In order to facilitate the prediction of some physical properties, we propose several simple formulas based on two parameters only, the metallic valence and metallic atomic radii. Knowing the composition, for single-phase alloys, the average parameters can be calculated by the rule of mixture. The input parameters can be obtained from tabulated databases. Adopting from the literature the results of Coulomb crystal model for metals and single-phase high-entropy alloys, we have derived formulas for the shear modulus (G) and the cohesion energy (E_coh). Based on these parameters separately, we set up two formulas to estimate the hardness in the case of pure metals. For single-phase (solid-solution) HEAs, by simplifying the Maresca and Curtin model, we obtained a formula for estimating the hardness, which takes into account the atomic misfit in addition to G. The maximal hardness for single-phase HEA is approximately 600 kg/mm² and is obtained for a composition with a valence electron concentration of approximately 6 ÷ 7.

The big advantage of refractory high entropy alloys (RHEAs) is their strength at high temperatures, but their big disadvantage is their brittleness at room temperature, which prevents their machining. There is a great need to classify the alloys in terms of brittle-ductile (B-D) properties, with easily obtainable ductility indices (DIs) ready to help design these refractory alloys. Usually, the DIs are checked by representing them as a function of fraction strain, ε. The critical values of DI and ε divide the DI—ε area into four squares. In the case of a successful DI, the points representing the alloys are located in the two diagonal opposite squares, well separating the alloys with (B-D) properties. However, due to the scatter of the data, the B-D separation is not perfect, and it is difficult to establish the critical value of DI. In this paper, we solve this problem by replacing the fracture strain parameter with new DIs that scale with the old DIs. These new DIs are based on the force constant and amplitude of thermal vibration around the Debye temperature. All of them are easily available and can be calculated from tabulated data.

The relationship between the tendencies towards rigidity (measured by shear modulus, G) and hardness (measured by Vickers hardness, HV) of early transition metal (ETM)-based refractory high-entropy alloys (RHEA) and bond parameters (i.e., valence electron concentration (VEC), enthalpy of mixing (ΔH_mix)) was investigated. These bond parameters, VEC and ΔH_mix, are available from composition and tabulated data, respectively. Based on our own data (9 samples) and those available from the literatures (47 + 27 samples), it seems that for ETM-based RHEAs the G and HV characteristics have a close correlation with the bonding parameters. The room temperature value of G and HV increases with the VEC and with the negative value of ΔH_mix. Corresponding equations were deduced for the first time through multiple linear regression analysis, in order to help design the mechanical properties of ETM refractory high-entropy alloys.

First, we discuss a common feature of single-phase pure metals and amorphous and high-entropy alloys: the maximum value of hardness corresponding to a valence electron count (VEC) value of around 6.5–7. This correlation is explained by the coincidence that by subtracting the number of sp valence electrons (Nsp = 2) from the VEC we obtain the maximal number of unpaired d electrons, N_d = 4.5–5 in the 3d, 4d, and 5d rows of transition elements. These unpaired d electrons form orbital overlap bonding, which is stronger than the isotropic metallic bonds of a delocalized electron cloud. The more unpaired d electrons there are, the higher the bonding strength. Second, we will discuss the hardness formulas derived from cohesion energy and shear modulus. We will demonstrate that both types of formulas originate in the electrostatic energy density of metallic bonds, expressing a 1/R⁴ dependence. Finally, we show that only two parameters are sufficient to estimate hardness: the atomic radius and the cohesion-based valence. In the case of alloys, our formula gives a lower bound on the hardness only. It is not suitable for calculation of the hardness increase caused by solid solution, grain size, precipitation, and phase mixture.

Correlations have been found for the base value of hardness (as the ratio between the heat of fusion and molar volume) and the softening temperature (as the ratio of heat of fusion and specific heat capacity). The relative change of bulk hardness as a function of temperature, H(T), is studied by three new parametric formulas beside the well-known exponential decay and Arrhenius-type expressions. Mathematically, two formulas can be considered as deriving from the exponential decay; the third one is a new rational fraction expression based on the power of normalized temperature. The normalizing temperature is taken as the softening temperature. In the Arrhenius expression, a temperature-dependent activation energy is introduced, which increases steadily with heating but never surpasses the value of self-diffusion. This rational fracture expression has been shown to be applicable to both pure metals and alloys with arbitrary H(T) curve shapes, from convex (pure metals) to concave (alloys). A detailed description of the fitting of these parametric formulas is given, applying the H(T) data from the literature and from our own measurements. Measuring our refractory high entropy alloy (RHEA) samples, an early softening temperature, smaller than the expected half of the melting point (Ts < Tm/2) was detected, signaling a phase instability in the case of Ti-, Zr- and Hf-containing alloys.

Home-built equipment will be presented able to measure the thermal expansion (with a flat indenter) and indentation depth (with a pointed indenter) up to 1100 °C. In dilatometer mode, the allotropic phase transformations can be studied. For hardness, a Rockwell-type measurement is adopted. First, we apply a small load and measure the displacement consisting of a dominant positive thermal expansion and a small negative indentation depth contribution. Then, we repeat the thermal cycle with such a high load that the compensation appears at around 250–300 °C. With increasing temperature, the indentation depth starts to dominate and we can notice a contraction. The indentation depth as a function of temperature, ID(T), will be obtained by subtracting the high load curve from the low load curve. A new rational fraction expression will be tested to describe the thermal softening of pure metals and refractory HEAs. Still, we are working on improving the equipment to extend the working temperature up to 1200 °C.