Kinematics¶

We represent the heart as a continuum body \(\mathfrak{B}\) embedded in \(\mathbb{R}^3\). A configuration of \(\mathfrak{B}\) is a mapping \(\chi: \mathfrak{B} \rightarrow \mathbb{R}^3\). We denote the reference configuration of the heart by \(\Omega \equiv \chi_0(\mathfrak{B})\), and the current configuration by \(\omega \equiv \chi(\mathfrak{B})\). The mapping \(\varphi : \Omega \rightarrow \omega\), given by the composition \(\varphi = \chi \circ \chi_0^{-1}\), is a smooth, orientation preserving (positive determinant) and invertible map. We denote the coordinates in the reference configuration by \(\mathbf{X} \in \Omega\), and the coordinates in the current configuration by \(\mathbf{x} \in \omega\). The coordinates \(\mathbf{X}\) and \(\mathbf{x}\) are commonly referred to as material and spatial points respectively, and are related through the mapping \(\varphi\), by \(\mathbf{x} = \varphi(\mathbf{X})\). For time-dependent problems it is common to make the time-dependence explicitly by writing \(\mathbf{x} = \varphi(\mathbf{X}, t)\). In the following we will only focus on the mapping between two configurations and therefore no time-dependence is needed. The deformation gradient is a rank-2 tensor, defined as the partial derivative of \(\varphi\) with respect to the material coordinates:

(1)¶\[\begin{align} \mathbf{F} = \nabla_{\mathbf{X}} \varphi= \nabla \mathbf{x}. \end{align}\]

Here we also introduce the notation \(\nabla\), which means derivative with respect to reference coordinates. The deformation gradient maps vectors in the reference configuration to vectors in the current configuration, and belongs to the space of linear transformations from \(\mathbb{R}^3\) to \(\mathbb{R}^3\) with strictly positive determinant, which we denote by \(\mathrm{Lin}^+\). Another important quantity is the displacement field

(2)¶\[\begin{align} \mathbf{u} = \mathbf{x}-\mathbf{X}, \end{align}\]

which relates positions in the reference configuration to positions in the current configuration. From (1) we see that

\[\begin{align} \mathbf{F} = \nabla \mathbf{x} = \nabla \mathbf{u} + \nabla \mathbf{X} = \nabla \mathbf{u} + \mathbf{I}. \end{align}\]

Some other useful quantities are the right Cauchy-Green deformation tensor \(\mathbf{C} = \mathbf{F}^T\mathbf{F}\), the left Cauchy-Green deformation tensor \(\mathbf{B} = \mathbf{F}\mathbf{F}^T\), the Green-Lagrange strain tensor \(\mathbf{E} = \frac{1}{2}(\mathbf{C} - \mathbf{I})\), and the determinant of the deformation gradient \(J = \det \mathbf{F}\).

An important concept in mechanics is the concept of stress, which is defined as force per area \(\left[\frac{\mathrm{N}}{\mathrm{m}^2}\right]\). When working with different configurations one needs to be careful with which forces and which areas we are talking about. Table 1 shows how forces and areas are related for the most important stress tensors used in this thesis. Note that the explicit form of the stress tensor requires a constitutive law for the material at hand. This will be discussed in more detail in Constitutive relations.

Table 1 Showing different stress tensors used in this thesis, and how they relate forces to areas trough different configurations.¶
Stress tensor	Forces	Area
Second Piola-Kirchhoff (\(\mathbf{S}\))	Reference configuration	Reference configuration
First Piola-Kirchhoff (\(\mathbf{P}\))	Current configuration	Reference configuration
Cauchy (\(\sigma\))	Current configuration	Current configuration

Code¶

In pulse there is a separate module for kinematics that contains the most relevant quantities, e.g

import dolfin
import pulse

# Some mesh
mesh = dolfin.UnitCubeMesh(3, 3, 3)

# Space for displacement
V = dolfin.VectorFunctionSpace(mesh, "CG", 1)
u = dolfin.Function(V)

F = pulse.kinematics.DeformationGradient(u)
E = pulse.kinematics.GreenLagrangeStrain(F)

pulse

Kinematics

Contents

Kinematics¶

Code¶