Elastic Orthotropic

Contents

Elastic Orthotropic#

class xara.MultiaxialMaterial("ElasticOrthotropic", Ex, Ey, Ez, vxy, vyz, vzx, Gxy, Gyz, Gzx, rho=0.0)
Parameters:
Ex: float

elastic modulus in x direction

Ey: float

elastic modulus in y direction

Ez: float

elastic modulus in z direction

vxy: float

Poisson’s ratio in xy plane

vyz: float

Poisson’s ratio in yz plane

vzx: float

Poisson’s ratio in zx plane

Gxy: float

shear modulus in xy plane

Gyz: float

shear modulus in yz plane

Gzx: float

shear modulus in zx plane

rho: float

mass density. optional default = 0.0

The response of an elastic orthotropic material is defined by the following constitutive relation:

\[\boldsymbol{\sigma} = \mathbf{C} \boldsymbol{\varepsilon}\]

where \(\boldsymbol{\sigma}\) and \(\boldsymbol{\varepsilon}\) are the stress and strain vectors in Voigt notation, and \(\mathbf{C}\) is the elastic stiffness matrix. The inverse of the stiffness matrix is given by:

\[\begin{split}\mathbf{C}^{-1} = \left[\begin{array}{cccccc} \frac{1}{E_1} & -\frac{\nu_{21}}{E_2} & -\frac{\nu_{31}}{E_3} & 0 & 0 & 0 \\ -\frac{\nu_{12}}{E_1} & \frac{1}{E_2} & -\frac{\nu_{32}}{E_3} & 0 & 0 & 0 \\ -\frac{\nu_{13}}{E_1} & -\frac{\nu_{23}}{E_2} & \frac{1}{E_3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{31}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{array}\right]\end{split}\]

with

\[E_1>0, E_2>0, E_3>0, G_{12}>0, G_{23}>0, G_{13}>0\]

References#

  • Zienkiewicz, Olgierd C., Robert L. Taylor, and Sanjay Govindjee. “The Finite Element Method: Its Basis and Fundamentals.” Eighth edition. Butterworth-Heinemann, 2025.

Code Developed by: Michael H. Scott

Contents