![Beagle Channel -k](https://upload.wikimedia.org/wikipedia/commons/7/74/Beagle_Channel_-k.jpg)
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 persistent mechanical
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
es. It can occur as a result of long-term exposure to high levels of stress that are still below the
yield strength
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
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.
The rate of deformation is a function of the material's properties, exposure time, exposure temperature and the applied
structural load
A structural load or structural action is a force, deformation, or acceleration applied to structural elements. A load causes stress, deformation, and displacement in a structure. Structural analysis, a discipline in engineering, analyzes the ef ...
. 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 could cause the blade to contact the casing, resulting in the
failure
Failure is the state or condition of not meeting a desirable or intended objective (goal), objective, and may be viewed as the opposite of Success (concept), success. The criteria for failure depends on context, and may be relative to a parti ...
of the blade. Creep is usually of concern to engineers and 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
Failure causes are defects in design, process, quality, or part application, which are the underlying cause of a failure or which initiate a process which leads to failure. Where failure depends on the user of the product or process, then human er ...
. For example, moderate creep in concrete is sometimes welcomed because it relieves
tensile stress
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
es that might otherwise lead to cracking.
Unlike
brittle fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displa ...
, creep deformation does not occur suddenly upon the application of stress. Instead,
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
accumulates as a result of long-term stress. Therefore, creep is a "time-dependent" deformation.
Temperature dependence
The temperature range in which creep deformation may occur differs in various materials. Creep deformation generally occurs when a material is stressed at a temperature near its melting point. While tungsten requires a temperature in the thousands of degrees before creep deformation can occur, lead may creep at room temperature, and ice will creep at temperatures below . Plastics and low-melting-temperature metals, including many solders, can begin to creep at room temperature. Glacier flow is an example of creep processes in ice. The effects of creep deformation generally become noticeable at approximately 35% of the melting point (in Kelvin) for metals and at 45% of melting point for ceramics.
Stages
Creep behavior can be split into three main stages.
In primary, or transient, creep, the strain rate is a function of time. In Class M materials, which include most pure materials, strain rate decreases over time. This can be due to increasing
dislocation density, or it can be due to
evolving grain size. In class A materials, which have large amounts of solid solution hardening, strain rate increases over time due to a thinning of solute drag atoms as dislocations move.
In the secondary, or steady-state, creep, dislocation structure and grain size have reached equilibrium, and therefore strain rate is constant. Equations that yield a strain rate refer to the steady-state strain rate. Stress dependence of this rate depends on the creep mechanism.
In tertiary creep, the strain rate exponentially increases with stress. This can be due to
necking phenomena, internal cracks, or voids, which all decrease the cross-sectional area and increase the true stress on the region, further accelerating deformation and leading to fracture.
Mechanisms of deformation
Depending on the temperature and stress, different deformation mechanisms are activated. Though there are generally many deformation mechanisms active at all times, usually one mechanism is dominant, accounting for almost all deformation.
Various mechanisms are:
* Bulk diffusion (
Nabarro–Herring creep
Nabarro–Herring creep is a mode of deformation of crystalline materials (and amorphous materials) that occurs at low stresses and held at elevated temperatures in fine-grained materials. In Nabarro–Herring creep (NH creep), atoms diffuse thr ...
)
* Grain boundary diffusion (
Coble creep
Coble creep, a form of diffusion creep, is a mechanism for deformation of crystalline solids. Contrasted with other diffusional creep mechanisms, Coble creep is similar to Nabarro–Herring creep in that it is dominant at lower stress levels and h ...
)
* Glide-controlled
dislocation creep Dislocation creep is a deformation mechanism in crystalline materials. Dislocation creep involves the movement of dislocations through the crystal lattice of the material, in contrast to diffusion creep, in which diffusion (of vacancies) is the dom ...
: dislocations move via glide and climb, and the speed of glide is the dominant factor on strain rate
* Climb-controlled dislocation creep: dislocations move via glide and climb, and the speed of climb is the dominant factor on strain rate
* Harper–Dorn creep: a low-stress creep mechanism in some pure materials
At low temperatures and low stress, creep is essentially nonexistent and all strain is elastic. At low temperatures and high stress, materials experience plastic deformation rather than creep. At high temperatures and low stress, diffusional creep tends to be dominant, while at high temperatures and high stress, dislocation creep tends to be dominant.
Deformation mechanism maps
Deformation mechanism maps provide a visual tool categorizing the dominant deformation mechanism as a function of
homologous temperature
Homologous temperature expresses the thermodynamic temperature of a material as a fraction of the thermodynamic temperature of its melting point (i.e. using the Kelvin scale):
T_H = \frac
For example, the homologous temperature of lead at room ...
