Four Hamiltonian Systems

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Without loss of generality, for this numerical illustration, it is assumed that the interfacial shear stress s is of order of the DFRC shear modulus (G) and G = {(EL + ET)/2}/(2 + 2v), where (EL + ET)/2 represents the ‘effective stiffness’ of the DFRC this section, simple arguments are presented to relate Vf to q. For simplicity, let q describes the centre-to-centre distance between adjacent fibres. For discontinuous fibres, one invokes two assumptions: (1) that there is a sufficiently large number of fibres, such that at the microscopic length scale, the lateral arrangement appears somewhat regular (as indicated previously in Sect.

5 shows a sketch of the stress generated in the fibre versus the displacement of two adjacent atomic planes. At equilibrium, the interatomic plane separation distance is denoted by b0; the net force is zero (and so is the corresponding stress). Of note, b0 has been identified with the length of a simple cubic closed-pack unit cell (Sect. 5). As the interatomic plane distance increases, the force rises to a maximum and then falls back to zero. To model this force–displacement relationship, one notes that the area under the curve is identified with the work of fracture.

Different bundles of the fibres are arranged laterally in the square packing configuration (Fig. 9b) and (2) that, longitudinally, most fibres would be lined with ends very close to the next fibre. Note that for the square packing configuration, the region abcd (area = q2) defined in Fig. 9b encloses one fibre; the cross-sectional area of the fibre is given by pr2f . 3 Structure 33 Fig. 8 Graphs of the a stress (r) versus strain (e) for varying fibre orientation angle, h, and b fracture strength, rU, versus fibre orientation angle, h, of discontinuous carbon fibres reinforced magnesium composites.

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