Leonov Model Research Articles

Theory construction and computation for polymer film melt stretching by modified Leonov model

Polymer film melt stretching exhibits a unique stress–strain behavior, featuring a stress plateau followed by a sudden increase with increasing the strain. This phenomenon poses a challenge to conventional melt-extension theories and methods. To address this, we derive a simplified flow theory according to the symmetric characteristics of thin melt-stretching, and propose a modified multi-mode Leonov model (m-Leonov) that accounts for molecular orientation effects, a crucial factor in accurately simulating the process. In order to improve the computational efficiency, an iterative algorithm was developed to decouple the interdependence between governing and constitutive equations. The computational results reveal that the viscoelastic nature of the film contributes to a delay in the transformation of stretching forces, leading to the observed stress plateau. Significantly, our m-Leonov model successfully predicts both the plateau and the subsequent abrupt increase in stress, across various temperatures and stretching rates. These simulations closely agree with experimental data, underlining the robustness of our approach. Our findings would enhance the understanding of rheological complexities in film stretching, offering potential applications in various materials and polymer processing.

Study of overflow effects on flow‐induced stress in injection molding

AbstractOverflow is usually used to keep fluid continuously flowing to avoid the fluid from changing flow direction and subsequently inducing large stresses. To explore the mechanism of flow‐induced stress evolution, a flow model was established in terms of compressible, non‐isothermal, and viscoelastic flow. Based on the Leonov model and the mid‐plane approach, a pseudo‐Poisson type equation for pressure was derived to decouple the interdependence between velocity and pressure. To improve computational efficiency, a numerical approach involving double iterations was proposed to solve the flow problem. Two kinds of mold cavities, with and without overflow, were designed and manufactured. Molding experiments were conducted, and corresponding simulations based on the proposed model and algorithm were performed. The results show that the simulated contours of flow‐induced stress are in good agreement with the photoelastic streaks. The use of overflow can significantly reduce flow‐induced stresses. If the slot thickness is set to a proper value, the flow‐induced stress can be reduced by more than 80%. On the other hand, a slot that is too thick does not have significant reducing effects. Therefore, using overflow to reduce flow‐induced stresses during transparent part production is a promising approach if the material cost is worthy of the quality.Highlights A double iteration method is proposed for viscoelastic flow problem. Overflow can reduce the stress as much as 80% if the dimension is properly set. The simulated stress streaks are in good agreement with photoelasticities.

Computational investigation into the effect of constitutive formulations for flow modeling of high elasticity

• Three constitutive formulations of the Leonov model are numerically investigated. • It is verified that keeping positive-definiteness of the conformation tensor is crucial for computational evolution while appropriate interpolation of exponentially convecting domain is essential for accuracy. • Swirling asymmetric corner vortices are described in 3D contraction flow modeling. For the purpose of defeating the high Deborah (or Weissenberg) number problems in viscoelastic flow modeling, the matrix-logarithmic formulation of the constitutive equations has been suggested and proven probably the most effective. This transformation of the conformation tensor endows proper resolution of the boundary layer with exponential convection as well as conservation of its positive definiteness. Herein we make an attempt to evaluate numerical consequences of these two features independently for the Leonov viscoelastic model mainly in the contraction flow. To this end, we from the viewpoint of computational stability and accuracy investigate three formulations written with variables such as the original conformation tensor c (c form), its matrix-logarithm (h form) and additive decomposition (b form). 2D planar 4:1 contraction flow in the limit of vanishing inertia in addition to its 1D toy problem is considered in the framework of finite element method. The h and b forms are designed to preserve the positive definiteness of c , and furthermore the h form grants efficient interpolation of exponentially convecting variables. The 1D analogy expresses the characteristics qualitatively equivalent to the original problem in a simpler fashion. The c form modeling of 2D flow exhibits typical nonevolutionary behavior when the Deborah number ( De ) exceeds a certain limit, and this degradation gets worse with finer mesh. While the h and b forms have always been computationally stable with proper description of steady behavior or elastic instability otherwise, the h form manifests faster mesh convergence. As a complementary study, the h form calculation of full 3D flow along an axisymmetric contraction channel has been performed, which also attests well-posedness without upper bound in De . Its steady result is corroborated with the one under axisymmetric flow approximation. Loss of convergence in the axisymmetric flow regime at high De is verified to result from the occurrence of 3D instability, which is then well portrayed in 3D modeling. In this instability, swirling of asymmetric structure for the corner vortices about the geometrical symmetry axis plausibly mimics experimental observation. Consequently, maintaining the positive definiteness in terms of h or b form is crucial for numerically stable evolution, and the h form interpolation delineating appropriately the exponential convection assures the fastest mesh convergence, hence enhancing the accuracy.

