# Appendix A: Basic Relations of Elasticity

## Isotropic Material

### Hooke Law for 2D Plan Stress

$\left\{\sigma \right\}=\left[H\right]\left\{\epsilon \right\}$

${\sigma }_{1}={H}_{11}{\epsilon }_{1}+{H}_{12}{\epsilon }_{2}$

$\left\{\sigma \right\}=\left[H\right]\left\{\epsilon \right\}$; $\left[H\right]=\frac{E}{1-{\nu }^{2}}\left[\begin{array}{ccccc}1& \nu & 0& 0& 0\\ \nu & 1& 0& 0& 0\\ 0& 0& \frac{1-\nu }{2}& 0& 0\\ 0& 0& 0& \frac{1-\nu }{2}& 0\\ 0& 0& 0& 0& \frac{1-\nu }{2}\end{array}\right]$

$\left\{\epsilon \right\}=\left[C\right]\left\{\sigma \right\}$; $\left[C\right]=\frac{1}{E}\left[\begin{array}{ccccc}1& -\nu & 0& 0& 0\\ -\nu & 1& 0& 0& 0\\ 0& 0& 2\left(1+\nu \right)& 0& 0\\ 0& 0& 0& 2\left(1+\nu \right)& 0\\ 0& 0& 0& 0& 2\left(1+\nu \right)\end{array}\right]$

### Hooke Law for 2D Plane Strain

$\left\{\sigma \right\}=\left[H\right]\left\{\epsilon \right\}$ ; $\left[H\right]=\frac{E}{\left(1+\nu \right)\left(1-2\nu \right)}\left[\begin{array}{ccc}1-\nu & \nu & 0\\ \nu & 1-\nu & 0\\ 0& 0& \frac{1-2\nu }{2}\end{array}\right]$

$\left\{\epsilon \right\}=\left[C\right]\left\{\sigma \right\}$; $\left[C\right]=\frac{1+\nu }{E}\left[\begin{array}{ccc}1-\nu & -\nu & 0\\ -\nu & 1-\nu & 0\\ 0& 0& 2\end{array}\right]$

Table 1. Material Constants Relations
E, $\nu$ E,G E,B G, $\nu$ G, B B, $\nu$ $\lambda ,\mu$
E E E E 2(1+v)G $\frac{9BG}{3B+G}$ 3(1-2v)B $\frac{\left(3\lambda +2\mu \right)\mu }{\lambda +\mu }$
$G=\mu$ $\frac{E}{2\left(1+v\right)}$ G $\frac{3EB}{9B-E}$ G G $\frac{3\left(1-2v\right)B}{2\left(1+v\right)}$ $\mu$
B=K $\frac{E}{3\left(1-2v\right)}$ $\frac{EG}{9G-3E}$ B $\frac{2\left(1+v\right)G}{3\left(1-2v\right)}$ B B $\frac{3\lambda +2\mu }{3}$
$\nu$ $\nu$ $\frac{E-2G}{2G}$ $\frac{3B-E}{6B}$ $\nu$ $\frac{3B-2G}{6B+2G}$ $\nu$ $\frac{\lambda }{2\left(\lambda +\mu \right)}$
$\lambda$ $\frac{Ev}{\left(1+v\right)\left(1-2v\right)}$ $\frac{\left(E-2G\right)G}{3G-E}$ $\frac{\left(3B-E\right)3B}{9B-E}$ $\frac{2Gv}{1-2v}$ $\frac{3B-2G}{3}$ $\frac{3Bv}{\left(1+v\right)}$ $\lambda$

## Orthotropic Material

### General 3D Orthotropic Case

The strain-stress relations are defined using 9 material constants:
• Three Young's modulus in orthotropic directions 1, 2 and 3: ${E}_{1}$, ${E}_{2}$, ${E}_{3}$
• Three shear modulus in planes 12, 13 and 23: ${G}_{12}$, ${G}_{13}$, ${G}_{23}$
• Three Poisson ratio's satisfying the relations:

$\frac{{\nu }_{12}}{{E}_{1}}=\frac{{\nu }_{21}}{{E}_{2}}\text{ };\text{ }\frac{{\nu }_{13}}{{E}_{1}}=\frac{{\nu }_{31}}{{E}_{3}}\text{ };\text{ }\frac{{\nu }_{23}}{{E}_{2}}=\frac{{\nu }_{32}}{{E}_{3}}$

$1-{\nu }_{12}{\nu }_{21}>0$ ;$1-{\nu }_{13}{\nu }_{31}>0$; $1-{\nu }_{23}{\nu }_{32}>0$

