Stress-strain curve

# Stress-strain curve

Overview

Discussion

Encyclopedia

During tensile testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the sample, and strain
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...

, derived from measuring the deformation of the sample, i.e. elongation, compression, or distortion. The slope of stress-strain curve at any point is called the tangent modulus
Tangent modulus
In solid mechanics, the tangent modulus is the slope of the compression stress-strain curve at any specified stress or strain. Below the proportional limit the tangent modulus is equivalent to Young's modulus. Above the proportional limit the tangent modulus varies with strain and is most...

; the slope of the elastic (linear) portion of the curve is a property used to characterize materials and is known as the Young's modulus
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

. The area under the elastic portion of the curve is known as the modulus of resilience.

The nature of the curve varies from material to material. The following diagrams illustrate the stress–strain behaviour of typical materials in terms of the engineering stress and engineering strain where the stress and strain are calculated based on the original dimensions of the sample and not the instantaneous values.

## Ductile materials

Low carbon steel generally exhibits a very linear stress–strain relationship up to a well defined yield point (Fig.2). The linear portion of the curve is the elastic region and the slope is the modulus of elasticity or Young's Modulus
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

. After the yield point, the curve typically decreases slightly because of dislocations escaping from Cottrell atmosphere
Cottrell atmosphere
In materials science, the concept of the Cottrell atmosphere was introduced by Cottrell and Bilby in 1949 to explain how dislocations are pinned in some metals by carbon or nitrogen interstitials....

s. As deformation continues, the stress increases on account of strain hardening until it reaches the ultimate strength. Until this point, the cross-sectional area decreases uniformly because of Poisson contractions
Poisson's ratio
Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

. The actual rupture point is in the same vertical line as the visual rupture point.

However, beyond this point a neck
Necking (engineering)
Necking, in engineering or materials science, is a mode of tensile deformation where relatively large amounts of strain localize disproportionately in a small region of the material. The resulting prominent decrease in local cross-sectional area provides the basis for the name "neck"...

forms where the local cross-sectional area decreases more quickly than the rest of the sample resulting in an increase in the true stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

. As shown in Fig.2, On an engineering stress–strain curve this is seen as a decrease in the apparent stress. However if the curve is plotted in terms of true stress and true strain the stress will continue to rise until failure. Eventually the neck becomes unstable and the specimen ruptures (fractures).

Less ductile materials such as aluminum and medium to high carbon steels do not have a well-defined yield point.There are generally two types of yield points,upper and lower yield point. For these materials the yield strength is typically determined by the "offset yield method", by which a line is drawn parallel to the linear elastic portion of the curve and intersecting the abscissa at some arbitrary value (generally from 0.1% to 0.2%). The intersection of this line and the stress–strain curve is reported as the yield point. The elastic region is the portion of the curve where the material will return to its original shape if the load is removed. The plastic region is the portion where some permanent deformation will occur, even if the load is removed. Failure point is when the object ruptures.

## Brittle materials

Brittle
Brittle
A material is brittle if, when subjected to stress, it breaks without significant deformation . Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Breaking is often accompanied by a snapping sound. Brittle materials include most ceramics and glasses ...

materials such as concrete
Concrete
Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...

and carbon fiber
Carbon fiber
Carbon fiber, alternatively graphite fiber, carbon graphite or CF, is a material consisting of fibers about 5–10 μm in diameter and composed mostly of carbon atoms. The carbon atoms are bonded together in crystals that are more or less aligned parallel to the long axis of the fiber...

do not have a yield point, and do not strain-harden. Therefore the ultimate strength and breaking strength are the same. A most unusual stress-strain curve is shown in Fig.3. Typical brittle materials like glass
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

do not show any plastic deformation but fail while the deformation is elastic
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

. 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 as there will not be a neck formation like in the case of ductile materials. A typical stress strain curve for a brittle material will be linear. Testing of several identical specimen, cast iron
Cast iron
Cast iron is derived from pig iron, and while it usually refers to gray iron, it also identifies a large group of ferrous alloys which solidify with a eutectic. The color of a fractured surface can be used to identify an alloy. White cast iron is named after its white surface when fractured, due...

, or soil
Soil
Soil is a natural body consisting of layers of mineral constituents of variable thicknesses, which differ from the parent materials in their morphological, physical, chemical, and mineralogical characteristics...

