~hnarayanan/phd-dissertation/trunk : revision 131

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%% \section{Leibniz's product rule}

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%% \label{product-rule}

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%% We begin with two differentiable functions $f(x)$ and $g(x)$ and show

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%% that their product is differentiable, and that the derivative of the

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%% product has the desired form.

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%% By simply calculating, we have for all values of $x$ in the domain of

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%% $f$ and $g$ that

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%% \begin{eqnarray*}

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%% \frac{\D{}}{\D{x}}\left[f(x)g(x)\right]

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%% & = & \lim_{h\to0}\frac{f(x+h)g(x+h) - f(x)g(x)}{h} \\

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%% & = & \lim_{h\to0}\frac{f(x+h)g(x+h) + f(x+h)g(x) - f(x+h)g(x) -

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%% f(x)g(x)}{h} \\

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%% & = & \lim_{h\to0}\left[f(x+h)\frac{g(x+h)-g(x)}{h} +

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%% g(x)\frac{f(x+h)-f(x)}{h}\right] \\

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%% & = & \lim_{h\to0}\left[f(x+h)\frac{g(x+h)-g(x)}{h}\right] +

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%% \lim_{h\to0}\left[g(x)\frac{f(x+h)-f(x)}{h}\right] \\

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%% & = & f(x)g'(x) + f'(x)g(x).

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%% \end{eqnarray*}

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%% The key argument here is the next to last line, where we have used the

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%% fact that both $f$ and $g$ are differentiable, hence the limit can be

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%% distributed across the sum to give the desired equality.

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%% \section{Source balance and the law of mass action}

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%% \label{law-of-mass-action}

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%% The conversion of precursors to tissue and the reverse process of

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%% its breakdown are governed by a series of chemical reactions. The

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%% stoichiometry of these reactions varies in a limited range.

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%% Continuing in the simple vein adopted above, it is assumed that

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%% the formation of tissue and byproducts from precursors, and the

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%% breakdown of tissue, are governed by the forward and reverse

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%% directions of a single reaction:

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%% \begin{equation}

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%% \sum\limits_{\iota=\alpha}^{\omega} n_\iota[\iota] \longrightarrow

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%% [\mathrm{s}]. \label{chemreac}

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%% \end{equation}

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%% \noindent Here, $n_\iota$ is the (possibly fractional) number of

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%% moles of species $\iota$ in the reaction. For a tissue precursor,

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%% $n_\iota > 0$, and for a byproduct, $n_\iota < 0$. By the Law of

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%% Mass Action for this reaction, the rate of the forward reaction

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%% (number of moles of $\mathrm{s}$ produced per unit time, per unit

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%% volume in $\Omega_0$) is

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%% $k_\mathrm{f}\prod\limits_{\iota=\alpha}^\omega

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%% [\rho_0^\iota]^{n_\iota}$, where $\prod$ on the right hand-side

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%% denotes a product, not to be confused with the source, $\Pi$. The

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%% rate of the reverse reaction (number of moles of $\mathrm{s}$

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%% consumed per unit time, per unit volume in $\Omega_0$) is

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%% $k_\mathrm{r}[\rho_0^\mathrm{s}]$, where $k_f$ and $k_r$ are the

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%% corresponding reaction rates. Assuming, for the purpose of this

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%% example, that the solid phase is a single compound, let the

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%% molecular weight of $\mathrm{s}$ be $\sM_\mathrm{s}$. From the

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%% above arguments the source term for $\mathrm{s}$ is

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%% \begin{equation}

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%% \Pi^\mathrm{s} =

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%% \left(k_\mathrm{f}\prod\limits_{\iota=\alpha}^\omega

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%% [\rho_0^\iota]^{n_\iota} -

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%% k_\mathrm{r}[\rho_0^\mathrm{s}]\right)\sM_\mathrm{s},

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%% \label{sourceA}

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%% \end{equation}

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%% \noindent Since the formation of one mole of $\mathrm{s}$ requires

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%% consumption of $n_\iota$ moles of $\iota$, we have

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%% \begin{equation}

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%% \Pi^\iota =

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%% -\left(k_\mathrm{f}\prod\limits_{\vartheta=\alpha}^\omega

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%% [\rho_0^\vartheta]^{n_\vartheta} -

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%% k_\mathrm{r}[\mathrm{s}]\right)n_\iota\sM_\iota, \label{sourceI}

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%% \end{equation}

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%% \noindent where $\sM_\iota$ is the molecular weight of species

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%% $\iota$. Since, due to conservation of mass, $\sM_\mathrm{s} =

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%% \sum\limits_{\iota=\alpha}^\omega n^\iota\sM_\iota$ the sources

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%% satisfy $\sum\limits_{\iota=\mathrm{s},\alpha}^\omega \Pi^\iota =

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%% 0$.

