homopy.elasticity module

Created on Wed Apr 27 21:09:24 2022

@author: nicolas.christ@kit.edu

Module that contains the linear elastic stiffness classes of Isotropy and Transverse Isotropy.

class homopy.elasticity.Elasticity

Bases: Tensor

Elasticity class to express generic elasitc stiffness tensors. The class inherits from the Tensor class.

The generic stiffness matrix has the following form in the normalized Voigt (Mandel) notation

\[\begin{split}\underline{\underline{C}} = \begin{pmatrix} C_{1111} & C_{1122} & C_{1133} & \sqrt{2}C_{1123} & \sqrt{2}C_{1131} & \sqrt{2}C_{1112} \\ C_{2211} & C_{2222} & C_{2233} & \sqrt{2}C_{2223} & \sqrt{2}C_{2231} & \sqrt{2}C_{2212} \\ C_{3311} & C_{3322} & C_{3333} & \sqrt{2}C_{3323} & \sqrt{2}C_{3331} & \sqrt{2}C_{3312} \\ \sqrt{2}C_{2311} & \sqrt{2}C_{2322} & \sqrt{2}C_{2333} & 2 C_{2323} & 2 C_{2331} & 2 C_{2312} \\ \sqrt{2}C_{3111} & \sqrt{2}C_{3122} & \sqrt{2}C_{3133} & 2 C_{3123} & 2 C_{3131} & 2 C_{3112} \\ \sqrt{2}C_{1211} & \sqrt{2}C_{1222} & \sqrt{2}C_{1233} & 2 C_{1223} & 2 C_{1231} & 2 C_{1212} \\ \end{pmatrix},\end{split}\]

which is a symmetric matrix, giving 21 parameters.

Variables:

• stiffness3333 (ndarray of shape (3, 3, 3, 3)) – Stiffness values in the regular tensor notation.

• stiffness66 (ndarray of shape (6, 6)) – Stiffness values in the normalized Voigt notation.

class homopy.elasticity.Orthotropy(E1, E2, E3, G12, G13, G23, nu12, nu13, nu23)

Bases: Elasticity

Orthotropy class to express orthotropic elastic stiffness tensors. The class inherits from the Elasticity class.

The orthotropic stiffness matrix has the following form in the normalized Voigt (Mandel) notation

\[\begin{split}\underline{\underline{C}} = \begin{pmatrix} \frac{1-\nu_{23}\nu_{32}}{D}E_{1}& \frac{\nu_{21}+\nu_{23}\nu_{31}}{D}E_{1}& \frac{\nu_{31}+\nu_{32}\nu_{21}}{D}E_{1} \\ \frac{\nu_{12}+\nu_{13}\nu_{32}}{D}E_{2} & \frac{1-\nu_{13}\nu_{31}}{D}E_{2}& \frac{\nu_{32}+\nu_{31}\nu_{12}}{D}E_{2} \\ \frac{\nu_{13}+\nu_{12}\nu_{23}}{D}E_{3} & \frac{\nu_{23}+\nu_{21}\nu_{13}}{D}E_{3} & \frac{1-\nu_{12}\nu_{21}}{D}E_{3} \\ & & &2G_{23} \\ & & & &2G_{13} \\ & & & & &2G_{12} \end{pmatrix},\end{split}\]

where

\[D=1-\nu_{12}\nu_{21}-\nu_{13}\nu_{31}-\nu_{23}\nu_{32}-2\nu_{12}\nu_{23}\nu_{31}.\]

The corresponding compliance matrix is

\[\begin{split}\underline{\underline{S}} = \begin{pmatrix} \frac{1}{E_1} & -\frac{\nu_{21}}{E_2} & -\frac{\nu_{31}}{E_3} & & & \\ -\frac{\nu_{12}}{E_1} & \frac{1}{E_2} & -\frac{\nu_{32}}{E_3} & & & \\ -\frac{\nu_{13}}{E_1} & -\frac{\nu_{23}}{E_2} & \frac{1}{E_3} & & & \\ & & & \frac{1}{2G_{23}} & & \\ & & & & \frac{1}{2G_{13}} & \\ & & & & & \frac{1}{2G_{12}} \end{pmatrix}.\end{split}\]

Parameters:

• E1 (float) – Young’s modulus in first principal direction.

• E2 (float) – Young’s modulus in second principal direction.

• E3 (float) – Young’s modulus in third principal direction.

• G12 (float) – Shear modulus in the first-second plane.

• G13 (float) – Shear modulus in the first-third plane.

• G23 (float) – Shear modulus in the second-third plane.

• nu12 (float) – Poisson’s ratio to express strain in second principal direction caused by load in first principal direction (dimensionless).

• nu13 (float) – Poisson’s ratio to express strain in third principal direction caused by load in first principal direction (dimensionless).

• nu23 (float) – Poisson’s ratio to express strain in third principal direction caused by load in second principal direction (dimensionless).

Variables:

• E1 (float) – Young’s modulus in first principal direction.

