Finite strain theory

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.in the reference and current configurations:This is the case where a specimen is stretched in 1-direction with a stretch ratio of α = α 1 {displaystyle mathbf {alpha =alpha _{1}} ,!} . If the volume remains constant, the contraction in the other two directions is such that α 1 α 2 α 3 = 1 {displaystyle mathbf {alpha _{1}alpha _{2}alpha _{3}=1} ,!} or α 2 = α 3 = α − 0.5 {displaystyle mathbf {alpha _{2}=alpha _{3}=alpha ^{-0.5}} ,!} . Then:In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines iswhere d x {displaystyle dx,!} is the deformed magnitude of the differential element d X {displaystyle dmathbf {X} ,!} . 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. 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. 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 P i {displaystyle P_{i},!} and deformed configuration p i {displaystyle p_{i},!} is called the displacement vector. Using X {displaystyle mathbf {X} ,!} in place of P i {displaystyle P_{i},!} and x {displaystyle mathbf {x} ,!} in place of p i {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: Where e i {displaystyle mathbf {e} _{i},!} are the orthonormal unit vectors that define the basis of the spatial (lab-frame) coordinate system.