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Open Access 2024 | OriginalPaper | Buchkapitel

6. Solid Solution Hardening

verfasst von : Rolf Sandström

Erschienen in: Basic Modeling and Theory of Creep of Metallic Materials

Verlag: Springer Nature Switzerland

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Abstract

The size and modulus misfit between solute and parent atoms gives rise to strengthening, solid solution hardening (SSH). With the development of Argon’s expression for the interaction energies for solute atoms and dislocations for size and modulus misfit, both effects can now be modeled without the introduction of adjustable or arbitrary parameters. These expressions are used to derive models for SSH during creep. Although the constants for the modulus misfit can be an order larger than those for size misfit, the latter effect is still dominating. The interaction energy gives a direct contribution to the activation energy for creep. The solutes form Cottrell atmospheres around the dislocations. For slowly diffusion elements, these atmospheres give rise to a drag force that slows down the motion of the dislocations. Fast diffusing elements have to break away from the dislocations to enable their motion. This creates a break stress that is the source of SSH in this case.

6.1 General

Elements in solid solution are used in many alloy systems to increase the strength and that is referred to as solid solution hardening (SSH). When the size of the solute atoms is different from that of the parent metal atoms, it makes it more difficult for the dislocations to propagate and that raises the strength. SSH is in fact one of the major ways to increase the strength of creep resistant alloys.

The size misfit is not the only way that solute atoms can affect the strength. If the shear modulus of the solutes is different from that of the parent metal, it is also of importance for SSH. This will be analyzed in the present chapter. This effect is less easy to understand intuitively than that of the size misfit. The most direct way to recognize the significance of this effect is to consider that the expression for interaction energy between a solute and a dislocation is proportional to the shear modulus. Any change in the value of shear modulus would affect the size of the interaction energy. There are a number of other mechanisms that can influence SSH. Examples are the presence of stacking faults, short range order of solutes (solutes are not fully randomly distributed), and the solutes from more or less complex defects. These cases will not be covered here. When the solutes agglomerate in particles, it is considered outside SSH and will be discussed in Chap. 7 on precipitation hardening.

SSH at ambient temperatures and at lower temperatures have been covered extensively in the literature. There are excellent reviews on SSH on this topic. The texts by Haasen [1], Suzuki et al. [2] and Argon [3] can be mentioned. At elevated temperatures and particularly for creep the number of publications is much more limited. Primarily solid solution hardened aluminum alloys have been studied. We will cover this literature below.

At low temperatures, SSH is assessed as the force that the solutes give on the dislocations. This is in principle the force acting on the dislocation at 0 K. In several papers, semi-empirical temperature dependencies have been introduced (summarized in [1, 3]) to find values at ambient temperatures. In some of the models for binary alloys, SSH is proportional to c^1/2 in some models, and to c^2/3 in others, where c is the concentration of the solute. In engineering applications where more than one alloying element is involved, SSH is often linear in c [4, 5]. At elevated temperatures, it is the interaction energy between the solutes and the dislocations that is of interest. This interaction energy has the consequence that the solutes gather around the dislocations and form so-called Cottrell atmospheres. The Cottrell atmospheres slow down the motion of the climbing dislocation and thereby strengthening the alloy. The mechanisms for SSH are different at ambient and high temperatures. We will focus on the creep case at high temperatures.

In many alloy systems, SSH is of high significance for raising the creep strength. Thus, SSH is extensively used in creep resistant austenitic stainless steels, superalloys (both Co and Ni based) and titanium alloys. Many experiments on the role of Mg in Al–Mg have been published. They have in general been analyzed according to Weertman’s original proposals [6, 7]. This is referred to as the classical picture, which is presented in Sect. 6.2. Basic models for the influence of lattice and shear modulus misfit on SSH are given in Sect. 6.3. In Sect. 6.4, the role of the drag stress is discussed. The mechanisms for slow and fast diffusion elements are different. The first case is covered in Sect. 6.3 and the latter case in Sect. 6.5.

