deletions | additions       diff --git a/method.tex b/method.tex  index 7849be6..27c9db3 100644  --- a/method.tex  +++ b/method.tex 
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The forces being applied in the global system (through the transformation of the elastic constants) results in stresses being calculated in the global system. As the experimental work was conducted in the local system the proportionality limit stresses are in the local system. Equation \ref{eq:stress_tans} was used in order to transform the stresses in the global system (calculated by the model) back into the local system in order to evaluate the state of failure.   The forces due to gravity fromthe  self weight of the stem and canopy are were  applied to appropriate domains. The force due to gravity from self weight of the stem was applied as a body force to the whole domain \(\Omega\) \(\omega\)  (i.e. the whole stem) in equation \ref{eq:stem_grav}. Equation --ref 3.11---.  Green density \(\rho\) is calculated as per Section \ref{sec:material_interpolation} --how?--  and g is gravity.   \begin{equation}\label{eq:stem_grav}  B_s = \rho g gravity (\(9.81 m/s \)). The canopy is assumed to take the geometry of the upper half of an ellipsoid described in Equation --3.12--, where \(z_0\) is the height at the bottom of the current element and \(z_1\) is the height at the top of the current element, with the top of the ellipsoid at the top of the stem, height \(h\). The start height \(S_c\) of the canopy was estimated using Equation --3.9-- which was derived from data presented by Waghorn et al. (2007a). In order to estimate crown radius, \(r_c\) , Equation --3.10-- is used, which was derived from the assumption that the maximum radius which can be achieved by the crown is half the distance between two trees evenly spaced in the stand. The weight of the canopy on the stem is applied as a body force described in Equation --3.14--.  \begin{equation}\label{eq:}  \end{equation}  The weight of the canopy on the stem is also applied as a body force described in Equation \ref{eq:canopy_grav}.   \begin{equation}\label{eq:canopy_vol}  V_c \begin{equation}\label{eq:Sc}  S_c  = \pi r_c^2 \frac{h_c}{3} −123.4r_s + 20.6  \end{equation}  \begin{equation}\label{eq:stem_vol}  V_s \begin{equation}\label{eq:rc}  r_c  = \pi r_s^2 \frac{h_c}{3} 34.4r_s − 2.07  \end{equation}  \begin{equation}\label{eq:canopy_grav}  B_c \text{\textbar}_{\boldsymbol{\Omega_1}} \begin{equation}\label{eq:Bs}  B_s  = V_c \rho_c \rho  g\frac{1}{V_{s}}  \end{equation}  \begin{equation}\label{eq:omega_1}  \boldsymbol{\Omega_1} \begin{equation}\label{eq:Vc}  V_c  = \{(x,y,z) \in \boldsymbol{\Omega}:z>S_c\} \pi r^2_c \Big(z_1 \Big( \frac{1-z^2_1}{3(h-S_c)^2}\Big) - z_0\Big(\frac{1-z^2_0}{3(h-S_c)^2}\Big)\Big)  \end{equation}  Where \(r_c\) is the radius of the canopy, \(r_s\) is the radius of the stem, \(h_c\) is the height of the canopy, \(\rho_c\) is the canopy density and \(g\) is gravity. However the canopy force due to gravity is only applied to the subdomain \(\Omega_1\), defined in Equation \ref{eq:omega_1}, where \(S_c\) is the starting height of the canopy and \(z\) is the vertical coordinate of any point. \(\frac{1}{V_s}\) is needed in order transform the canopy's gravitational force into a force per unit volume.  Where r s is the radius of the stem (from Equation 3.1), r z 0 is the radius of the stem at height z 0 , r z 1 is the radius of the stem at height z 1 and g is gravity. ρ c is the canopy density of 5.6 kg/m 3 estimated from data in Beets and Whitehead (1996) at a stocking of 741 stems per hectare. The canopy force due to gravity is only applied to the subdomain Ω 1 , defined in Equation 3.15, where z is the vertical coordinate of any point. V 1 s is needed in order transform the canopy’s gravitational force into a force per unit stem volume.