finitestrain

Finite strain, also referred to as large strain, describes deformation in bodies undergoing non-infinitesimal displacements and rotations. In this framework, the deformation gradient F maps an infinitesimal material vector dX in the reference configuration to the current configuration dx = F dX. The approach accounts for both stretches and rotations, making it suitable for problems where linearized, small-strain theories fail.

Key strain measures are built from F and its associated tensors. The right Cauchy-Green deformation tensor C = F^T F and the left Cauchy-Green tensor B = F F^T are commonly used because they are objective and invariant under rigid body motions. A standard finite-strain measure is the Green-Lagrange strain, E = 1/2 (F^T F − I). For small deformations, E reduces to the linearized strain ε ≈ 1/2 (∇u + ∇u^T). Other finite-strain measures include the Almansi strain in the current configuration, e = 1/2 (I − F^{−T} F^{−1}), and the logarithmic (Hencky) strain h = ln U, where F = R U is the polar decomposition into rotation R and stretch U.

Applications of finite-strain theory span metal forming, rubber elasticity, biomechanics, and geomechanics, where large displacements or rotations occur. Constitutive models are typically formulated in terms of invariants of C or E and are implemented within nonlinear finite element analyses. Challenges include choosing objective stress rates, avoiding numerical instabilities, and ensuring accurate convergence in nonlinear solvers.