Elasticity/Stress example 1

Example 1
For a unit cube that is in statical equilibrium and has a homogeneous state of stress, show that there are only six possible components of stress.

Solution
Since the stress in the cube is homogeneous, the forces on opposite faces of the cube should be equal and opposite. Hence, the possible components of stress are

\sigma_{xx},~\sigma_{yy},~\sigma_{zz},~\tau_{xy},~\tau_{yx},~ \tau_{yz},~\tau_{zy},~\tau_{zx},~\tau_{xz}. $$ The homogeneity of the stress also means that the three components of force on any face must pass through the center of the face.

Since the cube is in static equilibrium, the total moment about an axis passing through the center of the cube and parallel to the $$x\,$$ axis should be zero. (Recall that the moment of a force about an axis measures the tendency to rotate a rigid body about the axis. Also recall that the sign of a moment is positive or negative depending on the clockwise or anticlockwise direction of rotation due to the applied force).

Hence, $$\tau_{yz} = \tau_{zy}\,$$. Similarly, $$\tau_{zx} = \tau_{xz}\,$$, and $$\tau_{xy} = \tau_{yx}\,$$.

Therefore, the six possible components of stress are

\sigma_{xx},~\sigma_{yy},~\sigma_{zz},~\tau_{xy},~\tau_{yz},~\tau_{zx}. $$