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In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically deforming materials and other fluids and biological soft tissue.

Displacement field

This section is an excerpt from Displacement field (mechanics) § Decomposition.

The displacement of a body has two components: a rigid-body displacement and a deformation.

• A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.
• Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration ${\displaystyle \kappa _{0}({\mathcal {B}})}$ to a current or deformed configuration ${\displaystyle \kappa _{t}({\mathcal {B}})}$ (Figure 1).

A change in the configuration of a continuum body can be described by a displacement field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.

Deformation gradient tensor

The deformation gradient tensor ${\displaystyle \mathbf {F} (\mathbf {X} ,t)=F_{jK}\mathbf {e} _{j}\otimes \mathbf {I} _{K}}$ is related to both the reference and current configuration, as seen by the unit vectors ${\displaystyle \mathbf {e} _{j}}$ and ${\displaystyle \mathbf {I} _{K}\,\!}$, therefore it is a two-point tensor. Two types of deformation gradient tensor may be defined.

Due to the assumption of continuity of ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, ${\displaystyle \mathbf {F} }$ has the inverse ${\displaystyle \mathbf {H} =\mathbf {F} ^{-1}\,\!}$, where ${\displaystyle \mathbf {H} }$ is the spatial deformation gradient tensor. Then, by the implicit function theorem,^[1] the Jacobian determinant ${\displaystyle J(\mathbf {X} ,t)}$ must be nonsingular, i.e. ${\displaystyle J(\mathbf {X} ,t)=\det \mathbf {F} (\mathbf {X} ,t)\neq 0}$

The material deformation gradient tensor ${\displaystyle \mathbf {F} (\mathbf {X} ,t)=F_{jK}\mathbf {e} _{j}\otimes \mathbf {I} _{K}}$ is a second-order tensor that represents the gradient of the mapping function or functional relation ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector ${\displaystyle \mathbf {X} \,\!}$, i.e., deformation at neighbouring points, by transforming (linear transformation) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, i.e. differentiable function of ${\displaystyle \mathbf {X} }$ and time ${\displaystyle t\,\!}$, which implies that cracks and voids do not open or close during the deformation. Thus we have,