GreenLagrangestrain

GreenLagrange strain, often called Green-Lagrange strain, is a nonlinear measure of finite deformation used in continuum mechanics. It is defined with respect to the reference configuration by E = 1/2 (F^T F − I), where F is the deformation gradient mapping material points from the undeformed state X to the deformed position x = F X, and I is the identity tensor. The right Cauchy-Green deformation tensor is C = F^T F, so E = 1/2 (C − I). This makes Green-Lagrange strain a natural measure for large deformations in the Lagrangian (material) description.

In the limit of small deformations, Green-Lagrange strain reduces to the linearized small-strain tensor ε ≈ 1/2 (∇u + ∇u^T). It is frame-indifferent and defined relative to the original configuration, distinguishing it from Eulerian strains that reference the current configuration. E is symmetric and commonly used in nonlinear elasticity formulations to express strain energy in terms of C or E.

Applications of the Green-Lagrange strain are widespread in nonlinear solid mechanics and finite-strain analyses. It is used in constitutive models for hyperelastic materials and in finite element methods to describe large deformations. From E, principal Green-Lagrange strains are obtained as the eigenvalues of E, with principal stretches related by λ_i = sqrt(1 + 2 ε_i^G).

The term is named for George Green and the Lagrangian formulation of deformation, and it remains a standard tool for describing finite strains in engineering and physics.