Ericksen stress

R 9. — and J. C. Ericksen Stress-deformation ration for isotropic materials. 

Known as the molecular field, h tends to restore n to its former orientation. Minimizing the total free energy shows that an equilibrium is achieved when h is parallel to . Furthermore, when the LC is deformed, the variation of Fd n,Vn), with respect to the strain, gives rise to an elastic stress known as the Ericksen stress ... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

The present analysis was based on the lubrication approximation that is, we neglected changes in the x direction. If this assumption is lifted, we are faced with a flow field in which two nonvanishing velocities exist that are functions of two spatial coordinates, vx(x, y), vy (x, y). This is clearly a nonviscometric flow situation, and the Criminale-Ericksen-Filbey (CEF) equation is not applicable. White (13) made an order of magnitude evaluation of normal stress effects for this more realistic flow situation. In this case, the equation of motion reduces to... 

Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

The LE theory is rather complex since it contains both viscous and elastic stresses. It can best be understood by considering viscous and elastic effects separately. If elastic effects are neglected, the LE equations reduce to Ericksens transversely isotropic fluidy while in the absence of flow the elastic stresses are just those of the Frank-Oseen theory (discussed below in Section 10.2.2). ... 

In the absence of elastic stresses, the stress tensor for a flowing nematic is given by (Ericksen 1960, 1961)... 

In a flowing liquid crystal, both the viscous stresses and Frank elastic stresses are normally important. Thus, the Ericksen theory for the viscous stresses, must somehow be combined with the Frank theory for the elastic stresses. This was accomplished by Leslie, who... 

However, if the director field is not uniform, Frank distortional stresses influence the rate of rotation of the director, and a new equation for h (or, equivalently for N) is required to replace Ericksen s equation (10-3). This is obtained from the balance of angular momentum, which gives... 

The total stress tensor in the Leslie-Ericksen theory is the sum of the viscous stress of Eq. (10-10), an isotropic pressure, and the Frank distortional stress, given by... 

Here is a typical Leslie viscosity, AT is a Frank constant, V is a flow velocity, /i is a length scale of the flow geometry, such as the tube diameter in Poiseuille flow, and ftn is the average shear rate. The Ericksen number is the ratio of the flow-induced viscous stress 6Ye.fi = /h to the Frank stress K/h. The appropriate Leslie viscosity or Frank constant... 

Again, we express n in terms of the angle 9 using Eq. (AlO-22). We need the xy component of the stress tensor given by the Leslie-Ericksen equation, Eq. (10-10). We consider, in turn, the jry component of each term of this equation. We use Eq. (A 10-26) to evaluate the first term of Eq. (10-10) ... 

The Rivlin-Ericksen constitutive equation gives a good account of some characteristics of both the time dependence of the viscoelastic behavior and the normal stress effects. This relationship is based on the assumption that the stress depends not only on the velocity (x ) and the shear rate gradient (dxi/dx ) but also on derivatives of higher order (%, dXp/dXq. .. 8xf /8xi). As a consequence of the principle of material... 

For steady-state shearing flows, the relationship between the shear-stress tensor and the shear-rate tensor is given by Criminale-Ericksen-... 

The first normal stress difference exhibits a linear dependency on the shear rate in the region of constant viscosity for the two solutions in Figure 2. This proportionality is predicted by the Doi theory (10) and the Leslie-Ericksen theory (111 although the basic assumption in these theories, i.e. a monodomain structure, is not satisfied. 

Leslie recognized from early experiments that the anisotropy of the materials calls for multiple viscosity coefficients corresponding to different orientation of the LC relative to the flow. Combining this idea with the Ericksen theory leads to the Leslie-Ericksen (LE) theory, which comprises two elements one describing the evolution of n(r) in a flow field, and the other prescribing an extra stress tensor due to the evolving (r) field. 

The viscous stress of liquid crystals is defined as the Ericksen-Leslie stress tensor and is related to A, n and w or IV ... 

Figure 6.10 schematically shows the physical meaning of the six terms of the Ericksen-Leslie stress tensor. The correspondent flow fields and the viscous moments are depicted. It is seen that a4 is the term arisen from the conventional fluid. ot is symmetrical, representing an extension effect caused by a non-rotation flow, which produces a viscous stress while being without moment. arotation flow which produces a moment, as and a6 terms are non-symmetrical, same as a2 and 03, but they are associated with the stress and moment resulting from nonrot at ional flow. 

The negative first normal stress difference under a medium shear rate, characterized by liquid crystalline polymers, makes the material avoid the Barus effect—a typical property of conventional polymer melt or concentrated solution, i.e., when a polymer spins out from a hole, or capillary, or slit, their diameter or thickness will be greater than the mold size. The liquid crystalline polymers with the spin expansion effect have an advantage in material processing. This phenomenon is verified by the Ericksen-Leslie theory. On the contrary, the first normal stress difference for the normal polymers is always positive. 

We have seen that a simple apparent viscosity is used to characterize the stress-shear rate relationship for a non-Newtonian fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by the Criminale-Ericksen-Filbey equation... 

Unfortunately, because there are so many variables it is rarely possible to compare the observations made in different studies and the literature contains many conflicting results. For example, there is some contention as to whether the bands form during shear [ 109,119] or on stress relaxation afterwards [83, 120], Ernst and Navard [120] considered a rather pure relaxation process via a periodic distortion after shear flow has ceased. According to this model, the macromolecules are stretched in the flow and then relax back. It is not clear, however, which mechanism in the relaxation process is responsible for the observed periodic structure. There is also disagreement as to whether the shear threshold does [ 110,121 ] or does not [120, 122] depend on sample thickness. A shear threshold inversely proportional to the sample thickness, as reported in [121], would support the instability concept. Zielinska and Ten Bosch [124, 125] argued that, for simple shear flow, the band structures are due to an instability mechanism present in the Leslie-Ericksen... 

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