Mechanical models generalized Maxwell

When the examination was over and the report submitted, a new tumult was raised. Kelvin opposed the theory in general. He could -understand nothing, he said, which could not be translated into a mechanical model. For this reason he had likewise rejected Maxwell s electromagnetic theory of light. Only the Dane submitted an enthusiastic judgment of the... 

Equation (21.10) is the general equation for the Maxwell mechanical model analogy for viscoelastic behavior. 

The behavior of a polymer system is so complicated that we cannot represent it with the response time of a single Maxwell element. In other words, the simple model described above cannot approach the behavior of a real system. In 1893, Weichert showed that stress-relaxation experiments could be represented as a generalization of Maxwell s equation. The mechanical model according to Weichert s formulation is shown in Figure 3.11 it consists of a large number of Maxwell elements coupled in parallel. 

This result provides a general definition of the relaxation time of a polymer and allows the relaxation time to be found easily from experimental data without recourse to a mechanical model. It can be used as a material property to give an indication of the time scale associated with viscoelastic response in a polymer and is indicative of the intrinsic viscosity of the polymer. It should again be noted that the relaxation time for a Maxwell model is related to the viscosity through the equation, x = ji/E. In a sense, the Maxwell model provides a defining relationship for the viscosity of a material. It will be shown later that a polymer possesses a distribution of relaxation times and that an individual chain can be thought of as having various relaxation times. 

It is possible using transform methods to convert viscoelastic problems into elastic problems in the transformed domain, allowing the wealth of elasticity solutions to be utilized to solve viscoelastic boundary value problems. Although there are restrictions on the applicability of this technique for certain types of boundary conditions (discussed further in Chapter 9), the method is quite powerful and can be introduced here by building on the framework provided by mechanical models. Recall the differential equation for a generalized Maxwell or Kelvin model,... 

For the very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

This chapter is devoted to describe the impact of metallic nanosphere to the multi-photon excitation fluorescence of Tryptophan, and little further consideration to multi-photon absorption process will be given, as the reader can find several studies in [11-14]. In section II, the nonlinear light-matter interaction in composite materials is discussed through the mechanism of nonlinear susceptibilities. In section III, experimental results of fluorescence induced by multi-photon absorption in Tryptophan are reported and analyzed. Section IV described the main results of this chapter, which is the effect of metallic nanoparticles on the fluorescent emission of the Tryptophan excited by a multi-photon process. Influence of nanoparticle concentration on the Tryptophan-silver colloids is observed and discussed based coi a nonlinear generalization of the Maxwell Garnett model, introduced in section II. The main conclusion of the chapter is given in secticHi IV. 

The equations can be generalized for both shear and tension, and G can be replaced by E. The mechanical analogue for the Maxwell unit can be represented by a combination of a spring and a dashpot arranged in series so that the stress is the same on both elements. This means that the total strain is the sum of the strains on each element as expressed by Equation 13.19. A typical stress-strain curve predicted by the Maxwell model is shown in Figure 13.12(a). Under conditions of constant stress, a Maxwell body shows instantaneous elastic deformation first, followed by a viscous flow. 

Fig. 19 Mechanical-viscoelastic model of Lin and Chen (1999) with two Maxwell models to describe SME in segmented PUs. (a) General model, (b) Change of the model in the shape-memory cycle, (c) Shape-memory behavior for two PU samples. Solid lines indicate the recoverable ration curves of the model. Taken from ref. [36], Copyright 1999. Reprinted with permission of John WUey Sons, Inc.

Finally, more accurate differential and integral constitutive equations were presented, and their successes and failures in de-st bing experimental data, were discussed. No single nonlinear constitutive equation is best for all purposes, and thus one s choice of an appropriate constitutive equation must be guided by the problem at himd, the accuracy with which one wishes to solve the problem, and the effort one is willing to expend to solve it. Generally differential models of the Maxwell type are easier to implement numerically, and some are available in fluid mechanics codes. Also, some cmistitutive equations are better founded in molecular theory, as discussed in Chiqpter 11. 

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