Computational Modeling of Polymer Matrix Based Textile Composites.

A simple approach to the multiscale analysis of a plain weave reinforced composite made of basalt fabrics bonded to a high performance epoxy resin L285 Havel is presented. This requires a thorough experimental program to be performed at the level of individual constituents as well as formulation of an efficient and reliable computational scheme. The rate-dependent behavior of the polymer matrix is examined first providing sufficient data needed in the calibration step of the generalized Leonov model, which in turn is adopted in numerical simulations. Missing elastic properties of basalt fibers are derived next using nanoindentation. A series of numerical tests is carried out at the level of yarns to promote the ability of a suitably modified Mori–Tanaka micromechanical model to accurately describe the nonlinear viscoelastic response of unidirectional fibrous composites. The efficiency of the Mori–Tanaka method is then exploited in the formulation of a coupled two scale computational scheme, while at the level of textile ply the finite element computational homogenization is assumed, the two-point averaging format of the Mori–Tanaka method is applied at the level of yarn to serve as a stress updater in place of another finite element model representing the yarn microstructure as typical of FE based multiscale approach. Several numerical simulations are presented to support the proposed modeling methodology.

Quantitative insights into rheology of self-crosslinking dialdehyde carboxymethyl cellulose/collagen solution: A comprehensive experimental and modelling investigation

Rheological behaviors (fluid type, thixotropy, viscoelasticity and deformation resistance) of self-crosslinking dialdehyde carboxymethyl cellulose (DCMC)/collagen solutions were comprehensively investigated and quantitatively modelled. The formation of self-crosslinking bonds gave rise to the enhancement of entanglements among collagen molecules and aggregates, resulting in the improvement of collagen solution's mechanical properties. The collagen solution was typical pseudoplastic fluid with shear-thinning behavior, and DCMC did not alter its flow behavior; therefore, the flow curves for all samples could be well simulated by Ostwald-de Waele and Carreau models. However, the viscosity and pseudoplasticity increased, and the effect of DCMC became more pronounced at higher DCMC/collagen ratios. Similar upward variation was observable for thixotropy, detected by the increased area of hysteresis loops from 20.947 to 219.978 Watts/m3. Additionally, the negative values of apparent yield stress and flow index below 1 from Herschel–Bulkley model confirmed that the DCMC/collagen solutions remained pseudoplasticity without yield stress. The viscoelasticity was examined based on dynamic frequency sweep and creep–recovery measurement and then simulated using Leonov model, Burger model and a semi-empirical model, respectively. The increases in elastic moduli and proportion of recoverable compliance reflected the self-crosslinking DCMC/collagen solutions had stronger elasticity and exhibited more resistance to deformation than the native collagen solution. Nevertheless, the more complicated structures of the DCMC/collagen solutions required more time to recover after removing external force. These data can be directly applied to the utilization of DCMC/collagen solutions, whilst also provide a basis for the prediction of printability or spinnability of collagen-based materials.

Constitutive Modeling of Nonlinear Rheological Behavior of Carbon Nanotube-Filled Polypropylene Nanocomposites

Abstract The rheological behavior of multi-walled carbon nanotube (MWCNT)-filled polypropylene (PP) nanocomposites with different filler loadings was experimentally studied and simulated using constitutive modeling. Rheological behavior was characterized in small amplitude oscillatory shear (SAOS) flow, large amplitude oscillatory shear (LAOS) flow, startup of shear flow, steady shear flow, and stress relaxation after the imposition of a step shear strain. Virgin PP and PP with CNT loadings of 1, 3, and 5 wt% were used. The formation of a rheological percolation network was observed at these loadings. The Leonov and Simhambhatla-Leonov models were used to simulate the rheological behavior. In the linear region, the simulations provided good predictions of the experimental data for both the unfilled and filled PP. In the nonlinear region, the simulations also provided good results for the virgin PP and satisfactory results for the PP/1 wt%CNT nanocomposite under most flow conditions. However, for the other two nanocomposites the model showed mixed results.

