Transient and steady-state creep

6.9.1 Transient and steady-state creep

Creep is the process by which plastic flow occurs when a constant stress is applied to a metal for a prolonged period of time. After the initial strain ε 0 which follows the application of the load, creep usually exhibits a rapid transient period of flow (stage 1) before it settles down to the linear steady- stage stage 2, which eventually gives way to tertiary creep and fracture. Transient creep, sometimes referred to as β-creep, obeys a t 1/3 law. The linear stage of creep is often termed steady-state creep and obeys the relation

ε = κt. (6.49) Consequently, because both transient and steady-state creep usually occur together during creep at

high temperatures, the complete curve (Figure 6.59) during the primary and secondary stages of creep fits the equation

ε = βt 1/3 + κt (6.50) extremely well. In contrast to transient creep, steady-state creep increases markedly with both

temperature and stress. At constant stress the dependence on temperature is given by ˙ε ss = dε/dt = const. exp[−Q/kT ],

(6.51) where Q is the activation energy for steady-state creep, while at constant temperature the dependence

on stress σ (compensated for modulus E) is ˙ε ss = const.(σ/E) n .

(6.52) Steady-state creep is therefore described by the equation:

˙ε n ss = A(σ/E) exp( −Q/kT ). (6.53)

362 Physical Metallurgy and Advanced Materials The basic assumption of the mechanism of steady-state creep is that during the creep process the

rate of recovery r (i.e. decrease in strength, dσ/dt) is sufficiently fast to balance the rate of work hardening h = (dσ/dε). The creep rate (dε/dt) is then given by

dε/dt = (dσ/dt)/(dσ/dε) = r/h. (6.54) To prevent work hardening, both the screw and edge parts of a glissile dislocation loop must be able

to escape from tangled or piled-up regions. The edge dislocations will, of course, escape by climb, and since this process requires a higher activation energy than cross-slip, it will be the rate-controlling process in steady-state creep. The rate of recovery is governed by the rate of climb, which depends on diffusion and stress such that

= A(σ/E) p D = A(σ/E) D

0 exp[ −Q/kT ],

where D is a diffusion coefficient and the stress term arises because recovery is faster the higher the stress level and the closer dislocations are together. The work-hardening rate decreases from the

initial rate h 0 with increasing stress, i.e. h =h 0 (E/σ) q , thus

˙ε n ss = r/h = B(σ/E) D, (6.55)

where B ( =A/h 0 ) is a constant and n ( =p + q) is the stress exponent.

The structure developed in creep arises from the simultaneous work hardening and recovery. The dislocation density ρ increases with ε and the dislocation network gets finer, since dislocation spacing is proportional to ρ −1/2 . At the same time, the dislocations tend to reduce their strain energy by mutual annihilation and rearrange to form low-angle boundaries, and this increases the network spacing. Straining then proceeds at a rate at which the refining action just balances the growth of the network by recovery; the equilibrium network size being determined by the stress. Although dynamical recovery can occur by cross-slip, the rate-controlling process in steady-state creep is climb, whereby edge dislocations climb out of their glide planes by absorbing or emitting vacancies; the activation energy is therefore that of self-diffusion. Structural observations confirm the importance of the recovery process to steady-state creep. These show that sub-grains form within the original grains and, with increasing deformation, the sub-grain angle increases while the dislocation density within

them remains constant. 6 The climb process may, of course, be important in several different ways. Thus, climb may help a glissile dislocation to circumvent different barriers in the structure, such as

a sessile dislocation, or it may lead to the annihilation of dislocations of opposite sign on different glide planes. Moreover, because creep-resistant materials are rarely pure metals, the climb process may also be important in allowing a glissile dislocation to get round a precipitate or move along a grain boundary. A comprehensive analysis of steady-state creep, based on the climb of dislocations, has been given by Weertman.

The activation energy for creep Q may be obtained experimentally by plotting ln ˙ε ss versus 1/T , as shown in Figure 6.60. Usually above 0.5T m , Q corresponds to the activation energy for self-diffusion

E SD , in agreement with the climb theory, but below 0.5T m ,Q<E SD , possibly corresponding to pipe diffusion. Figure 6.61 shows that three creep regimes may be identified and the temperature range where Q =E SD can be moved to higher temperatures by increasing the strain rate. Equation

6 Sub-grains do not always form during creep and in some metallic solid solutions where the glide of dislocations is restrained due to the dragging of solute atoms, the steady-state substructure is essentially a uniform distribution of

dislocations.

Mechanical properties I 363

41.36 MN m

68.94 MN m ⫺ 2

10 ⫺ 8 /s

34.47 MN m ⫺ 2 ⫺ 2 ⫺ 2

27.58 MN m

55.15 MN m 20.68 MN m ⫺ 2 41.36 MN m ⫺ 2

10 ⫺ 9 /s

(1000/T) K ⫺ 1 (a)

96.52 MN m ⫺ 2 82.73 MN m ⫺ 2 ⫺ 2 10 ⫺ 7 /s 137.90 MN m

.. ε Log

10 ⫺ 8 /s

24.13 MN m ⫺ 2 68.94 MN m ⫺ 2

20.68 MN m ⫺ 2 55.15 MN m ⫺ 2 13.79 MN m ⫺ 2

10 ⫺ 9 /s

(1000/T) K ⫺ 1 (b)

Figure 6.60 Log ˙ε versus 1/T for Ni–Al 2 O 3 (a) and Ni–67Co–Al 2 O 3 (b), showing the variation in activation energy above and below 0.5T m (after Hancock, Dillamore and Smallman, 1972; courtesy of Institute of Materials, Minerals and Mining).

0.25 0.5 T/T m

Figure 6.61 Variation in activation energy Q with temperature for aluminum.

(6.55) shows that the stress exponent n can be obtained experimentally by plotting ln ˙ε ss versus ln σ, as shown in Figure 6.62, where n ≈ 4. While n is generally about 4 for dislocation creep, Figure 6.63 shows that n may vary considerably from this value depending on the stress regime; at low stresses (i.e. regime I) creep occurs not by dislocation glide and climb, but by stress-directed flow of vacancies.

364 Physical Metallurgy and Advanced Materials

Ni–Al 10 ⫺ 6 2 O 3 67% Co–Ni–Al 2 O /s 3

. ε 10 /s 1000 K

830 K 1000 K

Log stress (⫻14.5 MN m ⫺ 2 )

Figure 6.62 Log ˙ε versus log σ for Ni–Al 2 O 3 (a) and Ni–67Co–Al 2 O 3 (b) (after Hancock,

Dillamore and Smallman, 1972; courtesy of Institute of Materials, Minerals and Mining).

Worked example

Creep data for a light alloy are given in the table

Stress

Temperature

Minimum creep rate

Calculate the expected steady-state creep rate at a constant stress of 2.8 N mm −2 at (i) 600 K and (ii) 640 K.

Solution

The creep equation: ˙ε ss = Aσ n exp( −Q/RT ). At constant T = 600 K,

Therefore, at 2.8 MPa, 600 K, −5 − log ˙ε = 4 × ( log 8.9 − log 2.8)

∴ ˙ε = 0.98 × 10 −7 s −1 . At constant σ = 5.0 MPa, ln 10 −6

= ln(Aσ ) − RT

Mechanical properties I 365

Figure 6.63 Schematic diagram showing influence of stress on diffusion-compensated steady-state creep.

ln 5

× 10 −6 = ln(Aσ ) − RT

= 128.4 kJ mol .

At constant σ = 2.8 MPa, ln