Atomistic simulations of defects

7.5.2 Atomistic simulations of defects

While further breakthrough in the meso length scale and in bridging time scales is needed in order for the multiscale approach in Table 7.3 to work, CMS has been making important contributions in

Mechanical properties II – Strengthening and toughening 443

Sessile screw segment

Kink

a ⫽ atomic spacing (a)

(b)

Figure 7.42 (a) Differential displacement map of 1/2<1 1 1> screw dislocation in iron-like bcc metal, predicted by MS (Wen and Ngan, 2000). (b) Schematic showing the movement of the screw dislocation by the kink-pair mechanism (Li, Ngan and Gumbsch, 2003).

the understanding of material behaviors within the more traditional models or frameworks described earlier in this book, by providing reliable prediction of defect properties, such as stacking-fault ener- gies or activation energies for mobility, which are crucial input in these models. One area much investigated is dislocation core effects. In crystalline materials the yield properties are principally determined by the mobility of individual dislocations, rather than by their group behavior. While the elastic properties of a dislocation discussed in Section 3.3.5 determine its long-range interaction effects, the mobility of the dislocation is controlled by its ‘non-linear’ core (see equation (3.10)).

The geometry of the core governs whether the dislocation is mobile or not, and is determined by crystal symmetry as well as the energy of the planar fault(s) into which it is dissociated. Predicting the geometry of dislocation cores can be done by MS or MD methods. As an example, Figure 7.42a shows the core of screw dislocations in iron-like bcc materials computed by MS. In the differential displacement map shown in Figure 7.42a, the circles represent atoms viewed along the dislocation axis, and the length of each arrow drawn between a pair of atoms represents the magnitude of the relative displacement between the two atoms, due to the strain field of the dislocation. The Burgers vector content of the dislocation is evidently stored along three {1 0 1} planes separated by 120 ◦ , as schematized in Figure 7.42b, and so the ground-state configuration would be rather immobile or sessile. The dislocation can only move forward by forming a pair of kinks and propagating them in opposite directions along the dislocation line, as shown in Figure 7.42b. Since the enantiomer, or mirror-image form, of the configuration shown in Figure 7.41a can also exist, a variety of structurally different kinks can exist in the bcc structure, as shown in Figure 7.43a. However, MS calculations predict that all the other kinks have considerably higher energy than the ‘APB’ kink, and so only the

APB kink is favorable in practice. The only way to involve only the favorable APB kink in the dislo- cation’s motion is to have the dislocation changing path after each atomic step to an intersecting glide plane, as shown in Figure 7.43b. The zigzag motion of screw dislocations, known as ‘pencil glide’, is long thought to be responsible for the observation of wavy slip traces in deformed bcc metals. Such

a peculiar way of dislocation motion can only be interpreted by results from atomistic simulations.

A closely related application is to use atomistic simulations to predict energies of competing stack- ing faults in complicated structures. For example, dislocations in ordered structures (Section 3.6.5) may in principle dissociate into different faults, such as anti-phase domain boundaries, superlattice

444 Physical Metallurgy and Advanced Materials

B state

APB kink

Screw direction A state

A state

BPA kink

[121] B state

⫺ ve P-K force

A state

A (B)

APA kink B A (B)

A state

(a)

(b)

Figure 7.43 (a) Possible kink configurations in iron-like bcc (Li, Ngan and Gumbsch, 2003). (b) Zigzag motion of screw dislocation to involve only the lowest energy kink (Wen and Ngan, 2000).

intrinsic stacking faults or complex stacking faults. The understanding of the properties of structural intermetallics has benefitted enormously from accurate ab initio calculations of the various fault energies.

Problems

7.1 During age hardening of an aluminum alloy the maximum hardness could be achieved by ageing at 327 ◦

C. How long would it take at 257 ◦ C? If the alloy then contains precipitates 10 −7 m in diameter separated by 10 −6 m, estimate the tensile yield stress.

