7. Precipitation Hardening

At about half the absolute melting point T_m, many particle free materials have a stress exponent n_N for the creep rate in the interval 4–7. Particle strengthened alloys typically have a higher stress exponent. This can be rationalized if the creep rate \({\dot{\upvarepsilon }}\) is expressed asσ is the applied stress, σ_i the internal stress from the particles, and A_n and n_i constants. With this formulation, the stress exponent n_i is smaller than n_N. As can be seen from the analysis in Chap. 2, an equation of the type in Eq. (7.1) can be derived from basic principles so the equation has a good basis. In a number of papers, σ_i, A_n and n_i have been used as adjustable parameters to make Eq. (7.1) fit the experiments and to have n_i to fall in the range 4–7. A special procedure called the Lagneborg-Bergman plot was developed for this purpose [5].

Assuming that σ_i is constant which was frequently done, it implies that σ_i is a threshold stress and below this stress no creep will take place. For oxide dispersion strengthened alloys (ODS) such a threshold stress has been observed [6, 7], but there are also ODS where a threshold is not found. However, for most particle strengthened materials for example the common CrMo steels, no threshold stress has been recorded. By analyzing Eq. (7.1), it is found that the stress exponent n_N decreases with increasing stress. This behavior is observed for a few ODS alloys [7], but not for most particle strengthened alloys, which is a drawback of the model.

The reasons for the failure of the assumption with a constant threshold stress are now well understood. It is now possible to derive σ_i directly. In fact, it is shown later in this chapter that σ_i is not a constant. It depends on both temperature and stress and there is no indication that it will tend to a limiting value at low stresses.

Dislocations can pass particles by cutting through them, by flowing around them or climbing across them. At ambient temperatures only the first two are usually considered. Particle cutting will not be analyzed in the present text because as we will see later in this chapter, it is unlikely that it is of importance in creep exposed materials except in special cases. For a summary of mechanisms for particle cutting, see [8].

The Orowan model for dislocation looping of particles will briefly be described here because it is needed in Sect. 7.3. When the stress increases for a dislocation attached to particles, the dislocation will eventually almost meet itself around the particles. The maximum force F that the dislocation can take is 2τ_L ≈ 2Gb²/2 where τ_L is the dislocation line tension. The external force F on a dislocation segment of length λ is (Peach-Koehler formula)where σ is the external stress and m_T is the Taylor factor. By equating the two forces the critical stress is obtained. This is the stress for Orowan looping σ_O

Many refinements of this expression are available in the literature. However, they can approximately be taken into account by adding a factor C_O = 0.8 [9]. The precision in the prediction of PH does not justify that a more elaborate formulation is needed. λ is usually assumed to be taken as the nearest neighbor distance for randomly distributed spherical particles of radius r and a volume fraction of f_V. This distance is usually referred to as the planar lattice square spacing λ_s

As can be seen from Eqs. (7.3) and (7.4), the Orowan strength increases with decreasing particle radius and increasing volume fraction of particles.

As pointed out above, the Orowan strength has been used many times to estimate the influence of particles on the creep strength. This overestimates the strength contribution since Eq. (7.3) is only weakly temperature dependent through the shear modulus G, which decreases approximately linearly with temperature.

Many attempts were made in the past to generalize the Orowan model by taking climb into account. For a review, see [7, 10]. Initial attempts to determine the size of the energy barrier and the associated value for σ_i were made by Brown and Ham [9] and by Lagneborg [11]. They found a value for internal stress σ_i of about half the Orowan stress, which was in agreement with observations for some materials. However, they assumed the presence of local dislocations that were attached to the particles and that the dislocations remained in the glide planes between the particles. This introduces sharp bends on the dislocations that are easily relaxed. Further modeling therefore concentrated to general dislocations that are only attached to a single point on the particles and have more general degrees of freedom. The assumption about climb of general dislocations rather than of local dislocations is clearly more realistic. However, with this approach the values for σ_i turned out to be quite small and tended to decrease with each investigation [6, 7, 12]. The best estimate for the minimum climb stress σ_clmin is [6, 7]whereα_cl is called the climb resistance. This gives a value of 0.02–0.06 for σ_clmin of the Orowan stress for common particle volume fractions of f_V = 1 to 5%.