, shear modulus-normalized stress, and strain rate. Generally, two of these three properties (most commonly temperature and stress) are the axes of the map, while the third is drawn as
contours
Contour may refer to:
* Contour (linguistics), a phonetic sound
* Pitch contour
* Contour (camera system), a 3D digital camera system
* Contour, the KDE Plasma 4 interface for tablet devices
* Contour line, a curve along which the function has a ...
on the map.
To populate the map, constitutive equations are found for each deformation mechanism. These are used to solve for the boundaries between each deformation mechanism, as well as the strain rate contours.
Deformation mechanism maps can be used to compare different strengthening mechanisms, as well as compare different types of materials.
General equation
:
where ''ε'' 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
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
of the creep mechanism, ''σ'' is the applied stress, ''d'' is the grain size of the material, ''k'' is
Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and ''T'' is the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
.
Dislocation creep
At high stresses (relative to the
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackrel ...
), creep is controlled by the movement of
dislocation
In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sl ...
s.
For dislocation creep, ''Q'' = ''Q''(self diffusion), 4 ≤ ''m'' ≤ 6, and ''b'' < 1. Therefore, dislocation creep has a strong dependence on the applied stress and the intrinsic activation energy and a weaker dependence on grain size. As grain size gets smaller, grain boundary area gets larger, so dislocation motion is impeded.
Some alloys exhibit a very large stress exponent (''m'' > 10), and this has typically been explained by introducing a "threshold stress," ''σ''
th, below which creep can't be measured. The modified power law equation then becomes:
:
where ''A'', ''Q'' and ''m'' can all be explained by conventional mechanisms (so 3 ≤ ''m'' ≤ 10), and ''R'' is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
. The creep increases with increasing applied stress, since the applied stress tends to drive the dislocation past the barrier, and make the dislocation get into a lower energy state after bypassing the obstacle, which means that the dislocation is inclined to pass the obstacle. In other words, part of the work required to overcome the energy barrier of passing an obstacle is provided by the applied stress and the remainder by thermal energy.
Nabarro–Herring creep
![Nabarro Herring Creep1](https://upload.wikimedia.org/wikipedia/commons/f/f1/Nabarro_Herring_Creep1.png)
Nabarro–Herring (NH) creep is a form of
diffusion creep Diffusion creep refers to the deformation of crystalline solids by the diffusion of vacancies through their crystal lattice. Diffusion creep results in plastic deformation rather than brittle failure of the material.
Diffusion creep is more sens ...
, while dislocation glide creep does not involve atomic diffusion. Nabarro–Herring creep dominates at high temperatures and low stresses. As shown in the figure on the right, the lateral sides of the crystal are subjected to tensile stress and the horizontal sides to compressive stress. The atomic volume is altered by applied stress: it increases in regions under tension and decreases in regions under compression. So the activation energy for vacancy formation is changed by ±''σΩ'', where ''Ω'' is the atomic volume, the positive value is for compressive regions and negative value is for tensile regions. Since the fractional vacancy concentration is proportional to , where ''Q''
f is the vacancy-formation energy, the vacancy concentration is higher in tensile regions than in compressive regions, leading to a net flow of vacancies from the regions under compression to the regions under tension, and this is equivalent to a net atom diffusion in the opposite direction, which causes the creep deformation: the grain elongates in the tensile stress axis and contracts in the compressive stress axis.
In Nabarro–Herring creep, ''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 the grain size is increased.
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
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captur ...
) to the nearby
vacant site (another potential well). The general form of the diffusion equation is
:
where ''D''
0 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
A Schottky defect is an excitation of the site occupations in a crystal lattice leading to point defects named after Walter H. Schottky. In ionic crystals, this defect forms when oppositely charged ions leave their lattice sites and become incor ...
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.
Coble creep
Coble creep is the 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, thus, Coble creep will be more important in materials composed of very fine grains. 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) is less than ''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. Generally, the diffusional creep rate should be the sum of Nabarro–Herring creep rate and Coble creep rate. Diffusional creep leads to grain-boundary separation, that is, voids or cracks form between the grains. To heal this, grain-boundary sliding occurs. The diffusional creep rate and the grain boundary sliding rate must be balanced if there are no voids or cracks remaining. When grain-boundary sliding can not accommodate the incompatibility, grain-boundary voids are generated, which is related to the initiation of creep fracture.
Solute drag creep
Solute drag creep is one of the mechanisms for power-law creep (PLC), involving both dislocation and diffusional flow. Solute drag creep is observed in certain metallic
alloys
An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductility, ...
