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 $Ω$ , and current configuration $ω$ , are related via the deformation map $φ$ in the sense that a point $p \in B$ with reference coordinates $X$ and current coordinates $x$ satisfies $x = φ (X)$ . Likewise a vector in the reference configuration is related to a vector in the current configuration via the deformation gradient $F$ ; if $d X$ is a vector in the reference configuration it will transform to the vector $d x$ in the current configuration, and $d x = F d X$ . From this relation we also derive that the transformation of an infinitesimal volume element in the reference configuration, $d V$ is related to an infinitesimal volume element in the current configuration, $d v$ via the determinant of the deformation gradient,

(3)#

\begin{array}{r} d v = \det (F) d V . \end{array}

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

\begin{array}{r} \begin{aligned} d s n d x & = d v = \det (F) d V = \det (F) d S N d X \\ ⟹ (d s n F - d S \det (F) N) d X = 0 \\ ⟹ (d s F^{T} n - d S \det (F) N) d X = 0, \end{aligned} \end{array}

we get Nanson’s formula

(4)#

\begin{array}{r} d s n = \det (F) F^{- T} d S N, \end{array}

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 $ρ$ this implies that

(5)#

\begin{aligned} \int_{ω} ρ \dot{v} d v = f, & f = \int_{\partial ω} t d s + \int_{ω} b d v, \end{aligned}

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

\begin{array}{r} \int_{\partial ω} t d s = \int_{\partial ω} σ n d s = \int_{ω} \nabla \cdot σ d v, \end{array}

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

(6)#

\begin{array}{r} \nabla \cdot σ + b = ρ \dot{v} . \end{array}

The contribution from the body force ( $b$ ) and inertial term ( $ρ \dot{v}$ ) 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{array}{r} \nabla \cdot σ = 0 . \end{array}

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{array}{r} \nabla \cdot P = 0, \end{array}

where $P$ 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{array}{r} σ = σ^{T} . \end{array}

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.