Compressional Energy of the Random Fiber Assembly

Abstract

The compressional mechanism of a random fiber assembly is analyzed by an energy method. An infinitesimal fiber segment, which is bounded by two neighboring contact points, is chosen as the unit bending element. The geometry of this element is characterized by its arc length, curvature, and orientation. The change in bending energy of each fiber segment due to the compression of the assembly is derived in terms of the compressional strain and the Poisson's ratio of the assembly. The summation of each energy contribution is done using a continuous joint probability density function of the length and orientation of the segments.

Elasticity of planar fiber networks

“…In this model, the wool assembly was treated as a layered system with fiber bending deflection between neighboring contacts, while no fiber elongation, contraction, or torsion were considered. Based largely on this scheme, several refined models have been proposed [17][18][19][20][21][22][23][24][25][26] to account for fiber slippage, large deformation, and hybrid fiber systems. The fundamental assumption of these models is the constant force distribution across any FN section, which may conflict with the deformation compatibility of fiber segments in the assembly.…”

Cross-linked fiber network embedded in an elastic matrix

“…For a uniaxial deformation with ε 11 = ε and ε 22 = ε 33 = −ν m ε, and assuming that the fiber orientation distribution is uniform, one may write the strain energy density of an affinely deforming network as [20,21]: $$u_{aff}^n=\frac{1}{4}\rho E_f{A}_f{\int }_0^{\pi }\text{sin}\phi {true(\sqrt{{(1-{\nu }^m\epsilon )}^2\text{sin}{\phi }^2+{(1+\epsilon )}^2\text{cos}{\phi }^2}-1true)}^{2}d\phi =\frac{1}{4}\rho E_f{A}_{f}Ffalse(\epsilon false)$$ where φ is the angle made by the direction of given fiber with the x 1 axis and F (ε) represents the integral. Requiring that this energy density equals $1/2E_{aff}^n{\epsilon }^2$, leads to $$E_{aff}^n=\frac{1}{2}\rho E_f{A}_{f}Ffalse(\epsilon false)/{\epsilon }^2$$ …”

Softening in random networks of non-identical beams

“…Similar models based on the affine deformation assumption have been derived for athermal networks (e.g. Lee and Carnaby, 1992; Wu and Dzenis, 2005). Further developments of the theory of molecular networks partially relaxed the affine deformation restriction while considering a small number of chains as being representative for the entire network.…”