. In these alloys, the creep rate increases during the first stage of creep (Transient creep) before reaching a steady-state value. This phenomenon can be explained by a model associated with solid-solution strengthening. At low temperatures, the solute atoms are immobile and increase the flow stress required to move dislocations. However, at higher temperatures, the solute atoms are more mobile and may form atmospheres and clouds surrounding the dislocations. This is especially likely if the solute atom has a large misfit in the matrix. The solutes are attracted by the dislocation stress fields and are able to relieve the elastic stress fields of existing dislocations. Thus the solutes become bound to the dislocations. The concentration of solute, ''C'', at a distance, ''r'', from a dislocation is given by the
Cottrell atmosphere
In materials science, the concept of the Cottrell atmosphere was introduced by A. H. Cottrell and B. A. Bilby in 1949 to explain how dislocations are pinned in some metals by boron, carbon, or nitrogen interstitials.
Cottrell atmospheres occ ...
defined as
:
where ''C''
0 is the concentration at ''r'' = ∞ and ''β'' is a constant which defines the extent of segregation of the solute. When surrounded by a solute atmosphere, dislocations that attempt to glide under an applied stress are subjected to a back stress exerted on them by the cloud of solute atoms. If the applied stress is sufficiently high, the dislocation may eventually break away from the atmosphere, allowing the dislocation to continue gliding under the action of the applied stress. The maximum force (per unit length) that the atmosphere of
solute
In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent ...
atoms can exert on the dislocation is given by Cottrell and Jaswon
:
When the diffusion of solute atoms is activated at higher temperatures, the solute atoms which are "bound" to the dislocations by the misfit can move along with edge dislocations as a "drag" on their motion if the dislocation motion or the creep rate is not too high. The amount of "drag" exerted by the solute atoms on the dislocation is related to the diffusivity of the solute atoms in the metal at that temperature, with a higher diffusivity leading to lower drag and vice versa. The velocity at which the dislocations glide can be approximated by a power law of the form
:
where ''m'' is the effective stress exponent, ''Q'' is the apparent activation energy for glide and ''B''
0 is a constant. The parameter ''B'' in the above equation was derived by Cottrell and Jaswon for interaction between solute atoms and dislocations on the basis of the relative atomic size misfit ''ε''
a of solutes to be
[
:
where ''k'' is Boltzmann's constant, and ''r''1 and ''r''2 are the internal and external cut-off radii of dislocation stress field. ''c''0 and ''D''sol are the atomic concentration of the solute and solute diffusivity respectively. ''D''sol also has a temperature dependence that makes a determining contribution to ''Q''g.
If the cloud of solutes does not form or the dislocations are able to break away from their clouds, glide occurs in a jerky manner where fixed obstacles, formed by dislocations in combination with solutes, are overcome after a certain waiting time with support by thermal activation. The exponent ''m'' is greater than 1 in this case. The equations show that the hardening effect of solutes is strong if the factor ''B'' in the power-law equation is low so that the dislocations move slowly and the diffusivity ''D''sol is low. Also, solute atoms with both high concentration in the matrix and strong interaction with dislocations are strong gardeners. Since misfit strain of solute atoms is one of the ways they interact with dislocations, it follows that solute atoms with large atomic misfit are strong gardeners. A low ]diffusivity
Diffusivity is a rate of diffusion, a measure of the rate at which particles or heat or fluids can spread.
It is measured differently for different mediums.
Diffusivity may refer to:
*Thermal diffusivity, diffusivity of heat
*Diffusivity of mass: ...
''D''sol is an additional condition for strong hardening.
Solute drag creep sometimes shows a special phenomenon, over a limited strain rate, which is called the Portevin–Le Chatelier effect
The Portevin–Le Chatelier (PLC) effect describes a serrated stress–strain curve or jerky flow, which some materials exhibit as they undergo plastic deformation, specifically inhomogeneous deformation. This effect has been long associated wit ...
. When the applied stress becomes sufficiently large, the dislocations will break away from the solute atoms since dislocation velocity increases with the stress. After breakaway, the stress decreases and the dislocation velocity also decreases, which allows the solute atoms to approach and reach the previously departed dislocations again, leading to a stress increase. The process repeats itself when the next local stress maximum is obtained. So repetitive local stress maxima and minima could be detected during solute drag creep.
Dislocation climb-glide creep
Dislocation climb-glide creep is observed in materials at high temperature. The initial creep rate is larger than the steady-state creep rate. Climb-glide creep could be illustrated as follows: when the applied stress is not enough for a moving dislocation to overcome the obstacle on its way via dislocation glide alone, the dislocation could climb to a parallel slip plane by diffusional processes, and the dislocation can glide on the new plane. This process repeats itself each time when the dislocation encounters an obstacle. The creep rate could be written as:
:
where ''A''CG includes details of the dislocation loop geometry, ''D''L is the lattice diffusivity, ''M'' is the number of dislocation sources per unit volume, ''σ'' is the applied stress, and ''Ω'' is the atomic volume. The exponent ''m'' for dislocation climb-glide creep is 4.5 if ''M'' is independent of stress and this value of ''m'' is consistent with results from considerable experimental studies.
