<COEFFICIENT_MATRIX>

Description

This object is used to enter coefficients for a desired form of 4th order coefficient matrix. The use of these matrices is often as an elasticity matrix, so we sometimes refer to the object as ELASTICITY.

Syntax

These objects take a list of coefficient declarations after the type keyword is given. There are no explicit restrictions on the types of coefficients or their dependencies. The different types (keywords) available are the following:

CODE	DESCRIPTION
diagonal	1 or dim coefficients
isotropic	2 coefficients
cubic	3 coefficients
transverse	5 coefficients
orthotropic	9 coefficients
anisotropic	21 coefficients

The default type depends on its application. For elasticity objects this is isotropic. The anisotropic model represents the most general elasticity model. Coefficient names follow the same convention to describe components of the coefficient matrix. This sets the coefficient name for a component \(C_{ijkl}\) to Cijkl. For example, a component \(y_{1212}\) is named y1212.

Whenusing elasticity objects, special care has to be taken with respect to the nomenclature of the components. When representing a tensor of elasticity in 6\(\times\)6 form, the Voigt convention is assumed, but with one important difference: the order of storage is

(330)\[11 \rightarrow 1 \mbox{, } 22 \rightarrow 2 \mbox{, } 33 \rightarrow 3 \mbox{, } 12 \rightarrow 4 \mbox{, } 23 \rightarrow 5 \mbox{, } 31 \rightarrow 6 \mbox{ (Zebulon) } \nonumber\]

instead of the usual

(331)\[11 \rightarrow 1 \mbox{, } 22 \rightarrow 2 \mbox{, } 33 \rightarrow 3 \mbox{, } 23 \rightarrow 4 \mbox{, } 31 \rightarrow 5 \mbox{, } 12 \rightarrow 6 \mbox{ (Voigt) } \nonumber\]

For instance, a Voigt coefficient \(c_{66}\) becomes c44 or y1212 in Zmat/Zebulon.

The matrix forms are given below:

isotropic

The isotropic model allows several methods of entering the coefficients for convenience. The primary method gives the Young’s modulus (\(E\)) and Poisson coefficient (\(\nu\)) which correspond to the names young and poisson respectively.

(332)\[D_{ij} = {E\over1+\nu} + {\nu E\over (1-2\nu)(1+\nu)} \delta_{ij}\]

An alternate definition defines the shear modulus (\(G\) or \(\mu\)) and the bulk modulus (\(K\) or \(\kappa\)). These correspond to the coefficient names G or mu, and K or kappa. Transverse isotropy has two directions which are equivalent. There are two ways to specify transverse elasticity parameters.

• Elasticity modules

  Direction 1 is the longitudinal one and directions 2 and 3 are the two transverse ones. The elasticity is specified with 5 classical parameters : \(El\), \(Et\), \(nult\), \(nutt\) and \(Glt\). They define the Hooke relation such as :

  (333)\[\begin{split} \left( \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{22} \\\varepsilon_{33} \\\varepsilon_{23}\\\varepsilon_{13}\\\varepsilon_{12} \end{array}\right) = {\left\lgroup\begin{matrix} \frac{1}{El}&\frac{-nult}{Et}&\frac{-nult}{Et} &0 &0 &0 \\ &\frac{1}{Et}&\frac{-nutt}{Et} &0 &0 &0 \\ & &\frac{1}{Et} &0 &0 &0 \\ &&&\frac{1+nutt}{Et}&0&0 \\ &{\rm sym}&& &\frac{1}{2Glt}&0 \\ &&& & &\frac{1}{2Glt} \\ \end{matrix} \right\rgroup } \left( \begin{array}{c} \sigma_{11} \\ \sigma_{22} \\\sigma_{33} \\\sigma_{23}\\\sigma_{13}\\\sigma_{12} \end{array}\right)\end{split}\]

  An old parameter named \(glt\) (instead of \(Glt\)) is defined as :

  (334)\[\sigma_{12}=glt\, \varepsilon_{12}\]

  whereas \(Glt\) is defined as :

  (335)\[\sigma_{12}=2\, Glt\, \varepsilon_{12}\]

  This old parameter is still available for compatibility with old data sets.

