Nonlinear Viscoelasticity Research Articles

Modeling the viscoelastic behavior of wheat flour dough prepared from a wide range of formulations

The viscoelastic behavior of wheat flour dough has been characterized using constitutive models of varying degrees of complexity. Modeling characterizations of the viscoelasticity of dough have not always demonstrated the general applicability of their models to a wide range of dough strengths and formulations. Therefore, the objective of this study was to use two extrema in viscoelastic rheological models (i.e., a power-law gel and the four-element Burgers models) to characterize the linear and non-linear shear behavior of wheat flour doughs. A wide range of breadmaking strength and formulations were employed, particularly those with varying salt content (to ascertain model applicability to strategies to reduce sodium in bread). The power-law gel model fitted all the experimental data well for oscillatory rheometry and creep-recovery tests. The power-law gel model was better at describing the linear viscoelasticity of wheat flour dough compared to the Burgers model, but was poorer at characterization of the recovery phase. Gel strength parameters derived from power-law gel models are recommended for defining the effects of changes in wheat cultivar, water and salt content on both the linear and non-linear viscoelasticity of dough rheology.

Large amplitude oscillatory shear (LAOS) rheology of nixtamalized corn masa

Masa was prepared by mixing 63.0 g of commercial nixtamalized corn flour and 37.0 g of water, and subjected to large amplitude oscillatory shear (LAOS) for assessing nonlinear viscoelasticity. Fourier decomposition of the stress time response was conducted to extract information on nonlinearities. The results showed that nonlinearities were expressed for large deformations higher than 1%. In contrast, the mechanical response was essentially harmonic for small amplitude values, indicating the absence of nonlinearities and hence the dominance of linear viscoelastic response. In particular, nonlinearities quantified in terms of the harmonic contributions were stronger for the elastic response (about 75–100%) than for the viscous response (less than 70%). The results illustrated the viability of LAOS method for characterizing the nonlinear mechanical response of masa made with nixtamalized corn flour.

Characterization and validation of the nonlinear viscoelastic-viscoplastic with hardening-relaxation constitutive relationship for asphalt mixtures

Abstract This paper presents a systematic and straightforward analysis procedure to extract the material parameters associated with a coupled nonlinear viscoelastic (VE), viscoplastic (VP), and hardening-relaxation (HR) constitutive model for asphalt mixtures. The recoverable and irrecoverable strain are represented by Schapery nonlinear VE model and Perzyna-type VP theory, respectively. A HR model is proposed to consider the relaxation of hardening. Dynamic moduli and repeated creep-recovery tests are used to calibrate the coupled VE-VP-HR model. The model is then validated against independent experimental data. Results show that the VE-VP-HR model has the ability to capture the response of asphalt mixtures subjected to various loading paths.

A viscoelastic damage model for nanoparticle/epoxy nanocomposites at finite strain: A multiscale approach

Experimental tests show that nonlinear viscoelasticity characterizes the mechanical behavior of boehmite nanoparticle (BNP)/epoxy nanocomposites. This paper presents the development and numerical implementation of a physically based constitutive model for BNP/epoxy nanocomposites undergoing finite strain. The proposed constitutive model allows capturing the main features of the stress-strain relationship of BNP/epoxy nanocomposites, including the nonlinear hyperelastic, time-dependent and softening behavior. The characterizing feature of this study is to propose a methodological framework based on molecular dynamics simulations and experimental tests to identify the material parameters for the model. Molecular simulations in conjunction with the Eyring viscosity theory are used to characterize the viscoelastic deformation of epoxy resins under loading. The concept of strain amplification is also adopted to account for the effect of nanoparticles on the stress–strain response of the nanocomposites. The stress softening behavior is captured by a monotonically increasing function of deformation, so-called damage variable. The results show that the model predictions of stress-strain relationships are in good agreement with experimental data at different BNP weight fractions. Finally, the constitutive model is implemented in the finite element analysis and examined by means of a benchmark example. Experimental–numerical validation confirms the predictive capability of the present modeling framework, which provides a suitable tool for analyzing BNP/epoxy nanocomposites.