APPLICATION OF THE MORI-TANAKA METHOD TO DESCRIBE THE RATE-DEPENDENT BEHAVIOR OF UNIDIRECTIONAL FIBROUS COMPOSITES

This paper examines the possibility of using the Mori-Tanaka micromechanical model describe the rate dependent behavior of the polymer matrix based fibrous composites. The generalized Leonov model is adopted to capture the time and rate dependent character of the selected matrix, while fibers are assumed elastic. The performance of the Mori-Tanaka method is tested against the finite element simulations carried out in the framework of first-order homogenization. For simplicity, the periodichexagonal array model is chosen to represent the microstructural arrangement of fibers in the yarn cross-section. To match the predictions provided by the two approaches a suitable modification to the original Mori-Tanaka method is proposed. An extensive parametric study is presented to illustrate a considerable improvement of the predictive capability of the modified Mori-Tanaka method.

Correlation between linear and nonlinear material functions under large amplitude oscillatory shear

Fourier transform rheology is the most frequently used method to interpret the nonlinear rheological behavior of complex fluids under large amplitude oscillatory shear (LAOS). However, the unclear relationship between the higher harmonics and the fundamental harmonic obscures the physical meaning of the nonlinear functions. Here, we hypothesize that all the nonlinear oscillatory shear functions and normal stress functions can be expressed as linear combinations of linear viscoelastic functions or their derivatives at different frequencies under both strain-controlled LAOS (LAOStrain) and stress-controlled LAOS (LAOStress). We check this hypothesis using the time-strain separable Wagner model, Giesekus model, and modified Leonov model. We find such correlations between the nonlinear material functions and the linear material functions are intrinsic for viscoelastic liquids under LAOStrain, and for viscoelastic solids under LAOStress. Finally, these correlations are justified by a viscoelastic standard polydimethylsiloxane, an ethylene–octene multiblock copolymer melt, and a typical simple yield stress material (0.25 wt. % Carbopol).

Effect of heat transfer coefficient, draw ratio, and die exit temperature on the production of flat polypropylene membranes

In this work, a stable numerical scheme has been developed for the 1.5-dimensional film casting model of Silagy et al. [Polym. Eng. Sci. 36, “Study of the stability of the film casting process,” 2614–2625 (1996)] utilizing the viscoelastic modified Leonov model as the constitutive equation and energy equation coupled with the crystallization kinetics of semicrystalline polymers taking into account actual temperature as well as cooling rate. The model has been successfully validated on the experimental data for linear isotactic polypropylene taken from the open literature. Drawing distance, draw ratio, heat transfer coefficient, and die exit melt temperature were systematically varied in the utilized model in order to understand the role of process conditions in the neck-in phenomenon (unwanted film width shrinkage during stretching in the post die area) and crystalline phase development during flat film production. It is believed that the utilized numerical model together with the suggested stable numerical scheme as well as obtained research results can help to understand a processing window for the production of flat porous membranes from linear polypropylene considerably.

Effect of die exit stress state, Deborah number, uniaxial and planar extensional rheology on the neck-in phenomenon in polymeric flat film production

