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Official Journal of the Japan Wood Research Society

A mathematical verification of the reinforced-matrix hypothesis using the Mori-Tanaka theory

Abstract

This article presents a theoretical verification of the reinforced-matrix hypothesis derived from tensor equations, σ W = σ f + σ m and ε W = ε f = ε m (Wood Sci Technol 32:171–182, 1998; Wood Sci Technol 33:311–325, 1999; J Biomech Eng 124:432–440, 2002), using classical Mori-Tanaka theory on the micromechanics of fiber-reinforced materials (Acta Metall 21:571–574, 1973; Micromechanics — dislcation and inclusions (in Japanese), pp 141–147, 1976). The Mori-Tanaka theory was applied to a small fragment of the cell wall undergoing changes in its physical state, such as those arising from sorption of moisture, maturation of wall components, or action of an external force, to obtain 〈σ AD = ϕ·〈σ FI + (1−ϕ)·〈σ MD−I. When the constitutive equation of each constituent material was applied to the equation 〈σ AD = ϕ·〈σ FI + (1−ϕ)·〈σ MD−I, the equations σ W = σ f + σ m and ε W = ε f = ε m were derived to lend support to the concept that two main phases, the reinforcing cellulose microfibril and the lignin-hemicellulose matrix, coexist in the same domain. The constitutive equations for the cell wall fragment were obtained without recourse to additional parameters such as Eshelby’s tensor S and Hill’s averaged concentration tensors AF and AM. In our previous articles, the coexistence of two main phases and σ W = σ f + σ m and ε W = ε f =ε m had been taken as our starting point to formulate the behavior of wood fiber with multilayered cell walls. The present article provides a rational explanation for both concepts.

References

  1. Cave ID (1968) Anisotropic elasticity of the plant cell wall. Wood Sci Technol 6:284–292

    Article  Google Scholar 

  2. Page DH, El-Hosseiny F, Winkler K, Lancaster APS (1977) Elastic modulus of single wood pulp fibers. TAPPI 60:114–117

    Google Scholar 

  3. Salmen L, De Ruva A (1985) A model for the prediction of fiber elasticity. Wood Fiber Sci 17:336–350

    CAS  Google Scholar 

  4. Meylan BA (1972) The influence of microfibril angle on the longitudinal shrinkage-moisture content relationship. Wood Sci Technol 6:293–301

    Article  Google Scholar 

  5. Cave ID (1972) A theory of shrinkage of wood. Wood Sci Technol 6:284–292

    Article  Google Scholar 

  6. Yamamoto H, Okuyama T (1988) Analysis of the generation process of growth stresses in cell walls (in Japanese). Mokuzai Gakkaishi 34:788–793

    Google Scholar 

  7. Yamamoto H (1998) Generation mechanism of growth stresses in wood cell walls: roles of lignin deposition and cellulose microfibril during cell wall maturation. Wood Sci Technol 32:171–182

    Article  CAS  Google Scholar 

  8. Almeras T, Gril J, Yamamoto H (2005) Modelling anisotropic maturation strains in wood in relation with fibre boundary conditions, microstructure and maturation kinetics. Holzforschung 59:347–353

    Article  CAS  Google Scholar 

  9. Kojima Y, Yamamoto H (2005) Effect of moisture content on the longitudinal tensile creep behavior of wood. J Wood Sci 51:462–467

    Article  Google Scholar 

  10. Barber NF, Meylan BA (1964): The anisotropic shrinkage of wood. A theoretical model. Holzforschung 18:146–156

    Article  Google Scholar 

  11. Yamamoto H (1999) A model of anisotropic swelling and shrinking process of wood (part 1). Generalization of Barber’s wood fiber model. Wood Sci Technol 33:311–325

    Article  CAS  Google Scholar 

  12. Yamamoto H, Kojima Y, Okuyama T, Avasolo WP, Gril J (2002) Origin of the biomechanical properties of wood related to the fine structure of the multi-layered cell wall. J Biomech Eng 124:432–440

    Article  CAS  PubMed  Google Scholar 

  13. Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc London A 241:376–396

    Article  Google Scholar 

  14. Eshelby JD (1959) The elastic field outside an ellipsoidal inclusion. Proc R Soc London A 252:561–569

    Article  Google Scholar 

  15. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574

    Article  Google Scholar 

  16. Cave ID (1972) Swelling of a fiber reinforced composite in which the matrix is water reactive. Wood Sci Technol 6:157–161

    Article  Google Scholar 

  17. Mura T, Mori T (1976) Micromechanics — dislocation and inclusions (in Japanese). Baifukan, Tokyo, pp 141–147

    Google Scholar 

  18. Clair B, Almeras T, Yamamoto H, Okuyama T, Sugiyama J (2006) Mechanical behavior of cellulose microfibrils in tension wood, in relation with maturation stress generation. Biophys J 91:1128–1135

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Hill R (1963) Elastic properties of reinforced solids — some theoretical principles. J Mech Phys Solids 11:357–372

    Article  Google Scholar 

  20. Cave ID (1978) Modelling moisture-related properties of wood. Part I: properties of the wood constituents. Wood Sci Technol 12:75–86

    Article  Google Scholar 

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Correspondence to Hiroyuki Yamamoto.

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Yamamoto, H., Almèras, T. A mathematical verification of the reinforced-matrix hypothesis using the Mori-Tanaka theory. J Wood Sci 53, 505–509 (2007). https://doi.org/10.1007/s10086-007-0897-5

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  • DOI: https://doi.org/10.1007/s10086-007-0897-5

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