On the predictive/fitting capabilities of the advanced differential constitutive equations for linear polyethylene melts

This contribution examines the capabilities of three differential constitutive models (XPP, PTT-XPP, and modified Leonov) in predicting rheological properties of two virtually linear polyethylene materials (HDPE Tipelin FS 450-26, mLLDPE Exact 0201) with specific attention to both steady as well as transient shear and uniaxial elongational flow situations. For each situation the (dis)advantages of the individual models are discussed and both, qualitative and quantitative model efficiency evaluation has been carried out.     

A model of traps for yield in amorphous glassy polymers

Summary  Constitutive equations are derived for the viscoelastic and viscoplastic behavior of amorphous glassy polymers at isothermal loading with small strains. The model is based on the trapping concept: a disordered medium is treated as an ensemble of plastic flow units (with the characteristic size of micrometers), which, in turn, consist of a number of cooperative rearranging regions (with the characteristic length of nanometers). The viscoelastic response is described by rearrangement of relaxing regions, whereas the viscoplastic behavior is modeled as irreversible deformation of plastic units. Adjustable parameters are found by fitting observations for aromatic polyesters, nylon-66, polycarbonate block copolymers and an epoxy glass. Fair agreement is demonstrated between experimental data and results of numerical simulation. Received 17 November 1999; accepted for publication 23 March 2000     

Orthogonal and parallel superposition measurements on lyotropic liquid crystalline polymers

 Mechanical spectroscopy is used to probe the structure of lyotropic liquid crystalline polymers during flow and after the cessation of flow. The oscillatory flow is either parallel or perpendicular to the steady-state flow. The resulting moduli provide information about the time- and shear-dependent microstructure, including anisotropy. Two different concentrations of poly(benzylglutamate) (PBG) in m-cresol and a concentrated hydroxypropylcellulose (HPC) solution, also in m-cresol, are investigated. In all cases, the orthogonal superposition moduli evolve differently from the parallel ones. The former are less sensitive to the flow-induced changes in structure than the latter ones. Together with the lack of sensitivity of the superposition moduli to texture refinement during flow, this suggests a strong relation between director orientation and superposition moduli. After the cessation of flow the parallel moduli decrease for the PBG solutions, whereas the opposite is observed in the HPC solutions. A comparison with the orthogonal moduli provides a direct measure of anisotropy. At rest, the PBG solutions tend toward a higher degree of anisotropy while the HPC solutions become more isotropic. In the latter systems, all moduli are much larger, reflecting a larger contribution from the texture. Received: 8 July 1999/Accepted: 1 October 1999     

Shear-banding structure orientated in the vorticity direction observed for equimolar micellar solution

 The non-monotonic shear flow of a viscoelastic equimolar aqueous surfactant solution (cetylpyridinium chloride-sodium salicylate) is investigated rheologically and optically in a transparent strain-controlled Taylor Couette flow cell. As reported before, this particular wormlike micellar solution exhibits first a shear thinning and then a pronounced shear-thickening behavior. Once this shear-thickening regime is reached, a transient phase separation/shear banding of the solution into turbid and clear ring-like patterns orientated perpendicular to the vorticity axis, i.e., stacked like pancakes, is observed (Wheeler et al. 1998; Fischer 2000). The solution exhibit several unique features as no induction period of the shear induced phase, no structural build-up at the inner rotating cylinder, jumping pancake structure of clear and turbid ringlike phases, and oscillating shear stresses appear once the pancake structure is present. According to our analysis this flow phenomenon is not purely a mechanical or rheological driven hydrodynamic instability but one has to take into account structural changes of the oriented micellar aggregates (flow induced non-equilibrium phase transition) as proposed by several authors. Although this particular flow behavior and the underlying mixture of shear induced phases and mechanical instabilities is not fully understood yet, some classification characteristics based on a recent theoretical approach by Schmitt et al. (1995) and Porte et al. (1997) where a strong coupling between the flow instability (non-homogeneous flow profile due to the bands) and the structural changes causes the observed transient phenomena can be derived. In reference to the presented model the observed orientation of the rings is typical for complex fluids that undergo a spinodal phase separation coupled with a thermodynamic flow instability. In contrast to other shear banding phenomena, this one is observed in parallel plate, cone-plate, and Couette flow cell as well as under controlled stress and controlled rate conditions. Therefore, it adds an additional aspect to the present discussion on shear banding phenomena, i.e., the coupling of hydrodynamics and phase transition of rheological complex fluids. Received: 8 January 2001 Accepted: 15 May 2001     

