Creep - the plastic deformation of a material

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Introduction

When designing any system there are a number of mechanical behavior of materials questions that must be answered. Among them are fracture, yield, fatigue, stress corrosion cracking, and creep, just to scratch the surface. Some of these considerations, like fracture and yield, are well recognized mechanical behavior problems. The other phenomena are more subtle and happen over longer time periods. Creep is the plastic deformation of a material that is subjected to a stress below its yield stress when that material is at a high homologous temperature. Homologous temperature refers to the ratio of a materials temperature to its melting temperature. The homologous temperatures involved in creep processes are greater than 1/3.

Creep in general

Creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses. It occurs as a result of long term exposure to levels of stress that are below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods, and near the melting point. Creep always increases with temperature.

The term "creep" as applied to plasticity of materials of material likely arose from the observation that at modest and constant stress, at or even below the macroscopic yield stress of the metal (at a "conventional" strain rate).

Creep, of materials is classically associated with time-dependent plasticity under a fixed stress at an elevated temperature.

Stages of creep

Fig.1. Typical Creep Strain Curve for HP Alloys

· Primary creep: This is the deformation that occurs just after the load is applied. In this region, the curve is downward. This means the deformation rate is decreasing. During primary creep, the internal structure of the alloy is changing in response to the applied load.

· Secondary creep: There is often a stage where the slope of the creep curve remains approximately constant, like a straight line. This is the period of secondary creep (also called steady state creep). During secondary creep, the internal structure of the alloy remains approximately constant.

· Tertiary creep: At the end of secondary creep, the plot begins to curve upward. This signals the onset of failure for the alloy and is called tertiary (third stage) creep. During this period, small cavities begin to form and grow inside the alloy. Growth and inter-linkage of these cavities eventually lead to failure of the alloy.

When materials scientists study creep of metals and alloys, much more sophisticated experiments are usually conducted. The alloys are precisely machined into test specimens, the testing temperature is fully controlled, and the elongation of the material is recorded in detail. In addition, the samples are usually analyzed before and after creep testing to better understand the relationship between creep deformation and the internal structure of the material.

Characteristics of Creep

*Creep in service is usually affected by changing conditions of loading and temperature

*The number of possible stress-temperature-time combinations is infinite.

*The creep mechanisms is often different between metals, plastics, rubber, concrete.

Mechanisms of creep

There are three basic mechanisms that can contribute to creep in metals, namely:

(i) Dislocation slip and climb.

(ii) Grain boundary sliding.

(iii) Diffusional flow.

Dislocation slip and climb: Dislocations are line defects that slip through a crystal lattice when a minimum shear stress is applied. Dislocations initially slip along the closest packed planes in the closest packed directions since this requires the least energy or applied stress. An edge dislocation consists of an unfinished atomic plane, the edge of the plane being the line of the dislocation, and it is characterised by a Burgers vector, or 'misfit strain' at 90o to the dislocation line. Screw dislocations have a Burgers vector parallel to the dislocation line and can slip on any close packed plane containing both line and Burgers vector, whereas edge dislocations only slip on the plane defined by the line and the perpendicular Burgers vector. Figure 2. shows an edge dislocation slipping through a crystal lattice (simple cubic for simplicity) and producing a unit of slip called a slip step, on the surface of the crystal.

Figure 2: Showing slip of an edge dislocation.

Grain Boundary sliding: The onset of tertiary creep is a sign that structural damage has occurred in an alloy. Rounded and wedge shaped voids are seen mainly at the grain boundaries (Figure 3), and when these coalesce creep rupture occurs. The mechanism of void formation involves grain boundary sliding which occurs under the action of shear stresses acting on the boundaries. (Figures 4, 5 and 6).

Figure 3: Voids in creep ruptured Nimonic 80A. (1023 K and 154 MNm-2) X150

Evidence for grain boundary sliding is the displacement of scratch lines during creep testing. (Figure 21)

Figure 4: Showing scratch lines displaced across a grain boundary in aluminium.

Figure 5: A model for the formation of cracks due to grain boundary sliding.

Figure 6: The formation of wedge cracks during grain boundary sliding.

The more rounded voids are possibly produced where there is a step on the grain boundary which initially gives a square void as the grains slide, but this is rounded off by diffusion which reduces the surface area and energy of the void. The grain boundary sliding may account for 10% to 65% of the total creep strain, the contribution increasing with increasing temperature and stress and reducing grain size. Above about 0.6 TM the grain boundary region is thought to have a lower shear strength than the grains themselves, probably due to the looser atomic packing at grain boundaries. Boundaries lying at about 45o to the applied tensile stress experience the largest shear stress and will slide the most. Vacancy or atomic diffusion along the boundary is also easier, and may play a role in sliding since it has been observed that processes such as ordering and precipitation which makes slip within the grains more difficult, also decrease the grain boundary strain contribution by the same factor.

Diffusional Flow: The third distinct mechanism for creep is significant at low stress and high temperature. Under the driving force of the applied stress shown in Figure 24, atoms diffuse from the sides of the grains to the tops and bottoms. The grain becomes longer as the applied stress does work, and the process will be faster at high temperatures as there are more vacancies. (Atomic diffusion in one direction is the same as vacancy diffusion in the opposite direction). For diffusion paths through the grains the atoms have a slower jump frequency, but more paths, and is called Nabarro-Herring creep. Along the grain boundaries the jump frequency is higher, but fewer paths exist, and if this mechanism is called Coble creep. Needless to say, the rate controlling mechanism is again vacancy diffusion, or self-diffusion.

