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 ^A〉_D = ϕ·〈σ ^F〉_I + (1−ϕ)·〈σ ^M〉_D−I. When the constitutive equation of each constituent material was applied to the equation 〈σ ^A〉_D = ϕ·〈σ ^F〉_I + (1−ϕ)·〈σ ^M〉_D−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 A^F and A^M. 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.