6.2 The Classical Picture

6.2.1 Observations

In the analysis of the influence of solid solution hardening on creep much focus has been devoted to Al–Mg alloys. The reason is that there is a change in the stress exponent for the secondary creep rate with stress above 300 °C. As was discussed in Chap. 2, change in the stress exponent has often been associated with a change in creep mechanism. The effect of magnesium on the creep rate is illustrated in Fig. 6.1.

With increasing Mg content, the creep rate rapidly decreases. The creep exponent of pure aluminum is n = 5, as marked with the number 5 on the curve. At least for some of the Al–Mg alloys, the creep exponent at low stresses is 5 but it is reduced to 3 at slightly higher stresses. At still higher stresses the creep exponent increases again first in some cases to 5 and then to still higher values.

Al–Mg alloys are rarely used above 150 °C. All the interest in the Al–Mg alloys is thus mechanistic. For Al–Mg alloys the classic assumption is that the various creep exponents are due to different creep mechanisms. Two papers of Weertman from 1957 suggested that n = 3 and n = 5 correspond to dislocation glide and climb, respectively [6, 7].

The unusual transition from n = 5 and n = 3 is attributed to a sudden introduction of viscous glide of dislocations where solute atmospheres (Cottrell atmospheres) are dragged along with the dislocations. In general, dislocation climb is slower than glide, and is therefore expected to control the creep rate. However, the drag was assumed to slow down the gliding dislocations sufficiently to make them control the creep process. At high stresses, the gliding dislocations could break away from the solute atmospheres. This would make the gliding dislocations move faster and climb would become controlling. Hence, a transition from n = 3 and n = 5 is expected. Friedel has given a model for such a break away [12].

The distinction between glide and climb controlled alloys is considered so important that they are described as Class I and Class II alloys, respectively. In Class I alloys the dislocations interact with the solutes forming atmospheres around the dislocations. Since the solutes follow the dislocation through diffusion which is a slow process and slower than the glide velocity, the dislocations are slowed down and creep can be controlled by glide. The dislocations are dragged by the solutes and hence the designation solute drag.

When no solute atmospheres are formed, glide will take place without any effect of solutes, and the glide velocity will be high. Creep will be climb controlled in the same way as for pure metals. Such materials are referred to as Class II materials. Most alloys for example stainless steels and superalloys are of Class II type. In addition to Al–Mg some Fe–Mo alloys are of Class I type. Another distinction that is often mentioned is that Class II form substructure during creep whereas Class I do not. However, this is not general since Fe–Mo alloys also form substructure [13].

6.2.2 Issues with the Classical Picture

There are a number of problems with the classical pictures

We saw in Chap. 2 that in many cases there is a continuous increase in the creep exponent with stress if a sufficiently large interval of stresses is considered. This applies to aluminum as well [14]. The change in creep exponent is therefore not necessarily associated with a change in creep mechanism.
Even Weertman who was the first to suggest a difference in creep exponent between glide and climb, made the statement that practically all creep laws based on creep recovery give n = 3, which is referred to as the natural mode [8]. Although this statement is not correct, it illustrates that it is not directly possible to use the creep exponent to distinguish between glide and climb.
The Class I alloy Fe–1.8Mo shows the same type of behavior as Al–Mg [15]. However, creep exponent in the assumed glide region is n = 4, not n = 3.
The influence of solid solution on glide and climb is now well understood, see below. As demonstrated in Chap. 2, glide is always faster than climb. Solutes are now believed to influence climb and glide in the same way. Consequently, the transition from n = 5 to n = 3 cannot be explained by the presence of solutes.
If the dislocations move fast enough, they will break away from their Cottrell atmospheres. The required stress for the breakaway was first derived by Friedel [12]. A numerical more precise solution has then been given by Hirth and Lothe [16], Eqs. 18–131.
$$\upsigma _{\max } = \frac{{m_{{\text{T}}} c_{0}\upbeta ^{2} }}{{\Omega _{{\text{a}}} b^{2} k_{{\text{B}}} T}} $$

