Nonlinear Viscoelastic Model Research Articles

Static and Dynamic Tensile Mechanical Behavior of Polyvinyl Chloride Elastomers with Different Shore Hardness

An experiment on the static and dynamic tensile mechanical properties of polyvinyl chloride (PVC) elastomers is conducted using an Instron-5943 universal testing machine and an improved Split Hopkinson Tensile Bar to study the dynamic tensile mechanical properties of PVC elastomer materials. The stress-strain curves of PVC materials with three types of Shore hardness (57A, 52A, and 47A) under the strain rates of 0.1 s−1 and 400 ∼ 1800 s−1 are obtained. Results show that the mechanical behavior of PVC elastomer materials with different Shore hardness has remarkable linear elastic characteristics under the action of quasistatic tensile load. It has substantial sensitivity to strain rate and viscoelastic mechanical characteristics under the action of dynamic tensile load. The Zhu–Wang–Tang nonlinear viscoelastic constitutive model is used to characterize the viscoelastic mechanical characteristics with small error. This paper can provide theoretical model and method support for the design, development, production, and reliability analysis of PVC elastomers and other soft polymer materials.

Optimisation on Thermoforming of Biodegradable Poly (Lactic Acid) (PLA) by Numerical Modelling.

Poly (lactic acid) (PLA) has a broad perspective for manufacturing green thermoplastic products by thermoforming for its biodegradable properties. The mechanical behaviour of PLA has been demonstrated by its strong dependence on temperature and strain rate at biaxial deformation. A nonlinear viscoelastic model by the previous study was employed in a thermoforming process used for food packaging. An optimisation approach was developed by achieving the optimal temperature profile of specimens by defining multiple heating zones based on numerical modelling with finite element analysis (FEA). The forming process of a PLA product was illustrated by modelling results on shape evolution and biaxial strain history. The optimal temperature profile was suggested in scalloped zones to achieve more even thickness distribution. The sensitivity of the optimal results was addressed by checking the robustness under perturbation.

Constitutive modeling of bond breaking and healing kinetics of physical Polyampholyte (PA) gel

A three-dimensional finite strain nonlinear viscoelastic model is developed to study the mechanical behavior of a physically cross-linked Polyampholyte (PA) gel. The time dependent behavior of this gel is due to the ionic interactions of oppositely charged monomers randomly distributed along the chain backbone. In this work, we divide the physical cross-links broadly into weak and strong bonds, depending on their survival and reformation characteristics. Our constitutive model connects the strain dependent bond breaking and reforming kinetics in the microscopic regime to the deformation of the gel at the continuum level. We compare the predictions of our model with uniaxial tension, tensile-relaxation, cyclic, and small strain torsional relaxation tests. The material parameters in our model are obtained using least squares optimization. Our theory agrees well with the experimental behavior of the gel.

Mechanics of soft polymeric materials using a fractal viscoelastic model

Soft materials are known for their plethora of biomedical applications, intricate structure–property correlation and nonlinear mechanical response. Multiple length–time scale phenomena and hierarchical structure results in their nonlinearity. Phenomenological and continuum mechanical models have been developed to predict their mechanics, which have mostly been very material-specific with inability to predict the mechanics of different types of soft materials simultaneously. This shortcoming has been addressed in the present work, wherein a generic nonlinear viscoelastic model has been proposed to predict the mechanical response of hydrogels, sponges, and xerogels. A fractal derivative viscoelastic model is proposed considering a fractal Maxwell model in parallel with a nonlinear spring. In particular, this model is chosen to qualitatively mimic the material nonlinearity inherent in soft materials. The fractal dashpot in combination with the nonlinear spring accounts for the power law time-dependent rheology of generic soft materials. These two different aspects in the form of nonlinear stiffness and non-Newtonian rheology account for mechanics of most soft materials. The present model is shown to fit well the existing literature results for mechanical response of a multitude of soft material classes with different test conditions and loading rates, which is one of the salient features of the model, apart from its simplistic mathematical framework. Further, a parametric study is reported on the mechanics of nanocellulose loaded poly(vinyl alcohol) xerogel. The model predictions are observed to be in conjunction with the experimental observations.

