Effect of temperature on dynamic rheological properties of uncured rubber materials in both the linear and the nonlinear viscoelastic domains

Investigating how rubber materials are affected by strain and strain rate (or frequency) in the room‐to‐curing temperature range allows to fully characterize them not only with respect to their processing behavior but also in view of their likely performances after vulcanization. Using a closed cavity dynamic rheometer, the adequate test protocols and data treatment, three gum elastomers, and three carbon black filled compounds were investigated in the 60–180°C range. The article describes the test protocol and the associated data treatment that were developed to document the effects of strain and temperature in both the linear and the nonlinear domains. Complex modulus G* and third relative torque harmonic variations with strain amplitude and temperature are reported and discussed in details. A set of relatively simple mathematical equations are demonstrated to offer the possibility to summarize large quantities of experimental data through a limited number of parameters whose physical meaning can be explained with respect to known aspects of polymer and rubber sciences. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

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     

Constitutive modeling of visco-hyperelastic behavior of double-network hydrogels using long-term memory theory

Double-network hydrogels with viscoelastic behavior are appropriate materials for biomechanical applications. In this article, the standard linear solid (SLS) rheological model for the linear viscoelastic materials is generalized to the viscoelastic materials with large nonlinear deformations. Based on this viewpoint, the constitutive equation is proposed as sum of two parts including the strain-dependent elastic stress, and the viscous stress, which depends on the strain and strain rate. The elastic part of the stress is modeled via considering a hyperelastic strain energy function, while the main core of the viscous stress part requires a time-dependent weight function to satisfy the long-term memory fading principle. In addition, the weight function is proposed such that it can capture the mechanical behavior trend corresponding to the strain and strain rate for a double-network hydrogel in the relaxation test. Finally, to evaluate the performance of the proposed constitutive equation for the mechanical behavior modeling of double-network hydrogels, the tests on these materials have been used, and the material parameters are determined from fitting the experimental results to the theory. The agreement of test and theory results showed that the proposed model is capable to model the mechanical behavior of double-network hydrogels.     

The tensile behavior of off‐axis loaded plant fiber composites: An insight on the nonlinear stress–strain response

Composites in load‐bearing applications are often exposed to off‐axis loads. For plant fiber composites (PFCs) to be seriously and readily considered in structural applications, knowledge and reliable prediction of their response to off‐axis loads is critical. This article (i) characterizes the stress–strain response, (ii) investigates the tensile properties, and (iii) analyses the fracture modes, of unidirectional flax‐polyester composites subjected to off‐axis tensile loading. A key finding of this study is that due to the nonlinear stress–strain response of PFCs, the apparent stiffness of the composite reduces by ∼30% in the strain range of 0.05 to 0.25%. In addition, through cyclic tests on the composites, the elastic strain limit is found to be only ∼0.15%. This has major implications on the strain range to be used for the determination of the composite elastic Young's modulus. Consequently, it is proposed that the tensile modulus for PFCs should be measured in the strain range of 0.025 to 0.100%. Through comparison with experimental data, conventional composite micromechanical models are found to be adequate in quantitatively describing the tensile behavior of off‐axis loaded PFCs. The application of such models has also enabled the determination of, otherwise difficult to measure, material properties, such as fiber shear and transverse modulus. Off‐axis loaded PFCs fail by three distinct fracture modes in three different off‐axis ranges; each fracture mode produces a unique fracture surface. POLYM. COMPOS. 2012. © 2012 Society of Plastics Engineers     

