, Volume 24, Issue 1, pp 105–115

Numerical study of how creep and progressive stiffening affect the growth stress formation in trees


    • Department of Civil EngineeringTechnical University of Denmark
  • Ola Dahlblom
    • Division of Structural MechanicsLund University
  • Marie Johansson
    • School of Technology and DesignVäxjö University
Original Paper

DOI: 10.1007/s00468-009-0383-3

Cite this article as:
Ormarsson, S., Dahlblom, O. & Johansson, M. Trees (2010) 24: 105. doi:10.1007/s00468-009-0383-3


It is not fully understood how much growth stresses affect the final quality of solid timber products in terms of, e.g. shape stability. It is, for example, difficult to predict the internal growth stress field within the tree stem. Growth stresses are progressively generated during the tree growth and they are highly influenced by climate, biologic and material-related factors. To increase the knowledge of the stress formation, a finite element model was created to study how the growth stresses develop during the tree growth. The model is an axisymmetric general plane strain model where material for all new annual rings is progressively added to the tree during the analysis. The material model used is based on the theory of small strains (where strains refer to the undeformed configuration which is good approximation for strains less than 4%) where so-called biological maturation strains (growth-related strains that form in the wood fibres during their maturation) are used as a driver for the stress generation. It is formulated as an incremental material model that takes into account elastic strain, maturation strain, viscoelastic strain and progressive stiffening of the wood material. The results clearly show how the growth stresses are progressively generated during the tree growth. The inner core becomes more and more compressed, whereas the outer sapwood is subjected to slightly increased tension. The parametric study shows that the growth stresses are highly influenced by the creep behaviour and evolution of parameters such as modulus of elasticity, micro-fibril angle and maturation strain.


Growth stressesTreesFinite element simulationsWoodCreepDistortions


According to Plomion et al. (2001), the wood formation process is exceptionally complex and is far from being adequately investigated. Cell formation occurs in the cambium zone, which is made up of a layer of cells called the cambium initials. Both the wood (xylem) and the bark (phloem) cells are produced by division of these cambium initials. The formation process as a whole is driven by a large number of structural genes that govern cell-origination (by division), cell-differentiation, cell-maturation, cell-death and heartwood formation. The cell wall consists of different layers (middle lamella, primary wall and secondary wall) created at different periods during the differentiation process. At the beginning of this process, the middle lamella is developed further and the primary cell wall (which is highly deformable and attached to the middle lamella) is formed. Thereafter, disposition of the secondary cell wall occurs, i.e. layers S1, S2 and S3 are formed. When the cell maturation starts, first, the cellulose material is laid down (crystallization) and thereafter the lignin is produced and infiltrated into the newly formed cellulose material (lignification). These mechanisms are hypothesized to cause growth stresses to develop in the living tree. The disposition of lignin results in transversal expansion of the fibres (Boyd 1950), whereas crystallization of the cellulose leads to longitudinal shrinkage of the wood cells (Bamber 1978). During this phase the stiffness of the cell increases to its final value. Since the maturing cells are attached to the old and already matured cells, a strain constraint develops in the stem, the maturing cells becomes stretched longitudinally and compressed tangentially, whereas the matured cells are exposed to the opposite stress conditions (see, Fournier et al. 1994). During the progressive growth of a tree, the wood cells inside the trunk become increasingly compressed in the longitudinal direction. The material will with time be affected by creep/relaxation due to the stress state. The final growth stress field can cause significant distortion directly when the timber boards are sawn from the logs. Various models based on the microstructure of the cell have been developed for analysing how the maturation strains evoked in wood cells are affected by the microfibril angle (see, e.g. Yamamoto 1998; Guitard et al. 1999; Alméras et al. 2005). These models agree quite well with experimental results carried out by Yamamoto (1998).

Growth stresses caused at the cell level result in a growth stress variation over the stem cross-section. To confirm the radial growth stress distributions, some experimental studies have been performed (see, e.g. Archer 1986; Kübler 1987; Alhasani 1999; Raymond et al. 2004). One of the first to describe this variation mathematically was Kübler (1959a, b), whilst Archer (1987) presented an analytical solution for growth stresses based on the assumption of axisymmetry. Several growth stress models for trees based on the biological maturation strains have been reported (see, e.g. Archer and Byrnes 1974; Fournier et al. 1990; Skatter and Archer 2001; Fourcaud and Lac 2003; Ormarsson and Johansson 2006; Ormarsson et al. 2009). These authors used different types of finite elements in their work, i.e. axisymmetric solid elements, cylindrical shell elements, multi-layer beam elements and three-dimensional solid elements. In Fourcaud et al. (2003), the multi-layer beam model was implemented into the plant architecture simulation software AMAPpara. The main novelty of the present model is that it treats time-dependent behaviours (creep/relaxation), material inhomogeneity and progressive material stiffening. The present study shows the creep deformation, the material inhomogeneity and the stiffness properties used during the maturation period to be very important parameters for growth stress evolution in living trees.


