Nonlinear viscoelastic response in two dimensions—numerical modeling and experimental verification

Recently we developed a nonlinear viscoelastic constitutive model which called the Phases Model (PHM) [10]. The main advantages of the PHM over existing nonlinear viscoelastic models are that it lends itself to simple, straightforward calibration of the material functions, and that it is naturally suitable to step-by-step computer simulation. In this paper we apply the PHM to two dimensional (2-D) situations. To this end, we've developed a 2-D finite difference code in cylindrical geometry which is based on the PHM. We demonstrate the validity of our modeling by applying the code to a 2-D set-up developed for this purpose. The 2-D specimen is in the form of a disc of the test material compressed between two rigid metal anvils by a velocity-controlled Instron machine. We monitor the response of the viscoelastic disc in terms of the overall axial force history, and the axial stress component at the disc center. We get good agreement between theory and experiment.     

Constitutive modeling for mechanical behavior of PMMA microcellular foams

Constitutive equations for nonlinear tensile behavior of PMMA foams were studied. Five viscoelastic models composed of elastic and viscous components were accounted for the modeling of the constitutive equations. The developed constitutive equations are expressed in terms of material properties and foam properties such as strain, strain rate, elastic modulus, relative density of foam, and relaxation time constant. It was found that the stress-strain behaviors by Generalized Maxwell model, Three Element model and Burgers model could be described by the constitutive equation obtained from the Maxwell model. For the verification of the constitutive model, poly(methyl methacrylate) (PMMA) microcellular foams were manufactured using batch process method, and then uniaxial tensile tests were performed. The stress-strain curves by experiment were compared with the theoretical results by the constitutive equation. It was demonstrated that nonlinear tensile stress-strain behaviors of PMMA foams were well described by the constitutive equation.     

A Comparison of Linear with Nonlinear Viscoelastic Solutions for Shear Stress Concentration in Double Lap Joints

The correspondence principle based on the Maxwell model and a nonlinear viscoelastic solution involving an iterative scheme are used to describe the time dependent variation of the adhesive maximum shear stress in adhesively bonded double lap joints. The results indicate that if the correspondence principle is applied, the use of Maxwell chain is necessary to approximate the continuous change in the relaxation time and to coincide with the results calculated using the nonlinear viscoelastic theory.     

A Comparison of Linear with Nonlinear Viscoelastic Solutions for Shear Stress Concentration in Double Lap Joints

The correspondence principle based on the Maxwell model and a nonlinear viscoelastic solution involving an iterative scheme are used to describe the time dependent variation of the adhesive maximum shear stress in adhesively bonded double lap joints. The results indicate that if the correspondence principle is applied, the use of Maxwell chain is necessary to approximate the continuous change in the relaxation time and to coincide with the results calculated using the nonlinear viscoelastic theory.     

Time-dependent rheological properties and transient structural states of entangled polymeric liquids—A kinetic network model

A kinetic network model has been developed to describe the nonlinear rheological behavior of entangled polymer solutions and melts in either steady-state or transient flows. This model is based on the concept of flow-induced structure variation, controlled by the simultaneous existence of entanglement loss and regeneration. When steady-state is attained, both processes occur at the same rate and the structural state (entanglement density) of the fluid remains constant with time. Under transient flow conditions, the fluid structure becomes time-dependent as a result of unequal rates of the two competing processes affecting entanglement density. The viscoelastic response of the fluid is, in turn, influenced by the varying structural state. This is incorporated into the model through the use of structure-dependent coefficients in the contravariant Maxwell constitutive equation for entanglement stress. Model predictions are generated for a number of transient flow programs, including the stress growth and relaxation, interrupted flow, sudden change in shear rate, and nonlinear creep experiments. Comparison with literature data supports not only the qualitative success but also the quantitative ability of this model.     

A computer study of film blowing

This paper extends Petrie's work on film-blowing to cover viscoelastic non-isothermal flow for both the convected Maxwell and the Leonov models. Good agreement with experimental data is shown for the former. The calculations are highly unstable, however, and it is difficult to obtain convergence with arbitrary values of the film-blowing parameters. The general effect of viscoelasticity is to stiffen the film and restrain the increase in bubble diameter. There is some difference between bubble shapes generated using the Maxwell and Leonov constitutive models but the temperature variation of properties is seen as the dominant effect, The Leonov model tends not to be stiff enough in its response to the present flow, which is purely extensional and good agreement with experiment was not obtained. For the Maxwell model the relevant mean relaxation time must be used for good agreement between experiment and calculation.     

