Understanding Torsion Theory Assumptions for Circular Sections

Torsion theory helps us understand how shafts twist when subjected to a torque (twisting moment). This theory relies on certain assumptions to simplify the analysis, particularly for shafts with circular or axisymmetric cross-sections.

Let's examine the assumptions listed in the options:

1. Circular sections remain circular after twisting: This is a fundamental assumption for circular shafts under torsion. Unlike non-circular sections which may warp (sections perpendicular to the axis do not remain plane), circular sections do not warp. They remain plane and circular after twisting. This is because the shear stresses developed are purely tangential and radial deformation is uniform around the circumference.
2. The material is homogenous and isotropic: This is a common assumption in many mechanics of materials problems, including torsion theory.
   • Homogenous means the material has the same properties throughout its volume.
   • Isotropic means the material has the same properties in all directions at any given point.
   Assuming the material is homogenous and isotropic simplifies the relationship between stress and strain (Hooke's Law) and ensures uniform material behavior under torque.
3. Straight radial lines in a transverse section remain straight after twisting: Consider a line drawn from the center of a circular section to its circumference. After applying a torque, this line will rotate, but it will remain a straight radial line. It doesn't curve. This assumption is a direct consequence of circular sections remaining plane and the deformation being proportional to the distance from the center.

These three points are indeed the core assumptions made when applying the standard torsion formula ($\tau = \frac{Tr}{J}$) to circular shafts. The theory is specifically developed based on these behaviors observed in circular cross-sections made of homogenous, isotropic materials under elastic torsion.

Therefore, all the listed assumptions are considered true for the torsion theory applied to axisymmetric (circular) sections.