# Confused about shear elasticity and complementary shear stress

I am a self learner of continuum mechanic. I am confused about simple shear stress in situation similar to figure 1,

in case $F_\textrm{ext}$ is caused by external perturbation by i.e., human, what are the forces that involve in this scenario? In my mind, there are

1. $F_\textrm{ext}$ is caused by external perturbation

2. $F_\textrm{couple}$ which $F_\textrm{couple}=-F_\textrm{ext}$

3. $F_\textrm{com}$ which are the complementary shear stress multiply with area

4. $F_\textrm{comneg}$ which $F_\textrm{comneg}=-F_\textrm{com}$ however, these three forces could result in figure 2, not figure 1. Therefore, I think there should be constrained forces i.e., from human hand, surrounding material or floor that eliminate the force component in z axis by reaction forces.

5. $F_\textrm{reac}$ which equal to Force components in z axis, $F_\textrm{reac} =-F_\textrm{com}$

Is that right?

in the case of dynamic deformation,it will not necessary that $F_\textrm{couple}$ equal to $-F_\textrm{ext}$ and $F_\textrm{comneg}$ equal to $F_\textrm{com}$. $F_\textrm{reac}$ equal to force components in z axis (in case $F_\textrm{reac}$ is caused from surrounding continuum material) right?

I understand is, in this situation, the object's stiffness reacts to force in x-direction only ($F_\textrm{ext}$ and $F_\textrm{couple}$), not the force in z-direction ($F_\textrm{com}$ and $F_\textrm{comneg}$). Therefore $F_\textrm{reac} =-F_\textrm{com}$. Is this right?

• $F_{com}$ and $F_{comneg}$ aren't exist because of free edges. – lucas May 7 '16 at 18:55
• Ok, I get it now. Anyway, does it have Freac normal to surface AB and CD to stop the object from clockwise rotation? – user3188389 May 8 '16 at 9:47

The couple created by the shear stresses is opposed by the couple created by the complementary shear stresses, which prevents turning of the object. The object will not deform to the shape as shown in Figure 2 because $F_{com} = F_{comneg}$ and is actually a reaction couple to the couple produced by the shear stresses, thus causing deformation in only one direction, which is the direction in which the shear stress was applied.