A Nonlinear Viscoelastic Model for the Yielding of Gelled Waxy Crude Oil

We explore some rheological aspects of the yielding of gelled waxy crude oil on the basis of a fractal model for the structural description of the waxy gel and Marrucci’s model for the time evolution of the stress with mixed elastic and viscous effects. With some parameters of the model directly obtained from classic rheometry, and others by fitting the parameters to the experimental data of one shear-rate condition, the flow curves for another shear-rate condition are predicted. Both theoretical curves—the fitting and the predicted ones—share the basic features of the experimental ones. Comparison with results of Maxwell model shows that Marrucci’s model used here leads to much better results, as it incorporates nonlinear viscoelasticity of waxy crude gels in the stress evolution equation. The strain dependence of the elastic modulus also plays a relevant role on the prediction of the model, suggesting a double-network contribution for very small strain values. Due to the inertia of rheometric device, the actual shear rate is often found to depart from the setting one, and modification of shear rate history can be necessary in model validation.

The Scheme to Determine the Convergence Term of the Galerkin Method for Dynamic Analysis of Sandwich Plates on Nonlinear Foundations

The vibration of a plate resting on elastic foundations under a moving load is of great significance in the design of many engineering fields, such as the vehicle-pavement system and the aircraft-runway system. Pavements or runways are always laminated structures. The Galerkin truncation method is widely used in the research of vibration. The number of truncation terms directly affects the convergence and accuracy of the response results. However, the selection of the number of truncation terms has not been clearly stated. A nonlinear viscoelastic foundation model under a moving load is established. Based on the natural frequency of linear undisturbed derivative systems, the truncation terms are used to determine the convergence of vibration response. The criterion for the convergence of the Galerkin truncation term is presented. The scheme is related to the natural frequency with high efficiency and practicability. Through the dynamic response of the sandwich beam under a moving load, the feasibility of the scheme is verified. The effects of different system parameters on the scheme and the truncation convergence of dynamic response are presented. The research in this paper can be used as a reference for the study of the vibration of elastic foundation plates. Especially, the model established and the truncation analysis method proposed are helpful for studying the vibration of vehicle-pavement system and related systems.

Nonlinear Viscoelastic Modeling of Synthesized Silicate-Based Bioactive Glass/Polysulfone Composite: Theory and Medical Applications

Bioactive glass/Polysulfone composite has been considered as an appropriate material for biomechanical and medical applications. Time-dependency and nonlinear-elasticity are significant mechanical behaviors of these type of materials. Hence, in this research, the 20% vol. 58 s bioactive glass/polysulfone composite was prepared using the solvent casting method, and tensile tests with 3 different strain rates were performed on the samples. Also, a nonlinear-viscoelastic model based on the Ogden nonlinear-elastic strain energy was generalized to investigate the mechanical behavior of the 20 vol.% 58 s bioactive glass/polysulfone composite. To show the agreement of the model with the experimental results, a rate dependent tensile test was simulated by ABAQUS which validated the results. The tensile test showed a maximum stress of 15 ~ 25 MPa, based on the test speed; also, the simulation demonstrated a 16.2 MPa stress for the 5 mm/min test speed, which were in good agreement. Later, a model of a human tibia bone was constructed using MIMICS, SOLIDWORKS and ABAQUS to study the application of the 58 s bioactive glass/polysulfone composite as an implant. The results revealed that the stress tolerated by the implant was neglectable, and most of the stress was tolerated by the bone itself.

Electrokinetics of polymeric fluids in narrow rectangular confinements.

The flow of polymeric liquids in narrow confinements with a rectangular cross section, in the presence of electrical double layers is analyzed here. Our analysis is motivated by the fact that many of the previous studies on the flow of complex fluids tend to focus on highly idealized parallel plate channels, which are markedly different from the rectangular ducts, used in many experiments and devices. We consider the combined electroosmotic and pressure driven flows as well as the streaming potential resulting from a mechanically driven flow. We use two distinct constitutive relations to model the polymeric liquids, namely the simplified exponential Phan-Thien-Tanner (sePTT) model and the Giesekus model, both of which are non-linear viscoelastic models, capable of capturing the shear thinning behavior. We establish that the applied electric field may have a strong influence on the overall flow rate, which rapidly increases with the field strength as well as the extent of viscoelasticity of the fluid. Viscoelasticity and shear thinning behavior also enhance the streaming potential by several fold as compared to a Newtonian medium. We demonstrate that the aspect ratio of a channel has a bigger influence on the net throughput and the streaming potential, when the extent of viscoelasticity is relatively large. We illustrate that for sePTT fluids, the flow is strictly unidirectional, while for Giesekus fluids, secondary flows are inevitably present on account of their non-zero second normal stress coefficient. Although the electric field does not change the overall patterns of these secondary flows, their magnitude does depend on the imposed field strength for combined flows.

