LAW12 and LAW14

Describes orthotropic solid material which use the Tsai-Wu formulation. The materials are 3D orthotropic-elastic, before the Tsai-Wu criterion is reached. LAW12 is a generalization and improvement of LAW14.

Elastic Phase

Both material laws require Young's modulus, shear modulus and Poisson ratio (9 parameters) to describe the material orthotropic in elastic phase.

(1)

[\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{23} \\ γ_{31} \end{matrix}] = [\begin{matrix} \frac{1}{E_{11}} & - \frac{ν_{12}}{E_{11}} & - \frac{ν_{31}}{E_{33}} & 0 & 0 & 0 \\ \frac{1}{E_{22}} & - \frac{ν_{23}}{E_{22}} & 0 & 0 & 0 \\ \frac{1}{E_{33}} & 0 & 0 & 0 \\ \frac{1}{2 G_{12}} & 0 & 0 \\ s y m m . & \frac{1}{2 G_{23}} & 0 \\ \frac{1}{2 G_{31}} \end{matrix}] \cdot [\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{23} \\ σ_{31} \end{matrix}]

Stress Damage

Stress limits

σ_{t 1}, σ_{t 2} and σ_{t 3}

(in tensile/compression) are requested for damage. These stress limits could be observed from a tensile test in 3 related directions.

Once stress limit is reached, then damage to material begins (stress reduced with damage parameter

δ

). If Damage (

D_{i} = D_{i} + δ

) reaches D=1, then stress is reduced to 0.

Tsai-Wu Yield Criteria

In LAW12 (3D_COMP), the Tsai-Wu yield criteria is:(2)

\begin{array}{l} F (σ) = F_{1} σ_{1} + F_{2} σ_{2} + F_{3} σ_{3} + F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{33} σ_{3}^{2} + F_{44} σ_{12}^{2} + F_{55} σ_{23}^{2} \\ + F_{66} σ_{31}^{2} + 2 F_{12} σ_{1} σ_{2} + 2 F_{23} σ_{2} σ_{3} + 2 F_{13} σ_{1} σ_{3} \end{array}

The 12 coefficients of the Tsai-Wu criterion could be determined using the yield stress from the following tests:

Tensile/Compression Tests

Longitude tensile/compression (in direction 1):

Figure 5.
(3)
$F_{1} = - \frac{1}{σ_{1 y}^{c}} + \frac{1}{σ_{1 y}^{t}}$
(4)
$F_{11} = \frac{1}{σ_{1 y}^{c} σ_{1 y}^{t}}$
Transverse tensile/compression (in direction 2):

Figure 6.
(5)
$F_{2} = - \frac{1}{σ_{2 y}^{c}} + \frac{1}{σ_{2 y}^{t}}$
(6)
$F_{22} = \frac{1}{σ_{2 y}^{c} σ_{2 y}^{t}}$
Transverse tensile/compression (in direction 3):

Figure 7.
(7)
$F_{3} = - \frac{1}{σ_{3 y}^{c}} + \frac{1}{σ_{3 y}^{t}}$
(8)
$F_{33} = \frac{1}{σ_{3 y}^{c} σ_{3 y}^{t}}$

Then the interaction coefficients can be calculated as:(9)

F_{12} = - \frac{1}{2} \sqrt{(F_{11} F_{22})}

(10)

F_{23} = - \frac{1}{2} \sqrt{(F_{22} F_{33})}

(11)

F_{13} = - \frac{1}{2} \sqrt{(F_{11} F_{33})}

Shear Tests

Shear in plane 1-2 test:

Figure 8.
$σ_{12 y}^{t}$ and $σ_{12 y}^{c}$ can result from the sample tests below:

Figure 9.
(12)
$F_{44} = \frac{1}{σ_{12 y}^{c} σ_{12 y}^{t}}$
Shear in plane 1-3

Figure 10.
$σ_{31 y}^{t}$ and $σ_{31 y}^{c}$ can result from the sample tests below:

Figure 11.
(13)
$F_{66} = \frac{1}{σ_{31 y}^{c} σ_{31 y}^{t}}$
Shear in plane 2-3:

Figure 12.
(14)
$F_{55} = \frac{1}{σ_{23 y}^{c} σ_{23 y}^{t}}$

The parameters shown below in LAW12 and LAW14 are requested to calculate the Tsai-Wu criteria:

The yield surface for Tsai-Wu is $F (σ) = 1$ . As long as $(F (σ) \leq 1)$ , the material is in the elastic phase. Once $(F (σ) > 1)$ , the yield surface is exceeded and the material is in nonlinear phase.

In these two material laws, the following factors could also be considered for the yield surface.

Plastic work $W_{p}$ with parameter B and n
Strain rate $\dot{ε}$ with parameter ${\dot{ε}}_{0}$ and c. (15)
$F (W_{p}, \dot{ε}) = (1 + B W_{p}^{n}) (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{o}})$

Then the yield surface will be

F (σ) = F (W_{p}, \dot{ε})

Material will be in elastic phase, if $F (σ) \leq F (W_{p}, \dot{ε})$
Material will be in nonlinear phase, if $F (σ) > F (W_{p}, \dot{ε})$

This yield surface $F (W_{p}, \dot{ε})$ will be limited with $f_{\max}$ ( $F (W_{p}, \dot{ε}) \leq f_{\max}$ ), where $f_{\max}$ is the maximum value of the Tsai-Wu criterion limit.

$f_{\max} = {(\frac{σ_{\max}}{σ_{y}})}^{2}$

Depending on parameter B, n, c and

{\dot{ε}}_{0}

, the yield surface is between 1 and

f_{\max}