Symmetry Property Relationships

Symmetry Property Relationships

Properties of crystalline materials inherently depend on the crystal symmetry of the material and symmetry stemming from constitutive relationships (cause and effect) for a particular property. For example, Hooke’s law for anisotropic elasticity, \(\sigma_{ij}=c_{_2ijkl}E_{kl}\), is a constitutive relationship that imparts the minor symmetry on the indices of the second-order elastic moduli tensor, \(c_{_2ijkl}=c_{_2jikl}=c_{_2ijlk}=c_{_2jilk}\), which results from symmetric stress \(\boldsymbol{\sigma}\) and strain \(\textbf{E}\) tensors. On this page, symmetry relationships for property tensors like \(c_{_2ijkl}\) can be found for a variety of various material properties.

Property tables have been updated to include tensors found in Tiersten’s stored energy function for electroelastic materials. Stored energy functions like the one seen below are often used to describe the coupled nonlinear electric and elastic interactions in piezoelectric and dielectric materials.
\(\begin{eqnarray}
\label{Eq5}
U&=&\frac{1}{2}c_{_2ijkl}E_{ij}E_{kl}-e_{ijk}W_iE_{jk}-\frac{1}{2}\chi_{_2ij}W_iW_j+\frac{1}{6}c_{_3ijklmn}E_{ij}E_{kl}E_{mn}+\frac{1}{2}d_{_1ijklm}W_iE_{jk}E_{lm}\nonumber\\
&&-\frac{1}{2}b_{ijkl}W_iW_jE_{kl}-\frac{1}{6}\chi_{_3ijk}W_iW_jW_k+\frac{1}{24}c_{_4ijklmnpq}E_{ij}E_{kl}E_{mn}E_{pq}+\frac{1}{6}d_{_2ijklmnp}W_iE_{jk}E_{lm}E_{np}\nonumber\\
&&+\frac{1}{4}a_{_1ijklmn}W_iW_jE_{kl}E_{mn}-\frac{1}{6}d_{_3ijklm}W_iW_jW_kE_{lm}-\frac{1}{24}\chi_{_4ijkl}W_iW_jW_kW_l\nonumber
\end{eqnarray}\)
where \(\textbf{W}\) and \(\textbf{E}\) are an electric field vector and strain tensor, respectively.

Click on the tensor to see the symmetry table:
\(\chi_{_2ij}\), \(\chi_{_3ijk}\), \(\chi_{_4ijkl}\), \(e_{ijk}\), \(d_{_1ijklm}\), \(d_{_2ijklmnp}\), \(b_{ijkl}\), \(a_{_1ijklmn}\), \(d_{_3ijklm}\), \(c_{_2ijkl}\), \(c_{_3ijklmn}\), , \(c_{_4ijklmnpq}\)