, tensile strength is negligible compared to the compressive strength and it is assumed zero for many engineering applications. Glass fibers have a tensile strength
Tensile strength
Ultimate tensile strength , often shortened to tensile strength or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract...

stronger than steel, but bulk glass usually does not. This is because of the stress intensity factor
Stress Intensity Factor
The stress intensity factor, K, is used in fracture mechanics to predict the stress state near the tip of a crack caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion...

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 sum of the tensile strength of its individual fibers.

## Elastomers

An elastomer is a polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

with the property of viscoelasticity
Viscoelasticity
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just...

(colloquially "elasticity"), generally having notably low Young's modulus
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

and high yield strain compared with other materials. The term, which is derived from elastic polymer, is often used interchangeably with the term rubber
Rubber (disambiguation)
Natural rubber is a latex material, originally from the Para rubber tree.Rubber may also refer to:- Science and industry :*Synthetic rubber, general term for many types of man-made rubbers...

, although the latter is preferred when referring to vulcanisates
Vulcanization
Vulcanization or vulcanisation is a chemical process for converting rubber or related polymers into more durable materials via the addition of sulfur or other equivalent "curatives." These additives modify the polymer by forming crosslinks between individual polymer chains. Vulcanized material is...

. Each of the monomer
Monomer
A monomer is an atom or a small molecule that may bind chemically to other monomers to form a polymer; the term "monomeric protein" may also be used to describe one of the proteins making up a multiprotein complex...

s which link to form the polymer is usually made of carbon
Carbon
Carbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...

, hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

, oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

and/or silicon
Silicon
Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...

. Elastomers are amorphous polymers existing above their glass transition temperature, so that considerable segmental motion is possible. At ambient temperatures, rubbers are thus relatively soft (E
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

~3MPa) and deformable. Their primary uses are for seal
Seal (mechanical)
A mechanical seal is a device which helps join systems or mechanisms together by preventing leakage , containing pressure, or excluding contamination...

An adhesive, or glue, is a mixture in a liquid or semi-liquid state that adheres or bonds items together. Adhesives may come from either natural or synthetic sources. The types of materials that can be bonded are vast but they are especially useful for bonding thin materials...

s and molded flexible parts.

Elastomers are usually thermosets (requiring vulcanization
Vulcanization
Vulcanization or vulcanisation is a chemical process for converting rubber or related polymers into more durable materials via the addition of sulfur or other equivalent "curatives." These additives modify the polymer by forming crosslinks between individual polymer chains. Vulcanized material is...

) but may also be thermoplastic
Thermoplastic
Thermoplastic, also known as a thermosoftening plastic, is a polymer that turns to a liquid when heated and freezes to a very glassy state when cooled sufficiently...

(see thermoplastic elastomer
Thermoplastic elastomer
Thermoplastic elastomers , sometimes referred to as thermoplastic rubbers, are a class of copolymers or a physical mix of polymers which consist of materials with both thermoplastic and elastomeric properties...

). The long polymer chains cross-link
Cross-links are bonds that link one polymer chain to another. They can be covalent bonds or ionic bonds. "Polymer chains" can refer to synthetic polymers or natural polymers . When the term "cross-linking" is used in the synthetic polymer science field, it usually refers to the use of...

during curing, i.e., vulcanizing. The molecular structure of elastomers can be imagined as a 'spaghetti and meatball' structure, with the meatballs signifying cross-links. The elasticity is derived from the ability of the long chains to reconfigure themselves to distribute an applied stress. The covalent cross-linkages ensure that the elastomer will return to its original configuration when the stress is removed. As a result of this extreme flexibility, elastomers can reversibly extend from 5-900%, depending on the specific material. Without the cross-linkages or with short, uneasily reconfigured chains, the applied stress would result in a permanent deformation. The stress-strain curve is quite unique, and shows an S-shaped curve sloping upwards to the elastomeric limit.

• Elastomers
• Strength of materials
Strength of materials
In materials science, the strength of a material is its ability to withstand an applied stress without failure. The applied stress may be tensile, compressive, or shear. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a...

• Tensometer
Tensometer
A tensometer is a device used to evaluate the Young's modulus of a material and other tensile properties of materials, such as tensile strength. It is usually a universal testing machine loaded with a sample between 2 grips that are either adjusted manually or automatically to apply force to the...

• Universal testing machine
Universal Testing Machine
A universal testing machine, also known as a universal tester, materials testing machine or materials test frame, is used to test the tensile stress and compressive strength of materials...