Modulus of elasticity (Young's modulus): what it is, formula, units, table of materials and how to measure it

The modulus of elasticity, or Young's modulus (E), is the ratio of stress to strain in the linear elastic region of a material: E = σ/ε. It measures stiffness: the higher the E, the less the material deforms under the same load. Steel has about 200 GPa; aluminum, 69 GPa, one third of that.

Modulus of elasticity and Young's modulus are two names for the same quantity. In the tensile test, it is calculated as E = (F/A₀)/(ΔL/L₀), and in practice it is the slope of the straight line fitted to the initial part of the stress-strain curve. The article goes from the definition to the table of materials and ends with what determines the quality of the result: measurement with an extensometer.

What is the modulus of elasticity

When a bar is pulled with a small force, it stretches slightly and returns to its original length when the force is removed. In this regime, strain is proportional to stress, which is Hooke's law written for the material: σ = E·ε. The proportionality constant E is the modulus of elasticity. According to ASTM E111, it is used to calculate how much a structural part deforms under uniaxial load within the range of elastic deformation.

The photo shows bars of six materials with the same geometry: steel, aluminum, copper, polymer, glass and rubber. Under the same stress they would stretch very differently, because the modulus of elasticity ranges from a few megapascals in rubber to about 200 GPa in steel.

Are Young's modulus and modulus of elasticity the same thing?

Yes, when referring to tension or compression in a single direction. The name honors Thomas Young, who described the modulus of elasticity in 1807, although the idea already appears in Euler (1727) and in experiments by Riccati (1782). The name "longitudinal modulus of elasticity" distinguishes E from two other elastic moduli, the shear modulus (G) and the bulk modulus (K), discussed further on.

Stiffness is not strength

The modulus tells how much the material deforms for a given stress, not how much it can withstand. Ordinary glass has an E of about 69 GPa, the same as aluminum, and breaks without warning. A mild steel and a maraging steel have practically the same modulus, close to 200 GPa, with yield strengths ranging from about 300 MPa to about 2,000 MPa. That is why failures by excessive elastic deformation, such as large deflection of a beam or buckling of a column, are governed by the modulus and not by strength. To make a part stiffer without changing the material, the way forward is to change its shape or increase its cross section.

Where the value of the modulus comes from

The modulus reflects the strength of the bonds between atoms, that is, the resistance they offer to being pulled away from their equilibrium spacing. That is why it depends on the class of material and changes little within the same family. Dieter, in Mechanical Metallurgy, classifies the elastic constants among the structure-insensitive properties, alongside density and melting point. Heat treatment and cold working greatly change the strength of a steel and hardly change its E at all.

High modulus and low modulus

A high E indicates a stiff material that deforms little under load: diamond (700 to 1,200 GPa), tungsten (407 GPa), alumina (380 GPa) and steel. A low E indicates a flexible material that deforms a lot under little stress: polymers lie between about 0.4 and 4 GPa, and rubber between 0.01 and 0.1 GPa at small strains. A low modulus is not a defect: an implant with a modulus closer to that of bone shares the load better with the surrounding tissue.

Modulus of elasticity formula and how to calculate it

The formula starts from the two quantities measured in the test. Engineering stress is the force divided by the initial cross-sectional area, σ = F/A₀. Strain is the elongation divided by the initial gauge length, ε = ΔL/L₀. Since the modulus is the ratio between them:

• Definition: E = σ/ε.
• Test form: E = (F/A₀)/(ΔL/L₀) = F·L₀/(A₀·ΔL).
• Slope form: E = Δσ/Δε, between two points of the linear region or from the least-squares line fitted to it.
• Elongation of a bar: ΔL = F·L₀/(A·E), the same relationship solved for the deformation, used in design.

The slope form is the one used in testing, because it does not depend on the curve passing through the origin, which is always affected by the seating of the grips. This is how ISO 6892-1:2019 defines the modulus, and it recommends reporting the result in GPa to one decimal place.

On the stress-strain curve, the modulus is the slope of the initial straight portion: the steeper the line, the stiffer the material. The graph below shows this reading on the curve of a ductile metal. The angle α marks the slope; since the axes have different scales, what matters is the ratio Δσ/Δε, not the angle measured on the paper.

The formula only holds up to the end of the straight portion, the proportional limit. The elastic limit, the highest stress without permanent deformation, is equal to or slightly above it. The area under the elastic portion is the energy the material stores and gives back, the resilience, U_r = σ_e²/(2E): with the same elastic limit, the material with the lower modulus stores more energy.