Harper–Dorn creep
Harper–Dorn creep is a climb-controlled dislocation mechanism at low stresses that has been observed in aluminum, lead, and tin systems, in addition to nonmetal systems such as ceramics and ice. It was first observed by Harper and Dorn in 1957. It is characterized by two principal phenomena: a power-law relationship between the steady-state strain rate and applied stress at a constant temperature which is weaker than the natural power-law of creep, and an independent relationship between the steady-state strain rate and grain size for a provided temperature and applied stress. The latter observation implies that Harper–Dorn creep is controlled by dislocation movement; namely, since creep can occur by vacancy diffusion (Nabarro–Herring creep, Coble creep), grain boundary sliding, and/or dislocation movement, and since the first two mechanisms are grain-size dependent, Harper–Dorn creep must therefore be dislocation-motion dependent. The same was also confirmed in 1972 by Barrett and co-workers where FeAl3 precipitates lowered the creep rates by 2 orders of magnitude compared to highly pure Al, thus, indicating Harper–Dorn creep to be a dislocation based mechanism.
Equation
Harper–Dorn creep is typically overwhelmed by other creep mechanisms in most situations, and is therefore not observed in most systems. The phenomenological equation which describes Harper–Dorn creep is
:
where ''ρ''0 is dislocation density (constant for Harper–Dorn creep), ''D''v is the diffusivity through the volume of the material, ''G'' is the shear modulus and ''b'' is the Burgers vector, ''σ''s, and ''n'' is the stress exponent which varies between 1 and 3.
Later investigation of the creep region
Twenty-five years after Harper and Dorn published their work, Mohamed and Ginter made an important contribution in 1982 by evaluating the potential for achieving Harper–Dorn creep in samples of Al using different processing procedures. The experiments showed that Harper–Dorn creep is achieved with stress exponent ''n'' = 1, and only when the internal dislocation density prior to testing is exceptionally low. By contrast, Harper–Dorn creep was not observed in polycrystalline Al and single crystal Al when the initial dislocation density was high.
However, various conflicting reports demonstrate the uncertainties at very low stress levels. One report by Blum and Maier, claimed that the experimental evidence for Harper–Dorn creep is not fully convincing. They argued that the necessary condition for Harper–Dorn creep is not fulfilled in Al with 99.99% purity and the steady-state stress exponent ''n'' of the creep rate is always much larger than 1.
The subsequent work conducted by Ginter et al. confirmed that Harper–Dorn creep was attained in Al with 99.9995% purity but not in Al with 99.99% purity and, in addition, the creep curves obtained in the very high purity material exhibited regular and periodic accelerations. They also found that the creep behavior no longer follows a stress exponent of ''n'' = 1 when the tests are extended to very high strains of >0.1 but instead there is evidence for a stress exponent of ''n'' > 2.
;Requirements for the occurrence
* Harper–Dorn creep is usually regarded as a Newtonian viscous process with ''n'' = 1. Some very recent experimental evidence suggests that the stress exponent may be closer to ∼2. Harper–Dorn creep should be observed at a low-stress creep regime where the stress exponent is lower than in the conventional power-law regime where ''n'' ≈ 3–5.
* Unlike Nabarro–Herring diffusion depending on grain size, the Harper–Dorn flow process is independent of grain size. In the initial experiments of Harper and Dorn, identical creep rates are recorded either over a wide range of grain sizes in polycrystalline samples or in a combination of polycrystalline samples and single crystals.
* The measured creep rates should be significantly faster, typically by more than two orders of magnitude, than the creep rates anticipated for Nabarro–Herring diffusion creep. At very high testing temperatures, Coble diffusion creep will be of negligible significance under these conditions.
* The volumetric activation energy indicates that the rate of Harper–Dorn creep is controlled by vacancy diffusion to and from dislocations, resulting in climb-controlled dislocation motion. Unlike in other creep mechanisms, the dislocation density here is constant and independent of the applied stress.
* The dislocation density must be low for Harper–Dorn creep to dominate. The density has been proposed to increase as dislocations move via cross-slip from one slip-plane to another, thereby increasing the dislocation length per unit volume. Cross-slip can also result in jogs along the length of the dislocation, which, if large enough, can act as single-ended dislocation sources.
Future expectation
Harper–Dorn creep is considered as a distinct mechanism under some conditions. But more definitive experiments are needed both to more fully establish the precise requirements for observing this process and to provide detailed information used to develop a more appropriate theoretical flow mechanism.
Sintering
At high temperatures, it is energetically favorable for voids to shrink in a material. The application of tensile stress opposes the reduction in energy gained by void shrinkage. Thus, a certain magnitude of applied tensile stress is required to offset these shrinkage effects and cause void growth and creep fracture in materials at high temperature. This stress occurs at the ''sintering limit'' of the system.