Constitutive modeling of strain-dependent bond breaking and healing kinetics of chemical polyampholyte (PA) gel.

A finite strain nonlinear viscoelastic constitutive model is used to study the uniaxial tension behaviour of chemical polyampholyte (PA) gel. This PA gel is cross-linked by chemical and physical bonds. Our constitutive model attempts to capture the time and strain dependent breaking and healing kinetics of physical bonds. We compare model prediction by uniaxial tension, cyclic and relaxation tests. Material parameters in our model are obtained by least squares optimization. These parameters gave fits that are in good agreement with the experiments.

A Viscoelastic Model for Human Myocardium

Understanding the biomechanics of the heart in health and disease plays an important role in the diagnosis and treatment of heart failure. The use of computational biomechanical models for therapy assessment is paving the way for personalized treatment, and relies on accurate constitutive equations mapping strain to stress. Current state-of-the art constitutive equations account for the nonlinear anisotropic stress-strain response of cardiac muscle using hyperelasticity theory. While providing a solid foundation for understanding the biomechanics of heart tissue, most current laws neglect viscoelastic phenomena observed experimentally. Utilizing experimental data from human myocardium and knowledge of the hierarchical structure of heart muscle, we present a fractional nonlinear anisotropic viscoelastic constitutive model. The model is shown to replicate biaxial stretch, triaxial cyclic shear and triaxial stress relaxation experiments (mean error ~7.5%), comparing well with its hyperelastic (mean error ~25%) counterparts. Model sensitivity, fidelity and parameter uniqueness are demonstrated. The model is also compared to rate-dependent biaxial stretch as well as different modes of biaxial stretch, illustrating extensibility of the model to a range of loading phenomena.

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

Dielectric elastomers are employed for a wide variety of adaptive structures. Many of these soft elastomers exhibit significant rate-dependencies in their response. Accurately quantifying this viscoelastic behavior is non-trivial and in many cases a nonlinear modeling framework is required. Fractional-order operators have been applied to modeling viscoelastic behavior for many years, and recent research has shown fractional-order methods to be effective for nonlinear frameworks. This implementation can become computationally expensive to achieve an accurate approximation of the fractional-order derivative. Accurate estimation of the elastomer’s viscoelastic behavior to quantify parameter uncertainty motivates the use of Markov Chain Monte Carlo (MCMC) methods. Since MCMC is a sampling based method, requiring many model evaluations, efficient estimation of the fractional derivative operator is crucial. In this paper, we demonstrate the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of estimating the uncertainty of a nonlinear viscoelastic model. We also demonstrate the use of parameter subset selection techniques to isolate parameters that are identifiable in the sense that they are uniquely determined by measured data. For those identifiable parameters, we employ Bayesian inference to compute posterior distributions for parameters. Finally, we propagate parameter uncertainties through the models to compute prediction intervals for quantities of interest.

Free volume based nonlinear viscoelastic model for polyurea over a wide range of strain rates and temperatures

A series of quasi-static and dynamic uniaxial compression experiments over a wide range of temperatures were conducted on polyurea elastomer. The stress-strain response of polyurea exhibits extreme nonlinearity, temperature and strain-rate sensitivity. How to comprehensively describe these complex behaviors regulated by the unique micro-structure of polyurea is still an open question. In this paper, a nonlinear viscoelastic constitutive model for polyurea is developed. The nonlinear viscoelasticity, entropic and energetic elasticity are taken into account, correlated with the interaction of chain segments, the stretch of molecular chains and covalent bonds from the perspective of molecular structure. The viscoelastic response is formulated by the hereditary integral of relaxation modulus and strain rate with time. The real time is accelerated by shift factor, which depends on the free volume. A new mechanism of shear-induced increase of free volume, which gives rise to the nonlinearity of viscoelasticity, is proposed, and the corresponding rate-dependent evolution equation of fractional free volume is developed. Parameters in the model are determined through the compression experiments and DMA test. Comparison with experimental data, as well as the data from references, verifies the ability of the model to predict the nonlinear stress-strain response of polyurea over a wide range of temperatures and strain rates.

