MAT9OR

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Example

Definitions

Comments

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:

   $$E_i>{\nu }_{\mathit{ij}}^2{E}_j\hspace{0.17em}\text{and}\hspace{0.17em}1-{\nu }_{12}{\nu }_{21}-{\nu }_{23}{\nu }_{32}-{\nu }_{31}{\nu }_{13}-2{\nu }_{21}{\nu }_{32}{\nu }_{13}>0$$

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:

   $$\begin{array}{l}{E}_1=E_2=E_p\hfill \\ {\nu }_{31}={\nu }_{32}={\nu }_{\mathit{np}}\hfill \\ {\nu }_{13}={\nu }_{23}={\nu }_{\mathit{pn}}\hfill \\ G_{13}=G_{23}=G_n\hfill \end{array}$$
   with $\frac{{\nu }_{\mathit{np}}}{E_n}=\frac{{\nu }_{\mathit{pn}}}{E_p}$ and $G_p=\frac{E_p}{2(1+{\nu }_p)}$

   There are five independent material constants for normal anisotropy ( $E_p$ , $E_n$ , $v$ _pn, $v$ _np, and G_n).

   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:
   $$G_{11}=\frac{1-{\nu }_{23}{\nu }_{32}}{E_2{E}_3\Delta }$$
   $$G_{22}=\frac{1-{\nu }_{31}{\nu }_{13}}{E_3{E}_1\Delta }$$
   $$G_{33}=\frac{1-{\nu }_{12}{\nu }_{21}}{E_1{E}_2\Delta }$$
   $$G_{44}=G_{12}$$
   $$G_{55}=G_{23}$$
   $$G_{66}=G_{31}$$
   $$G_{12}=G_{21}=\frac{{\nu }_{21}+{\nu }_{31}{\nu }_{23}}{E_2{E}_3\Delta }=\frac{{\nu }_{12}+{\nu }_{13}{\nu }_{32}}{E_1{E}_3\Delta }$$
   $$G_{13}=G_{31}=\frac{{\nu }_{31}+{\nu }_{21}{\nu }_{32}}{E_2{E}_3\Delta }=\frac{{\nu }_{13}+{\nu }_{12}{\nu }_{23}}{E_1{E}_2\Delta }$$
   $$G_{23}=G_{32}=\frac{{\nu }_{32}+{\nu }_{31}{\nu }_{12}}{E_3{E}_1\Delta }=\frac{{\nu }_{23}+{\nu }_{13}{\nu }_{21}}{E_1{E}_2\Delta }$$

   Where,

   $\Delta =\frac{1}{E_1{E}_2{E}_3}\left|\begin{array}{ccc}1& -{\nu }_{21}& -{\nu }_{31}\\ -{\nu }_{12}& 1& -{\nu }_{32}\\ -{\nu }_{13}& -{\nu }_{23}& 1\end{array}\right|$

   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.

7. For material-dependent Rayleigh damping, the equivalent viscous damping, $C$ , is defined as:
   $$C=\text{ALPHA}*M+\text{BETA}*K$$
   Where,

   ALPHA and BETA

   Defined on the RAYL continuation line on the material entry

   $M$

   Mass matrix

   $K$

   Stiffness matrix

   Supported solutions for material-dependent Rayleigh damping on MAT9OR:
   • Direct Frequency Response
   • Modal Frequency Response
   • Direct Transient Response
   • Modal Transient Response
   • Nonlinear Transient Analysis