Passive Tissue Response

The formulation is based on that proposed by Holzapfel and Ogden.

The following strain energy potential governs the deviatoric response:

Ψ_{d e v} = \frac{a}{2 b} \exp [b (I_{1} - 3)] + \sum_{i = f, s}^{} \frac{a_{i}}{2 b_{i}} {\exp [b_{i} ({(I_{4 i} - 1)}^{2})] - 1} + \frac{a_{f s}}{{2 b}_{f s}} [\exp (b_{f s} I_{8 f s}^{2}) - 1],

where the following table defines all the parameters:

Table 1. Holzapfel Deviatoric Parameters
Parameters	Description
$a, b$	Governs the isotropic response of the tissue
$a_{f}, b_{f}$	Governs the additional stiffness in the fiber direction
$a_{s}, b_{s}$	Governs the additional stiffness in the sheet direction
$a_{f s}, b_{f s}$	Governs the coupling stiffness in the $f$ - and $s$ -directions
$I_{1}$	The first deviatoric strain invariant
$I_{4 i}$	A pseudo-invariant defined as $A_{i} \cdot C \cdot A_{i}$
$I_{g f s}$	A pseudo-invariant defined as $A_{f} \cdot C \cdot A_{s}$
$C$	Right Cauchy-Green deformation tensor
$A_{i}$	Vector in direction $i$

The following equation governs the volumetric response:

Ψ_{v o l} = \frac{1}{D} (\frac{(J^{2} - 1)}{2} - \ln (J)),

where the following table defines all the parameters:

Table 2. Holzapfel Volumetric Parameters
Parameter	Description
$D$	Multiple of bulk modulus $(K = 2 / D)$
$J$	The third deformation gradient invariant

The Heart Model calibrates the values of the parameters such that the myocardial strains predicted by the simulation compare favorably to those experimentally measured by Genet et al. (See the figure below.) Because Genet et al. assumes a transversely isotropic response, the material response of the sheet and normal directions is identical.