Analysis of elastic moduli

Overview

The analysis of the second-order elastic moduli fourth-rank tensor of a material may provide many useful information. In the following, a brief description of the theory and capabilities of Quantas within this framework is provided.

Last update

Jul 25, 2024

Author

Gianfranco Ulian

In the theory of linear elasticity, the stress tensor can be expressed in terms of strain by:

\[\sigma_{ij} = C_{ijkl} \epsilon_{kl}\]

with \(C_{ijkl}\) the components of the fourth-rank modulus tensor, whose coordinates depend on the choice of the axes. These components are also referred by physicists and engineers as elastic moduli and second-order elastic constants (SOECs), respectively. The previous equation can be inverted:

\[\epsilon_{ij} = S_{ijkl} \sigma_{kl}\]

where \(S_{ijkl}\) are the component of the compliance tensor, the inverse of the modulus tensor. A fourth-rank tensor has 81 components, but a maximum of 21 independent values, for triclinic crystals. It is also common to express the stiffness and compliance tensor components using the (engineering) notation of Voigt 1, which is often adopted because of its simplicity. It is worth noting that the notation of Voigt is not a tensor, but a matrix representation of it.

Several mechanical properties can be calculated from the elastic tensor. Those mainly reported in literature regards the polycrystalline (isotropic) elastic behaviour of a material, by means of the Voigt, Reuss and Hill averages:

\[K_V = \frac{1}{9} \big[ C_{11} + C_{22} + C_{33} + 2\big( C_{12} + C_{13} + C_{23} \big) \big]\]

\[K_R = \big[ S_{11} + S_{22} + S_{33} + 2\big( S_{12} + S_{13} + S_{23} \big) \big]^{-1}\]

\[\mu_V = \frac{1}{15} \big[ C_{11} + C_{22} + C_{33} + 3\big( C_{44} + C_{55} + C_{66} \big) - \big( C_{11} + C_{13} + C_{23} \big) \big]\]

\[\mu_R = \frac{15}{4} \Big[ S_{11} + S_{22} + S_{33} - \big( S_{11} + S_{13} + S_{23} \big) + 3\big( S_{44} + S_{55} + S_{66} \big) \Big]^{-1}\]

\[K_{VRH} = \frac{K_V + K_R}{2}\]

\[\mu_{VRH} = \frac{\mu_V + \mu_R}{2}\]

\[E_{VRH} = \frac{9 K_{VRH} \mu_{VRH}}{3K_{VRH} + \mu_{VRH}}\]

\[\nu_{VRH} = \frac{3 K_{VRH} - 2\mu_{VRH}}{2(3K_{VRH} + \mu_{VRH})}\]

where \(K\) is the bulk modulus, \(\mu\) is the shear modulus, \(E\) is the elastic (Young) modulus and \(\nu\) is the Poisson’s ratio. The subscripts R and V indicate the Reuss (lower) and Voigt (upper) bounds, respectively.

The mean shear, \(v_s\), and longitudinal, \(v_p\), wave velocities of a polycrystal with no preferred orientation of the grains depend on the coupling between grains, and can range from the Reuss limit (with free grain boundaries) to the Voigt limit (with locked grain boundaries). Most randomly oriented polycrystals have shear and Youngs moduli close to, but not identical to, the VRH averages, for which the following approximation of the mean wave velocities is valid:

\[v_s = \sqrt{\frac{\mu_{VRH}}{\rho}}\]

\[v_p = \sqrt{\frac{4 K_{VRH} + 3\mu_{VRH}}{3\rho}}\]

The calculation of the six eigenvalues of the second-order elastic tensor allows to define the mechanical stability of the solid: if any of the eigenvalues is negative, the system is unstable. 2

If the system is mechanically stable, the spatial variation of the cited properties (Young’s, bulk and shear moduli, Poisson’s ratio and seismic wave velocities) can be calculated. Young’s modulus, \(E\) and linear compressibility, \(\beta = 1 / K\) are obtained by considering a single unit vector \(\vec{a}\), which can be parametrized in spherical coordinates by considering two angles, \(0 \leq \theta \leq \pi\) and \(0 \leq \phi \leq 2\pi\):

\[\begin{split}\vec{a} = \begin{bmatrix} \sin(\theta)\cos(\phi) \\ \sin(\theta)\sin(\phi) \\ \cos(\theta) \end{bmatrix}\end{split}\]

The shear modulus, \(\mu\) and the Poisson’s ratio, \(\nu\) depends on two unit vectors, \(\vec{a}\) (stress direction) and \(\vec{b}\) (measurement direction). In this case, according to literature, 3 \(,\) 4 three angles are required, with \(\theta\) and \(\phi\) the same as above and the third angle \(0 \leq \chi \leq 2\pi\). This leads to the parametrization of the \(\vec{b}\) vector as :

\[\begin{split}\vec{b} = \begin{bmatrix} \cos(\theta)\cos(\phi)\cos(\chi) - \sin(\phi)\sin(\chi) \\ \cos(\theta)\sin(\phi)\cos(\chi) - \cos(\phi)\sin(\chi) \\ -\sin(\theta)\cos(\chi) \end{bmatrix}\end{split}\]

Workflow of elastic moduli analysis

The following picture briefly shows the workflow followed by Quantas for calculating the elastic properties from any input SOEC matrix.

References

1

Nye, J.F., 1957. Physical properties of crystals. Oxford University Press, Oxford.

2

Mouhat, F., Coudert, F.X., 2014. Necessary and sufficient elastic stability conditions in various crystal systems. Phys. Rev. B 90.

3

Marmier, A., Lethbridge, Z.A.D., Walton, R.I., Smith, C.W., Parker, S.C., Evans, K.E., 2010. ElAM: A computer program for the analysis and representation of anisotropic elastic properties. Comput. Phys. Commun. 181, 2102-2115.

4

Mouhat, F., Coudert, F.X., 2014. Necessary and sufficient elastic stability conditions in various crystal systems. Phys. Rev. B 90, 224104, 224104.