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%% \section{Transport of a the fluid species: the example of an ideal

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%% fluid}

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%% \label{ideal-fluid-transport}

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%% Consider the stress divergence term

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%% $\bF^\mathrm{T}\Bnabla\cdot\bP^\iota$. An elementary calculation

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%% gives

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%% \begin{equation}

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%% \bF^\mathrm{T}\Bnabla\cdot\bP^\iota =

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%% \Bnabla\cdot\left(\bF^\mathrm{T}\bP^\iota\right) -

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%% \Bnabla\bF^\mathrm{T}\colon\bP^\iota. \label{stressdivI}

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%% \end{equation}

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%% \noindent In indicial form, where lower/upper case indices are for

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%% components of quantities in the current/reference configuration

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%% respectively, this relation is

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%% \begin{displaymath}

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%% F_{iK}P^\iota_{iJ,J} = \left(F_{iK}P^\iota_{iJ}\right)_{,J} -

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%% F_{iK,J}P^\iota_{iJ}.

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%% \end{displaymath}

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%% \noindent For an ideal fluid, supporting only an isotropic Cauchy

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%% stress, $p\bone$, we have $\bP^\mathrm{f} =

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%% \mathrm{det}(\bF)p\bF^{-\mathrm{T}}$, where $p$ is positive in

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%% tension. The arguments that follow assume this case. (The more

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%% general case of a non-ideal, viscous fluid will merely have

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%% additional terms from the viscous Cauchy stress.) The stress

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%% divergence term is

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%% \begin{equation}

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%% \bF^\mathrm{T}\Bnabla\cdot\bP^\mathrm{f} =

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%% \Bnabla\left(\mathrm{det}(\bF)p\right) -

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%% \Bnabla\bF^\mathrm{T}\colon\bF^{-\mathrm{T}}\mathrm{det}(\bF)p,

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%% \end{equation}

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%% \noindent demonstrating the appearance of a hydrostatic

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%% stress-driven contribution to $\sF^\mathrm{f}$. This

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%% is Darcy's Law for transport of a fluid down a pressure gradient.

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%% For the special case of a compressible, ideal fluid we have

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%% $e^\mathrm{f} =

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%% \bar{e}^\mathrm{f}(\eta^\mathrm{f},\bar{\rho}^\mathrm{f})$; i.e.,

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%% the fluid stores strain energy as a function of its \emph{current,

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%% intrinsic} density. Fluid saturation conditions hold in biological

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%% tissue, for which case the fluid volume fraction, $f^\mathrm{f}$,

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%% is simply the pore volume fraction. Recall from Section

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%% \ref{sect2} that the individual species deform with the common

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%% deformation gradient $\bF$. Therefore the pores deform

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%% \emph{homogeneously} with the surrounding solid phase. Physically

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%% this corresponds to the pore size being smaller than the scale at

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%% which the homogenization assumption of a continuum theory holds.

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%% Momentarily ignoring changes in reference concentration of the

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%% fluid, we have $\bF^{\mathrm{e}^\mathrm{f}} = \bF$. Then, since

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%% $\rho^\mathrm{f}_0 = \bar{\rho}^\mathrm{f}_0 f^\mathrm{f}$, we can

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%% write $\hat{e}^\mathrm{f}(\bF,\eta^\mathrm{f},\rho^\mathrm{f}_0) =

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%% \hat{e}^\mathrm{f}(\bF,\eta^\mathrm{f},\bar{\rho}^\mathrm{f}_0

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%% f^\mathrm{f}) =

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%% \bar{e}^\mathrm{f}(\eta^\mathrm{f},\bar{\rho}^\mathrm{f}_0/\mathrm{det}\bF)=

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%% \bar{e}^\mathrm{f}(\eta^\mathrm{f},\bar{\rho}^\mathrm{f})$. In

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%% this case a simple calculation shows that the hydrostatic pressure

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%% is

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%% \begin{displaymath}

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%% p =

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%% -\frac{\bar{\rho}^\mathrm{f}}{\mathrm{det}(\bF)}\frac{\partial\bar{e}^\mathrm{f}}{\partial\bar{\rho}^\mathrm{f}},

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%% \end{displaymath}

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%% \noindent and the stress divergence term is

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%% \begin{displaymath}

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%% \bF^\mathrm{T}\Bnabla\cdot\bP^\mathrm{f} =

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%% -\Bnabla\left(\bar{\rho}^\mathrm{f}\frac{\partial\bar{e}^\mathrm{f}}{\partial\bar{\rho}^\mathrm{f}}\right)

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%% +

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%% \Bnabla\bF^\mathrm{T}\colon\bF^{-\mathrm{T}}\bar{\rho}^\mathrm{f}\frac{\partial\bar{e}^\mathrm{f}}{\partial\bar{\rho}^\mathrm{f}}.

