Finite strain theory

Finite strain theory describes material deformation without assuming that changes in shape or size are small. It accounts for geometric nonlinearities that infinitesimal-strain measures neglect.

Deformation gradient: A tensor that maps infinitesimal material line elements in a reference configuration to their deformed counterparts. It is the central local quantity from which finite strain measures are constructed.
Continuum mechanics: The mechanics of materials modeled as continuous fields of mass, motion, stress, and deformation. Finite strain theory is a geometric framework within continuum mechanics.
Infinitesimal strain theory: A linearized framework that approximates deformation using small displacement gradients. It omits geometric terms that finite strain theory retains when deformation is large.
Finite element method: A numerical method that approximates solutions to field equations using finite-dimensional elements. Nonlinear finite element simulations use finite strain kinematics for large-deformation problems.
Right Cauchy–Green deformation tensor: The tensor formed as the transpose of the deformation gradient times the deformation gradient. It measures changes in material lengths while removing rigid rotation.
Lagrangian description: A description of motion that tracks material points from a chosen reference configuration. Reference-based strain measures compare current geometry with labeled material points.
Engineering strain: A change in length divided by the original length, commonly used for small uniaxial deformation. Its reference-length ratio can misrepresent large stretches and does not capture general deformation geometrically.
Finite-strain plasticity: Plasticity theory formulated for material deformation large enough to require finite kinematics. It combines finite strain measures with evolving irreversible deformation.
Green–Lagrange strain tensor: A strain tensor defined as half the difference between the right Cauchy–Green tensor and the identity tensor. It measures finite changes in squared material lengths in the reference configuration.
Eulerian description: A description of motion and fields in terms of positions in the current spatial configuration. It provides the spatial viewpoint that contrasts with reference-based formulations.

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