GreenLagrangeVerzerrungstensor

The Green-Lagrange Verzerrungstensor, commonly called the Green-Lagrange strain tensor, is a measure of finite (large) deformation defined with respect to the reference configuration. It is defined by E = 1/2 (F^T F − I), where F is the deformation gradient that maps differential material vectors from the reference configuration to the current configuration, and I is the identity tensor. If C = F^T F is the right Cauchy–Green deformation tensor, then E = 1/2 (C − I).

Properties of the tensor include symmetry and objectivity: E vanishes for a rigid-body motion and encodes stretch and shear relative to the reference state. Its principal strains are the eigenvalues of E. The determinant J = det(F) describes volume change, and invariants of C (or E) relate to material response in constitutive models. Since C = I + 2E, many hyperelastic formulations express energy densities as functions of C or equivalently of E.

Relation to displacement fields: Let u(X) be the displacement field in the reference configuration, so F = I + ∇u. Then E = ε + 1/2 (∇u)^T ∇u, where ε = sym(∇u) is the linear small-strain tensor. In the small-strain limit (|∇u| small), E ≈ ε, recovering the linear strain measure.

Applications: E is used in nonlinear elasticity and finite-deformation analyses to quantify strain and to formulate constitutive models and strain-energy densities that account for large deformations. It provides a direct link between the geometry of deformation (via F and C) and material response.