Creep (deformation)

Deformation mechanism maps provide a visual tool categorizing the dominant deformation mechanism as a function of homologous temperature, 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 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.[6]

At high stresses (relative to the shear modulus), creep is controlled by the movement of dislocations. For dislocation creep, Q = Q(self diffusion), m = 4–6, and b is less than 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:

${\displaystyle {\frac {\mathrm {d} \varepsilon }{\mathrm {d} t}}=A\left(\sigma -\sigma _{\rm {th}}\right)^{m}e^{\frac {-Q}{{\bar {R}}T}}}$

where A, Q and m can all be explained by conventional mechanisms (so 3 ≤ m ≤ 10), R is the gas constant. 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.

A diagram showing the diffusion of atoms and vacancies under Nabarro–Herring Creep.

Nabarro–Herring (NH) creep is a form of diffusion creep, 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 ±${\displaystyle \sigma \Omega }$, where ${\displaystyle \Omega }$ is the atomic volume, the "${\displaystyle +}$" sign is for compressive regions and "${\displaystyle -}$" sign is for tensile regions. Since the fractional vacancy concentration is proportional to ${\textstyle \exp(-{\frac {Q_{f}\pm \sigma \Omega }{RT}})}$, where ${\textstyle Q_{f}}$ 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 tension to the regions under compression, 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) to the nearby vacant site (another potential well). The general form of the diffusion equation is ${\displaystyle D=D_{0}e^{\frac {E}{KT}}}$ where D₀ 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.

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) < 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 is one kind of mechanism for power law creep (PLC), involving both dislocation and diffusional flow. Solute drag creep is observed in certain metallic alloys. Their creep rate increases during the first stage of creep before a steady-state, which can be explained by a model associated with solid-solution strengthening. The size misfit between solute atoms and edge dislocations could restrict dislocation motion. The flow stress required for dislocations to move is increased at low temperatures due to immobility of the solute atoms. But solute atoms are mobile at higher temperatures, so the solute atoms could move along with edge dislocations as a "drag" on their motion, if the dislocation motion or the creep rate is not too high. The solute drag creep rate is:

${\displaystyle {\frac {d\varepsilon }{dt}}=C{\frac {D_{\rm {sol}}\sigma ^{3}}{\varepsilon _{b}^{2}c_{0}}}}$

where C is a constant, D_sol is the solute diffusivity, ${\displaystyle c_{0}}$ is the solute concentration, and ${\displaystyle \varepsilon _{b}}$ is the misfit parameter, ${\displaystyle \sigma }$ is the applied stress. So it could be seen from the equation above, m is 3 for solute drag creep. Solute drag creep shows a special phenomenon, which is called the Portevin-Le Chatelier effect. 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 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:

${\displaystyle {\frac {d\varepsilon }{dt}}={\frac {A_{\rm {CG}}D_{\rm {L}}}{M^{1/2}}}\left({\frac {\sigma \Omega }{kT}}\right)^{4.5}}$

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, ${\displaystyle \sigma }$ is the applied stress, and ${\displaystyle \Omega }$ 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 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 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.[8] The same was also confirmed in 1972 by Barrett and co-workers[9] 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.

However, 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:

${\displaystyle {\frac {\mathrm {d} \varepsilon }{\mathrm {d} t}}=\rho _{0}{\frac {D_{v}Gb^{3}}{kT}}\left({\frac {\sigma _{s}^{n}}{G}}\right)}$

where: ${\displaystyle \rho _{0}}$ is dislocation density (constant for Harper–Dorn creep), ${\displaystyle D_{v}}$ is the diffusivity through the volume of the material, ${\displaystyle G}$ is the shear modulus and ${\displaystyle b}$ is the Burger's vector, ${\displaystyle \sigma _{s}}$,and n is the creep exponent which varies between 1 and 3.[10]

