Modified Cross Model Research Articles

Compressive Creep Behaviour of Asphalt Mixtures

The uniaxial and triaxial steady-state deformation behaviour of a gap graded and a continuously graded asphaltic mixture have been studied. Uniaxial creep and constant strain rate tests have been conducted over a range of stress levels, strain rates and temperatures, and triaxial creep tests have been conducted over a range of deviatoric stress levels and confining pressures. The steady-state deformation behaviour of both mixtures is found to be well captured by the Modified Cross Model which predicts linear viscous behaviour at low stress levels and non-linear viscous behaviour at high stress levels. The temperature dependency of the steady-state deformation behaviour is found to be well captured by the WLF equation. It is found that confinement has a stiffening effect on the steady-state deformation behaviour which is found to be higher for the continuously graded mixture. The volumetric strain is found to increase linearly with the shear strain with a gradient that is independent of stress level and confining pressure.

Compressive Deformation Behaviour of Asphalt Mixtures Containing Incinerator Bottom Ash Aggregate

ABSTRACT This paper forms part of a programme to study the mechanical and environmental properties of asphaltic mixtures containing Incinerator Bottom Ash Aggregate (IBAA). The Compressive Deformation Behaviour (CDB) of four mixtures containing 0, 30, 60 and 80% IBAA was investigated under uniaxial and triaxial compression conditions. The steady state axial strain rates and stresses were evaluated and used to identify the CDB of the tested mixtures. Mixtures exhibited linear and non linear viscous behaviour at low and high stress levels, respectively. IBAA content was found to have a noticeable effect on the mixtures' steady state CDB. The dilation gradient was found to increase with temperature. The CDB of the tested mixtures was captured by the Modified Cross Model and was found to be linear at low stress levels while at high stress levels, the materials exhibited nonlinear power-law viscous behaviour.

Numerical simulation of non‐Newtonian blood flow in bypass models

AbstractThe article presents the numerical investigation of non–Newtonian effects of steady blood flow in complete idealized 3–D bypass models, whose native artery is either coronary or femoral with average physiological parameters. Considering the blood to be a generalized Newtonian fluid, the shear–dependent viscosity is described by two well–known macroscopic non–Newtonian models (the Carreau–Yasuda model and the modified Cross model). The results were obtained by own developed computational software based on the pseudo–compressibility approach and on the cell–centred finite volume method defined on unstructured hexahedral grids. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computer Aided Engineering Design of Powder Injection Molding Process for a Dental Scaler Tip Mold Design

Powder Injection Molding (PIM) has recently been recognized as an advanced manufacturing technology for low-cost mass production of metal or ceramic parts of complicated geometry. With this regards, design technology of dental scaler tip PIM mold, which has complex shape and a slim core pin of 0.6 mm diameter, with the help of computer-aided analysis for powder injection molding process was developed. Computer-aided analysis for dental scaler tip mold was implemented by finite element method with non-Newtonian fluid, modified Cross model viscosity, PvT data of powder/binder mixture. The core deflection analysis of dental scaler tip PIM mold during PIM filling process was also investigated. Compter-aided analysis results, such as filling pattern, weldline formation, and air vent position prediction were investigated and eventually showed good agreements with experimental results.

치과용 스케일러 금형의 분말사출성형 CAE 해석설계

Powder Injection Molding(PIM) has recently been recognized as an advanced manufacturing technology for low-cost mass production of metal or ceramic parts of complicated geometry With this regards, design technology of dental scaler tip PIM mold, which has complex shape and small core pin (diameter=0.6mm), with the help of computer-aided analysis of powder injection molding process was developed. Computer-aided analysis for dental scaler tip mold was implemented by finite element method with non-Newtonian fluid, modified Cross model viscosity, PvT data of powder/binder mixture. Compter-aided analysis results, such as filling pattern, weldline formation, air vent position prediction were compared with experimental result, and eventually have been shown good agreement. The core pin (diameter=0.6mm) deflection analysis of dental scaler tip PIM mold during PIM filling process was also investigated before mold fabrication.

