Continuous formation of vacancies is required in order to maintain the cavity growth rate, and these vacancies form predominantly as a result of dislocation ingress at grain boundaries. This means that the vacancy formation is the rate-limiting step for the diffusional growth of creep cavities. The ingress of dislocations to a grain boundary can cause excess volume and stress concentrations to accumulate in the grain boundary.
The relaxation of the excess volume and the stress concentration on the grain boundary can happen by draining vacancies from the grain boundary to the creep cavities. In the present models that describe creep cavity growth [ 9 ], the implicit assumption is that the vacancy concentration remains at equilibrium values at a characteristic distance from the creep cavity at all times. It is not a priori obvious that this should be true. In fact, the concepts of Ishida and co-workers [ 12 , 36 ] that a grain boundary requires the ingress of dislocations in order to be able to slide can be combined with the proposal of Dyson [ 13 , 37 ] that grain boundary sliding is a constraint for the growth of creep cavities.
This means that the movement of dislocations, which controls the strain rate of metal that deforms under creep conditions, is also the rate-determining factor for cavity growth. This sheds some light on the Monkman—Grant relationship: the strain rate determines the time to failure by the formation of vacancies on the grain boundaries close to the diffusion zone of the creep cavities. When a dislocation network has developed and the steady-state strain rate causes a certain number of dislocations per second to reach a grain boundary, each of them carries an open volume [ 38 ], part of which is transferred to the grain boundary when the dislocation impinges.
For climb-controlled creep, the strain rate of a metal depends on the mobile dislocation density. The strain rate depends on the stress through the dislocation density [ 40 ] and the climb velocity [ 41 ].
The stress dependence of the dislocation climb velocity [ 42 ] can be approximated by:. During stage II creep with a constant strain rate, the average collective dislocation movement is of interest for the deformation rate. The drift velocity of the dislocation network can be correlated with the individual movements of dislocations [ 43 ]. The collective climbing or gliding rate of dislocations in a dislocation network is unknown, but as an approximation the individual movement can be considered.
The strain rate according to the Orowan equation Eq. This value is of similar magnitude compared to the values found by Caillard for single dislocation kink movement, in the presence of solute [ 41 ]. When a dislocation impinges on or near a grain boundary, it will provide a back stress on the following dislocations.
The character of a grain boundary is altered by the absorption of a dislocation and its associated volume [ 45 , 46 ]. This change in character, in the form of a stress concentration, provides a repulsive barrier for the influx of the next dislocation [ 47 ].
The increase in volume in the grain boundary leads to a more disordered structure and an excess vacancy concentration. It has been observed that the formation and growth of creep cavities are highly dependent on the grain boundary character of the surrounding grain boundaries [ 48 , 49 ].
We postulate that the relaxation of the excess volume and the stress concentration on certain grain boundaries can happen by draining vacancies from these grain boundary to the creep cavities. This flux of vacancies from a disordered section of grain boundary to the creep cavities leads to a less disordered grain boundary and allows new dislocation to ingress into the grain boundary.
This link of the grain deformation rate and the creep cavity growth rate causes the Monkman—Grant relation. Experimentally, it has been observed that the presence of supersaturated solute can result in an autonomous filling of creep cavities and a significant extension of the creep lifetime [ 25 — 27 ].
It is found that the self-healing mechanism does not significantly affect the critical strain at rupture, but does reduce the steady-state strain rate, as schematically illustrated in Fig. The solute that segregates at the free creep cavity surfaces is found to be transported along the grain boundaries from the supersaturated bulk. This flux of segregating solute competes with the vacancy flux and thereby reduces both the cavity growth rate and the vacancy flux away from grain boundaries under stress towards the creep cavities.
This process is known as the Kirkendall effect. Evolution of strain with time for a non-self-healing and a self-healing alloy. The time to failure is predominantly controlled by the strain rate in stage II. After nucleation, a diffusional growth of the precipitate initiates a flux of solute, driven by a chemical potential;. The difference in chemical potential of solute atoms between precipitation in the bulk and on creep cavity surface causes a preference for precipitate growth in the creep cavities.
