Zener ratio

The Zener ratio is a dimensionless number that is used to quantify the anisotropy for cubic crystals. It is sometimes referred as anisotropy ratio and is named after Clarence Zener.^[1] Conceptually, it quantifies how far a material is from being isotropic (where the value of 1 means an isotropic material).

Its mathematical definition is^[1][2]

${\displaystyle a_{r}={\frac {2C_{44}}{C_{11}-C_{12}}},}$

where ${\displaystyle C_{ij}}$ refers to Elastic constants in Voigt notation.

Cubic materials[]

Cubic materials are special orthotropic materials that are invariant with respect to 90° rotations with respect to the principal axes, i.e., the material is the same along its principal axes. Due to these additional symmetries the stiffness tensor can be written with just three different material properties like

${\displaystyle {\underline {\underline {\mathsf {C}}}}={\begin{bmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\C_{12}&C_{11}&C_{12}&0&0&0\\C_{12}&C_{12}&C_{11}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&C_{44}\end{bmatrix}}\quad .}$

The inverse of this matrix is commonly written as^[3]

${\displaystyle {\underline {\underline {\mathsf {S}}}}={\begin{bmatrix}{\tfrac {1}{E}}&-{\tfrac {\nu }{E}}&-{\tfrac {\nu }{E}}&0&0&0\\-{\tfrac {\nu }{E}}&{\tfrac {1}{E}}&-{\tfrac {\nu }{E}}&0&0&0\\-{\tfrac {\nu }{E}}&-{\tfrac {\nu }{E}}&{\tfrac {1}{E}}&0&0&0\\0&0&0&{\tfrac {1}{G}}&0&0\\0&0&0&0&{\tfrac {1}{G}}&0\\0&0&0&0&0&{\tfrac {1}{G}}\\\end{bmatrix}}\quad .}$

where ${\displaystyle {E}\,}$ is the Young's modulus, ${\displaystyle G\,}$ is the shear modulus, and ${\displaystyle \nu \,}$ is the Poisson's ratio. Therefore, we can think of the ratio as the relation between the shear modulus for the cubic material and its (isotropic) equivalent:

${\displaystyle a_{r}={\frac {G}{E/[2(1+\nu )]}}={\frac {2(1+\nu )G}{E}}\equiv {\frac {2C_{44}}{C_{11}-C_{12}}}.}$

Universal Elastic Anisotropy Index[]

The Zener ratio is only applicable to cubic crystals. To overcome this limitation, a 'Universal Elastic Anisotropy Index (AU)' ^[4] was formulated from variational principles of elasticity and tensor algebra. The AU is now used to quantify the anisotropy of elastic crystals of all classes.

See also[]

• Anisotropy
• Orthotropic material
• Linear elasticity

References[]