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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
Article

Abstract

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.

Keywords

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

\({\mathbb{A}_r}\)

Concentration tensor of phase r

ar

Pore pressure-related influence tensor of phase r

amocel

Amorphous cellulose

bhom

Homogenized (‘macroscopic’) Biot tensor r

br

Biot tensor of phase r

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

Homogenized (‘macroscopic’) stiffness tensor

\({\mathbb{c}_r}\)

Stiffness tensor of phase r

Cr,i jkl

Components of the stiffness tensor of phase r

cel

Cellulose

crycel

Crystalline cellulose

cwm

Cell wall material

E

Homogenized (‘macroscopic’) strain tensor

E

Fictitious strain tensor at infinity

ext

Extractives

\({\tilde{f}_r,\bar{f}_r,f_r,\hat{f}_r}\)

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

H2O

Water

hemcel

Hemicellulose

kr

Bulk modulus of phase r

1

Second-order unity tensor

\({\mathbb{I}}\)

Fourth-order unity tensor

\({\mathbb{J}}\)

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

\({\mathbb{K}}\)

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

L

Longitudinal direction

lR

Mean radial lumen diameter of softwood unit cell

lT

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

lig

Lignin

lum

Lumen

Nhom

Homogenized (‘macroscopic’) Biot modulus

Nr

Biot modulus of phase r

p

Pore pressure within cell wall

p

Eigenstress, eigenstrain (superscript)

polynet

Polymer network

\({\mathbb{P}_r^s}\)

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

R

Radial direction

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

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

SW

Softwood

T

Tangential direction

t

Cell wall thickness

VRVE

Volume of representative volume element (RVE)

Vr

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

x

Position vector inside a representative volume element

α

Inclination angle of the radial cell walls

δij

Kronecker delta

εr

average (‘microscopic’) strain tensor of phase r

\({\phi}\)

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

\({\phi_0}\)

Initial porosity

\({\theta,{\varphi}}\)

Latitudinal and longitudinal angles (spherical coordinates)

\({\bar{\theta}}\)

Microfibril angle

λ

Cross-sectional aspect ratio of softwood unit cell

μr

Shear modulus of phase r

ξ

Microscopic displacement vector

Σ

Homogenized (‘macroscopic’) stress tensor

σr

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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References