The terms playing a role are the possibility for the precipitate to reduce the surface energy of the free surface of bulk material in the creep cavity, the possibility of reducing the surface energy of a precipitate, and the reduction in stress concentration between the precipitate and the bulk material.
The driving force for precipitation is then given by this chemical potential, but also by the supersaturated solute which remains in solution during service life. This is assumed to be the largest contribution to the self-healing process in metals, and it is measurable with atom probe tomography [ 25 ]; the solute is then depleted from the grain boundary and neighbouring bulk as a result of the diffusion towards the precipitate Fig.
The flux of vacancies through a grain boundary towards a creep cavity during stage II creep causes this creep cavity to grow. The net vacancy flux can be zero, preventing the creep cavity to grow.
The difference in diffusivity of host and substitutional solute causes a net diffusion of vacancies in the direction opposite to the faster species. The flux balance of this process can be approximated with the following Darken equation which, in the dilute limit, can be simplified to:. This approximation is valid in the dilute limit, with negligible off-diagonal terms of the Onsager matrix [ 50 ]. The opposite vacancy fluxes caused by the gradient in stress-induced chemical potential and by the solute gradient result in a net vacancy flux, either towards or from the cavity.
Self-healing can be achieved when. When the two fluxes are equal, a critical stress can be defined below which diffusional creep can be self-healed. Combining Eqs. The length l is the diffusion length of the supersaturated solute in the bulk towards the grain boundary.
Creep cavity growth rate can be estimated from the net vacancy flux integrated over the creep void area connecting the grain boundary:. The rate-limiting factor for the void growth is the formation of vacancies, which is linked to the strain rate.
For stage II creep where the supersaturated solute is transported exclusively to the creep cavities, it is possible to write the constrained cavity growth rate as:. The depletion of supersaturated solute from the bulk close to the grain boundaries is clearly observed by Zhang and co-workers [ 25 ].
This depleted zone points to a diffusion-controlled process. This proves that grain boundary sliding is not rate-limiting to the deformation.
Using Eq. The model is applied to the critical stress for the experimentally studied binary alloys: Fe—1at. The relevant part of the phase diagram between and K and between 0 and 4 at. The magnetic Curie temperature T C of pure iron is indicated for reference. All fractions indicated in the figure are in at. The grain boundary self-diffusivity of iron was measured over a wide temperature range [ 55 ]. The bulk diffusivities and the influences of magnetic ordering on their activation energy for the substitutional elements used in bcc iron are obtained from the manuscript of Versteylen and co-workers [ 56 , and references therein].
The diffusivity parameters, the vacancy concentrations, the volume of a vacancy, the thickness of a grain boundary, the considered creep cavity radius and spacing, and the applied stress that are used as modelling parameters to obtain the critical stresses for self-healing and the efficiency for self-healing are gathered in Table 1.
These model parameters are used to estimate the critical stress for self-healing of diffusional creep damage. The self-healing process in Fe—1at. At high temperatures, the efficiency drops quickly, which is caused by: 1 the decrease in amount of supersaturated solute available for self-healing and 2 the diffusivities of solute and host are getting closer to each other at high temperatures.
In addition, the activation energy for grain boundary diffusion shows a considerable temperature evolution close to the Curie temperature [ 55 ]. The nominal solute concentration in at. Molybdenum Mo and tungsten W are more soluble at high temperatures and therefore analysed for higher solute contents.
The mechanism that reduces the growth rate of the creep cavities also functions at stresses higher than the previously determined critical stress for self-healing. In Fig. The addition of molybdenum and tungsten in solid solution is common for creep steels [ 58 ] and is generally related to the formation of nanoprecipitates in creep steels. In a recent article by Fedoseeva and co-workers however, it was shown that a commercial alloy with added tungsten content loses creep strength after prolonged creep times, which coincides with the depletion of solute tungsten [ 59 ].