Semi-analytical solution for electro-thermo-mechanical non-stationary creep behaviour of rotating disk made of nonlinear piezoelectric polymer

Electro-thermo-mechanical non-stationary creep response of a rotating disk made of nonlinear polymeric piezoelectric material has been investigated. The viscoelastic properties of the material are time, stress and temperature dependent which vary along radius. The long-term creep constitutive equation is the Burgers viscoelastic model. A non-homogeneous differential equation with variable coefficients is derived using stress-displacement relations, equilibrium equation, charge equation of electrostatics and the Maxwell equation. Time-dependent creep strains are involved in the non-homogeneous term of the differential equation. A semi-analytical solution has been developed to obtain displacement, stresses, strains and electric potential in terms of creep strains. Then, Prandtl–Reuss relations and the creep constitutive model are employed in a novel numerical procedure based on the Mendelson method to obtain history of displacement, stresses, electric potential and strains. It has been concluded that the displacement is increasing with time while effective stresses are decreasing. The results are validated by finite element methods modelling using ABAQUS software. A very good agreements between the results can be observed.     

2-D Hydro-Viscoelastic Model for Convective Drying of Highly Deformable Saturated Product

The aim of this work was to simulate in two-dimensions the spatio-temporal evolution of the moisture content, the temperature, and the mechanical stress within a highly deformable and water saturated product during convective drying. The material under study was an elongated potato sample with a square section placed in hot air flow. A comprehensive hydro-thermal model had been merged with a mechanical model, assuming a viscoelastic material, a plane deformation, and an isotropic linear hydric-shrinkage of the sample. This model was validated on the basis of the average water content and core temperature curves for drying trials under different operating conditions. The material viscoelastic properties were measured by means of stress relaxation tests at different water contents. The viscoelastic behavior was described by a generalized Maxwell model whose parameters were correlated to water content. The simulations of the spatio-temporal distributions of mechanical stress were performed and interpreted in terms of product potential damage. The sample shape was also predicted all aver the drying process with reasonable accuracy.     

Simulation of nonisothermal flow of melt during melting process of vibration‐induced polymer extruder

A simplified 2D melt film model was established to simulate the nonisothermal melt flow during the melting process of the vibration‐induced polymer extruder of which the screw can vibrate axially. Since polymer has time‐dependent nonlinear viscoelastic characteristic with vibration force filed (VFF), a self‐amended nonisothermal Maxwell constitutive equation that can reflect the relaxation time spectrum of polymer was adopted. Using the 2D melt film model, melt films of two kinds of thickness representing different melting stages were simulated to investigate the influence tendency of the same VFF on the different melting stage. Special flow patterns and temperature distribution of melt in the melt film between the driving wall and the solid/melt interface with various vibration force fields were systematically simulated. It is found out that within a certain range of vibration strength, the application of vibration can optimize the time‐averaged shear‐rate distribution, improve the utilization efficiency of energy, and promote melting process; and the thinner the melt film is, the more intense the nonlinear viscoelastic response becomes with the same VFF; moreover, there exists optimum vibration strength to make the melting process fastest, which is in accord with the visualization experimental results. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 102: 5825–5840, 2006     

Investigation and prediction on the nonlinear viscoelastic behaviors of nylon1212 toughened with elastomer

Studies on the nonlinear viscoelastic behaviors of nylon1212 toughened with styrene‐[ethylene‐(ethylene‐propylene)]‐styrene block copolymer (SEEPS) were carried out. The linear relaxation curves at relatively low shear strains show good overlap, the relaxation time and modulus corresponding to the characteristic relaxation modes were also acquired through simulating the linear relaxation modulus curves using Maxwell model. The nonlinear relaxation curves of nylon1212 blends at different shear strains have been obtained and their damping functions were evaluated. Meanwhile, it is found that most blends in the experimental windows follow the strain‐time separation principle and Laun double exponential model can predict damping curves well. The successive start‐up of shear behavior was investigated. The results showed that Wagner model, derived from the K‐BKZ (Kearsley‐Bernstein, Kearsley, Zapas) constitutive equation, could simulate the experiment data of nylon 1212 blend with 10 wt % SEEPS well, but there exists some deviation for experiment data of nylon1212 blends with high SEEPS concentrations. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Modeling the viscoelasticity of polyetherimide

