Generalized Maxwell Model Research Articles

Automated identification of linear viscoelastic constitutive laws with EUCLID

We extend EUCLID, a computational strategy for automated material model discovery and identification, to linear viscoelasticity. For this case, we perform a priori model selection by adopting a generalized Maxwell model expressed by a Prony series, and deploy EUCLID for identification. The methodology is based on four ingredients: i. full-field displacement and net force data; ii. a very wide material model library — in our case, a very large number of terms in the Prony series; iii. the linear momentum balance constraint; iv. the sparsity constraint. The devised strategy comprises two stages. Stage 1 relies on sparse regression; it enforces momentum balance on the data and exploits sparsity-promoting regularization to drastically reduce the number of terms in the Prony series and identify the material parameters. Stage 2 relies on k-means clustering; starting from the reduced set of terms from stage 1, it further reduces their number by grouping together Maxwell elements with very close relaxation times and summing the corresponding moduli. Automated procedures are proposed for the choice of the regularization parameter in stage 1 and of the number of clusters in stage 2. The overall strategy is demonstrated on artificial numerical data, both without and with the addition of noise, and shown to efficiently and accurately identify a linear viscoelastic model with five relaxation times across four orders of magnitude, out of a library with several hundreds of terms spanning relaxation times across seven orders of magnitude.

Rate-dependent properties of resistance-welded carbon fiber-reinforced polyetheretherketone joints: Simulations and experiments

The rate-dependent characteristics of the resistance-welded joints of thermoplastic composites are investigated via simulations and experiments. A rate-dependent cohesive zone model based on the Kohlrausch–William–Watts formulation is proposed to capture the mechanical response of the joints at different loading rates, which is validated by the experimental results in terms of stiffness, strength, and failure. The proposed model is equivalent to the generalized Maxwell model with only eight input parameters to describe the welding interface's time- and history-dependent mechanical response. The stress distribution, crack emergence, and damage evolution of welded joints at different loading rates are revealed through numerical calculations. Additionally, the experimental results indicate that the strength of the joint is enhanced with the increase in the loading speed, as the strengthening effect caused by the high strain rate and the shorter loading time results in a tightly bonded stainless-steel/matrix interface and a rougher fracture surface.

Experimental realization of rate-independent linear damping using a semiactive damper to enhance the seismic performance of low-frequency structures

Rate-independent linear damping (RILD) has been shown to simultaneously reduce the displacement and acceleration responses of low-frequency structures. However, previous studies on RILD have focused on its theoretical aspects and have almost excluded experimental approaches. Among the existing causal RILD models, the tuned Maxwell–Wiechert model can provide constant loss stiffness in a specified frequency range, thus offering a viable option as a control device intended to be incorporated into a base-isolated structure. A challenge remains regarding its undesired storage stiffness that compromises the flexibility of the base-isolated structure. This study eliminates the undesired storage stiffness via a negative stiffness-tuned Maxwell-Wiechert (NSTMW) model and experimentally investigates its feasibility to enhance the seismic performance of low-frequency structures. However, it is impossible to realize the NSTMW model using mechanical devices owing to the need for the negative damping coefficient of the Maxwell elements in some special cases. Therefore, the NSTMW model's frequency-domain expressions were transformed into time-domain differential equations using a bilinear transformation, and a small-scale magnetorheological damper was designed and fabricated to realize the NSTMW model. A series of real-time hybrid simulations were then conducted to examine the effectiveness of the NSTMW model for the protection of low-frequency structures. The experimental results confirmed that the NSTMW model could effectively control the dynamic response of low-frequency structures, compared with ideal RILD and linear viscous damping.

Rheological properties and microstructure of thermodynamically stable microemulsions as factors influencing the release rate of liposoluble vitamins

Microemulsions (ME) are stable nano-sized emulsions that constitute one of the most effective ways to incorporate lipophilic active compounds into water-based food matrices. Great stability, colloidal domain droplets, and optical isotropy are some of the advantages of ME. The present work analyzed the influence of rheological behavior and microstructural characteristics of food microemulsions on the release rate of lipophilic vitamins E and D. A fluid ME was compared to a ME with a higher dispersed phase content and a bicontinuous structure (gel-ME). Additionally, carboxymethyl cellulose (CMC) was added to fluid ME (3.5 and 10 g/100 g), obtaining systems with different apparent viscosities and microstructure. The percentage of released vitamins was determined by HPLC, and mathematically modeled. All ME were characterized using TEM microscopy FTIR, rheological and viscoelastic analysis. Fluid and gel-ME reached higher percentages of vitamin release in a very fast manner, while systems containing CMC showed a matrix-driven nature of the release. Those systems with similar zero-shear viscosity and different microstructure exhibited significantly different viscoelastic behavior. Microemulsions with CMC exhibited a viscoelastic solid type behavior with G’ > G’’. Mechanical spectra were satisfactorily fitted with the Generalized Maxwell model and relaxation time spectra were determined. In the quiescent state, gel-ME exhibited a higher plateau modulus than those containing large amount of thickener in the continuous phase. The relaxation time behavior of the structure could explain the kinetic release of liposoluble vitamins from the inner lipid phase of the microemulsions.