(6.1)
where m_T is the Taylor factor, c₀ the concentration of the solute in at.%, Ω_a is atomic volume, β is the solute strengthening parameter (defined below), b is Burgers’ vector and k_B T has its usual meaning. The values of c₀ and β are given in Sect. 6.4. For the alloys in Fig. 6.1, the following values are obtained from Eq. (6.1): 55, 120, 226, 359 and 640 MPa. They are all outside the range of the experimental data and cannot explain the observed change in stress exponent.

One can conclude that many of the classical assumptions about creep of Al–Mg alloys are questionable. There are clear distinctions between pure aluminum and Al–Mg alloys with respect to creep. The origin of these differences is less clear than what has been assumed in the past. An alternative way of explaining the observations will be presented below.

6.3 Modeling of Solid Solution Hardening. Slowly Diffusing Elements

For slowly diffusing solutes, the solutes form Cottrell atmosphere that follow the dislocations as described above. This creates a back stress on the dislocations that slows down their motion. This is referred to as solute drag. Slowly diffusing elements are often major alloying elements that are in solid solution.

Fast diffusing elements like the interstitial elements C and N in steel also raise the creep strength but the mechanism is different. These elements can lock the dislocations and they have to break away to contribute to the straining. The critical quantity is the break stress which is needed to make the dislocations move. The break stress is derived in Sect. 6.5.

6.3.1 Lattice and Modulus Misfit

If the atomic radius of solutes that are present in the lattice is different from that in the matrix, it creates an interaction between the solutes and the dislocations. This is referred to as lattice misfit. It generates a friction stress that influences the motion of the dislocations and thereby increases the strength of the alloy. A difference in the modulus also influences the forces on the dislocations.

The lattice misfit can be expressed in terms of the difference in atomic volume between the solute and the host matrix. The atomic volumes can be obtained from the lattice parameters ${a}_{i}$ as ${\Omega }_{i}={a}_{i}^{3}/4$ (for fcc). These volumes are linearly expanded (as a function of concentration) around the host composition. The linear misfit δ_i is given by:

$$ \delta_{i} = \frac{1}{3\Omega }\frac{\partial \Omega }{{\partial c_{i} }} \approx \frac{1}{3\Omega }\frac{{\Omega_{i} - \Omega_{0} }}{{c_{i} }} $$

(6.2)

where Ω₀ is the atomic volume of the matrix, Ω_i the atomic volume of the solute of element i and c_i the concentration of the solute. It can also be related to the change in the lattice parameter,

$$\updelta _{i} = \frac{1}{a}\frac{\partial a}{{\partial c_{i} }} \approx \frac{1}{a}\frac{{a_{i} - a}}{{c_{i} }} $$

(6.3)

where a_i and a are the lattice parameters of the solute and the matrix, respectively. The linear misfit is a third of the volume misfit as shown in Eq. (6.2). The modulus misfit μ_i can be expressed as

$$\upmu _{i} = \frac{1}{G}\frac{\partial G}{{\partial c_{i} }} \approx \frac{1}{G}\frac{{G_{i} - G}}{{c_{i} }} $$

(6.4)

where G_i and G are the shear modulus in the solute and the matrix, respectively.

Values for experimental lattice spacings and misfit parameters can be found in Pearson’s handbook [17] and in King’s paper [18]. The interaction of solutes with stacking faults (Suzuki effect) can also contribute to the creep strength. However, models for these interactions are currently unavailable. For example, for aluminum King gives the following values for the linear misfit δ_Cu = −0.13, δ_Mg = 0.12, and δ_Mn = −0.19. Experimental modulus misfit values are less readily available in the literature, and in general modulus misfits have to be computed with ab initio methods.