In this section, the finite element formulation used to simulate progressive stress formation during tree growth is described. The tree is considered as a very long solid cone and in order to simplify modelling, spiral grain is neglected, see Fig. 1. Based on this, the growth stress generation is modelled as a one-dimensional axisymmetric generalised strain problem where the progressive increases in the gravity load and the orthotropic maturation strains are used as a driver for the growth stress generation. The local and the global coordinate systems used to handle the conical shape of the stem are shown in Fig. 1. The angle α represents the conical angle that defines the taper of the studied annual ring. The local coordinate system (\( \bar{r},\bar{t},\bar{l} \)) refers to the material directions of the wood material, whereas the global one (\( r,t,z \)) refers to a fixed coordinate system. The figure also shows a schematic illustration of the rectangular axisymmetric finite elements used in this work. The element representing the newest annual ring shall illustrate how all elements are integrated around the pith. Each element has four degrees of freedom (two radial and two vertical). Each cross-section of the tree is assumed to be plane after the deformation, the vertical displacements are represented with just one degree of freedom, see Fig. 1. For more detailed description of this finite element, see Ormarsson et al. (2009).
Fig. 1

Coordinate systems and finite element used in modelling of axisymmetric tree growth

Equilibrium relations (strong and weak form)

Derivation of differential equations for an axisymmetric stress problem (strong form) is based on requirements of static equilibrium in the r and z directions for a curved incremental element shown in Fig. 2.
Fig. 2

Traction and body forces acting on a curved incremental element in a tree stem (normal stresses: \( \sigma_{r} ,\sigma_{t} ,\sigma_{z} , \) shear stress: \( \tau_{rz} , \) body forces: \( b_{r} ,b_{z} \))

The stresses and body forces acting on the incremental element in Fig. 2 give the static equilibrium equations with respect to the r and z directions as
$$ \begin{gathered} - \sigma_{r} {{\uptheta}}r{\kern 1pt} {\text{d}}z + (\sigma_{r} + {\text{d}}\sigma_{r} ){{\uptheta}}{\kern 1pt} {\kern 1pt} (r + {\text{d}}r){\text{d}}z - \sigma_{t} \sin \left( {{\frac{{{\uptheta}}}{2}}} \right){\text{d}}r{\kern 1pt} {\kern 1pt} {\text{d}}z - (\sigma_{t} + {\text{d}}\sigma_{t} )\sin \left( {{\frac{{{\uptheta}}}{2}}} \right){\text{d}}r{\kern 1pt} {\kern 1pt} {\text{d}}z - \tau_{rz} {{\uptheta}}\left( {r + {\frac{{{\text{d}}r}}{2}}} \right){\text{d}}r + \hfill \\ (\tau_{rz} + {\text{d}}\tau_{rz} ){{\uptheta}}\left( {r + {\frac{{{\text{d}}r}}{2}}} \right){\text{d}}r + b_{r} {{\uptheta}}\left( {r + {\frac{{{\text{d}}r}}{2}}} \right){\text{d}}r\,{\text{d}}z = 0 \hfill \\ \end{gathered} $$
$$ - \sigma_{z} {{\uptheta}}\left( {r + {\frac{{{\text{d}}r}}{2}}} \right){\text{d}}r + (\sigma_{z} + {\text{d}}\sigma_{z} )\theta \left( {r + {\frac{{{\text{d}}r}}{2}}} \right){\text{d}}r - \tau_{rz} {{\uptheta}}r{\kern 1pt} {\rm d}z + (\tau_{rz} + {\text{d}}\tau_{rz} ){{\uptheta}}(r + {\text{d}}r){\text{d}}z + b_{z} {{\uptheta}}\left( {r + {\frac{{{\text{d}}r}}{2}}} \right){\text{d}}r\,{\text{d}}z = 0 $$
where \( {\text{d}}r,{\text{d}}z \) and \( {{\uptheta}}r,{{\uptheta}}(r + {\text{d}}r) \) are the dimensions of the incremental element. For an infinitely small element, Eqs. 1 and 2 result in the following differential equations (strong form)
$$ {\frac{1}{r}}\,{\frac{\partial }{\partial r}}(r\sigma_{r} ) - {\frac{{\sigma_{t} }}{r}} + {\frac{\partial }{\partial z}}(\tau_{rz} ) + b_{r} = 0 $$
$$ {\frac{\partial }{\partial z}}(\sigma_{z} ) + {\frac{\partial }{\partial r}}(\tau_{rz} ) + b_{z} = 0 $$
where \( \sigma_{r} ,{}_{{}}\sigma_{t} ,_{{}} \sigma_{z} {}_{{}}\;{\text{and}}\;_{{}} \tau_{rz} \) are the stress components in the global coordinate system (r, t, z), whilst br and bz are the body forces in the r and z directions, respectively. The integral form (weak form) of these equations can be expressed as
$$ - 2\pi \int\limits_{A} \left({\frac{\partial }{\partial r}} (v_{r} ) \sigma_{r} + {\frac{\partial }{\partial z}}(v_{r} ) \tau_{rz} \right)r\,{\text{d}}A + 2\pi \int\limits_{\mathcal{L}} {v_{r} } (\sigma_{r} n_{r} + \tau_{rz} n_{z} )r\,{\text{d}}{\mathcal{L}} - 2\pi \int\limits_{A} {v_{r} \sigma_{t} {\text{d}}A + 2\pi \int\limits_{A} {v_{r} } } b_{r} r\,{\text{d}}A = 0 $$
$$ - 2\pi \int\limits_{A} \left({{\frac{\partial }{\partial z}}(v_{z} )} \sigma_{z} + {\frac{\partial }{\partial r}}(v_{z} )\tau_{rz}\right )r\,{\text{d}}A + 2\pi \int\limits_{\mathcal{L}} {v_{z} (\sigma_{z} n_{z} + \tau_{rz} n_{r} )r\,{\text{d}}{\mathcal{L}} + 2\pi \int\limits_{A} {v_{z} b_{z} r\,{\text{d}}A} } = 0 $$
where \( v_{r} \,{\text{and}}\,v_{z} \) are arbitrary weight functions in the (r, z) coordinate system. The area A is the studied region within the boundary \( {\mathcal{L}}\), where \( n_{r} \,{\text{and}}\,n_{z} \) represent the components of the normal vectors for the boundary \( {\mathcal{L}}.\) Adding and differentiating these two expressions with respect to time yields the final (incremental) weak formulation in a matrix form as
$$ - 2\pi \int\limits_{A} {(\tilde{\nabla}{\varvec{v}})}^{T} {\dot{\varvec{\sigma}}}r\,{\text{d}}A + 2\pi \int\limits_{\mathcal{L}} {}{\varvec{v}}^{T} {\dot{\varvec{t}}r\,{\text{d}}{\mathcal{L}} + 2\pi \int\limits_{A} {{\varvec{v}}^{T} {\dot{\varvec{b}}}r\,{\text{d}}A = 0} } $$
where \( \tilde{\nabla} = \left[ {\begin{array}{*{20}c} {{\frac{\partial }{\partial r}}} & 0 \\ {{\frac{1}{r}}} & 0 \\ 0 & {{\frac{\partial }{\partial z}}} \\ {{\frac{\partial }{\partial z}}} & {{\frac{\partial }{\partial r}}} \\ \end{array} } \right] \); \({\dot{\sigma}} = \left[ {\begin{array}{*{20}c}{\dot{\sigma}}_{r} \\ {\dot{\sigma}}_{t} \\{\dot{\sigma}}_{z} \\ {\dot{\varvec{\tau }}}_{rz} \\\end{array} } \right]\); \( {\varvec{v} }= \left[ {\begin{array}{*{20}c} {v_{r} } \\ {v_{z} } \\ \end{array} } \right] \); \({\dot{\varvec{b}}} = \left[ {\begin{array}{*{20}c} {\dot{b}_{r} } \\ {\dot{b}_{z} } \\ \end{array} } \right] \) and \({\dot{\varvec{t}}} = \left[ {\begin{array}{*{20}c} {\dot{t}_{r} } \\ {\dot{t}_{z} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\dot{\sigma }_{r} n_{r} + \dot{\tau }_{rz} n_{z} } \\ {\dot{\sigma }_{z} n_{z} + \dot{\tau }_{rz} n_{r} } \\ \end{array} } \right] .\) The vectors \({\dot{\varvec{\sigma }}}, {\dot{\varvec{b}}}\; {\text{and}}\;{\dot{\varvec{t}}} \) represent the stress rate, the body force rate and the traction force rate, respectively, in the rz plane.