When molybdenum or tungsten is added in excess, keeping a percentage in solution, the solubility can extend to high temperatures. The temperature reach for self-healing can therefore be much higher than for copper or gold see Fig. The efficiency of self-healing is therefore analysed for higher nominal concentrations or molybdenum and tungsten see Fig.
The Fe—Au healing is indicated as a function of concentration between 0. The difference in bulk diffusivities between Cu and Au predicts that the self-healing process will work much more efficiently for Fe—Au than for Fe—Cu at the same degree of supersaturation. This is in concurrence with what was found in experiments [ 60 ]. As with plastic deformation, creep is a complex process that is strongly affected by the microstructure of the material. Some of the microstructural effects that influence plasticity are summarised in the TLP on Mechanical Testing.
As with plasticity, however, guidelines can be identified concerning features likely to affect inhibit creep, and some of these are similar for the two. For example, a fine array of precipitates , which will inhibit dislocation glide and hence raise the yield stress, is also likely to inhibit dislocation creep.
However, there are limits to such linkage. For example, precipitates might dissolve at the high temperatures involved in creep. More fundamentally, some features can affect creep and plasticity quite differently. For example, while a fine grain size tends to raise the yield stress, as a result of grain boundaries acting as obstacles to dislocation glide, it can cause accelerated creep in a diffusion-dominated regime, since such boundaries also constitute fast diffusion paths.
Nevertheless, as with plasticity, empirical constitutive laws can be used to model and predict creep behaviour. There is sometimes scope for interpreting the values of parameters in these laws in terms of the dominant mechanisms involved.
In particular, if rates of creep are measured over a range of temperature, then it may be possible to evaluate the activation energy , Q in an Arrhenius expression , which could in turn provide information about the type of diffusional process that is rate-determining. The theoretical basis for such conclusions may sometimes be questioned, and in any event these laws must be recognised as essentially empirical, but it is certainly important to be able to characterize the creep response of a material, and to be clear about the regime of temperature and stress for which a particular law is valid.
The creep strain rate rate of change of the von Mises plastic strain in the steady state Stage II regime is often written. This is a relatively simple equation, but several caveats should be added.
The most important of these are apparent in Fig. It is sometimes stated that the overall creep life is often dominated by this regime. In practice, this may or may not be true. Depending on a number of factors, simply ignoring primary creep may be highly inappropriate.
Plot of self-diffusion activation energy vs. It may be noted that the activation energy for creep at high temperatures is often found to agree closely with that for bulk diffusion - see, for example, the data in the figure. This is consistent with the concept of N-H creep diffusion through the lattice dominating Coble Creep grain boundary diffusion at high temperatures. In view of the potential importance of the primary regime, there is strong interest in using modelling approaches that incorporate it.
Several expressions have been proposed for capture of both primary and secondary regimes, and of the transition between them. One that can be taken as representative is the following equation, which is sometimes termed the Miller-Norton law. The simulation below, in which this equation is plotted, can be used to explore Miller-Norton creep strain plots as the 6 parameters involved are varied.
Simulation showing behaviour predicted by M-N law with constant true stress. A number of features should quickly become apparent, such as the high sensitivity to temperature and that the sensitivity to the applied stress increases as the value of n is raised. One issue here is whether the applied stress is a nominal or a true value.
It certainly should be a true value, as this is implicit in the M-N law. However, it is common during testing to fix the load often in the form of a dead weight , rather than the true stress. Also, most uniaxial creep tests tend to be carried out in tension. Neglecting any inhomogeneity that might arise, such as a necking effect - which is not common during creep testing - the true stress will rise as straining occurs and the cross-sectional area reduces.
Depending on the value of n , this could have the effect of causing the strain rate to rise whereas it would otherwise be falling and approaching a constant value. This effect is modelled in the simulation below. The M-N law can be differentiated with respect to time, to give. Therefore, by stepping in time and repeatedly re-evaluating the true stress, and hence the strain rate, the full creep strain curve can be built up although it can no longer be expressed as a single analytical equation.
In order to implement this, the relationship between true and nominal stresses is needed:. The above plot can therefore be modified using Eqns. This is done by stepping forward in small increments of time ie this is a numerical procedure, rather than just the plotting of an analytical equation.
