Acta Mechanica

, Volume 217, Issue 1–2, pp 75–100 | Cite as

The poroelastic role of water in cell walls of the hierarchical composite “softwood”

  • Thomas K. BaderEmail author
  • Karin Hofstetter
  • Christian Hellmich
  • Josef Eberhardsteiner


Wood is an anisotropic, hierarchically organized material, and the question how the hierarchical organization governs the anisotropy of its mechanical properties (such as stiffness and strength) has kept researchers busy for decades. While the honeycomb structure of softwood or the chemical composition of the cell wall has been fairly well established, the mechanical role of the cell wall water is less understood. The question arises how its capability to carry compressive loads (but not tensile loads) and its pressurization state affect mechanical deformations of the hierarchical composite “wood”. By extending the framework of poro-micromechanics to more than two material phases, we here provide corresponding answers from a novel hierarchical set of matrix-inclusion problems with eigenstresses: (i) Biot tensors, expressing how much of the cell wall water-induced pore pressure is transferred to the boundary of an overall deformation-free representative volume element (RVE), and (ii) Biot moduli, expressing the porosity changes invoked by a pore pressure within such an RVE, are reported as functions of the material’s composition, in particular of its water content and its lumen space. At the level of softwood, where we transform a periodic homogenization scheme into an equivalent matrix-inclusion problem, all Biot tensor components are found to increase with decreasing lumen volume fraction. A further research finding concerns the strong anisotropy of the Biot tensor with respect to the water content: Transverse components increase with increasing water content, while the relationship “longitudinal Biot tensor component versus volume fraction of water within the wood cell wall” exhibits a maximum, representing a trade-off between pore pressure increase (increasing the longitudinal Biot tensor component, dominantly at low water content) and softening of the cell wall (reducing this component, dominantly at high water contents). Soft cell wall matrices reinforced with very stiff cellulose fibers may even result in negative longitudinal Biot tensor components. The aforementioned maximum effect is also noted for the Biot modulus.


Lignin Hemicellulose Pore Pressure Representative Volume Element Crystalline Cellulose 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Concentration tensor of phase r


Pore pressure-related influence tensor of phase r


Amorphous cellulose


Homogenized (‘macroscopic’) Biot tensor r


Biot tensor of phase r

\({{\mathbb C}^{\rm hom}}\)

Homogenized (‘macroscopic’) stiffness tensor


Stiffness tensor of phase r

Cr,i jkl

Components of the stiffness tensor of phase r




Crystalline cellulose


Cell wall material


Homogenized (‘macroscopic’) strain tensor


Fictitious strain tensor at infinity




Volume fraction of phase r in a representative volume element (RVE) of polymer network, cellulose, cell wall material, or softwood, respectively






Bulk modulus of phase r


Second-order unity tensor


Fourth-order unity tensor


Volumetric part of \({\mathbb{I}}\)


Deviatoric part of \({\mathbb{I}}\)


Longitudinal direction


Mean radial lumen diameter of softwood unit cell


Mean tangential lumen diameter of softwood unit cell

lI, lII, lIII

Lengths of the cell wall axes

lI,r, lII,r, lIII,r

Free lengths of the cell walls






Homogenized (‘macroscopic’) Biot modulus


Biot modulus of phase r


Pore pressure within cell wall


Eigenstress, eigenstrain (superscript)


Polymer network


Hill tensor of inclusion or phase r embedded in a matrix material s


Radial direction

\({\mathbb{S}_r^{{\rm Esh},s}}\)

Eshelby tensor of inclusion or phase r embedded in a matrix material s




Tangential direction


Cell wall thickness


Volume of representative volume element (RVE)


Volume inside representative volume element (RVE), occupied by phase r


Position vector inside a representative volume element


Inclination angle of the radial cell walls


Kronecker delta


average (‘microscopic’) strain tensor of phase r


Lagrangian porosity, i.e. volume of pores over volume of RVE of porous material in the (undeformed) reference condition


Initial porosity


Latitudinal and longitudinal angles (spherical coordinates)


Microfibril angle


Cross-sectional aspect ratio of softwood unit cell


Shear modulus of phase r


Microscopic displacement vector


Homogenized (‘macroscopic’) stress tensor


average (‘microscopic’) stress tensor of phase r

\({\langle (\cdot) \rangle}\)

Volume average of quantity (.)

\({\langle (\cdot) \rangle_{V_r}}\)

Volume average over phase r, of quantity (.)


First-order tensor contraction (inner product)


Second-order tensor contraction


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Thomas K. Bader
    • 1
    Email author
  • Karin Hofstetter
    • 1
  • Christian Hellmich
    • 1
  • Josef Eberhardsteiner
    • 1
  1. 1.Institute for Mechanics of Materials and StructuresVienna University of TechnologyViennaAustria

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