shearstress

Shear stress is the component of stress that acts parallel to a surface, in contrast to normal stress which acts perpendicular to the surface. In a three‑dimensional stress state, the traction vector t on a surface with outward normal n is given by t = σ · n, and the shear (tangential) portion of t is the component lying in the plane of the surface. A common way to express it for engineering purposes is tau = F_t / A, where F_t is the tangential force and A is the contact area. Shear stress is measured in pascals (N/m^2).

In fluids, shear stress arises from viscosity and relates to the rate of deformation. Newton’s law of viscosity states tau = mu * (du/dy), where mu is the dynamic viscosity and du/dy is the velocity gradient perpendicular to the flow direction. This relationship underpins the behavior of Newtonian fluids and governs boundary-layer flows.

In solids, shear is related to deformation through the shear strain gamma and the shear modulus G, with tau = G * gamma for small deformations. The shear modulus describes a material’s resistance to shape changes at constant volume.

Special cases include torsion of circular shafts, where the shear stress is tau = T * r / J, with T the applied torque, r the radial coordinate, and J the polar moment of inertia.

The maximum shear stress on a plane is related to principal stresses by tau_max = (sigma1 − sigma3)/2, a result visualized by Mohr’s circle. Applications span structural design, geophysics, and materials science, where controlling shear stress is key to preventing failure and understanding deformation.