Rubber modeling using uniaxial test data

Accurate modeling of large rubber deformations is now possible with finite‐element codes. Many of these codes have certain strain‐energy functions built‐in, but it can be difficult to get the relevant material parameters and the behavior of the different built‐in functions have not been seriously evaluated. In this article, we show the benefits of assuming a Valanis–Landel (VL) form for the strain‐energy function and demonstrate how this function can be used to enlarge the data set available to fit a polynomial expansion of the strain‐energy function. Specifically, we show that in the ABAQUS finite‐element code the Ogden strain‐energy density function, which is a special form of the VL function, can be used to provide a planar stress–strain data set even though the underlying data used to determine the constants in the strain‐energy function include only uniaxial data. Importantly, the polynomial strain‐energy density function, when fit to the uniaxial data set alone, does not give the same planar stress–strain behavior as that predicted from the VL or Ogden models. However, the polynomial form does give the same planar response when the VL‐generated planar data are added to the uniaxial data set and fit with the polynomial strain‐energy function. This shows how the VL function can provide a reasonable means of estimating the three‐dimensional strain‐energy density function when only uniaxial data are available. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 81: 837–848, 2001     

A new development of the strain energy function for hyperelastic materials using a logarithmic strain approach

The development of a new strain energy function for hyperelastic solids based on the logarithmic strain measure is the objective of the present article. For all possible types of deformation it was shown that the proposed energy function is based on three independent material parameters. Using available experimental data for rubber‐like materials from the literature, one may determined the materials parameters by a nonlinear fitting. The available domain of the strain energy function can be determined by plotting the third invariant of logarithmic strain vs the second one. The numerical integration of the experimental data of true stress as a function of the logarithmic strain for various types of deformation yields the strain energy function W, for rubber‐like solids. The proposed model involves only one parameter that must be determined by fitting with the experimental data. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 660–672, 2000     

A Nonlinear Viscoelastic Theory for Solid Rocket Propellants Based on a Cumulative Damage Approach

A uniaxial nonlinear viscoelastic constitutive equation incorporating cumulative damage was developed and used to successfully model three highly filled composite solid propellants, two based on hydroxy-terminated polybutadiene with an ammonium perchlorate oxidizer and the third on a glycidyl azide polymer with a phase stabilized ammonium nitrate oxidizer. The nonlinear component of the model consists of a strain rate term, a damage term and a nonlinear exponent. The cumulative damage function itself is calibrated using data from two constant strain rate tests at opposing extreme values of strain rate. Four parameters in the nonlinear viscoelastic model are calibrated at a reference strain rate(low rate of strain) for various conditions of cumulative damage and two parameters are calibrated at the opposing extreme rate of strain for the condition of total damage(i.e., when cumulative damage equals unity). The theoretical predictions at intermediate strain rates are very encouraging, providing a good correlation with the experimental stress--strain data measured under uniaxial constant strain rate loading conditions. The incorporation of cumulative damage theory into the nonlinear viscoelastic constitutive equation enables the strength and failure time to be quantitatively defined in terms of the damage history. Predicted values of strength versus failure time and strength versus constant strain rate at the condition when the cumulative damage equals unity, agree reasonably well with the observed experimental results.     

A linear model predictive control algorithm for nonlinear large‐scale distributed parameter systems

This work provides a framework for linear model predictive control (MPC) of nonlinear distributed parameter systems (DPS), allowing the direct utilization of existing large‐scale simulators. The proposed scheme is adaptive and it is based on successive local linearizations of the nonlinear model of the system at hand around the current state and on the use of the resulting local linear models for MPC. At every timestep, not only the future control moves are updated but also the model of the system itself. A model reduction technique is integrated within this methodology to reduce the computational cost of this procedure. It follows the equation‐free approach (see Kevrekidis et al., Commun Math Sci. 2003;1:715–762; Theodoropoulos et al., Proc Natl Acad Sci USA. 2000;97:9840‐9843), according to which the equations of the model (and consequently of the simulator) need not be given explicitly to the controller. The latter forms a “wrapper” around an existing simulator using it in an input/output fashion. This algorithm is designed for dissipative DPS, dissipativity being a prerequisite for model reduction. The equation‐free approach renders the proposed algorithm appropriate for multiscale systems and enables it to handle large‐scale systems. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Strain rate influence on nonlinear response of polymer matrix composites

