Orthotropic linear elastic material - OrthoLE

Orthotropic, linear elastic material model. The model parameters are summarized in Tab. 2. Local coordinate system, which determines axes of material orthotrophy can by specified using lcs array. This array contains six numbers, where the first three numbers represent directional vector of a local x-axis, and next three numbers represent directional vector of a local y-axis. The local z-axis is determined using the vector product. The right-hand coordinate system is assumed.

Local coordinate system can also be specified using scs parameter. Then local coordinate system is specified in so called ``shell'' coordinate system, which is defined locally on each particular element and its definition is as follows: principal z-axis is perpendicular to shell mid-section, x-axis is perpendicular to z-axis and normal to user specified vector (so x-axis is parallel to plane, with being normal to this plane) and y-axis is perpendicular both to x and z axes. This definition of coordinate system can be used only with plates and shells elements. When vector is parallel to z-axis an error occurs. The scs array contain three numbers defining direction vector . If no local coordinate system is specified, by default a global coordinate system is used.

For 3D case the material compliance matrix has the following form

$\displaystyle \mbox{\boldmath$C$}$$\displaystyle =\left[\begin{array}{cccccc} 1/E_X & -\nu_{xy}/E_x& -\nu_{xz}/E_x...
... 0 & 0 & 0 & 0 & 1/G_{xz} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/G_{xy} \end{array}\right]$ (1)

By inversion, the material stiffness matrix has the form

$\displaystyle \mbox{\boldmath$D$}$$\displaystyle =\left[\begin{array}{cccccc} d_{xx} & d_{xy} & d_{xz} & 0 & 0 & 0...
... 0\\ 0 & 0 & 0 & 0 & G_{xz} & 0\\ 0 & 0 & 0 & 0 & 0 & G_{xy} \end{array}\right]$ (2)

where $ \xi=1-(\nu_{xy}*\nu_{yx}+\nu_{yz}*\nu_{zy}+\nu_{zx}*\nu_{xz})-(\nu_{xy}*\nu_{yz}*\nu_{zx}+\nu_{yx}*\nu_{zy}*\nu_{xz})$ and
$\displaystyle d_{xx}$ $\displaystyle =$ $\displaystyle E_X(1-\nu_{yz}*\nu_{zy})/\xi,$ (3)
$\displaystyle d_{xy}$ $\displaystyle =$ $\displaystyle E_y*(\nu_{xy}+\nu_{xz}*\nu_{zy})/\xi,$ (4)
$\displaystyle d_{xz}$ $\displaystyle =$ $\displaystyle E_z*(\nu_{xz}+\nu_{yz}*\nu_{xy})/\xi,$ (5)
$\displaystyle d_{yy}$ $\displaystyle =$ $\displaystyle E_y*(1-\nu_{xz}*\nu_{zx})/\xi,$ (6)
$\displaystyle d_{yz}$ $\displaystyle =$ $\displaystyle E_z*(\nu_{yz}+\nu_{yx}*\nu_{xz})/\xi,$ (7)
$\displaystyle d_{zz}$ $\displaystyle =$ $\displaystyle E_z*(1-\nu_{yx}*\nu_{xy})/\xi.$ (8)

$ E_i$ is Young's modulus in the $ i$-th direction, $ G_{ij}$ is the shear modulus in $ ij$ plane, $ \nu_{ij}$ is the major Poisson ratio, and $ \nu_{ji}$ is the minor Poisson ratio. Assuming that $ E_x>E_y>E_z$, $ \nu_{xy} > \nu_{yx}$ etc., then $ \nu_{xy}$ is referred to as the major Poisson ratio, while $ \nu_{yx}$ is referred as the minor Poisson ratio. Note that there are only nine independent material parameters, because of symmetry conditions. The symmetry condition yields

$\displaystyle \nu_{xy}E_y=\nu_{yx}E_x,\ \ \nu_{yz}E_z=\nu_{zy}E_y,\ \ \nu_{zx}E_x=\nu_{xz}E_z$

The model description and parameters are summarized in Tab. 2.

Table 2: Orthotropic, linear elastic material - summary.
Description Orthotropic, linear elastic material
Record Format OrthoLE num(in) # d(rn) # Ex(rn) # Ey(rn) # Ez(rn) # NYyz(rn) # NYxz(rn) # NYxy(rn) # Gyz(rn) # Gxz(rn) # Gxy(rn) # tAlphax(rn) # tAlphay(rn) # tAlphaz(rn) # [ lcs(ra) #] [ scs(ra) #]
Parameters - num material model number
  - d material density
  - Ex, Ey, Ez Young moduli for x,y, and z directions
  - NYyz, NYxz, NYxy major Poisson's ratio coefficients
  - Gyz, Gxz, Gxy shear moduli
  - tAlphax, tAlphay, tAlphaz thermal dilatation coefficients in x,y,z directions
  - lcs Array defining local material x and y axes of orthotrophy
  - scs Array defining a normal vector n. The local x axis is parallel to plane with n being plane normal. The material local z-axis is perpendicular to shell mid-section.
Supported modes 3dMat, PlaneStress, PlaneStrain, 1dMat, 2dPlateLayer, 2dBeamLayer, 3dShellLayer, 2dPlate, 2dBeam, 3dShell, 3dBeam, PlaneStressRot

Borek Patzak