6.3.2 Solute Atmospheres

The dislocation-solute interaction can be estimated from elasticity theory by assuming the solute to be a dilation center. Due to the interaction, solute agglomerates around the dislocation. The concentration around a dislocation can be expressed as [16]

$$ c_{i} = c_{i}^{0} \exp \left( { - \frac{{U_{i} ({\mathbf{r}})}}{{k_{B} T}}} \right) $$

(6.5)

where $c_{i}^{0}$ is the mean concentration of the solute i, c_i the local concentration of the solute and U_i(r) at position r from the dislocation. The interaction energies take the values [3]

$$ U_{i}^{{e\updelta }} = \frac{1}{\uppi }\frac{{(1 +\upnu _{{\text{P}}} )}}{{(1 -\upnu _{{\text{P}}} )}}G {\Omega \updelta }_{i} \frac{by}{{r^{2} }} $$

(6.6)

$$ U_{i}^{{s\updelta }} = \frac{1}{{2\uppi ^{2} }}\frac{{(1 +\upnu _{{\text{P}}} )B}}{{(1 - 2\upnu _{{\text{P}}} )}}G {\Omega \updelta }_{i} \frac{{b^{2} }}{{r^{2} }} $$

(6.7)

$$ U_{i}^{e\upmu } = \frac{1}{{8\uppi ^{2} }}\frac{1}{{(1 -\upnu _{{\text{P}}} )^{2} }}G {\Omega \upmu }_{i} \frac{{b^{2} z^{2} }}{{r^{4} }} $$

(6.8)

$$ U_{i}^{s\upmu } = \frac{1}{{8\uppi ^{2} }}G {\Omega \upmu }_{i} \frac{{b^{2} }}{{r^{2} }} $$

(6.9)

where $G$ is the shear modulus, and ν_P is the Poisson’s ratio of the material. The indices for U_i refer to edge (e) and screw (s) dislocations and size (δ) and modulus (μ) misfit. r is the overall distance from the dislocation core, y the distance above the dislocation, and z the distance in the plane. The expressions (6.6)–(6.9) are illustrated in Fig. 6.2. The value of y is set to correspond to the distance to the second plane of atoms, in fcc $y = \sqrt {3/2} b$. The first atom plane cuts the dislocation core and gives a weaker interaction.

In Fig. 6.2, the interaction energies are normalized with respect to GΩ and the misfit parameters δ_i or μ_i. In this way the result is material independent. In spite of the fact that μ_i is often 10 times larger than δ_i, it is evident from Fig. 6.2 that the lattice misfit for edge dislocations, Eq. (6.6), gives the largest interaction energy

$$ U_{i}^{\max } = \frac{1}{\uppi }\frac{{(1 +\upnu _{{\text{P}}} )}}{{(1 -\upnu _{{\text{P}}} )}}G\Omega _{0}\updelta _{i} $$

(6.10)

The dislocations that are slowed down mostly by the interaction with the solutes will also control the magnitude of SSH. Consequently, it is the interaction energy in Eq. (6.10) that is the important quantity.

Due to the interaction energy, it is energetically favorably for the solutes to be located close to the dislocations. Therefore, atmospheres of solutes are created around the dislocations. If the dislocations are not moving (they are static), the concentration of solute atoms $c_{i}^{stat}$ is given by Eq. (6.5). The concentration of solutes around a moving dislocation can be derived from the following diffusion equations [16], Eqs. 18–10

$$ D_{i}^{{{\text{sol}}}} \frac{{\partial^{2} c_{i} }}{{\partial y^{2} }} + \frac{{D_{i}^{{{\text{sol}}}} }}{{k_{B} T}}\frac{\partial }{\partial y}c_{i} \frac{{\partial U_{i} (y,z)}}{\partial y} + v\frac{{\partial c_{i} }}{\partial y} = 0 $$

(6.11)

This is just Fick’s second law taking into account the dislocation-solute interaction in the second and the moving frame with a velocity v in the third term. $D_{i}^{sol}$ is the diffusion coefficient for the solute i in the matrix. The Cartesian coordinates represent the position of the solute relative to an edge dislocation that is climbing in the y-direction. Equation (6.11) can be integrated directly.