Based on the weak formulation given in Eq. 7 and the approximations v = Nc, the global FE-formulation can be expressed as
$$ - 2\pi \int\limits_{A} {{\varvec{B}}^{T}{\dot{\varvec{\sigma }}}r} \,{\text{d}}A + 2\pi \int\limits_{\mathcal{L}} {{\varvec{N}}^{T} {\dot{\varvec{t}}}r\,{\text{d}}{\mathcal{L}} + 2\pi \int\limits_{A} {{\varvec{N}}^{T} {\dot{\varvec{b}}}r\,{\text{d}}A = 0} } $$
where N is the shape function matrix, c is an arbitrary constant vector and \( {\varvec{B}} = \tilde{\nabla}{\varvec{N}}. \) The global constitutive relation is given in rate form as
$$ {\dot{\varvec{\sigma }}} = {\varvec{D}}({\dot{\varvec{\varepsilon }}} - {\dot{\varvec{\varepsilon}}}_{m} - {\dot{\varvec{\varepsilon }}}_{c} ) $$
where \( {\varvec{D}} \) is the (global) constitutive matrix. The column matrices \( {\dot{\varvec{\varepsilon}}},\,{\dot{\varvec{\varepsilon}}}_{m} \;{\text{and}}\;{\dot{\varvec{\varepsilon}}}_{c} \) represent the total strain rate, the maturation strain rate and the creep strain rate in the global coordinate system (r, t, z). The approximation for the total strain rate is given by
$$ {\dot{\varvec{\varepsilon}}} = \tilde{\nabla }{\dot{\varvec{u}}} = \tilde{\nabla}{\varvec{N}}{\dot{\varvec{a}}} = {{\varvec{B}}\dot{\varvec{a}}} $$
where u = [uruz]T is the displacement vector and a is the global nodal displacement vector. The (global) matrices \( {\varvec{D}}, \)\( {\dot{\varvec{\varepsilon }}}_{m} \) and \( {\dot{\varvec{\varepsilon }}}_{c} \) can be calculated through transformation of the (local) matrices \( {\bar{\varvec{D}}}, \)\( {\dot{\bar{\varvec{\varepsilon }}}}_{m} \) and \( {\dot{\bar{\varvec{\varepsilon }}}}_{c} \) which represent the material behaviour and strain state in the orthotropic coordinate system (\( \bar{r}, \bar{t}, \bar{l} \)), see Fig. 1.
$$ {\varvec {D}} = {\varvec {G}^{\varvec {T}}} {\bar {\varvec {D}} {\varvec {G}}} $$
$$ {\dot{\varvec{\varepsilon }}}_{m} = \left[ {\begin{array}{*{20}c} {\dot{\varepsilon }_{m,r} } \\ {\dot{\varepsilon }_{m,t} } \\ {\dot{\varepsilon }_{m,z} } \\ {\dot{\gamma }_{m,rz} } \\ \end{array} } \right] = {\varvec{G}}^{ - 1} {\dot{\bar{\varvec{\varepsilon }}}}_{m} = {\varvec{G}}^{ - 1} \left[ {\begin{array}{*{20}c} {\dot{\varepsilon }_{{m,\bar{r}}} } \\ {\dot{\varepsilon }_{{m,\bar{t}}} } \\ {\dot{\varepsilon }_{{m,\bar{l}}} } \\ 0 \\ \end{array} } \right] = {\frac{{{\varvec{G}}^{ - 1} }}{100}}\left[ {\begin{array}{*{20}c} {\dot{\xi }_{{\bar{r}}} (\gamma (r,g))} \\ {\dot{\xi }_{{\bar{t}}} (\gamma (r,g))} \\ {\dot{\xi }_{{\bar{l}}} (\gamma (r,g))} \\ 0 \\ \end{array} } \right] $$
$$ {\dot{\varvec{\varepsilon }}}_{c} = \left[ {\begin{array}{*{20}c} {\dot{\varepsilon }_{c,r} } \\ {\dot{\varepsilon }_{c,t} } \\ {\dot{\varepsilon }_{c,z} } \\ {\dot{\gamma }_{c,rz} } \\ \end{array} } \right] = {\varvec{G}}^{ - 1} {\dot{\bar{\varvec{\varepsilon }}}}_{c} = {\varvec{G}}^{ - 1} \left[ {\begin{array}{*{20}c} {\dot{\varepsilon }_{{c,\bar{r}}} } \\ {\dot{\varepsilon }_{{c,\bar{t}}} } \\ {\dot{\varepsilon }_{{c,\bar{l}}} } \\ {\dot{\gamma }_{{c,\bar{r}\bar{l}}} } \\ \end{array} } \right] $$
where the transformation matrix \( {\varvec{G}} \) and the local constitutive matrix \( {\bar{\varvec{D}}} \) are
$$ {\varvec{G}} = \left[ {\begin{array}{*{20}c} {{ \cos }^{2} \alpha (r)} & 0 & { - { \sin }^{2} \alpha (r)} & { - { \cos }\alpha (r){ \sin }\alpha (r)} \\ 0 & 1 & 0 & 0 \\ {{ \sin }^{2} \alpha (r)} & 0 & {{ \cos }^{2} \alpha (r)} & {{ \cos }\alpha (r){ \sin }\alpha (r)} \\ { 2 {\text{cos}}\alpha (r)sin\alpha (r)} & 0 & { - 2 {\text{cos}}\alpha (r){ \sin }\alpha (r)} & {{ \cos }^{2} \alpha (r) - { \sin }^{2} \alpha (r)} \\ \end{array} } \right] $$
