Third-order stiffnesses

The general strategy follows ideas by Bragger (1964), Tiersten (1971), Nelson (1979), and Adam et al. (1988) which is to start from the idea of stieking to the symbols of the second-order quantities arrd just augment them by additional indices according to the variable irrvolved in the additional derivative. The procediue is most easily demonstrated with the third-order stiffnesses... 

The second derivatives of H2 or G2 with respect to the strain coordinates still are functions of all the variables and might be called stiffness functions since their values at the reference state are per definition the second-order stiffness constants. In this sense the third-order stiffnesses are a measure of the strain dependence of the stiffnesses. The symbol A stands for a or and the symbol + for E or Dio indicate the thermal and electrical constraints in the manner used in Table 4.3. 

Deviations from the linearity of Hooke s law are represented to the next higher order by the third-order stiffnesses compliances Th se are... 

In principle, this has important consequences for experimenters who in one instance may find it easier to measure third-order stiffnesses and then have a means to calculate compliances from them. It has already been pointed ont that by this formalism it becomes possible to measure the influence of an electric field on elastic properties of a material and learn something about the deformation dependence of piezoelectricity by this process. 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

Neither the Runge-Kutta nor tiie Adams Bashforth methods can handle stiff differential equations well. The Adams-Moulton method is an implicit multistep method that can handle stiff problems better (stiff problems are dicussed later in this chapter). The two-step Adams-Moulton method (third-order accurate) is... 

Lee et al. [33] measured the elastic properties and intrinsic breaking strength of a freestanding monolayer graphene ribbon (GNR) by nanoindentation in an AFM. They found that the force-displacement behavior is nonUnearly elastic and yields second- and third-order elastic stiffness of 340 Nm (Nm = TPa nm) and... 

On the day of attack, 15 September 1944, the elaborate plan for using the flame tractors completely broke down. The ist Marine Regiment was stopped by stiff resistance just beyond the beach, and its flame tractor waited five hours for some kind of order. The flame vehicles with the other regiments were told to stand offshore out of danger. When the three flame tractors eventually landed they stood idle on the beaches, a result no doubt of extreme confusion and the unfamiliarity of the marines with the weapon. Inactivity on the second day was caused by the fact that the air compressor had not yet landed. The flame vehicles saw action on the third day, and from then on their commitment was regular. ... 

A relatively new class of polymers, the liquid-crystalline polymers, exhibits orientational order, i.e. alignment of molecules along a common director in the molten state. Liquid-crystalline polymers are used, after solidification, as strong and stiff engineering plastics and fibres. Functional liquid-crystalline polymers with unique electrical and optical properties are currently under development. The fundamental physical and rheological aspects of liquid-crystalline polymers are the third subject of this chapter (section 6.5). 

It is important to observe that only the first term in equation (137) depends on chain architecture and the lattice coordination number, and that both it and the second term cancel out of the entropy of mixing, leaving only the third term equal to Q/Qq. Consequently the theory would apply alike to solutions of flexible and stiff-chain polymers (except that above a certain concentration the latter would respond to packing constraints and minimize the free energy by separation of an ordered phase). The factorization of the combinatorial factors in equation (137) is the fundamental reason why the lattice calculation works at all, despite the extreme artificiality of picturing the chain as fitting a sequence of regular lattice sites with a definite coordination number. These aspects of the model simply disappear in the final result. The independence of the intermolecular factor also implies that the chain conformation should be independent of dilution Rq should be the same in pure liquid polymer as in solution. Naturally this rationale would not hold for dilute solutions, for which the intermolecular factor in equation (137) is not valid. 

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