Stress and strain

When a body is deformed but kept in equilibrium, a restoring force appears inside it, equal in size and opposite in direction to the applied force. Stress is this restoring force per unit area: magnitude F/A.

Stress has the SI unit N m⁻², called the pascal (Pa), and the dimensional formula [ML⁻¹T⁻²], the same as pressure.

Tensile stress: equal and opposite pulls normal to the cross-section stretch a cylinder. Compressive stress: pushes shorten it. Both are longitudinal stress, and the resulting longitudinal strain is ΔL/L.

Shearing (tangential) stress: equal and opposite forces act parallel to the cross-section, so opposite faces slide by Δx relative to each other. Shearing strain = Δx/L = tan θ, where θ is the tilt from the vertical.

θ is usually tiny, so tan θ ≈ θ (in radians). Even at θ = 10°, θ and tan θ differ by only about 1%. Pushing the top cover of a thick book sideways shows shear.

Hydraulic stress: a solid held in a fluid under high pressure is squeezed normal to its surface at every point. Its volume falls with no change of shape; the internal restoring force per unit area equals the hydraulic pressure.

Volume strain = ΔV/V. Every strain is a ratio of a change in a dimension to the original dimension, so strain has no unit and no dimensions.