In this article, influence of strain rate on nonlinear response of unidirectional fiber‐reinforced composites is studied. The fibers are assumed to be periodic arrays in composite structures. By studying a representative volume element, a microscopic constitutive model for characterizing macro‐mechanical response of polymer matrix composites is developed. Viscoplastic material parameters of polymer matrix are acquired by axial tension and pure shear experiment, and the proposed method is validated by experimental data. The results showed that mechanical behavior of composites, which is affected by strain rate, can be ignored in the linear stage of loading. Furthermore, with the increase in strain rate, stiffness behavior of composites tends to be stiffer at the stage of nonlinear deformation. POLYM. COMPOS., 36:800–810, 2015. © 2014 Society of Plastics Engineers     

Modulus recovery kinetics and other insights into the payne effect for filled elastomers

The nonlinear viscoelastic behavior of filled elastomers is examined in detail using a variety of samples including carbon‐black filled natural rubbers and fumed silica filled silicone elastomers. New insights into the Payne effect are provided by examining the generic results of sinusoidal dynamic and constant strain rate tests conducted in true simple shear both with and without static strain offsets. The effect of deformation history is explored by probing the low amplitude modulus recovery kinetics resulting from a perturbation by a large strain deformation such as a sinusoidal pulse or the application or removal of a static strain. It is found that a static strain has no effect on either the fully equilibrated dynamic (storage and loss) moduli or the incremental stress‐strain curves taken at constant strain rate. The reduction in low amplitude dynamic modulus and subsequent recovery kinetics due to a perturbation is found to be independent of the type of perturbation. Modulus recovery is complete but requires thousands of seconds, and is independent of the static strain. The results suggest that deformation sequence is as critical as strain amplitude in determining the properties, and that currently available theories are inadequate to describe these phenomena. The distinction between fully equilibrated dynamic response and transitory response is critical and must be considered in the formulation of any constitutive equation to be used for design purposes with filled elastomers.     

Thermoforming of a PMMA transparency near glass transition temperature

To simulate the thermoforming of a transparency, constitutive equations are proposed for the nonlinear viscoelastic behavior of poly(methyl methacrylate) near glass transition temperature, which include large deformations. In a first step, they are fitted on a set of uniaxial tension‐relaxation tests at various strain levels and strain rates. In a second step, their implementation in a finite element code is performed. Finally, the thermoforming of a transparency at a constant and uniform temperature is simulated and compared with experimental results. POLYM. ENG. SCI., 50:2004–2012, 2010. © 2010 Society of Plastics Engineers     

A visco‐hyperelastic constitutive model to characterize both tensile and compressive behavior of rubber

The strain rate–dependent finite deformation behavior of three types of rubber under tension and compression are experimentally characterized using a Hopkinson bar. Based on the measured data, a frame‐independent incompressible visco‐hyperelastic constitutive equation is proposed to describe the tensile and compressive responses of rubber under high strain rates. The equation comprises two parts: a three‐parameter component based on an elastic strain energy potential, to characterize static hyperelastic behavior, and another with four parameters, developed from the BKZ model, to define rate sensitivity and strain history dependence. Established static and dynamic experimental techniques are employed to determine the seven parameters in the constitutive relationship. Comparison of predictions based on the proposed model with experiments shows that it is able to describe the visco‐hyperelastic behavior of rubber‐like materials under high strain rates. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 92: 523–531, 2004     

Decomposition strategy for the global optimization of flexible energy polygeneration systems