Exercise 1: modulus from two points

In a tensile test, two points of the linear region are σ₁ = 40 MPa with ε₁ = 0.00020 and σ₂ = 240 MPa with ε₂ = 0.00120. The modulus is E = (240 − 40)/(0.00120 − 0.00020) = 200/0.00100 = 200,000 MPa, or 200 GPa, a value consistent with a steel. Note the scale: the entire strain of this region is one thousandth, which anticipates the difficulty of measuring it.

Unit of the modulus of elasticity and conversions

Strain is dimensionless (mm/mm), so the modulus has the same unit as stress: the pascal in SI. Since the values are large, engineering uses GPa for metals and ceramics and MPa, which is the same as N/mm², for polymers and design calculations. Older tables and American literature still use kgf/cm² and psi. The table converts the units using aluminum at 69 GPa as an example (1 kgf = 9.80665 N; 1 psi = 6,894.76 Pa):

Unit	Equivalent to	Aluminum (69 GPa) in this unit
GPa	10⁹ Pa = 1,000 MPa	69 GPa
MPa	10⁶ Pa = 1 N/mm²	69,000 MPa
N/mm²	1 MPa	69,000 N/mm²
kN/cm²	10 MPa	6,900 kN/cm²
kgf/cm²	0.0980665 MPa (1 GPa ≈ 10,197 kgf/cm²)	≈ 7.04 × 10⁵ kgf/cm²
psi	6,894.76 Pa	≈ 10.0 × 10⁶ psi (10 Msi)

Two mistakes are common: rounding 1 kgf/cm² to 0.1 MPa, which introduces a 2% error, and confusing kN/cm² with kN/m² or with MPa, which shifts the result by powers of ten. The complete conversion for steel, with the values from design codes, is in the article on the modulus of elasticity of steel.

Modulus of elasticity of materials: table

The modulus varies by more than five orders of magnitude across engineering materials, depending on the type of bonding. In metals, metallic bonding and an ordered crystal structure give moduli from 45 GPa (magnesium) to 407 GPa (tungsten). In ceramics, ionic and covalent bonds are even stronger; the stiffness comes with brittleness, because a ceramic does not yield to relieve stress concentrations. In polymers, part of the load is carried by the weak forces between long chains, and the modulus drops to a few GPa, depending on crystallinity, chain entanglement and additives.

In composites, the modulus depends on the proportion and orientation of the matrix and the reinforcement and lies between the two: close to that of the reinforcement along the fibers, much lower in the transverse direction. Wood behaves this way, and the tabulated values apply along the grain.

The table gathers reference values at room temperature. The ranges cover the variation between alloys, grades or sources; for polymers, also the temperature and test speed.

Material	E (GPa)	Source
Natural diamond	700 to 1,200	Sonelastic (ASM data)
Tungsten	407	Callister, Table 6.1
Alumina (Al2O3) 99.9%	380	Sonelastic (ASM data)
Silicon nitride	304	Sonelastic (ASM data)
Steel (carbon and alloy)	190 to 210 (textbook value 207)	Callister, Table 6.1
Zirconia (3 mol% Y2O3)	205	Sonelastic (ASM data)
Nickel	207	Callister, Table 6.1
Copper	110 to 117	Callister; Engineering ToolBox
Titanium and alloys	105 to 120 (pure Ti 107)	Callister; Engineering ToolBox
Brass	97 to 125	Callister; Engineering ToolBox
Dental enamel	84 (range 41 to 100)	Koç, review of dental FEA
Aluminum and alloys	69 to 70	Callister, Table 6.1
Soda-lime / borosilicate glass	69 / 70	Sonelastic (ASM data)
Magnesium	45	Callister, Table 6.1
Structural concrete	25.4 to 36.6	Sonelastic (ASM data)
Dentin	18.6 (hydrated, 18 to 25)	Koç; Kinney et al., 2003
Cortical bone	13.7 (range 10 to 20)	Koç, review of FEA
Wood, along the grain	pine ≈ 9; oak ≈ 11	Engineering ToolBox
PVC	2.41 to 4.14	Callister, Table B.2
PMMA (acrylic)	2.24 to 3.24	Callister, Table B.2
Polycarbonate	2.38	Callister, Table B.2
Epoxy	2.41	Callister, Table B.2
Nylon 6,6	1.59 to 3.79	Callister, Table B.2
Polypropylene	1.14 to 1.55	Callister, Table B.2
High-density polyethylene (HDPE)	≈ 0.8 to 1.1	Engineering ToolBox; Callister
PTFE	≈ 0.4	Engineering ToolBox
Rubber (small strain)	0.01 to 0.1	Engineering ToolBox

On a linear scale, polymers and rubber would be invisible next to steel. The chart below uses a logarithmic scale, in which each division multiplies the modulus by ten, and shows the classes with almost no overlap: ceramics and metals at the top, tissues, wood and concrete in the middle, polymers at the bottom.