$$ {\bar{\varvec{D}}} = {\bar{\varvec{C}}}^{ - 1} = \left[ {\begin{array}{*{20}c} {{\frac{1}{{E_{{\bar{r}}} (r,\hat{t})}}}} & { - {\frac{{\nu_{{\bar{t}\bar{r}}} }}{{E_{{\bar{t}}} (r,\hat{t})}}}} & { - {\frac{{\nu_{{\bar{l}\bar{r}}} }}{{E_{{\bar{l}}} (r,\hat{t})}}}} & 0 \\ { - {\frac{{\nu_{{\bar{r}\bar{t}}} }}{{E_{{\bar{r}}} (r,\hat{t})}}}} & {{\frac{1}{{E_{{\bar{t}}} (r,\hat{t})}}}} & { - {\frac{{\nu_{{\bar{l}\bar{t}}} }}{{E_{{\bar{l}}} (r,\hat{t})}}}} & 0 \\ { - {\frac{{\nu_{{\bar{r}\bar{l}}} }}{{E_{{\bar{r}}} (r,\hat{t})}}}} & { - {\frac{{\nu_{{\bar{t}\bar{l}}} }}{{E_{{\bar{t}}} (r,\hat{t})}}}} & {{\frac{1}{{E_{{\bar{l}}} (r,\hat{t})}}}} & 0 \\ 0 & 0 & 0 & {{\frac{1}{{G_{{\bar{r}\bar{l}}} (r,\hat{t})}}}} \\ \end{array} } \right]^{ - 1} $$
The \( \alpha (r) \) represents the radial function for the conical angle (the angle of the last annual ring represents the taper of the tree stem). The elastic moduli \( E_{i} (r,\hat{t}) \) and the shear modulus \( G_{{\bar{r}\bar{l}}} (r,\hat{t}) \) are allowed to vary with the radius r (inhomogeneous material in matured condition) and during the maturation time \( \hat{t} \) of each individual annual ring. The data used to estimate the local maturation strain \( {\dot{\bar{\varvec{\varepsilon }}}}_{m} \) are based on experimental work (on mature wood) presented by Yamamoto (1998), which describes the relationship between the so-called released maturation strains \( \xi_{{\bar{r}}} ,\,\xi_{{\bar{t}}} ,\,\xi_{{\bar{l}}} \) and the micro-fibril angle γ. The micro-fibril angle \( \gamma (r,g) \) is assumed to vary both with the radius r and with the growth index g as shown in Ormarsson et al. (2009). The viscoelastic behaviour related to the progressive growth stress generation in the stem was modelled with the hereditary approach where the principle of superposition is an essential issue. The hereditary approach results in a series-coupled Kelvin-model that can be studied in more detail in, e.g. Ottosen and Ristinmaa (2005). Based on this approach, the local creep strains can be calculated by the following integral expression
$${\bar{\varvec{\varepsilon}}}_{c} = \int\limits_{0}^{t}{\bar{\varvec{C}}}_{c} (t,t^{\prime}){\frac{{{\rm d}\bar{\varvec{\sigma} }(t^{\prime} )}}{{{\rm d}t^{\prime}}}}{\rm d}t^{\prime} = \int\limits_{0}^{t} {\left[ {\begin{array}{*{20}c}{{\frac{1}{{E_{{\bar{r}}} }}}\,\phi_{{\sigma_{{\bar{r}}} }} } & {- {\frac{{\nu_{{\bar{t}\bar{r}}} }}{{E_{{\bar{t}}}}}}\,\phi_{{\nu_{{\bar{t}\bar{r}}} }} } & { -{\frac{{\nu_{{\bar{l}\bar{r}}} }}{{E_{{\bar{l}}}}}}\,\phi_{{\nu_{{\bar{l}\bar{r}}} }} } & 0 \\ { -{\frac{{\nu_{{\bar{r}\bar{t}}} }}{{E_{{\bar{r}}}}}}\,\phi_{{\nu_{{\bar{r}\bar{t}}} }} } &{{\frac{1}{{E_{{\bar{t}}} }}}\,\phi_{{\sigma_{{\bar{t}}} }} } & {- {\frac{{\nu_{{\bar{l}\bar{t}}} }}{{E_{{\bar{l}}}}}}\,\phi_{{\nu_{{\bar{l}\bar{t}}} }} } & 0 \\ { -{\frac{{\nu_{{\bar{r}\bar{l}}} }}{{E_{{\bar{r}}}}}}\,\phi_{{\nu_{{\bar{r}\bar{l}}} }} } & { -{\frac{{\nu_{{\bar{t}\bar{l}}} }}{{E_{{\bar{t}}}}}}\,\phi_{{\nu_{{\bar{t}\bar{l}}} }} } &{{\frac{1}{{E_{{\bar{l}}} }}}\,\phi_{{\sigma_{{\bar{l}}} }} } & 0\\ 0 & 0 & 0 & {{\frac{1}{{G_{{\bar{r}\bar{l}}}}}}\,\phi_{{\tau_{{\bar{r}\bar{l}}} }} } \\ \end{array} }\right]\left[ {\begin{array}{*{20}c} {{\frac{{{\rm d}\sigma_{{\bar{r}}}}}{{{\rm d}t^{\prime} }}}} \\ {{\frac{{{\rm d}\sigma_{{\bar{t}}} }}{{{\rm d}t^{\prime} }}}} \\{{\frac{{{\rm d}\sigma_{{\bar{l}}} }}{{{\rm d}t^{\prime} }}}} \\{{\frac{{{\rm d}\tau_{{\bar{r}\bar{l}}} }}{{{\rm d}t^{\prime} }}}} \\ \end{array} }\right]{\rm d}t^{\prime} } $$
