Modelling of the influence of alloy composition on flow stress in high-strength nickel-based superalloys
Introduction
Nickel-based superalloys display excellent high-temperature mechanical properties, particularly in creep and fatigue [1], [2], [3]. But why is this? The overarching reason is that deformation on the microscale is restricted by the precipitation hardening conferred by a significant fraction of the Ni3(Al, Ti, Ta) phase, which is commonly referred to as gamma prime (). The origin of this effect cannot be explained by differences in the elastic moduli of matrix and precipitate, since these do not differ greatly from phase to phase. Instead, it is the fault energies associated with anti-phase boundaries (APBs) and stacking faults (SFs) which are responsible for it [4], [5]. Their magnitudes severely limit the penetration of the phase achieved by the dislocations introduced during deformation, and thus bulk plastic flow.
It follows that the fault energies are of great significance to the physical metallurgy of Ni-based superalloys. This is emphasized further by considering theoretical expressions which describe the how yield strength varies in the alloy with respect to precipitate size and distribution. Take the case of so-called weakly and strongly coupled dislocations; Eqs. (1a), (1b) describe permeation of a superalloy containing a fraction f of precipitates. The shear yield stress varies according to [6]:for the cases of weakly and strongly coupled dislocations, respectively. Here, is the APB energy, b is the Burgers vector, r is the mean precipitate radius, is the line tension, G is the shear modulus of the phase and w is a dimensionless constant to account for uncertainties, which is expected to be of the order of unity. One can see that the APB energy is expected to influence the flow properties substantially. Moreover, some of the dependence of mechanical strength on alloy chemistry is due to differences in the APB energy, which depends upon the chemistry of the phase. Elements such as Ti and Ta which partition strongly to , which thus influence the APB energy, might therefore be expected to enhance the yield stress. The stoichiometry of Ni3(Al, Ti, Ta) is also likely to be important. However, calculations are needed to deduce the precise details of these quantitative relationships.
The work reported in this paper was motivated with the above in mind. We set out to identify quantitative alloy composition/mechanical property relationships in polycrystalline Ni-based superalloys. Emphasis is placed on the APB energy and its role in conferring static strength in these materials. The paper is structured in the following way. First, the experimental and theoretical evidence for the magnitude of the APB energy is considered. Next, the dependence of the APB energy on changes in chemistry of Ni3(Al, Ti, Ta) is quantified, using methods based upon computational thermodynamics and electron structure calculations. Finally, models for the yield stress which contain a composition-dependent APB energy are considered. These are used to rationalize the role played by the composition dependence of the APB energy in conferring strength.
Section snippets
Background
In Ni-based superalloys, precipitates of the ordered phase () reside within a coherent face-centred cubic (fcc) matrix, commonly referred to as the phase. The dislocations in have a Burgers vector of on the plane, but this is not a full dislocation in the phase. Two dislocations with Burgers vector , or superlattice partial dislocations, are required to generate a full dislocation within the precipitate. In practice, a superlattice partial dislocation can
Modelling approach
The CALPHAD method for calculating the APB energy using a thermodynamic database is described in Section 3.1. The DFT calculations performed as part of this work are described in Section 3.2; these were used to validate the thermodynamic database. It is necessary to validate the thermodynamic database—which is empirically based—using ab initio calculations to ensure reasonable agreement between both methods in regions where the CALPHAD may rely on a high degree of interpolation or possibly even
Results
This section is divided into three. The first subsection compares the internal energy—in ternary Ni–Al–X alloys with both the and fcc configuration—calculated using the thermodynamic database and ab initio methods. Following this, the APB energy for Ni–Al–X alloys with the crystal structure is considered. The final section focuses on the application of the model for APB energy within the context of designing high-strength polycrystalline Ni-based superalloys for turbine disc
Discussion
The results confirm that the composition dependence of the APB can be accounted for in multicomponent Ni-based superalloys. Furthermore, for a number of Ni–Al–X alloys, modelling predictions using CALPHAD have been shown to be broadly comparable with ab initio calculations. The first subsection of this Discussion compares the APB energies determined using CALPHAD and DFT methods, with emphasis placed on some of the apparent limitations of the thermodynamic database used in this work. In Section
Conclusions
- 1.
A direct relationship for the determination of the APB energy from a CALPHAD database for the crystal structure in superalloys has been established. Predictions of APB energies on for ternary and multicomponent alloys have been obtained.
- 2.
DFT calculations, using an SQS approach, have been used to study the accuracy of the CALPHAD predictions for simple ternaries. The method employed differs from that usually employed, which makes use of supercells.
- 3.
Good correlation was observed between
Acknowledgements
The financial support of this work from the Engineering and Physical Sciences Research Council (EPSRC) and Rolls-Royce Strategic Partnership in Structural Metallic Systems for Advanced Gas Turbine Applications is greatly appreciated.
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