Viscoelasticity is a mechanical phenomenon where the material modulus varies with time and temperature. Modern experimental methods can determine material properties within certain time and temperature ranges, but modeling the viscoelastic behavior remains challenging, mainly because the data processing is complex and different materials have distinct properties. Using polyetherimide as an example and based on the change in the secondary bonds of polyetherimide in different viscoelastic stages, we proposed a new shift factor model in Arrhenius format with alterable activation energy. We also used two methods based on nonlinear least squares to obtain the Maxwell model of the polyetherimide, and we then used a novel method integrated with Laplace transforms and partial fraction decomposition to convert the Maxwell model into the Voigt model. The results of our model are reliable and self‐consistent, showing its potential for modeling the viscoelasticity of other materials. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018 , 135, 46102.     

Modeling and simulation of conditionally volume averaged viscoelastic two‐phase flows

Conditional volume averaging is used to develop a model capable of simulating two‐phase flows of viscoelastic fluids with surface tension effects. The study is started with the single‐phase mass and momentum balances, which are subsequently conditionally volume averaged. In doing so, we arrive at a set of equations having unclosed interfacial terms, for which closure relations for viscoelastic fluids are presented. The resulting equations possess a structure similar to the single‐phase equations; however, separate conservation equations are solved for each phase. As a result, each phase has its own pressure and velocity over the entire domain. Next, our numerical implementation is briefly outlined. We find that a Poiseuille single‐phase flow is predicted correctly. The closure terms are examined by considering a two‐phase shearing flow and a quiescient cylinder with surface tension. A convergence analysis is performed for a steady stratified two‐phase flow with both phases being viscoelastic. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3914–3927, 2013     

Nonlinear viscoelastic model for the prediction of double yielding in a linear low‐density polyethylene film

Tensile testing and tensile creep experiments for linear low‐density polyethylene in a thin‐film form were examined and analyzed in terms of a nonlinear viscoelastic model. The proposed model, based on two distinct thermally activated rate processes (Eyring models), was proved to describe the double‐yield‐point tensile behavior of the material tested. The required model parameters were evaluated from the corresponding creep‐strain curves, and this revealed the relationship between the main aspects of the inelastic behavior of polymers, that is, the monotonic loading and creep response. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 91: 3519–3527, 2004     

黏弹流体动态黏度数学模型与实验研究

针对经典Maxwell本构模型的局限性进行了相应的非线性修正,提出了一种新的黏弹本构模型,利用修正模型和基于黏性耗散机理建立的黏滞动力方程,得到了振动剪切流场黏弹流体的动态黏度函数.通过将动态黏度函数应用于简单振动剪切流,得到了与传统计算方法相一致的结果.最后,进行了同轴圆筒动态流变实验.实验数据与垂直叠加振动剪切流场中LDPE熔体黏度的理论值进行了对比,发现动态黏度函数在小振幅范围内具有较好的预测能力.     

Rheological properties of aqueous α A12O3 suspensions: Influence of dispersant concentration

The effects of concentration of polyacrylic acid as a dispersant on rheological properties of aqueous alumina suspensions have been investigated under steady and oscillatory shear conditions. At solid volume fractions between 0.45 and 0.6, a high degree of particle stabilization is achieved when 0.2 wt% of polyacrylic acid is added. At lower dispersant concentrations, suspensions exhibit pronounced irreversible thixotropic behaviour, whereas at higher dispersant concentrations, time dependent effects on the flow properties are not detectable. When the saturation adsorption limit of the polyelectrolyte on Al₂O₃ is reached, further addition of the dispersant appreciably changes the flow behaviour, as well as the viscoelastic response of investigated suspensions. The data under steady shear are described by application of the generalized Casson model, and for the analysis of viscoelastic data the generalized Maxwell model is used.     