Nonlinearity in the Mechanical Response of Rubber as Investigated by High-Frequency DMA.

Nonlinear material response is analysed with the Fourier transform (FT) of the raw signal measured by a high-frequency dynamic mechanical analyzer (HF DMA). It is known from rheological behaviour of elastomers that reinforcing fillers additionally induce nonlinearity in an already inherently nonlinear system. This behaviour is often described in terms of a mechanical response of strain sweeps, essentially the transition from the linear viscoelastic (LVE) to the nonlinear viscoelastic (NVE) region. In the current investigation, the NVE region could be observed with respect to frequency under low-amplitude deformation. A foldover effect was observed, whereby the material exhibited a nonlinear dependency in relation to the increment of the filler amount above the percolation threshold. In addition, an apparent superharmonic resonance was observed within higher orders of vibrational modes which is further indication of nonlinearity. In this paper, the analytical approach is presented as a novel method to characterise the behaviour of the polymer–filler interaction by HF DMA.

Rotationally Symmetric Motions and Their Blowup for Incompressible Nonlinearly Elastic and Viscoelastic Annuli

This paper treats rotationally symmetric motions of incompressible, transversely isotropic, nonlinearly elastic and viscoelastic annuli subjected to the live loads of (time-dependent) centrifugal force and hydrostatic pressure. The incompressibility is responsible for some simplification of the analysis of such motions, which is nevertheless more complicated than that for radially symmetric motions. The theory is illustrated with blowup theorems, which have proofs different from those used for compressible media.

Comparing Predictive Accuracy and Computational Costs for Viscoelastic Modeling of Spinal Cord Tissues.

The constitutive equation used to characterize and model spinal tissues can significantly influence the conclusions from experimental and computational studies. Therefore, researchers must make critical judgments regarding the balance of computational efficiency and predictive accuracy necessary for their purposes. The objective of this study is to quantitatively compare the fitting and prediction accuracy of linear viscoelastic (LV), quasi-linear viscoelastic (QLV), and (fully) nonlinear viscoelastic (NLV) modeling of spinal-cord-pia-arachnoid-construct (SCPC), isolated cord parenchyma, and isolated pia-arachnoid-complex (PAC) mechanics in order to better inform these judgements. Experimental data collected during dynamic cyclic testing of each tissue condition were used to fit each viscoelastic formulation. These fitted models were then used to predict independent experimental data from stress-relaxation testing. Relative fitting accuracy was found not to directly reflect relative predictive accuracy, emphasizing the need for material model validation through predictions of independent data. For the SCPC and isolated cord, the NLV formulation best predicted the mechanical response to arbitrary loading conditions, but required significantly greater computational run time. The mechanical response of the PAC under arbitrary loading conditions was best predicted by the QLV formulation.

Mechanical Model of a Hybrid Non-linear Viscoelastic Material Damping Device With Its Verifications

This paper proposes a new viscoelastic (VE) material damping device with hybrid nonlinear properties. Compared with traditional linear material dampers, the new nonlinear VE material damping device is characterized by its better energy dissipation and deformation capability. The series performance tests of the VE device are conducted, based on which the sources and variation law of material nonlinearities are analyzed. Five aspects of material nonlinearities are summarized, including the shape change of hysteresis loop caused by phase difference, the initial stiffness caused by large loading rate, the softening effect caused by high temperature and fatigue, and the softening and stiffening effect under large strain deformation. Then, a mechanic model for this device is proposed which considers multiple nonlinear effects of the material. For the verifications of the proposed mechanic model, a shaking table test on a steel frame equipped with the new VE devices is designed and performed. Based on the proposed mathematic mechanic model, the numerical mechanic model is implemented in the OpenSees software. The accuracy of the mechanic model is firstly verified by comparing the performance tests data with the simulation results. Then numerical model is also used to calculate the time history response of the shake table tested steel frame under earthquake loading. It is concluded that the mechanic model can well depict several nonlinear material behaviors of the new VE device, and the corresponding numerical model created in the open source calculation platform is reliable to be used to calculate nonlinear time history response of a structure equipped with the new VE material damping devices.