where \( {\bar{\varvec{C}}}_{c} (t,t^{\prime} ) \) is the creep compliance matrix and \( \phi_{{\sigma_{{\bar{r}}} }} , \phi_{{\sigma_{{\bar{t}}} }} , \phi_{{\nu_{{\bar{t}\bar{r}}} }} , \) etc. are experimentally observed curves (the relative creep curves) that describe the viscoelastic material behaviour of wood. Lacking experimental data, the same creep curve \( \phi_{{\bar{\sigma }}} \) is used in all directions. It is estimated with the series
$$ \phi_{{\bar{\sigma }}} = \sum\limits_{n = 1}^{N} {\phi_{{\bar{\sigma }}}^{n} } = \sum\limits_{n = 1}^{N} {\phi_{n} } (1 - e^{{ - {\frac{{t - t^{\prime} }}{{\tau_{n} }}}}} ) $$
where \( \phi_{n} {\text\;{and}}\; \tau_{n} \) are N-number of constants that describe the shape of the relative creep curve, i.e. the different exponential functions become active during different time periods in the series. After insertion of Eq. 17 into Eq. 16, the creep strain rate is given by
$$ {\dot{\bar{\varvec{\varepsilon }}}}_{c} = \sum\limits_{n = 1}^{N} {{\frac{1}{{\tau_{n} }}}}\; e^{{ - {\frac{t}{{\tau_{n} }}}}} \bar{\gamma }_{n} (t) $$
where the creep driver \( \bar{\varvec{\gamma }}_{n} (t) \) is given by the integral
$$ \begin{gathered} \bar{\varvec{\gamma }}_{n} (t) = \int\limits_{0}^{t} {e^{{{\frac{{t^{\prime} }}{{\tau_{n} }}}}} } {\bar{\varvec{C}}}_{{c_{n} }} {\frac{{{\text{d}}{\bar{\varvec{\sigma }}}(t^{\prime} )}}{{{\text{d}}t^{\prime} }}}\;{\text{d}}t^{\prime} = \hfill \\ \int\limits_{0}^{t} {e^{{{\frac{{t^{\prime} }}{{\tau_{n} }}}}} } \left[ {\begin{array}{*{20}c} {{\frac{1}{{E_{{\bar{r}}} (r,\hat{t})}}}\,\phi_{n} } & { - {\frac{{\nu_{{\bar{t}\bar{r}}} }}{{E_{{\bar{t}}} (r,\hat{t})}}}\,\phi_{n} } & { - {\frac{{\nu_{{\bar{l}\bar{r}}} }}{{E_{{\bar{l}}} (r,\hat{t})}}}\,\phi_{n} } & 0 \\ { - {\frac{{\nu_{{\bar{r}\bar{t}}} }}{{E_{{\bar{r}}} (r,\hat{t})}}}\,\phi_{n} } & {{\frac{1}{{E_{{\bar{t}}} (r,\hat{t})}}}\,\phi_{n} } & { - {\frac{{\nu_{{\bar{l}\bar{t}}} }}{{E_{{\bar{l}}} (r,\hat{t})}}}\,\phi_{n} } & 0 \\ { - {\frac{{\nu_{{\bar{r}\bar{l}}} }}{{E_{{\bar{r}}} (r,\hat{t})}}}\,\phi_{n} } & { - {\frac{{\nu_{{\bar{t}\bar{l}}} }}{{E_{{\bar{t}}} (r,\hat{t})}}}\,\phi_{n} } & {{\frac{1}{{E_{{\bar{l}}} (r,\hat{t})}}}\,\phi_{n} } & 0 \\ 0 & 0 & 0 & {{\frac{1}{{G_{{\bar{r}\bar{l}}} (r,\hat{t})}}}\,\phi_{n} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\frac{{{\text{d}}\sigma_{{\bar{r}}} }}{{{\text{d}}t^{\prime} }}}} \\ {{\frac{{{\text{d}}\sigma_{{\bar{t}}} }}{{{\text{d}}t^{\prime} }}}} \\ {{\frac{{{\text{d}}\sigma_{{\bar{l}}} }}{{{\text{d}}t^{\prime} }}}} \\ {{\frac{{{\text{d}}\tau_{{\bar{r}\bar{l}}} }}{{{\text{d}}t^{\prime} }}}} \\ \end{array} } \right]{\text{d}}t^{\prime} . \hfill \\ \end{gathered} $$
The time t is the total time of the analysis whereas \( t^{\prime} \) represents different times when the new stress increments (from gravity load and maturation strains) are applied on the tree stem. Insertion of Eqs. 913 into Eq. 8 yields the final FE-equation
$$ {{{\varvec{K}}}{\dot{\varvec{a}}}} = {\dot{\varvec{f}}_{b} + {\dot{\varvec{f}}}_{l} + {\dot{\varvec{f}}}}_{m} + {\dot{\varvec{f}}}_{c} $$
$${\varvec{K}} = 2\pi \int\limits_{A} {{\varvec{B}}^{T} {\varvec{G}}^{T} {\bar{\varvec{D}}}}{\varvec {GB}}r\,{\text{d}}A \quad ({\text{Stiffness\;matrix}}) $$
$$ {\dot{\varvec{f}}}_{b} = 2\pi \int\limits_{\cal{L}} {\varvec{N}}^{T} {\dot{\varvec{t}}}r\,{\text{d}}{\mathcal{L}}\quad (\text{Boundary vector}) $$
$$ {\dot{\varvec{f}}}_{l} = 2\pi \int\limits_{A} {{\varvec{N}}^{T} {\dot{\varvec{b}}}r\,{\text{d}}A} \quad ({\text{Load vector}}) $$
$${\dot{\varvec{f}}}_{m} = 2\pi \int\limits_{A} {{\varvec{B}}^{T} {\varvec{G}}^{T} {\bar{\varvec{D}}}} {\dot{\bar{\varvec{\varepsilon }}}}_{m} r\,{\text{d}}A \quad ({\text{Maturation strain vector}}) $$
$$ {\dot{\varvec{f}}}_{c} = 2\pi \int\limits_{A} {{\varvec{B}}^{T} {\varvec{G}}^{T} {\bar{\varvec{D}}}} {\dot{\bar{\varvec{\varepsilon }}}}_{c} r\,{\text{d}}A \quad ({\text{Creep strain vector}}). $$

