Constitutive viscoelastic

The viscoelastic nature of polymers generally determines rate and temperature dependence of their mechanical properties. At low strain levels, i.e. in a linear regime, this dependence is well described by intrinsic material properties defined within constitutive viscoelastic laws [1]. At high strains, in presence of failure processes, such as yielding or fracture, it is more difficult to establish a constitutive behaviour as well as to define material properties able to intrinsically characterise the failure process and its possible viscoelastic features. 

Three-Dimensional Constitutive Viscoelastic Model for Isotropic Materials... 

Larobina D, Greco F (2012) Prediction of the effects of constitutive viscoelasticity on stress-diffusion coupling in gels. J Chem Phys 136 134904. doi 10.1063/1.3699978 Li W, Zhao H, Teasdale PR et al (2002) Synthesis and characterisation of a polyacrylamide-po-lyacrylic acid copolymer hydrogel for environmental analysis of Cu and Cd Reaet Funct Polym 52 31-41. doi 10.1016/S1381-5148(02)00055-X... 

The constitutive viscoelastic law is developed for the rheological model presented in Figure 2, where masses oscillate together with the springs. Separating the stress tensor into pressure and deviatoric parts. 

Work by Schapery, Saxena, Wilhams, and others details the analysis of cracks in creeping, strain rate dependent materials, and provides a predictive basis for the apparently brittle nature of FCP in UHMWPE [43-48]. Particularly usefid are the models developed by Schapery and Wilhams, which directly link the intrinsic, constitutive viscoelastic relaxation behavior of the material to the advance of a stable crack tip [46,48]. The power of these models is the predictive nature of the mechanics in relating ECP dynamics to the material s viscoelastic behavior that is eashy measured in a simple one-dimensional creep test The elementary consequences of the models result in the static mode fatigue crack propagation behavior that is observed in UHMWPE, and thus potentially provide a first-principles explanation of the fatigue and fracture behavior by the material. 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Model (material) parameters used in viscoelastic constitutive equations... 

Depending on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

Single-integral constitutive equations for viscoelastic fluids... 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

Equations of continuity and motion in a flow model are intrinsically connected and their solution should be described simultaneously. Solution of the energy and viscoelastic constitutive equations can be considered independently. 

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. 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

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 integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

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