Appendix A: Basic Relations of Elasticity

Isotropic Material

Hooke Law 3D (Principal Stress and Strain)

$\begin{array}{l} {σ} = [D] {ε} \\ σ_{1} = D_{11} ε_{1} + D_{12} ε_{2} + D_{13} ε_{3} \\ σ_{1} = (λ + 2 μ) ε_{1}^{} + λ (ε_{2} + ε_{3}) \\ σ_{1} = λ (ε_{1} + ε_{2} + ε_{3}) + 2 μ ε_{1} \\ σ_{1} = K ε_{k k} + 2 μ e_{1} with ε_{k k} = (ε_{1} + ε_{2} + ε_{3}) {and e}_{1} = ε_{1} - 1 / 3 (ε_{1} + ε_{2} + ε_{3}) \\ {σ} = [D] {ε}; [D] = \frac{E}{(1 + ν) (1 - 2 ν)} [\begin{matrix} 1 - ν & ν & ν & 0 & 0 & 0 \\ 1 - ν & ν & 0 & 0 & 0 \\ 1 - ν & 0 & 0 & 0 \\ \frac{1 - 2 ν}{2} & 0 & 0 \\ S y m m . & \frac{1 - 2 ν}{2} & 0 \\ \frac{1 - 2 ν}{2} \end{matrix}] \\ {ε} = [C] {σ}; [C] = [\begin{matrix} \frac{1}{E} & \frac{- ν}{E} & \frac{- ν}{E} & 0 & 0 & 0 \\ \frac{1}{E} & \frac{- ν}{E} & 0 & 0 & 0 \\ \frac{1}{E} & 0 & 0 & 0 \\ \frac{2 (1 + ν)}{E} & 0 & 0 \\ S y m m . & \frac{2 (1 + ν)}{E} & 0 \\ \frac{2 (1 + ν)}{E} \end{matrix}] \end{array}$

Hooke Law for 2D Plan Stress

${σ} = [H] {ε}$

$σ_{1} = H_{11} ε_{1} + H_{12} ε_{2}$

${σ} = [H] {ε}$ ; $[H] = \frac{E}{1 - ν^{2}} [\begin{matrix} 1 & ν & 0 & 0 & 0 \\ ν & 1 & 0 & 0 & 0 \\ 0 & 0 & \frac{1 - ν}{2} & 0 & 0 \\ 0 & 0 & 0 & \frac{1 - ν}{2} & 0 \\ 0 & 0 & 0 & 0 & \frac{1 - ν}{2} \end{matrix}]$

${ε} = [C] {σ}$ ; $[C] = \frac{1}{E} [\begin{matrix} 1 & - ν & 0 & 0 & 0 \\ - ν & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 (1 + ν) & 0 & 0 \\ 0 & 0 & 0 & 2 (1 + ν) & 0 \\ 0 & 0 & 0 & 0 & 2 (1 + ν) \end{matrix}]$

Hooke Law for 2D Plane Strain

${σ} = [H] {ε}$ ; $[H] = \frac{E}{(1 + ν) (1 - 2 ν)} [\begin{matrix} 1 - ν & ν & 0 \\ ν & 1 - ν & 0 \\ 0 & 0 & \frac{1 - 2 ν}{2} \end{matrix}]$

${ε} = [C] {σ}$ ; $[C] = \frac{1 + ν}{E} [\begin{matrix} 1 - ν & - ν & 0 \\ - ν & 1 - ν & 0 \\ 0 & 0 & 2 \end{matrix}]$

Table 1. Material Constants Relations
	E, $ν$	E,G	E,B	G, $ν$	G, B	B, $ν$	$λ, μ$
E	E	E	E	2(1+v)G	$\frac{9 B G}{3 B + G}$	3(1-2v)B	$\frac{(3 λ + 2 μ) μ}{λ + μ}$
$G = μ$	$\frac{E}{2 (1 + v)}$	G	$\frac{3 E B}{9 B - E}$	G	G	$\frac{3 (1 - 2 v) B}{2 (1 + v)}$	$μ$
B=K	$\frac{E}{3 (1 - 2 v)}$	$\frac{E G}{9 G - 3 E}$	B	$\frac{2 (1 + v) G}{3 (1 - 2 v)}$	B	B	$\frac{3 λ + 2 μ}{3}$
$ν$	$ν$	$\frac{E - 2 G}{2 G}$	$\frac{3 B - E}{6 B}$	$ν$	$\frac{3 B - 2 G}{6 B + 2 G}$	$ν$	$\frac{λ}{2 (λ + μ)}$
$λ$	$\frac{E v}{(1 + v) (1 - 2 v)}$	$\frac{(E - 2 G) G}{3 G - E}$	$\frac{(3 B - E) 3 B}{9 B - E}$	$\frac{2 G v}{1 - 2 v}$	$\frac{3 B - 2 G}{3}$	$\frac{3 B v}{(1 + v)}$	$λ$