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%% \end{displaymath}

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%% \chapter{Material models and parameters}

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%% \label{model-parameters}

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%% \todo{See if other things can be moved/added to the appendix.}

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%% The engineered tendon construct is $12$ mm in length and

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%% $1\;\mathrm{mm}^2$ in area. In this paper an internal energy

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%% density for the solid phase based upon the worm-like chain model

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%% is used. The reader is directed to \citet{Riefetal:97} and

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%% \cite{Bustamanteetal:2003} where the one-dimensional version of

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%% this model has been applied to long chain molecules. It has been

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%% described and implemented into an anisotropic representative

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%% volume element by \citet{Bischoffetal:2002}, and is summarized

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%% here. The internal energy density of a single constituent chain of

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%% an eight-chain model (Figure \ref{eightchain}) is,

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%% \begin{eqnarray}

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%% \bar{\rho}_0^\mathrm{s}\hat{e}^\mathrm{s}(\bF^{\mathrm{e}^\mathrm{s}},\rho_0^\mathrm{s},\eta^\mathrm{s})

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%% &=& \frac{N k \theta}{4 A}\left(\frac{r^2}{2L} +

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%% \frac{L}{4(1-r/L)} -

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%% \frac{r}{4}\right)\nonumber\\

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%% &-&\frac{N k \theta}{4\sqrt{2L/A}}\left(\sqrt{\frac{2A}{L}} +

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%% \frac{1}{4(1 - \sqrt{2A/L})} -\frac{1}{4} \right)\log(\lambda_1^{a^2}\lambda_2^{b^2}\lambda_3^{c^2})\nonumber\\

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%% &+& \frac{\gamma}{\beta}({J^{\mathrm{e}^\mathrm{s}}}^{-2\beta} -1)

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%% + 2\gamma{\bf 1}\colon\bE^{\mathrm{e}^\mathrm{s}} \label{wlcmeq}

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%% \end{eqnarray}

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%% \noindent Here, $N$ is the density of chains, $k$ is the Boltzmann

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%% constant, $r$ is the end-to-end length of a chain, $L$ is the

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%% fully-extended length, and $A$ is the persistence length that

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%% measures the degree to which the chain departs from a straight

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%% line. The preferred orientation of tendon collagen is described by

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%% an anisotropic unit cell with sides $a,b$ and $c$---see Figure

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%% \ref{eightchain}. All lengths in this model have been rendered

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%% non-dimensional (Table \ref{mattab}) by dividing by the link

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%% length in a chain.

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%% \begin{figure}[ht]

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%% \psfrag{A}{$a$} \psfrag{B}{$b$} \psfrag{C}{$c$} \centering

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%% {\includegraphics[width=3.5cm]{images/elucidation/wlcm-cuboid.eps}} \caption{Worm-like

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%% chains grouped into an initially anisotropic eight-chain model.}

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%% \label{eightchain}.

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%% \end{figure}

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%% The elastic stretches along the unit cell axes are respectively

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%% denoted $\lambda^\mathrm{e}_1,\lambda^\mathrm{e}_2$ and

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%% $\lambda^\mathrm{e}_3$, and $\bE^{\mathrm{e}^\mathrm{s}} =

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%% \frac{1}{2}(\bC^{\mathrm{e}^\mathrm{s}} - {\bf 1})$ is the elastic

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%% Lagrange strain. The factors $\gamma$ and $\beta$ control bulk

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%% compressibility. The end-to-end length is given by

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%% \begin{equation}

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%% r =

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%% \frac{1}{2}\sqrt{a^2\lambda_1^{\mathrm{e}^2}+b^2\lambda_2^{\mathrm{e}^2}+c^2\lambda_3^{\mathrm{e}^2}},\quad

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%% \lambda_I^{\mathrm{e}} = \sqrt{\bN_I\cdot\bC^{\mathrm{e}}\bN_I}

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%% \label{rwlcm}

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%% \end{equation}

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%% Preliminary mechanical tests of the engineered tendon have been

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%% carried out in our laboratory but, at this stage, the worm-like

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%% chain model has not been calibrated to these tests. Instead,

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%% published data for the worm-like chain, obtained by calibrating

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%% against rat cardiac tissue \citep{Bischoffetal:2002}, has been

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%% employed.