Shape optimization in unsteady blood flow: A numerical study of non-Newtonian effects

This paper presents a numerical study of non-Newtonian effects on the solution of shape optimization problems involving unsteady pulsatile blood flow. We consider an idealized two dimensional arterial graft geometry. Our computations are based on the Navier–Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. Using a gradient-based optimization algorithm, we compare the optimal shapes obtained using both the Newtonian and generalized Newtonian constitutive equations. Depending on the shear rate prevalent in the domain, substantial differences in the flow as well as in the computed optimal shape are observed when the Newtonian constitutive equation is replaced by the modified Cross model. By varying a geometric parameter in our test case, we investigate the influence of the shear rate on the solution.

Phenomenological Model for Monotonic and Cyclic Behavior of Pure Bitumen

An extensive experimental study of the monotonic, recovery, and cyclic behavior of two bitumens commonly used for pavement construction in the U.K. was performed over a wide range of strain rates, stresses, and temperatures. The steady-state monotonic and cyclic behavior of the materials followed the modified Cross model with the bitumen exhibiting linear and nonlinear viscous behavior at low and high stresses, respectively. Continuous and pulse-loading cyclic tests revealed that the behavior of the bitumen is mean-stress governed in the continuous cyclic tests while strain recovery is important in the pulse-loading tests. A phenomenological model calibrated by a minimum of four tensile experiments is proposed and shown to capture the bitumen response accurately over all the loading histories (monotonic, continuous cyclic, and pulse loading) and temperatures investigated.

Shape optimization in steady blood flow: A numerical study of non-Newtonian effects

We investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks, motivated by optimal design problems in biomedical engineering. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. We apply sensitivity-based optimization procedure to a flow through an idealized arterial graft. For this problem we study the influence of the inflow velocity, and thus the shear rate. Furthermore, we introduce an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.

Prediction and measurement of residual stresses in injection molded parts

Residual stresses were predicted by a flow analysis in the mold cavity and residual stress distribution in the injection molded product was measured. Flow field was analyzed by the hybrid FEM/FDM method, using the Hele Shaw approximation. The Modified Cross model was used to determine the dependence of the viscosity on the temperature and the shear rate. The specific volume of the polymer melt which varies with the pressure and temperature fields was calculated by the Tait’s state equation. Flow analysis results such as pressure, temperature, and the location of the liquid-solid interface were used as the input of the stress analysis. In order to calculate more accurate gap-wise temperature field, a coordinate transformation technique was used. The residual stress distribution in the gap-wise direction was predicted in two cases, the free quenching and the constrained quenching, under the assumption that the shrinkage of the injection molded product occurs within the mold cavity and that the solid polymer is elastic. Effects of the initial flow rate, packing pressure, and mold temperature on the residual stress distribution was discussed. Experimental results were also obtained by the layer removal method for molded polypropylene.

Numerical analysis of a reactive extrusion process. Part I: Kinetics study on grafting of vinylsilane to polyethylene

AbstractFor controlling a reactive extrusion process in the subsequent study. the model equations of reaction kinetics and shear viscosity were studied. We focused on a free radical reaction between the molten polyethylene and vinylsilane. The kinetics model was expressed as a reaction rate equation with an apparent rate constant. The shear dependent of reacted polyethylene was formulated by employing the modified Cross model proposed in our early study. In addition, the average molecular weight was considered to correlate the shear viscosity with the reaction kinetics, leading to a series of rheo‐kinetics formulas. The experiments were carried out in a specific batch mixer suitably designed for sampling in arbitrary periods. Their reaction conversions, molecular weight distributions, and shear viscosity were measured, respectively, with an induced coupled plasma (ICP) emission spectrochemical analyzer, a high temperature gel permeation chromatography (GPC), and a capillary rheometer. Determining the parameters in each model, a simulator is set to investigate an engineering extrusion process.