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:
$\frac{ν_{12}}{E_{1}} = \frac{ν_{21}}{E_{2}}; \frac{ν_{13}}{E_{1}} = \frac{ν_{31}}{E_{3}}; \frac{ν_{23}}{E_{2}} = \frac{ν_{32}}{E_{3}}$
$1 - ν_{12} ν_{21} > 0$ ; $1 - ν_{13} ν_{31} > 0$ ; $1 - ν_{23} ν_{32} > 0$
$1 - ν_{12} ν_{21} - ν_{13} ν_{31} - ν_{23} ν_{32} - ν_{12} ν_{23} ν_{31} - ν_{21} ν_{13} ν_{32} > 0$
${ε} = [C] {σ}$ ; $[C] = [\begin{matrix} \frac{1}{E_{1}} & \frac{- ν_{21}}{E_{2}} & \frac{- ν_{31}}{E_{3}} & 0 & 0 & 0 \\ \frac{- ν_{12}}{E_{1}} & \frac{1}{E_{2}} & \frac{- ν_{32}}{E} & 0 & 0 & 0 \\ \frac{- ν_{13}}{E_{1}} & \frac{- ν_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{12}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{23}} \end{matrix}]$

2D In-plane Orthotropic Material

Orthotropic plane 1-2, isotropic plane 2-3
- Orthotropy coefficients in the plane 1-2: $E_{1}, E_{2}, ν_{12}, G_{12}$
- Isotropy coefficients in plane 2-3: $E_{2}, ν$
Five independent coefficients
${ε} = [C] {σ}$ ; $[C] = [\begin{matrix} \frac{1}{E_{1}} & \frac{- ν_{12}}{E_{1}} & \frac{- ν_{12}}{E_{1}} & 0 & 0 & 0 \\ \frac{- ν_{12}}{E_{1}} & \frac{1}{E_{2}} & \frac{- ν}{E_{2}} & 0 & 0 & 0 \\ \frac{- ν_{12}}{E_{1}} & \frac{- ν}{E_{2}} & \frac{1}{E_{2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{12}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{12}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{2 (1 + ν)}{E_{2}} \end{matrix}]$

Stiffness Matrix of Beam Element

Terms of the stiffness matrix:

$[k] = [\begin{matrix} \frac{E A}{L} & 0 & 0 & 0 & 0 & 0 & - K_{11} & 0 & 0 & 0 & 0 & 0 \\ \frac{12 E I_{3}}{L^{3} (1 + ϕ_{2})} & 0 & 0 & 0 & \frac{L}{2} K_{22} & 0 & - K_{22} & 0 & 0 & 0 & K_{26} \\ \frac{12 E I_{2}}{L^{3} (1 + ϕ_{2})} & 0 & - \frac{L}{2} K_{33} & 0 & 0 & 0 & - K_{33} & 0 & K_{35} & 0 \\ \frac{G J}{L} & 0 & 0 & 0 & 0 & 0 & - K_{44} & 0 & 0 \\ \frac{(4 + ϕ_{3}) E I_{2}}{L (1 + ϕ_{3})} & 0 & 0 & 0 & - K_{35} & 0 & \frac{2 - ϕ_{3}}{4 + ϕ_{3}} K_{55} & 0 \\ \frac{(4 + ϕ_{2}) E I_{3}}{L (1 + ϕ_{2})} & 0 & - K_{26} & 0 & 0 & 0 & \frac{2 - ϕ_{2}}{4 + ϕ_{2}} K_{66} \\ K_{11} & 0 & 0 & 0 & 0 & 0 \\ K_{22} & 0 & 0 & 0 & - K_{26} \\ S y m m . & K_{33} & 0 & - K_{35} & 0 \\ K_{44} & 0 & 0 \\ K_{55} & 0 \\ K_{66} \end{matrix}]$

For a rectangle cross-section:

$ϕ_{2} = \frac{144 (1 + ν) I_{3}}{5 A L^{2}}$

$ϕ_{3} = \frac{144 (1 + ν) I_{2}}{5 A L^{2}}$

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