smalldeformation

Small deformation, also called infinitesimal deformation, is a regime in continuum mechanics in which displacements are small compared with the body's dimensions. In this regime strains are small and rotations of material fibers are negligible, allowing linearized descriptions of motion. The displacement field u(x) maps reference points to their current positions, and the linearized strain tensor ε is defined by ε = 1/2 (∇u + ∇u^T). The deformation gradient F = ∂x/∂X is approximated by F ≈ I + ∇u, so geometric nonlinearities are neglected.

Under small strains, the stress–strain relation is typically linear. In linear elastic materials, stress σ is related to ε by σ = C:ε, where C is the stiffness tensor. For isotropic materials, this reduces to σ = 2μ ε + λ tr(ε) I, with μ and λ the Lamé parameters (equivalently expressed in terms of Young's modulus E and Poisson's ratio ν). The equations of motion (or equilibrium in static problems) are ∇·σ + b = 0, where b represents body forces, together with boundary conditions on displacements or tractions. Problems often assume plane strain or plane stress to reduce dimensionality.

Limitations of the small-deformation framework include its inapplicability to large rotations or large strains, where the Green–Lagrange strain and nonlinear kinematics are required. It also assumes material behavior can be captured by linear or weakly nonlinear constitutive laws. In such cases finite-deformation theories provide more accurate descriptions.

Applications of small-deformation theory include structural analysis of beams, plates, and shells under small loads, linear elasticity problems in civil and mechanical engineering, vibrations, and acoustics.