Prev Subsection 2.3.1: The deformation gradient tensor Up Section 2.3: Deformation, Strain and Stress Subsection 2.3.3: Stress Next
The right Cauchy-Green deformation tensor is defined as follows: This tensor is an example of a material tensor and is typically expressed a function of the material coordinates . The left Cauchy-Green deformation tensor is defined as follows: This tensor is an example of a spatial tensor and is typically expressed as a function of the spatial coordinates . The implementation of the updated Lagrangian finite element method used by FEBio is described in the spatial configuration.
The left and right deformation tensors can also be split into volumetric and deviatoric components. With the use of,(2.3.1-7) the deviatoric deformation tensors are: The deformation tensors defined above are not good candidates for strain measures since in the absence of strain they become the identity tensor . However, they can be used to define strain measures. The Green-Lagrange strain tensor is defined as: This tensor is a material tensor. Its spatial equivalent is known as the Almansi strain tensor and is defined as: In the limit of small displacement gradients, the components of both strain tensors are identical, resulting in the small strain tensor or infinitesimal strain tensor: Note that the small strain tensor is also the linearization of the Green Lagrange strain,