTM D4918, which applied the same principle to writing and printing paper, was withdrawn in 2010 with no replacement.

Rolling friction is not sliding friction

The rolling resistance of a wheel is a different phenomenon, with a different and much smaller coefficient: for a steel wheel it is on the order of 0.001. It should not be confused with the sliding μ discussed here.

Coefficient of friction table

Coefficient of friction tables compile values from handbooks and serve as an order of magnitude: since μ depends on surface finish, cleanliness, humidity, temperature and method, the same pair varies from one source to another. To specify a material or accept a batch, the value must come from testing the actual pair, according to the product standard.

Some patterns stand out: lubrication lowers the coefficient (steel on steel drops to 0.16 with grease), PTFE has low friction against almost everything, which explains its use in bearings and guides, and ice shows the effect of temperature. The following table comes from the Engineering ToolBox compilation (accessed in 2026); "n/a" indicates a value not available in the source.

Surface pair	Condition	Static μ	Kinetic μ
Steel × steel	clean and dry	0.5 to 0.8	0.42
Steel × steel	greased	0.16	n/a
Steel × steel	light mineral oil	0.23	n/a
Aluminum × mild steel	clean and dry	0.61	0.47
Aluminum × aluminum	clean and dry	1.05 to 1.35	0.4
Brass × steel	clean and dry	0.51	0.44
Copper × steel	clean and dry	0.53	0.36
Rubber × asphalt	dry	0.9	0.5 to 0.8
Rubber × asphalt	wet	n/a	0.25 to 0.75
Rubber × concrete	dry / wet	n/a	0.6 to 0.85 / 0.45 to 0.75
Wood × wood	clean and dry	0.25 to 0.5	n/a
Oak × oak	dry, parallel to the grain	0.62	0.48
PTFE × steel	clean and dry	0.05 to 0.2	n/a
PTFE × PTFE	clean and dry	0.04	0.04
Polyethylene × steel	clean and dry	0.2	n/a
Nylon × steel	clean and dry	0.4	n/a
Glass × glass	clean and dry	0.9 to 1.0	0.4
Ice × ice	0 °C	0.1	0.02
Ice × ice	−12 °C	0.3	0.035
Ice × steel	clean	0.03	n/a

The method also changes the number: in the ASTM G99-23 interlaboratory study (ball-on-disk, 28 laboratories), steel on steel gave μ of 0.60 ± 0.11, above the 0.42 in the table, because the geometry and contact regime are different.

Worked examples

The three examples below apply the physics formulas, with g = 9.81 m/s². The calculations for the laboratory test appear after the test is presented.

Example 1: block on a horizontal plane

A 10 kg box rests on the floor, with μ_s = 0.5 and μ_k = 0.4. (a) What is the minimum force to start motion? (b) And to keep it moving at constant speed? (c) With a 45 N push, what is the friction force?

• Normal force: N = 10 × 9.81 = 98.1 N.
• (a) F_at,e,máx = 0.5 × 98.1 = 49.05 N.
• (b) F_at,c = 0.4 × 98.1 = 39.24 N.
• (c) Since 45 N is less than 49.05 N, the box does not move, and static friction equals 45 N: it adjusts to the applied force up to the limit.

Example 2: inclined plane

A block starts to slide when the ramp reaches 22°. What is the μ_s of the pair?

• μ_s = tan 22° = 0.404.
• Conversely, a pair with μ_s = 0.4 starts to slide at θ = arctan 0.4 = 21.8°.

Example 3: what μ = 0.4 means for a pallet

A pallet with a total mass of 25 kg sits on a floor with μ_s = 0.4. What horizontal force starts it sliding?

• N = 25 × 9.81 = 245.25 N.
• F = 0.4 × 245.25 = 98.1 N, that is, 40% of the weight.

