Modulus series model

In this work the mechanical model proposed by Takayanagi (9) will be used. Two variants were proposed, both assuming that the two phases are connected partly in parallel and partly in series (Figures 7a and 7b). Kaplan and Tschoegl (10) have shown that the two variants of the Takayanagi model are equivalent. The series model (Figure 7b) will be used for our calculations. The modulus is given by ... 

In a subsequent investigation, with Roos and Kampschreur (1989), Northolt extended the modified series model to include viscoelasticity. For that an additional assumption was made, viz. that the relaxation process is confined solely to shear deformation of adjacent chains. The modified series model maybe applied to well-oriented fibres having a small plastic deformation (or set). In particular it explains the part of the tensile curve beyond the yield stress in which the orientation process of the fibrils takes place. The main factor governing this process is the modulus for shear, gd, between adjacent chains. At high deformation frequencies yd attains its maximum value, ydo at lower frequencies or longer times the viscoelasticity lowers the value of gd, and it becomes a function of time or frequency. Northolt s relations, that directly follow from his theoretical model for well-oriented fibres, are in perfect agreement with the experimental data if acceptable values for the elastic parameters are substituted. 

The early attempts to interpret the dynamic mechanical behaviour in structural terms include that of Smith et al. where the plateau modulus was correlated with the fraction of non-crystalline material f, determined by NMR. Plots of the plateau compliances at —60 °C and —160 °C as a function of f suggested a modified Takay-anagi series model, with a constant amount of non-crystalline material in parallel with the simple series model. The modd showed good internal consistency, with values for the compliances of the non-ciystalline regions which were acceptable in physical terms. 

Deduce the equations corresponding to equations (11.16) and (11.17) for the Takayanagj models shown in figs 11.10(b) and (c). A polymer that is 75% crystalline by volume has amorphous and crystalline moduli E3 and equal to 1.0 x 10 and 1.0 x lO Pa, respectively, parallel to the draw direction. Calculate the modulus parallel to the draw direction for the simple series and simple parallel Takayanagi models and for the series-parallel and parallel-series models, assuming that in the last two models a = ft = 0.5. 

This is a very powerful method when applied to oriented materials, since it yields information about the modulus of the crystalline region, i.e. information on a molecular scale. The only drawback is that, whilst the strain can be measured accurately, the stress can only be measured if a basic assumption is made—that the stress is homogeneous. This assumption amounts to the statement that the system can be represented by the series model, in which the strain is inhomogeneous and the stress homogeneous. Evidence for this has been set out in the paper of Holliday and White, and can be regarded as satisfactory for the fibre direction. 

If we assume that the model in Figure 2.12 is modified into a parallel rather than a series model (see Figure 2.12b), then the modulus of the system is represented by the following equation (Takayanagi, 1972) ... 

Figure 8.25. Plots of the dynamic modulus E at 23°C against the polyacrylate content. The upper and the lower solid lines are calculated with a parallel and a series model, respectively, using following equations (parallel) = (1 - A) J + AFJ (series) = [(1 - A)/Fa] + (Klempner et al, 1970.)...

These two simple models have been used to predict different thermal and mechanical properties of a two-state material such as thermal conductivity and elastic modulus, where they are also called parallel and series models respectively. These two models are further able to define the upper and lower bounds for the effective properties of the mixture [1], as follows ... 

Fig. 7 Young s modulus versus volume fraction PPO for the incompatible PpClS/PPO blends. Curve 1, the series model (Eq. 3) curve 2, rule of mixtures (Eq. 2) curve 3, modified rule of mixtures (Eq. 4).

Both compatible and semi-compatible PPO blends that show blend densification exhibit a small synergistic maximum in their modulus-composition plots which can be modeled by the classical rule of mixtures for composites with an additional interaction term. The incompatible PPO blends exhibit no blend densification and can be modeled adequately by the series model for composites. 

Figure 2.1 Effect of polyurethane (PU)/polycaprolactone flbn structure on the modulus of 65 layer and blend films in comparison with values predicted from the parallel and series models. From J. Du, S.R. Armstrong, E. Baer, Co-extruded multilayer shape memory materials comparing layered and blend architectures. Polymer (United Kingdom) 54 (20) (2013) 5399-5407.

Values of storage modulus at 23 C are plotted as a function of composition in Figure 6.29. Takayanagi s parallel model for the mechanical behavior of a two-component system corresponds to the case in which the stiffer component is continuous, while his series model corresponds to the case in which the softer component is continuous. Clearly the experimental results agree best with the parallel model over most of the concentration... 

Figure 6.29. Plot of the dynamic modulus E at 23°C against the polyacrylate content. The upper and the lower solid lines are calculated with a parallel and a series model, respectively. ...

Takayanagi [17] devised series-parallel and parallel-series models as an aid to understanding the viscoelastic behaviour of a blend of two isotropic amorphous polymers in terms of the properties of the individual components. For an A phase dispersed in a B phase there are two extreme possibilities for the stress transfer. For efficient stress transfer perpendicular to the direction of tensile stress we have the series-parallel model (Figure 8.9(a)) in which the overall modulus is given by the contribution for the two lower components in parallel (as in Equation (8.3)) in series with the contribution for the upper component (as in Equation (8.5)) ... 

A model referred to as the equivalent box model (EBM) has shown promise in the ability to predict the modulus behavior over the entire composition. This model has similarities to earlier models by Takayangi et al. [6,7]. This model is a combination of the parallel and series models and has been developed by Kolarik [8-10]. This model will be described in detail because of its versatility for phase separated blends. This mechanical model is illustrated in Fig. 6.5. The modulus is calculated from... 

The prediction of conductivity, a (electrical, ionic or proton), based on pure component values for miscible and phase separated systems can employ the same models as used for modulus and permeability, where a (S/cm) can be substituted for modulus (E) or permeability (P) in the log additivity relationship for miscible systems and the parallel model, series model or equivalent box model for phase separated systems. While these expressions have not been generally employed for conductivity modeling, the principles on which they are based are analogous to modulus and permeability values. 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

An estimate of the shear modulus is also given by an expression based on the series spring model... 

A recent series of papers [18, 24, 32-34] substantially clears up the three-dimensional polymerization mechanism in the AAm-MBAA system. Direct observation of the various types of acrylamide group consumption using NMR technique, analysis of conversion at the gel-point, and correlation of the elastic modulus with swelling indicate a considerable deviation of the system from the ideal model and a low efficiency of MBAA as a crosslinker. Most of these experimental data, however, refer to the range of heterogeneous hydrogels where swelling is not more than 80 ml ml-1 [24]. 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

Fig. 58 The rheological model of a polymer fibre consists of a series arrangement of an elastic tensile spring representing the chain modulus, ec, and a shear spring, g(t), with viscoelastic and plastic properties representing the intermolecular bonding...

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

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