@@ -199,18 +199,14 @@ Film stiffness has the largest effect (considering the orders of magnitude of th

\subsection{Assumption of no contraction}

For all simulations, we calculated the actual length $L_d$ of the deformed, curled, films to validate the assumption of no contraction. We use the mean of the lengths $L_d$ of the left and right bottom edge $\overline{L_d}$to define the film length stretch $\lambda_L$ (figure \ref{isospace_dL_fig}):

For all simulations, we calculated the actual length $L_d$ of the deformed, curled, films to validate the assumption of no contraction. We use the mean of the lengths $L_d$ of the left and right bottom edge to define the film length stretch $\lambda_L$ (figure \ref{isospace_dL_fig}):

Again with $\sigma_x =0.025$\,MPa as a reference, we find:

\input{isospace_dL_eq.tex}

for the model of equation \ref{mregress} with $y =\lambda_L$ (table \ref{isospace_dL_tab}).

The intercept is 1. Which is \textsl{very} encouraging for the assumption of no contraction. The range of effects is small, and if anything: films seems to \textsl{elongate} rather than contract, in the simulations ($\lambda_L > 1$, figure \ref{isospace_dL_fig}).

\begin{table}[p]

\center\small

\caption{Multiple linear regression model fitted to $\Deltaup L$ for $\sigma_x =0.025$\,MPa for the 132 random models. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{isospace_dL_tab}}

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@@ -221,7 +217,11 @@ The intercept is 1. Which is \textsl{very} encouraging for the assumption of no

\caption{Distribution of $\lambda_L$ (equation \ref{eq:lambdal}) for the 132 successfully build models. \label{isospace_dL_fig}}

\end{figure}

\begin{figure}[p]

The intercept is 1. Which is \textsl{very} encouraging for the assumption of no contraction. In fact, films seem to \textsl{elongate} as often as that they contract, in the simulations (figure \ref{isospace_dL_fig})\footnote{Yep, the joke's on me.}. The range of effects is small, give or take few percent, and together they explain only $\frac{1}{3}$ of the variation in de model ($R_a^2=0.30$, SS$_\text{error}$/SS$_\text{tot}=0.67$). Of this 33\,\% of variation, 28\,\% is explained by three only parameters (table \ref{isospace_dL_tab}): $t_\text{cells}$ (SS/SS$_\text{tot}=0.11$, $p < 0.001$), $E_\text{film}$ (SS/SS$_\text{tot}=0.11$, $p < 0.001$), and $L$ (SS/SS$_\text{tot}=0.06$, $p =0.004$). \\

\noindent We investigated the coefficients in equation \ref{isospace_dL_eq} over the entire range for $\sigma_x$ for the simulations (figure \ref{fig:pspaceiso_dL_as}) as we did with $\frac{l}{L}$ and equation \ref{isospace_lL_eq}. In general, the both magnitude of the coefficient as well as the associated 95\,\% confidence interval increase over the range of $\sigma_x$. And even if the effects are are small, the coefficients of some parameters do show a clear response to $\sigma_x$.

\caption{Parameters of equation \ref{isospace_dL_eq} (red) with their 95\,\% confidence intervals (blue) as function of $\sigma_x$. With the coefficients for \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} film length $L$$(a_1)$, \textbf{(c)} film width $w$$(a_2)$, \textbf{(d)} film stiffness $E_\text{film}$$(a_3)$, \textbf{(e)} cell layer stiffness $E_\text{cells}$$(a_4)$, \textbf{(f)} film Poisson ratio $\nu_\text{film}$$(a_5)$, \textbf{(g)} cell layer Poisson ratio $\nu_\text{cells}$$(a_6)$, \textbf{(h)} film thickness $t_\text{film}$$(a_7)$, and \textbf{(i)} cell layer thickness $t_\text{cells}$$(a_8)$. \label{fig:pspaceiso_dL_as}}

\caption{Coefficients of equation \ref{isospace_dL_eq} ($\lambda_L$, red) with their 95\,\% confidence intervals (blue) as function of $\sigma_x$. With the coefficients for \textbf{(a)} the intercept $(a_0)$, \textbf{(b)} film length $L$$(a_1)$, \textbf{(c)} film width $w$$(a_2)$, \textbf{(d)} film stiffness $E_\text{film}$$(a_3)$, \textbf{(e)} cell layer stiffness $E_\text{cells}$$(a_4)$, \textbf{(f)} film Poisson ratio $\nu_\text{film}$$(a_5)$, \textbf{(g)} cell layer Poisson ratio $\nu_\text{cells}$$(a_6)$, \textbf{(h)} film thickness $t_\text{film}$$(a_7)$, and \textbf{(i)} cell layer thickness $t_\text{cells}$$(a_8)$. \label{fig:pspaceiso_dL_as}}

\end{figure}

The coefficient for the intercept suggests that `on average' these film elongated for small $\sigma_x$ with a maximum at about $\sigma_x =0.01$\,MPa, and that this average elongation decreases to `no contraction' ($a_0=1$, $L_d = L$) at $\sigma_x =0.025$\,MPa (figure \ref{pspaceiso_dL_a0}, see also figure \ref{isospace_dL_fig}). The coefficient for $E_\text{film}$ has minimum at about $\sigma_x =0.006$\,MPa and a strong overall increase that clearly shows beyond the confidence intervals (figure \ref{pspaceiso_dL_a3}). The coefficient for $t_\text{cells}$ has maximum at about $\sigma_x =0.004$\,MPa and a strong overall decrease that clearly shows beyond the confidence intervals (figure \ref{pspaceiso_dL_a8}). These seem to be `real' effects, even if they are still small.

There is a trend for increasing magnitude of the coefficient for $L$, but it is matched by an equal increase in the width of the confidence interval. And that 95\,\% confidence interval rarely excludes zero (figure \ref{pspaceiso_dL_a1}). Trends for $\nu_\text{film}$ (figure \ref{pspaceiso_dL_a5}) and $t_\text{film}$ are obfuscated by their trend for larger confidence intervals, and for $\nu_\text{film}$ it is hard to see if there is a trend at all, or if we are just looking at the effect of increasing confidence intervals with increasing $\sigma_x$ (figure \ref{pspaceiso_dL_a6}).

The coefficient for $E_\text{cells}$ has a minimum and maximum (figure \ref{pspaceiso_dL_a4}), as with $\frac{l}{L}$ (figure \ref{pspaceiso_lL_a4}). And there is again evidence for minima and maxima in the behaviour of the coefficients, e.g.\ $E_\text{film}$ ($a_3$, figure \ref{pspaceiso_dL_a3}), $t_\text{film}$ ($a_7$, figure \ref{pspaceiso_dL_a7}), and $t_\text{cells}$ ($a_8$, figure \ref{pspaceiso_dL_a8}).\\

\noindent\textbf{round up} The assumption of no contraction is not violated to a degree to care about in isotropic films.