In this work, effect of the second to first normal stress difference ratio at the die exit, −N2/N1, uniaxial extensional strain hardening, ηE,U,max3η0, planar-to-uniaxial extensional viscosity ratio, ηE,PηE,U, and Deborah number, De, has been investigated via viscoelastic isothermal modeling utilizing 1D membrane model and a single-mode modified Leonov model as the constitutive equation. Based on the performed parametric study, it was found that there exists a threshold value for De and ηE,U,max3η0, above which, the neck-in starts to be strongly dependent on −N2/N1. It was found that such critical De decreases if −N2/N1, ηE,U,max3η0 increases and/or ηE,PηE,U decreases. Numerical solutions of the utilized model were successfully approximated by a dimensionless analytical equation relating the normalized maximum attainable neck-in with ηE,U,max3η0, ηE,PηE,U, −N2/N1 and De. Suggested equation was tested by using literature experimental data considering that −N2/N1 depends on die exit shear rate, temperature and utilized constitutive model parameters for given polymer melt. It was found that approximate model predictions are in a very good agreement with the corresponding experimental data for low as well as very high Deborah numbers, at which neck-in strongly depends on −N2/N1. It is believed that the obtained knowledge together with the suggested simple model can be used for optimization of the extrusion die design (influencing flow history and thus die exit stress state), molecular architecture of polymer melts and processing conditions to suppress neck-in phenomenon in production of very thin polymeric flat films.

Natural rubber/styrene-butadiene rubber blends prepared by solution mixing: Influence of vulcanization temperature using a Semi-EV sulfur curing system on the microstructural properties

Blends of natural rubber (NR) and styrene-butadiene rubber (SBR) were prepared by solution mixing and vulcanized with sulfur and accelerator in a Semi-EV system at 433 K and 443 K in order to study the vulcanization kinetic and the influence of vulcanization temperature on final structure of the blends. The vulcanization kinetic studied through the variation in rheometer curves was analyzed using the Ding and Leonov model, which takes into account the reversion effect during the cure process. The average free nanohole volume and the fractional free volume of samples with different NR/SBR ratio were estimated using positron annihilation lifetime spectroscopy (PALS). Also, the crosslink density was determined by means of swelling tests in a solvent. For all the compounds, a correlation between the free nanohole volume and the delta torque obtained from the respective rheometer curves was established.

On the role of extensional rheology and Deborah number on the neck-in phenomenon during flat film casting

In this work, viscoelastic, isothermal extrusion film casting simulations have been performed utilizing a 1D membrane model and the viscoelastic modified Leonov model as the constitutive equation in order to elucidate the role of planar to uniaxial extensional viscosity ratio, extensional strain hardening and Deborah number on the neck-in phenomenon. Based on the performed theoretical parametric study, it has been found that neck-in can be correlated to all the above mentioned variables via a simple dimensionless analytical equation. This correlation can provide detailed view into the complicated relationship between polymer melt rheology, die design, process conditions and undesirable neck-in phenomenon. Obtained results have been validated against literature experimental data for different polyethylene melts and processing conditions.

Insights into the rheological behaviors evolution of alginate dialdehyde crosslinked collagen solutions evaluated by numerical models

The elaboration of the rheological behaviors of alginate dialdehyde (ADA) crosslinked collagen solutions, along with the quantitative analysis via numerical models contribute to the controllable design of ADA crosslinked solution system's fluid mechanics performance during manufacturing process for collagen biomaterials. In the present work, steady shear flow, dynamical viscoelasticity, creep-recovery, thixotropy tests were performed to characterize the rheological behaviors of the collagen solutions incorporating of ADA from the different aspects and fitted with corresponding numerical models. It was found that pseudoplastic properties of all samples enhanced with increasing amounts of ADA, which was confirmed by the parameters calculated from the Ostwald-de Waele model, Carreau and Cross model, for instance, the non-Newtonian index (n) decreased from 0.786 to 0.201 and a great increase by 280 times in value of viscosity index (K) was obtained from Ostwald-de Waele model. The forth-mode Leonov model was selected to fit all dynamic modulus-frequency curves due to its higher fitting precision (R2>0.99). It could be found that the values of correlation shear viscosity (ηk) increased and the values of relaxation time (θk) decreased with increasing ADA at the fixed k value, suggesting that the incorporation of ADA accelerated the transformation of the collagen solutions from liquid-like to gel-like state due to more formation of CN linkages between aldehyde groups and lysine residues. Also, the curves of creep and recovery phase of the native and crosslinked collagen solutions were simulated well using Burger model and a semi-empirical model, respectively. The ability to resist to deformation and elasticity strengthened for the samples with higher amounts of ADA, accompanied with the important fact that compliance value (J50) decreased from 56.317Pa−1 to 2.135Pa−1 and the recovery percentage (Rcreep) increased from 2.651% to 28.217%. Finally, it was found that the area of thixotropic loop increased from 8.942Pa/s to 17.823Pa/s with increasing introduction of ADA, suggesting that stronger thixotropic behavior was associated with higher amount of ADA. Furthermore, Herschel–Bulkley model was employed to describe the up and down curves of all samples and it was confirmed that all collagen solutions belonged to pseudoplastic fluid (the flow index<1) without apparent yield stress and shear-thinning behaviors were more obvious with increasing additions of ADA according to the increasing consistency coefficient K values. Overall, this work contributed a new insight into the interactions between collagen and ADA based on quantitative rheological methods reflecting the different rheological properties and the results obtained should be of great utility in the extensive application of ADA crosslinked collagen solutions into diverse collagen-based materials.