$$ D_{i}^{{{\text{sol}}}} \frac{{\partial c_{i} }}{\partial y} + \frac{{D_{i}^{{{\text{sol}}}} }}{{k_{{\text{B}}} T}}c_{i} \frac{{\partial U_{i} (y,z)}}{\partial y} + v(c_{i} - c_{0} ) = 0 $$

(6.12)

By solving Eq. (6.12), the concentration $c_{i}^{dyn}$ of solutes around a dislocation moving in the y-direction can be obtained [16, 19]

$$ c_{i}^{{{\text{dyn}}}} = \frac{{vc_{i}^{0} }}{{D_{i}^{{{\text{sol}}}} }}\left( {e^{{ - \frac{{U_{i} (y,z)}}{kT} - \frac{vy}{{D_{i}^{{{\text{sol}}}} }}}} } \right)\int\limits_{ - \infty }^{y} {e^{{\frac{{U_{i} (y^{\prime },z)}}{kT} + \frac{{vy^{\prime } }}{{D_{i}^{{{\text{sol}}}} }}}} } dy^{\prime } $$

(6.13)

The quantities β and r_i are introduced that will be used below

$$\upbeta _{i} = bU_{i}^{\max } \quad r_{i} = \frac{{\upbeta _{i} }}{{k_{{\text{B}}} T}} $$

(6.14)

β_i is the maximum force of the dislocation from individual solutes, and r_i the radius of the Cottrell atmosphere or cloud of solutes around the dislocations. The velocity of a climbing dislocation v is given by

$$ v_{{{\text{climb}}}} = M_{{{\text{climb}}}} b\upsigma $$

(6.15)

M_climb is the climb mobility and σ the applied stress. M_climb can be expressed in terms of the coefficient of self-diffusion D_self, Eq. (2.34), and the activation energy for the solutes Q_sol.

$$ M_{{{\text{climb}}}} = \frac{{D_{{{\text{self}}}} b}}{kT}e^{{\frac{{\upsigma b^{3} }}{{k_{{\text{B}}} T}}}} \exp ( - \frac{{Q_{{{\text{sol}}}} }}{kT}) $$

(6.16)

The size of Q_sol is taken as $U_{i}^{\max }$, Eq. (6.10), for the element that has the largest solid solution hardening effect on the creep strength. At lower temperatures, the climb enhancement factor g_climb, Eq. (2.37) should be taken into account. For gliding dislocations, the velocity is given by Eqs. (2.39) and (2.42)

$$ v_{{{\text{glide}}}} = M_{{{\text{glide}}}} b\upsigma = M_{{{\text{climb}}}} g_{{{\text{glide}}}} b\upsigma = M_{{{\text{climb}}}} \frac{1}{{b\sqrt\uprho }}b\upsigma $$

(6.17)

again ignoring the climb enhancement factor. ρ is the dislocation density. Since g_glide is much larger than unity, the glide velocity is always higher than the climb velocity. The distribution of solutes around dislocations is illustrated in Fig. 6.3.

For a climbing dislocation there is agglomeration of solutes on one side and depletion on the other side. For a gliding dislocation the static distribution is symmetric in the direction of the motion. The concentration in the static model is slightly higher than according to the dynamic model for climbing dislocations, Fig. 6.3a. For glide the dynamic concentration is much lower than the static one, Fig. 6.3b. The reason is the much higher glide velocity in comparison to the climb velocity.

By integrating over the profiles, the agglomeration of solutes can be determined. For the case corresponding to Fig. 6.3a, the agglomeration is 25 and 27 for the static and dynamic distribution, respectively. For the gliding dislocation, the agglomeration is 128 and 1.2 for the static and dynamic distribution, respectively. The agglomeration factor can be interpreted in two ways. If all the additional atoms are placed at the dislocation core over a distance of a Burgers vector, the concentration there of the solute would be enhanced by the agglomeration factor. Alternatively, it can be taken as the distance in terms of Burgers vectors over which the concentration is more than twice the average solute concentration.