Teeth and bone are included in the table because they are input data for finite element models in dentistry. The scatter between studies is large, and the modulus of dentin changes depending on whether it is dry or hydrated.

Steel

Steel has a modulus between 190 and 210 GPa depending on the type, and the most cited textbook value is 207 GPa (Callister). The range is narrow because the modulus hardly changes with composition or heat treatment: an ordinary structural steel and a quenched alloy steel deform the same amount under the same elastic stress. The values from design codes, the table by type and stainless steel are covered in the article on the modulus of elasticity of steel.

Concrete

Concrete is the best-known example of a nonlinear material. Its modulus depends on compressive strength, aggregate and age, with a reference range of 25.4 to 36.6 GPa. It is measured in the compression test of cylindrical specimens with extensometers, according to ABNT NBR 8522-1:2021 (the Brazilian standard for determining the elastic modulus of concrete), and estimated in design using the expressions of NBR 6118 (the Brazilian code for the design of concrete structures). The calculation and the table by strength class are in the article on the modulus of elasticity of concrete.

Young's modulus, shear modulus (G), bulk modulus (K) and Poisson's ratio

A body under tension does not only stretch: it also gets thinner. The ratio between transverse strain and axial strain, with the sign reversed, is Poisson's ratio, ν = −ε_transverse/ε_axial, which is close to 0.33 in metals, 0.25 in ceramics and 0.40 in polymers. The other two moduli describe other loading modes:

• Shear modulus or modulus of rigidity (G): ratio between shear stress and angular distortion (shear strain), G = τ/γ. It is measured in the torsion test and is related to the shear test.
• Bulk modulus (K): ratio between a pressure applied equally in all directions and the relative change in volume.

In an isotropic material, with the same properties in all directions, only two of these constants are independent: E = 2G(1 + ν) = 3K(1 − 2ν). Hence G = E/[2(1 + ν)] and K = E/[3(1 − 2ν)], and anyone who knows E and ν can estimate G and K without another test. The table gives measured values for some metals (Callister and Rethwisch, Table 6.1):

Metal	E (GPa)	G (GPa)	ν
Aluminum	69	25	0.33
Magnesium	45	17	0.29
Brass	97	37	0.34
Titanium	107	45	0.34
Copper	110	46	0.34
Nickel	207	76	0.31
Tungsten	407	160	0.28

Exercise 2: G and K of aluminum from E and ν

With E = 69 GPa and ν = 0.33, the shear modulus is G = 69/(2 × 1.33) = 69/2.66 = 25.9 GPa, compared with 25 GPa in the table. The bulk modulus is K = 69/[3 × (1 − 0.66)] = 69/1.02 = 67.6 GPa. The small difference from the tabulated value is normal, because real alloys are not perfectly isotropic. In composites, wood and single crystals the relationship does not apply.

Tangent, secant and chord modulus

In metals, the initial portion of the curve is straight, and its slope is the modulus without ambiguity. Polymers, rubber, concrete and soils have no straight portion, and the slope changes from point to point. For these materials there are conventional definitions, and a result only makes sense if it states which one was used:

• Tangent modulus: the slope of the curve at a specified point. ASTM E111 includes it among its methods.
• Chord modulus: the slope of the straight line joining two specified points on the curve. ISO 527-1:2019 defines the tensile modulus of plastics as the chord between 0.05% and 0.25% strain, or the linear regression over that interval. ASTM C469 uses the chord for concrete, with the upper point at 40% of the compressive strength.
• Secant modulus: the slope of the straight line from the origin to a point on the curve. ASTM E111 leaves it outside its scope, because the origin is hard to define precisely.
• Flexural modulus: obtained in the flexural test, common for polymers and composites, and not necessarily equal to the tensile modulus.

The chart below applies the four slopes to the same illustrative polymer curve, σ = 60·(1 − e^−ε/0.04) MPa. The tangent at the origin gives 1.50 GPa. The ISO 527-1 chord, between 0.75 MPa at 0.05% and 3.64 MPa at 0.25%, gives 1.44 GPa. The secant up to 2% strain, where the stress is 23.6 MPa, gives 1.18 GPa, and the tangent at that same point gives 0.91 GPa. The same test specimen yields "moduli" that differ by up to 40%, all of them correct within their own definition.

ASTM E111 also points out that, in many materials, the modulus in tension differs from the modulus in compression, and that it should be measured in the loading mode in which the part will work.