Measurements and modeling of temperature-strain rate dependent uniaxial and planar extensional viscosities for branched LDPE polymer melt

Abstract In this work, novel rectangle and circular orifice (zero-length) dies have been utilized for temperature-strain rate dependent planar and uniaxial extensional viscosity measurements for the LDPE polymer melt by using standard twin bore capillary rheometer and Cogswell model and the capability of five different constitutive equations (novel generalized Newtonian model, original Yao model, extended Yao model, modified White-Metzner model, modified Leonov model) to describe the measured experimental data has been tested. It has been shown that chain branching causes the strain hardening occurrence in both uniaxial and planar extensional viscosities and its maximum is shifted to the higher strain rates if the temperature is increased. The level of uniaxial extensional strain hardening for the branched LDPE sample has been found to be higher in comparison with the planar extensional viscosity within wide range of temperatures.

Rheological Modeling of Polymeric Melts in Extruders upon Two-Dimensional Shear Flow

Abstract The rheological modeling of a polymer melt in the course of its extrusion through a circular cylindrical channel inside a shaping die equipped with a rotating mandrel and a rotating or stationary nozzle is investigated. This paper attempts to provide a detailed rheological description of the physical processes of a polymer melt based on Leonov model inside the channel of an extruder under its kinematic deformation upon two-dimensional shear flow. Based on the obtained rheological equations of state, the flow process can be described in a form of dependencies reflecting kinematics of complex polymer deformation in the die channel. These expressions allow estimation of the velocity distribution for the flow elements within the channel as well as of the flow-pressure characteristics of the process. A quantitative relation between flow characteristics, rheological properties of polymer, and technical parameters of the channel of an extruder was determined. Comparison of this theoretical model with experimental results validates the suggested model.

Rheological behavior of acylated pepsin-solubilized collagen solutions: Effects of concentration

Effects of concentration on the rheological behavior of acylated pepsin-solubilized collagen solutions were investigated by steady shear tests, dynamic frequency sweep, creep tests and thixotropic loop measurements in this paper. The results showed that both acylated collagen and native collagen solutions exhibited the typical pseudoplastic behavior and displayed shear thinned behavior with the increase of shear rate. With the increase of acylated collagen concentrations from 5 to 10 mg/mL, shear viscosity, elasticity modulus (G′), viscous modulus (G″), complex viscosity (η*), and the ability to resist deformation increased due to the physical entanglement, whilst loss tangent (tan δ) decreased. Additionally, with the increase of acylated collagen concentrations, the area of thixotropic loop increased from 6.94 to 44.40 watts/m3, indicating that the thixotropy of acylated collagen increased. Compared with native collagen solution, acylated collagen solution had stronger shear viscosity, η*, thixotropy, and ability to resist deformation. Furthermore, Power law model, Carreau model, Cross model, Leonov model and Burger model, were suitable for the fitting of the experimental data.