Figure 7: Diffusional flow.

General creep equation

where 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, у is the applied stress, d is the grain size of the material, k is Boltzmann's 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), m = 4-6, and b = 0. Therefore, dislocation creep has a strong dependence on the applied stress and no grain size dependence.

Some alloys exhibit a very large stress exponent (n > 10), and this has typically been explained by introducing a "threshold stress," у_th, below which creep can't be measured. The modified power law equation then becomes:

where A, Q and n can all be explained by conventional mechanisms (so 3 ? n ? 10).

Nabarro-Herring creep

Nabarro-Herring creep is a form of diffusion controlled creep. In Nabarro-Herring creep, atoms diffuse through the lattice causing grains to elongate along the stress axis; 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 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 = D₀exp(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 a 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. 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.

Creep of polymers

Fig.8. 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:

where:

· у = applied stress

· C₀ = instantaneous creep compliance

· C = creep compliance coefficient

· ф = retardation time

· f(ф) = 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. 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.

Both polymers and metals can creep. Polymers experience significant creep at all temperatures above ca. -200°C; however, there are three main differences between polymetric and metallic creep.

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. [1,2,3]

Creep of concrete

creep polymer molecular deformation

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, but it is next to impossible to dry concrete fully without severe cracking.

Application

Fig.9. Creep on the underside of a cardboard box: a largely empty box was placed on a smaller box, and fuller boxes were placed on top of it. Due to the weight, the portions of the empty box not sitting on the lower box gradually crept downward.

Though mostly due to the reduced yield strength at higher temperatures, the Collapse of the World Trade Center was due in part to creep from increased temperature operation.

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 was blamed for the Big Dig tunnel ceiling collapse in Boston, Massachusetts that occurred in July 2006.

An example of an application involving creep deformation is the design of tungsten light bulb filaments. Sagging of the filament coil between its supports increases with time due to creep deformation caused by 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 grain boundaries is used to slow the rate of Coble creep.

In steam turbine power plants, pipes carry steam at high temperatures (566°C or 1050°F) and pressures (above 24.1 MPa or 3500 psi). In jet engines, temperatures can reach up to 1400°C (2550°F) and initiate creep deformation in even advanced coated turbine blades. Hence, it is crucial for correct functionality to understand the creep deformation behavior of materials. [4]

Conclusion

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 around 500°C (900°F). 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, for example, Maxwell, Kelvin-Voigt, Standard Linear Solid and Jeffrey media.[5,6,7]

References

1. Jump up Rosato, D. V. et al. (2001) Plastics Design Handbook. Kluwer Academic Publishers. pp. 63-64. ISBN 0792379802.

2. Jump up M. A. Meyers and K. K. Chawla (1999). Mechanical Behavior of Materials. Cambridge University Press. p. 573. ISBN 978-0-521-86675-0.

3. Jump up^ McCrum, N.G, Buckley, C.P; Bucknall, C.B (2003). Principles of Polymer Engineering. Oxford Science Publications. ISBN 0-19-856526-7.

4. Jump up Zdenмk Baћant and Yong Zhu, Why Did the World Trade Center Collapse?--Simple Analysis,Journal of Engineering Mechanics, January 2002

5. Jump up Lakes, Roderic S. (1999). Viscoelastic Solids. p. 476. ISBN 0-8493-9658-1.

6. Jump up "Is glass liquid or solid?". University of California, Riverside. Retrieved 2008-10-15.

7. Jump up Osama Abuzeid, Anas Al-Rabadi, Hashem Alkhaldi. Recent advancements in fractal geometric-based nonlinear time series solutions to the micro-quasistatic thermoviscoelastic creep for rough surfaces in contact, Mathematical Problems in Engineering, Volume 2011, Article ID 691270

Bibliography

1. Ashby, Michael F.; Jones, David R. H. (1980). Engineering Materials 1: An Introduction to their Properties and Applications. Pergamon Press. ISBN 0-08-026138-8..

2. Frost, Harold J.; Ashby, Michael F. (1982). Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics. Pergamon Press. ISBN 0-08-029337-9..

3. Turner, S (2001). Creep of Polymeric Materials. Oxford: Elsevier Science Ltd. pp. 1813-1817. ISBN 0-08-043152-6

4. Grain-boundary sliding and diffusion creep in polycrystalline solids, DOI:10.1080/14786437108216383 R. N. Stevens^a pages 265-283

5. Diffusion creep, grain-boundary sliding and superplasticity pp. 194-212, Cambridge University Press, 1985

6. High-Temperature Deformation Processes in Metals, Ceramics and Minerals, Cambridge Earth Science series Mineralogy, petrology and volcanology, Materials Science 1985

7. Creep and Fatigue in Polymer Matrix Composites, Book, November 2010, by Guedes

8. Creep of Soils, Book, June 1992, by Feda, Creep-Resistant Steels, Book, March 2008, by Abe

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