A dislocation density based material model to simulate the anisotropic creep behavior of single-phase and two-phase single crystals

The primary and secondary creep behavior of single crystals is observed by a material model using evolution equations for dislocation densities on individual slip systems. An interaction matrix defines the mutual influence of dislocation densities on different glide systems. Face-centered cubic (fcc), body-centered cubic (bcc) and hexagonal closed packed (hcp) lattice structures have been investigated. The material model is implemented in a finite element method to analyze the orientation dependent creep behavior of two-phase single crystals. Three finite element models are introduced to simulate creep of a γ′ strengthened nickel base superalloy in 〈1 0 0〉, 〈1 1 0〉 and 〈1 1 1〉 directions. This approach allows to examine the influence of crystal slip and cuboidal microstructure on the deformation process.     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001     

The implicit standard material theory for modelling the nonassociative behaviour of metals

Summary  Modelling the elastoplastic or elastoviscoplastic behaviour of metallic materials exhibiting strain hardening and damage leads to complex nonassociative constitutive equations, sources of many theoretical and numerical troubles. The usual modelling of a nonassociative constitutive equation leads to the loss of the interesting and very useful properties of generalised standard materials deriving from the key concepts of convexity and normality. The argument that will be developed is that the bipotential concept is an appropriate answer. In the first part, after introducing the state variables generally used to describe the behaviour of metallic materials, the constitutive equations subjected to the principles of thermodynamics are derived from two potentials. The state potential gives the state laws, and the bipotential of dissipation delivers the evolution laws for state variables, through the implicit normality assumption. The second part is devoted to several particular applications to metal elastoplasticity and elastoviscoplasticity models. Received 29 March 2000; accepted for publication 26 September 2000     

Effect of melt viscosity of polypropylene on fibrillation of thermotropic liquid crystalline polymer in in situ composite film

 Various grades of polypropylene were melt blended with a thermotropic liquid crystalline polymer, a block copolymer of p-hydroxy benzoic acid and ethylene terephthalate (60/40 mole ratio). The blends were extruded as cast films at different values of draw ratio (slit width/film thickness). Fibrillation of TLCP dispersed phase with high fiber aspect ratio (length/width) was obtained with the matrix of low melt flow rate, i.e., high viscosity and with increasing film drawing. Melt viscosities of pure components and blends measured using capillary rheometer were found to decrease with increasing shear rate and temperature. Viscosity ratios (dispersed phase to matrix phase) of the systems being investigated at 255 °C at the shear rate ranged from 10² to 10⁴ s⁻¹, were found to lie between 0.04 and 0.15. The addition of a few percent of elastomeric compatibilizers; a tri-block copolymer SEBS, EPDM rubber and maleated-EPDM, was found to affect the melt viscosity of the blend and hence the morphology. Among these three compatibilizers, SEBS was found to provide the best fibrillation. Received: 10 January 2000/Accepted: 24 January 2000     

A potential technique in the theory of interaction between a structure and a viscous, compressible fluid

Summary  Based on the previous analysis, [4], a slightly generalised fluid-structure interaction problem is examined. A closed-form solution is constructed to be compared with the `thin-layer' approach of [4]. It is shown that the `thin-layer' results are a rough approximation which are significantly improved by the results of the method presented here for a large gap length. Received 21 April 1999; accepted for publication 20 July 2000     

Dynamic melt flow of nanocomposites based on poly-ɛ-caprolactam

 The dynamic flow behavior of polyamide-6 (PA-6) and a nanocomposite (PNC) based on it was studied. The latter resin contained 2 wt% of organoclay. The two materials were blended in proportions of 0, 25, 50, 75, and 100 wt% PNC. The dynamic shear rheological properties of well-dried specimens were measured under N₂ at T=240 °C, frequency ω=0.1–100 rad/s, and strains γ=10 and 40%. At constant T, γ, and ω the time sweeps resulted in significant increases of the shear moduli. The γ and ω scans showed a complex rheological behavior of all clay-containing specimens. At γ=10% the linear viscoelasticity was observed for all compositions only at ω>1 rad/s, while at γ=40% only for 0 and 25 wt% of PNC. However, the effect was moderate, namely decreasing G′ and G′′ (at ω=6.28 rad/s; γ=50%) by 15 and 7.5%, respectively. For compositions containing >25 wt% PNC two types of non-linearity were detected. At ω≤ω_c=1.4 ± 0.2 rad/s yield stress provided evidence of a 3-D structure. At ω > ω_c, G′ and G′′ were sensitive to shear history – the effect was reversible. From the frequency scans at ω > ω_c the zero-shear relative viscosity vs concentration plot was constructed. The initial slope gave the intrinsic viscosity from which the aspect ratio of organoclay particles, p=287 ± 9 was calculated, in agreement with the value calculated from the reduced permeability data, p=286. Received: 24 May 2001 Accepted: 27 August 2001     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