• E2 (float) – Young’s modulus in second principal direction.

• E3 (float) – Young’s modulus in third principal direction.

• G12 (float) – Shear modulus in the first-second plane.

• G13 (float) – Shear modulus in the first-third plane.

• G23 (float) – Shear modulus in the second-third plane.

• nu12 (float) – Poisson’s ratio to express strain in second principal direction caused by load in first principal direction (dimensionless).

• nu21 (float) – Poisson’s ratio to express strain in first principal direction caused by load in second principal direction (dimensionless).

• nu13 (float) – Poisson’s ratio to express strain in third principal direction caused by load in first principal direction (dimensionless).

• nu31 (float) – Poisson’s ratio to express strain in third principal direction caused by load in first principal direction (dimensionless).

• nu23 (float) – Poisson’s ratio to express strain in third principal direction caused by load in second principal direction (dimensionless).

• nu32 (float) – Poisson’s ratio to express strain in third principal direction caused by load in second principal direction (dimensionless).

_get_stiffness()

Calculate the stiffness parameters for both notations.

class homopy.elasticity.TransverseIsotropy(E1, E2, G12, G23, nu12)

Bases: Elasticity

Transverse Isotropy class to express transverse-isotropic elasitc stiffness tensors. The class inherits from the Elasticity class. The convention in HomoPy is that the first principal direction is orthogonal to the isotropic plane.

The transverse-isotropic stiffness matrix has the following form in the normalized Voigt (Mandel) notation

\[\begin{split}\underline{\underline{C}} = \begin{pmatrix} C_{1111} & 2\nu_{12}(\lambda+G_{23}) & 2\nu_{12}(\lambda+G_{23}) & 0 & 0 & 0 \\ & \lambda+2G_{23} & \lambda & 0 & 0 & 0 \\ & & \lambda+2G_{23} & 0 & 0 & 0 \\ & & & 2G_{23} & 0 & 0 \\ & \mathrm{sym} & & & 2G_{12} & 0 \\ & & & & & 2G_{12} \\ \end{pmatrix},\end{split}\]

where

\[\begin{split}\begin{array}{lcl} \lambda&=&\dfrac{\nu_{12} \nu_{21} + \nu_{23}} {( 1-\nu_{23} - 2 \nu_{12}\nu_{21} ) (1+\nu_{23})} E_2 \\ C_{1111} &=& \dfrac{1 - \nu_{23}}{1 - \nu_{23} - 2 \nu_{12} \nu_{21}} E_1 \end{array}\end{split}\]

The corresponding compliance matrix is

\[\begin{split}\underline{\underline{S}} = \begin{pmatrix} \frac{1}{E_1} & -\frac{\nu_{12}}{E_1} & -\frac{\nu_{12}}{E_1} & 0 & 0 & 0 \\ & \frac{1}{E_2} & -\frac{\nu_{23}}{E_2} & 0 & 0 & 0 \\ & & \frac{1}{E_2} & 0 & 0 & 0 \\ & & & \frac{1}{2G_{23}} & 0 & 0 \\ & \mathrm{sym} & & & \frac{1}{2G_{12}} & 0 \\ & & & & & \frac{1}{2G_{12}} \\ \end{pmatrix}.\end{split}\]

Parameters:

• E1 (float) – Young’s modulus in longitudinal direction.

• E2 (float) – Young’s modulus in transverse direction.

• G12 (float) – Shear modulus in the longitudinal-transverse plane.

• G23 (float) – Shear modulus in the transverse-transverse plane.

• nu12 (float) – Poisson’s ratio to express strain in transverse direction caused by load in longitudinal direction (dimensionless).

Variables:

• E1 (float) – Young’s modulus in longitudinal direction.

• E2 (float) – Young’s modulus in transverse direction.

• G12 (float) – Shear modulus in the longitudinal-transverse plane.

• G23 (float) – Shear modulus in the transverse-transverse plane.

• nu12 (float) – Poisson’s ratio to express strain in isotropic plane caused by load in longitudinal direction (dimensionless).

• nu23 (float) – Poisson’s ratio to express strain in isotropic plane caused by load in transverse direction (dimensionless).

_get_stiffness()

Calculate the stiffness parameters for both notations.

class homopy.elasticity.Isotropy(E, nu)

Bases: TransverseIsotropy

Isotropy class to express isotropic elasitc stiffness tensors. The class inherits from the Transverse Isotropy class.

Parameters:

• E (float) – Young’s modulus.

• nu (float) – Poisson’s ratio (dimensionless).

Variables:

• E (float) – Young’s modulus.

• nu (float) – Poisson’s ratio (dimensionless).

• lam (float) – First Lamé constant.

• mu (float) – Second Lamé constant.

_get_lambda()

Return the first Lamé constant from other material parameters.

Returns:

float – First Lamé constant.

_get_mu()

Return the second Lamé constant from other material parameters.

Returns:

float – Second Lamé constant.