Abstract
A simple statistical analysis of elastic modulus field of paper sheet has been made on the basis of the staler stress-strain equation proposed by C.T.J. Dodson7)8).
Paper sheet was assumed to be an inhomogeneous body whose elastic state at a point x is characterized by the equation σ(x)=Kl(x)ε(x), where σ is the isotropic (two dimensional) stress, ε the areal dilatation and Kl the local elastic modulus for areal dilatation. Under the condition of statistical homogeneity9) of the elastic modulus field, the effective modulus (overall modulus10)11)) K, defined as the ratio of mean stress to mean strain, was shown to be equal to the reciprocal of the mean elastic compliance as the zero-th order approximation. By assuming that the local modulus Kl(x) is proportional to the mass density at a point x, the effective modulus K can be expressed as
K=MH2/H+1=Mf(H),
where M is the proportional constant and H the mean number of overlapping fibers12) which corresponds to the basis weight. The function f(H) is a nonlinear function of H, but in the range of sufficiently high H(>2) it shows almost linear dependence on H, and the gradient of f(H) is approximately unity.
The areal dilatation modulus K and other in-plane moduli were determined as a function of the basis weight by use of a biaxial tensile tester for the sheet made from the bleached kraft pulp (spruce) beaten in various degrees. The results showed that with increasing the revolution number of PFI mill the gradient of the linear regression equation of K against the basis weight increases, while the corresponding intercept on the basis weight axis, which is caused by the sheet inhomogeneity, does not vary markedly.
The specific elastic modulus defined as the gradient of the regression line, which can be regarded as the mean local modulus per unit basis weight, was proposed as a new characteristic quantity of paper elasticity, and the effect of beating on in-plane specific moduli was also discussed.
