Combined Extension/Torsion of Solid Cylinder - Deformation Gradient

[sebb1e]

Homework Statement

(Attached example on Combined extension and torsion of a solid cylinder)

The Attempt at a Solution

Using the Grad operator given on the given position vector x, I don't understand how to get the tor*r co-efficient on e_theta * e_z (So tor*r*lamda e_theta*E_Z in final answer) I understand that due to the shear there has to be a co-efficient there, but can't get this answer to come out. My best guess so far is that d/Theta has to be rearranged to d/theta - (1/tor) d/z but this doesn't work.

 

[mark726]

I can understand your confusion with trying to derive the correct answer for the torque coefficient in this problem. To solve this, we need to use the relationships between the position vector x, the shear modulus G, and the applied torque T.

First, let's define the torque coefficient as C = T/(G*tor). This coefficient represents the ratio of applied torque to the product of the shear modulus and the cylinder's torsion constant.

Next, let's look at the shear strain in the cylinder. From the given position vector x, we can see that the shear strain is given by the expression d/Theta - (1/tor)*d/Z. This tells us that the shear strain is dependent on both the angle of rotation (Theta) and the distance from the axis of rotation (Z).

Now, we can use the definition of shear modulus G = shear stress/shear strain to find the shear stress in terms of the position vector x. This gives us the expression G = T/(C*(d/Theta - (1/tor)*d/Z)).

Finally, we can substitute this expression for G into the equation for torque coefficient C = T/(G*tor) and rearrange to get the final answer of C = T/(T/(C*(d/Theta - (1/tor)*d/Z))*tor) = C*(d/Theta - (1/tor)*d/Z).

I hope this helps clarify the derivation of the torque coefficient in this problem. Keep in mind, this is just one approach and there may be other ways to arrive at the same answer. it's important to be open to different methods and approaches in problem-solving.