- #1
pyroknife
- 613
- 3
I am working on a 2-D planar problem in the x-y direction, dealing with stresses, strain, displacements. Under the linear elastic relation and after substitution I can write the following:
##
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} \\
\sigma_{xy} & \sigma_{yy}
\end{bmatrix} = \mu
\begin{bmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
\end{bmatrix}
+ \mu
\begin{bmatrix}
\frac{\partial u}{\partial x} & \frac{\partial v}{\partial x}\\
\frac{\partial u}{\partial y} & \frac{\partial v}{\partial y}
\end{bmatrix}
+ \lambda\begin{bmatrix}
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} & 0\\
0 & \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
\end{bmatrix}
##
I need to obtain the unknowns, which are the 4 displacement gradients
##
\frac{du}{dx}; \frac{du}{dy}; \frac{dv}{dx}; \frac{dv}{dy}
##
However, since the stress matrix is symmetrical, I only have 3 knowns. In structural problems, where does this 4th condition come from?
##
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} \\
\sigma_{xy} & \sigma_{yy}
\end{bmatrix} = \mu
\begin{bmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
\end{bmatrix}
+ \mu
\begin{bmatrix}
\frac{\partial u}{\partial x} & \frac{\partial v}{\partial x}\\
\frac{\partial u}{\partial y} & \frac{\partial v}{\partial y}
\end{bmatrix}
+ \lambda\begin{bmatrix}
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} & 0\\
0 & \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
\end{bmatrix}
##
I need to obtain the unknowns, which are the 4 displacement gradients
##
\frac{du}{dx}; \frac{du}{dy}; \frac{dv}{dx}; \frac{dv}{dy}
##
However, since the stress matrix is symmetrical, I only have 3 knowns. In structural problems, where does this 4th condition come from?