Numerical implementation

The progressive growth stress formation was modelled incrementally in three main steps, which were repeated for each individual annual ring, shown in Fig. 3.
Fig. 3

Illustration of the modelling steps in progressive tree growth. a Formation of the top shoot and the new annual ring (geometry and material properties). b Gravity load and biological maturation strains (spring and summer period). c Creep deformation and additional stiffness evolution (autumn and winter period)

Step 1: formation of a new volume and material properties

This is a pre-step that corresponds to the longitudinal formation of the top shoot and the diametrical formation of a new annual ring. Based on the annual ring number and the growth rate, the density, the stiffness properties, the fibre orientation and the maturation strain properties of the new annual ring are decided. The constitutive assumption here requires that the new material has to be added to the structure without changing the stress state within the already matured annual rings. The volume and density are needed for calculation of the gravity force and the gravity traction acting on the element boundaries. Both geometrical and material data used here are based on experimental work on Norway spruce.

Step 2: gravity load and biological maturation of the growing annual ring

This step represents simulation of stresses caused by the gravity load and the maturation strains in the growing annual ring. The maturation strain is modelled as a biologically driven strain field that causes internal constraints in the wood material that in turn results in a new stress state in the stem. For an annual ring (number n) created at the time t, the new geometry (the maturated rings and the new ring) is applied to an incremental gravity load \( \Updelta {\varvec{f}}_{i = 1\, \to \,n}^{t} \) and maturation strain \( \Updelta {\varvec{\varepsilon}}_{m,n}^{t} \) generated by the growing annual ring. The incremental displacements \( \Updelta {\varvec{a}}_{j = 1\, \to \,n}^{t} \) can be solved for the new geometry and added to the total displacement vector from the previous time step \( {\varvec{a}}_{i = 1 \to (n - 1)}^{t - 1} . \) Note that the new ring has to be added to the current (deformed) geometry which means that the displacement vector at the time t becomes
$$ {\varvec{a}}_{n}^{t} = {\varvec{a}}_{n - 1}^{t - 1} + \Updelta {\varvec{a}}_{n}^{t} $$
$$ {\varvec{a}}_{i = 1 \to (n - 1)}^{t} = {\varvec{a}}_{i = 1 \to (n - 1)}^{t - 1} + \Updelta {\varvec{a}}_{i = 1 \to (n - 1)}^{t} $$
where \( {\varvec{a}}_{n}^{t} \) and \( {\varvec{a}}_{i = 1 \to (n - 1)}^{t} \) are the total displacements at time t for the new annual ring (number n) and the already maturated annual rings number 1 to n − 1. The stresses can be calculated as
$$ {\varvec{\sigma}}_{i = 1 \to n}^{t} = {\varvec{\sigma}}_{i = 1 \to n}^{t - 1} + \Updelta {\varvec{\sigma}}_{i = 1 \to n}^{t} $$
$$ \Updelta {\varvec{\sigma}}_{i = 1 \to (n - 1)}^{t} = {\varvec{D}}({\varvec{B}}\Updelta {\varvec{a}}_{i = 1 \to (n - 1)}^{t} - \Updelta {\varvec{\varepsilon}}_{c,i = 1 \to (n - 1)}^{t} ) $$
$$ \Updelta {\varvec{\sigma}}_{n}^{t} = {\varvec{D}}({\varvec{B}}\Updelta {\varvec{a}}_{n}^{t} - \Updelta {\varvec{\varepsilon}}_{m,n}^{t} - \Updelta {\varvec{\varepsilon}}_{c,n}^{t} ). $$

