Ab Initio Elastic And Mechanical Properties

By Author: Rakheehkr
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Ab Initio Study of Elastic and Mechanical Properties in FeCrMn Alloys
Abstract: Mechanical properties of FeCrMn-based steels are of major importance for practical applications. In this work, we investigate mechanical properties of disordered paramagnetic fcc FeCr10–16Mn12–32 alloys using density functional theory. The effects of composition and temperature changes on the magnetic state, elastic properties and stacking fault energies of the alloys are studied. Calculated dependencies of the lattice and elastic constants are used to evaluate the effect of the solid solution strengthening by Mn and Cr using a modified Labusch-Nabarro model and a model for concentrated alloys. The effect of Cr and Mn alloying on the stacking fault energies is calculated and discussed in connection to possible deformation mechanisms. Keywords: first principles calculations; austenitic steels; mechanical properties; elastic constants; disordered alloys; paramagnetism; solid solution strengthening; stacking fault energy.

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1. Introduction:
High ...
... strength in steels is typically achieved at the cost of reduced ductility. However, this is not the case for transformation induced plasticity (TRIP) and twinning induced plasticity (TWIP) steels. These materials achieve high strength levels through enhanced strain hardening which arises mainly from the interaction of dislocations with stacking faults, twins or e-martensite. In addition, the interaction of dislocations with solute atoms gives rise to solid solution strengthening (SSS) which is normally difficult to investigate experimentally as its influence on the strength is difficult to separate from the TWIP and TRIP effects. Therefore, the role of SSS in these classes of steel remains a largely under-investigated phenomenon. Manganese and chromium are considered to be the main alloying elements in TWIP and TRIP steels characterized by high strength and high ductility. Manganese is known as an austenite stabilizer and is used instead of nickel to lower the costs. Chromium provides corrosion resistance if used above 12 wt.% and increases the nitrogen solubility, which in its turn is used together with carbon to stabilize the austenitic matrix and also the strength of steels. Presence of C and N in steel makes experimental investigation of the effect of the key alloying elements Mn and Cr on the mechanical properties and possible deformation mechanisms in high strength and high ductility steels a challenging task. The problem arises from the fact that even high purity steels often contain impurities and other allying elements that may have a sizeable effect on SSS and the stacking fault energy (SFE) and thus influence the deformation mechanism. Another problem is the phase stability that is affected by alloying and even small fractions of C and N may change microstructure and phase composition of an austenitic steel. These problems can be avoided in predictive first principles calculations where the crystal structure and chemical composition of an alloy of interest is fully controlled and such properties as SSS and SFE can be investigated rather precisely.

In this paper, we use density functional theory (DFT) to estimate the SSS of FeCrMn alloys based on mechanical models. To this end, we calculate the equilibrium lattice constant and the bulk modulus as a function of Cr and Mn content which serve as input parameters to the existing SSS-models. The FeCrMn alloys are magnetically and chemically disordered which poses special requirements to the used DFT methodology. To the best of our knowledge, such ab initio informed modeling of SSS has not been carried out so far. Previous works have relied either on experimental input or on semi-empirical potential modeling. In addition, we investigate the SFE of FeCrMn alloys, which is a crucial design parameter in austenitic steels with regard to the occurring deformation mechanisms (TRIP, TWIP or dislocation slip).

2. Methodology

2.1. Electronic Structure Calculations
The exact muffin-tin orbital (EMTO) method implemented in the Green’s function formalism and combined with the full charge density (FCD) technique has been used in the coherent potential approximation (CPA) calculations of disordered alloys. The paramagnetic state of these alloys has been modeled by the disordered local moment (DLM) model. All self-consistent DLM-CPA calculations have been performed using the orbital momentum cut-off of lmax = 3 for partial waves. The integration over the Brillouin zone has been performed using 37 × 37 × 37 Monkhorst-Pack of k-points grids [29] for the fcc, 37 × 37 × 23 for the hcp and 27 × 27 × 13 for the double hexagonal close packed (dhcp) structures, respectively. The core states have been recalculated at each self-consistent iteration. The screening constants for the screened Coulomb interactions have been obtained by the EMTO-LSGF (locally self-consistent Green function) method.
2.2. Elastic Constants Calculations

