Elastic Constants
Elastic constants (the components Cij of the stiffness tensor) describe how a crystal’s stress responds to small strains, and are computed from first principles by applying small deformations and fitting the stress or energy response. A cubic crystal has three independent constants (C11, C12, C44). Averages such as the Voigt-Reuss-Hill bulk (K) and shear (G) moduli summarize them, and derived quantities such as Pugh’s ratio K/G and the Cauchy pressure C12 - C44 are widely used as empirical descriptors of intrinsic ductility, with higher values associated with more ductile behavior.
Pugh ratio as a predictor of cracking in printed W–Ta–Nb alloys (2026)
Abdelmaqsoud and coworkers computed the elastic tensors of Wx Ta(90-x) Nb10 solid solutions with the UMA potential and DFT and found that the averaged shear modulus plateaus below 50% W because the two BCC shear constants diverge (C44 rises while the tetragonal shear C’ falls). Arguing that tungsten fails by cleavage along the tetragonal path, they redefined the Pugh ratio as K/C’. This ratio drops sharply between 50 and 60% W, which they linked to the Fermi level entering a pseudogap in the density of states. In alloys printed by directed energy deposition, intergranular crack density tracked the inverse ratio C’/K, and the two lowest-W alloys showed no visible intergranular cracking, an experimental check of an ab initio ductility descriptor that is rarely made for alloys (Abdelmaqsoud et al. 2026). Summary: abdelmaqsoud2026.
Electronic origin of the shear modulus anomaly in Ta-W (2026)
Abdelmaqsoud, Kitchin and Widom computed the cubic elastic constants of Ta(1-x)Wx with PBE DFT and traced their composition dependence to the electronic structure. The tetragonal shear modulus C’ is nearly flat up to x = 1/2 and then rises steeply, C44 follows a muted version of the same trend, and the bulk modulus increases steadily, so the Pugh ratio shows a broad peak for Ta-rich compositions and the Cauchy pressure, positive everywhere, peaks over the same range. Crystal orbital Hamilton populations and wave function plots show that electrons added up to VEC 5.5 fill T2g orbitals that bond nearest neighbors along <111> but are antibonding along the <001> next-nearest-neighbor bonds that feed only C11; once the Fermi level passes a T2g degeneracy 0.12 eV above the equiatomic Fermi level, both effects weaken, C11 rises faster and C12 flattens. Similar Pugh ratio peaks below x = 1/2 appear in VCr, NbMo, NbW and TaMo, supporting valence electron count as the controlling variable (Abdelmaqsoud et al. 2026). Summary: abdelmaqsoud2026a.