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| {{Mechanical failure modes}}
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| In [[materials science]], '''creep''' (sometimes called '''cold flow''') is the tendency of a [[solid]] material to move slowly or deform permanently under the influence of mechanical [[stress (physics)|stress]]es. It can occur as a result of long-term exposure to high levels of stress that are still below the [[yield strength]] of the material. Creep is more severe in materials that are subjected to [[heat]] for long periods, and generally increases as they near their [[melting point]]. Creep always increases with temperature.{{fact|date=August 2013}}
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| The rate of deformation is a function of the material properties, exposure time, exposure [[temperature]] and the applied [[structural load]]. Depending on the magnitude of the applied stress and its duration, the deformation may become so large that a component can no longer perform its function — for example creep of a [[turbine blade]] will cause the blade to contact the casing, resulting in the [[Structural failure|failure]] of the blade. Creep is usually of concern to [[engineering|engineers]] and [[metallurgy|metallurgists]] when evaluating components that operate under high stresses or high temperatures. Creep is a deformation mechanism that may or may not constitute a [[failure mode]]. For example, moderate creep in [[concrete]] is sometimes welcomed because it relieves [[tensile stress]]es that might otherwise lead to [[Fracture|cracking]].
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| Unlike [[brittle fracture]], creep deformation does not occur suddenly upon the application of stress. Instead, [[Strain (materials science)|strain]] accumulates as a result of long-term stress. Therefore, creep is a "time-dependent" deformation.
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| The temperature range in which creep deformation may occur differs in various materials. For example, [[tungsten]] requires a temperature in the thousands of degrees before creep deformation can occur, while [[ice]] will creep at temperatures near 0 °C (32 °F).<ref>{{cite web|title = Rheology of Ice|accessdate = 2008-10-16|url = http://www.geo.brown.edu/People/Postdocs/goldsby/Icephysics.htm}} {{Dead link|date=September 2010|bot=H3llBot}}</ref> As a general guideline, the effects of creep deformation generally become noticeable at approximately 30% of the [[melting point]] (as measured on a [[thermodynamic temperature]] scale such as kelvin or rankine) for [[metals]], and at 40–50% of melting point for [[ceramic]]s. Virtually any material will creep upon approaching its melting temperature. Since the creep minimum temperature is related to the melting point, creep can be seen at relatively low temperatures for some materials. Plastics and low-melting-temperature metals, including many [[solder]]s, can begin to creep at [[room temperature]], as can be seen markedly in old [[lead]] hot-water pipes. [[Glacier]] flow is an example of creep processes in ice.
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| ==Stages of creep==
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| [[Image:3StageCreep.svg|right|thumb|250px| Strain as a function of time due to constant stress over an extended period for a viscoelastic material.]]
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| In the initial stage, or primary creep, the strain rate is relatively high, but slows with increasing time. This is due to [[work hardening]]. The strain rate eventually reaches a minimum and becomes near constant. This is due to the balance between work hardening and [[annealing (metallurgy)|annealing]] (thermal softening). This stage is known as secondary or steady-state creep. This stage is the most understood. The characterized "creep strain rate" typically refers to the rate in this secondary stage. Stress dependence of this rate depends on the creep mechanism. In tertiary creep, the strain rate exponentially increases with stress because of [[Necking (engineering)|necking]] phenomena.
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| ==Mechanisms of creep==
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| The mechanism of creep depends on temperature and stress. Various mechanisms are:
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| *Bulk diffusion (Nabarro-Herring creep)
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| *[[Dislocation#Dislocation climb|Climb]] — here the strain is actually accomplished by climb
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| *Climb-assisted glide — here the climb is an ''enabling'' mechanism, allowing dislocations to get around obstacles
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| *Grain boundary diffusion ([[Coble creep]])
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| *Thermally activated glide — e.g., via cross-[[Slip (materials science)|slip]]
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| ==General creep equation==
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| : <math> \frac{\mathrm{d}\varepsilon}{\mathrm{d}t} = \frac{C\sigma^m}{d^b} e^\frac{-Q}{kT}</math>
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| where <math>{\varepsilon}</math> is the creep strain, ''C'' is a constant dependent on the material and the particular creep mechanism, ''m'' and ''b'' are exponents dependent on the creep mechanism, ''Q'' is the activation energy of the creep mechanism, σ is the applied stress, ''d'' is the grain size of the material, ''k'' is [[Boltzmann's constant]], and ''T'' is the absolute temperature.
