A stress-strain curve shows the mechanical response of a material during a tensile test. It provides important engineering properties including Young's modulus, yield or proof strength, tensile strength, elongation at fracture and other measures of ductility.
The shape of the curve also shows the transition from elastic behaviour to plastic deformation, strain hardening, necking and eventual fracture. Different materials can exhibit substantially different stress-strain behaviour, so the diagrams on this page should be regarded as representative rather than universal material curves.
A conventional tensile test is performed by applying an increasing axial tensile force to a prepared test specimen. The applied force and extension over a defined gauge length are measured as the test proceeds, normally until the specimen fractures.
The measured force and extension are converted into engineering stress and engineering strain:
Engineering stress
Engineering strain
where S0 is the original cross-sectional area and L0 is the original gauge length.
The resulting stress-strain curve is used to determine several important mechanical properties of the material.
Representative stress-strain curves for several engineering metals
A tensile test characterises material behaviour under a controlled tensile loading condition. It does not by itself establish fatigue strength, fracture toughness, impact resistance, creep strength or behaviour under complex multiaxial loading. Separate tests and design methods are required where those properties are important.
| Symbol | Description | Typical Units |
|---|---|---|
| A | Percentage elongation after fracture | % |
| E | Young's modulus | MPa or GPa |
| F | Applied tensile force | N |
| Fm | Maximum force during test | N |
| L0 | Original gauge length | mm |
| Lu | Final gauge length after fracture | mm |
| Lc | Parallel length of test piece | mm |
| Lt | Total length of test piece | mm |
| S0 | Original cross-sectional area | mm2 |
| S | Instantaneous cross-sectional area | mm2 |
| Su | Minimum cross-sectional area after fracture | mm2 |
| σ | Normal tensile stress | MPa |
| ε | Normal strain | dimensionless |
| ReH | Upper yield strength | MPa |
| ReL | Lower yield strength | MPa |
| Rm | Tensile strength | MPa |
| Rp | Proof strength based on specified plastic extension | MPa |
| Rr | Strength at specified permanent set | MPa |
| Rt | Proof strength based on specified total extension | MPa |
Depending on the material and test procedure, a tensile stress-strain curve can provide or assist in determining:
The exact properties reported depend on the material specification and tensile-test standard.
The following diagram represents the characteristic engineering stress-strain behaviour of a ductile steel having a pronounced yield phenomenon.
At relatively low stress the relationship between stress and strain is approximately linear. For a linear-elastic material:
and therefore:
Young's modulus E is the slope of the linear portion of the stress-strain curve. It is a measure of elastic stiffness and should not be confused with material strength.
If the specimen is unloaded while its behaviour remains elastic, it returns approximately to its original dimensions.
The proportional limit is the stress above which stress is no longer directly proportional to strain.
The elastic limit is the greatest stress that can be applied without producing permanent deformation after unloading.
These concepts are related but are not necessarily represented by exactly the same stress on a real test curve.
Some low-carbon steels exhibit a distinct yield phenomenon consisting of an upper yield strength followed by a lower yield strength. Significant plastic strain can then occur with relatively little change in engineering stress.
These values may be reported as:
Many engineering materials do not exhibit a sharply defined yield point. For these materials a specified proof strength is normally used.
A commonly specified value is the 0.2% proof strength, usually written Rp0.2. It is obtained by drawing a line parallel to the elastic portion of the stress-strain curve but offset by an engineering strain of 0.002.
The intersection of this offset line with the stress-strain curve defines the 0.2% proof strength.
After yielding, permanent plastic deformation occurs. If the specimen is unloaded from the plastic region, the elastic portion of the strain is recovered but a permanent plastic strain remains.
For many ductile metals the stress required to continue plastic deformation initially increases. This behaviour is known as strain hardening or work hardening.
The engineering tensile strength, or ultimate tensile strength Rm, is based on the maximum measured tensile force:
For a conventional ductile tensile specimen, the maximum engineering stress normally corresponds approximately to the onset of localised necking.
Before necking, reduction in cross-sectional area is distributed relatively uniformly along the gauge length. Once necking begins, deformation becomes concentrated in a local region and the minimum cross-sectional area decreases rapidly.
Because engineering stress continues to use the original area S0, the engineering stress normally falls after the maximum load has been reached. This does not mean that the local true stress at the neck necessarily falls.
Percentage elongation after fracture is commonly used as a measure of tensile ductility:
The reported elongation depends on the original gauge length and specimen geometry because deformation after necking is strongly localised. Consequently, percentage elongation values should only be compared when the test-piece geometry and gauge-length convention are compatible.
Reduction of area after fracture provides another useful measure of ductility:
The diagram below compares representative ductile and brittle tensile behaviour.
A ductile material can undergo substantial plastic deformation before fracture. A brittle material fractures with relatively little plastic deformation under the test conditions.
Grey cast iron is a familiar engineering example of comparatively brittle tensile behaviour, while many low-carbon steels exhibit substantial ductility. These descriptions should not be treated as universal classifications: composition, microstructure, temperature, strain rate, section size and processing can all influence fracture behaviour.
Some materials also behave very differently in tension and compression. Consequently a tensile stress-strain curve should not automatically be used to infer compressive or multiaxial failure behaviour.
The conventional stress and strain plotted during a standard tensile test are normally engineering stress and engineering strain.
True stress is based on the instantaneous cross-sectional area:
True strain, also called logarithmic strain, is obtained by summing incremental changes in length relative to the instantaneous length:
Before necking, where deformation remains approximately uniform, engineering and true strain are related by:
If plastic deformation is assumed approximately volume conserving and remains uniform:
Tensile specimens are prepared with a defined gauge length and parallel section so that extension can be measured under controlled conditions. Specimens may be round, flat, rectangular or tubular depending on the form of the material being tested and the requirements of the applicable standard.
The dimensions and proportions of the test piece are important because they affect the measurement of elongation after fracture. Standardised specimens therefore allow tensile properties from different tests to be compared on a consistent basis.
Historically, typical proportional round specimens have used combinations such as:
| Diameter d | Original gauge length L0 | Parallel length Lc |
|---|---|---|
| 5 mm | 25 mm | 28 mm |
| 10 mm | 50 mm | 55 mm |
| 14 mm | 70 mm | 77 mm |
These values are included as representative historical specimen proportions. For testing or specification work, specimen dimensions should be taken from the applicable edition of the relevant tensile-test standard.
The principal international standard for room-temperature tensile testing of metallic materials is:
ISO 6892-1 – Metallic materials — Tensile testing — Part 1: Method of test at room temperature.
European and British implementations are published as EN ISO 6892-1 and BS EN ISO 6892-1 respectively. RoyMech's original notes referenced BS EN ISO 6892-1:2009. Users preparing test specifications or demonstrating compliance should check the currently applicable edition and any relevant material or product standard.
A stress-strain curve contains considerably more information than a single strength value. The appropriate property depends on the engineering problem.
| Design Question | Relevant Property or Behaviour |
|---|---|
| How much will the component elastically deform? | Young's modulus, E |
| When will permanent deformation begin? | Yield strength or specified proof strength |
| What is the maximum engineering tensile stress reached in the test? | Tensile strength, Rm |
| How much plastic deformation can occur before fracture? | Elongation and reduction of area |
| How does the material behave during large plastic deformation? | True stress-strain relationship |
| Will the component survive cyclic loading? | Fatigue data – not determined by the tensile curve alone |
| How resistant is the material to crack propagation? | Fracture-toughness data – not determined by the tensile curve alone |