C for 10 hours or 280 hours at 227 ◦

7.2 In certain Al–Cu alloys, enhanced diffusion occurs following quenching from an elevated temperature. From the data given below for an Al–4 wt% Cu alloy, calculate the energies for formation, E f , and motion, E m , of vacancies by assuming that the activation energy for diffusion

C, show graphically how the factor ℜ, by which quenching increases diffusivity, varies with quenching temperature, T q , between 25 and 550 ◦ C.

in an annealed alloy equals E f +E m . For an alloy at 25 ◦

Diffusion coefficients of Cu in Al–4 wt% Cu:

T( ◦ C)

D (m 2 s −1 )

Heat treatment

Quenched from 500 ◦ C

Mechanical properties II – Strengthening and toughening 445

7.3 A steel is strengthened by a dispersion of chromium carbides of initial mean radius 0.25 µm. The operating temperature is to be 650 ◦

C and material strength is insufficient for particles in excess of 0.75 µm radius. Estimate the component lifetime if the interfacial energy is 500 mJ m −2 and coarsening is controlled by carbon diffusion. Solubility of carbon is 0.05 wt% in the alloy steel,

D o = 2 × 10 −5 m 2 s −1 ,Q = 130 kJ mol −1 for carbon diffusion, R = 8.31 J mol −1 K −1 . How would the lifetime be affected if Cr diffusion was limiting? Solubility of Cr is

20 wt%, D o = 1.6 × 10 −4 m 2 s −1 and Q = 240 kJ mol −1 for Cr diffusion, Boltzmann’s constant k = 1.38 × 10 −23 JK −1 .

7.4 The kinetics of the austenite-to-pearlite transformations obey the Avrami relationship f =1− exp (Kt n ) where f is the fraction transformed in time t and K , n are constants. Using the fraction transformed–time data given below, determine the total time required for 95% of the austenite to transform to pearlite:

Fraction transformed ( f ) Time (t)

0.2 280 s

0.6 425 s

7.5 In order to strengthen a ferrous alloy it is possible to refine the grain size or the second-phase dispersion. Which route would you choose and why?

7.6 A thick steel plate had a microcrack of 5 mm and a fracture toughness K c ∼ 40 MN m −3/2 . Determine the stress at which fast fracture takes place.

7.7 The crack growth rate equation in a steel component da/dN m has values of

a = 0.1 mm, c = 2 × 10 −13 (MN m −2 ) −4 m −1 ,m = 4 and K c = 54 MN m −3/2 . Calculate the life- time to failure if the component is subjected to an alternating stress from 0 to 180 MN m −2

7.8 Explain what is meant by the terms elastic stress concentration factor, as relating to a circular hole in a plate subjected to uniform tensile stress, and stress intensity factor, as relating to a sharp

crack similarly subjected to uniform stress. State the dimensions associated with these terms. (a) A cylindrical pressure vessel contains large circular openings into which nozzles are

welded. If the welding process induces cracks of length 10 mm, lying normal to the hoop stress, indicate how the critical pressure at which the vessel would fracture could

be estimated. (b) A thin steel sheet, of dimensions 1 m × 1 m, contains a central hole of diameter 20 mm and is coated with a brittle lacquer, which fractures at a tensile strain of 0.1%. If Young’s modulus for the steel is 210 GPa, explain carefully what you would observe as a stress applied to the top and bottom edges of the plate is increased to 75 MPa. How would the observations differ if the hole were 80 mm in diameter? A second, similar plate contains not a hole, but a central crack, of total length 80 mm oriented normal to the applied stress. If the applied stress is 75 MPa, estimate the distance ahead of the crack tip over which the lacquer would be observed to fracture.

Further reading

Ashby, M. F. and Jones, D. R. H. (1980). Engineering Materials – An introduction to their properties and applications. Pergamon. Bilby, B. A. and Christian, J. W. (1956). The Mechanism of Phase Transformations in Metals. Institute of Metals.

446 Physical Metallurgy and Advanced Materials Bowles and Barrett, Progress in Metal Physics, 3, 195. Pergamon Press.

Charles, J. A., Greenwood, G. W. and Smith, G. C. (1992). Future Developments of Metals and Ceramics. Institute of Materials, London. Honeycombe, R. W. K. (1981). Steels, Microstructure and Properties. Edward Arnold, London. Kelly, A. and MacMillan, N. H. (1986). Strong Solids. Oxford Science Publications, Oxford. Kelly, A. and Nicholson, R. B. (eds) (1971). Strengthening Methods in Crystals. Elsevier, New York. Knott, J. (1973). Fundamentals of Fracture Mechanics. Butterworths, London. Knott, J. F. and Withey, P. (1993). Fracture Mechanics, Worked Examples. Institute of Materials, London. Pickering, F. B. (1978). Physical Metallurgy and the Design of Steels. Applied Science Publishers, London. Porter, D.A. and Easterling, K. E. (1992). PhaseTransformations in Metals andAlloys, 2nd edn. Chapman &

Hall, London. Raabe, D. (1998). Computational Materials Science. Wiley-VCH, Weinheim.

Chapter 8

Advanced alloys