Theory Background

The strain-energy method to calculate TOECs is based on the conituumn elasticity theory. The base equation is as follows (note: the Einstein summation notation is used in the following equations):

\[E=E_0+\frac{1}{2!} V_0 C_{ijkl} \eta_{ij} \eta_{kl} + \frac{1}{3!} V_0 C_{ijklmn} \eta_{ij} \eta_{kl} \eta_{mn} + ...\]

Where

• \(E\) is the energy of the deformed structure
• \(E_0\) is the energy of the initial structure
• \(V_0\) is the volume of the initial structure
• \(C_{ijkl}\) is SOECs
• \(C_{ijklmn}\) is TOECs
• \(\eta_{ij}\) is the Lagrangian strain

And the Lagrangian strain can be written as follows:

\[\eta_{ij} = \frac{1}{2} (F_{ki} F_{kj} - \delta_{ij})\]

The F is the deformation gradient, and can be expressed by the lattic vector of deformed structure(r’) and intial structure (r).

\[r^' = Fr\]

when using symmetrical strain, the deformation gradient can be expressed by the Lagrangian strain.

\[ \begin{align}\begin{aligned}F = Q \sqrt{\lambda} Q^T\\Y = 2\eta + I\\Q = (y_1, y_2, y_3)\\\lambda = diag(\lambda_1, \lambda_2, \lambda_3)\end{aligned}\end{align} \]

where \(y_i\) and \(\lambda_i\) are the eigenvector and eigenvalue of Y, I is the identity matrix.

For each strain mode, we only adjust the amplitude. Hence, we can express the strain as a function of strain amplitude \(\eta\), then we have

\[\frac{E-E_0}{V_0} = \frac{1}{2} A_2 \eta^2 + \frac{1}{6} A_3 \eta^3 + O(\eta^4)\]

Finally, we can choose different strain modes, then get different equations about TOECs and SOECs. By solving the equations, we can get the value of SOECs and TOECs.