The optimal design and operation of flexible energy polygeneration systems using coal and biomass to coproduce power, liquid fuels, and chemicals are investigated. This problem is formulated as a multiperiod optimization problem, which is a potentially large‐scale nonconvex mixed‐integer nonlinear program (MINLP) and cannot be solved to global optimality by state‐of‐the‐art global optimization solvers, such as BARON, within a reasonable time. A duality‐based decomposition method, which can exploit the special structure of this problem, is applied. In this work, the decomposition method is enhanced by the introduction of additional dual information for faster convergence. The enhanced decomposition algorithm (EDA) guarantees to find an ε‐optimal solution in a finite time. The case study results show that the EDA achieves much faster convergence than both BARON and the original decomposition algorithm, and it solved the large‐scale nonconvex MINLPs to ε‐optimality in practical times. © 2011 American Institute of Chemical Engineers AIChE J, 58: 3080–3095, 2012     

A new equation for predicting electrical conductivity of carbon‐filled polymer composites used for bipolar plates of fuel cells

In this article, a statistical‐thermodynamic formula based on a new approach has been developed to predict electrical conductivity of carbon‐filled composites used for bipolar plate of proton exchange membrane fuel cell. In this model, based on percolation threshold phenomenon, it is assumed that the relationship between electrical conductivity of composite and filler volume fraction follows a sigmoidal equation. Afterwards, the four effective factors on composite conductivity including filler electrical conductivity, filler aspect ratio, wettability, as well as interface contact resistance are replaced upon constant parameters of sigmoidal function. In order to test the model, some single‐filler composites have been manufactured by using the phenolic resin as binder and graphite (G), expanded graphite (EG), and carbon fiber (CF) as fillers. The fitting quality is measured by R‐square, adjusted R‐square, SSE, and RMSE parameters. The results showed that there is a noteworthy agreement between the model and the experimental data. Compared to the other models, this model can be used for more types of fillers. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013     

Modeling of visco‐hyperelastic behavior of transversely isotropic functionally graded rubbers

In this article, visco‐hyperelastic constitutive model is developed to describe the rate‐dependent behavior of transversely isotropic functionally graded rubber‐like materials at finite deformations. Zener model that consists of Maxwell element parallel to a hyperelastic equilibrium spring is used in this article. Steady state response is described by equilibrium hyperelastic spring and rate‐dependence behavior is modeled by Maxwell element that consists of a hyperelastic intermediate spring and a nonlinear viscous damper. Modified and reinforced neo‐Hookean strain energy function is proposed for the two hyperelastic springs. The mechanical properties and material constants of strain energy function are graded along the axial direction based on exponential function. A history‐integral method has been used to develop a constitutive equation for modeling the behavior of the model. The applied history integral method is based on the Kaye‐BKZ theory. The material constant parameters appeared in the formulation have been determined with the aid of available uniaxial tensile experimental tests for a specific material and the results are compared to experimental results. It is then concluded that, the proposed constitutive equation is quite proficient in forecasting the behavior of rubber‐like materials in different deformation and wide ranges of strain rate. POLYM. ENG. SCI., 56:342–347, 2016. © 2016 Society of Plastics Engineers     

Fatigue response and constitutive behavior modeling of poly(ethylene terephthalate) unreinforced and nanocomposite fibers using genetic neural networks

The constitutive behavior of poly(ethylene terephthalate) (PET) unreinforced (control) and PET fibers reinforced with 5 wt% vapor‐grown carbon nanofibers (VGCNFs) under uniaxial tension and subsequent to fatigue loading has been evaluated utilizing various analytical models. Two types of fatigue tests were performed: (1) Long cycle fatigue at 50 Hz (glassy fatigue) to evaluate fatigue resistance and (2) fatigue at 5 Hz (rubbery fatigue) to evaluate residual strength performance. The long cycle fatigue results at 50 Hz indicated that the PET‐VGCNF sample exhibited an increased fatigue resistance of almost two orders of magnitude when compared to the PET unreinforced filament. The results of the fatigue tests at 5 Hz indicated that the constitutive response of both the PET control and PET‐VGCNF samples changed subsequent to fatigue loading. The large deformation uniaxial constitutive response of the PET and PET‐VGCNF fibers was modeled utilizing genetic‐algorithm (GA) based training neural networks. The results showed that the large deformation uniaxial tension constitutive behavior of both PET unreinforced and PET‐VGCNF samples with and without prior fatigue can be represented with good accuracy utilizing neural networks trained via genetic‐based backpropagation algorithms, once the appropriate post‐fatigue constitutive behavior is utilized. Experimental data of uniaxial tensile tests and experimental postfatigue constitutive data have been implemented into the networks for adequate training. The fatigue tests were conducted under tension‐tension fatigue conditions with variations in the stress ratio (R), maximum stress (σ_max), number of cycles (N), and the residual creep strain (ε_R). POLYM. COMPOS., 2012. © 2012 Society of Plastics Engineers     

