Lodge model

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

Additional complexity can he brought to the constitutive equation in its integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in a large class of constitutive equations. In a perfect body, the strain energy W may be linked to strain and stress by ... 

The Lodge model is a special case of this class of models, where the material functions are selected as ... 

In terms of network description, Wagner considers that the damping fimction may reflect an additional process of destruction of network junctions by strain effects, as described by the generalized invariant, thus involving some peculiarities of the flow such as its geometry and the strsdn intensity. As regards the rate of creation of network junctions, it is assumed to remain constant as in the Lodge model. 

In terms of network analogy, the damping function may be viewed as the expression of the retraction of the strands as compared to the continuum. The Lodge model thus corresponds to no retraction (affine deformation, a=0 in equation (30) ), the Doi-Edwards equation corresponds to complete retraction (a=0.2), whereas incomplete retraction makes the damping function more softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine since there is a difference between that of the continuum and that of the network strands. Wagner [33] showed that the Doi Edwards strain function... 

The first kind of modification to the UCM model that may be conceivable is that of the convected derivative. This leads one to consider that the motion of the network junctions is no more that of the continuum and thus, the afiine assumption of the Lodge model is removed. Among the various possibilities, Phan Thien and Tanner suggested the use of the (Jordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative ... 

As a general rule in this case, it is obvious that the higher the value, the less pronoimced the maximum, the extreme case being that of e = 0 (Lodge model and divergence of the viscosity). 

In the work of Piau and colleagues [23,24], the Lodge elastic liquid constitutive model was used directly. These authors found that the Lodge model provided reasonable agreement with their uniaxial extension results, showing slightly more strain hardening than was observed in the data. 

Next, we consider the second class of experiments and check the predictions of the Lodge model with regard to extensional flows. Using again an equation from the ideal rubbers we can directly write down the time-dependent Finger tensor B(t, t ). It has the form... 

It is seen from Eq. (3.50) that the Lodge model predicts virtually the same form for material functions as the upper convected Maxwell model does (see Eq. (3.6)). 

Other possibilities exist to solve the frame invariant problem Cauchy-Maxwell equation uses the Cauchy tensor, C, which is also independent of the system of reference, the Lodge rubber-like liquid model uses the Finger tensor but contrarily to the Lodge model, it uses a generalized memory function ... 

Production QA is the process whereby the manufacturer adopts a quality system for the manufacture and final inspection of a device. ISO 13485 may also be used as the model in these circumstances, as the design and development elements may be omitted from this standard. The manufacturer must lodge an application with a Notified Body to have his/her system examined. The application must be accompanied by documentation on the quality system and information on any relevant EC type-examined devices. 

Produced process models can be used for the design of measuring devices based on electromagnetic oscillation effect in the first case and based on charged particle lodging area definition in second. The equations decribing the motions in thermoelectric field have the following form ... 

This equation has recently been derived in rather complete fashion for bead-spring models by Lodge and Wu (101). Derivation and application to models other than spring-bead systems is given by Bird and co-workers (102). 

Constitutive equations for the Rouse and Zimm models have been derived, and are found to be expressible in the form of Lodge s elastic liquid equation [Eq.(6.15)], with memory function given by (101) ... 

The relaxing Gaussian network of Green and Tobolsky (4) is the earliest version of this model. Lodge (12) and Yamamoto (J5) independently derived constitutive equations for similar systems, based on a stress-free state for each newly created strand and a distribution of junction lifetimes which is independent of flow history. For Gaussian strands in an incompressible system ... 

Lodge,A.S., Wu,Y.-J. Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm. Rheol. Acta 10,539-553 (1971). 

Fig. 4.27 SAXS intensity as a function of wavevector for a PS-P1 diblock (Mw = 60 kg mol-1, 17wt% PS) (points) in dibutyl phthalate with a polymer volume fraction

---