Viscoelasticity integral model

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. 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

Integral viscoelastic models. Integral models with a memory function have been widely used to describe the viscoelastic behavior of polymers and to interpret their rheological measurements [37, 41, 43], In general one can write the single integral model as... 

Liu K, Ovaert TC (2011) Poro-viscoelastic constitutive modeling of unconfined creep of hydrogels using finite element analysis with integrated optimization method. J Mech Behav Biomed Mater 4 440-450... 

Johnson [186] have shown that, if the relaxation property can be described by a scalar function k(t), the single integral representation (103) is equation Viscoelastic (QLV) model first introduced by Fung [164], i.e.,... 

Figure 4.12 Viscoelastic simulation of flow patterns for the melts of Figure 4.11 using the K-BKZ integral model (Eq. (4.12)) [34].

Figure4.21 Simulation ofextrudate swell from viscoelastic properties of the melts, hence coextrusion circular dies for a combination of different stress ratio, Sr. Viscoelastic HDPE and PS melts (PS/HDPE = inner/outer). simulations with the K-BKZ integral model A different configuration produces different (Eq. (4.12)) [46]. swelling for the same flow rate due to different...

The finite element description of the nonlinear viscoelastic behavior of technical fabric was presented by Klosowski et al. [65]. The technical fabric called Panama used in this model was made of two polyester thread families woven perpendicularly to each other with the 2/2 weave. The long term uniaxial creep laboratory tests in directions were conducted at five different constant stress levels. The dense net model [66] together with the Schapery one-integral viscoelastic constitutive model [67] was assumed for the fabric behavior characterization and the least square method in the Levenberg-Marquardt variant was used for the parameters identification. 

The comparison of calculated and experimental data of the creep curves showed a good correlation. After comparing the calculated results it can be concluded that the viscoelastic behavior of the technical fabric can be described by the one-integral model. In warp and weft directions the numerical curve fitting resulted in a difference of 3.5-9.9% and 0.3-2.6% respectively. Therefore, the Schapery model with the power function characterized more accurate creep behavior in weft direction than warp direction. Also the power function described the strain evaluation better than the exponential function. This research concluded that both the linear and nonlinear viscoelastic identifications based on different material models can be brought together and the results of linear characterization can be applied to the nonlinear description of the material. 

The basis for the gmeral mathematical description of flow conditions are the conservation equations as the eontinuity equation, mommtum equation and energy equation and a rheological material law. The rheologieal material law or constitutive equation specifies the relation between the velocity field and the acting stresses. The known viscoelastic material models can be classified in differential and integral models. 

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

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 the following section representative examples of the development of finite element schemes for most commonly used differential and integral viscoelastic models are described. 

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)... 

As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow... 

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. 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

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