Strain and small-deformation kinematics

Strain measures relative shape change

Strain compares deformation with original size, so it is dimensionless. Normal strain measures length change; shear strain measures angle change; volumetric strain measures local volume change.

Linearize the squared length

For small displacement u(x), a material vector dX changes approximately to (I + ∇u)dX. The change in squared length is 2 dX·ε·dX, where ε = ½(∇u + ∇uᵀ). The symmetric part measures stretch; the antisymmetric part is local rigid rotation.

A rod of original length 2 m extending by 1 mm has normal strain ε = ΔL/L = 0.001/2 = 5×10⁻⁴. It is dimensionless, often reported as 500 microstrain.

Small strain has a small-geometry limit

For large rotations or elongations, the linear strain tensor can misrepresent finite changes. Use a finite-deformation measure when displacement gradients are not small.

Work it: A displacement field u = (0.002x, 0.003y, 0) produces ε_xx = 0.002 and ε_yy = 0.003. Calculate first-order area strain ε_xx + ε_yy and state assumptions.