Elastic constitutive equations

Linear elastic constituve equations are supported in both forward and inverse (or reverse) form.

Transversely isotropic

Symmetries	Stiffness matrix \({\bf C}\) for \(\bf{m}=\hat{\bf{z}}\)
	\(\begin{bmatrix}\gamma & \lambda & \hat{\lambda}\lambda & 0 & 0 & 0 \\\lambda & \gamma & \hat{\lambda}\lambda & 0 & 0 & 0 \\\hat{\lambda}\lambda & \hat{\lambda}\lambda & \hat{\gamma}\gamma & 0 & 0 & 0 \\0&0&0& \hat{\mu}\mu & 0 & 0\\0&0&0& 0 & \hat{\mu}\mu & 0\\0&0&0& 0 & 0 & \mu\\\end{bmatrix}\)

API

E = sf.elas_fwd_tranisotropic(S, lam, mu, Elam, Emu, Egam, m)

S = sf.elas_rev_tranisotropic(E, lam, mu, Elam, Emu, Egam, m)

where

Arguments	Type
S, E	Stress and strain tensor (3x3)
lam, mu	Isotropic Lamé parameters \(\lambda\) and \(\mu\)
Elam, Emu, Egam	Anisotropic enhancement factors \(\hat{\lambda}\), \(\hat{\mu}\), and \(\hat{\gamma}\)
m	Rotational symmetry axis \(\bf{m}\)

Orthotropic

Symmetries	Stiffness matrix \({\bf C}\) for \(({\bf m}_1,{\bf m}_2,{\bf m}_3)=(\hat{{\bf x}},\hat{{\bf y}},\hat{{\bf z}})\)
	\(\small\begin{bmatrix} \lambda_{11} + 2\mu_1 & \lambda_{12} & \lambda_{13} & 0 & 0 & 0 \\ \lambda_{12} & \lambda_{22} + 2\mu_2 & \lambda_{23} & 0 & 0 & 0 \\ \lambda_{13 }& \lambda_{23} & \lambda_{33} + 2\mu_3 & 0 & 0 & 0 \\ 0 & 0 & 0 & \dfrac{\mu_2+\mu_3}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \dfrac{\mu_3+\mu_1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \dfrac{\mu_1+\mu_2}{2} \end{bmatrix}\)

API

Not yet made available

E = sf.elas_fwd_orthotropic(S, lame, m1,m2,m3)

S = sf.elas_rev_orthotropic(E, lame, m1,m2,m3)

where

Arguments	Type
S, E	Stress and strain tensor (3x3)
lame	Tuple of anisotropic Lamé parameters \((\lambda_{11},\lambda_{22},\lambda_{33}, \lambda_{23},\lambda_{13},\lambda_{12}, \mu_{1}, \mu_{2}, \mu_{3})\)
m1, m2, m3	Reflection symmetry axes \(\bf{m}_1\), \(\bf{m}_2\), and \(\bf{m}_3\)