6.4 Drag Stress

For slowly moving dislocations, the solutes exert a drag stress on the dislocations that is the source of SSH. The drag stress can be derived numerically from Eq. (6.13) [16].

$$\upsigma _{i}^{{{\text{drag}}}} = \frac{{k_{B} Tv_{{{\text{climb}}}} }}{{b^{2} D_{i} }}\int {c_{i}^{{{\text{dyn}}}} } \,dz $$

(6.18)

Alternatively the drag stress can be expressed as

$$\upsigma _{i}^{{{\text{drag}}}} = - \int {c_{i}^{{{\text{dyn}}}} \frac{\partial U(y,z)}{{\partial z}}} \,dz $$

(6.19)

It is important that the dynamic solution is used in Eqs. (6.18) and (6.19). The static solution in Eq. (6.5) cannot be utilized because it does not give the correct behavior at large z. The need to use the dynamic expression makes the full solution fairly complex. An approximate solution was derived in [16]. The approximate solution illustrates important features of SSH and will be summarized below. A common form of the drag stress $\upsigma _{i}^{drag}$ for element i is

$$\upsigma _{i}^{{{\text{drag}}}} = \frac{{v_{{{\text{climb}}}} c_{i0}\upbeta _{i}^{2} }}{{bD_{i} k_{B} T}}I(z_{0} ) $$

(6.20)

where

$$ I(z_{0} ) = \int\limits_{1}^{{z_{0} }} {\frac{{2\sqrt {2\uppi } }}{{3z^{2.5} }}} e^{z} \,dz $$

(6.21)

v_climb is the dislocation climb speed, cf. Equation (6.15), c_i0 is the concentration of solute i, and D_i the diffusion constant for solute i. I(z₀) is an integral of z₀ = b/r₀k_BT where r₀ is the dislocation core radius. I(z₀) often has values of about 3. β_i is the force in Eq. (6.14).

If the radius of the static cloud r_i is less than the burgers’ vector b or if D_i/v_climb is larger than the average distance between the dislocations R_disl, Eq. (6.20) is replaced by

$$\upsigma _{i}^{{{\text{drag}}}} = \frac{{v_{{{\text{climb}}}} c_{i0}\upbeta _{i}^{2} }}{{bD_{i} k_{{\text{B}}} T}}\log \left( {\frac{{D_{i} }}{{v_{{{\text{climb}}}} b}}} \right)\quad r_{i} < b\;{\text{or}}\;D_{i} > v_{{{\text{climb}}}} R_{{{\text{disl}}}} $$

(6.22)

Finally, if r_i > R_disl the expression for $\upsigma _{i}^{drag}$ takes the form

$$\upsigma _{i}^{{{\text{drag}}}} = \frac{{v_{{{\text{climb}}}} c_{i0}\upbeta _{i}^{2} }}{{bD_{i} k_{{\text{B}}} T}}\log \left( {\frac{{D_{i} }}{{v_{{{\text{climb}}}} r_{i} }}} \right)\quad D_{i} > v_{{{\text{climb}}}} R_{{{\text{disl}}}} $$

(6.23)

The four alternative expressions are not very different. Only the final (logarithmic) factor varies. The situation is another if the dislocation speed v_disl is high and the motion of the solute cloud is no longer diffusion controlled.

$$\upsigma _{i}^{{{\text{drag}}}} = \frac{{\pi D_{i} c_{i0}\upbeta _{i}^{2} }}{{k_{B} Tb^{2} v_{{{\text{climb}}}} }}\quad v_{disl} > 4D_{i} k_{B} T/\upbeta _{i} $$

(6.24)

$$\upsigma _{i}^{{{\text{drag}}}} = \pi c_{i0}\upbeta _{i} \quad r_{i} > \sqrt {D_{i}\upbeta _{i} /v_{{{\text{disl}}}} k_{B} T} $$

(6.25)

The dependence of v_disl/D_i is inverted in Eq. (6.24) and absent in Eq. (6.25).