Static modulus and dynamic modulus

So far we have dealt with the static modulus, obtained from the stress-strain curve under slow loading, which is the value used by design codes. The dynamic modulus comes from vibration tests: the natural resonant frequency of a bar excited by an impulse (ASTM E1876 for materials in general; ASTM C215 and ABNT NBR 8522-2:2021, the Brazilian standard for the dynamic modulus of concrete, for concrete) or the speed of sound in the material, through the relation E = ρ·c², where ρ is the density and c is the propagation speed in a thin bar.

Dynamic methods are fast, non-destructive and show less scatter than the tensile test, according to the good practice guide of the UK's NPL; Hessel and co-authors (Revista Brasileira de Ensino de Física, 2016) measured 198 GPa for 304 stainless steel from the time of flight of sound. The two moduli, however, are not interchangeable: in concrete, the dynamic modulus is usually higher than the static one, and ASTM C469 notes that the static value tends to be lower than the one obtained under rapid loading.

How to measure the modulus of elasticity in the tensile test

The modulus is measured on a universal testing machine, which applies the force and measures it with a load cell, while an extensometer attached to the test specimen measures the elongation within the gauge length L₀. For metals, the methods are those of ASTM E111 and Annex G of ISO 6892-1:2019; for plastics, those of ISO 527-1 and ASTM D638. The procedure has six steps:

1. Prepare the test specimen according to the standard and measure the area A₀.
2. Grip the test specimen aligned with the load axis; misalignment introduces bending, which distorts the reading.
3. Mount the extensometer on the parallel length, preferably an averaging extensometer that reads on both sides.
4. Load within the elastic range, at the rate set by the standard and at controlled temperature. ISO 6892-1 allows 10 °C to 35 °C, or 23 ± 5 °C when the temperature is controlled.
5. Convert and fit: divide the force by A₀ and the elongation by L₀, and fit a straight line to the linear portion. Annex G of ISO 6892-1 uses R² and the relative standard deviation of the slope as quality criteria.
6. Report the modulus in GPa to 0.1 GPa, the interval used in the fit and, for a nonlinear material, the type of modulus (tangent, chord or secant).

Above 0.25% strain, ASTM E111 requires the area and the gauge length to be corrected, because they are no longer the initial values.

Exercise 3: real test with an extensometer

A steel test specimen with d₀ = 10 mm is fitted with a clip-on extensometer with L₀ = 50 mm. In the linear portion, the force goes from 5 kN to 15 kN, and the extensometer reads 0.0160 mm and 0.0468 mm.

1. Area: A₀ = π × 10²/4 = 78.54 mm².
2. Stress change: Δσ = 10,000 N/78.54 mm² = 127.3 MPa.
3. Strain change: ΔL = 0.0468 − 0.0160 = 0.0308 mm, so Δε = 0.0308/50 = 6.16 × 10⁻⁴.
4. Modulus: E = 127.3/(6.16 × 10⁻⁴) ≈ 206,700 MPa ≈ 207 GPa.

The measured elastic elongation was only 0.03 mm over 50 mm. A reliable measurement requires resolution on the order of a micrometer, which is what the extensometer provides.

Exercise 4: the same calculation from the crosshead

Suppose, with illustrative numbers, that in the same test the crosshead moved 0.20 mm between 5 and 15 kN. Dividing this displacement by the 50 mm gauge length gives ε = 0.004 and E = 127.3/0.004 ≈ 32 GPa, about six times lower than the real value. The crosshead displacement adds up the deformation of the frame, the load cell, the grips and the couplings, the seating of the test specimen and the elongation outside the gauge length. Instron calls this share the machine compliance and points to the extensometer as the solution, because it measures directly on the test specimen. ISO 6892-1 itself warns that the slope of the elastic portion of the recorded curve does not always represent the modulus.

Extensometer class and uncertainty

Even with an extensometer, the uncertainty depends on the class of the instrument, calibrated according to ISO 9513. The NPL shows that a class 1 extensometer, with a permitted error of ±1% or 3 µm (whichever is greater), on a 25 mm gauge length and at 0.1% strain, can lead to an error of up to ±12% in the modulus. The guide recommends a class 0.2 or 0.5 averaging extensometer for modulus measurement. The same guide notes that ASTM E111 requires the strains on opposite sides of the test specimen not to differ by more than 3%, precisely to limit the effect of bending.

