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On completion of this tutorial package, you should:
It may be helpful to complete the TLPs Introduction to Dislocations and Introduction to Mechanical Testing, although these go into more detail than is necessary for the purposes of this TLP.
When a material is subjected to a stress that is greater than or equal to its yield stress, the material deforms plastically. When the stress is below this level, then in principle it should only deform elastically.
However, provided the temperature is relatively high (see later for the meaning of this), plastic deformation can occur even when the stress is lower than the yield stress. This deformation is time-dependent and is known as creep.
During loading under a constant stress, the strain often varies as a function
of time in the manner shown below:
This TLP focuses primarily on steady-state creep. In practice, this often dominates the creep behaviour – for example, the period during which it occurs is usually much greater than those for primary or tertiary creep.
There are two broad mechanisms by which steady state creep takes place: diffusion creep and dislocation creep.
Diffusion creep occurs by transport of material via diffusion of atoms within a grain. Like all diffusional processes, it is driven by a gradient of free energy (chemical potential), created in this case by the applied stress. For example, an applied tensile stress creates regions of high hydrostatic tension at the extremities of each grain, along the loading direction. In what might be termed the “equatorial” regions of the grain, the hydrostatic stress is lower. Since atoms have a lower free energy in these “polar” regions of high hydrostatic stress (ie “low pressure” regions), they will tend to diffuse towards such regions and this motion will lead to elongation of the grain along the loading direction. Since this occurs on the scale of the individual grains, diffusion distances are shorter in fine-grained materials, which thus tend to be more susceptible to creep.
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There are two types of diffusion creep, depending on whether the diffusion paths are predominantly through the grain boundaries, termed Coble creep (favoured at lower temperatures) or through the grains themselves, termed Nabarro-Herring creep (favoured at higher temperatures).
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
Dislocation creep is a mechanism involving motion of dislocations. This mechanism of creep tends to dominate at high stresses and relatively low temperatures.
Dislocations can move by gliding in a slip plane, a process requiring little thermal activation.
This is discussed in the Introduction to Dislocations TLP.
However, the rate-determining step for their motion is often a climb process, which requires diffusion and is thus time-dependent and favoured by higher temperatures. Obstacles in the slip plane, such as other dislocations, precipitates or grain boundaries, can lead to such situations.
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
Diffusion is governed by an Arrhenius equation:
Since all mechanisms of steady-state creep are in some way dependent on diffusion, we expect that creep rate will have this exponential dependence on temperature
Creep occurs faster at higher temperatures. However, what constitutes a high temperature is different for different metals. When considering creep, the concept of an homologous temperature is useful.
The homologous temperature is the actual temperature divided by the melting point of the metal, with both being expressed in K. In general, creep tends to occur at a significant rate when the homologous temperatures is 0.4 or higher.
The applied stress provides a driving force for dislocation movement and diffusion of atoms. As the stress is increased, the rate of deformation also increases. In general, it is found that
where n is termed the stress exponent. Prediction of the value of n from first principles is not easy, but its value does depend on which mechanism of creep is operating. For example, for diffusion creep its value is approximately 1, while for dislocation creep it is usually in the range 3-8.
The equation governing the rate of steady state creep is:
Q = activation energy; n = stress exponent; A = constant;
This can be rearranged into the form: 
The activation energy Q can be determined experimentally, by plotting the natural log of creep rate against the reciprocal of temperature.
The stress exponent n can be determined by plotting the strain rate as a function of stress.
In order to determine the value of n from a single experiment, it is necessary to have a range of stress levels acting within a single specimen. This is achieved by making the sample into a coil. The stress is provided by the weight of the coil itself, so that the upper part of the coil experiences more stress than the lower parts.
The stress in a particular turn of the coil is proportional to its number, N, where the turns are numbered beginning from the bottom turn and ending at the top. The shear stress τ in each turn varies from zero at the centre of the turn (axis of the coil) to a maximum value at the edge of the coil, given by:
The coil is then allowed to creep over a fixed amount of time (e.g. one minute) and at the end of this time the spacings, s, between the turns are measured giving information about the dependence of strain rate on stress.
