deletions | additions       diff --git a/method.tex b/method.tex  index 2b96365..a4eda93 100644  --- a/method.tex  +++ b/method.tex 
...
Hooks law can be used to characterise elastic relationships in mathematical terms. Generalised Hook’s law (Equation ---3.28---) is used here to describe the mechanical elastic characteristics of wood. Because the characterisation of the proportionality limit is the point at which the stress strain curve is no longer linear, stress is only proportional to the strain and not the strain rate. Above this point non-linear elastic and plastic effects need to be considered, which was beyond the scope of this study.  Stress, σ, \(\boldsymbol{\sigma}\)  is calculated from strain \(\boldsymbol{\epsilon}\)  and the stiffness matrix C. \(\boldsymbol{C}\).  When growth stresses were not considered Equation 3.28 equation \ref{eq:hooks}  was used, when growth stresses were considered σ gs \(\boldsymbol{\sigma_{gs}}\)  was defined by Equation 3.26 equation \ref{eq:stress_gs}  and σ \(\boldsymbol{\sigma}\)  was calculated through Equation 3.29.  1 equation \ref{eq:stress}.  \begin{equation}\label{eq:E}  \boldsymbol{\epsilon}  = ( u +  2  u T )  (3.27) \frac{1}{2}(\bigtriangledown \boldsymbol{u}+\bigtriangledown \boldsymbol{u}^T)  \end{equation}  \begin{equation}\label{eq:hooks}  \boldsymbol{\sigma = C\epsilon}  \end{equation}  \begin{equation}\label{eq:stress}  \boldsymbol{\sigma = C\epsilon+\sigma_{gs}}  \end{equation}  Strain energy density can then be calculated  \begin{equation}\label{eq:W}  W = \frac{1}{2} \boldsymbol{\sigma\epsilon}  \end{equation}  and the total potential energy found  \begin{equation}\label{eq:psi}  \boldsymbol{\prod} = \boldsymbol{\int_\Omega} W \mathit{d}\boldsymbol{\Omega} - \boldsymbol{\int_\Omega B_s.u} \mathit{d}\boldsymbol{\Omega} - \boldsymbol{\int_{\Omega_1} B_c.u }\mathit{d}\boldsymbol{\Omega_1} - \boldsymbol{\int_{\Gamma_1}T_s.u} \mathit{d}\boldsymbol{\Gamma_1}   \end{equation}  \begin{equation*}  \boldsymbol{\Omega_1} = \{(x,y,z) \in \boldsymbol{\Omega}:z>S_c\}  \end{equation*}  \begin{equation*}  \boldsymbol{\Gamma_1} = \{(x,y,z) \in \boldsymbol{\Gamma}:x>0,z>S_c\}  \end{equation*}  By taking the directional derivative of --- \(\prod\)  with respect to the change in u \(u\)  and setting it to zero the displacement field u \(u\)  can be calculated at the minimum potential energy. \begin{equation}\label{eq:F_new}  F = \boldsymbol{\bigtriangledown_u} \prod(\boldsymbol{u}) = 0  \end{equation}  Subject to the Dirichlet boundary conditions, Equations 3.33 equations \ref{eq:BC1}  and 3.34. \ref{eq:BC2}.  \begin{equation}\label{eq:BC1}  \boldsymbol{u} \text{\textbar}_{\boldsymbol{\Omega_{bc1}}} = \begin{bmatrix}  u_x\\  u_y\\  0  \end{bmatrix} \hspace{0.2cm}  \end{equation}  \begin{equation*}  \boldsymbol{\Omega_{bc1}} = \{(x,y,z) \in \boldsymbol{\Omega}:z>tol_z\}  \end{equation*}  and  \begin{equation}\label{eq:BC2}  \boldsymbol{u} \text{\textbar}_{\boldsymbol{\Omega_{bc2}}} = \begin{bmatrix}  0\\  0\\  0  \end{bmatrix} \hspace{0.2cm}  \end{equation}  \begin{equation*}  \boldsymbol{\Omega_{bc2}} = \{(x,y,z) \in \boldsymbol{\Omega_{bc1}}:(x,y)-tol\}  \end{equation*}  Where tol z \(tol_z\)  is the tolerance required to select all points at the boundary between the base of the stem and the ground, and where tol \(tol\)  is the tolerance sufficient to select only the points at the boundary between the base of the stem and the ground for the initial seedlings’ radius. seedling's radius.\\  By solving Equation 3.32 for displacement u subject to the boundary conditions (Equations 3.33 and 3.34) and internal and external loadings (Equations 3.11, 3.14, 3.19 and 3.25) the displacement field was obtained. Hook’s law (Equation 3.28) was used to find the stresses. Solving of the system is achieved through the FEM.