Step 3: creep (and some stiffness evolution of the new annual ring)

This step corresponds to redistribution of stresses due to the creep deformations that occur during the autumn and the winter period. If the stiffness properties are not assumed to be fully matured in the step 2, they will continue to mature during this step but without generation of biologic strain. The displacement and stresses at the end of this step (time t + 1) become
$$ {\varvec{a}}_{i = 1 \to n}^{t + 1} = {\varvec{a}}_{i = 1 \to n}^{t} + \Updelta {\varvec{a}}_{i = 1 \to n}^{t + 1} $$
$$ {\varvec{\sigma}}_{i = 1 \to n}^{t + 1} = {\varvec{\sigma}}_{i = 1 \to n}^{t} + \Updelta {\varvec{\sigma}}_{i = 1 \to n}^{t + 1} $$
$$ \Updelta {\varvec{\sigma}}_{i = 1 \to n}^{t + 1} = {\varvec{D}}({\varvec{B}}\Updelta {\varvec{a}}_{i = 1 \to n}^{t} - \Updelta {\varvec{\varepsilon}}_{c,i = 1 \to n}^{t + 1} ). $$

For more detailed descriptions of the numerical implementation of an axisymmetric general strain element see Ormarsson et al. (2009).

Numerical example

Stress simulation of tree growth differs significantly from applications in which the model geometry is defined from the beginning of the analysis. The volume of the tree has to be created progressively during the analysis and all the material parameters (and fibre orientations) need to be determined progressively during the analysis as well. The above theory for axisymmetric tree growth was implemented into CALFEM which is a FE-toolbox in the Matlab environment (see CALFEM 2004). How the material properties mature during the annual ring evolution has not yet been adequately investigated, but it has been observed experimentally for Norway spruce that parameters such as annual ring width, wood density, micro-fibril angle and stiffness properties, etc. vary more or less predictably from pith to bark (see, e.g. Persson 2000; Dahlblom et al. 1999a, b). The material data used here for spruce (see Table 1) represent a Norway spruce tree grown under normal growth conditions (growth index g = 0.5). In Ormarsson et al. (2009), a more detailed description of how the growth index g is defined and used to create the material property evolution during a progressive tree growth. The radial variation for the micro-fibril angle, the longitudinal elastic modulus and annual ring width is based on average data from different heights up the tree. The radial variation for elastic moduli representing the radial and tangential direction is based on relationships between material properties created with a micro-mechanical model developed by Persson (2000). The remaining parameters used are the same as used in Ormarsson (1999). The radial variation in maturation strain data is based on experimental data presented by Yamamoto (1998) where the released strains of an individual fibre are shown to be strongly influenced by the micro-fibril angle. The creep data used are based on experimental work presented by Gressel (1984). To explain better the coefficients in Table 1, the expressions (and curves) used to create the relative creep curve \( \phi_{{\bar{\sigma }}} \) are shown in Fig. 4. Because of the lack of creep data for living trees, the relative creep curve \( \phi_{{\bar{\sigma }}} \) is assumed to represent all the orthotropic material directions and to be independent of the growth rate.
Table 1

Material parameters used in the simulations for normal grown trees (g = 0.5), r = distance from pith in (m) and n = annual ring number

Annual ring evolution (m)

r = (2.2n − 0.013n2)10−3 for n ≤ 60

r = (46.8 + 0.64n)10−3 for n > 60

Density of growing wood (kg/m3)

ρ = 513.9 + 1143r

Conical angle (degrees)

α = −0.5

Micro-fibril angle (degrees)

γ = 24.0 − 200r for r ≤ 0.05

γ = 14.0 for r > 0.05

Elastic moduli (MPa)

\( E_{{\bar{l}}} = 120000r + 7 5 0 0 {\text{ for }}r \, \le 0 . 0 5 \)

\( E_{{\bar{l}}} {\text{ = 13500 for }}r > 0 . 0 5 \)

\( E_{{\bar{r}}} = 1510r + 1030 \)

\( E_{{\bar{t}}} = 5570r + 530 \)

Shear moduli (MPa)

\( G_{{\bar{l}\bar{r}}} = 600 \)

Poisson’s ratio

\( \nu_{{\bar{l}\bar{r}}} = 0. 35 \)

\( \nu_{{\bar{l}\bar{t}}} = 0. 60 \)

\( \nu_{{\bar{r}\bar{t}}} = 0. 55 \)

Maturation strain (%)

\( \xi_{{\bar{l}}} \) = 0.0001γ2 + 0.00062γ − 0.1;

\( \xi_{{\bar{r}}} = \xi_{{\bar{t}}} = \) (−0.10071γ2 + 1.1714γ + 88.9)10−3

Creep data (N = 2 and τ in years)

\( \phi_{1} = 0.2;{\text{ and }}\tau_{ 1} = 10 \)

\( \phi_{2} = 0.3;{\text{ and }}\tau_{ 2} = 100 \)
Fig. 4

Expressions and curves used to determine the relative creep curve \( \phi_{{\bar{\sigma }}} \)