Characterization of short glass fiber filled polystyrene by fiber orientation and mechanical properties

AbstractA quasi‐steady state, non‐isothermal, compressible, inelastic, and creeping flow of polymer melt into a thin cavity is analyzed to predict fiber orientation states. Modified Cross model and Tait's state equation are adopted to consider shear‐thinning behavior and compressibility of the polymer melt. Second order tensors are introduced to describe 3‐dimensional fiber orientation. Flow‐induced fiber orientation can be predicted by solving the equations of change for the orientation tensor with a suitable closure approximation. The orthotropic closure is applied except for the case of low interaction coefficient. Fiber orientation develops mainly due to shear flow in the skin layer and due to stretching effect in the core layer. It turns out that the compressibility, which induces additional velocity gradients during packing, reduces development of the fiber orientation. Results are dependent upon the magnitude of the interaction coefficient. The larger the interaction coefficient, the smaller the orientation development and the effect of compressibility. To predict orientation dependent mechanical properties, the orientation averaging for an arbitrary orientation is carried out from the properties of a transversely isotropic unit cell. The compressibility reduces the axial modulus and increases the transverse modulus. Opposite trends are observed for thermal expansion coefficients. It is also observed that the consideration of compressibility reduces the overall anisotropy of the molded product. Effects of compressibility on mechanical properties of the parts are reduced as the interaction coefficient becomes larger.

Characterization of deformation in injection molded parts after packing and cooling

AbstractResidual stresses which are developed in injection molded parts affect dimensional accuracy and mechanical properties of the final products. To predict the residual stresses in injection molded parts, three stages of injection molding, i. e., filling, packing, and cooling, must be taken into consideration for the thermal and flow analyses. Flow field anaysis for filling and postfilling stages has been carried out by using the control volume based FEM/FDM hybrid method. The generalized Hele‐Shaw flow is assumed. Compressibility of the Polymer melt is considered during packing and cooling stages. Modified Cross model is employed to reflect the dependency of the viscosity upon shear rate and temperature. An equation of state proposed by Tait offers an efficient means to describe pvt‐relationship of the polymer. Variations in temperature and pressure fields are obtained over all stages by the numerical flow analysis and used as input data for the stress analysis of the part. Plane stress elements, such as shell elements, are used for finite element stress analysis of injection molded parts with appropriate boundary conditions both in the mold and after ejected from the mold. The numerical analysis yields useful information which is relevant to the mechanical properties of the final products, e. g., residual stress distribution, shape of deformation, displacement field, and strain distribution.

Crystallinity and microstructure in injection moldings of isotactic polypropylenes. Part 1: A new approach to modeling and model parameters

AbstractThe present study is an attempt to formulate a new approach to the modeling of the injection moldings of semicrystalline polymers. A phenomenological approach is employed to develop the qualitative description of crystallization phenomena under nonisothermal flow occurring in various polymer processing operations. The unified crystallization model, which is suitable to describe the multilayer microstructure arising in the injection moldings of semicrystalline polymers, is proposed. The “competing mechanisms” of crystallization in terms of the induction time indices for introducing various microstructure layers are utilized to quantitatively distinguish between the highly oriented skin layer and spherulitic core in the moldings. The application of the equations of continuity, momentum and energy, along with the rheological model, the equation of state and the unified crystallization model, to the injection molding process leads to the modeling of crystallinity and microstructure development in the moldings. The material parameters to be used in the modeling of the crystallinity and microstructure development in the injection moldings of isotactic polypropylenes (i‐PP's) with various molecular weights are determined. The melt viscosity as a function of shear rate and temperature is measured and fitted to the modified Cross model. Quiescent crystallization kinetics, including the induction time and the heat release due to crystallization, is characterized using differential scanning calorimetry. The spherulite growth rate of i‐PP's under isothermal and nonisothermal quiescent crystallization conditions is measured by optical microscopy. The molecular deformation factor during the induction period of shear‐in‐duced crystallization is obtained from the extrusion‐quenching and extrusion‐relaxation‐quenching experiments with a slit die. The variation of the induction time with shear rate obtained by Lagasse and Maxwell from the shearing‐crystallization experiments is used to determine the shear enhancement coefficient of crystallization.