The local shear strain γ in each turn is given by:
The average local strain rate is thus related to the spacing between turns, s, and the time, t, by:
It should be noted that, strictly, the above analysis applies only while the material remains elastic. As with all cases in which a moment (bending or twisting) is applied, such that the stress distribution is non-uniform, the situation becomes more complex after the onset of plastic deformation. The distribution of strain remains linear along the radius of the wire, but the associated distribution of stress tends to become more complex. For the creeping coil geometry, papers have been published covering various aspects – eg see IG Crosland et al, “The Use of Helically Coiled Springs in Creep Experiments with Special Reference to the Case of Bingham Flow”, J. Phys. D: Applied Physics, vol.6 (1973) p.1040-1046, and Measurements of Creep at High Temperatures using Helical Springs, FD Boardman et al, J. Strain Analysis, vol. 1 (1966) p.140-144. In fact, provided primary creep can be ignored (which is often doubtful), the procedure described here should be reasonably accurate as a method of estimating the stress exponent, as long as it is of the order of unity, ie not very large.
In order to create a temperature-controlled environment, the coil is placed inside a Perspex tube. Once the temperature in the tube has stabilised, the coil is allowed to creep under its own weight for one minute. The videos below show qualitatively how creep rate varies with temperature.
Creep deformation of a coil of solder at 28oC
View video of the demonstration (2.96 MB) ... in separate window ... video alone
Creep deformation of a coil of solder at 65oC
View video of the demonstration (3.72 MB) ... in separate window ... video alone
Creep deformation of a coil of solder at 85oC
View video of the demonstration (3.81 MB) ... in separate window ... video alone
.
The spacings between the turns of the coil can be obtained from photos, as here, or more simply by just rotating the support cylinder until it is horizontal (so that creep will stop) - spacings are then readily measured directly with a ruler (or on a piece of paper which has been held against the coil and imprints of the positions of the turns obtained by light use of a pencil - ie using a "brass rubbing" technique).
In the basic creep rate equation
substitution of previously-derived expressions for the strain rate within the wire
where C is a constant, and the average shear stress in the wire
leads to
and hence an expression for the velocity of the Nth turn of the coil
in which K is a constant. This equation can be used to obtain various parameters from experimental data. For example, by plotting ln(s/t) against ln(N), the value of the stress exponent, n, can be obtained as the gradient.
It can be seen that the calculated stress exponent is about 1, which is consistent with the fact that it is diffusion creep which is the main mechanism in these solder specimens over this temperature range. (Diffusion creep tends to predominate at relatively high homologous temperatures.)
Also, by plotting ln(s/t) against (1/T), for a given value of N, the activation energy Q can be found from the gradient (-Q/R).The photos below are taken after creeping for one minute. The spacings between the 11th and 12th coils (in mm) are marked.
It may be noted that the calculated activation energy of about 30 kJ mole-1 is approximately the value for atomic diffusion in solder, as expected.
To control the creep rate, vary the parameters T, Q and n using the sliders. The timer in the top right hand corner shows how much time has passed since start of creep. In this simulation, the real time duration is less than half a minute, but the simulation time elapsed may vary over a wide range, depending on the creep rate.
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
As a general rule, creep starts to become significant when the homologous temperature is greater than 0.4. Most metals do not suffer from creep at room temperature, since they have much higher melting points than solder. However, creep can still be a major concern when designing metallic components that have to function at high temperatures.
An example of one such engineering challenge is in the design of turbine blades for use in jet engines. The blades in these engines can be exposed to hot gases at up to about 1400°C. They are also under stress, as a result of the high centrifugal forces. These blades must withstand this environment without excessive creep, which would cause them to strike the turbine enclosure.
The materials used to make aero-engine turbine blades are nickel-based superalloys. These not only have relatively high melting points, reducing the homologous temperature, but also have microstructures designed to impart high creep resistance at any given homologous temperature. Furthermore, relatively cold (by-pass) air is ducted through channels in the blades, to help keep them cool, and they are sometimes also coated with insulating ceramic layers (thermal barrier coatings). These measures ensure that the blades are kept below 1000°C, even when the turbine entry temperature is as high as 1400°C. It can be seen from the creep rate information in the figure that this is essential if the blades are to perform adequately.
In order to obtain creep rates as low as those shown in the figure, considerable
evolution towards an optimal microstructure has been necessary. For example,
as creep behaviour became better understood, it became clear that grains should
be elongated in the direction of the applied load - in fact single crystals
give the best creep resistance. The grain structure development shown in the
figure therefore took place during the 1960s and 1970s.