Coupling of adsorption and diffusion in porous and granular materials. A 1-D example of the boundary value problem

Summary  In the paper, we present a macroscopic continuum model of adsorption in porous materials consisting of three components. We consider the flow of a fluid component through channels of the skeleton. It serves as carrier for an adsorbate whose mass balance equation contains a source term. The source consists of two parts: a Langmuir contribution, connected with bare sites on internal surfaces, which becomes in equilibrium the Langmuir isotherm, and changes of the internal surface driven by the source of porosity. The model for the latter contribution is new. Parameters of this model are analyzed by means of an example of solution of a boundary value problem for the full set of field equations, which describes the transport of pollutants in soils. Received 27 May 1999; accepted for publication 20 October 1999     

A note on probabilistic contractor couple and solutions for a system of nonlinear equations in n a menger pn-spaces

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Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables

In the vicinity of the glass transition, glass-forming materials exhibit pronounced frequency-dependent changes in the mechanical material properties, the thermal expansion behaviour and the specific heat. The frequency dependence becomes apparent under harmonic stress, strain or temperature excitations. The Prigogine-Defay ratio is a characteristic number which connects the changes in magnitude of these quantities at the glass transition. In order to represent the thermoviscoelastic properties of glass-forming materials in continuum mechanics, a three-dimensional approach which is based on the Gibbs free energy as thermodynamic potential is developed in this article. The Gibbs free energy depends on the stress tensor, the temperature and a set of internal variables which is introduced to take history-dependent phenomena into account. In the vicinity of an equilibrium reference state, the specific Gibbs free energy is approximated up to second order terms. Evaluating the Clausius-Duhem inequality, the constitutive relations for the strain tensor, the entropy and the internal variables are derived. In comparison with other approaches, the entropy, the strain tensor and the internal variables are functionals not only of the stress tensor but also of the temperature. Applying harmonic temperature- or stress-controlled excitations, complex frequency-dependent relations for the specific heat under constant stress, for the thermal expansion coefficients as well as for the dynamic mechanical compliance are obtained. The frequency-dependence of these quantities depicts the experimentally observed behaviour of glass-forming materials as published in literature. Under the assumption of isotropic material behaviour, it is shown that the developed theory is compatible with the Prigogine-Defay inequality for arbitrary values of the material parameters.     

An efficient algorithm based on the differential quadrature method for solving Navier–Stokes equations

In this paper, an approach to improve the application of the differential quadrature method for the solution of Navier–Stokes equations is presented. In using the conventional differential quadrature method for solving Navier–Stokes equations, difficulties such as boundary conditions' implementation, generation of an ill conditioned set of linear equations, large memory storage requirement to store data, and matrix coefficients, are usually encountered. Also, the solution of the generated set of equations takes a long running time and needs high computational efforts. An approach based on the point pressure–velocity iteration method, which is a variant of the Newton–Raphson relaxation technique, is presented to overcome these problems without losing accuracy. To verify its performance, four cases of two‐dimensional flows in single and staggered double lid‐driven cavity and flows past backward facing step and square cylinder, which have been often solved by researchers as benchmark solution, are simulated for different Reynolds numbers. The results are compared with existing solutions in the open literature. Very good agreement with low computational efforts of the approach is shown. It has been concluded that the method can be applied easily and is very time efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM⁺ ? up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ ? up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.     

Gelation study of high processability and high reliability ternary systems based on benzoxazine, epoxy, and phenolic resins for an application as electronic packaging materials

 Fourier transform mechanical spectroscopy technique (FTMS) is utilized as a powerful tool to study the sol-gel transition of covalent bonded polymeric network. Winter and Chambon criteria resulting from the fractal-geometry characteristic of the gel networks allow the determination of the gel point with only single experiment using this technique. The gelation behaviors of low melt viscosity ternary systems of benzoxazine, epoxy, and phenolic resins are investigated and analyzed by the technique in order to study the effect of epoxy diluent on the rheological property development before and after the gel points. The gel time at 140 °C ranges from 5 min to 30 min and less than 5 min at 180 °C for all tested ternary system compositions. The gelation of the ternary mixture shows an Arrhenius-type behavior and the gel time can be well-predicted by the Arrhenius equation. Received: 23 November 1999 Accepted: 2 January 2001