• Matrix coefficients

  The program allows the user to specify which terms in the elastic matrix will be set equal, with a tag input after the transverse keyword. It is best to explain this syntax through an example:

  (336)\[\begin{split}D = {\left\lgroup\begin{matrix} c_{11}&c_{12}&c_{13}&0&0&0 \\ &c_{11}&c_{13}&0&0&0 \\ &&c_{33}&0&0&0 \\ &&&{1\over 2}(c_{11}-c_{12})&0&0 \\ &{\rm sym}&&&c_{55}&0 \\ &&&&&c_{55} \\ \end{matrix} \right\rgroup }\end{split}\]

  which is created by **elasticity transverse and the coefficients c11, c12, c13, c33 and c55 (or the corresponding y1111 etc.).

  For the moment it is only possible to work with c11 and c22 equal instead of, for instance, equal c22 and c33.

  The following relations are also given for reference in the case of \(c_{11} = c_{22}\):

  (337)\[\begin{split}\begin{aligned} c_{11} &= \frac{(1-n\nu_{zx}^2)E_{x}}{AB} \\ c_{12} &= \frac{(\nu_{xy}+n\nu_{zx}^2)E_{x}}{AB} \\ c_{13} &= \frac{\nu_{zx}E_{x}}{B} \\ c_{33} &= \frac{(1-\nu_{xy})E_{z}}{B} \\ c_{44} &= \dfrac{1}{2}(C_{11} - C_{12}) \\ c_{55} &= C_{66} = G_{xy} = G_{xz} \\ c_{xy} &= \frac{E_x}{2(1+\nu_{xy})}\\ \end{aligned} n = E_x / E_z \qquad A = 1+\nu_{xy} \qquad B = 1 - \nu_{xy} - 2n\nu_{zx}^2\end{split}\]

cubic

(338)\[\begin{split}D = {\left\lgroup \begin{matrix} y_{1111}&y_{1122}&y_{1122}&0&0&0 \\ &y_{1111}&y_{1122}&0&0&0 \\ &&y_{1111}&0&0&0 \\ &&&y_{1212}&0&0 \\ &{\rm sym}&&&y_{1212}&0 \\ &&&&&y_{1212} \\ \end{matrix} \right\rgroup }\end{split}\]

orthotropic

(339)\[\begin{split}D = {\left\lgroup \begin{matrix}y_{1111}&y_{1122}&y_{3311}&0&0&0 \\ &y_{2222}&y_{2233}&0&0&0 \\ &&y_{3333}&0&0&0 \\ &&&y_{1212}&0&0 \\ &{\rm sym}&&&y_{2323}&0 \\ &&&&&y_{3131} \\ \end{matrix}\right\rgroup }\end{split}\]

An alternative naming uses c11, c22, c33, c44, c55, c66, c12, c13, and c23 (or their symmetric counterparts).

anisotropic

(340)\[\begin{split}D = {\left\lgroup\begin{matrix} y_{1111}&y_{1122}&y_{1133}&y_{1112}&y_{1123}&y_{1131} \\ &y_{2222}&y_{2233}&y_{2212}&y_{2223}&y_{2231} \\ &&y_{3333}&y_{3312}&y_{3323}&y_{3331} \\ &&&y_{1212}&y_{1223}&y_{1231} \\ &{\rm sym}&&&y_{2323}&y_{2331} \\ &&&&&y_{3131} \\ \end{matrix}\right\rgroup }\end{split}\]

Example

**elasticity isotropic
   young   200000.
   poisson 0.3

**elasticity cubic
   y1111 162321.0
   y1122  78075.0
   y1212 110615.0

**elasticity
   young     T
     200000. 0.
     200000. 1000.
   poisson  0.3

**elasticity transverse
   El     115000.
   Et       8500.
   Glt      4500.
   nult        0.32
   nutt        0.40