LAW25 (Tsai-WU and CRASURV)

LAW25 is the most commonly used composite material in Radioss. It can be used with shell and solid elements. The two formulations available in LAW25 are the Tsai-Wu and CRASURV formulations.

Elastic Phase

In the elastic phase, Young's modulus (3 parameters), shear modulus (3 parameters) and one parameter for Poisson ratio are required to describe the orthotropic material.(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_{12}}{E_{33}} & 0 & 0 & 0 \\ \frac{1}{E_{22}} & - \frac{ν_{12}}{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}]

Tsai-Wu Yield Criteria for `I`_form=0 and =1

The Tsai-Wu yield surface in LAW25 is defined with 6 coefficient:

I_form=0: Tsai-Wu $(F (σ) \leq F (W_{p}^{*}, \dot{ε}))$: I_form=1: CRASURV $(F (W_{p}^{*}, \dot{ε}, σ) \leq 1)$
$F (σ) = F_{1} σ_{1} + F_{2} σ_{2} + F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{44} σ_{12}^{2} + 2 F_{12} σ_{1} σ_{2}$: $\begin{array}{l} F (W_{p}^{*}, \dot{ε}, σ) = F_{1} (W_{p}^{*}, \dot{ε}) σ_{1} + F_{2} (W_{p}^{*}, \dot{ε}) σ_{2} + F_{11} (W_{p}^{*}, \dot{ε}) σ_{1}^{2} \\ + F_{22} (W_{p}^{*}, \dot{ε}) σ_{2}^{2} + F_{44} (W_{p}^{*}, \dot{ε}) σ_{12}^{2} + 2 F_{12} (W_{p}^{*}, \dot{ε}) σ_{1} σ_{2} \end{array}$

To check if material in yield, in Tsai-Wu (I_form=0)

F (σ)

will be compared with

F (W_{p}^{*}, \dot{ε})

at each stress state and in CRASURV (I_form=1)

F (W_{p}^{*}, \dot{ε})

will be simply compared with 1 at each stress state.

These 6 coefficients could be determined with yield stress from these tests:

Tensile/compression tests
Longitudinal tensile/compression tests (in direction 1 which is fiber direction):

I_form = 0: Tsai-Wu $(F (σ) \leq F (W_{p}^{*}, \dot{ε}))$

I_form = 1: CRASURV $(F (W_{p}^{*}, \dot{ε}, σ) \leq 1)$

$F_{1} = - \frac{1}{σ_{1 y}^{c}} + \frac{1}{σ_{1 y}^{t}}$

$F_{11} = \frac{1}{σ_{1 y}^{c} σ_{1 y}^{t}}$

$F (W_{p}^{*}, \dot{ε}) = (1 + b {(W_{p}^{*})}^{n}) \cdot (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

Here, $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$

$F_{1} (W_{p}^{*} \cdot \dot{ε}) = - \frac{1}{σ_{1 y}^{c} (W_{p}^{*} \cdot \dot{ε})} + \frac{1}{σ_{1 y}^{t} (W_{p}^{*} \cdot \dot{ε})}$
$F_{11} (W_{p}^{*} \cdot \dot{ε}) = \frac{1}{σ_{1 y}^{c} (W_{p}^{*} \cdot \dot{ε}) \cdot σ_{1 y}^{t} (W_{p}^{*} \cdot \dot{ε})}$

In tension:

$σ_{1 y}^{t} (W_{p}^{*} \cdot \dot{ε}) = σ_{1 y}^{t} (1 + b_{1}^{t} {(W_{p}^{*})}^{n_{1}^{t}}) (1 + c_{1}^{t} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

In compression:

$σ_{1 y}^{c} (W_{p}^{*} \cdot \dot{ε}) = σ_{1 y}^{c} (1 + b_{1}^{c} {(W_{p}^{*})}^{n_{1}^{c}}) (1 + c_{1}^{c} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

Here $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$

Transverse tensile/compression tests (in direction 2)

I_form = 0: Tsai-Wu $(F (σ) \leq F (W_{p}^{*}, \dot{ε}))$

I_form = 1: CRASURV $(F (W_{p}^{*}, \dot{ε}, σ) \leq 1)$

$F_{2} = - \frac{1}{σ_{2 y}^{c}} + \frac{1}{σ_{2 y}^{t}}$

$F_{22} = \frac{1}{σ_{2 y}^{c} σ_{2 y}^{t}}$

Here, $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$

$F_{2} (W_{p}^{*} \cdot \dot{ε}) = - \frac{1}{σ_{2 y}^{c} (W_{p}^{*} \cdot \dot{ε})} + \frac{1}{σ_{2 y}^{t} (W_{p}^{*} \cdot \dot{ε})}$
$F_{22} (W_{p}^{*} \cdot \dot{ε}) = \frac{1}{σ_{2 y}^{c} (W_{p}^{*} \cdot \dot{ε}) \cdot σ_{2 y}^{t} (W_{p}^{*} \cdot \dot{ε})}$

In tension:

$σ_{2 y}^{t} (W_{p}^{*} \cdot \dot{ε}) = σ_{2 y}^{t} (1 + b_{2}^{t} {(W_{p}^{*})}^{n_{2}^{t}}) (1 + c_{2}^{t} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