Coefficient of friction test for films: ASTM D1894 and ISO 8295

In the laboratory, the friction of plastic film and sheeting is measured on a horizontal plane. ASTM D1894 determines the static and kinetic coefficients of film and sheeting sliding over itself or over another material; ISO 8295 does the same for non-blocking films up to about 0.5 mm thick. It is a quality control test: ISO 8295 warns that it does not fully evaluate how the film behaves on a packaging machine. Both standards allow a tensile testing machine, with the crosshead motion redirected to the horizontal by a pulley, as in the fixture in the photo at the top. It is one of the mechanical tests performed on a universal testing machine with a dedicated accessory.

Test procedure, step by step

1. Condition the samples in the atmosphere specified by the standard: per ASTM D618 for D1894; for at least 16 h at 23 °C and 50% relative humidity (ISO 291) for ISO 8295.
2. Cut the test specimens without touching the face to be tested. In D1894, the sled film is about 120 mm long; in ISO 8295, the pairs are 80 × 200 mm, in the machine direction.
3. Fix one specimen to the horizontal table and wrap the sled with the other, noting which faces are in contact.
4. Connect the sled to the load cell with the cord running over the pulley, keeping the cord horizontal and aligned with the direction of motion.
5. Pull the sled at constant speed: 150 ± 30 mm/min in D1894; 100 ± 10 mm/min in ISO 8295.
6. Record the force with data acquisition fast enough to capture the initial peak.
7. Calculate μ_s from the peak and μ_k from the average of the uniform sliding portion, and repeat on the remaining specimens (5 in D1894; at least 3 pairs in ISO 8295).

D1894 also allows the reverse arrangement, with the sled stationary and the plane moving, which gives the same results, and provides for a heated or cooled plane as an option.

How to read the force vs. displacement curve

As in the stress-strain curve of a tensile test, each portion of the curve corresponds to a regime. In the sled test there are three:

• Ramp: the force rises while the cord (or the spring, in ISO 8295) stretches; the sled has not moved yet.
• Initial peak: this is the static friction force F_s. ISO 8295 states explicitly that the first force peak is caused by static friction.
• Plateau: after the peak, the force drops and oscillates around a steady value; this is the kinetic regime. D1894 averages over at least 130 mm of sliding, discarding the peak.

The graph below shows these portions in a D1894 test with a 200 g sled. In the upper trace, in orange, the 0.78 N peak and the plateau averaging 0.55 N within the 130 mm blue window. In the lower trace, the same test on a pair that shows stick-slip.

Stick-slip is the regular sawtooth pattern produced when the sled alternates between sticking (force rises) and slipping (force drops suddenly). ISO 8295 does not accept this portion for the dynamic coefficient and requires repeating the test with a rigid connection instead of the spring; the rigid measurement, in turn, is not valid for the static coefficient because of the inertia error. The peak also grows with the time the faces remain at rest in contact, which is why ISO 8295 sets a 15 s rest before pulling.

Worked example: calculation per ASTM D1894

A PE film is tested against itself with the 200 g sled at 150 mm/min. The software records an initial peak of 0.78 N and an average of 0.55 N over the uniform sliding portion (130 mm, peak discarded), which are the values in the graph above.

• Normal force: N = 0.200 kg × 9.80665 m/s² = 1.961 N.
• μ_s = 0.78 / 1.961 = 0.40.
• μ_k = 0.55 / 1.961 = 0.28.

Since the sled mass is fixed, every 0.1 N of force corresponds to about 0.05 of coefficient. How to interpret this result for packaging is covered in the section on films, just ahead.

ASTM D1894 or ISO 8295

Both standards use the same principle, a sled of about 200 g pulled over the film, but they change parameters that affect the result: speed, sled-to-load-cell connection, rest time, averaging portion and conditioning. The comparison below shows where they differ:

Parameter	ASTM D1894-24	ISO 8295:1995
Sled	200 g, 63.5 mm face	200 ± 2 g (1.96 N), 40 cm² base (63 mm) with felt
Speed	150 ± 30 mm/min	100 ± 10 mm/min (500 mm/min for special films, to be reported)
Static measurement	initial force peak	initial peak, with a 2 ± 1 N/cm spring and 15 s rest
Kinetic average	at least 130 mm, excluding the peak	portion following the peak, about 6 cm
Test specimens	5	at least 3 pairs of 80 × 200 mm
Conditioning	ASTM D618	at least 16 h at 23 °C and 50% RH (ISO 291)

That is why the results are not interchangeable: a film with μ_k = 0.28 under one standard will not necessarily give 0.28 under the other. The test report and the purchase specification must state the standard.