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.[11][12] Unlike in other creep mechanisms, the dislocation density here is constant and independent of the applied stress.[8] Moreover, 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.[13] According to a recent study,[10] the extremely low dislocation density in a single crystalline material will result in the natural power law (n~3). On the other hand the n might vary between 1 and 3 based on the initial dislocation densities of crystals which is slightly higher.[14]

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.[15]

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 r₁ and r₂, the sintering limit stress is:[16]

${\displaystyle \sigma _{\rm {sint}}={\gamma \over r_{1}}+{\gamma \over r_{2}}}$

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.[17] 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:[18][19]

${\displaystyle {\dot {\rho }}={3A \over 2}{\rho (1-\rho ) \over [1-(1-\rho )^{1 \over n}]^{n}}\left({3 \over 2}{P_{e} \over n}\right)^{n}}$

In which ${\displaystyle {\dot {\rho }}}$ is the densification rate, ${\displaystyle \rho }$ is the density, ${\displaystyle P_{e}}$ is the pressure applied, ${\displaystyle n}$ 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:

${\displaystyle {\dot {\epsilon }}=A\sigma ^{n}}$

Where ${\displaystyle {\dot {\epsilon }}}$ is the strain rate, and ${\displaystyle \sigma }$ is the tensile stress. For the purposes of this mechanism, the constant ${\displaystyle A}$ comes from the following expression, where ${\displaystyle A'}$ is a dimensionless, experimental constant, μ is the shear modulus, ${\displaystyle b}$ is the Burgers' Vector, ${\displaystyle k}$ is Boltzmann's Constant, ${\displaystyle T}$ is absolute temperature, ${\displaystyle D_{0}}$ is the diffusion coefficient, and ${\displaystyle Q}$ is the diffusion activation energy:[18]

${\displaystyle A=A^{'}{D_{0}\mu b \over kT}\exp \left(-{Q \over kT}\right)}$

a) Applied stress and b) induced strain as functions of time over a short period for a viscoelastic material.

Creep can occur in polymers and metals which are considered 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 Hookean spring and a Newtonian dashpot in parallel. The creep strain is given by the following convolution integral:

${\displaystyle \varepsilon (t)=\sigma C_{0}+\sigma C\int _{0}^{\infty }f(\tau )(1-\exp[-t/\tau ])\,\mathrm {d} \tau }$

where:

• σ = applied stress
• C₀ = instantaneous creep compliance
• C = creep compliance coefficient
• ${\displaystyle \tau }$ = retardation time
• ${\displaystyle f(\tau )}$ = 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 t₀, 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 t₁ 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.[20] 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.

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.[21]

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.[22]

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 aramids (Twaron, Kevlar) will show a time-logarithmic creep.

Wood is considered as an orthotropic material, 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.[23][24] 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 has shown that certain Poisson's ratios gradually go from positive to negative values during the duration of the compression creep test, which does not occur in tension.[23]

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 as well as polymers. 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 micro-stresses in the nano-porous 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.

Materials operating at high temperatures, such as this nickel superalloy jet engine (RB199) turbine blade, must be able to withstand creep present at these temperatures.

Materials operating in high-performance systems, such as jet engines, often reach extreme temperatures surpassing 1000 °C, necessitating specialized material design. Superalloys based on Co, Ni, and Fe have been engineered to be highly resistant to creep. The term ‘superalloy’ generally refers to austenitic Ni, Fe, or Co - 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.[32] 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 650 °C, 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.[33]

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. Ni-based superalloys have found widespread use in high-temperature, low stress applications. Fe-based superalloys are generally not used at high temperatures as the γ’-phase is not stable in the Fe matrix, but are sometimes used at moderately high temperatures, as iron is significantly cheaper than nickel. Co-based γ’ structure was found in 2006, allowing the development of Co-based superalloys, which are superior to Ni-based superalloys in corrosion resistance. However, in the base (Co-W-Al) system, γ’ is only stable below 900 °C, and Co-based superalloys tend to be weaker than their Ni counterparts.[34]