(
ε) as a function of time due to constant stress over an extended period for a Class M material 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, primary 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) • Grain boundary diffusion (
Coble creep) • Glide-controlled
dislocation creep: 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, 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. \frac{\mathrm{d}\varepsilon}{\mathrm{d}t} = \frac{C\sigma^m}{d^b} e^\frac{-Q}{kT} 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 of the creep mechanism,
σ is the applied stress,
d is the grain size of the material,
k is the
Boltzmann constant, and
T is the
absolute temperature.
Dislocation creep At high stresses (relative to the
shear modulus), creep is controlled by the movement of
dislocations. For dislocation creep,
Q =
Q(self diffusion), 4 ≤
m ≤ 6, and
b 10), and this has typically been explained by introducing a "threshold stress,"
σth, below which creep can't be measured. The modified power law equation then becomes: \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), and
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.
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 ±
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
Qf 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 D = D_0e^{\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 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. 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 defined as C_r = C_0 \exp\left(-\frac{\beta\sin\theta}{rKT}\right) where
C0 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 atoms can exert on the dislocation is given by Cottrell and Jaswon \frac{F_{\rm max}}{L} = \frac{C_0 \beta^2}{bkT} 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 v = B {\sigma^*}^m B =B_0 \exp\left(\frac{-Q_{\rm g}}{RT}\right) where
m is the
effective stress exponent,
Q is the apparent activation energy for glide and
B0 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
εa of solutes to be Solute drag creep sometimes shows a special phenomenon, over a limited strain rate, 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 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: \frac{\mathrm{d}\varepsilon}{\mathrm{d}t} = \frac{A_{\rm CG}D_{\rm L}}{\sqrt M}\left(\frac{\sigma\Omega}{kT}\right)^{4.5} where
ACG includes details of the dislocation loop geometry,
DL 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. 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 \frac{\mathrm{d}\varepsilon}{\mathrm{d}t} = \rho_0 \frac{D_{\rm v} G b^3}{k T} \left(\frac{\sigma_{\rm s}^n} G \right) where
ρ0 is dislocation density (constant for Harper–Dorn creep),
Dv is the diffusivity through the volume of the material,
G is the shear modulus and
b is the
Burgers vector,
σs, 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, Unlike in other creep mechanisms, the dislocation density here is constant and independent of the applied stress.
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
r1 and
r2, the sintering limit stress is \sigma_{\rm sint} = \frac{\gamma}{r_1}+\frac{\gamma}{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. 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: \dot{\rho}=\frac{3A}{2}\frac{\rho(1-\rho)}{\left(1-(1-\rho)^\frac1n\right)^n}\left(\frac32\frac{P_{\rm e}}{n}\right)^n in which
ρ̇ is the densification rate,
ρ is the density,
Pe 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, \dot{\varepsilon}=A\sigma^n 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 the Boltzmann constant,
T is absolute temperature,
D0 is the diffusion coefficient, and
Q is the diffusion activation energy: A = A'\frac{D_0 \mu b}{kT} \exp\left(-\frac{Q}{kT}\right) == Examples ==