Coefficient of friction in packaging

In flexible packaging, the coefficient of friction determines how the film runs through filling machines and how bags stack: too much friction makes the film stick on forming collars and guides; too little makes the stack slide off the pallet. According to ISO 8295, the COF of films ranges from about 0.2 to 1, and the industry classifies it as low (up to 0.20), medium (0.20 to 0.40) and high (above 0.40), according to Mazo (Plastics Engineering, 2025). A COF of 0.4 to 0.7 is high: typical of film without slip additive or of an anti-slip face, which for stacking targets μ_s above 0.45.

On vertical form-fill-seal (VFFS) machines, the reference is a COF of 0.200 to 0.300 on the inner face and 0.300 to 0.400 on the outer face, with the outside about 0.100 higher (Viking Masek, 2020). The static coefficient matters for stacked bags; the kinetic one, for the roll running over the metal parts. Since the faces may differ (treated or untreated, inner or outer), ISO 8295 requires identifying them and testing the combination of interest: 1 against 1, 2 against 2 or 1 against 2. The PE film in the worked example, with μ_k = 0.28, has a medium COF, within the VFFS inner-face range; the value for each line, however, should be agreed with the machine manufacturer.

The COF of polyolefins is controlled by amide slip additives, such as erucamide and oleamide, which migrate to the surface after extrusion: with 200 to 2000 ppm in the film, the COF falls between 0.1 and 0.4 (Plastics Technology, 2006). Since migration depends on time and temperature, the age of the film changes the result. Lamination can undo the effect: in the same source, films with erucamide jumped from about 0.2 to 0.68 to 0.85 after being laminated to PET with a polyurethane adhesive. The table summarizes these references:

Reference	Value	Source
Typical COF range of films	0.2 to 1	ISO 8295:1995
Low / medium / high	up to 0.20 / 0.20 to 0.40 / above 0.40	Mazo, Plastics Engineering, 2025
Anti-slip for stacking	μs above 0.45	Mazo, 2025
VFFS: inner / outer face	0.200 to 0.300 / 0.300 to 0.400	Viking Masek, 2020
Amide additives	COF 0.1 to 0.4, with 200 to 2000 ppm	Plastics Technology, 2006
Non-migratory slip additive	COF 0.20 to 0.40	Plastics Technology, 2006
Erucamide after lamination to PET	from about 0.2 to 0.68 to 0.85	Plastics Technology, 2006

COF is rarely the only test run on a film. The same universal testing machine performs the tensile test of films (strength and elongation per ASTM D882) and the peel test of tapes and laminates, which measures adhesion between layers.

Coefficient of friction standards by material

Beyond films, each family of materials has its own method, and the setup changes with it. Since the value depends on the method, the standard is part of the result.

• Paper and board: ISO 15359 and TAPPI T 549 use the horizontal plane, like films; ISO 15359 measures the static coefficient before, and the static and kinetic coefficients after, a specified wear. TAPPI T 815 measures the slide angle on an inclined plane. This matters for feeding printers and converting machines.
• Ceramic floor tiles: ABNT NBR 16919:2020 (a Brazilian standard) determines the dynamic coefficient of friction of ceramic floor tiles, related to slip safety. For floors and pavements the British pendulum, described in ASTM E303, is also used; it measures slip resistance by a different principle.
• Textiles: ASTM D3108 measures the friction of yarn against a solid surface, such as guides and needles; D3412, of yarn against yarn.
• Tribology: ASTM G99 and ISO 20808 (advanced ceramics) measure friction and wear with a pin or ball on a rotating disk. ASTM G115 guides the choice of method and the minimum content of the report.