Accelerated characterization of a chopped fiber composite using a strain energy failure criterion

The creep and creep rupture response of a chopped fiber composite material (SMC-R50) were investigated experimentally and analytically. The goal of this research was to use the short time laboratory data to predict long time creep and creep rupture behavior. The creep response data up to 200 min duration were obtained at various constant temperature and stress levels. The short time creep data were then modeled using a modified power law equation. The modified power law equation contains the parameters of the so-called accelerated characterization procedure. Using this power law equation, the short time creep response at the elevated temperatures were able to successfully predict the long time creep response at a lower temperature and stress level. To predict the creep rupture behavior, the modified power law equation was then coupled with a strain energy based failure criterion. It was found that the same parameters that were used in the prediction of the long-time creep response can also be used to predict the creep rupture. At a given temperature level, the strain energy density related to creep rupture was found to be a constant. Furthermore, this strain energy density was found to increase with an increase in temperature. With a limited amount of data, it was found that the strain energy based failure criterion coupled with the modified power law equation can be used to predict long time creep rupture behavior.     

Engineering performance and material viscoelastic analyses along a compounding line for silica‐based compounds,part 2: Nonlinear viscoelastic analysis

In the first part of this article, a mixing line for silica‐based compounds was thoroughly described and its performance was studied in terms of mixing fingerprints. Along the mixing line, the compound experiences a large spectrum of strain, stress and temperature conditions such that important “cascade” or “stream” effects occur. At a given point of the process, the material has an important strain‐stress‐temperature history that is obviously affecting its behavior during the subsequent steps, and the situation is further complicated by the in situ silanisation, which obviously must be complete for stabilized rheological properties to be obtained at the end of the line. In this second part of the article, results are reported which were obtained with a new and promising rheometrical technique, i.e., the so‐called Fourier Transform rheometry, implemented on a commercial torsional dynamic rheometer. As shown, Fourier Transform (FT) rheometry provides a number of information about the complex set of events that occur along the silica‐silane mixing line. Odd torque harmonics become significant as strain increases, and therefore the variation of torque harmonics with strain amplitude can be considered as the nonlinear viscoelastic “signature” of tested materials. Silica filled materials exhibit also a typical pattern, with a “bump” appearing in the 500% strain range, essentially on the third relative harmonic versus strain curves and changing as optimal silica dispersion and silinisation are achieved. The appropriate modeling of experimental results provide parameters with typical variations along the mixing line which are interpreted with respect to current views on the in situ silanisation process and are found in line with the mixing signatures analysis. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

A phenomenological constitutive model for foams under large deformations

A nonlinear phenomenological constitutive model applicable to structural foams subjected to large deformations is proposed. The five‐parameter model can fully capture the three fundamental features of stress‐strain response, i.e., linearity, plasticity‐like stress plateau, and densification phases, when subjected to compressive loads. Moreover, the parameters of the model can be systematically varied to capture the influence of initial density of foams that may be responsible for changes in yield stress and hardening‐like or softening‐like behavior under various confinement conditions. The model was successfully applied to capture the stress‐strain response of two structural foams of different initial densities when subjected to uniaxial compression without lateral confinement and unixial compression under rigid confinement. Polym. Eng. Sci. 44:463–473, 2004. © 2004 Society of Plastics Engineers.