Finitastrainsteorier

Finitastrainsteorier, or finite strain theories, are frameworks within continuum mechanics for describing materials undergoing large deformations. They extend small-strain theories by employing nonlinear kinematics and strain measures that remain meaningful when displacements are not small. The term is used in several European languages to denote these theories and their applications to solid mechanics, rubber, and soft tissues.

Key concepts include the deformation gradient F, which maps a reference configuration to the current configuration; right and left Cauchy–Green tensors C = F^T F and B = F F^T; and strain measures such as the Green–Lagrange strain E = 1/2 (F^T F − I) and the Euler–Almansi strain e = 1/2 (I − (F F^T)^{-1}). These measures allow constitutive models to describe large rotations and stretches. Material behavior is often formulated through hyperelastic energy functions W(F) or through rate-based plasticity with multiplicative decomposition F = F_e F_p.

Applications of finite strain theories include metal forming, rubber elasticity, geomechanics, and biomechanics, where deformations are not small and nonlinear effects are important. Numerical implementation is typically via the finite element method, with implicit or explicit time integration and considerations for stability and convergence under strong nonlinearities and large rotations.

History and development of finite strain theory emerged in the 20th century with work by Green, Lagrange, Euler, and Almansi, among others, leading to a variety of strain measures and constitutive frameworks. Ongoing research continues to address anisotropy, viscoelasticity, and coupled multi-physics phenomena within the finite strain paradigm.