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.

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

The deformation gradient tensor is a quantity related to both the reference and current configuration, and expresses motion locally around a point. Two types of deformation gradient tensor may be defined.

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 smooth and invertible mapping function ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, which describes the motion of a continuum. In particular, the continuity of the mapping function ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$ implies that cracks and voids do not open or close during the deformation. 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. Thus we have, $${\displaystyle {\begin{aligned}d\mathbf {x} &={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}\,d\mathbf {X} \qquad &{\text{or}}&\qquad dx_{j}={\frac {\partial x_{j}}{\partial X_{K}}}\,dX_{K}\\&=\nabla \chi (\mathbf {X} ,t)\,d\mathbf {X} \qquad &{\text{or}}&\qquad dx_{j}=F_{jK}\,dX_{K}\,.\\&=\mathbf {F} (\mathbf {X} ,t)\,d\mathbf {X} \end{aligned}}}$$

Assuming that ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$ has a smooth inverse, ${\displaystyle \mathbf {F} }$ has the inverse ${\displaystyle \mathbf {H} =\mathbf {F} ^{-1}={\frac {\partial \mathbf {X} }{\partial \mathbf {x} }}\,\!}$, which is the spatial deformation gradient tensor. ${\displaystyle \mathbf {F} }$ being invertible is equivalent to ${\displaystyle {\text{det}}\mathbf {F} \neq 0}$, which corresponds to the notion that the material cannot be infinitely compressed.

Consider a particle or material point ${\displaystyle P}$ with position vector ${\displaystyle \mathbf {X} =X_{I}\mathbf {I} _{I}}$ in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by ${\displaystyle p}$ in the new configuration is given by the vector position ${\displaystyle \mathbf {x} =x_{i}\mathbf {e} _{i}\,\!}$. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.

Consider now a material point ${\displaystyle Q}$ neighboring ${\displaystyle P\,\!}$, with position vector ${\displaystyle \mathbf {X} +\Delta \mathbf {X} =(X_{I}+\Delta X_{I})\mathbf {I} _{I}\,\!}$. In the deformed configuration this particle has a new position ${\displaystyle q}$ given by the position vector ${\displaystyle \mathbf {x} +\Delta \mathbf {x} \,\!}$. Assuming that the line segments ${\displaystyle \Delta X}$ and ${\displaystyle \Delta \mathbf {x} }$ joining the particles ${\displaystyle P}$ and ${\displaystyle Q}$ in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as ${\displaystyle d\mathbf {X} }$ and ${\displaystyle d\mathbf {x} \,\!}$. Thus from Figure 2 we have $${\displaystyle {\begin{aligned}\mathbf {x} &=\mathbf {X} +\mathbf {u} (\mathbf {X} ),\\\mathbf {x} +d\mathbf {x} &=\mathbf {X} +d\mathbf {X} +\mathbf {u} (\mathbf {X} +d\mathbf {X} ),\\{\text{and}}{\text{ therefore}}&\\d\mathbf {x} &=\mathbf {X} -\mathbf {x} +d\mathbf {X} +\mathbf {u} (\mathbf {X} +d\mathbf {X} )\\&=d\mathbf {X} +\mathbf {u} (\mathbf {X} +d\mathbf {X} )-\mathbf {u} (\mathbf {X} )\\&=d\mathbf {X} +d\mathbf {u} \\\end{aligned}},}$$

where ${\displaystyle \mathbf {du} }$ is the relative displacement vector, which represents the relative displacement of ${\displaystyle Q}$ with respect to ${\displaystyle P}$ in the deformed configuration.