# MAT9OR

Bulk Data Entry Defines the material properties for linear, temperature-independent, and orthotropic materials for solid elements in terms of engineering constants.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MAT9OR MID E1 E2 E3 NU12 NU23 NU31 RHO
G12 G23 G31 A1 A2 A3 TREF GE
RAYL ALPHA BETA

## Example

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MAT9OR 21 1e6 1e3 1e3 0.1 0.1   1e5
1e3 1e3   1e-6 1e-6 1e-6

## Definitions

Field Contents SI Unit Example
MID Material identification number. Must be unique with respect to other MAT1, MAT2, MAT8, MAT9, and MAT9OR definitions.

No default (Integer > 0)

E1 Elastic modulus in 1-direction.

No default (Real)

E2 Elastic modulus in 2-direction.

No default (Real)

E3 Elastic modulus in 3-direction.

No default (Real)

NU12 Poisson's ratio. It is the strain in the 2-direction due to a unit strain in the 1-direction. 3

No default (Real)

NU23 Poisson's ratio. It is the strain in the 3-direction due to a unit strain in the 2-direction. 3

No default (Real)

NU31 Poisson's ratio. It is the strain in the 1-direction due to a unit strain in the 3-direction. This field can be interpreted either as NU31 (default) or NU13, based on the optional SYSSETTING(MAT9ORT). If NU13, then this Poisson’s ratio is the strain in the 1-direction due to a unit strain in the 3-direction. 3

Default = Value of NU23 field (Real)

RHO Mass density.

No default (Real)

G12 Shear modulus on plane 1-2.
G23 Shear modulus on plane 2-3.
G31 Shear modulus on plane 3-1.
Ai Coefficient of thermal expansion in the i-direction

Default =0.0 (Real)

TREF Reference temperature for the calculation of thermal loads.

Default = blank (Real or blank)

GE Structural element damping coefficient. 5

Default = 0.0 (Real)

RAYL Continuation line flag for material-dependent Rayleigh damping.
ALPHA Material-dependent Rayleigh Damping coefficient for the mass matrix.

Default = blank (Real ≥ 0.0)

BETA Material-dependent Rayleigh Damping coefficient for the stiffness matrix.

Default = blank (Real ≥ 0.0)

1. This input definition is internally converted to an equivalent MAT9 definition on reading (Comment 6). This is reflected in echoed (ECHO / ECHOON / ECHOOFF) input data and all messaging.
2. The material identification number must be unique for all MAT1, MAT2, MAT8, MAT9 and MAT9OR entries.
3. In general, $v$ 12 is not the same as $v$ 21, but they are related by $\frac{{\nu }_{\mathit{ij}}}{{E}_{i}}=\frac{{\nu }_{\mathit{ji}}}{{E}_{j}}$ .
Furthermore, material stability requires that:(1)
${E}_{i}>{\nu }_{\mathit{ij}}^{2}{E}_{j} \text{and} 1-{\nu }_{12}{\nu }_{21}-{\nu }_{23}{\nu }_{32}-{\nu }_{31}{\nu }_{13}-2{\nu }_{21}{\nu }_{32}{\nu }_{13}>0$
4. It may be difficult to find all nine orthotropic constants. In some practical problems, the material properties may be reduced to normal anisotropy in which the material is isotropic in a plane (for example, plane 1-2), and has different properties in the direction normal to this plane.
In the plane of isotropy, the properties are reduced to:(2)
$\begin{array}{l}{E}_{1}={E}_{2}={E}_{p}\hfill \\ {\nu }_{31}={\nu }_{32}={\nu }_{\mathit{np}}\hfill \\ {\nu }_{13}={\nu }_{23}={\nu }_{\mathit{pn}}\hfill \\ {G}_{13}={G}_{23}={G}_{n}\hfill \end{array}$
with $\frac{{\nu }_{\mathit{np}}}{{E}_{n}}=\frac{{\nu }_{\mathit{pn}}}{{E}_{p}}$ and ${G}_{p}=\frac{{E}_{p}}{2\left(1+{\nu }_{p}\right)}$

There are five independent material constants for normal anisotropy ( ${E}_{p}$ , ${E}_{n}$ , $v$ pn, $v$ np, and Gn).

In case the material has a planar anisotropy, in which the material is orthotropic only in a plane, the elastic constants are reduced to seven (E1, E2, E3, $v$ 12, G12, G23, and G31).

5. To obtain the damping coefficient GE, multiply the critical damping ratio, $C/{C}_{0}$ by 2.0.
6. Internal conversion from MAT9OR to MAT9. The material property fields of the MAT9 entry are calculated internally from the MAT9OR entry using:(3)
${G}_{11}=\frac{1-{\nu }_{23}{\nu }_{32}}{{E}_{2}{E}_{3}\Delta }$
(4)
${G}_{22}=\frac{1-{\nu }_{31}{\nu }_{13}}{{E}_{3}{E}_{1}\Delta }$
(5)
${G}_{33}=\frac{1-{\nu }_{12}{\nu }_{21}}{{E}_{1}{E}_{2}\Delta }$
(6)
${G}_{44}={G}_{12}$
(7)
${G}_{55}={G}_{23}$
(8)
${G}_{66}={G}_{31}$
(9)
${G}_{12}={G}_{21}=\frac{{\nu }_{21}+{\nu }_{31}{\nu }_{23}}{{E}_{2}{E}_{3}\Delta }=\frac{{\nu }_{12}+{\nu }_{13}{\nu }_{32}}{{E}_{1}{E}_{3}\Delta }$
(10)
${G}_{13}={G}_{31}=\frac{{\nu }_{31}+{\nu }_{21}{\nu }_{32}}{{E}_{2}{E}_{3}\Delta }=\frac{{\nu }_{13}+{\nu }_{12}{\nu }_{23}}{{E}_{1}{E}_{2}\Delta }$
(11)
${G}_{23}={G}_{32}=\frac{{\nu }_{32}+{\nu }_{31}{\nu }_{12}}{{E}_{3}{E}_{1}\Delta }=\frac{{\nu }_{23}+{\nu }_{13}{\nu }_{21}}{{E}_{1}{E}_{2}\Delta }$

Where,

$\Delta =\frac{1}{{E}_{1}{E}_{2}{E}_{3}}|\begin{array}{ccc}1& -{\nu }_{21}& -{\nu }_{31}\\ -{\nu }_{12}& 1& -{\nu }_{32}\\ -{\nu }_{13}& -{\nu }_{23}& 1\end{array}|$

The values of $v$ , E and G for the expressions in the above equations are taken from the NUij , Ei and Gij fields respectively of this MAT9OR entry; where i, j € {1,2,3} and the values of G11, G22, G33, G44, G55, G66, G12, G13, and G23 (see above equations) are used to populate the G11, G22, G33, G44, G55, G66, G12, G13, and G23 fields (G12=G21, G13=G31 and G23=G32 due to symmetry) of the MAT9 entry. The remaining elements of the MAT9 entry (that is G14, G15, G24, and so on) are equal to zero.