$1-{\nu }_{12}{\nu }_{21}-{\nu }_{13}{\nu }_{31}-{\nu }_{23}{\nu }_{32}-{\nu }_{12}{\nu }_{23}{\nu }_{31}-{\nu }_{21}{\nu }_{13}{\nu }_{32}>0$

$\left\{\epsilon \right\}=\left[C\right]\left\{\sigma \right\}$;$\left[C\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{1}}& \frac{-{\nu }_{21}}{{E}_{2}}& \frac{-{\nu }_{31}}{{E}_{3}}& 0& 0& 0\\ \frac{-{\nu }_{12}}{{E}_{1}}& \frac{1}{{E}_{2}}& \frac{-{\nu }_{32}}{E}& 0& 0& 0\\ \frac{-{\nu }_{13}}{{E}_{1}}& \frac{-{\nu }_{23}}{{E}_{2}}& \frac{1}{{E}_{3}}& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{12}}& 0& 0\\ 0& 0& 0& 0& \frac{1}{{G}_{13}}& 0\\ 0& 0& 0& 0& 0& \frac{1}{{G}_{23}}\end{array}\right]$

### 2D In-plane Orthotropic Material

• Orthotropic plane 1-2, isotropic plane 2-3
• Orthotropy coefficients in the plane 1-2: ${E}_{1},\text{\hspace{0.17em}}{E}_{2},\text{\hspace{0.17em}}{\nu }_{12},\text{\hspace{0.17em}}{G}_{12}$
• Isotropy coefficients in plane 2-3: ${E}_{2},\text{\hspace{0.17em}}\nu$
• Five independent coefficients

$\left\{\epsilon \right\}=\left[C\right]\left\{\sigma \right\}$ ; $\left[C\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{1}}& \frac{-{\nu }_{12}}{{E}_{1}}& \frac{-{\nu }_{12}}{{E}_{1}}& 0& 0& 0\\ \frac{-{\nu }_{12}}{{E}_{1}}& \frac{1}{{E}_{2}}& \frac{-\nu }{{E}_{2}}& 0& 0& 0\\ \frac{-{\nu }_{12}}{{E}_{1}}& \frac{-\nu }{{E}_{2}}& \frac{1}{{E}_{2}}& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{12}}& 0& 0\\ 0& 0& 0& 0& \frac{1}{{G}_{12}}& 0\\ 0& 0& 0& 0& 0& \frac{2\left(1+\nu \right)}{{E}_{2}}\end{array}\right]$

## Stiffness Matrix of Beam Element

Terms of the stiffness matrix:

$\left[k\right]=\left[\begin{array}{cccccccccccc}\frac{EA}{L}& 0& 0& 0& 0& 0& -{K}_{11}& 0& 0& 0& 0& 0\\ & \frac{12E{I}_{3}}{{L}^{3}\left(1+{\varphi }_{2}\right)}& 0& 0& 0& \frac{L}{2}{K}_{22}& 0& -{K}_{22}& 0& 0& 0& {K}_{26}\\ & & \frac{12E{I}_{2}}{{L}^{3}\left(1+{\varphi }_{2}\right)}& 0& -\frac{L}{2}{K}_{33}& 0& 0& 0& -{K}_{33}& 0& {K}_{35}& 0\\ & & & \frac{GJ}{L}& 0& 0& 0& 0& 0& -{K}_{44}& 0& 0\\ & & & & \frac{\left(4+{\varphi }_{3}\right)E{I}_{2}}{L\left(1+{\varphi }_{3}\right)}& 0& 0& 0& -{K}_{35}& 0& \frac{2-{\varphi }_{3}}{4+{\varphi }_{3}}{K}_{55}& 0\\ & & & & & \frac{\left(4+{\varphi }_{2}\right)E{I}_{3}}{L\left(1+{\varphi }_{2}\right)}& 0& -{K}_{26}& 0& 0& 0& \frac{2-{\varphi }_{2}}{4+{\varphi }_{2}}{K}_{66}\\ & & & & & & {K}_{11}& 0& 0& 0& 0& 0\\ & & & & & & & {K}_{22}& 0& 0& 0& -{K}_{26}\\ & & Symm.& & & & & & {K}_{33}& 0& -{K}_{35}& 0\\ & & & & & & & & & {K}_{44}& 0& 0\\ & & & & & & & & & & {K}_{55}& 0\\ & & & & & & & & & & & {K}_{66}\end{array}\right]$

For a rectangle cross-section:

${\varphi }_{2}=\frac{144\left(1+\nu \right){I}_{3}}{5A{L}^{2}}$

${\varphi }_{3}=\frac{144\left(1+\nu \right){I}_{2}}{5A{L}^{2}}$

$I=\frac{b{h}^{3}}{12}$