Modified Single-Mode Leonov Rheological Equations for Polymer Melts and Solutions

The shear viscosity and the normal stress coefficients are important parameters in the flow of polymer melts and polymer solutions. Based on the Leonov model, modified single-mode rheological equations are presented by introducing relaxation time and temperature functions, and the shear viscosity and the normal stress coefficients are predicted. Without a complex statistical simulation, the experimental data of a low-density polyethylene melt, a poly(ethylene oxide) solution and a mixed decalin/polybutene oil solution were compared to verify the modified equations in very wide range of deformation rates. Furthermore, based on the equations, the relationship between the stress overshoot and the temperature is discussed. In addition, the predicted shear thinning behavior for the modified equations is also compared with other single-mode models.

Predictive/fitting capabilities of differential constitutive models for polymer melts—reduction of nonlinear parameters in the eXtended Pom-Pom model

A phenomenological modification of the eXtended Pom-Pom (XPP) model is proposed with the aim to reduce the number of free nonlinear parameters. The modified XPP model includes three parameters per mode in total (two linear viscoelastic parameters—linear relaxation time λ and shear modulus G, and one nonlinear parameter). The original XPP model contains five parameters (two linear viscoelastic parameters and three nonlinear ones, one nonlinear parameter participates in the second normal stress difference prediction). The predictive/fitting capabilities of the modified model are compared with the Giesekus, eXtended Pom-Pom, and modified Leonov models using various low-density PE materials in steady and transient shear and uniaxial elongational flows. It has been found that the modified model is capable of predicting/fitting the rheological properties, with the exception of the second normal stress difference, for studied LDPE materials with sufficient accuracy, including strain hardening in uniaxial elongational flow.

Numerical aspects in modeling high Deborah number flow and elastic instability

Investigating highly nonlinear viscoelastic flow in 2D domain, we explore problem as well as property possibly inherent in the streamline upwinding technique (SUPG) and then present various results of elastic instability. The mathematically stable Leonov model written in tensor-logarithmic formulation is employed in the framework of finite element method for spatial discretization of several representative problem domains. For enhancement of computation speed, decoupled integration scheme is applied for shear thinning and Boger-type fluids. From the analysis of 4:1 contraction flow at low and moderate values of the Deborah number (De) the solution with SUPG method does not show noticeable difference from the one by the computation without upwinding. On the other hand, in the flow regime of high De, especially in the state of elastic instability the SUPG significantly distorts the flow field and the result differs considerably from the solution acquired straightforwardly. When the strength of elastic flow and thus the nonlinearity further increase, the computational scheme with upwinding fails to converge and evolutionary solution does not become available any more. All this result suggests that extreme care has to be taken on occasions where upwinding is applied, and one has to first of all prove validity of this algorithm in the case of high nonlinearity. On the contrary, the straightforward computation with no upwinding can efficiently model representative phenomena of elastic instability in such benchmark problems as 4:1 contraction flow, flow over a circular cylinder and flow over asymmetric array of cylinders. Asymmetry of the flow field occurring in the symmetric domain, enhanced spatial and temporal fluctuation of dynamic variables and flow effects caused by extension hardening are properly described in this study.

Simulation Study of Thermotropic LCPs and Prediction of Normal Stress Difference at High Shear Rate

Abstract The shear viscosity and normal stress difference of two filled and two unfilled thermotropic liquid crystal polymer (TLCPs) were studied. The rigid and rod like molecules of TLCPs orientate differently at different shear rates. Under low shear rate, the molecules tend to align in the direction of the flow but also tumble and wagging on their own axis. The abnormal orientation of the molecules also depends upon temperature, fillers contents, aspect ratio and elastic nature of LCP molecules. These behaviors lead to unusual rheological properties of LCPs, such as negative first normal stress difference for filled LCPs at low shear rates. But with the increase of shear rate, the molecules are oriented in the direction of flow, which lead to isotropic flow at high shear rates. The complicated rheological properties and characteristic anisotropic properties of LCPs are modelled by recently developed Leonov's viscoselastic constitutive equations. Simulation has been carried out using Mathematica software and the characteristic anisotropic properties of LCPs have been identified. The experimentally measured viscosities at high shear rate have been compared with model predictions. Moreover, the normal stress differences using at high shear rates have been estimated using Leonov's model, which is experimentally not accessible.