The Tensor Diffusion approach for simulating viscoelastic fluids

In this paper, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluids is introduced based on the idea, that the extra-stress tensor in the momentum equation of the flow model can be replaced by the product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity. As potential advantage (which can be demonstrated for fully developed channel, resp., pipe flows), this approach offers the possibility to reduce the full nonlinear viscoelastic three-field model to a generalized Tensor Stokes problem, avoiding the need of considering a separate stress tensor in the solution process. Moreover, an (artificial) diffusive operator is introduced into the momentum equation in the case of “no solvent” viscoelastic fluids, which is related to the physics of the problem leading to an improved numerical behaviour w.r.t. approximation and convergence properties of the iterative solvers. After validating the Tensor Diffusion approach for fully developed channel flow configurations, it is evaluated in the context of more general two-dimensional flow settings of benchmarking character, too, taking into account a symmetrized formulation. Numerical simulations for several prototypical flow configurations illustrate the mathematical and numerical properties of this new approach and can be viewed as proof-of-concept for further development.

A nonlinear viscoelastic constitutive model taking into account of physical aging

Polymers exhibit viscoelastic behavior: their mechanical response depends on the loading time, or on the loading frequency. In addition, if a polymer structure has a long service life, the mechanical behavior can also depend on physical aging and chemical degradation. This paper describes a thermodynamically consistent constitutive law taking into account the viscoelastic phenomena and the physical aging. First, an original nonlinear viscoelastic law, depending on the physical aging time, is developed. Then, considering experimental values of dynamic modulus from the literature, the model parameters are identified, using a new method based on the discrete form of the spectrum of relaxation time. The obtained model is numerically implemented and compared to experimental results.

Fracture of warm S2 columnar freshwater ice: size and rate effects

Large scale laboratory experiments on size and rate effects on the fracture of warm columnar freshwater ice have been conducted with floating edge-cracked rectangular plates loaded at the crack mouth. The largest test plate size had dimensions of 19.5m x 36m. The overall crack-parallel dimension covered a size range of 1:39, possibly the largest for ice tested under laboratory conditions. The loading rates applied led to test durations from fewer than 2 seconds to more than 1000 seconds, leading to an elastic response at the highest rates to a viscoelastic response at the lower rates. Methods for both the linear elastic fracture mechanics (LEFM) and a non-linear viscoelastic fictitious crack model (VFCM) were derived to analyze the data and calculate values for the apparent fracture toughness, crack opening displacement, stress-separation curve, fracture energy, and size of the process zone near a crack tip. Issues of notch sensitivity and minimum size requirements for polycrystalline homogeneity were addressed. Both size and rate effects were observed, as well as how these two factors are interrelated in the fracture of columnar freshwater ice. There was a size effect at low rates but no size effect at high rates. There was a rate effect for the larger test sizes but a weaker or no rate effect for the smallest test size.

Effect of temperature and filler volume fraction on the creep and recovery behaviour of MWCNT–COOH–reinforced polypropylene films

Nonlinear viscoelastic behaviour of –COOH functionalized multi-walled carbon nanotubes reinforced polypropylene nanocomposites is examined up to 2% MWCNT concentration. Isochronal creep and strain recovery experiments are conducted in dynamic mechanical analyser with oscillatory inputs. Creep stress level of 5 MPa is selected and successive creep and recovery experiments are performed at $$-\,20\,^\circ $$ C, $$25\,^\circ $$ C and $$50\,^\circ $$ C to analyse the effect of temperature and MWCNT concentration on strain developed during creep loading and strain recovery stages. A well-known Schapery nonlinear viscoelastic solid model with Zapas–Crissman viscoplastic term is fitted to the experimental observations. Stress-dependent nonlinear model parameters are obtained under temperature and MWCNT loading. Viscoelastic and viscoplastic strains are predicted at the end of recovery period. High temperatures and low concentrations of MWCNT are observed to cause higher viscoplastic strain. Strengthening MWCNT up to 1% improves the thermomechanical behaviour and viscoelastic recovery and decreases the viscoplastic strain in nanocomposites. Temperature-dependent compliance behaviour of nanocomposites is investigated with Williams–Landel–Ferry model and Arrhenius model using time–temperature superposition principle.

Mechanical behavior and dynamic damage constitutive model of glass fiber-reinforced Vinyl Ester with different fiber contents.