On the Development of Creep Laws for Rubber in the Parallel Rheological Framework

ABSTRACT It is widely known that filler-reinforced rubber material in tires shows a very complicated material behavior when subjected to cyclic loadings. One of the most interesting effects for rolling tires is the nonlinear rate-dependent behavior, which is implicitly linked to the amplitude dependency of dynamic stiffness (Payne effect) at a given frequency and temperature. This effect, however, cannot be described by a conventional linear viscoelastic constitutive law, e.g., the Prony series model. Several nonlinear viscoelastic material models have been proposed in the last decades. Among others, Lapczyk et al. (Lapczyk, I., Hurtado, J. A., and Govindarajan, S. M., “A Parallel Rheological Framework for Modeling Elastomers and Polymers,” 182nd Technical Meeting of the Rubber Division of the American Chemical Society, Cincinnati, Ohio, October 2012) recently proposed a quite general framework for the class of nonlinear viscoelasticity, called parallel rheological framework (PRF), which is followed by Abaqus. The model has an open option for different types of viscoelastic creep laws. In spite of the very attractive nonlinear rate-dependency, the identification of material parameters becomes a very challenging task, especially when a wide frequency and amplitude range is of interest. This contribution points out that the creep law is numerically sound if it can be degenerated to the linear viscoelastic model at a very small strain amplitude, which also significantly simplifies model calibration. More precisely, the ratio between viscoelastic stress and strain rate has to converge to a certain value, i.e., the viscosity in a linear viscoelastic case. The creep laws implemented in Abaqus are discussed in detail here, with a focus on their fitting capability. The conclusion of the investigation consequently gives us a guideline to develop a new creep law in PRF. Here, one creep law from Abaqus that meets the requirements of our guideline has been selected. A fairly good fit of the model is shown by the comparison of the simulated complex modulus in a wide frequency and amplitude range with experimental results.

Review on tube model based constitutive equations for polydisperse linear and long-chain branched polymer melts

Rheological equations of state are of great importance for characterization of polymer melts and for simulation of polymer processing. This concise review considers tube model based constitutive equations developed in the last 40 years since the original publication of Doi and Edwards in 1978. The emphasis is on the concepts, assumptions, and material parameters introduced to model nonlinear viscoelasticity of polydisperse linear and long-chain branched polymer melts. Linear viscoelasticity is assumed to be known, either from linear-viscoelastic modeling or by experimental characterization. The scope is limited to constitutive equations which are based on the linear-viscoelastic relaxation modulus and can be expressed in terms of integral, differential, or integro-differential equations. Multimode models based on the concept of preaveraged stretch require a large number of nonlinear model parameters. Relaxing the assumption of a constant tube diameter, the number of nonlinear model parameters can be drastically reduced to 2 or 3, independent of the number of Maxwell modes needed to represent the linear viscoelasticity.

Application of nanotubes in conveying nanofluid: a bifurcation analysis with consideration of internal energy loss and geometrical imperfection

This article deals with developing a coupled scale-dependent model to explore the nonlinear bifurcation response of initially imperfect nanotubes conveying nanofluid flow taking into consideration the influences of nonlinear viscoelasticity. Furthermore, the influences of both centrifugal and Coriolis forces are considered. The Beskok–Karniadakis model is employed to capture the influences of slip at the interface between the imperfect viscoelastic nanotube and the nanofluid. A refined combination of nonlocal and strain gradient elasticities is employed for taking into consideration size influences. After formulating the kinetic energy, elastic energy, viscous work and external work, the nonlinear coupled equations are derived for the nanofluid-conveying nanosystem, which simultaneously vibrates along the transverse and longitudinal directions. The nonlinear dynamical characteristics are calculated via utilising a Galerkin procedure and a direct-time-integration technique. It is found that chaotic regions can be removed by imposing a proper geometric imperfection.