In compression:

$σ_{2 y}^{c} (W_{p}^{*} \cdot \dot{ε}) = σ_{2 y}^{c} (1 + b_{2}^{c} {(W_{p}^{*})}^{n_{2}^{c}}) (1 + c_{2}^{c} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

Here $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$
Shear tests
Shear in plane 1-2

I_form = 0: Tsai-Wu $(F (σ) \leq F (W_{p}^{*}, \dot{ε}))$

I_form = 1: CRASURV $(F (W_{p}^{*}, \dot{ε}, σ) \leq 1)$

$F_{44} = \frac{1}{σ_{12 y}^{c} σ_{12 y}^{t}}$

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

$F (W_{p}^{*}, \dot{ε}) = (1 + b {(W_{p}^{*})}^{n}) \cdot (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

Here, $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$

$F_{44} (W_{p}^{*}, \dot{ε}) = \frac{1}{σ_{12 y}^{} (W_{p}^{*}, \dot{ε}) \cdot σ_{12 y}^{} (W_{p}^{*}, \dot{ε})}$

In shear:

$σ_{12 y}^{} (W_{p}^{*} \cdot \dot{ε}) = σ_{12 y}^{} (1 + b_{12}^{} {(W_{p}^{*})}^{n_{12}^{}}) (1 + c_{12}^{} \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

$σ_{12 y}$ can result from the sample test:
Interaction coefficients

I_form = 0: Tsai-Wu $(F (σ) \leq F (W_{p}^{*}, \dot{ε}))$

I_form = 1: CRASURV $(F (W_{p}^{*}, \dot{ε}, σ) \leq 1)$

$F_{12} = - \frac{α}{2} \sqrt{F_{11} F_{22}}$

The default reduction factor, $α = 1$ , is typically used.

$F (W_{p}^{*}, \dot{ε}) = (1 + b {(W_{p}^{*})}^{n}) \cdot (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$

Here, $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$

$F_{12} (W_{p}^{*}, \dot{ε}) = - \frac{α}{2} \sqrt{F_{11} (W_{p}^{*}, \dot{ε}) F_{22} (W_{p}^{*}, \dot{ε})}$

The default reduction factor, $α = 1$ , is typically used.

Note that the relative plastic work

W_{p}^{*}

is used in Tsai-Wu to calculate the yield surface; whereas in CRASURV, the relative plastic work is used to calculate the yield stress.

I_form = 0: Tsai-Wu $(F (σ) \leq F (W_{p}^{*}, \dot{ε}))$: I_form = 1: CRASURV $(F (W_{p}^{*}, \dot{ε}, σ) \leq 1)$
$F (σ) \leq F (W_{p}^{*}, \dot{ε})$
With $F (W_{p}^{*}, \dot{ε}) = (1 + b {(W_{p}^{*})}^{n}) * (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}}))$
Here $W_{p}^{*} = \frac{W_{p}}{W_{p}^{r e f}}$: $F (W_{p}^{*}, \dot{ε}, σ) \leq 1$
Material in elastic phase, if $F (σ) \leq F (W_{p}^{*}, \dot{ε})$
Material in nonlinear phase, if $F (σ) > F (W_{p}^{*}, \dot{ε})$

The yield stress limit $F (W_{p}^{*}, \dot{ε})$ is in range of 1 and $f_{\max}$: Material in elastic phase, if $F (W_{p}^{*}, \dot{ε}, σ) \leq 1$; Material in nonlinear phase, if $F (W_{p}^{*}, \dot{ε}, σ) > 1$

In LAW25 (Tsai-Wu and CRASURV) damage is a function of the total strain and the maximum damage factor.

If the total strain

ε > ε_{t}

or out of plane strain

γ_{i n i} < γ < γ_{\max}

, then the material is softened using the following method:(2)

σ^{r e d u c e} = σ \cdot (1 - d_{i})

with i=1,2,3

Where, d_i is the damage factor and is defined as:(3)

d_{i} = \min (\frac{ε_{i} - ε_{t i}}{ε_{i}} \cdot \frac{ε_{m i}}{ε_{m i} - ε_{t i}}, d_{\max})

with i=1,2(4)

d_{3} = \min (\frac{γ - γ_{i n i}}{γ_{\max} - γ_{i n i}} \cdot \frac{γ_{\max}}{γ}, d_{3 \max})

in direction 3 (delamination)

If the total strain is between $ε_{t} < ε < ε_{f}$ , the material begins to soften, but this damage is reversible. Once $ε > ε_{f}$ , then the damage is irreversible and if $ε \geq ε_{m}$ , then stress in material is reduced to 0.
Damage could be in elastic phase or in plastic phase. It depends on which phase $ε_{t}$ and $ε_{f}$ are defined in.
Element deletion is controlled by I_off. Select a different I_off option to control the criteria of element deletion. For additional information, refer to I_off in LAW25 in the Reference Guide.

LAW25 (Tsai-WU and CRASURV)

Elastic Phase

Tsai-Wu Yield Criteria for Iform=0 and =1

Tsai-Wu Yield Criteria for `I`_form=0 and =1