Tensile strength
Tensile strength \sigma_{UTS}, or S_U measures the stress required to pull something such as rope, wire, or a structural beam to the point where it breaks.  It is an intensive property of the material.

# Explanation
The tensile strength of a material is the maximum amount of tensile stress that it can be subjected to before failure.  The definition of failure can vary according to material type and design methodology. This is an important concept in engineering, especially in the fields of material science, mechanical engineering and structural engineering.
There are three typical definitions of tensile strength:

# Concept
The various definitions of tensile strength are shown in the following stress-strain graph for low-carbon steel:
Metals including steel have a linear stress-strain relationship up to the yield point, as shown in the figure.  In some steels the stress falls after the yield point.  This is due to the interaction of carbon atoms and dislocations in the stressed steel.  Cold worked and alloy steels do not show this effect.  For most metals yield point is not sharply defined. Below the yield strength all deformation is recoverable, and the material will return to its initial shape when the load is removed.  For stresses above the yield point the deformation is not recoverable, and the material will not return to its initial shape.  This unrecoverable deformation is known as plastic deformation.  For many applications plastic deformation is unacceptable, and the yield strength is used as the design limitation.
After the yield point, steel and many other ductile metals will undergo a period of strain hardening, in which the stress increases again with increasing strain up to the ultimate strength.  If the material is unloaded at this point, the stress-strain curve will be parallel to that portion of the curve between the origin and the yield point.  If it is re-loaded it will follow the unloading curve up again to the ultimate strength, which has become the new yield strength.
After a metal has been loaded to its yield strength it begins to "neck" as the cross-sectional area of the specimen decreases due to plastic flow.  When necking becomes substantial, it may cause a reversal of the engineering stress-strain curve, where decreasing stress correlates to increasing strain because of geometric effects. This is because the engineering stress and engineering strain are calculated assuming the original cross-sectional area before necking. If the graph is plotted in terms of true stress and true strain the curve will always slope upwards and never reverse, as true stress is corrected for the decrease in cross-sectional area.  Necking is not observed for materials loaded in compression. The peak stress on the engineering stress-strain curve is known as the ultimate strength. After a period of necking, the material will rupture and the stored elastic energy is released as noise and heat.  The stress on the material at the time of rupture is known as the tensile strength.
Ductile metals do not have a well defined yield point.  The yield strength is typically defined by the "0.2% offset strain".  The yield strength at 0.2% offset is determined by finding the intersection of the stress-strain curve with a line parallel to the initial slope of the curve and which intercepts the abscissa at 0.002.  A stress-strain curve typical of aluminum along with the 0.2% offset line is shown in the figure below.
Brittle materials such as concrete and carbon fiber do not have a yield point, and do not strain-harden which means that the ultimate strength and breaking strength are the same.  A most unusual stress-strain curve is shown in the figure below. Typical brittle materials do not show any plastic deformation but fail while the deformation is elastic. One of the characteristics of a brittle failure is that the two broken parts can be reassembled to produce the same shape as the original component.  A typical stress strain curve for a brittle material will be linear.  Testing of several identical specimens will result in different failure stresses.  The curve shown below would be typical of a brittle polymer tested at very slow strain rates at a temperature above its glass transition temperature.  Some engineering ceramics show a small amount of ductile behaviour at stresses just below that causing failure but the initial part of the curve is a linear.
Tensile strength is measured in units of force per unit area.  In the SI system, the units are newtons per square metre (N/m2) or pascals (Pa), with prefixes as appropriate.  The non-metric units are pounds-force per square inch (lbf/in2 or PSI). Engineers in North America usually use units of ksi which is a thousand psi.
The breaking strength of a rope is specified in units of force, such as newtons, without specifying the cross-sectional area of the rope.  This is often loosely called tensile strength, but this is not a strictly correct use of the term.
In brittle materials such as rock, concrete, cast iron, or soil, tensile strength is negligible compared to the compressive strength and it is assumed zero for many engineering applications. Glass fibers have a tensile strength stronger than steel, but bulk glass usually does not.  This is due to the Stress Intensity Factor associated with defects in the material.  As the size of the sample gets larger, the size of defects also grows.
In general, the tensile strength of a rope is always less than the tensile strength of its individual fibers.
Tensile strength can be defined for liquids as well as solids.  For example, when a tree draws water from its roots to its upper leaves by transpiration, the column of water is pulled upwards from the top by capillary action, and this force is transmitted down the column by its tensile strength.  Air pressure from below also plays a small part in a tree's ability to draw up water, but this alone would only be sufficient to push the column of water to a height of about ten metres, and trees can grow much higher than that. (See also cavitation, which can be thought of as the consequence of water being "pulled too hard".)

# Typical tensile strengths
Some typical tensile strengths of some materials:
- Note: Multiwalled carbon nanotubes have the highest tensile strength of any material yet measured, with labs producing them at a tensile strength of 63 GPa, still well below their theoretical limit of 300 GPa.  However as of 2004, no macroscopic object constructed of carbon nanotubes has had a tensile strength remotely approaching this figure, or substantially exceeding that of high-strength materials like Kevlar.
- Note: many of the values depend on manufacturing process and purity/composition.
(Source: A.M. Howatson, P.G. Lund and J.D. Todd, "Engineering Tables and Data" p41)