Analytical modelling and characterisation of an infrasound generator in the air

This paper presents an accurate characterisation based on an experimental set-up and mathematical models of a dynamic pressure infrasound generator. The Commissariat à l’énergie atomique (CEA) has developed microbarometers to measure infrasound waves in the atmosphere. To characterise its sensors and validate their requirements, an infrasound generator has been designed, which covers a frequency range from 4.0 10−4 Hz to 300 Hz. This pressure generator still needs accurate characterisation as there is no standard reference in the infrasound frequency range for such sensor calibration. The research focused on 17 parameters that affect the behaviour of this infrasound pressure generator. Two analytical models of the sound pressure in the cylindrical cavity of the calibrator are presented. An experimental characterisation was also performed to adjust the model parameters with genetic algorithms to the measurements, and the results of the comparison between the measurements and the models are discussed. This study highlights the influence of the thermodynamic transition from isothermal to adiabatic transformation and the influence of viscoelastic non-linearities of a loudspeaker membrane. It aims to characterise the amplitude and phase responses of the generator in order to develop an improved infrasound calibration device for microbarometers with a similar technology.

Interfacial Fourier transform shear rheometry of complex fluid interfaces

Nonlinear rheometry of interfaces is very challenging because of the limits of current day instrumentation and the intricate coupling of flows at interfaces and in the bulk. The use of time periodic flows may constitute a first step in addressing this issue. Fourier transform rheology (FTR) measurements with quasi-monolayers at the air-water interface are used in order to assess the suitability of the different devices to investigate nonlinear interfacial shear viscoelasticity. The probe material was a poly (methyl methacrylate) forming a soft glassy interface, whereas complementary measurements were performed with a polystyrene latex suspension forming a predominantly viscous interface at intermediate surface coverages. The obtained data with the magnetic rod rheometer (ISR) were compared against those obtained with the bicone and the double wall ring geometries attached to standard rotational rheometers. In particular, an unexpected appearance of even in addition to odd harmonics is discussed in terms of flow field asymmetry.

Mechanical guidelines on the properties of human healthy arteries in the design and fabrication of vascular grafts: experimental tests and quasi-linear viscoelastic model.

Knowledge of mechanical behavior of healthy human arteries as the guidelines to target properties of vascular grafts deserves special attention. There is a lack of mathematical model to characterize mechanical behavior of biomaterial while many mathematical models to reflect mechanics of human arteries have been proposed. The objective of this paper was set to measure mechanical properties of healthy human arteries including Common Carotid Artery (CCA), Abdominal Aorta Artery (AAA), Subclavian Artery (SA), Common Iliac Artery (CIA) and Right and Left Iliac Artery (RIA and LIA) and compare them to those of commercial ePTFE and Dacron®. Series of stress relaxation and strain to failure tests vere performed on all samples. The experimental data was utilized to develop quasi-linear viscoelastic (QLV) model of both natural and artificial arteries. ePTFE is the stiffest sample, while the CCA is the most compliant one among all. RIA and CIA are more viscous than the other natural arteries, while AA and CCA are less viscous. The proposed model demonstrated an accurate fit to the experimental results, a proof of its ability to model both nonlinear elasticity and viscoelasticity of the human arteries and commercial ones. ePTFE and Dacron® are much stiffer than human arteries that may lead to the disruption of blood hemodynamic and may not be biomechanically feasible as a replacement.