What changes the modulus and what does not

Because it depends on atomic bonds, the modulus responds to few variables, and knowing which ones avoids specifying a useless treatment or overlooking a service condition:

• Hardly change the modulus: heat treatment, work hardening and composition variations within the same family. They change yield strength, tensile strength and hardness, not stiffness.
• Temperature: the modulus drops as temperature rises, in practically all materials. In metals the drop is gradual up to a few hundred degrees and accelerates afterwards; for carbon steel, Eurocode 3 gives 70% of the room temperature value at 400 °C and 31% at 600 °C. Polymers are far more sensitive, which is why standards specify the test temperature.
• Direction: wood, composites and single crystals are anisotropic. In single-crystal silicon, for example, the modulus ranges from 129 GPa in the (100) direction to 187 GPa in the (111) direction.
• Porosity: pores reduce the modulus of ceramics and sintered metals.
• Moisture: dentin loses about 35% of its modulus when hydrated, according to a review in the International Journal of Oral Science (2014); moisture also affects wood and polyamides.
• Time and loading rate: polymers are viscoelastic, and the measured modulus depends on the strain rate. In concrete, age also matters.

Design exercise: steel and aluminum

The previous exercises showed how to obtain the modulus. This one uses it as design data, to compare two materials.

Exercise 5. A bar 2 m long and 20 mm in diameter (A = 314.16 mm²) is loaded in tension with 20 kN, in the elastic range.

1. Stress: σ = 20,000/314.16 = 63.66 MPa, below the yield strength of common structural alloys.
2. Aluminum (E = 69 GPa): ε = 63.66/69,000 = 9.23 × 10⁻⁴, and ΔL = 9.23 × 10⁻⁴ × 2,000 mm = 1.85 mm.
3. Steel (E = 207 GPa): ε = 63.66/207,000 = 3.08 × 10⁻⁴, and ΔL = 0.62 mm.

Since 207/69 = 3, under the same load and with the same geometry, aluminum elongates three times as much as steel. To match the stiffness, the aluminum bar would need three times the area.

Standards for determining the modulus of elasticity

The right standard depends on the material and the method. Those for metals and plastics cover the static modulus in tension; those for concrete, the modulus in compression; and the resonance standards, the dynamic modulus. The table lists the standards cited in this article:

Standard	Scope	Edition
ASTM E111	Young's modulus, tangent modulus and chord modulus	E111-17(2025)e1
ISO 6892-1	tensile testing of metals at room temperature; Annex G, determination of the modulus	2019
ABNT NBR ISO 6892-1	Brazilian adoption of ISO 6892-1	2024
ASTM E8/E8M	tensile testing of metals	E8/E8M-25
ISO 9513	calibration and classes of extensometers	2012
ISO 527-1	tensile testing of plastics; chord modulus between 0.05% and 0.25%	2019
ASTM D638	tensile testing of plastics	D638-22
ISO 178	flexural properties of plastics	not specified
ASTM D3039 and ISO 527-5	tensile testing of polymer matrix composites	not specified
ASTM C848	moduli of whiteware ceramics by resonance	not specified
ASTM E1876	dynamic moduli by impulse excitation	not specified
ASTM C469/C469M	static modulus and Poisson's ratio of concrete in compression	C469/C469M-22
ABNT NBR 8522-1	static moduli of concrete in compression (Brazilian standard)	2021
ASTM C215 and ABNT NBR 8522-2	dynamic modulus of concrete by resonant frequencies	NBR 8522-2:2021

Machine for measuring the modulus of elasticity

Measuring the modulus requires applying the force at a controlled rate with good alignment, measuring it with a load cell of suitable class and, above all, measuring the elongation with an extensometer on the test specimen, as exercises 3 and 4 showed. The Biopdi M2 universal testing machine is electromechanical and comes in single-column models, from 200 N to 10 kN, and dual-column models, from 5 kN to 1,000 kN, with class 1 load cells according to ISO 7500-1 (ABNT NBR ISO 7500-1:2023 in Brazil). The extensometers come in four families: clip-on, with gauge lengths from 25 to 100 mm, long-travel, deflectometer and COD clip gauge. The BioLab software plots the Young's modulus line and the parallel line with a 0.2% offset, reads up to four extensometer channels and switches to the encoder at a defined strain point, which allows the clip-on extensometer to be removed before fracture. The panel below shows the machine, the role of each part in measuring the modulus and the grips used.

For metals and sheet, like the test specimen in exercise 3, the 5 to 50 kN dual-column machine is the starting point; for polymers, which need lower forces and the ISO 527-1 chord, the 2 to 10 kN single-column machine. The modulus of concrete is measured in compression, on a high-capacity machine, with extensometers attached to the cylindrical test specimen. Dynamic methods, by resonance or ultrasound, are an alternative that uses dedicated equipment. The other capacities and the BioLab software are on the universal testing machine page.