PlanarBiaxialExtension and LayeredPlanarBiaxial

We consider a planar tissue sample in the \(xy\) plane (i.e., the third coordinate direction is along the small tissue thickness). The sample has reference dimensions \(L_{10}\), \(L_{20}\), and \(t\) along the coordinates 1, 2, and 3, respectively. Furthermore, we assume that the material is incompressible. Thus, the deformation gradient is:

\[\mathbf{F} = \mathop{\mathrm{diag}}\left[\lambda_1,\lambda_2,\frac{1}{\lambda_1\lambda_2}\right].\]

The first two directions are in the tissue’s plane, and the third direction stress will be zero. Hence, we can calculate the Lagrange multiplier \(p\) by equating \(\sigma_{33}=0\).

If \(\boldsymbol{M}=\left[\cos(\theta),\sin(\theta),0 \right]\) lies in 1-2 plane (i.e., planar), then the third term does not produce stress in the third direction. However, it may produce shear stress in 1-2 plane (if the fibers are not symmetric about the first axis). The shear stress components are neglected when returning the force measure, which can be either the first two diagonal components of the stress tensor (Cauchy, 1st PK, or 2nd PK), membrane tension in the two directions (calculated as \(P_{11}t\) and \(P_{22}t\)), or forces in the two directions (calculated as \(P_{11}L_{20}t\) and \(P_{22}L_{10}t\)).

From the two stretches, other deformation metrics can be calculated. Deformed length \(l_\alpha=\lambda_\alpha L_{\alpha0}\), change in length \(\Delta l_\alpha = (\lambda_\alpha-1)L_{l_\alpha 0}\), and strain \(\epsilon_\alpha = (\lambda_\alpha-1)\). Given deformation (in terms of stretch/change in length, strain, or deformed length), the force measure can be calculated using SampleExperiment.disp_controlled() function. Conversely, given stress or force, any of the deformation metric are solved iteratively via SampleExperiment.force_controlled() function.

PlanarBiaxialExtension samples can be “layered” via LayeredPlanarBiaxial. Such a setup can be used for representing, for example, tissues that have multiple layers with different material models and possibly even different reference lengths. The result would be that there is no zero stress state for the layered sample. One has to be careful with the inputs and outputs of the layered samples though. It is required to use lengths (rather than stretches or strains) as the deformation metric. Similarly, the force measure should not be stresses since the stresses would not simply add up (instead they would be weighted by each layer’s thickness).