Finite strain theory

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which both rotations and strains are arbitrarily large, i.e. invalidates the assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made 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.

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. Relative displacement between particles occurs if and only if deformation has occurred. If displacement occurs without deformation, then it is deemed a rigid-body displacement.

The displacement of particles indexed by variable i may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration ${\displaystyle P_{i}\,\!}$ and deformed configuration ${\displaystyle p_{i}\,\!}$ is called the displacement vector. Using ${\displaystyle \mathbf {X} \,\!}$ in place of ${\displaystyle P_{i}\,\!}$ and ${\displaystyle \mathbf {x} \,\!}$ in place of ${\displaystyle p_{i}\,\!}$, both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector:

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