Smallstrain

Small-strain refers to a simplifying assumption used in continuum mechanics and solid mechanics, applicable when deformations are small compared with the characteristic dimensions of the body. Under this assumption, strains are small and the kinematic relations between displacements and strains can be linearized. It is widely used for linear elastic materials and many engineering problems where deformations are modest, such as metal structures under service loads.

Kinematics and strain measures: Let u(x) be the displacement field. The infinitesimal (small) strain tensor is defined as ε_ij = 1/2 (∂u_i/∂x_j + ∂u_j/∂x_i). Terms of second order in the displacement gradient ∇u are neglected. For contrast, finite-strain theory uses measures such as the Green-Lagrange strain E = 1/2 (F^T F − I) with F = I + ∇u, which retain nonlinear terms.

Constitutive relations: In the small-strain regime, linear elasticity is commonly assumed, with stress linearly related to strain via Hooke’s law σ = C : ε. In isotropic form, σ_ij = λ δ_ij ε_kk + 2μ ε_ij, described by Lamé parameters λ and μ (or equivalently Young’s modulus E and Poisson’s ratio ν). Governing equations combine the equilibrium of linear momentum, ∂σ_ij/∂x_j + b_i = 0, with appropriate boundary conditions and the compatibility of strains.

Limitations and scope: The small-strain framework is valid when strains are small, typically well below about 1%, and may be insufficient for large deformations, plastic flow, or nonlinear material behavior. It provides a foundational, linearized description for many structural and mechanical analyses and serves as a baseline against more general finite-strain theories.