The stress tending to shrink voids that must be overcome is related to the surface energy and surface area-volume ratio of the voids. For a general void with surface energy ''γ'' and principle radii of curvature of ''r''1 and ''r''2, the sintering limit stress is
:
Below this critical stress, voids will tend to shrink rather than grow. Additional void shrinkage effects will also result from the application of a compressive stress. For typical descriptions of creep, it is assumed that the applied tensile stress exceeds the sintering limit.
Creep also explains one of several contributions to densification during metal powder sintering by hot pressing. A main aspect of densification is the shape change of the powder particles. Since this change involves permanent deformation of crystalline solids, it can be considered a plastic deformation process and thus sintering can be described as a high temperature creep process. The applied compressive stress during pressing accelerates void shrinkage rates and allows a relation between the steady-state creep power law and densification rate of the material. This phenomenon is observed to be one of the main densification mechanisms in the final stages of sintering, during which the densification rate (assuming gas-free pores) can be explained by:
:
in which ''ρ̇'' is the densification rate, ''ρ'' is the density, ''P''e is the pressure applied, ''n'' describes the exponent of strain rate behavior, and ''A'' is a mechanism-dependent constant. ''A'' and ''n'' are from the following form of the general steady-state creep equation,
:
where ''ε̇'' is the strain rate, and ''σ'' is the tensile stress. For the purposes of this mechanism, the constant ''A'' comes from the following expression, where ''A''′ is a dimensionless, experimental constant, ''μ'' is the shear modulus, ''b'' is the Burgers vector, ''k'' is Boltzmann's constant, ''T'' is absolute temperature, ''D''0 is the diffusion coefficient, and ''Q'' is the diffusion activation energy:
:
Examples
Polymers
Creep can occur in polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s and metals which are considered viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
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 Hookean spring and a Newtonian dashpot
A dashpot, also known as a damper, is a mechanical device that resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly us ...
in parallel. The creep strain is given by the following convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
integral:
:
where ''σ'' is applied stress, ''C''0 is instantaneous creep compliance, ''C'' is creep compliance coefficient, ''τ'' is retardation time, and ''f''(''τ'') is the distribution of retardation times.
When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain. This phenomenon is known as viscoelastic creep.
At a time ''t''0, 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''1 at which the stress is relieved, at which time the strain immediately decreases (discontinuity) then continues decreasing gradually to a residual strain.
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.[Rosato, D. V. ''et al''. (2001) ''Plastics Design Handbook''. Kluwer Academic Publishers. pp. 63–64. .] Below its critical stress, the viscoelastic creep modulus is independent of the 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.
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.
Both polymers and metals can creep. Polymers experience significant creep at temperatures above around ; however, there are three main differences between polymeric and metallic creep. In metals, creep is not linearly viscoelastic, it is not recoverable, and it is only present at high temperatures.
Polymers show creep basically in two different ways. At typical work loads (5% up to 50%) ultra-high-molecular-weight polyethylene
Ultra-high-molecular-weight polyethylene (UHMWPE, UHMW) is a subset of the thermoplastic polyethylene. Also known as high-modulus polyethylene, (HMPE), it has extremely long chains, with a molecular mass usually between 3.5 and 7.5 million amu. T ...
(Spectra, Dyneema
Ultra-high-molecular-weight polyethylene (UHMWPE, UHMW) is a subset of the thermoplastic polyethylene. Also known as high-modulus polyethylene, (HMPE), it has extremely long chains, with a molecular mass usually between 3.5 and 7.5 million amu. T ...
) will show time-linear creep, whereas polyester
Polyester is a category of polymers that contain the ester functional group in every repeat unit of their main chain. As a specific material, it most commonly refers to a type called polyethylene terephthalate (PET). Polyesters include natural ...
or aramid
Aramid fibers, short for aromatic polyamide, are a class of heat-resistant and strong synthetic fibers. They are used in aerospace and military applications, for ballistic-rated body armor fabric and ballistic composites, in marine cordage, ma ...
s (Twaron
Twaron (a brand name of Teijin Aramid) is a para-aramid. It is a heat-resistant and strong synthetic fibre developed in the early 1970s by the Dutch company Akzo Nobel's division Enka BV, later Akzo Industrial Fibers. The research name of the para ...
, Kevlar
Kevlar (para-aramid) is a strong, heat-resistant synthetic fiber, related to other aramids such as Nomex and Technora. Developed by Stephanie Kwolek at DuPont in 1965, the high-strength material was first used commercially in the early 1970s a ...
) will show a time-logarithmic creep.
Wood
Wood is considered as an orthotropic material
In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength can b ...
, exhibiting different mechanical properties in three mutually perpendicular directions. Experiments show that the tangential direction in solid wood tend display a slightly higher creep compliance than in the radial direction. In the longitudinal direction, the creep compliance is relatively low and usually do not show any time-dependency in comparison to the other directions.
