解説

異方性線形弾性材料の弾性定数テンソル

については、まず、応力、歪みとも対称テンソルなので、

次に、３つの主軸まわりの回転について弾性挙動が同じになる材料を直交異方性材料と呼ぶ。この場合、弾性定数テンソル

、弾性コンプライアンステンソル

の独立な成分は９つとなる。

弾性歪み ${}^{t} [ \epsilon^e ]$ を応力 ${}^{t} [ \sigma ]$ で表わすと、垂直成分は、

$\displaystyle {}^{t} \epsilon^e_{xx}$	$\textstyle =$	$\displaystyle B^e_{xxxx} {}^{t} \sigma_{xx} + B^e_{xxyy} {}^{t} \sigma_{yy} + B^e_{xxzz} {}^{t} \sigma_{zz}$
$\displaystyle {}^{t} \epsilon^e_{yy}$	$\textstyle =$	$\displaystyle B^e_{yyxx} {}^{t} \sigma_{xx} + B^e_{yyyy} {}^{t} \sigma_{yy} + B^e_{yyzz} {}^{t} \sigma_{zz}$
$\displaystyle {}^{t} \epsilon^e_{zz}$	$\textstyle =$	$\displaystyle B^e_{zzxx} {}^{t} \sigma_{xx} + B^e_{zzyy} {}^{t} \sigma_{yy} + B^e_{zzzz} {}^{t} \sigma_{zz}$	(8.11)

$\displaystyle {}^{t} \epsilon^e_{xy}$	$\textstyle =$	$\displaystyle B^e_{xyxy} {}^{t} \sigma_{xy}$
$\displaystyle {}^{t} \epsilon^e_{yz}$	$\textstyle =$	$\displaystyle B^e_{yzyz} {}^{t} \sigma_{yz}$
$\displaystyle {}^{t} \epsilon^e_{zx}$	$\textstyle =$	$\displaystyle B^e_{zxzx} {}^{t} \sigma_{zx}$
$\displaystyle {}^{t} \epsilon^e_{yx}$	$\textstyle =$	$\displaystyle B^e_{yxyx} {}^{t} \sigma_{yx}$
$\displaystyle {}^{t} \epsilon^e_{zy}$	$\textstyle =$	$\displaystyle B^e_{zyzy} {}^{t} \sigma_{zy}$
$\displaystyle {}^{t} \epsilon^e_{xz}$	$\textstyle =$	$\displaystyle B^e_{xzxz} {}^{t} \sigma_{xz}$	(8.12)

次に、この逆関係をとる。応力 ${}^{t} [ \sigma ]$ を弾性歪み ${}^{t} [ \epsilon^e ]$ で表わすと、垂直成分は、

$\displaystyle {}^{t} \sigma_{xx}$	$\textstyle =$	$\displaystyle C^e_{xxxx} {}^{t} \epsilon^e_{xx} + C^e_{xxyy} {}^{t} \epsilon^e_{yy} + C^e_{xxzz} {}^{t} \epsilon^e_{zz}$
$\displaystyle {}^{t} \sigma_{yy}$	$\textstyle =$	$\displaystyle C^e_{yyxx} {}^{t} \epsilon^e_{xx} + C^e_{yyyy} {}^{t} \epsilon^e_{yy} + C^e_{yyzz} {}^{t} \epsilon^e_{zz}$
$\displaystyle {}^{t} \sigma_{zz}$	$\textstyle =$	$\displaystyle C^e_{zzxx} {}^{t} \epsilon^e_{xx} + C^e_{zzyy} {}^{t} \epsilon^e_{yy} + C^e_{zzzz} {}^{t} \epsilon^e_{zz}$	(8.13)

$\displaystyle {}^{t} \sigma_{xy}$	$\textstyle =$	$\displaystyle C^e_{xyxy} {}^{t} \epsilon^e_{xy}$
$\displaystyle {}^{t} \sigma_{yz}$	$\textstyle =$	$\displaystyle C^e_{yzyz} {}^{t} \epsilon^e_{yz}$
$\displaystyle {}^{t} \sigma_{zx}$	$\textstyle =$	$\displaystyle C^e_{zxzx} {}^{t} \epsilon^e_{zx}$
$\displaystyle {}^{t} \sigma_{yx}$	$\textstyle =$	$\displaystyle C^e_{yxyx} {}^{t} \epsilon^e_{yx}$
$\displaystyle {}^{t} \sigma_{zy}$	$\textstyle =$	$\displaystyle C^e_{zyzy} {}^{t} \epsilon^e_{zy}$
$\displaystyle {}^{t} \sigma_{xz}$	$\textstyle =$	$\displaystyle C^e_{xzxz} {}^{t} \epsilon^e_{xz}$	(8.14)

$\displaystyle C^e_{ijkl} = C^e_{jikl}$
$\displaystyle C^e_{ijkl} = C^e_{ijlk}$			(8.8)