탄성학 기본구성 방정식 (응력평형방정식,변형률-변위 관계식, 응력-변형률 관계식)

1) 응력 평형 방정식

$\frac{\partial \sigma_{xx}}{\partial x}+ \frac{\partial \sigma_{yx}}{\partial y}+ \frac{\partial \sigma_{zx}}{\partial z}+ f_{x}=0$

 

$\frac{\partial \sigma_{xy}}{\partial x}+ \frac{\partial \sigma_{yy}}{\partial y}+ \frac{\partial \sigma_{zy}}{\partial z}+ f_{y}=0$

 

$\frac{\partial \sigma_{xz}}{\partial x}+ \frac{\partial \sigma_{yz}}{\partial y}+ \frac{\partial \sigma_{zz}}{\partial z}+ f_{z}=0$

 

간단한 형태로 표현하면 아래와 같습니다. 

 

$\sigma_{ji,j}+f_{i}=0$

 

아인슈타인 표기법입니다. 

 

2) 변형률-변위 관계식

$\varepsilon_{x}=\frac{\partial u}{\partial x}$

$\varepsilon_{y}=\frac{\partial v}{\partial y}$

$\varepsilon_{z}=\frac{\partial w}{\partial z}$

 

$\gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}$

$\gamma_{yz}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}$

$\gamma_{xz}=\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}$

 

 

3) 응력-변형률 관계식

$\varepsilon _{x}=\frac{1}{E}\left [ \sigma_{x}-\nu (\sigma_{y}+\sigma_{z}) \right ]$
$\varepsilon _{y}=\frac{1}{E}\left [ \sigma_{y}-\nu (\sigma_{x}+\sigma_{z}) \right ]$
$\varepsilon _{z}=\frac{1}{E}\left [ \sigma_{z}-\nu (\sigma_{x}+\sigma_{y}) \right ]$

 

$\gamma _{xy}=\frac{1}{G}\tau_{xy}$

$\gamma _{xz}=\frac{1}{G}\tau_{xz}$

$\gamma _{yz}=\frac{1}{G}\tau_{yz}$

 

G와 E의 관계는 아래와 같습니다. 

 

$G=\frac{E}{2(1+\nu)}$