deformationgradienten

In continuum mechanics, the deformation gradient, denoted F, describes the local deformation of a material element as it moves from a reference configuration X to a current configuration x. It is defined by F = ∂x/∂X, so differential line elements satisfy dx = F dX. F is a second-order tensor that encodes rotation, stretch and shear of material elements. For a smooth deformation, F is invertible and its determinant J = det F represents local volume change.

Derived quantities associated with F include the right Cauchy–Green deformation tensor C = F^T F and the Green–Lagrange strain E = 1/2 (F^T F − I). The polar decomposition of F expresses it as F = R U = V R, where R is a rotation and U, V are the right and left stretch tensors. The determinant J = det F > 0 ensures non-interpenetration of matter in finite strains and quantifies local volume change.

Small-strain theory uses the linear approximation F ≈ I + ∇u, with u the displacement field. The corresponding linearized strain is ε = 1/2 (∇u + ∇u^T). This simplification is valid for small displacements and rotations.

Applications of the deformation gradient are central to constitutive modeling and numerical simulation. In hyperelastic, elastoplastic, and viscoelastic models, F serves as the primary kinematic variable from which stresses or strain energies are derived. In finite element analysis, F relates material points to current positions and ensures frame-independence of physical laws, commonly written in index form as F_iJ = ∂x_i/∂X_J.