Injection Molding of a Natural Rubber Compound: Simulation and Experimental Studies

Abstract Experimental and theoretical results are obtained for the cavity filling of a natural rubber (NR) compound. The shear rate and temperature dependent viscosity function based on the modified Cross model is fitted to experimentally measured viscosity data. A nonisothermal vulcanization model with nonisothermal induction time is fitted to the nonisothermal curing kinetic data obtained by differential scanning calorimetry (DSC). These data are utilized in the simulation of the cavity filling and curing stages. The pressure traces at various locations in the mold during cavity filling are obtained at different inlet and mold temperatures. The predicted results for the pressure traces are in good agreement with those measured in the cavity, but significantly lower than those measured in the nozzle. Evolution of the state of cure in the post-filling stage is predicted and compared with experimental data measured in moldings obtained at various cycle times. The measured states of cure in the moldings are in good agreement with their predicted values. Based on earlier flow studies of SBR compounds, it is shown that the discrepancy between the experimental and predicted pressure traces in the nozzle for the NR compound is due to high local pressure drops arising during the flow in juncture regions of the delivery system and the possibility of flow-induced crystallization. The importance of taking into account the entry pressure losses, viscoelastic effects and flow-induced crystallization in the injection molding process is strongly emphasized. The complexity of the issues related to this subject is discussed.

Rheological properties of guar galactomannan and rice starch mixtures II. Creep measurements

The effect of particulate inclusions, or ‘fillers’, on the rheological properties of a typical polysaccharide entanglement solution (guar galactomannan/water) have been studied. Earlier steady shear experiments have shown an apparent upswing of the viscosity-shear rate profiles at low shear rates, with higher filler concentrations (Rayment et al., 1995). In creep experiments, a low constant stress is applied to the sample. In the present paper, the creep experimental data have been compared to the steady shear flow behaviour fitted by the yield stress modified Cross equation (introduced in Part I of this series). The apparent zero-shear viscosity and yield stress parameters have been constrained in a number of ways in the model to establish the effect of these parameters on the modified Cross model. Although the creep data appears to alter the precision of the modified Cross equation, due to the low shear rates accessed, the apparent upswing of the steady shear data at low shear rates appear to be supported by the data described in this paper. This lends further credibility to the scaled filler models introduced previously.

Compressible flow analysis of filling and postfilling in injection molding with phase-change effect

In order to predict the shrinkage, warpage and mechanical properties of the injection molded parts, it is necessary to know the history of the flow field during injection-molding processes. In the present investigation a numerical simulation program was developed to predict the flow field in filling and post-filling stages of injection molding. To simulate the real molding conditions more accurately, a generalized Hele-Shaw model for a non-Newtonian fluid was assumed considering the effects of phase change and compressibility of the resin. A finite-element-finite-difference (FEM-FDM) hybrid scheme with control volume approach was employed as the solving technique. For modeling the viscosity of the resin, a modified Cross model was used with a double-domain Tait equation of state being employed in describing the compressibility of the resin during molding. The energy balance equation, including latent-heat dissipation for semicrystalline materials, was solved in order to predict the solidified layer and temperature profile in detail. For verification of the numerical results obtained from the developed program, the simulation results were compared with the experimental results obtained from the test mold set designed in the current study using commercial-grade PP and the data available in the literature. Based on a comparison between experiments and simulations, it was found that the currently developed program was useful in unified simulations of filling and post-filling in injection-molding processes when considering the phase-change effect.

