__The -updated Lagrange formulation__

Deformations (i.e. the incremental deformation matrix ) refers to the previous time point. TOCHNOG decomposes the incremental deformation tensor with a polar decomposition into with the incremental deformation matrix, the incremental rotation matrix and the incremental stretch matrix. The incremental stretch matrix is used to determine the incremental strain matrix with the identity tensor. The velocities between the two time points are the unknowns to be solved. Stresses are calculated from adding incremental stresses to the old stresses. Incremental stresses are caused by the incremental strain matrix and a rigid body rotation by the incremental rotation matrix of the old stresses.

__The -updated_without_rotation Lagrange formulation__

Deformations (i.e. the incremental deformation matrix ) refers to the previous time point. Any rigid body rotation between the two time points are neglected, so TOCHNOG decomposes the incremental deformation tensor with a polar decomposition into with the incremental deformation matrix, and the incremental stretch matrix. The velocities between the two time points are the unknowns to be solved. Stresses are calculated from adding incremental stresses to the old stresses. Incremental stresses are caused by the incremental strain matrix.

__The -total Lagrange formulation__

Deformations (i.e. the total deformation matrix ) refers to the time 0. TOCHNOG decomposes the total deformation tensor with a polar decomposition into with the total deformation matrix, the total rotation matrix and the total stretch matrix. The total stretch matrix is used to determine the total strain matrix with the identity tensor. The displacements between the current time and time 0 are the unknowns to be solved. Stresses are calculated from the total displacements.

__The -total_piola Lagrange formulation__

Deformations (i.e. the total deformation matrix ) refers to the time 0. TOCHNOG decomposes the total deformation tensor with a polar decomposition into with the total deformation matrix, the total rotation matrix and the total stretch matrix. The total stretch matrix is used to determine the total Green-Lagrange strain matrix Stresses are calculated from the total displacements.

__The -total_linear Lagrange formulation__

Deformations (i.e. the total deformation matrix ) refers to the time 0. TOCHNOG neglects any rigid body rotations and uses linear engineering strains . The displacements between the time current time and time 0 are the unknowns to be solved. Stresses are calculated from the total displacements.

A remark on the total Lagrange models.
Normally stresses are calculated from the total displacements (and thus the total strains).
The old stresses are not used.
This means that any initial stresses are neglected.
This type of stress calculation for the total Lagrange models
is used whenever **materi_strain_total** is not initialised.
However, in case **materi_strain_total** is initialised, the difference
between the total displacements (and thus strains) between two time points
is used to determine incremental stresses, which are added to the stresses at
the previous time point.
And thus, in case **materi_strain_total** is initialised, the old
stresses are used and any initial stresses are not neglected.

See also **group_materi_memory**.