Torsions

Torsion is a general term describing twisting about an axis. In engineering and physics it most often refers to the deformation produced when a torque is applied to a body. In mathematics and theoretical physics it also denotes twisting properties of curves, spacetime, or algebraic structures that quantify non-planarity or finite-order behavior.

In mechanical engineering, torsion occurs when a shaft or bar is subjected to a torque, causing shear stresses and an angular twist along its length. For a circular shaft of radius r, the polar moment of inertia is J = π d^4 / 32 (d = diameter). The shear stress at the outer surface is τ = T r / J, where T is the applied torque. The angle of twist is θ = T L / (G J), with L the length and G the shear modulus. The maximum shear stress is at the outer surface, τ_max = T r / J, and the power transmitted is P = T ω, where ω is angular velocity.

In differential geometry, torsion measures how a space curve departs from planarity. Along a curve parameterized by arc length s, curvature κ and torsion τ describe the Frenet-Serret frame through dT/ds = κ N, dN/ds = −κ T + τ B, and dB/ds = −τ N. A common explicit formula for torsion is τ = ( (r' × r'') · r''' ) / |r' × r''|^2. Nonzero torsion indicates the curve twists out of a plane.

In algebra, an element x of a group G is called torsion if some positive integer n satisfies x^n = e (multiplicative notation) or n x = 0 (additive). The torsion elements form the torsion subgroup T(G). For modules, a torsion element is annihilated by a nonzero ring element; examples include cyclic groups Z/nZ.

Other usages include torsion tensors in geometry and spacetime theories, and Reidemeister torsion in topology, which generalize twisting invariants.