The torsional rigidity shaft is expressed by:
A. Maximum torque it can transmit
B. Number of cycles it undergoes before failure
C. Elastic limit up to which it can resist torsion, shear and bending stresses
D. The torque required to produce a twist of one radian per unit length of the shaft

Hint: First understand the concept of torsional rigidity and shafts by defining them. Then find out about the torque. Then develop a relation between torsional rigidity, shafts, and torque and, you will reach the correct answer.

Complete step by step answer:
Shafts are a type of mechanical component having a circular cross-section. They rotate and through this motion they emit power or in other words, we can say torque.
Torque is the force required for rotation.
Torsional rigidity is the minimum amount of force required to change the shape of an object. This is to be done by twisting through the unit dimension. Mathematical expression for torsion can be given as:
\[\dfrac{T}{J}=\dfrac{G\times \theta }{L}=\dfrac{\tau }{r}\]
where,
T is torque
J is the polar moment of inertia of the shaft
$\tau $ is the shear stress
r is the radius of the shaft
G is the shear modulus
L length of the shaft
$\theta $ is the angle of twist
Thus, from the above equation we can conclude that torsional rigidity of the shaft is expressed as the torque required to produce a twist of one radian per unit length of the shaft. So, option D is correct.

Note: Torsional rigidity is the minimum force thus option A is incorrect. Also, it does not depend on the number of cycles before failure, thus option B is incorrect. And also, it does not resist torsion, shear, and bending stress thus, option C is also incorrect.
The question can also be solved using elimination methods.