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Balance laws and transformations

Balance laws and transformations

In this section we will cover some basic transformations used to derive the fore-balance equations for the mechanics of the heart.

Transformations between reference and current configuration

By definition, the reference configuration \(\Omega\), and current configuration \(\omega\), are related via the deformation map \(\varphi\) in the sense that a point \(\mathfrak{p} \in \mathfrak{B}\) with reference coordinates \(\mathbf{X}\) and current coordinates \(\mathbf{x}\) satisfies \(\mathbf{x} = \varphi(\mathbf{X})\). Likewise a vector in the reference configuration is related to a vector in the current configuration via the deformation gradient \(\mathbf{F}\); if \(\mathrm{d}\mathbf{X}\) is a vector in the reference configuration it will transform to the vector \(\mathrm{d}\mathbf{x}\) in the current configuration, and \(\mathrm{d}\mathbf{x} = \mathbf{F} \mathrm{d}\mathbf{X}\). From this relation we also derive that the transformation of an infinitesimal volume element in the reference configuration, \(\mathrm{d}V\) is related to an infinitesimal volume element in the current configuration, \(\mathrm{d}v\) via the determinant of the deformation gradient,

(3)\[\begin{align} \mathrm{d}v =\mathrm{det} \; (\mathbf{F}) \mathrm{d}V. \end{align}\]

Another important transformation is the transformation of normal vectors. By noting that we can write (3) using surface elements

\[\begin{split}\begin{align*} \mathrm{d}s \mathbf{n} \mathrm{d}\mathbf{x} &= \mathrm{d}v = \mathrm{det} \; (\mathbf{F}) \mathrm{d}V = \mathrm{det} \; (\mathbf{F}) \mathrm{d}S \mathbf{N} \mathrm{d}\mathbf{X}\\ &\implies \left( \mathrm{d}s \mathbf{n} \mathbf{F} - \mathrm{d}S \mathrm{det} \; (\mathbf{F}) \mathbf{N} \right) \mathrm{d}\mathbf{X} = 0\\ &\implies \left( \mathrm{d}s \mathbf{F}^T \mathbf{n} - \mathrm{d}S \mathrm{det} \; (\mathbf{F}) \mathbf{N} \right) \mathrm{d}\mathbf{X} = 0,\\ \end{align*}\end{split}\]

we get Nanson’s formula

(4)\[\begin{align} \mathrm{d}s \mathbf{n} = \mathrm{det} \; (\mathbf{F}) \mathbf{F}^{-T} \mathrm{d}S \mathbf{N}, \end{align}\]

which relates the normal vector in the current configuration to the normal vector in the reference configuration.

Conservation of linear momentum

Newton’s seconds law states that the change in linear momentum equals the net impulse acting on it. For a continuum material with constant mass density \(\rho\) this implies that

(5)\[\begin{align} \int_{\omega} \rho \dot{\mathbf{v}} \mathrm{d}v = \mathbf{f}, && \mathbf{f} = \int_{\partial \omega} \mathbf{t} \mathrm{d}s + \int_{\omega} \mathbf{b} \mathrm{d}v, \end{align}\]

where \(\mathbf{v}\) is the spatial velocity field, \(\mathbf{t}\) is the traction acting on the boundary, and \(\mathbf{b}\) is the body force. From Cauchy’s stress theorem we know that there exists a second order tensor \(\sigma\), known as the Cauchy stress tensor that is related to the traction vector by \(\mathbf{t} = \sigma \mathbf{n}\), where \(\mathbf{n}\) is the unit normal in the current configuration. Using the divergence theorem we get

\[\begin{align*} \int_{\partial \omega} \mathbf{t} \mathrm{d}s = \int_{\partial \omega} \sigma \mathbf{n} \mathrm{d}s = \int_{\omega} \nabla \cdot \sigma \mathrm{d}v, \end{align*}\]

and by collecting the terms from (5) we arrive at Cauchy’s momentum equation

(6)\[\begin{align} \nabla \cdot \sigma + \mathbf{b} = \rho \dot{\mathbf{v}}. \end{align}\]

The contribution from the body force (\(\mathbf{b}\)) and inertial term (\(\rho \dot{\mathbf{v}}\)) can be considered negligible compared to the stresses [Costa et al., 1996, Moskowitz, 1981, Tallarida et al., 1970], which is why the force balance equations is typically only stated as

(7)\[\begin{align} \nabla \cdot \sigma = \mathbf{0}. \label{momentum_simple_current} \end{align}\]

Note that we have formulated the balance law in the current configuration. An equivalent statement can be formulated in terms of the reference configuration

(8)\[\begin{align} \nabla \cdot \mathbf{P} = \mathbf{0}, \label{momentum_simple_reference} \end{align}\]

where \(\mathbf{P}\) is the first Piola-Kirchhoff stress tensor. Note that the operator \(\nabla \cdot\) acting on the Cauchy stress tensor represents differentiation with respect to coordinates in the current configuration, while when acting on the first Piola-Kirchhoff stress tensor represent differentiation with respect to coordinates in the reference configuration.

Conservation of angular momentum

Just like linear momentum, the angular momentum is also a conserved quantity. We will not go through the derivation, but state that as a consequence, the Cauchy stress tensor is symmetric

(9)\[\begin{align} \sigma = \sigma^T. \end{align}\]

References

CHR+96

KD Costa, PJ Hunter, JM Rogers, Julius M Guccione, LK Waldman, and Andrew D McCulloch. A three-dimensional finite element method for large elastic deformations of ventricular myocardium: i—cylindrical and spherical polar coordinates. Journal of biomechanical engineering, 118(4):452–463, 1996.

Mos81

Samuel E Moskowitz. Effects of inertia and viscoelasticity in late rapid filling of the left ventricle. Journal of biomechanics, 14(6):443–445, 1981.

TRL70

Ronald J Tallarida, Ben F Rusy, and Michael H Loughnane. Left ventricular wall acceleration and the law of laplace. Cardiovascular research, 4(2):217–223, 1970.

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