Bridge laws

If the crack is considered to be a wake zone and p = p(u, t) in Eqns. (19) and (21) then, for a stationary crack, both the crack shape and Keff are time dependent, given respectively by u(x,t) and KejAt) from Eqns. (19) and (21) for the case of partial shielding. Cox and Rose35 recently considered an elastic time-dependent bridging law of the form... 

Through-thickness reinforcement, z-pinning, bridging law, mode 1 delamination, Double-Cantilever-Beam (DCB), pull-out. 

In Eq (7), the bridging force, Pi, is a function of the flexural displacement z(xi) which is given by the bridging law. Thus, it is mathematically difficult to obtain a closed-form solution of Eq. (7). Instead, an iteration method is used to obtain a numerical solution. In the iterative calculation, we add the displacement, 5, and the crack length, h, step by step by a tiny increment. In the first step, we give a tiny increase in the crack length. 

Substituting the obtained displacement, z(xi), into the pin bridging law, the bridging force. Pi, can be obtained. Increasing the applied displacement, S=5 a5, and applying the bridging force. Pi from the previous step to Eqs. (7) and (8), the deflection of the beam under present applied displacement, S=5 AS, can be approximately calculated ... 

The new displacement of the first pin z(x[) ean be obtained from Eq. (11). Substituting z(x ) into the bridging law, a new bridging force at this step can be obtained. This bridging force is subsequently applied to caleulate the defleetion of the beam in the next step. If the number of the bridging z-pins is N when the delamination crack length is Lc, the differential equation of the deflection curve in each interval beeomes ... 

The main novelties introduced in the DCB testing concern a) the loading device, b) the specimen geometry, c) the measurement scheme, and d) the determination of Gic and the bridging law from experimental data. 

Fracture toughness, delamination, 3D composite, bridging law, crack opening displacement. 

In [3], it was noticed that DCB specimens with a higher bending stiffness required a longer crack extension before the steady-state crack growth resistance was attained. In [ I], the concept of a bridging law was introduced to characterise the f -curve for DCB specimens. 

In [1, 2], it was shown that the bridging law for a UD composite in the presence of large-scale bridging can be obtained by the formula... 

Fig, 12. Bridging law (traction at the crack faces versus the crack opening)... 

The additional parameter — the crack opening displacement at the initial precrack (ICOD) — was measured to obtain the bridging law, which is a material characteristic independent of the specimen geometry. It is recommended to use the graphs Gic (ICOD) instead of the traditional / -curves Gic(Afl) for characterizing the delamination fracture resistance of composites with extensive bridging. 

The connection with older ideas about reduction will be obvious [SI] and [S2] are the generalized successors of the "bridge laws" or "linking propositions" that limit the relation to a token-token instead of a type-type relationship. The hope was that supervenience relations would have all the advantages of reduction with no disadvantages. Reduction has the advantage that it explains why events are not causally overdetermined. The disadvantage of this picture is that it seems to undermine the autonomy of the S-discourse unduly. The literature on supervenience is dominated by this tension between determinacy and autonomy. 

Interdiscourse relations (bridge laws, supervenience, or emergence relations) are empirical ceteris paribus regularities, not metaphysical necessities—the only circumstances in which they might apply strictly, are model situations, completely isolated from the rest of the universe... 

Because of the difficulties associated with Nagel s bridge-law model of reduction and the success of multiple realization arguments against type-identity materialism, Kim has come to reject type-identity materialism in favor of what he ealls functional reduction. The functional reduction of a property requires three steps ... 

Lindhagen JE, Berglund LA. Application of bridging law concepts to short fiber composites Part 1 DCB test procedures for bridging law and fiacture energy. Compos Sci Technol 2000 60 871-83. http //dx.doi.org/10.1016/S0266-3538(00)00004-X. 

Subsequently, the terms in the specialised theory 7 are replaced with their correspondents via bridge laws, generating a theory 7. ... 

Horgan T (1978) Supervenient bridge laws. Philos Sci 45(2) 229-249 Kemeny JG, Oppenheim P (1956) On reduction. Philos Stud VII 6-19... 

The existence of functional properties in chemistry also impacts reductionism about chemical laws. On Nagel s concept of reduction (1961), chemistry would reduce to physics if one could derive all the laws of chemistry from the laws of physics together with bridge laws connecting the terms in the vocabularies of the two sciences. The philosophical literature on Nagelian reduction has long debated... 

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