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| ===Dislocation creep===
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| {{Main|Dislocation creep}}
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| At high stresses (relative to the [[shear modulus]]), creep is controlled by the movement of [[dislocation]]s.
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| For dislocation creep, ''Q = Q(self diffusion), m'' = 4-6, and ''b'' = 0. Therefore, dislocation creep has a strong dependence on the applied stress and no grain size dependence.
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| Some alloys exhibit a very large stress exponent (''n'' > 10), and this has typically been explained by introducing a "threshold stress," σ<sub>th</sub>, below which creep can't be measured. The modified power law equation then becomes:
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| : <math>\frac{\mathrm{d}\epsilon}{\mathrm{d}t} = A \left(\sigma-\sigma_{th}\right)^n e^\frac{-Q}{\bar R T}</math>
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| where ''A, Q'' and ''n'' can all be explained by conventional mechanisms (so 3 ≤ ''n'' ≤ 10).
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| ===Nabarro-Herring creep===
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| Nabarro-Herring creep is a form of [[diffusion creep]]. In Nabarro-Herring creep, atoms diffuse through the lattice causing grains to elongate along the stress axis; ''k'' is related to the diffusion coefficient of atoms through the lattice, ''Q = Q(self diffusion), m'' = 1, and ''b'' = 2. Therefore Nabarro-Herring creep has a weak stress dependence and a moderate grain size dependence, with the creep rate decreasing as grain size is increased.
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| Nabarro-Herring creep is strongly temperature dependent. For lattice diffusion of atoms to occur in a material, neighboring lattice sites or interstitial sites in the crystal structure must be free. A given atom must also overcome the energy barrier to move from its current site (it lies in an energetically favorable [[potential well]]) to the nearby [[Vacancy (chemistry)|vacant site]] (another potential well). The general form of the diffusion equation is ''D = D<sub>0</sub>exp(E/KT)'' where ''D<sub>0</sub>'' has a dependence on both the attempted jump frequency and the number of nearest neighbor sites and the probability of the sites being vacant. Thus there is a double dependence upon temperature. At higher temperatures the diffusivity increases due to the direct temperature dependence of the equation, the increase in vacancies through [[Schottky defect]] formation, and an increase in the average energy of atoms in the material. Nabarro-Herring creep dominates at very high temperatures relative to a material's melting temperature.
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| ==Coble creep==
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| {{Main|Coble creep}}
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| Coble creep is a second form of diffusion controlled creep. In Coble creep the atoms diffuse along grain boundaries to elongate the grains along the stress axis. This causes Coble creep to have a stronger grain size dependence than Nabarro-Herring creep. For Coble creep ''k'' is related to the diffusion coefficient of atoms along the grain boundary, ''Q = Q(grain boundary diffusion), m = 1'', and ''b = 3''. Because ''Q(grain boundary diffusion) < Q(self diffusion)'', Coble creep occurs at lower temperatures than Nabarro-Herring creep. Coble creep is still temperature dependent, as the temperature increases so does the grain boundary diffusion. However, since the number of nearest neighbors is effectively limited along the interface of the grains, and thermal generation of vacancies along the boundaries is less prevalent, the temperature dependence is not as strong as in Nabarro-Herring creep. It also exhibits the same linear dependence on stress as Nabarro-Herring creep.
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| ===Creep of polymers===
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| [[Image:Creep.svg|right|thumb|250px| a) Applied stress and b) induced strain as functions of time over a short period for a viscoelastic material.]]
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| Creep can occur in [[polymer]]s and [[metal]]s which are considered [[Viscoelasticity|viscoelastic]] materials. When a polymeric material is subjected to an abrupt force, the response can be modeled using the [[Kelvin-Voigt model]]. In this model, the material is represented by a [[Hooke's law|Hookean]] spring and a [[Newtonian fluid|Newtonian]] [[dashpot]] in parallel. The creep strain is given by the following [[convolution]] integral:
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| :<math>\varepsilon(t) = \sigma C_0 + \sigma C \int_0^\infty f(\tau)(1-\exp[-t/ \tau]) \,d \tau</math>
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| where:
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| *σ = applied stress
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| *''C<sub>0</sub> = instantaneous creep compliance
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| *''C'' = creep compliance coefficient
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| *<math>\tau</math> = retardation time
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| *<math>f(\tau)</math> = distribution of retardation times
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| When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain. This phenomenon is known as viscoelastic creep.