Immersed-finite-element method for deformable particle suspensions in viscous and viscoelastic media

Deformable elastic bodies in viscous and viscoelastic media constitute a large portion of synthetic and biological complex fluids. We present a parallelized 3D-simulation methodology which fully resolves the momentum balance in the solid and fluid domains. An immersed boundary algorithm is exploited known as the immersed finite element method (IFEM) which accurately determines the internal forces in the solid domain. The scheme utilized has the advantages of requiring no costly re-meshing, handling finite Reynold's number, as well as incorporating non-linear viscoelasticity in the fluid domain. Our algorithm is designed for computationally efficient simulation of multi-particle suspensions with mixed structure types. The internal force calculation in the solid domain in the IFEM is coupled with a finite volume based incompressible fluid solver, both of which are massively parallelized for distributed memory architectures. We performed extensive case studies to ensure the fidelity of our algorithm. Namely, a series of single particle simulations for capsules, red blood cells, and elastic solid deformable particles were conducted in viscous and viscoelastic media. All of our results are in excellent quantitative agreement with the corresponding reported data in the literature which are based on different simulation platforms. Furthermore, we assess the accuracy of multi-particle simulation of blood suspensions (red blood cells in plasma) with and without platelets. Finally, we present the results of a novel simulation of multiple solid deformable objects in a viscoelastic medium.

A viscoelastic beam theory of polymer jets with application to rotary jet spinning

Complex deformation of a polymer jet appears in many manufacturing processes, such as 3D printing, electrospinning, blow spinning, and rotary jet spinning. In these applications, a polymer melt or solution is first extruded through an orifice and forms a jet. The polymer jet is then dynamically deformed until the polymer solidifies. The final product is strongly affected by the deformation of the polymer jet. And the deformation is strongly affected by the viscoelasticity of the polymer. Here we develop a beam theory to incorporate both the nonlinear viscoelasticity and the bending/twisting stiffness of a polymer jet. As a demonstration, we study the formation of a polymer fiber under strong centrifugal force, a fiber manufacturing process known as rotary jet spinning.

Analysis of steady-state frictional rolling contact problems in Schapery–nonlinear viscoelasticity

In the context of an updated Lagrangian formulation, a computational model is developed for analyzing the steady-state frictional rolling contact problems in nonlinear viscoelastic solids. Schapery's nonlinear viscoelastic model is adopted to simulate the viscoelastic behavior. In addition to the material nonlinearity, the model accounts for geometrical nonlinearities, large displacements, and rotations with small strains. To satisfy the steady-state rolling contact condition, a spatially dependent incremental form of the viscoelastic constitutive equations is derived. Consequently, the dependence on the past history of the strain rate in the stress–strain law is expressed in terms of the spatial variation of the strain. The contact conditions are exactly satisfied by employing the Lagrange multiplier approach to enforce the contact constraints. The classical Coulomb's friction law is used to simulate friction. The developed model is verified and compared and good agreement is obtained. The applicability of the developed model is demonstrated by analyzing the steady-state rolling contact response of viscoelastically walled-wheel over rigid foundation. Moreover, the obtained results show remarkable effects of the rotational velocity and the viscoelastic material parameters on the mechanical response of steady-state frictional rolling contact.

Nonlinear viscoelasticity of a dilute suspension of Brownian spheroids in oscillatory shear flow

The nonlinear viscoelasticity of a dilute suspension of Brownian spheroids subject to an oscillatory shear deformation is calculated numerically. This is achieved by determining the suspension microstructure, parameterized via the orientation distribution function. Specifically, the long-time periodic orientation distribution function is obtained via a numerical solution to the Fokker–Planck equation by combining a finite-difference approximation in space with a Fourier series in time. From an ensemble average of the particle stresslet, weighted by the orientation distribution function, the entire stress tensor and relevant birefringence parameters, namely, the average orientation angle and linear dichroism, are calculated; this is done over a range of the Weissenberg number (Wi) and the Deborah number (De), or dimensionless strain-rate amplitude and oscillation frequency, respectively. Detailed calculations are presented for prolate spheroids of aspect ratio r=20; however, our methodology is general and can be applied to spheroids of arbitrary aspect ratio. We provide results in four viscoelastic regimes: linear viscoelastic (Wi≪1), quasilinear viscoelastic (Wi>1 and Wi/De≪1), quasisteady viscoelastic (De→0), and finally, the nonlinear viscoelastic regime (Wi ≳ 1 and Wi/De ≳ 1), which is our main emphasis. In this last regime, where the nonlinear and unsteady viscoelasticity of the material is probed, multiple overshoots are observed in the shear stress and first normal stress difference. The mechanistic origin of these overshoots can be understood from the periodic orientation dynamics (i.e., Jeffery orbits) of a particle under steady shear in the absence of Brownian rotation (Wi→∞). This is achieved by simultaneously analyzing the microstructure, shear stress, first normal stress difference, and birefringence parameters specifically at Wi=20 and De=1. For these values of Wi,De, and r, the period of a Jeffery orbit is comparable to the period of an oscillation cycle, allowing sufficient time for a single Jeffery orbit (and subsequent overshoot) to occur during an oscillation cycle. We contrast this behavior to recent work by Khair [J. Fluid. Mech. 791, R5 (2016)] on nearly spherical particles, for which many more overshoots are observed during an oscillation cycle, due to the shorter period of the Jeffery orbit. We briefly provide results for oblate spheroids of aspect ratio r=0.05 and compare to the results for r=20. Finally, we comment on the relevance of the present micro-mechanical analysis to the nonlinear viscoelasticity of other complex fluid materials.

