4.1.2 Orthotropic Elasticity

The required input for orthotropic elasticity depends on the dimensionality of the element.

Table 4.2: LINEAR ORTHOTROPIC ELASTICITY

	pl. stress	plate bend.	pl. strain	axisymm.	fl. shell	cu. shell	solid
Young's modulus	Ex	Ex	Ex	Ex	Ex	Ex	Ex
	Ey	Ey	Ey	Ey	Ey	Ey	Ey
			Ez	Ez	Ez	Ez	Ez
Poisson's ratio	$\nu_{{xy}}^{}$	$\nu_{{xy}}^{}$	$\nu_{{xy}}^{}$	$\nu_{{xy}}^{}$	$\nu_{{xy}}^{}$	$\nu_{{xy}}^{}$	$\nu_{{xy}}^{}$
			$\nu_{{yz}}^{}$	$\nu_{{yz}}^{}$	$\nu_{{yz}}^{}$	$\nu_{{yz}}^{}$	$\nu_{{yz}}^{}$
			$\nu_{{xz}}^{}$	$\nu_{{xz}}^{}$	$\nu_{{xz}}^{}$	$\nu_{{xz}}^{}$	$\nu_{{xz}}^{}$
Shear modulus	Gxy	Gxy	Gxy	Gxy	Gxy	Gxy	Gxy
					Gyz	Gyz	Gyz
					Gxz	Gxz	Gxz
Thermal exp.	$\alpha_{{x}}^{}$	$\alpha_{{x}}^{}$	$\alpha_{{x}}^{}$	$\alpha_{{x}}^{}$	$\alpha_{{x}}^{}$	$\alpha_{{x}}^{}$	$\alpha_{{x}}^{}$
	$\alpha_{{y}}^{}$	$\alpha_{{y}}^{}$	$\alpha_{{y}}^{}$	$\alpha_{{y}}^{}$	$\alpha_{{y}}^{}$	$\alpha_{{y}}^{}$	$\alpha_{{y}}^{}$
			$\alpha_{{z}}^{}$	$\alpha_{{z}}^{}$		$\alpha_{{z}}^{}$	$\alpha_{{z}}^{}$
Concentr. exp.	$\gamma_{{x}}^{}$	$\gamma_{{x}}^{}$	$\gamma_{{x}}^{}$	$\gamma_{{x}}^{}$	$\gamma_{{x}}^{}$	$\gamma_{{x}}^{}$	$\gamma_{{x}}^{}$
	$\gamma_{{y}}^{}$	$\gamma_{{y}}^{}$	$\gamma_{{y}}^{}$	$\gamma_{{y}}^{}$	$\gamma_{{y}}^{}$	$\gamma_{{y}}^{}$	$\gamma_{{y}}^{}$
			$\gamma_{{z}}^{}$	$\gamma_{{z}}^{}$		$\gamma_{{z}}^{}$	$\gamma_{{z}}^{}$

Table 4.2 shows this for the various element families. Note that orthotropic material models cannot be applied in a single dimension, i.e., not for truss and beam elements and not for reinforcement.

Plane stress and plate bending    (syntax)

Plane strain and axisymmetry    (syntax)

Flat shell    (syntax)

Curved shell and solid    (syntax)

YOUNG

ex, ey and ez are the Young's moduli E_x , E_y , and E_z respectively (16.1). (E > 0 )

POISON

nuxy, nuyz and nuxz are the Poisson's ratios $\nu_{{xy}}^{}$ , $\nu_{{yz}}^{}$ , and $\nu_{{xz}}^{}$ respectively. Conditions are:

$$\nu_{{xy}}^{{2}}$$ < $${\frac{{E_{x}}}{{E_{y}}}}$$    ,        $$\nu_{{yz}}^{{2}}$$ < $${\frac{{E_{y}}}{{E_{z}}}}$$    ,        $$\nu_{{xz}}^{{2}}$$ < $${\frac{{E_{x}}}{{E_{z}}}}$$    ,

2$$\nu_{{xy}}^{}$$$$\nu_{{yz}}^{}$$$$\nu_{{xz}}^{}$$$${\frac{{E_{z}}}{{E_{x}}}}$$ < 1 - $$\nu_{{xy}}^{{2}}$$$${\frac{{E_{y}}}{{E_{x}}}}$$ - $$\nu_{{yz}}^{{2}}$$$${\frac{{E_{z}}}{{E_{y}}}}$$ - $$\nu_{{xz}}^{{2}}$$$${\frac{{E_{z}}}{{E_{x}}}}$$ $$\leq$$ 1

SHRMOD

gxy, gyz and gzx are the shear moduli G_xy , G_yz and G_zx , respectively. (G > 0 )

THERMX

alphax, alphay and alphaz are the thermal expansion coefficients $\alpha_{{x}}^{}$ , $\alpha_{{y}}^{}$ and $\alpha_{{z}}^{}$ respectively.

CONCEX

gammax, gammay and gammaz are the concentration expansion coefficients $\gamma_{{x}}^{}$ , $\gamma_{{y}}^{}$ and $\gamma_{{z}}^{}$ respectively.

Orthotropic material properties per layer.

The default xyz directions for orthotropic properties are the same as for the element (you may overrule the default directions by specifying an $\bar{{x}}$ axis in table 'GEOMET' [Vol. Element Library]). However, if the material xyz directions for a layer in a layered shell element do not coincide with the element axes, then you may specify these directions explicitly in addition to the orthotropic material properties according to the following syntax.

    (syntax)

XAXIS

is a user-specified $\bar{{x}}$ axis where x, y and z are vector components in the model XYZ coordinate system. The specified direction of the $\bar{{x}}$ axis may not coincide with any of the local z axes (perpendicular to the plane of the layer). DIANA uses the $\bar{{x}}$ axis to setup local $\hat{{x}}$$\hat{{y}}$$\hat{{z}}$ axes in the same way as for the element axes [Vol. Element Library].

For layered orthotropic material, the primary strains and stresses are oriented in the local $\hat{{x}}$$\hat{{y}}$$\hat{{z}}$ of each layer.

    (file.dat)

'ELEMEN'
CONNEC
 67   CQ40L 228 456 327 112 536 89 116 92
MATERI
 67   LAYERS 5 8 4
'MATERI'
  4  YOUNG  10000.0  100.0   1.0
     POISON    0.10   0.25   0.0
     SHRMOD   100.0  100.0 100.0
     XAXIS  2. 1. 0.
  5  YOUNG  10000.0  100.0   1.0
     POISON    0.10   0.25   0.0
     SHRMOD   100.0  100.0 100.0
     XAXIS  1. 1. 0.
  8  YOUNG  10000.0  100.0   1.0
     POISON    0.10   0.25   0.0
     SHRMOD   100.0  100.0 100.0
     XAXIS  1. 2. 0.

In this example, element 67 comprises three layers with orthotropic Young's modulus. The first layer (nr.5) has E_x = 10⁴ , E_y = 100 and E_z = 1 and also three different $\nu$ 's and shear moduli, where the x axis makes an angle of 45° with the global XY axes. The second and the third layer each have the same stiffness properties as layer 5, yet with a different orientation.