In the computation of the secondary creep, the drag stress is added to the internal stress σ_i in Eq. (2.29). An application of the drag stress is illustrated in Fig. 6.4. The contribution from Q_sol in Eq. (6.15) to the creep activation energy is also taken into account. This increases the activation energy by $U_{i}^{\max }$. i is the element Mg in this case. In Fig. 6.4, Eq. (6.20) for the drag stress was used.

Three stages of stress dependence can be found in Fig. 6.4. At low stresses there is a slight increase in the stress exponent n_N due to the presence of the Peierls stress. Its value is the same as for pure aluminum used in Fig. 2.9. For stresses in the middle range a power-law behavior is observed. At higher stresses the stress exponent increases and a tendency to power-law break down is found.

The modeling in Fig. 6.4 is based on climb and it is assumed that climb is the controlling mechanism. It has been suggested many times in the literature that glide should be controlling for Al–Mg in part of the studied stress range, see Sect. 6.2. The background is that the stress exponent in the middle stress range is about three and that is what Weertman suggested for glide control in his original paper on the topic. However, it is evident from the analysis in Chap. 2 that climb control often gives the same stress exponent at modest stress levels. According to the classical picture, see Sect. 6.2, two changes in models and mechanisms have to be assumed to represent the stress dependence in Fig. 6.4. The absence of substructure in Al–Mg has been taken as one reason for not considering climb as the controlling mechanism. But that could also be a consequence of the presence of the alloying element. Increasing amounts of alloying elements tend to reduce the stacking fault energy and give a more planar dislocation structure [3]. It is demonstrated in Fig. 6.4 that the present model can describe the experimental data. In the same way as for other comparison with experiments in this book no adjustable parameters are involved. In summary, a single climb based model can accurately reproduce the creep data for Al–Mg. There is no need to assume that glide is controlling in part of the stress range which avoids a number of the difficulties discussed in Sect. 6.2.

6.5 Modeling of Solid Solution Hardening. Fast Diffusing Elements

Fast diffusing elements can have a dramatic effect on the creep rate and the rupture strength. Addition of 50 ppm phosphorus to pure copper reduces the creep rate and increases the creep rupture strength. Phosphorus reduces the experimentally observed creep rate by about a factor of 100 at 75 °C. This will be illustrated below. At the same time the creep strength at 10000 h rupture time is raised from about 140 to 170 MPa at the same temperature [21]. Another example is nitrogen in solid solution in austenitic stainless steel. An addition of 0.1% N can reduce the creep rate by an order of magnitude and increase the rupture strength at 650 °C for a rupture time of 10000 h by about 40 MPa [22].

These pronounced effects of small additions of alloying elements cannot be explained by solute drag. To get a significant contribution from solute drag to SSH, fairly large amounts of alloying elements are needed. We will concentrate on the influence of P on Cu. There are two reasons for that. The influence of P on creep in copper has been analyzed in detail [19]. In addition the low amount of P is clearly in solid solution so there are no particles present that can disturb the analysis. In [19] it has been demonstrated that the solute drag stress is at most 1 MPa. If the accurate expression in Eq. (6.18) is evaluated numerically, the solute drag value is even several orders of magnitude below 1 MPa. It can be concluded that solute drag cannot explain the influence of P on creep in Cu.

In [19] a model is presented that can explain the effect of P on creep quantitatively. It is assumed that the P atoms are agglomerated at the dislocations in the same way as for elements in solute drag and that the distribution of P atoms can be described by Eq. (6.5) for the static distribution and by Eq. (6.13) for the dynamic distribution. These distributions are illustrated in Fig. 6.5.