  1. 1.
    Auriault J.L., Sanchez-Palencia E.: Etude du comportement macroscopique d’un milieu poreux saturè dèformable [study of macroscopic behavior of a saturated deformable medium]. J. de Mé 16, 575–603 (1977) in FrenchMathSciNetzbMATHGoogle Scholar
  2. 2.
    Benveniste Y.: A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157 (1987)CrossRefGoogle Scholar
  3. 3.
    Böhm H.: A short introduction to continuum micromechanics. In: Böhm, H. (eds) Mechanics of Microstructure Materials. CISM Lecture Notes No. 464, pp. 1–40. Springer, New York (2004)Google Scholar
  4. 4.
    Böhm H., Han W., Eckschlager A.: Multi-inclusion unit cell studies of reinforcement stresses and particle failure in discontinuously reinforced ductile matrix composites. Comput. Meth. Eng. Sci. 5(1), 5–20 (2004)zbMATHGoogle Scholar
  5. 5.
    Cecchi A., Sab K.: Out of plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A Solids 21(5), 715–746 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Chateau X., Dormieux L.: Approche microméchanique du comportement d’un milieu poreux non saturé [Micromechanical approach for the behavior of a non-saturated porous medium]. Comptes Rendus de l’Académie des Sciences Série IIb 326, 533–538 (1998) in FrenchzbMATHGoogle Scholar
  7. 7.
    Chateau X., Dormieux L.: Micromechanics of saturated and unsaturated porous media. Int. J. Numer. Anal. Meth. Geomech. 26, 831–844 (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cousins W.: Elastic modulus of lignin as related to moisture content. Wood Sci. Technol. 10, 9–17 (1976)CrossRefGoogle Scholar
  9. 9.
    Cousins W.: Young’s modulus of hemicellulose as related to moisture content. Wood Sci. Technol. 12, 161–167 (1978)CrossRefGoogle Scholar
  10. 10.
    Cousins W., Armstrong R., Robinson W.: Young’s modulus of lignin from a continuous indentation test. J. Mater. Sci. 10, 1655–1658 (1975)CrossRefGoogle Scholar
  11. 11.
    Coussy O.: Poromechanics. Wiley, Chistester (2004)Google Scholar
  12. 12.
    Da Silva A., Kyriakides S.: Compressive response and failure of balsa wood. Int. J. Solids Struct. 44, 8685–8717 (2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dormieux L., Kondo D., Ulm F.J.: Microporomechanics. Wiley, Chichester (2006)CrossRefzbMATHGoogle Scholar
  14. 14.
    Dormieux, L., Ulm, F.J. (eds.): Applied Micromechanics of Porous Materials (CISM Courses and Lectures No. 480), Springer, Wien, New York (2004)Google Scholar
  15. 15.
    Dormieux L., Molinari A., Kondo D.: Micromechanical approach to the behavior of poroelastic materials. J. Mech. Phys. Solids 50, 2203–2231 (2004)CrossRefGoogle Scholar
  16. 16.
    Dvorak G.J., Benveniste Y.: Transformation field analysis of inelastic composite materials. Proc. Royal Soc. Lond. A 437, 291–310 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Easterling K., Harryson R., Gibson L., Ashby M.: On the mechanics of balsa and other woods. Proc. Royal Soc. Lond. A 383, 31–41 (1982)CrossRefGoogle Scholar
  18. 18.
    Eichhorn S., Young R.: The Young’s modulus of a microcrystalline cellulose. Cellulose 8, 197–207 (2001)CrossRefGoogle Scholar
  19. 19.
    Eshelby J.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Royal Soc. Lond. A 241, 376–396 (1957)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Fengel D., Wegener G., Wood G.: Wood–Chemistry, Ultrastructure, Reactions. 2nd edn. De Gruyter, Berlin (1984)Google Scholar
  21. 21.
    Friebel C., Doghri L., Legat V.: General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions. Int. J. Solids Struct. 43(9), 2513–2541 (2006)CrossRefzbMATHGoogle Scholar
  22. 22.
    Fritsch A., Hellmich C., Dormieux L.: Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: Experimentally supported micromechanical explanation of bone strength. J. Theor. Biol. 260, 230–252 (2009)CrossRefGoogle Scholar
  23. 23.
    Gibson L., Ashby M.: Cellular Solids, Structure and Properties. 2nd edn. Cambridge University Press, Cambridge (1997)Google Scholar
  24. 24.
    Gillis P.P.: Orthotropic elastic constants of wood. Wood Sci. Technol. 6, 138–156 (1972)CrossRefGoogle Scholar
  25. 25.
    Harada H.: Cellular ultrastructure of woody plants. In: Côté, W. (eds) Ultrastructure and organization of gymnosprem cell walls, pp. 215–233. Syracuse University Press, Syracuse (1965)Google Scholar
  26. 26.
    Hashin Z., Rosen B.W.: The elastic moduli of fiber-reinforced materials. J. Appl. Mech. 31, 223–232 (1964)Google Scholar
  27. 27.
    Hellmich C., Barthélémy J.-F., Dormieux L.: Mineral-collagen interactions in elasticity of bone ultrastructure—a continuum micromechanics approach. Eur. J. Mech. A Solids 23, 783–810 (2004)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hellmich C., Ulm F.-J.: Drained and undrained poroelastic properties of healthy and pathological bone: A poro-micromechanical investigation. Transp. Porous Media 58(3), 243–268 (2005)CrossRefGoogle Scholar
  29. 29.
    Hill R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)CrossRefzbMATHGoogle Scholar
  30. 30.
