On the Rheological Properties of Paper

Part II Vibrating reed method

Abstract

From previous report, by vibrating reed method, we can also obtain Young's modulus and viscosity of paper'
According to simple mathematical consideration, we can correlate it with theoretical equation derived from parallel linked Maxwell element, assuming distribution function of relaxation time at logarithmic scale is constant.
From this consideration
1) Youngs'modulus at vibrating reed method is apperent.Youngs' modulus or Youngs modulus at strain is zero in stress-strain curve with constant rate of elongation.
2) η from vibrating reed method from 20 c.p.s.to 100 c.p.s.is propartional to second elastic constant K in stress-strain curve with constant rate of elongation within 1 % strain.
In this equation 12 of 10⁶∼10⁷ order poise at 20 c.p.s.∼100 c.p.s.correspond to 10¹².∼10¹³ poise order viscosity at 10-⁵/sec order rate of elongation assuming 3 element model which is composed of 2 springs and one daslipot for paper.
3) η change at vibrating reed metho d varying frequency corresponds to change of K varying rate of elongation, that is K=B/2α (1/τ₁-1/τ₂) where B is contant α, rate of elongation τ₁, τ₂ relaxation time but it does not fit quantitatively experimental data
Let us take following two assumption
a) τ₂/τ₁=k ατ₁=constant when α is varied
b) τ₂/τ₁=k ατ₁=k₁α1/n when α is varied
k, k₁ constant n, constan n>1
from a) stress is independent of a it does not fit experimental data.
from b) S=γBl_nk-B2k₁α¹/n k-1/k
when τ₁>t<<τ₂ t=τ/α
S=γB((1-C+l_nk+l_nk₁)-k_nγ+1/n l_nα)
when τ₁<<t<<τ₂
these equations fit experimental data considerably qualitatively from Andersson and sjoberg, and from b) η (reed method) -υ (frequency) relation explains well experimental data.
4) from b) S-α plot represent the flow type of paper and it is concluded that paper has flow type of pseudo-plastic or plastic flow where strain is small
1) Y. Fujii : Japanese Tappi Vol.10.No.1
2) O. Andersson and Sjoberg : Svensk Paperstidning 1953 16 Aug.