The sequence is as follows. After an initial small time increment, true stress and true strain are taken to be equal to the nominal values.
For the next increment, the strain rate is obtained using Eqn. In order to use Eqn. This is obtained using Eqn. This operation is repeated after every time step, with the true stress progressively rising. In the simulation below, the stress selected is a nominal value. Depending on several factors particularly the value of n , a significant rise in the creep strain rate can be seen.
In the case of uniaxial compressive testing, the strain rate will tend to fall, due to a drop in true stress, and hence no such regime will be seen.
This figure compares the tensile creep strain plot of the standard M-N equation, using a constant nominal stress, with that obtained by stepping through time, repeatedly evaluating the true stress and then using that in the differential form of the equation.
It should also be noted that this analysis is based on the Miller-Norton equation being valid, over the range of stresses being considered. In fact, if the stress rises significantly, then this may not be the case. In particular, if the true stress starts to reach the yield stress, then conventional plasticity may be stimulated, which would probably be apparent in a creep strain plot as a sharply increasing strain rate.
For tensile testing, the gauge length must have a smaller sectional area than the region that is gripped, such that the latter undergoes only elastic deformation.
This is not the case for compressive testing, where the sample usually has a uniform section along its length, and is located between hard platens. For Creep Testing, however, there are additional challenges.
For example, it is commonly carried out at high temperature, so a furnace with good thermal stability is needed, and all of the sample must be held at the selected temperature.
Also, the load must be sustained for long periods - perhaps just a few hours, but in some cases periods of many weeks or even many months might be needed. Such conditions bring slightly different challenges from those of conventional stress-strain testing.
As for conventional testing, there is the issue of true or nominal versions of both stresses and strains. However, there is a difference with Creep Testing. For Stress-Strain Testing, the applied load is being ramped up continuously, so the issue reduces to that of converting any particular load level to a stress in the sample.
This chapter begins with a section on creep curves, covering the three distinct stages: primary, secondary, and tertiary. It then provides information on the stress-rupture test used to measure the time it takes for a metal to fail at a given stress at elevated temperature.
The major classes of creep mechanism, namely Nabarro-Herring creep and Coble creep, are then covered. The chapter also provides information on three primary modes of elevated fracture, namely, rupture, transgranular fracture, and intergranular fracture. The next section focuses on some of the metallurgical instabilities caused by overaging, intermetallic phase precipitation, and carbide reactions.
Subsequent sections address creep life prediction and creep-fatigue interaction and the approaches to design against creep. Sign In or Create an Account. User Tools. Luo A, Pekguleryuz MO. Cast magnesium alloys for elevated temperature applications. Mordike BL. Creep-resistant magnesium alloys. Creep resistant magnesium alloys for powertrain applications. Microstructure and creep behavior of high-pressure die-cast magnesium alloy AE A review of different creep mechanisms in Mg alloys based on stress exponent and activation energy.
Dieter, G. Mechanical Metallurgy , Vol. Creep-strengthening of steel at high temperatures using nano-sized carbonitride dispersoids. Ardell AJ. Precipitation hardening. Atomic pillar-based nanoprecipitates strengthened AlMgSi alloys. Nie JF. Precipitation and hardening in magnesium alloys.
The relationship between microstructure and creep resistance in die-cast magnesium-rare earth alloys. Role of Zn in enhancing the creep resistance of Mg-RE alloys. Enhanced age hardening response and creep resistance of Mg—Gd alloys containing Zn.
Solute segregation and precipitation in a creep-resistant Mg—Gd—Zn alloy. Periodic segregation of solute atoms in fully coherent twin boundaries. An a priori hot-tearing indicator applied to die-cast magnesium-rare earth alloys.
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Shewmon, P. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. Interactions between hydrogen impurities and vacancies in Mg and Al: a comparative analysis based on density functional theory. Interfacial structures and energetics of the strengthening precipitate phase in creep-resistant Mg-Nd-based alloys.
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