Stress–strain curve

σ=F/A

ε=Δl/L0.2%

1

2

EE

FA

FL Δl

Fig.1 'Stress–strain curve showing typical yield behavior fornonferrous alloys. Stress ( σ ) is shown as a function of strain( ϵ )1: Elastic (proportionality) limit2: Offset yield strength (0.2% proof strength)

The relationship between the stress and strain that a par-ticular material displays is known as that particular ma-terial’s stress–strain curve. It is unique for each mate-rial and is found by recording the amount of deformation(strain) at distinct intervals of tensile or compressive load-ing (stress). These curves reveal many of the propertiesof a material (including data to establish the Modulus ofElasticity, E).[1]

Stress–strain curves of various materials vary widely, anddifferent tensile tests conducted on the same materialyield different results, depending upon the temperatureof the specimen and the speed of the loading. It is possi-ble, however, to distinguish some common characteristicsamong the stress–strain curves of various groups of mate-rials and, on this basis, to divide materials into two broadcategories; namely, the ductile materials and the brittlematerials.[2]

Consider a bar of cross sectional area A being subjectedto equal and opposite forces F pulling at the ends so thebar is under tension. The material is experiencing a stressdefined to be the ratio of the force to the cross sectionalarea of the bar:

stress = FA

This stress is called the tensile stress because every partof the object is subjected to tension. The SI unit of stressis the newton per square meter, which is called the pascal.1 pascal = 1 Pa = 1 N/m2

Now consider a force that is applied tangentially to anobject. The ratio of the shearing force to the area A is

called the shear stress.If the object is twisted through an angle q, then the shearstrain is:shear strain = tan qFinally, the shear modulus MS of a material is defined asthe ratio of shear stress to shear strain at any point in anobject made of that material. The shear modulus is alsoknown as the torsion modulus.

1 Ductile materials

1

2

3

3

4 5

A

B

Stress

Strain

A stress–strain curve typical of structural steel.

• 1: Ultimate strength

• 2: Yield strength (yield point)

• 3: Rupture

• 4: Strain hardening region

• 5: Necking region

• A: Apparent stress (F/A0)

• B: Actual stress (F/A)

Ductile materials, which includes structural steel andmany alloys of other metals, are characterized by theirability to yield at normal temperatures.[3]

Low carbon steel generally exhibits a very linear stress–strain relationship up to a well defined yield point (Fig.2).

1

2 3 SEE ALSO

The linear portion of the curve is the elastic region andthe slope is the modulus of elasticity or Young’s Modulus( Young’s Modulus is the ratio of the compressive stressto the longitudinal strain). After the yield point, the curvetypically decreases slightly because of dislocations es-caping from Cottrell atmospheres. As deformation con-tinues, the stress increases on account of strain harden-ing until it reaches the ultimate tensile stress. Until thispoint, the cross-sectional area decreases uniformly andrandomly because of Poisson contractions. The actualfracture point is in the same vertical line as the visualfracture point.However, beyond this point a neck forms where the localcross-sectional area becomes significantly smaller thanthe original. The ratio of the tensile force to the truecross-sectional area at the narrowest region of the neckis called the true stress. The ratio of the tensile force tothe original cross-sectional area is called the engineeringstress. If the stress–strain curve is plotted in terms of truestress and true strain the stress will continue to rise un-til failure. Eventually the neck becomes unstable and thespecimen fractures.If the specimen is subjected to progressively increasingtensile force it reaches the ultimate tensile stress and thennecking and elongation occur rapidly until fracture. If thespecimen is subjected to progressively increasing lengthit is possible to observe the progressive necking and elon-gation, and to measure the decreasing tensile force in thespecimen. Civil structures such as bridges are mostlyloaded in a manner that applies progressively increasingforces. Sheet-metal pressing operations subject the work-piece to progressively increasing deformation. In order toaccurately record the behavior of materials between ulti-mate tensile stress and fracture, most tensile-testing ma-chines load the specimen with a progressively increasinglength.Less ductile materials such as medium to high carbonsteels do not have a well-defined yield point. There aregenerally two types of yield point - upper and lower yieldpoints. For these materials the yield stress is typically de-termined by the “offset yield method”, by which a line isdrawn parallel to the linear elastic portion of the curve andintersecting the abscissa at some arbitrary value (gener-ally from 0.1% to 0.2%). The intersection of this line andthe stress–strain curve is reported as the yield point. Theelastic region is the portion of the curve where the mate-rial will return to its original shape if the load is removed.The plastic region is the portion where some permanentdeformation will occur, even if the load is removed. Fail-ure point is when the specimen fractures.