DoITPoMS standard terms of use
However, the microstructure within a grain is actually quite complex. Superalloys usually have two distinct phases, γ and γ', with coherent interfaces and an orientation relationship between them. This leads to lattice strain and resistance to dislocation motion (through the γ phase), particularly when the γ' precipitates are fine - see the figure.
DoITPoMS standard terms of use
Steady-state creep occurs via atomic diffusion, either as a mass transport mechanism in itself (“diffusion creep”) or as a rate determining step in dislocation motion (“dislocation creep”). In diffusion creep, transport occurs on the scale of the grain, so coarser grains lead to slower creep rates. Like diffusion, the creep rate has an Arrhenius dependence on temperature. It also has a power-law dependence on stress. The value of the stress exponent indicates which mechanism of creep is acting. For diffusion creep, the exponent is around unity, while for dislocation creep it is usually higher (~3-8).
You should now be familiar with a particular experimental set-up to measure creep-rate as a function of stress and temperature. You should also understand how to use experimental data to calculate the stress exponent and the activation energy of creep.
This TLP has largely focused on the creep of solder, since this can be observed on a short timescale at quite low temperatures. It is important to realise that most metals do not experience creep at room temperature, but that it can be a problem at high temperatures even if the material is operating under stresses much lower than its yield stress.
Materials can be engineered to resist creep. Removing grain boundaries increases
the material’s resistance to diffusional creep. An array of finely spaced
precipitates can inhibit the motion of dislocations. These methods are used
in the manufacture of nickel-based superalloys for use in aero-engine turbine
blades.
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You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!
The gradient of which of these plots will give the stress exponent?
Which plot would you draw to calculate the activation energy?
Which of these grain structures would exhibit the highest resistance to diffusional creep?
The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.
Which of the following features of Ni superalloy turbine blades helps to make them resistant to creep? (There may be more than one)
Which of these components do you think is likely to be most susceptible to creep?
Diffusion is the movement of atoms between sites in a structure. There are two types: substitutional and interstitial.
Solute atoms that are much smaller than the parent atoms may occupy interstitial sites, and can jump between them.
An atom can also move through the lattice by jumping into an adjacent vacant site. This creates a new vacancy in the position previously occupied by the atom.
In both cases, in order to jump to a different site, the atom must pass through a region of high energy (surmount an energy barrier, q1) and must therefore possess a minimum (thermal) vibrational energy. The probability of this is given by a Boltzmann expression (exp(-q1/kT)). The jump frequency is therefore given by this expression times the vibration frequency (Debye frequency). In the case of a substitutional solute, the jump frequency is further reduced by a factor equal to the probability of the adjacent site being vacant, which is (exp(-q2/kT)), where q2 is the energy associated with a vacancy. The upshot of this is that, for both types of solute, the diffusion coefficient is given by an expression of the type
although q represents only the energy barrier (q1) in the case of interstitial solutes, whereas it also includes the effect of the need for an adjacent site to be vacant (q=q1+q2) in the case of substitutional solutes (and hence its value is always appreciably higher for such solutes). It’s also common to express q in molar terms, in which case it’s usually written as Q, and the Boltzmann constant k is replaced by the gas constant R.
F = total weight of coil = volume x density x acceleration due to gravity
Torsional shear stress, 
where r is the radial distance from the centre of the
wire (
)
and I is the second moment of area ( 
)
Therefore the shear stress varies from zero at the centre of the wire to a maximum value at its surface:
This small element of the wire subtends an angle dβ at the centre of the coil.
The maximum local strain in the material (at the surface of the wire) is:
where θL is the angular twist per unit length induced by the torque.
The torque acting on this element causes it to twist through an angle dθ.
Therefore, the twist per unit length is:
The vertical deflection of the coil due to the twisting of this element is:
The vertical deflection associated with the twist in one turn of the coil is:
The spacings between adjacent turns in the coil is thus related to the peak local shear strain in the wire by:
It may also be noted that the average local strain, γ, is proportional to the peak strain, so that:
Academic consultant: Bill Clyne (University of Cambridge)
Content development: Hannah Whiteoak and David Brook
Photography and video: Brian Barber and Carol Best
Web development: Lianne Sallows
DoITPoMS is funded by the UK Centre for Materials Education and the Department of Materials Science and Metallurgy, University of Cambridge
Additional support for the development of this TLP came from the Worshipful Company of Armourers and Brasiers'
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