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**Extra info for Four Hamiltonian Systems**

**Example text**

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.