2 Brittle materials

Brittle materials, which includes cast iron, glass, andstone, are characterized by the fact that rupture oc-

Fig.3 Stress–strain curve for brittle materials.

curs without any noticeable prior change in the rate ofelongation.[4]

Brittle materials such as concrete or carbon fiber do nothave a yield point, and do not strain-harden. Therefore,the ultimate strength and breaking strength are the same.A typical stress–strain curve is shown in Fig.3. Typicalbrittle materials like glass do not show any plastic defor-mation but fail while the deformation is elastic. One ofthe characteristics of a brittle failure is that the two brokenparts can be reassembled to produce the same shape as theoriginal component as there will not be a neck formationlike in the case of ductile materials. A typical stress–strain curve for a brittle material will be linear. For somematerials, such as concrete, tensile strength is negligiblecompared to the compressive strength and it is assumedzero for many engineering applications. Glass fibers havea tensile strength stronger than steel, but bulk glass usu-ally does not. This is because of the stress intensity factorassociated with defects in the material. As the size of thesample gets larger, the size of defects also grows. In gen-eral, the tensile strength of a rope is always less than thesum of the tensile strengths of its individual fibers.

3 See also

• Elastomers

• Strength of materials

• Tensometer

• Universal testing machine

• Stress–strain index

• Stress–strain analysis

3

4 References[1] Luebkeman, C., & Peting, D. (2012, 04

28). Stress–strain curves. Retrieved fromhttp://pages.uoregon.edu/struct/courseware/461/461_lectures/461_lecture24/461_lecture24.html.

[2] Beer, F, Johnston, R, Dewolf, J, & Mazurek, D. (2009).Mechanics of materials. New York: McGraw-Hill com-panies. P 51.

[3] Beer, F, Johnston, R, Dewolf, J, & Mazurek, D. (2009).Mechanics of materials. New York: McGraw-Hill com-panies. P 58.

[4] Beer, F, Johnston, R, Dewolf, J, & Mazurek, D. (2009).Mechanics of materials. New York: McGraw-Hill com-panies. P 59.

5 External links• British Society for Strain Measurement

• Stress–strain diagram

• Engineering stress–strain curve

4 6 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

6 Text and image sources, contributors, and licenses

6.1 Text• Stress–strain curve Source: https://en.wikipedia.org/wiki/Stress%E2%80%93strain_curve?oldid=684262684 Contributors: Patrick,Michael Hardy, Stone, Toiyabe, Sonett72, M1ss1ontomars2k4, Freakofnurture, Root4(one), Robotje, Duk, Slinky Puppet, Merenta, PAR,Moondoggy, TVBZ28, Karnesky, AshishG, Chobot, Sanpaz, RussBot, Gaius Cornelius, CambridgeBayWeather, NawlinWiki, Dhollm,Tony1, JLaTondre, Sardanaphalus, SmackBot, Slashme, Firey man, Bluebot, Shortiewaswo, Peterlewis, Wizard191, Iridescent, Foscoe,Mattbr, Basar, Alucard (Dr.), Rmcgiv, MSBOT, Recurring dreams, Glrx, R'n'B, Barbed, Bcartolo, Chemart, Joeinwap, Fantasi, AnnaLincoln, Kalivd, SieBot, Pkgx, Faradayplank, BenBritton, Mostafasaad~enwiki, Dolphin51, ClueBot, Browni1992, DragonBot, Lartoven,Tpkaplan, Dekisugi, Ottawa4ever, Saeed.Veradi, Hermanoere, Addbot, Queenmomcat, Nerak99, Fgnievinski, Yinweichen, Yobot, Ptbot-gourou, Dronkit, Jim1138, Xqbot, Erud, Gray91, J04n, Erik9bot, Thehelpfulbot, I dream of horses, HRoestBot, Corinne68, ZéroBot,Caspertheghost, ClueBot NG, A520, Dhh787, Manas-NJITWILL, Widr, Gur.sin007, Jitesh-NJITWILL, DarafshBot, AK456, Bright-StarSky, Kevmeister68, Jiyepp, Babitaarora, FrB.TG, Aditya kale 25 and Anonymous: 120

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