sandbox/huet/src/lagrangian_caps/skalak-ft.h

Skalak law for biological membranes

In this file we define the Skalak strain energy function, used to compute the elastic force on each Lagrangian node of the membrane.

The strain energy function is given by, e.g., Barthès-Biesel:

\displaystyle W^{\text{Skalak}} = \frac{E_s}{4} \left( I_1^2 + 2I_1 - 2I_2 + CI_2^2 \right),

where E_s is the elastic modulus of the membrane, C > -\frac{1}{2} is the area dilatation modulus preventing strong area changes, and I_{1,2} are two strain invariants: I_1 = \lambda_1^2 + \lambda_2^2 - 2 and I_2 = \lambda_1^2 \lambda_2^2 - 1 = J^2 - 1, with \lambda_{1,2} the principal stretches and J the surface Jacobian representing the ratio of deformed over undeformed area.

Differentiating W^{\text{Skalak}} with respect to the principal stretches yields: \displaystyle \frac{\partial W^{\text{Skalak}}}{\partial \lambda_1} = E_s \left( \lambda_1 ( \lambda_1^2 - 1) + C \lambda_1 \lambda_2^2 (\lambda_1^2 \lambda_2^2 - 1) \right) and \frac{\partial W^{\text{Skalak}}}{\partial \lambda_2} is obtained by switching \lambda_1 and \lambda_2.

Implementation

In this file we only have to specify the derivatives of the strain function with respect to the stretches. The computation of the membrane tensions is implemented in elasticity-ft.h.

#ifndef E_S
  #define E_S 1.
#endif
#ifndef AREA_DILATATION_MODULUS
  #define AREA_DILATATION_MODULUS 1.
#endif

#define DWDL1(L1, L2) (E_S*(L1*(sq(L1) - 1.) \
  + AREA_DILATATION_MODULUS*L1*sq(L2)*(sq(L1*L2) - 1.)))
#define DWDL2(L1, L2) (E_S*(L2*(sq(L2) - 1.) \
  + AREA_DILATATION_MODULUS*L2*sq(L1)*(sq(L1*L2) - 1.)))

#include "elasticity-ft.h"

References