Orthotropic Elasticity Based on Engineering Constants

Orthotropic elasticity can be defined by providing the engineering constants: the Young's moduli, Poisson's ratios, and shear moduli associated with the three principal material directions.

{\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = [\begin{matrix} 1 / E_{1} & - ν_{21} / E_{2} & - ν_{31} / E_{3} & 0 & 0 & 0 \\ - ν_{12} / E_{1} & 1 / E_{2} & - ν_{32} / E_{3} & 0 & 0 & 0 \\ - ν_{13} / E_{1} & - ν_{23} / E_{2} & 1 / E_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 / G_{12} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 / G_{13} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 / G_{23} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} .

The quantity

ν_{i j}

has the physical interpretation of the Poisson's ratio that characterizes the transverse strain in the j-direction, when the material is stressed in the i-direction. In general,

ν_{i j}

is not equal to

ν_{j i}

: they are related by

ν_{i j} / E_{i} = ν_{j i} / E_{j}

. The engineering constants can also be given as functions of temperature and other predefined fields, if required.

The restrictions on the elastic constants due to material stability are

E_{1}, E_{2}, E_{3}, G_{12}, G_{13}, G_{23} > 0

| ν_{12} | < {(E_{1} {/ E}_{2})}^{1 / 2}

| ν_{13} | < {(E_{1} {/ E}_{3})}^{1 / 2}

| ν_{23} | < {(E_{2} {/ E}_{3})}^{1 / 2}

\det (D^{e l}) = {1 - ν}_{12} ν_{21} {- ν}_{23} ν_{32} {- ν}_{31} ν_{13} - 2 ν_{21} ν_{32} ν_{13} > 0.

Using the relations

ν_{i j} / E_{i} = ν_{j i} / E_{j}

the second, third, and fourth restrictions in the above set can also be expressed as

| ν_{21} | < {(E_{2} {/ E}_{1})}^{1 / 2}

| ν_{31} | < {(E_{3} {/ E}_{1})}^{1 / 2}

| ν_{32} | < {(E_{3} {/ E}_{2})}^{1 / 2} .

Parameters


Input Data	Description
E1	Young's modulus in the first local direction, $E_{1}$ .
E2	Young's modulus in the second local direction, $E_{2}$ .
E3	Young's modulus in the third local direction, $E_{3}$ .
Nu12	Poisson's ratio in the plane defined by the first and second local directions, $ν_{12}$ .
Nu13	Poisson's ratio in the plane defined by the first and third local directions, $ν_{13}$ .
Nu23	Poisson's ratio in the plane defined by the second and third local directions, $ν_{23}$ .
G12	Shear modulus in the plane defined by the first and second local directions, $G_{12}$ .
G13	Shear modulus in the plane defined by the first and third local directions, $G_{13}$ .
G23	Shear modulus in the plane defined by the second and third local directions, $G_{23}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.