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Incompressibility

Incompressibility

The myocardium contains small blood vessels that supply the myocardial cells with oxygen. When the myocardium contracts, this perfused blood is squeezed out, resulting in an overall loss of 2-4% volume [Yin et al., 1996]. A material that change its volume in response to applied loads is referred to as compressible. When the volume is preserved we say that the material is incompressible. Since 2-4% is very little, a common assumption in cardiac mechanical modeling, which has also been made in the work conducted in this thesis, is to assume the myocardium to be incompressible. The reason for this choice is purely numerical.

For an incompressible material, only isochoric motions are possible. This means that the volume of the material does not change during any deformation, and hence we have the constraint

(14)\[\begin{align} J = \det(\mathbf{F}) = 1. \end{align}\]

The constraint (14) can be imposed by considering the modified strain energy function

(15)\[\begin{align} \Psi = \Psi(\mathbf{F}) + p(J-1), \end{align}\]

where \(p\) is a scalar which serves as a Lagrange multiplier, but which can be identified as the hydrostatic pressure. If we differentiate (15) with respect to \(\mathbf{F}\) we get the First Piola-Kirchhoff stress tensor for an incompressible material

\[\begin{align} \mathbf{P} = \frac{\partial \Psi(\mathbf{F})}{\partial \mathbf{F}} + J p \mathbf{F}^{-T}. \end{align}\]

Likewise the Cauchy stress tensor is given by

(16)\[\begin{align} \sigma = J^{-1} \frac{\partial \Psi(\mathbf{F})}{\partial \mathbf{F}}\mathbf{F}^{T} + p \mathbf{I}. \end{align}\]

Note

The sign of \(p\) is determined by whether you add or subtract the term \( p(J-1)\) to the total strain energy in (15). For all practical purposes, it does not matter if you add or subtract the term as long as you are consistent.

Uncoupling of volumetric and isochoric response

The total strain energy function in (15) can be written as a sum of isochoric and volumetric components. Let

\[\begin{align} \mathbf{F} = \mathbf{F}_{\mathrm{vol}} \mathbf{F}_{\mathrm{iso}}, \end{align}\]

then \( \mathbf{F}_{\mathrm{vol}} = J^{1/3}\mathbf{I}\) and \(\mathbf{F}_{\mathrm{iso}} = J^{-1/3}\mathbf{F}\). For compressible materials (i.e with \(J \neq 1\)) it is important to consider only deviatoric strains in the strain-energy density function, so that \(\Psi = \Psi_{\mathrm{iso}}(\mathbf{F}_{\mathrm{iso}}) + \Psi_{\mathrm{vol}}(J)\). For incompressible material (\(J = 1\)), we have \(\mathbf{F}_{\mathrm{vol}} = \mathbf{I}\) so that such a decomposition seems unnecessary. However, a similar decomposition has shown to be numerically beneficial [Weiss et al., 1996]. Note that, in this case, a similar decoupling of the stress tensors has to be done.

In the kinematics module you can specify whether if want the isochoric version

F_iso = pulse.kinematics.DeformationGradient(u, isochoric=True)

References

WMG96

Jeffrey A Weiss, Bradley N Maker, and Sanjay Govindjee. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Computer methods in applied mechanics and engineering, 135(1-2):107–128, 1996.

YCJ96

FC Yin, CC Chan, and Robert M Judd. Compressibility of perfused passive myocardium. American Journal of Physiology-Heart and Circulatory Physiology, 271(5):H1864–H1870, 1996.

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