How the composite answers the fibre

Uniaxial tension along x. Rotate the fibres in the xy-plane. A reduced fibre is unstrained at an angle that depends only on the matrix Poisson ratio; there E_x^c = E_m. A solid fibre is neutral only for d_f/L ≲ 0.01. Fibres at 90° can still stiffen slightly. A random packing is Poisson-neutral at ν_m ≈ 0.25. The plots follow Goudarzi and Simone, Int. J. Solids Struct. 156–157 (2019).

Fig. 7a · E_x^c / E_m vs θ_z

Fig. 7b · θ_n(ν_m)

Figs. 4, 11 & 13 · E_x^c vs d_f/L at θ_n

Figs. 5 & 10 · slip and axial strain

Fig. 8 · G_xy^c / G_m vs θ_z

Fig. 12 · G_xy vs d_f/L at 0°

Fig. 15 · 2d single vs double interface

Figs. 20–21 · Poisson neutrality

ε_f / ε_x = cos²θ − ν sin²θ

θ_z ν_m v_f E_f/E_m d_f/L interface packing

Navy: reduced / single-interface. Orange: solid fibre or double-interface. On θ_n(ν): navy is 3d reduced = Wang plane stress; gray is plane strain. At θ_n a reduced fibre sits on E_m; a solid fibre does so only for d_f/L ≲ 0.01. A double interface can drop below E_m. Random packing: ν_iso is independent of volume fraction at ν_m ≈ 0.25.

How the composite answers the fibre

Fig. 7a · Exc / Em vs θz

Fig. 7b · θn(νm)

Figs. 4, 11 & 13 · Exc vs df/L at θn