Model prediction of non-symmetric normal stresses under oscillatory squeeze flow

The non-symmetric responses of normal stresses in oscillatory squeeze flow have been investigated with model calculations. The simplest and most widely used constitutive equations were employed to predict the non-symmetric normal stresses, which is a distinctive feature of oscillatory squeeze flow. The model prediction was compared with experimental data of polymer solution in terms of stress shape, Lissajous plot, stress decomposition, and Fourier transformation. The upper-convected Maxwell, Giesekus, and exponential Phan-Thien Tanner models predicted the nonsymmetric characteristics of normal stresses under oscillatory squeeze flow. The predictions showed fairly good agreement with experimental data. However, the upper-convected Maxwell model showed unrealistic result in the Lissajous plot of [stress vs. strain] and [stress vs. strain rate]. From stress decomposition, it could be confirmed that the non-symmetric nature arises from the elastic contribution of the normal stress, which was verified in both experiment and model calculation. This study is expected to provide useful insights for further understanding of the nonlinear and non-symmetric characteristics of oscillatory squeeze flow.     

Some new operational modes and parameters of stress relaxation for the viscoelastic characterization of solid polymers. III. The area ratio mode and the intrinsic “strain–clock” function

The introduced area ratio mode of operation with its corresponding parameters seems to have a fairly high sensitivity to the viscoelastic response of the solid polymer. This appeared from the fact that a good distinction among the linear viscoelastic, the nonlinear viscoelastic, and the viscoplastic ranges of behavior can be made. By using a relevant rheological modeling and its corresponding algorithmical approach, in the case of isotactic polypropylene, this material can be characterized as a morphological three‐phase material consisting of an intraspherulitic crystalline, an amorphous phase, and a interspherulitic para‐crystalline phase. In this sense, the material was simulated using two models: the Poynting–Thomson and the Maxwell–Wierchert, from where a good response of the material to the first model appeared. The so‐called intrinsic “strain–clock” function and its corresponding coefficient of strength of nonlinear viscoelastic behavior, which were relieved by the experimental data, seem to be some powerful and very practical “tools” that can give a proven suplementary characterization of the material. Finally, by this intrinsic function, the existence of permanent internal stresses, was confirmed, in an indirect way, which was mentioned in part II of this study. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 87: 149–158, 2003     

Determination of parameters in convected Maxwell model from linear viscoelastic parameters

Owing to its simplicity and general ability to correctly portray a number of important flow phenomena, the two-parameter Maxwell model has been employed in a number of important engineering studies. The relation of this equation to linear viscoelasticity and to some molecular theories is considered. A new rule is developed which shows how the shear-dependent relaxation time and viscosity of the Maxwell model can be determined from linear viscoelastic parameters. It is thus shown that the two-parameter Maxwell model may be more general than earlier anticipated.     

Tensile creep of a long‐fibre glass mat thermoplastic (GMT) composite. II. Viscoelastic‐viscoplastic constitutive modeling

In Part I of this article, the short‐term tensile creep of a 3‐mm‐thick continuous long‐fibre glass mat thermoplastic composite was characterized and found to be linear viscoelastic up to 20 MPa. Subsequently, a nonlinear viscoelastic model has been developed for stresses up to 60 MPa for relatively short creep durations. The creep response was also compared with the same composite material having twice the thickness for a lower stress range. Here in Part II, the work has been extended to characterize and model longer term creep and recovery in the 3‐mm composite for stresses up to near failure. Long‐term creep tests consisting of 1‐day loading followed by recovery were carried out in the nonlinear viscoelastic stress range of the material, i.e., 20–80 MPa in increments of 10 MPa. The material exhibited tertiary creep at 80 MPa and hence data up‐to 70 MPa has been used for model development. It was found that viscoplastic strains of about 10% of the instantaneous strains were developed under load. Hence, a non‐linear viscoelastic–viscoplastic constitutive model has been developed to represent the considerable plastic strains for the long‐term tests. Findley's model which is the reduced form of the Schapery non‐linear viscoelastic model was found to be sufficient to model the viscoelastic behavior. The viscoplastic strains were modeled using the Zapas and Crissman viscoplastic model. A parameter estimation method which isolates the viscoelastic component from the viscoplastic part of the nonlinear model has been developed. The model predictions were found to be in good agreement with the average experimental curves. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers