# LAW25（Tsai-WUおよびCRASURV）

## 弾性相

$\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\gamma }_{12}\\ {\gamma }_{23}\\ {\gamma }_{31}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{11}}& -\frac{{\nu }_{12}}{{E}_{11}}& -\frac{{\nu }_{12}}{{E}_{33}}& 0& 0& 0\\ & \frac{1}{{E}_{22}}& -\frac{{\nu }_{12}}{{E}_{22}}& 0& 0& 0\\ & & \frac{1}{{E}_{33}}& 0& 0& 0\\ & & & \frac{1}{2{G}_{12}}& 0& 0\\ & symm.& & & \frac{1}{2{G}_{23}}& 0\\ & & & & & \frac{1}{2{G}_{31}}\end{array}\right]\cdot \left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{23}\\ {\sigma }_{31}\end{array}\right]$

## Iform=0 および =1の場合のTsai-Wu降伏基準

LAW25におけるTsai-Wu降伏曲面は、次のように6つの係数で定義されます：
Iform=0: Tsai-Wu $\left(F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\right)$
Iform=1: CRASURV $\left(F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1\right)$
$F\left(\sigma \right)={F}_{1}{\sigma }_{1}+{F}_{2}{\sigma }_{2}+{F}_{11}{\sigma }_{1}^{2}+{F}_{22}{\sigma }_{2}^{2}+{F}_{44}{\sigma }_{12}^{2}+2{F}_{12}{\sigma }_{1}{\sigma }_{2}$
$\begin{array}{l}F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)={F}_{1}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){\sigma }_{1}+{F}_{2}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){\sigma }_{2}+{F}_{11}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){\sigma }_{1}^{2}\\ +{F}_{22}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){\sigma }_{2}^{2}+{F}_{44}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){\sigma }_{12}^{2}+2{F}_{12}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){\sigma }_{1}{\sigma }_{2}\end{array}$

これら６つの係数は、次の試験の降伏応力により決定することもできます：
• 引張り / 圧縮試験

縦方向引張り / 圧縮試験（方向1、繊維方向）

Iform = 0：Tsai-Wu $\left(F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\right)$
Iform = 1：CRASURV $\left(F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1\right)$
${F}_{1}=-\frac{1}{{\sigma }_{1y}^{c}}+\frac{1}{{\sigma }_{1y}^{t}}$
${F}_{11}=\frac{1}{{\sigma }_{1y}^{c}{\sigma }_{1y}^{t}}$
$F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)=\left(1+b{\left({W}_{p}^{*}\right)}^{n}\right)\cdot \left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$
${F}_{1}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)=-\frac{1}{{\sigma }_{1y}^{c}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)}+\frac{1}{{\sigma }_{1y}^{t}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)}$

${F}_{11}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)=\frac{1}{{\sigma }_{1y}^{c}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\cdot {\sigma }_{1y}^{t}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)}$

引張の場合：

${\sigma }_{1y}^{t}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\text{=}{\sigma }_{1y}^{t}\left(1+{b}_{1}^{t}{\left({W}_{p}^{*}\right)}^{{n}_{1}^{t}}\right)\left(1+{c}_{1}^{t}\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$

圧縮の場合：

${\sigma }_{1y}^{c}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\text{=}{\sigma }_{1y}^{c}\left(1+{b}_{1}^{c}{\left({W}_{p}^{*}\right)}^{{n}_{1}^{c}}\right)\left(1+{c}_{1}^{c}\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$

ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$

横方向引張り / 圧縮試験（方向2）

Iform = 0：Tsai-Wu $\left(F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\right)$
Iform = 1：CRASURV $\left(F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1\right)$
${F}_{2}=-\frac{1}{{\sigma }_{2y}^{c}}+\frac{1}{{\sigma }_{2y}^{t}}$
${F}_{22}=\frac{1}{{\sigma }_{2y}^{c}{\sigma }_{2y}^{t}}$
ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$
${F}_{2}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)=-\frac{1}{{\sigma }_{2y}^{c}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)}+\frac{1}{{\sigma }_{2y}^{t}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)}$

${F}_{22}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)=\frac{1}{{\sigma }_{2y}^{c}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\cdot {\sigma }_{2y}^{t}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)}$

引張の場合：

${\sigma }_{2y}^{t}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\text{=}{\sigma }_{2y}^{t}\left(1+{b}_{2}^{t}{\left({W}_{p}^{*}\right)}^{{n}_{2}^{t}}\right)\left(1+{c}_{2}^{t}\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$

圧縮の場合：

${\sigma }_{2y}^{c}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\text{=}{\sigma }_{2y}^{c}\left(1+{b}_{2}^{c}{\left({W}_{p}^{*}\right)}^{{n}_{2}^{c}}\right)\left(1+{c}_{2}^{c}\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$

ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$

• せん断試験

平面1-2でのせん断

Iform = 0：Tsai-Wu $\left(F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\right)$
Iform = 1：CRASURV $\left(F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1\right)$
${F}_{44}=\frac{1}{{\sigma }_{12y}^{c}{\sigma }_{12y}^{t}}$
${\sigma }_{12y}^{t}$ および ${\sigma }_{12y}^{c}$ はサンプル試験の結果であり得る：

$F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)=\left(1+b{\left({W}_{p}^{*}\right)}^{n}\right)\cdot \left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$
${F}_{44}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)=\frac{1}{{\sigma }_{12y}^{}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\cdot {\sigma }_{12y}^{}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)}$
せん断の場合：
${\sigma }_{12y}^{}\left({W}_{p}^{*}\cdot \stackrel{˙}{\epsilon }\right)\text{=}{\sigma }_{12y}^{}\left(1+{b}_{12}^{}{\left({W}_{p}^{*}\right)}^{{n}_{12}^{}}\right)\left(1+{c}_{12}^{}\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
${\sigma }_{12y}$ はサンプル試験の結果であり得る：

• 相互作用係数
Iform = 0：Tsai-Wu $\left(F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\right)$
Iform = 1：CRASURV $\left(F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1\right)$
${F}_{12}=-\frac{\alpha }{2}\sqrt{{F}_{11}{F}_{22}}$
一般には、デフォルトの減少係数、 $\alpha =1$ が使用されます。
$F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)=\left(1+b{\left({W}_{p}^{*}\right)}^{n}\right)\cdot \left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$
${F}_{12}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)=-\frac{\alpha }{2}\sqrt{{F}_{11}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right){F}_{22}\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)}$
一般には、デフォルトの減少係数、 $\alpha =1$ が使用されます。
Tsai-Wuでは、相対塑性仕事 ${W}_{p}^{*}$ を使用して降伏曲面を計算します。CRASURVでは、相対塑性仕事は降伏応力の計算に使用されます。
Iform = 0：Tsai-Wu $\left(F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)\right)$
Iform = 1：CRASURV $\left(F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1\right)$
$F\left(\sigma \right)\le F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)$
ここで、 $F\left({W}_{p}^{*},\stackrel{˙}{\epsilon }\right)=\left(1+b{\left({W}_{p}^{*}\right)}^{n}\right)\ast \left(1+c\mathrm{ln}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
ここで、 ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$
$F\left({W}_{p}^{*},\stackrel{˙}{\epsilon },\sigma \right)\le 1$

LAW25（Tsai-WuおよびCRASURV）では、損傷が全ひずみと最大損傷係数の関数になります。

${\sigma }^{reduce}=\sigma \cdot \left(1-{d}_{i}\right)$
で、i=1,2,3
ここで、diは損傷係数で、以下のように定義されます：(3)
ここで、i=1,2(4)
${d}_{3}=\mathrm{min}\left(\frac{\gamma -{\gamma }_{ini}}{{\gamma }_{\mathrm{max}}-{\gamma }_{ini}}\cdot \frac{{\gamma }_{\mathrm{max}}}{\gamma },{d}_{3\mathrm{max}}\right)$
in direction 3 (delamination)
• If the total strain is between ${\epsilon }_{t}<\epsilon <{\epsilon }_{f}$ , the material begins to soften, but this damage is reversible. Once $\epsilon >{\epsilon }_{f}$ , then the damage is irreversible and if $\epsilon \ge {\epsilon }_{m}$ , then stress in material is reduced to 0.
• Damage could be in elastic phase or in plastic phase. It depends on which phase ${\epsilon }_{t}$ and ${\epsilon }_{f}$ are defined in.
• Element deletion is controlled by Ioff. Select a different Ioff option to control the criteria of element deletion. For additional information, refer to Ioff in LAW25 in the Reference Guide.