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| At a time ''t''<sub>0</sub>, a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period. The material responds to the stress with a strain that increases until the material ultimately fails. When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time ''t''<sub>1</sub> at which the stress is relieved, at which time the strain immediately decreases (discontinuity) then continues decreasing gradually to a residual strain.
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| Viscoelastic creep data can be presented in one of two ways. Total strain can be plotted as a function of time for a given temperature or temperatures. Below a critical value of applied stress, a material may exhibit linear viscoelasticity. Above this critical stress, the creep rate grows disproportionately faster. The second way of graphically presenting viscoelastic creep in a material is by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time.<ref name=Rosato>Rosato, et al. (2001): "Plastics Design Handbook," 63-64.</ref> Below its critical stress, the viscoelastic creep modulus is independent of stress applied. A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied stresses are below the material's critical stress value.
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| Additionally, the molecular weight of the polymer of interest is known to affect its creep behavior. The effect of increasing molecular weight tends to promote secondary bonding between polymer chains and thus make the polymer more creep resistant. Similarly, aromatic polymers are even more creep resistant due to the added stiffness from the rings. Both molecular weight and aromatic rings add to polymers' thermal stability, increasing the creep resistance of a polymer.<ref>{{cite book|author=M. A. Meyers and K. K. Chawla|title=Mechanical Behavior of Materials|publisher=Cambridge University Press|year=1999|page=573|url=http://www.toodoc.com/Mechanical-Behavior-of-Materials-ebook.html|isbn=978-0-521-86675-0}}</ref>
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| Both polymers and metals can creep. Polymers experience significant creep at temperatures above ca. –200°C; however, there are three main differences between polymeric and metallic creep.<ref name=McCrum>{{cite book|author=McCrum, N.G, Buckley, C.P; Bucknall, C.B|title=Principles of Polymer Engineering| publisher=Oxford Science Publications|year=2003|isbn=0-19-856526-7}}</ref>
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| Polymers show creep basically in two different ways. At typical work loads (5 up to 50%) [[ultra high molecular weight polyethylene]] (Spectra, [[Dyneema]]) will show time-linear creep, whereas [[polyester]] or [[aramid]]s ([[Twaron]], [[Kevlar]]) will show a time-logarithmic creep.
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| ===Creep of concrete===
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| {{main|Creep and shrinkage of concrete}}
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| The creep of concrete, which originates from the calcium silicate hydrates (C-S-H) in the hardened Portland cement paste (which is the binder of mineral aggregates), is fundamentally different from the creep of metals
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| as well as polymers. Unlike the creep of metals, it occurs at all stress levels and, within the service stress
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| range, is linearly dependent on the stress if the pore water content is constant. Unlike the creep of polymers and metals, it exhibits multi-months aging, caused by chemical hardening due to hydration which stiffens the microstructure, and multi-year aging, caused by long-term relaxation of self-equilibrated micro-stresses in the nano-porous microstructure of the C-S-H. If concrete is fully dried, it does not creep, but it is next to impossible to dry concrete fully without severe cracking.
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| ==Applications==
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| [[File:Creep in cardboard.jpg|thumb|right|250px|Creep on the underside of a cardboard box: a largely empty box was placed on a smaller box, and fuller boxes were placed on top of it. Due to the weight, the portions of the empty box not sitting on the lower box gradually crept downward.]]
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| Though mostly due to the reduced yield strength at higher temperatures, the [[Collapse of the World Trade Center]] was due in part to creep from increased temperature operation.<ref>[[Zdeněk Bažant]] and Yong Zhu, ''[http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/405.pdf Why Did the World Trade Center Collapse?—Simple Analysis]'',''Journal of Engineering Mechanics'', January 2002</ref>
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| The creep rate of hot pressure-loaded components in a [[nuclear reactor]] at power can be a significant design constraint, since the creep rate is enhanced by the flux of energetic particles.
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| Creep was blamed for the [[Big Dig ceiling collapse|Big Dig tunnel ceiling collapse]] in [[Boston, Massachusetts]] that occurred in July 2006.