It has also been shown that there is a substantial difference in viscoelastic properties of wood depending on loading modality (creep in compression or tension). Studies have shown that certain Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
s gradually go from positive to negative values during the duration of the compression creep test, which does not occur in tension.
Concrete
The creep of concrete, which originates from the calcium silicate hydrate
Calcium silicate hydrate (or C-S-H) is the main product of the hydration of Portland cement and is primarily responsible for the strength in cement based materials (e.g. concrete).
Preparation
When water is added to cement, each of the compounds u ...
s (C-S-H) in the hardened Portland cement
Portland cement is the most common type of cement in general use around the world as a basic ingredient of concrete, mortar, stucco, and non-specialty grout. It was developed from other types of hydraulic lime in England in the early 19th c ...
paste (which is the binder of mineral aggregates), is fundamentally different from the creep of metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s as well as polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s. Unlike the creep of metals, it occurs at all stress levels and, within the service stress 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 microstresses in the nanoporous microstructure of the C-S-H. If concrete is fully dried it does not creep, though it is difficult to dry concrete fully without severe cracking.
Applications
Though mostly due to the reduced yield strength at higher temperatures, the collapse of the World Trade Center
The collapse of the World Trade Center occurred during the terrorist attacks of September 11, 2001, after the Twin Towers were struck by two hijacked commercial airliners. One World Trade Center (WTC 1, or the North Tower) was hit at 8:46 ...
was due in part to creep from increased temperature.
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.
Creep in epoxy anchor adhesive was blamed for the Big Dig tunnel ceiling collapse in Boston, Massachusetts, that occurred in July 2006.
The design of tungsten light bulb filaments attempts to reduce creep deformation. Sagging of the filament coil between its supports increases with time due to 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
A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains.
Bacillite is a type of crystallite. It is rodlike with parallel longulites.
Stru ...
grain boundaries
In materials science, a grain boundary is the interface between two grains, or crystallites, in a polycrystalline material. Grain boundaries are two-dimensional crystallographic defect, defects in the crystal structure, and tend to decrease the ...
is used to slow the rate of Coble creep
Coble creep, a form of diffusion creep, is a mechanism for deformation of crystalline solids. Contrasted with other diffusional creep mechanisms, Coble creep is similar to Nabarro–Herring creep in that it is dominant at lower stress levels and h ...
.
Creep can cause gradual cut-through of wire insulation, especially when stress is concentrated by pressing insulated wire against a sharp edge or corner. Special creep-resistant insulations such as Kynar (polyvinylidene fluoride
Polyvinylidene fluoride or polyvinylidene difluoride (PVDF) is a highly non-reactive thermoplastic fluoropolymer produced by the polymerization of vinylidene difluoride.
PVDF is a specialty plastic used in applications requiring the highest pur ...
) are used in wirewrap
Wire wrap is an electronic component assembly technique that was invented to wire telephone crossbar switches, and later adapted to construct electronic circuit boards. Electronic components mounted on an insulating board are interconnected by l ...
applications to resist cut-through due to the sharp corners of wire wrap terminals. Teflon insulation is resistant to elevated temperatures and has other desirable properties, but is notoriously vulnerable to cold-flow cut-through failures caused by creep.
In steam turbine power plants, pipes carry steam at high temperatures () and pressures (above ). In jet engines, temperatures can reach up to and initiate creep deformation in even advanced-design coated turbine blades. Hence, it is crucial for correct functionality to understand the creep deformation behavior of materials.
Creep deformation is important not only in systems where high temperatures are endured such as nuclear power plants, jet engines and heat exchangers, 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
The glass–liquid transition, or glass transition, is the gradual and reversible transition in amorphous materials (or in amorphous regions within semicrystalline materials) from a hard and relatively brittle "glassy" state into a viscous or rubb ...
around . 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, which resulted in inconsistent thickness.
Fractal geometry, using a deterministic Cantor structure, is used to model the surface topography, where recent advancements in thermoviscoelastic creep contact of rough surfaces are introduced. Various viscoelastic idealizations are used to model the surface materials, including the Maxwell, Kelvin–Voigt, standard linear solid and Jeffrey models.
Nimonic 75 has been certified by the European Union as a standard creep reference material.
The practice of tinning stranded wire
Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample diameter 40 mm
A wire is a flexible strand of metal.
Wire is co ...
s to facilitate the process of connecting the wire to a screw terminal
A screw terminal is a type of electrical connection where a wire is held by the tightening of a screw.
Description
The wire may be wrapped directly under the head of a screw, may be held by a metal plate forced against the wire by a screw, or ...