    Hofstetter K., Hellmich C., Eberhardsteiner J.: Development and experimental validation of a continuum micromechanics model for the elasticity of wood. Eur. J. Mech. A Solids 24, 1030–1053 (2005)CrossRefzbMATHGoogle Scholar
  31. 31.
    Hofstetter K., Hellmich C., Eberhardsteiner J.: The influence of the microfibril angle on wood stiffness: a continuum micromechanics approach. Comput. Assist. Mechan. Eng. Sci. 13, 523–536 (2006)zbMATHGoogle Scholar
  32. 32.
    Hofstetter K., Hellmich C., Eberhardsteiner J.: Micromechanical modeling of solid-type and plate-type deformation patterns within softwood materials. A review and an improved approach. Holzforschung 61, 343–351 (2007)CrossRefGoogle Scholar
  33. 33.
    Hofstetter K., Hellmich C., Eberhardsteiner J., Mang H.A.: Micromechanical estimates for elastic limit states in wood materials, revealing nanostructural failure mechanisms. Mechan. Adv. Mater. Struct. 15(6–7), 474–484 (2008)CrossRefGoogle Scholar
  34. 34.
    Holmberg S., Persson K., Peterson H.: Nonlinear mechanical behavior and analysis of wood and fibre materials. Comput. & Struct. 72, 459–480 (1999)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kahle E., Woodhouse J.: The influence of cell geometry on the elasticity of softwood. J. Mater. Sci. 29, 1250–1259 (1994)CrossRefGoogle Scholar
  36. 36.
    Kaminski M.M.: Computational Mechanics of Composite Materials: Sensitivity, Randomness, and Multiscale Behaviour. Springer, Berlin (2005)Google Scholar
  37. 37.
    Kanit T., Forest S., Galliet I., Mounoury V., Jeulin D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679 (2003)CrossRefzbMATHGoogle Scholar
  38. 38.
    Kollmann F.: Technologie des Holzes und der Holzwerkstoffe [Technology of Wood and Wood Products], 2nd Edition. Vol. 1. Springer Verlag, Berlin Heidelberg New York (1982) in GermanGoogle Scholar
  39. 39.
    Kollmann F., Côté W.: Principles of Wood Science and Technology, Vol. 1. Springer, Berlin (1968)Google Scholar
  40. 40.
    Laws N.: The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. J. Elast. 7(1), 91–97 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Mark R.: Cell Wall Mechanics of Tracheids. 2nd edn. Yale University Press, New Haven (1967)Google Scholar
  42. 42.
    Michel J.C., Moulinec H., Suquet P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Meth. Appl. Mechan. Eng. 172, 109–143 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Mori T., Tanaka K.: Average stress in the matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973)CrossRefGoogle Scholar
  44. 44.
    O’Sullivan A.: Cellulose: the structure slowly unravels. Cellulose 4, 173–207 (1997)CrossRefGoogle Scholar
  45. 45.
    Ostoja-Starzewski M.: Material spatial randomness: from satistical to representative volume element. Probab. Eng. Mechan. 21, 112–132 (2006)CrossRefGoogle Scholar
  46. 46.
    Papka S.D., Kyriakides S.: In-plane biaxial crushing of honeycombs–Part II: analysis. Int. J. Solids Struct. 36, 4397–4423 (1999)CrossRefzbMATHGoogle Scholar
  47. 47.
    Pedersen O.B.: Thermoelasticity and plasticity of composites-I. Mean field theory. Acta Metall. 31, 1795–1808 (1983)CrossRefGoogle Scholar
  48. 48.
    Salmen L., Burgert I.: Cell wall features with regard to mechanical performance. A review, COST Action E35 2004–2008: wood machining micromechanics and fracture. Holzforschung 63, 121–129 (2009)CrossRefGoogle Scholar
  49. 49.
    Scheiner St., Hellmich C.: Continuum microviscoelasticity model for aging basic creep of early-age concrete. J. Eng. Mech. (ASCE) 135(4), 307–323 (2009)CrossRefGoogle Scholar
  50. 50.
    Stamm A.J.: Wood and Cellulose Science. Roland Press, New York (1964)Google Scholar
  51. 51.
    Suquet P.: Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palencia, E., Zaoui, A. (eds) Homogenization Techniques for Composite Media. Lecture Notes in Physics. No. 272, pp. 193–278. Springer, Wien (1987)CrossRefGoogle Scholar
  52. 52.
    Suquet, P. (eds): Continuum Micromechanics. Springer, Wien (1997)zbMATHGoogle Scholar
  53. 53.
    Tang R.: The microfibrillar orientation in cell-wall layers of virginia pine tracheids. Wood Sci. Technol. 5, 181–186 (1973)Google Scholar
  54. 54.
    Tashiro K., Kobayashi M.: Theoretical evaluation of three-dimensional elastic constants of native and regenerated celluloses: role of hydrogen bonds. Polymer 32(8), 1516–1526 (1991)CrossRefGoogle Scholar
  55. 55.
    Thompson M., Willis J.: A reformation of the equations of anisotropic poroelasticity. J. Appl. Mechan. 58, 612–616 (1991)CrossRefzbMATHGoogle Scholar
  56. 56.
    Young R., Lovell P.: Introduction to Polymers. 2nd edn. Chapman & Hall, London (1991)Google Scholar
  57. 57.
    Zaoui A.: Continuum micromechanics: survey. J. Eng. Mechan (ASCE). 128(8), 808–816 (2002)CrossRefGoogle Scholar
  58. 58.
    Zhang K., Duan H., Karihaloo B.L., Wang J.: Hierarchical, multilayered cell walls reinforced by recycled silk cocoons enhance the structural integrity of honeybee combs. Proc. Nat. Acad. Sci. 107(21), 9502–9506 (2010)CrossRefGoogle Scholar

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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