The table lists the standards cited, with the current edition:

Standard	Material	Method
ASTM D1894-24	plastic film and sheeting	sled on horizontal plane, static and kinetic
ISO 8295:1995	plastic film and sheeting	sled on horizontal plane, static and kinetic
ISO 15359:1999	paper and board	horizontal plane, static and kinetic
TAPPI T 549 om-25	writing and printing paper	horizontal plane, static and kinetic
TAPPI T 815 om-24	packaging materials	inclined plane (slide angle), static
ABNT NBR 16919:2020	ceramic floor tiles	dynamic coefficient of friction
ASTM D3108/D3108M-13(2020)	yarn against solid	kinetic friction of running yarn
ASTM D3412/D3412M-13(2020)	yarn against yarn	yarn-to-yarn friction
ASTM G99-23	materials in general	pin or ball on disk, friction and wear
ISO 20808:2016	advanced ceramics	ball on disk, dry
ASTM G115-24	solids in general	guide for measuring and reporting friction coefficients

Common errors in the coefficient of friction test

With forces ranging from tenths of a newton to a few newtons, setup and handling details weigh on the result. The most frequent errors:

• Load cell too large for forces of a few newtons (see the calculation in the example in the machine section).
• Cord not horizontal or misaligned: a force component enters the reading (ISO 8295).
• Table or sled out of level. The normal force is no longer equal to the sled weight.
• Touching the test face: grease, dust and creases change the friction.
• Mixing standards: 150 mm/min is D1894; 100 mm/min is ISO 8295.
• Slow data acquisition: flattens the peak and underestimates μ_s; ISO 8295 requires a response time of 0.5 s or less.
• Not discarding the ramp and the peak when calculating μ_k.
• Uncontrolled rest time: inflates the peak; ISO 8295 sets 15 s.
• Using the sawtooth stick-slip portion for μ_k.
• Not identifying the faces and the combination (1/1, 2/2, 1/2; film against film or film against metal).
• Ignoring the age of the film and the migration of slip additive.
• Short conditioning, below the 16 h of ISO 8295 or what ASTM D618 specifies.
• Forgetting the inertia error: in ISO 8295 it raises μ by about 0.005, or 2.5% of a μ of 0.2.

How to run the test on the Biopdi universal testing machine

At Biopdi, the coefficient of friction test is run on the universal testing machine (UTM) with the coefficient of friction fixture per ASTM D1894. The horizontal table with the sled and the pulley replaces the grips, the crosshead pulls the sled at constant speed and the load cell measures the force. The panel below shows the machine, what each part does in the friction test and the fixture used.

For films, tapes and paper, the recommended model is the MBIOI-1P-M2 portable universal testing machine, from 200 N to 1 kN, with load cells from 50 N to 1 kN. Its speed range covers the 150 mm/min of D1894 and the 100 mm/min of ISO 8295, and the BioLab software calculates the static peak and the kinetic average over the window set by the operator, with a report and curve export. The report should state the standard, the faces in contact, the sled mass, the speed, the conditioning and the age of the film.

Worked example: choosing the load cell

In the example test, the forces stay below 1 N (peak of 0.78 N). Compare a 1 kN load cell with a 50 N one.

• 0.78 N is 0.078% of 1 kN and 1.56% of 50 N.
• With the 1 kN load cell, the measured force falls far below the low end of the working range. The accuracy class from verification per NM ISO 7500-1 (the Mercosur/Brazilian adoption of ISO 7500-1) only applies within the verified range of the load cell, so the measurement loses reliability.
• Answer: use the smallest load cell that covers the maximum force with margin, here the 50 N one, with fast data acquisition to capture the static peak.

The same machine runs tensile tests on films and tapes and peel tests on adhesives, changing only the fixture. To configure the machine for your material, request a quote.