The general behavior of these profiles is the same as for Al–Mg in Fig. 6.3. However, there are differences. The diffusion rate for P is so fast that even in the glide case, the dynamic distribution is virtually identical to the static one. In the climb case the agglomeration is even higher in the dynamic distribution than in the static one.

The main difference to the solute drag model is that the P atoms are assumed to lock the dislocations [19]. For the dislocations to move they must break away from the P atmospheres. But since the P atoms are rapidly moving, they will immediately catch up with the dislocations and lock them again. So there is a continuously repeating break away and locking process. When dislocations are more permanently breaking away from solute locking, serrated yielding is often observed. However, since the breakaway—locking is taking place continuously for P in Cu no serrated yielding is observed.

The stress needed to move a dislocation σ_break can be determined from an energy balance [19]. According to Peach-Koehler’s formula the force F on a dislocation length segment 2L is $F =\upsigma _{{{\text{break}}}} 2Lb$. If the dislocation is moved by one burgers’ vector, the consumed energy is Fb/2. This energy corresponds to the maximum binding energy $U_{i}^{\max }$

$$\upsigma _{{{\text{break}}}} Lb^{2} = U_{i}^{\max } $$

(6.26)

The average distance L between solute pinning points on a dislocation is

$$ L = \frac{b}{{\int {c_{i}^{{{\text{dyn}}}} dz} }} $$

(6.27)

Combining Eqs. (6.26) and (6.27) gives an expression for σ_break

$$\upsigma _{{{\text{break}}}} = \frac{{U_{i}^{\max } }}{{b^{3} }}\int {c_{i}^{{{\text{dyn}}}} \,dz} $$

(6.28)

The index i refers in this case to the element P.

To find the influence of σ_break on the creep rate, σ_break is added to the internal stress in Eq. (2.29). The creep rate versus stress for oxygen free copper with (Cu-OFP) and without P (Cu-OF) is shown in Fig. 6.6.

As stated above, the presence of 50 ppm P reduces the creep rate by two orders of magnitude and this can be fully accounted for by the model for the break stress in Eq. (6.28).

6.6 Summary

Solid solution hardening (SSH) is a result of the misfit between solutes and the matrix with respect to lattice parameters and elastic moduli. This makes it more difficult for the dislocation to move in the lattice which results in a hardening effect. The interaction energies between dislocations and solutes are proportional to the misfit in the lattice parameters and in the elastic moduli. In spite of the fact that the misfit parameters are larger for the elastic moduli than for the lattice parameters, the former give lower interaction energies and can in general be neglected.
The interaction energies between dislocations and solutes give a direct contribution to the activation energy for creep. This is the main reason why alloys typically have higher activation energy for creep than that for self-diffusion.
Slowly diffusion solute elements give rise to a drag stress. This drag stress is proportional to the interaction energies squared and inversely proportional to the diffusion coefficient of the solutes.
Since the diffusion coefficient appears in the denominator in the expression for solute drag, the effect is small or negligible for fast diffusion elements like interstitial elements. Instead the dislocations must break away from the fast diffusing elements to be able to move.
To verify the models it is suitable to study systems with only one main solute that contributes to the creep strength and without particles present. For slowly diffusion solutes the system Al–Mg alloys at around 300 °C has been chosen. The model can accurately describe the complex dependence on the creep stress and the Mg content. In the past it was necessary to involve several changes in the creep mechanisms which is no longer the case.
For fast diffusion elements, P in copper has been considered. The addition of 50 ppm P raises the creep strength significantly and that is possible to model quite well.

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Vorheriges Kapitel Creep with Low Stress Exponents

Nächstes Kapitel Precipitation Hardening

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Titel: Solid Solution Hardening
verfasst von: Rolf Sandström
Verlag: Springer Nature Switzerland
Buch: Basic Modeling and Theory of Creep of Metallic Materials
Print ISBN: 978-3-031-49506-9

Electronic ISBN: 978-3-031-49507-6

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-49507-6_6

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