A simple formulation for large-strain cyclic hyperelasto-plasticity using elastic correctors. Theory and algorithmic implementation

Proper finite element modelling of elastoplastic behavior under cyclic and multiaxial loading requires the consideration of nonlinear kinematic hardening. Popular models available for nonlinear kinematic hardening are based on multiple additive backstresses, whose evolution include a dynamic recovery term and follow the Armstrong-Frederick proposal; among them, the Ohno-Wang models. Whereas the small strain theory and its numerical implementation are satisfying, large strain extensions are more controversial, specially regarding the mathematical treatment of flow kinematics and the numerical implementation. In this work we present a new approach for modelling nonlinear kinematic hardening at large strains, reproducing the Ohno-Wang model at small strains without explicitly employing the backstress concept. The formulation uses only the classical Kröner-Lee multiplicative decomposition. It avoids the Lion decomposition and it is fully hyperelastic, employing only elastic variables both in the elastic and hardening parts, as well as in the flow equations. The theory has no restriction on the form of stored energies or in the amount of elastic strains so it can be used in soft materials, and it is weak-invariant and volume-preserving by construction. Furthermore, it has the same additive structure of classical small strain algorithms. Geometrical mapping tensors are systematically employed to account for large strain kinematics whereas the iterative algorithmic part is identical to the small strains model, which is recovered bypassing the geometrical mappings. The modelling of visco-hyperelastoplasticity is also straightforward by combining the present theory with finite nonlinear viscoelasticity formulations based on the same framework previously developed by our group.

Intraplate Strike‐Slip Faulting, Stress Accumulation, and Shear Localization of a Crust‐Upper Mantle System With Nonlinear Viscoelastic Material

AbstractWe investigate the evolution of tectonic background stress and elastic as well as inelastic strain in a crust‐upper mantle system around an infinitely long vertical strike‐slip fault in a self‐consistent mechanical earthquake cycle model. In the early stage of the stress evolution, deformation of the crust and the upper mantle is dominated by a uniform simple shear. Shear localization in the lower crust starts when coseismic rupture extends to the entire brittle upper crust. Together with this transition, the earthquake recurrence intervals decrease by an order of magnitude due to a basal drag originated from a localized plastic flow of the lower crust. After the shear zone is fully developed in the lower crust, the fault slip rate catches up with the far‐field velocity and earthquakes starts to occur periodically. Such a steady state can be reached in several hundred thousand years from the beginning, which includes few hundreds of earthquake cycles. A shear zone with large cumulative strain needs a few million years to develop under an intraplate strike‐slip fault, which is much longer than the time for shear strain rate to be localized. The model successfully reproduced evolution of tectonic stress around an intraplate strike‐slip fault, interacting with the development of localized shear zone in the lower crust. The model demonstrates the importance of considering the whole mechanical system in which rheological structure and fault activities interacting with each other for the better understanding of the intraplate earthquakes.