Based on the material data presented in Table 1, numerical simulations have been performed to study how stresses develop during 120 years growth of normal growing Norway spruce. To illustrate some simulation results, Figs. 5, 6 and 7 show how longitudinal, radial and tangential growth stresses vary along a radial path after thirteen different years (2, 10, 20, 30,…,120) of growing. The first subfigures noted (a) show stress distribution where the creep deformation is neglected whereas in subfigures b and c the creep is included. In subfigure b, final stiffness is assumed to work within the new annual ring during the maturation period (step 2 in Fig. 3), whereas in subfigure c only 70% of the final stiffness is assumed during the maturation period.
Fig. 5

Evolution of the longitudinal growth stresses during 120 years growth. a Elastic behaviour. b Creep deformation and full stiffness during the maturation period (step 2). c Creep deformation and 70% stiffness in step 2 and full stiffness in step 3
Fig. 6

Evolution of the radial growth stresses during 120 years growth. a Elastic behaviour. b Creep deformation and full stiffness during the maturation period (step 2). c Creep deformation and 70% stiffness in step 2 and full stiffness in step 3
Fig. 7

Evolution of the tangential growth stresses during 120 years growth. a Elastic behaviour. b Creep deformation and full stiffness during the maturation period (step 2). c Creep deformation and 70% stiffness in step 2 and full stiffness in step 3

For subfigure c, the stiffness properties become fully developed during the autumn/winter period (step 3). For stresses in the longitudinal direction (Fig. 5), the results show that the inner core of the log becomes increasingly compressed during growth, whereas the tension stresses at the surface increase in the early ages and become more constant after 30 years of growth. It may be noted that the shape of these stress curves is influenced by the strong radial variation in the material properties, micro-fibril angle, maturation strain and the annual ring width. The results in the subfigures b and c show that both the creep and the reduced stiffness (70% of the final stiffness during the maturation period) have a significant influence on the stress pattern, the compression stress becomes less in the inner core and the shape of the stress curves change. The tension stresses at the surface are also highly reduced in the diagram for reduced stiffness. The radial and tangential stresses in Figs. 6 and 7 show relatively high tensile values (ca. 1.5–3.0 MPa) in the log centre. In the area close to the pith the radial stress decreases very rapidly down to small values continuing with a slow change down to zero at the log surface. The tangential stress in Fig. 7 shows a similar tendency as observed for the radial stress component except that it changes to compression at a certain distance from the pith. The maximum tensile stresses close to the pith are so large that they can result in internal checking in radial and tangential direction of the stem. Since the conical angle is small (0.5°) in the studied application, the shear stresses become quite small (max 0.1 MPa). The shear stress varies from a (maximum) positive value close to the pith to a (maximum) negative value close to the bark where the positive stress value is slightly higher than the negative one.

Discussion and conclusions

The growth stress results show that the finite element theory for a rectangular axisymmetric general strain element can successfully be used to simulate progressive growth stress generation in a stem of normally grown tree. However, good knowledge of the stiffness properties, creep behaviour, micro-fibril angle, annual ring growth and maturation strain properties is needed to get good simulation results concerning the growth stress formation. The results of the simulation show that the growth stresses have a significant radial variation from pith to bark as well as substantial differences in the stress profiles during growing. For longitudinal stresses, the inner core becomes increasingly compressed during the tree growth whereas the outer layer is subjected to increasing tension stresses. The creep and the reduced stiffness (of the growing annual ring during the maturation period) significantly reduce the stresses in the inner core. The final stress patterns for just elastic behaviour (Figs. 5a, 6a 7a) are quite similar in shape as the stress patterns presented, e.g. by Kübler (1959a, b) and Skatter and Archer (2001). But the magnitude of the numerical values, especially for longitudinal stresses, differs quite markedly. The reason is that input data for radial variation in material properties, micro-fibril angle and maturation strains were very different to those used in this study. Ring width and stiffness of a new annual ring also has a significant influence on how the stress field changes during the year. The final stress pattern in Fig. 5c is quite similar to an experimental pattern presented by Alhasani (1999) for normally grown Norway spruce, but the simulated compression stresses in the inner core are still higher than the experimentally observed stresses. This indicates a need for better creep data for long-term behaviour in living trees and also better data for the progressive stiffening during the maturation period. The results have also shown that gravity load has almost no influence on the growth stress evolution. It may be noted that the density profile used here for calculation of the gravity load was slightly underestimated since it did not include weight from branches and leaves.

The radial variation in the longitudinal stress can cause significant bow or crook deformation of timber boards when they are created through the sawing of logs. How bow and crook distortion in timber boards is influenced by material properties, growth stresses and sawing patterns is studied in Johansson and Ormarsson (2009). The results also show relatively high radial and tangential tensile stresses in the inner core area which can result in internal checking in radial and tangential direction. For old trees where the possible damage zone can become relatively large, the bending capacity of the tree stem will be reduced and it will increase the risk of bending failure when the trees are exposed to a high wind load.

Future work

The axisymmetric simulation model can be used to study how growth stresses in normally grown trees are influenced by different wood properties and growth conditions. To reach a better understanding of the processes of growth stresses formation, it is necessary to gain a more thorough knowledge of the cell generation and maturation process. Studies of material properties of green wood as well as evolution of the material properties are important.

To study how growth stresses develop in trees which have reaction wood (a result from the adaptive reaction of the tree to mechanical disturbance), this model needs to be expanded to a full 3D, or to a 2D-model based on general plane strain assumptions that are able to handle a constant curvature as well as twist deformation caused by radial variation in spiral grain angle. If the material properties do not vary in the longitudinal direction, the 2D-model can additionally be used to simulate drying distortions (twist and bow) in sawn timber boards.

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© Springer-Verlag 2009