Multi-relaxation time model

Maxwell sub-models in parallel. In this model, the total stress is the sum of the stresses experienced by each Maxwell sub-model, while the total strain is the same in each of the sub-models, i.e.  

The total modulus of the Maxwell-Wiechert model also can be obtained  

Research shows by selectively adjusting the relaxation times of appropriate number of sub-models, Equations 16.43 and 16.44 can give a reasonably good representation of the stress relaxation behavior of polymer fibers over a wide time period. This indicates the stress relaxation of polymer fibers can be described as a superposition of a large number of independent modes of relaxation. Each 

The Maxwell-Wiechert model also can be used to describe the creep behavior of polymer fibers. However, for the creep behavior, it is mathematically more convenient to create a model involving a range of retardation times by connected a number of Kelvin-Voigt sub-models in series. 

The simple mechanical models discussed above only allow us to describe the viscoelastic behavior phenomenologically. The in-depth fundamental understanding of the viscoelastic behavior of polymer fibers still requires more complex microscopic or molecular models. The interested reader is referred to the works by De Geimes (1971), Ferry (1980), and Rouse (1953). 

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Premnath, K.N. and J. Abraham. Three-Dimensional Multi-Relaxation Time (MRT) Lattice-Boltzmann Models for Multiphase Flow. J. Comput. Phys. 224 539-559 (2007). 

Fig.20a,b. Transient data in a extension b shear on LDPE from [81] modelled with a multi-mode version of the pom-pom constitutive equation. The linear modes (of weights and relaxation times T, ) have been decorated with the structure of pom-pom elements (adding values of and q to each mode)... 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

Let the number of individual units in multi-element models tend to infinity. For creep, an infinite number of Kelvin units gives an infinite number of retardation times this is called the spectrum of retardation times. The analagous development for stress relaxation leads to the spectrum of relaxation times. 

Ha systems (jl). Care must be exercised in setting up multi-component simulations in order to insure that the composition dependent relaxation times and reaction rates can be directly intercompared. Unless density reduced time units have been utilized, the modeled systems must correspond to conditions... 

While the exponential stress relaxation predicted by the viscoelastic analog of the Mawell element, ie., a single exponential, is qualitatively similar to the relaxation of polymeric liquids, it does not describe the detailed response of real materials. If, however, it is generalized by assembling a number of Maxwell elements in parallel, it is possible to fit the behavior of real materials to a level of accuracy limited only by the precision and time-range of the experimental data. This leads to the generalized, or multi-mode. Maxwell model for linear viscoelastic behavior, which is represented mathematically by a sum of exponentials as shown by Eq. 4.16. 

Recently NOESY MAS was used to study molecular motions in technically relevant materials such as rubbers [46, 47]. For the evaluation of these parameters, it is necessary to understand the cross-relaxation process in the presence of anisotropic motions and under sample spinning. Such a treatment is provided in [47] and the cross-relaxation rates were found to weakly depend on fast motions in the Larmor-frequency range and strongly on slow motions of the order of the spinning frequency vR. Explicit expressions for the vR dependent cross-relaxation rates were derived for different motional models. Examples explicitly discussed were based on a heterogeneous distribution of correlation times [1,8,48] or on a multi-step process in the most simple case assuming a bimodal distribution of correlation times [49-51]. 

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