Mechanical properties of two-dimensional sheets of TiO$$_2$$: a DFT study

Screening rare‐earth aluminates as promising thermal barrier coatings by high‐throughput first‐principles calculations

“…The convergence criteria for optimizing the internal freedoms of atoms were the same as those used before. The second‐order elastic constants ( c ij ) were calculated by fitting the stress–strain ( σ i – ε i ) relationship 32,33 : σ i = c ij · ε j , in which ε was the applied strain tensor. $$$$\begin{equation} \varepsilon = \left( { \def\eqcellsep{&}\begin{array}{ccc} {{\varepsilon _{11}}}&{{\varepsilon _{12}}/2}&{{\varepsilon _{13}}/2}\\[8pt] {{\varepsilon _{12}}/2}&{{\varepsilon _{22}}}&{{\varepsilon _{23}}/2}\\[8pt] {{\varepsilon _{13}}/2}&{{\varepsilon _{23}}/2}&{{\varepsilon _{33}}} \end{array} } \right) \end{equation}$$$$…”

“…The convergence criteria for optimizing the internal freedoms of atoms were the same as those used before. The second-order elastic constants (c ij ) were calculated by fitting the stress-strain (σ i -ε i ) relationship 32,33 : σ i = c ij ⋅ε j , in which ε was the applied strain tensor.…”