Glass fiber-reinforced plastics (GFRP) based on polymer materials are widely used in lightweight impac-resistant structure design. In the process of design and development, it is very important to clarify the mechanical behavior under dynamic load to improve the product performance. Therefore, in this paper, Quasi-stati and dynamic compression behaviors of unidirectional continuous glass fiber-reinforced vinyl ester (GF/VE) composites with five kinds of fiber contents in the fiber direction were measured by an electro-hydraulic servo experiment system and a split Hopkinson pressure bar, and damage evolution of the material is analyzed by observing the microstructure of the cross section of the material. Results show that: The content of glass fiber affects the wettability between fiber and matrix, and the failure mechanism of material at high strain rate; Under quasi-static conditions, higher glass fiber content yields greater failure strength; Under dynamic conditions, as glass fiber content increases, toughness decreases, and the peak stress first increases and then decreases. Finally, the nonlinear viscoelastic constitutive model with damage evolution is derived, which can be used to predict the impact resistance of new composite structures in the product development and design stage and reduce the development cycle.

Analysis of reclaimed asphalt blended binders using linear and nonlinear viscoelasticity frameworks

This paper presents a comprehensive analysis of blended asphalt binders extracted from mixtures that contain various amounts of reclaimed asphalt pavements (RAP). The analysis is performed within a thermodynamically consistent non-linear viscoelastic (NVE) modeling framework, which has the advantage of accounting for the contributions of each of the constituents (i.e. virgin binder and RAP binder) to the response of the blended binder. In order to calibrate and validate the NVE model, the RAP blended binders were subjected to different testing protocols: frequency sweep, multiple stress creep and recovery, repeated creep and recovery with multiple stress levels, random creep and recovery, and stress relaxation. For the binders used in this study, linear viscoelasticity (LVE) was suitable to model the frequency sweep and multiple stress creep and recovery data, but it did not capture the repeated creep and recovery with multiple stress levels results that involved higher stress and strain levels. The NVE model, on the other hand, was successful in describing the repeated creep and recovery with multiple stress levels results. The validation was achieved by comparing the NVE model predictions with the random creep and recovery and stress relaxation test results. Finally, the paper recommends a new method, based on the parameters of the NVE model, to evaluate the rutting resistance of blended asphalt binders.

Modeling lung tissue dynamics and injury under pressure and impact loading.

A nonlinear viscoelastic model for the lung is implemented and evaluated for high-rate loading. Principal features of the model include a closed-cell approximation of the bulk compressibility accounting for air inside the lung and a damage-injury component by which local trauma is induced by cumulative normalized internal energy and amplified by gradients of energy density. The latter feature is adapted for use in standard numerical (i.e., explicit finite element) simulations in terms of the local rate of strain energy density and the longitudinal wave speed. Injury predictions for direct loading of a block of extracted lung material, rather than the entire thorax, via pressure pulses are in reasonably close agreement with experimental observations for an extracted rabbit lung: a threshold applied pressure exists above which edema is observed experimentally, correlating with low but non-negligible damage in the numerical results. Responses to impact by cylindrical and spherical projectiles are also interrogated. Penetration depths are comparable to those observed experimentally, as is drastically increasing damage with increasing impact velocity. Damage initiates and propagates from the impact surface, with local severity of injury decreasing with distance from the impact zone, in agreement with some empirical evidence. The model predicts more severe local injury, relative to the aforementioned surface pressure loading, than what is observed experimentally. Possible reasons for the discrepancy are analyzed, and adjustments to the model, with caveats, are suggested accordingly.

A micromechanical model for loading and unloading behavior of fiber reinforced plastics under cyclic loading

AbstractThe high fatigue resistance and low weight features make the unidirectional‐carbon fiber reinforced plastics (UD‐CFRP) attractive for cyclic loading applications. However, the complex damage mechanisms within the composite hinder the understanding of its fatigue response. An energy‐based fatigue model has been proposed for the fatigue behavior of epoxy polymers. However, this kind of approaches dependent on the accurate description of the material's behavior. Our work aims to build a representative volume element model (RVE) for the simulation of the cyclic shear loading behavior of UD‐CFRP, which could support the later extension of an energy‐based fatigue model. To precisely reproduce the micromechanical stress‐strain distribution within the matrix, a nonlinear viscoelastic model has been implemented and validated against experimental data. As a result, the simulated UD‐CFRP hysteresis, as well as the stiffness and strain energy density were accurately reproduced for two load ratios (R = 0.1 and R = − 1). Additionally, the cyclic strain accumulation observed in the UD‐CFRP can be accurately described by the proposed model.