The predicted energy barriers are so low that they are of little practical value to describe the significant PH in engineering alloys. One possibility is that there is an attractive interaction between the dislocations and the particles. Such an interaction has been observed for ODS [13, 14] but not in general for other PH systems. This behavior has also been modeled [15, 16]. The dilemma with this model is that the interaction strength is considered as an adjustable parameter and this means that the model is not predictive. The nature of the interaction cannot therefore be ascertained.

Clearly, the energy barriers against climb cannot be used as a basis for explaining PH. Instead another approach will be presented in Sect. 7.3 that is based on the time it takes for a dislocation to climb across a particle.

The model in Sect. 7.3 was first published in 2000 and was successfully applied both to Cr–Mo steels [2] and to austenitic stainless steels [1, 3, 17]. In these applications, PH is not the dominating contribution to the creep strength. Therefore they could not be considered as full verification of the PH model. In this section results on creep in Cu–Co alloys published by Wilshire and coworkers will be analyzed [20, 21]. This system has the advantage that the particles have a large influence on the creep rate. In addition, several ageing conditions with different particle size distributions were studied, so the influence of the particles can safely be ascertained. A valuable feature is that the effect of solid solution hardening in these studies is negligible small as will be shown and will not interfere with the analysis.

The analyzed alloys and conditions are summarized in Table 7.1.

Three alloys were included in the study with 0.88, 2.48 and 4.04 wt.% Co. In their main condition the alloys were fully aged at 600 and 700 °C generating a stable particle structure. These temperatures were sufficiently high that no particle coarsening took place during the creep testing at 439 °C. In addition, underaged and overaged conditions were covered for the 0.88 wt.% alloy.

The volume fraction of particles for the different alloys and the amount of Co were calculated with the thermodynamic software Thermo-Calc, see Table 7.1. The particle sizes were measured with transmission electron microscopy [20, 21]. Using the expression for the square lattice spacing, Eq. (7.4), the particle spacing was determined. With the help of Eq. (7.3) the Orowan strength for the alloys was computed, see Table 7.1. The Orowan strength significantly exceeds the creep strength of the alloys as will be seen.

The amount of solid solution hardening of the alloys was evaluated with the help of Eq. (6.​22). Some of the Co is in solid solution, Table 7.1. The computed solute drag stress was in the interval from 0.15 to 0.25 MPa, which is negligible in the context.

To demonstrate the validity of PH model, it is essential to verify that strength contribution from the dislocations can be described with the model in Chap. 2. This is tested for pure copper. Some creep data for pure copper can be found in [20, 21]. In addition creep data have been taken from [22] where tests were performed at 400–500 °C that are close to the test temperature for the Cu–Co alloys. The creep rate versus applied stress is shown in Fig. 7.1.

A scatterplot of creep rate versus stress plots datasets for experimental temperatures of 400, 439, 450, and 500 degrees Celsius. A concave upward-ascending curve fits each.

From Fig. 7.1 it can be seen that the predicted temperature dependence is larger than the observed one. The reason is that the activation energy for creep is lower than the activation energy of self-diffusion which is used in the prediction. That the activation energy for creep is lower than that for self-diffusion is unusual. This has been discussed by Raj and Langdon [22]. The stress exponent, i.e. the slope of the curves in Fig. 7.1 is in acceptable agreement with the observations. The model and experimental values at 439 and 450 °C are in close agreement.

Only particles larger than the critical size contribute to the creep strength. Such particles have to be passed by Orowan bowing. The critical particles radius is given by Eq. (7.12). The critical particle radii for the three Cu–Co alloys is illustrated in Fig. 7.2.

