纖維物性論(IB)纖維集合体の弾性論的考察

第12報　繊維束の引張リ現象に対する弾性論的取扱の総括

抄録

For quatitative information as to the elastic behaviour of fibre bundles, for example card sliver, top and roving, the present study was undertaken to establish an elastic model for a fibre system built from an elementary unit such as bar-like fibre which may be in itself deformable and mutually slippery. The treatments for a bundle (length L, bulk density ρ, linear density k) of parallel units (elementary fibres of length l, diameter b, density ρ, Young's modulus ε_??_, Poisson's ratio σ_??_ and coefficient of friction f, in the sense of all “effective”) are developed on assumptions that, -
(1) When stress S in a stretched bundle is elastic equilibrium to the external force P, the apparent extension of bundle γ would occur in such a fashion that γ becomes linear combination of the elongation of bar-like unit e as the strain of a line element in the elastic medium of constants ε, σ subject to stress S, and the mutual slippage of the centre of each unit, e as the displacement of a point in elastic medium of constants ε, σ subject to the same stress S,
(2) Distribution of the position of elementary units is uniform over the full range, and digstribution of their direction θ is symmetrical about the long axis of bundle.
Then from assumption (1) we obtaine the following relations: where ε, ε are Young's moduli, σ, σ are Poissonn's ratios in two sets of assumed elastic medie respectively. Here, specific length of bar-like unit to gauge length of bundle λ, fullness of units in the cross-section ω and frictional factor of inter-units α are defined by assumption (2):
Then, according to the Hooke's law appropriate for this bundle with two sets of assumed elastic constants, the observed Young's modulus E and Poisson's ratio ∑ may be expressed as:
Now, when we expand E and ∑ in terms of power of extension γ by means of above relations in which ε, α and λ are the functions of γ, we may derive the analytical formulae of stressstrain curve S(γ), lateral contraction curve β(γ), change of sectional area A(γ), and change of bulk density ρ(γ) by the definitions of eqs. (12.12) to (12.15) respectively. We therefore obtain load-elongation curve from expression P(γ)=A(γ)S(γ), and find the breaking strength P_M, . breaking elongatiou γ_M as the maximum values of P(γ) curve. These computed analytical formulae mainly depend upon the particular values of λ and α to which their numerical values of bundle's charactor correspond. Several examples given are by the eqs. (12.19) to (12.38) for the typical region of values of λ and α.
Comparisons of the computed and experimental values as to the nature of unit fibre, for various viscose staple slivers with charactors given in Table II, are made in Table I and III. A slight disagreement of theoretical and experimental values of unit fibre properties is introduced by regarding the elementary bar-like unit as coressponding to only one fibre in bundle. Therefore we see that by the interpretation in term of an elementary unit as a group of fibres, the assumptions upon which the computations were based are essentially correct to the elastic model of fibre bundle.
Other interesting expressions deduced from this model are those of hysteresis energy loss H(γ_r) and elastic resilience φ(γ_r) which appeared with cyclic elongation γ_r. In the range of small values of γ_r, H(γ_r) and φ(γ_r) are given by eqs, (12.43) to (12.46)