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%% The fluid phase was modelled as an ideal, nearly-incompressible

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%% fluid:

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%% \begin{equation}

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%% \bar{\rho}^\mathrm{f}_0\hat{e}^\mathrm{f}(\bF^{\mathrm{e}^\mathrm{f}},\rho_0^\mathrm{f},\eta^\mathrm{f})

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%% =

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%% \frac{1}{2}\kappa(\mathrm{det}(\bF^{\mathrm{e}^\mathrm{f}})-1)^2,

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%% \end{equation}

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%% \noindent where $\kappa$ is the fluid bulk modulus.

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%% Only a solid and a fluid phase were included for the tissue. Low

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%% values were chosen for the mobilities of the fluid

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%% \citep{Swartzetal:99} with respect to the solid phase (see Table

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%% \ref{mattab}). In order to demonstrate growth, the solid phase

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%% must have a source term, $\Pi^\mathrm{s}$ (Section \ref{sect2}),

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%% and the only other phase, the fluid, must have $\Pi^\mathrm{f} =

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%% -\Pi^\mathrm{s}$. Therefore, contrary to the case made in Section

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%% \ref{sect2}, a non-zero value of the fluid source,

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%% $\Pi^\mathrm{f}$, was assumed. A form motivated by first-order

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%% reactions was used:

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%% \begin{equation}

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%% \Pi^\mathrm{f} = -k^\mathrm{f}(\rho_0^\mathrm{f} -

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%% \rho_{0_\mathrm{ini}}^\mathrm{f}),\quad \Pi^\mathrm{s} =

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%% -\Pi^\mathrm{f}, \label{piform}

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%% \end{equation}

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%% \noindent where $k^\mathrm{f}$ is the reaction rate, and

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%% $\rho_{0_\mathrm{ini}}^\mathrm{f}$ is the initial fluid

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%% concentration. This term acts as a source for the solid when

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%% $\rho_0^\mathrm{f}

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%% > \rho_{0_\mathrm{ini}}^\mathrm{f}$, and a sink when

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%% $\rho_0^\mathrm{f} < \rho_{0_\mathrm{ini}}^\mathrm{f}$.

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%% In a very simple approximation, the fluid's mixing entropy was

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%% written as

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%% \begin{equation}

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%% \eta^\mathrm{f}_\mathrm{mix} =

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%% -\frac{k}{\sM^\mathrm{f}}\log\frac{\rho_0^\mathrm{f}}{\rho_0}.

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%% \label{mixentropy}

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%% \end{equation}

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%% \noindent Recall that in the notation of Section \ref{sect2},

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%% $\sM^\mathrm{f}$ is the fluid's molecular weight.

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%% \begin{table}[ht]

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%% \caption{Material parameters used in the analysis} \label{mattab}

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%% \begin{tabular}{lcll}

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%% \hline

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%% \multicolumn{1}{c}{Parameter} & Symbol & Value & Units\\

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%% \hline

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%% Chain density & $N$ & $7\times 10^{21}$ & $\mathrm{m}^{-3}$\\

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%% Temperature& $\theta$ & $310.0$ & K\\

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%% Persistence length & $A$ & $1.3775$ & --\\

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%% Fully-stretched length & $L$ & $25.277$ & --\\

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%% Unit cell axes & $a,\;b,\;,c$ & $9.2981,\;12.398,\;6.1968$ & --\\

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%% Bulk compressibility factors & $\gamma,\;\beta$ & $1000,\; 4.5$ & --\\

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%% Fluid bulk modulus &$\kappa$ & $1$ & GPa\\

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%% Fluid mobility tensor components& $D_{11},\;D_{22},\;D_{33}$ & $1\times 10^{-8},\;1\times 10^{-8},\;1\times 10^{-8}$ &$\mathrm{m}^{-2}\mathrm{sec}$\\

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%% Fluid conversion reaction rate & $k^\mathrm{f}$ & $-1.\times 10^{-7}$ & $\mathrm{sec}^{-1}$\\

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%% Gravitational acceleration & $\bg$ & $9.81$ & $\mathrm{m}.\mathrm{sec}^{-2}$\\

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%% Molecular weight of fluid &$\sM^\mathrm{f}$& $2.9885\times 10^{-23}$ & $\mathrm{kg}$\\

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%% \hline

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%% \end{tabular}

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%% \end{table}