, though having been prevalent and considered standard practice for quite a while, has been discouraged by professional electricians, owing to the fact that the solder
Solder (; NA: ) is a fusible metal alloy used to create a permanent bond between metal workpieces. Solder is melted in order to wet the parts of the joint, where it adheres to and connects the pieces after cooling. Metals or alloys suitable ...
is likely to creep under the pressure exerted on the tinned wire end by the screw of the terminal, causing the joint to lose tension and hence create a loose contact over time. The accepted practice when connecting stranded wire to a screw terminal is to use a wire ferrule on the end of the wire.
Prevention
Generally, materials have better creep resistance if they have higher melting temperatures, lower diffusivity, and higher shear strength. Close-packed
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
structures are usually more creep resistant as they tend to have lower diffusivity than non-close-packed structures. Common methods to reduce creep include:
* Solid solution strengthening
In metallurgy, solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal. The technique works by adding atoms of one element (the alloying element) to the crystalline lattice of another element (t ...
: adding other elements in solid solution can slow diffusion, as well as slow dislocation motion via the mechanism of solute drag.
* Particle dispersion strengthening: adding particles, often incoherent oxide or carbide particles, block dislocation motion.
* Precipitation hardening
Precipitation hardening, also called age hardening or particle hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel, titanium, and so ...
: precipitating a second phase out of the primary lattice blocks dislocation motion.
* Grain size: increasing grain size decreases the amount of grain boundaries, which results in slower creep due to the high diffusion rate along grain boundaries. This is opposite low-temperature applications, where increasing grain size decreases strength by blocking dislocation motion. In very high temperature applications such as jet engine turbines, single crystals are often used.
Superalloys
Materials operating in high-performance systems, such as jet engines, often reach extreme temperatures surpassing , necessitating specialized material design. Superalloys based on cobalt
Cobalt is a chemical element with the symbol Co and atomic number 27. As with nickel, cobalt is found in the Earth's crust only in a chemically combined form, save for small deposits found in alloys of natural meteoric iron. The free element, pr ...
, nickel
Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow to ...
, and iron
Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in f ...
have been engineered to be highly resistant to creep. The term ‘superalloy’ generally refers to austenitic nickel-, iron-, or cobalt-based alloys that use either γ′ or γ″ precipitation strengthening to maintain strength at high temperature.
The γ′ phase is a cubic L12-structure Ni3(Al,Ti,Ta,Nb) phase that produces cuboidal precipitates. Superalloys often have a high (60–75%) volume fraction of γ′ precipitates. γ′ precipitates are coherent with the parent γ phase, and are resistant to shearing due to the development of an anti-phase boundary when the precipitate is sheared. The γ″ phase is a tetragonal Ni3Nb or Ni3V structure. The γ″ phase, however, is unstable above , so γ″ is less commonly used as a strengthening phase in high temperature applications. Carbides are also used in polycrystalline superalloys to inhibit grain boundary sliding.
Many other elements can be added to superalloys to tailor their properties. They can be used for solid solution strengthening, to reduce the formation of undesirable brittle precipitates, and to increase oxidation or corrosion resistance. Nickel-based superalloys have found widespread use in high-temperature, low stress applications. Iron-based superalloys are generally not used at high temperatures as the γ′ phase is not stable in the iron matrix, but are sometimes used at moderately high temperatures, as iron is significantly cheaper than nickel. Cobalt-based γ′ structure was found in 2006, allowing the development of cobalt-based superalloys, which are superior to nickel-based superalloys in corrosion resistance. However, in the base (cobalt–tungsten–aluminum) system, γ′ is only stable below , and cobalt-based superalloys tend to be weaker than their Ni counterparts.
Contributing factors in creep resistivity
1. Stages of creep
Based on the description of creep mechanisms and its three different stages mentioned earlier, creep resistance generally can be accomplished by using materials in which their tertiary stage is not active since, at this stage, the strain rate increases significantly by increasing stress. Therefore, a sound component design should satisfy the primary stage of creep, which has a relatively high initial creep rate that decreases with increasing exposure time, leading to the second stage of creep, in which the creep rate in the material is decelerated and reaches its minimum value via work hardening. The minimum value of creep rate is actually a constant creep rate, which plays a crucial role in designing a component, and its magnitude depends on temperature and stress. The minimum value of creep rate that is commonly applied to alloys is based on two norms: (1) the stress required to produce a creep rate of and (2) the stress required to produce a creep rate of , which takes roughly about 11.5 years. The former standard has widely been used in the component design of turbine blades, while the latter is frequently used in designing steam turbines. One of the primary goals of creep tests is to determine the minimum value of the creep rate at the secondary stage and also to investigate the time required for an ultimate failure of a component. However, when it comes to ceramic applications, there are no such standards to determine their minimum creep rates, but it's safe to say that ceramics are usually picked up to operate at high-temperature applications under loads mainly because they possess a long lifetime. However, by acquiring information obtained from creep tests, proper ceramic material(s) can be selected for desired application to assure the safe service and evaluate the time period of secure service in high-temperature environments that the structural thermostability is essential. Therefore, by making the proper choice, suitable ceramic components may be selected, capable of operating at various conditions of high temperature and creep deformation.
