Non viscoelasticity

Viscoelastic fluids have elastic properties in addition to their viscous properties. When under shear, such fluids exhibit a normal stress in addition to a shear stress. For example, if a vertical rod is partly immersed and rotated in a non-viscoelastic liquid the rod s rotation will create a centrifugal force that drives liquid outwards toward the container walls, as shown in Figure 6.16(a). If, on the other hand, the liquid is viscoelastic then as the liquid is sheared about the rod s axis of rotation, a stress normal to the plane of rotation is created which tends to draw fluid in towards the centre. At some rotational speed, the normal force will exceed the centrifugal force and liquid is drawn towards and up along the rod see Figure 6.16(b). This is called the Weissenberg effect. Viscoelastic fluids flow when stress is applied, but some of their deformation is recovered when the stress is removed [381]. 

The proportionality constant can be measured directly using a non-viscoelastic sample such as spring steel. This is, in fact, done in the procedure for calibrating the DMA s tan delta constant (C ) which relates the moment amplitude in the sine wave drive, to the rms motor current signal "B" in... 

The Nagdi and Murch theory of viscoelastic-plasticity contains many of the same caveats as in plasticity theory and, in fact, reduces to the two limiting cases of plasticity for non-viscoelastic materials and linear viscoelasticity for non-yielded materials. The only portions needed here are the linear viscoelastic constitutive equations given in Chapter 6 and a generic failure law given by,... 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

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... 

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

The theoretical description of a non-isothermal viscoelastic flow presents a conceptual difficulty. To give a brief explanation of this problem we note that in a non-isothennal flow field the evolution of stresses will be affected by the... 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

Marchal, J. M. and Crochet, M.J., 1987. A new mixed finite element for calculating viscoelastic flow. J. Non-Newtonian Pluid Mech. 20, 77-114. 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Viscous Hquids are classified based on their rheological behavior characterized by the relationship of shear stress with shear rate. Eor Newtonian Hquids, the viscosity represented by the ratio of shear stress to shear rate is independent of shear rate, whereas non-Newtonian Hquid viscosity changes with shear rate. Non-Newtonian Hquids are further divided into three categories time-independent, time-dependent, and viscoelastic. A detailed discussion of these rheologically complex Hquids is given elsewhere (see Rheological measurements). 

The mechanical properties of LDPE fall somewhere between rigid polymers such as polystyrene and limp or soft polymers such as polyvinyls. LDPE exhibits good toughness and pHabiUty over a moderately wide temperature range. It is a viscoelastic material that displays non-Newtonian flow behavior, and the polymer is ductile at temperatures well below 0°C. Table 1 fists typical properties. 

Rheological Properties. Materials must have suitable flow properties in order to be used in production. Both Newtonian and non-Newtonian fluids and their viscoelasticity properties play a critical role in the performance of the embedding materials. 

Eig. 7. Viscoelastic behavior of encapsulant materials (a) Newtonian fluid (b) non-Newtonian fluid. 

Contact mechanics, in the classical sense, describes the behavior of solids in contact under the action of an external load. The first studies in the area of contact mechanics date back to the seminal publication "On the contact of elastic solids of Heinrich Hertz in 1882 [ 1 ]. The original Hertz theory was applied to frictionless non-adhering surfaces of perfectly elastic solids. Lee and Radok [2], Graham [3], and Yang [4] developed the theories of contact mechanics of viscoelastic solids. None of these treatments, however, accounted for the role of interfacial adhesive interactions. 

Viscoelastic contact problems have drawn the attention of researchers for some time [2,3,104,105]. The mathematical peculiarity of these problems is their time-dependent boundaries. This has limited the ability to quantify the boundary value contact problems by the tools used in elasticity. The normal displacement (u) and pressure (p) fields in the contact region for non-adhesive contact of viscoelastic materials are obtained by a self-consistent solution to the governing singular integral equation given by [106] ... 

Other ideas proposed to explain the 3/4 power-law dependence include effects due to viscoelasticity, non-linear elasticity, partial plasticity or yielding, and additional interactions beyond simply surface forces. However, none of these ideas have been sufficiently developed to enable predictions to be made at this time. An understanding of this anomalous power-law dependence is not yet present. 

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