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\begin{equation} \label{eq:rcore} \label{eq:rcore}r_{core}=Y_{l}R\\ \end{equation}

\begin{equation} \label{eq:ITstress} \label{eq:ITstress}IT_{stress}=\frac{Y_{cs}}{\frac{R}{r_{core}}\big{(}1-\frac{r_{core}}{R}\big{)}}\\ \end{equation}

\(IT_{stress}\) is the initial longitudinal tensile stress and \(Y_{cs}\) is the compressive stress at the limit of proportionality calculated as per Equation \ref{eq:constants}.

\begin{equation} \label{eq:GS} \label{eq:GS}G_{s}=\begin{cases}0,&\text{if }r_{core}=0\\ -IT_{stress}\frac{R}{r_{core}}(1-\frac{r_{core}}{R}),&\text{else if }r\leq r_{core}\\ -IT_{stress}\frac{R}{r_{core}}\frac{(\frac{r_{core}}{r}-2\frac{r_{core}}{R}+\frac{r_{core}^{2}}{R^{2}})}{1-\frac{r_{core}}{R}},&\text{otherwise}\end{cases}\\ \end{equation} \begin{equation} \label{eq:stress_gs} \label{eq:stress_gs}\boldsymbol{\sigma_{gs}}=\begin{bmatrix}0\\ 0\\ G_{s}\\ 0\\ 0\\ 0\end{bmatrix}\\ \end{equation}

Hooks law can be used to characterise elastic relationships in mathematical terms. Generalised Hook’s law (Equation \ref{eq:hooks}) 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, \(\boldsymbol{\sigma}\) is calculated from strain \(\boldsymbol{\epsilon}\) and the stiffness matrix \(\boldsymbol{C}\). When growth stresses were not considered Equation \ref{eq:hooks} was used, when growth stresses were considered \(\boldsymbol{\sigma_{gs}}\) was defined by Equation \ref{eq:stress_gs} and \(\boldsymbol{\sigma}\) was calculated through Equation \ref{eq:stress}.

\begin{equation} \label{eq:E} \label{eq:E}\boldsymbol{\epsilon}=\frac{1}{2}(\bigtriangledown\boldsymbol{u}+\bigtriangledown\boldsymbol{u}^{T})\\ \end{equation}