Active tension

Currently in CMISS we assume that the active tension component of the stress tensor is given as a true (Cauchy) stress. Thus, the standard 2nd Piola-Kirchhoff stress tensor is transformed into the Cauchy stress tensor and then the active tension (possibly from your cellular model) is added to the 1,1 entry of that tensor (corresponding to the fibre material axis). This is currently implemented in CMISS in the routine ZGTG5A.

The current publications from the institute have incorrect expressions of the rotation of the active stress from a Cauchy to a 2nd PK stress component of the full stress tensor. Typically the expression attempts to simplify the rotation through use of the (1,1) component of the contravariant metric tensor - which has recently been shown to be incorrect (by Peter and Martyn). The correct expression for the full stress tensor can be found in Andre's PhD thesis.

Two points to note:

• Depending on the experiments used in the definition of the active tension model the actual active tension component may be a 1st Piola-Kirchhoff stress rather than a Cauchy stress.
• We may become interested in the case when we use "active tension" components in the sheet and normal material directions (fairly easy to incorporate into the current code).

See also the page material constitutive laws using CellML.

Andre's notes

A Cauchy stress is measured as the deformed force over the deformed (instantaneous) cross-sectional area.

The 1st Piola-Kirchhoff stress is the deformed force over the original cross-sectional area.