A scatterplot of critical radius in micrometers versus applied stress in megapascals plots datasets for alloys C u 0.88 C o, C u 2.48 C o, and C u 4.04 C o. A concave upward-descending curve fits each dataset.

In Fig. 7.2 in addition to the critical radii, the average particle sizes are shown. Both the typical radii and the critical radii increase with the Co content. At low stresses the critical radii are about a factor of four larger than the average particle radii. At larger stresses the difference is smaller. For the two lower Co contents it is a factor of two. For the highest Co content, there is no longer any difference anymore. This means that all particles contribute to the creep strength.

When the volume fraction and the average radius are known, the size distribution can be estimated with the help of Eq. (7.15). The result is illustrated in Fig. 7.3.

A scatterplot of cumulative size distribution versus radius plots datasets for alloys C u 0.88 C o, C u 2.48 C o, and C u 4.04 C o. A linearly descending line fits each dataset.

Exponential size distributions imply that the number of particles per unit area can decrease quite rapidly with increasing radius. The Co content has a large impact on the slope of the size distributions. The critical radii are marked in Fig. 7.3 for different applied stresses. As is evident from Fig. 7.2, the largest critical radii correspond to the lowest stresses.

The influence of particles on the creep strength is via the critical Orowan strength in Eq. (7.17). This strength is added to internal stress in Eq. (2.​29). The critical Orowan strength is shown in Fig. 7.4.

A line graph of critical Orowan stress versus applied stress plots concave downward ascending curves for applied stress and alloys C u 0.88 C o, C u 2.48 C o, and C u 4.04 C o.

The critical strength increases with the applied stress. It might be thought that if the Co content is raised it would automatically enhance PH. From Fig. 7.4 it is clear that this is not the case. In fact, the alloy with 2.48 wt.% Co gives the highest strength. The predicted creep rate as a function of applied stress is shown in Fig. 7.5.

A scatterplot of creep rate versus stress plots experimental datasets for alloys C u 0.88 C o, C u 2.48 C o, and C u 4.04 C o. A concave upward-ascending curve fits each dataset.

The Co particles reduce the creep rate by about two orders of magnitude in relation to that of pure copper. It is evident that the model can reproduce this behavior quite well. The ranking of the three Cu–Co alloys is also in accordance with experiments. The stress dependence is handled in an acceptable way.

If Figs. 7.4 and 7.5 are compared, the difference in critical Orowan stress is directly related to the observed relations between the creep rates as proposed by the model. If the values for the Orowan strength at room temperature (Table 7.1) would be used to rank the creep strength it would suggest that Cu0.88Co and Cu2.48Co would have about the same creep strength and would be significantly better than Cu4.04Co. This is clearly not consistent with the model or with the experimental results. Another way of demonstrating that creep strength is not close related to the Orowan strength is illustrated in Fig. 7.6.

A scatterplot of critical to R T Orowan stress versus applied stress. It plots concave down downward ascending curves for alloys C u 0.88 C o, C u 2.48 C o, and C u 4.04 C o.

It can be seen that the ratio between the critical Orowan strength and room temperature Orowan strength varies with applied stress and alloy composition.

In [20] there is also a comparison between underaged, aged and overaged conditions for the Cu0.88Co alloy. The computed size distributions for these conditions are shown in Fig. 7.7.

A scatterplot of cumulative size distribution versus radius plots datasets for underaged, aged and overaged alloy C u 0.88 C o. A linearly descending line fits each dataset.

There is obviously a large difference between these conditions which makes it an interesting additional test of the model. Furthermore the size distributions are different from those in Fig. 7.3. The strain rate versus stress curves are presented in Fig. 7.8.

A scatterplot of creep rate versus stress plots experimental datasets for alloy C u 0.88 C o in underaged, aged, and overaged conditions. A concave upward-ascending curve fits each dataset.

As can be seen in the Figure, the model can describe the experimental data fairly well. The difference between the conditions can be accounted for, and the stress dependence is well reproduced.