Injection Molding of Rubber Compound with Rheology Affected by Vulcanization: Part II. Modeling and Experiment

Abstract Experimental and theoretical results are reported on the effect of vulcanization on viscosity and yield stress during cavity filling with an SBR compound. Rheological and vulcanization models were utilized in simulation of the cavity filling process. Numerical schemes was based on the hybrid approach of control volume finite-element and finite-difference techniques. The pressure profiles at various locations along the flow path in the mold during cavity filling were obtained by conducting the experiments at various mold and inlet temperatures and mold injection speeds. It was found that the predicted pressure profiles followed the trend determined in the experiments. Comparing two versions of the modified Cross viscosity models — with and without the effect of cure on viscosity — it appears that use of a viscosity model which accounts for the cure-effect improves the accuracy, and thus the usefulness, of the cavity pressure prediction models. Evolution of the state-of-cure in the postfilling process are predicted and compared with the state-of-cure in moldings. The predicted results on the development of the state-of-cure during the postfilling stage show good agreement with the experimental data. The rubber compound exhibits yield behavior during cavity filling at very low injection speeds. The modified Cross model was further extended to include the yield stress and was implemented in the simulation program. In the case of low injection speed, the results showed a significant improvement in the prediction of cavity pressure, compared with the earlier reported predictions.

Numerical nonisothermal flow analysis of non‐Newtonian fluid in a nonintermeshing counter‐rotating twin screw extruder

AbstractA simulation technique for nonisothermal flow analysis of non‐Newtonian fluid in a nonintermeshing counter‐rotating twin screw extruder was developed by modifying the flow analysis network (FAN) method. The local shear viscosity in the flow field was calculated by an iteration method combing the three types of mean shear rate functions. The modified Cross model and an Arrhenius‐type function for temperature dependence were also introduced. Streamlines in the flow field are represented by computing the movements of fluid particles based on the flux fields. The computed residence time from the streamlines led us to solve the energy equation by replacing the coordinate system. The profiles of pressure, shear rate, shear viscosity, temperature, and residence time were simulated. The influence of operating and geometrical parameters on the screw characteristics are discussed. Further, residence time distribution (RTD), strain distribution function (SDF), and interfacial area growth are predicted from the computation of streamlines to analyze the mixing capabilities of the extruder.

Numerical simulation and experimental investigation of injection mold filling with melt solidification

AbstractA two‐phase model is presented for simulating the injection mold filling process including the effect of transient melt solidification, i.e., the phase change effect. The liquid region is governed by Hele‐Shaw flow for a non‐Newtonian fluid using a modified Cross model to describe viscosity under non‐isothermal conditions. Further, the energy equation of the solid phase is dominated by a transient condition. The interfacial energy balance equation is also proposed to predict the solidified layer thickness and temperature profile. Two well‐characterized semicrystalline materials, polypropylene and polyethylene, were used in the present work. Good agreement is obtained between the predicted results and experimental observations from this study and the previous literature concerning the thickness of solid layer, the shape of, advancing melt front, and the pressure traces. In particular, the predicted pressure based upon the two‐phase model is higher than that in terms of the single‐phase model by about 13 percent. Finally, the semicrystalline structure of the frozen skin layer and the central core were investigated with a scanning electron microscope to verify the two‐phase model.

Two-Dimensional Simulation of Injection Molding of Rubber Compounds

Abstract A general finite-element and finite-difference computer program has been developed for filling a thin cavity during injection molding of rubber compounds. This program employs a finite-element representation of the cavity in the planar dimensions and a finite-difference representation in the gapwise direction and is capable of accommodating any type of rheological equation and vulcanization model and incorporates both viscous heating and heat of vulcanization. A detailed example of an application of this program has been presented for filling a quarter-disk cavity with rubber compounds having a viscosity function obeying a modified Cross model and curing kinetics according to our recently proposed nonisothermal vulcanization model. The pressure, velocity, and temperature fields and melt-front locations have been calculated during the filling stage. The evaluation of state of cure distribution in the rubber moldings has been predicted in the postfilling stage. Comparisons of the prediction of the model with experimental data regarding pressure distribution, melt front position, and distribution of degree of cure in molded parts will be presented in the near future.