Elastic constants and elastic moduli in this work have been calculated following the methodology described in detail. Three independent elastic constants for a cubic system can be represented by the bulk modulus B, C 0=(C11 − C12)/2 and C44. B has been calculated using the Birch-Murnaghan fit of the equation of state (EQOS). The EQOS has been calculated using the Wigner-Seiz radii (RWS) of the fcc alloys from 2.50 to 2.70 a.u. with a step of 0.02 a.u. C 0 and C44 have been calculated using volume conserving orthorhombic and monoclinic strains, respectively. The value of distortion x in C 0 and C44 calculations has been varied from zero (for the equilibrium state) to 0.05 with a step size of 0.01, in accordance with Mehl et al.’s prescription.

2.3. Solute Solution Strengthening Model
We use two different approaches for the calculation of SSS, the model by Labusch-Nabarro (LN), which we call the Labusch-Nabarro model in the following, and the model by Varvenne et, named as the VC model in the following.
Within the LN model, solid solution strengthening for a multi-component alloy can be calculated as
τLN = AG∑n (e n L 2 cn) 2/3,
where cn is the concentration of solute n and G is the isotropic shear modulus which we obtain from the elastic constants by Voigt averaging. The constant factor A is equal to 101/3/2 · 120−4/3 assuming that the parameter w in the LN model is equal to 5b where b is the Burgers vector. Misfit parameters treat the two main types of interactions between dislocations and solute atoms. The first misfit parameter (e n b , Equation (2)) arises due to different sizes of the alloying elements compared to the matrix elements, which leads to the strain field around solute atoms (lattice misfit). Materials 2019, 12, 1129 3 of 16 The strain field of a dislocation interacts with the strain field of solute atoms, more energy is required to move the dislocation further (paraelastic interaction).
e n b = 1 b db dcn
The second misfit parameter (e n G , Equation (3)) arises because solute atoms have a different shear modulus than the matrix atoms (modulus misfit). Therefore, dislocations containing solute atoms have a different elastic energy compared to dislocations containing only matrix atoms and more energy is required for the movement of a dislocation containing solute atoms (dielastic interaction).
e n G = 1 G dG dcn
The two misfit parameters are combined into the single misfit parameter e n L [17,21]:
where α 0 = 0.123, ν is the Poisson ratio, f1 is a parameter which equals to 0.35 provided the SFE is below 100 mJ/m2 , and ∆Vn is the volume mismatch between the effective medium and element n. The volume mismatch is the key quantity for strengthening. In the VC model, no distinction between lattice misfit and modulus misfit is made. The quantities τLN, τVC correspond to the critical resolved shear stress (CRSS) required to move dislocations through the solid solution at 0 K. For the VC model we also calculate the temperature dependence of τVC following the method outlined in Ref. [15].
2.4. Stacking Fault Energy Calculations

The intrinsic stacking-fault (SF) is one of the simplest planar defects of the crystal lattice. It is characterized by a fault in the usual ABC planar stacking sequence of the fcc structure, ...ABCAB|ABC..., which resembles locally the stacking sequence of the hcp structure. In the framework of the axial Ising model (AIM) [42,43], the SFE γ can be determined in terms of the total energies of the fcc, hcp, and dhcp structure.
where Ff cc, Fhcp and Fdhcp are the free energies of the fcc, hcp and dhcp phases and T is temperature. This formulation accounts for the interactions between the next nearest neighbor stacking plains and is knowns as the axial next nearest neighbor Ising model (ANNNI). The ANNNI has been shown to be a reasonable choice in terms of the accuracy and computational costs of required DFT calculation in the case of fcc Fe and FeMn alloys [13] and therefore has been selected as the method of choice in our study.