The superposition of strengthening contributions in engineering alloys
Introduction
In metallic alloys, the yield stress and work hardening are governed by the interaction between mobile dislocations and obstacles. Common obstacles include solute atoms, solute clusters, precipitates and forest dislocations. The appropriate way to add the strengths of multiple obstacle to determine the macroscopic flow stress is complex owing to the statistical nature by which a dislocation samples the distribution of obstacle in the glide plane [1].
The simplest approach to addition of flow stress components can be traced back to the case of grain size strengthening examined by Cottrell [2]. Cottrell assumed that the contributions from the intrinsic resistance to dislocation motion and the long range stress contribution from grain boundaries could be added in a linear manner
Subsequently, Koppenall and Kuhlmann-Wilsdorf [3] considered the superposition of flow stress components for the case of irradiated copper single crystals. They suggested that the strengthening contributions from point defects generated from irradiation damage and the pre-existing dislocations could be added geometrically (a formulation often referred as Pythagorean superposition)
On the other hand, Labusch [4] proposed a statistical theory for strengthening of solid solutions that suggested yet another way to superimpose two strengthening contributions, i.e.
Given the statistical nature of the problem, Foreman and Makin [5] were the first to use areal glide simulations to examine the motion of a dislocation across a glide plane populated with either one or two sets of obstacles. Based on their results, they found that the Pythagorean superposition, Eq. (2) gave good agreement with exceptions (e.g. when a few strong obstacles are added to many weak obstacles); a conclusion supported by the early review from Brown and Ham [1].
The seminal review of Kocks, Argon and Asbhy [6] re-examined the question of flow stress superposition laws. General qualitative guidance was provided to the reader according to what they refer to as “increasing structural scale”. They examined a number of scenarios , for example: (i) linear superposition of lattice resistance and the resistance due to strong obstacles such as forest dislocations or non-shearable precipitates, (ii) linear superposition of long range barriers (e.g. grain boundaries) and “small scale glide resistance” (lattice resistance, discrete obstacles such as small precipitates or forest dislocations) and (iii) the case of two sets of discrete obstacles of identical strength, but with different densities where the density of obstacles is simply summed, i.e. the Pythagorean addition law applies. Clearly, while these general guidelines are valuable, they leave many scenarios where ambiguity exists in which value of the superposition exponent to use, i.e. 1, 3/2 or 2.
In his review of precipitation hardening, Ardell [7], following Niete et al. [8], suggested that the superposition law could be generalized aswhere the exponent q is a parameter varying between 1 and 2. Ardell discussed specific examples were q could have values other than 1, 3/2 or 2. The more recent work of Bütter and Nembach [9], Neite et al. [8] and Zhu et al. [10] considered q as a parameter fit to simulation results without any explicit guidance on how its value should physically related to obstacle strength or density. The analysis of experimental data consequently, has employed empirically fit values for q [11], [12].
Recently, there has been a renewed interest in this question. Dong et al. [13] conducted areal glide simulations to examine in detail the situation for combinations of extremely weak obstacle (breaking angles larger than 176°). In that case, it was found that the Pythagorean addition was appropriate. However, they also did note that when the density of the weakest set of obstacles is less than 0.06%, this result breaks down. In these conditions, q was found to be a function of the density. It is worth noting, however, that this is a rather restrictive regime which is not of practical significance for engineering alloys.
As noted above, given only approximate theoretical recommendations, experimental studies have proposed empirical fits to select the best value for q [11], [12], [14], [15]. Alternatively, single values of q (q=1 or q=2) have been chosen to fit a wide range of data [16], [17], [18]. The goal of this paper is to use results from areal glide simulations to derive a mathematical expression which could be used to determine the addition law parameter, q, for a wide range of obstacle strengths and densities. Due to the nature of the areal glide model, this expression would be valid for alloys featuring obstacles of which dimensions are small compared to their average spacing and thus could be reasonably modeled as point obstacles. Further, the question of how to sum flow stresses when more than two sets of obstacles is considered.
To simulate the contribution from different sets of obstacles to the critical resolved shear stress (CRSS), a 2D areal glide model based on the method proposed by Nogaret and Rodney has been implemented [19]. The model is first applied to simulate a single set of randomly distributed obstacles and compared with the results from the classic “circle rolling” method first used by Foreman and Makin [20]. Next, the case of two sets of randomly distributed obstacles is examined and an expression for q proposed. Then, the case of three sets of obstacles is considered and one approach to implement the addition law is proposed. Finally, the developed formulae are applied to experimental data for aged Al–Mg–Sc specimens [12] where three sets of obstacles contribute to the strength.
Section snippets
Computational model
The glide of a single dislocation through an array of randomly distributed obstacles is simulated with a two-dimensional areal glide model similar to the one developed by Nogaret and Rodney [19]. This model differs from the “circle rolling” method previously used to simulate the glide of a single dislocation through an array of point obstacles. It assumes a constant dislocation line tension with and models obstacles as points characterized by their breaking angle . Breaking angles vary
Simulations of dislocation glide through a single set of obstacles
For a single set of obstacles having a breaking angle distributed in the glide plane, Foreman and Makin [5] proposed an empirical law for the normalized CRSS
Owing to the difference between the “circle rolling” method used by Foreman and Makin and the algorithm used here, it was decided to check the applicability of Eq. (11). Fig. 3 shows that the present simulations match very well with Eq. (11) for weak obstacles but that the results diverge for
Addition law for two sets of obstacles
In many practical situations, moving dislocations interact with more than one type of obstacles. In the following, a normalized version of the commonly used addition law (Eq. 4) is used. If N1 and N2 are respectively the number of obstacles of types 1 and 2 (N=N1+N2), from Eqs. (6), (7), the normalized critical resolved shear stresses are thenUsing the densities of obstacles, X1=N1/N and X2=N2/N, the normalized addition law is
Simulations were
Addition law for three sets of obstacles
There are a number of practical scenarios where one must consider three sets of obstacles. For example, in the next section the case of an Al–Mg–Sc alloy will be considered where solute, precipitates and forest dislocations all act as obstacles. Extending the addition law from two to three (or more) sets of obstacles is non-trivial. Thus the understanding of how these obstacles contribute to the CRSS requires simulations which, to the best of our knowledge, have never been done before. Similar
An example of application of the three obstacle addition law for an Al–Mg–Sc alloy
To test the methodology developed above, the experimental stress–strain response of a precipitation hardened Al–Mg–Sc has been re-examined using the proposed addition law. This particular alloy is a Al-2.8 wt% Mg-0.16 wt% Sc alloy that was originally studied by Fazeli et al. [12]. Strengthening contributions in this alloy comes from solid solution strengthening due to Mg , Al3Sc precipitates and work hardening due to the accumulation of dislocations [24], [25], [26].
In the
Conclusion
Areal glide simulations have been used to study the motion of a dislocation across a glide plane populated with randomly distributed obstacles of different strengths. These results were utilized to derive a new expression for the exponent q in the addition law for two sets of obstacles. This expression is an analytical function of only the breaking angle of the respective sets. The addition law was then extended to the case of three sets of obstacles. These new laws were then used to re-examine
Acknowledgement
The authors would like to gratefully acknowledge the financial support from Natural Sciences and Engineering Research Council of Canada (NSERC). A. de Vaucorbeil would like to acknowledge Henry Proudhon and the École Normale Supérieure de Cachan for their support during the final year of his Master's program.
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