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| An example of an application involving creep deformation is the design of tungsten light bulb filaments. Sagging of the filament coil between its supports increases with time due to creep deformation caused by the weight of the filament itself. If too much deformation occurs, the adjacent turns of the coil touch one another, causing an electrical short and local overheating, which quickly leads to failure of the filament. The coil geometry and supports are therefore designed to limit the stresses caused by the weight of the filament, and a special tungsten alloy with small amounts of [[oxygen]] trapped in the [[crystallite]] [[grain boundaries]] is used to slow the rate of [[Coble creep]].
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| In steam turbine power plants, pipes carry steam at high temperatures (566°C or 1050°F) and pressures (above 24.1 MPa or 3500 psi). In jet engines, temperatures can reach up to 1400°C (2550°F) and initiate creep deformation in even advanced coated turbine blades. Hence, it is crucial for correct functionality to understand the creep deformation behavior of materials.
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| Creep deformation is important not only in systems where high temperatures are endured such as [[nuclear power plant]]s, [[jet engine]]s and [[heat exchanger]]s, but also in the design of many everyday objects. For example, metal paper clips are stronger than plastic ones because plastics creep at room temperatures. Aging [[glass]] windows are often erroneously used as an example of this phenomenon: measurable creep would only occur at temperatures above the [[glass transition temperature]] around 500°C (900°F). While glass does exhibit creep under the right conditions, apparent sagging in old windows may instead be a consequence of obsolete manufacturing processes, such as that used to create [[Crown glass (window)|crown glass]], which resulted in inconsistent thickness.<ref>{{Cite book|isbn = 0-8493-9658-1|page = 476|last = Lakes|first = Roderic S.|title = Viscoelastic Solids|year = 1999}}</ref><ref>{{cite web|title = Is glass liquid or solid?|accessdate = 2008-10-15|url = http://math.ucr.edu/home/baez/physics/General/Glass/glass.html}}</ref>
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| ==See also==
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| {{colbegin|3}}
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| *[[Biomaterial]]
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| *[[Biomechanics]]
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| *[[Ductility#Ductile-brittle_transition_temperature|Ductile Brittle Transition Temperature in materials science]]
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| *[[Deformation mechanism]]
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| *[[Hysteresis]]
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| *[[Larson-Miller Parameter]]
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| *[[Stress relaxation]]
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| *[[Viscoelasticity]]
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| *[[Viscoplasticity]]
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| {{colend}}
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{cite book
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| |last1 = Ashby|first1 = Michael F.|author1-link = M. F. Ashby
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| |last2 = Jones|first2 = David R. H.
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| |isbn = 0-08-026138-8
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| |publisher = Pergamon Press
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| |title = Engineering Materials 1: An Introduction to their Properties and Applications
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| |year = 1980}}.
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| *{{cite book
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| |last1 = Frost|first1 = Harold J.
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| |last2 = Ashby|first2 = Michael F.|author2-link = M. F. Ashby
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| |isbn = 0-08-029337-9
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| |publisher = Pergamon Press
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| |title = Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics
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| |year = 1982}}.
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| *{{cite book
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| |last = Turner|first = S
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| |isbn = 0-08-043152-6
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| |location = Oxford
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| |pages = 1813–1817
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| |publisher = Elsevier Science Ltd.
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| |title = Creep of Polymeric Materials
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| |year = 2001}}.
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| ==External links==
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| *[http://www.polito.it/Creepanalysis Creep Analysis Research Group - Politecnico di Torino]
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| *[http://thayer.dartmouth.edu/~defmech/ Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics]
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| *[http://www.nist.gov/public_affairs/releases/wtc_briefing_april0505.htm The National Institute of Standards and Technology - WTC Briefing]
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| *{{cite web|title = Introduction to Creep|accessdate = 2008-10-16|url = http://www.nuc.berkeley.edu/thyd/ne161/jlrhoads/creep.html |archiveurl = http://web.archive.org/web/20080617053739/http://www.nuc.berkeley.edu/thyd/ne161/jlrhoads/creep.html <!-- Bot retrieved archive --> |archivedate = 2008-06-17}}
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| [[Category:Elasticity (physics)]]
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| [[Category:Materials degradation]]
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| [[Category:Deformation]]
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