2. Materials selection
Generally speaking, the structure of materials is different from one another. Metallic materials have different structures compared to polymer or ceramics, and even within the same class of materials, different structures might have existed at different temperatures. The difference in the structure includes the difference in grains (for example, their sizes, shapes, distributions), their crystalline or amorphous nature, and even dislocation and/or vacancy contents that are prone to change following a deformation. So, since creep is a time-dependent process and differs from one material to another, all these parameters must be considered in materials selection for a specific application. For instance, materials with low-dislocation content are among the suitable candidates for creep resistance. In other words, the dislocation glide and climb can be reduced if the proper material is selected (for example, ceramic materials are very popular in this case). In terms of vacancies, it's noteworthy to mention that the vacancy content not only depends on the chosen material but also on the component's service temperature. Vacancy-diffusion-controlled processes that promote creep can be categorized into grain boundary diffusion (Coble creep) and lattice diffusion (Nabarro–Herring creep). Therefore, the properties of dislocations and vacancies, their distributions throughout the structure, and their potential change due to long-time exposure to stress and temperature must be considered seriously in materials selection for components design. Therefore, the role of dislocations, vacancies, wide range of obstacles that can retard the dislocation motion, including grain boundaries in polycrystalline materials, solute atoms, precipitates, impurities, and strain fields originating from other dislocations or their pile-ups which increase the lifetime of materials and make them more resistive with respect to creep are always at the top list of materials selection and component design. For instance, different types of vacancies in ceramic materials have different charges stemming from their dominant chemical boding. So, existing or newly-formed vacancies must be charge-balanced to maintain the overall neutrality of the final structure. Besides paying attention to individual dislocation and vacancy contents, the correlation between them is also worth exploring since dislocation's ability to climb depends heavily on how many vacancies are available in its vicinity. To recap, materials must be selected and developed that possess low dislocation and vacancy contents to have a practical creep resistance component.
3. Various working conditions
● Temperature
Creep is a phenomenon that is related to a material's melting point (Tm). Generally, a high lifetime is expected when we have a higher melting point, and the reason behind selecting materials with high melting points originates from the fact that diffusion processes are related to temperature-dependent vacancy concentrations. The diffusion rate is slower in high-temperature materials. Ceramic materials are well known for having high melting points, and that's the reason why ceramics gravitated considerable attention to creep resistance applications. Although it's widely known that creep starts at a temperature equal to 0.5Tm, the safe temperature to avoid the initiation of creep is 0.3Tm. Creep that starts below or at 0.5Tm is called "low temperature creep" because diffusion is not very progressive at such low temperatures, and the kind of creep that occurs is not diffusion-dominant and is related to other mechanisms.
● Time
As mentioned previously, creep is a time-dependent deformation. Fortunately, creep doesn't occur suddenly in brittle materials as it does under tension and other forms of deformation, and it's an advantage for designers. Over time, creep strain develops in a material that is exposed to stress at the temperature of the application, and it depends on the duration of the exposure. Thus, besides temperature and stress, the creep rate is also a function of time, and it can be generalized as this function ε = F(t, T, σ) that tells the designer all the three parameters, including time, temperature, and stress acting in concert, and all of them must be considered if a successful creep-resistance component is to be attained.
See also
* Biomaterial
A biomaterial is a substance that has been engineered to interact with biological systems for a medical purpose, either a therapeutic (treat, augment, repair, or replace a tissue function of the body) or a diagnostic one. As a science, biomateria ...
* Biomechanics
Biomechanics is the study of the structure, function and motion of the mechanical aspects of biological systems, at any level from whole organisms to organs, cells and cell organelles, using the methods of mechanics. Biomechanics is a branch of ...
* Ductile–brittle transition temperature in materials science
* Deformation mechanism
A deformation mechanism, in geology, is a process occurring at a microscopic scale that is responsible for changes in a material's internal structure, shape and volume. The process involves planar discontinuity and/or displacement of atoms from th ...
* Downhill creep
Downhill creep, also known as soil creep or commonly just creep, is a type of creep characterized by the slow, downward progression of rock and soil down a low grade slope; it can also refer to slow deformation of such materials as a result of p ...
* Hysteresis
Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...
* Larson–Miller parameter
* Stress relaxation In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing some ...
* Viscoelasticity
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
* Viscoplasticity
Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The ine ...
* Brittle–ductile transition zone
__NOTOC__
The brittle-ductile transition zone (hereafter the "transition zone") is the zone of the Earth's crust that marks the transition from the upper, more brittle crust to the lower, more ductile crust. For quartz and feldspar-rich rocks in c ...
References
Further reading
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Elasticity (physics)
Materials degradation
Deformation (mechanics)
Rubber properties