A theoretical model for two‐way bending behavior of bilayer shape memory polymer plate

AbstractThis study proposes a simple theoretical model based on the principles of phase transition and strain energy balance theories (during the thermo‐mechanical cycle process) to predict the two‐way bending behavior of a bilayer plate composed of two types of shape memory polymers (SMPs) with different glass transition temperatures under axial strains from 5% to 20%. The two‐way bending behavior observed from the theoretical results was verified via comparison with the finite‐element method (FEM) simulation based on the Generalized Maxwell model, wherein good agreement was demonstrated. The maximum errors for the critical bending angle and deflection were 6.82% and 5.36%, respectively. In addition, the two‐way bending behavior of a bilayer plate, including SMP and elastic polymer layers, was also predicted by the theoretical model, where good agreement was observed. The maximum errors for the critical bending angle and deflection were 15.20% and 12.82%, respectively. The theoretical model can predict the thermo‐mechanical behavior of smart actuators and help design the geometry configuration based on bilayer SMP plates in the future, such as smart hands or grippers.

Numerical investigation on meshing temperature of multicomponent nylon gears by incorporating viscoelasticity

This study concerns a numerical analysis of the meshing temperature for a multicomponent nylon gear by incorporating typical viscoelastic responses of polymers. Firstly, the generalized Maxwell model parameters were calibrated by the dynamic mechanical analysis to capture the viscoelasticity of polyamide 66. Next, the finite element model predicting the meshing temperature was established, including the heat generation analysis model, which calculates the generation of the frictional and hysteretic heat, and the heat propagation analysis model, which simulates the heat transmission. Different material models were used for polyamide 66 to investigate the influences of temperature dependence, strain rate dependence, and hysteretic effect. The effect of the iteration sequence on the meshing temperature calculation was also analyzed, i.e., how the heat generation should be updated as the loading cycles increase. A reasonable iteration scheme was determined by balancing the computational precision and consumption. It is noteworthy that the evolution of the ambient temperature around the gear and the worm within the housing was also considered. Finally, the accuracy of such meshing temperature analysis was validated on a self-developed gear test rig, in terms of the peak temperature evolution on the gear side surface. The experimental results and the simulation with the ambient temperature correction match well and the maximum error is 5 °C.

Numerical and experimental study of the dynamic behaviour of a polymer-metal worm drive

Polymer-metal worm drives are common in automotives and mechatronic systems. Despite this, there is a lack of studies on this type of transmission, especially when it comes to their dynamic behaviour. With the modern orientation of Industry 4.0 towards advancing technology in predictive maintenance, it is of a great importance to consider the study of the dynamic behaviour of this mechanism. It is first proposed to introduce an appropriate dynamic model to correctly simulate, by numerical means, the behaviour of a non-defective polymer-metal worm drive. For this purpose, it is necessary to correctly model the Gear Mesh Stiffness (GMS) of the gearing system. The GMS depends on the nature of the worm and worm gear materials. It is assumed that no deformation occurs in the steel worm, in contrast to the polymer worm whose viscoelastic behaviour must be accurately modelled. Generalized Maxwell Model (GMM) is chosen to model this behaviour. Eventually, the vibration signals from the numerical model are compared with those determined by the experimental tests. To obtain more similarities between the numerical and experimental signals, it is proposed to perform an optimisation. The procedure consists in using the Nelder-Mead simplex method to obtain a minimum residual objective function. After optimisation, an accuracy of 94% between the experimental and numerical results is achieved.

Effects of material memory on acoustic absorption

Acoustic absorption may be expressed as a sum of energy dissipated in the fluid and energy dissipated in the material in contact with the fluid. This presentation explores the physical mechanisms of energy dissipation in the material, which is assumed here to be viscoelastic. The constitutive law of a viscoelastic material depends on the strain history of the material, thus giving the material a memory. The energy absorbed by the material is dependent on this strain history. Relationships between material memory and dissipation have been investigated [Harry H. Hilton and Sung Yi, Smart Materials and Structures, 1(2), 113–122 (1992)], leading to ‘viscoelastic designer materials' for vibration damping. Starting from this foundation, analysis and numerical simulations are used to investigate the relationships between material memory and acoustic absorption. This investigation establishes the influence of parameters in a generalized Maxwell model on the acoustic absorption. The results inform the creation of metamaterials with microstructures designed to produce specific memory characteristics. [Work supported by ONR.]

Linear and nonlinear thermoviscoelastic behavior of polyamide 6

AbstractThermoplastic polyamides are used in many industrial areas due to their potential in lightweight applications. Polyamides serve as matrix material in fiber reinforced thermoplastics, for instance. The mechanical behavior of polyamides is characterized by pronounced viscoelastic effects that are strongly affected by environmental conditions like temperature or humidity. In this work, linear thermoviscoelastic behavior of polyamide 6 is considered. Viscoelastic behavior is modeled by the generalized Maxwell model while extended time‐temperature superposition is used to model temperature dependency. A temperature‐frequency sweep conducted by dynamic mechanical analysis serves as input for the model. By horizontal and vertical shifting, master curves of the loss factor, storage modulus, and loss modulus are obtained. Based on this, limitations of time‐temperature superposition and linear thermoviscoelastic modeling are discussed. Furthermore, it is shown that the horizontal shifts can be well approximated by the Williams‐Landel‐Ferry equation for temperatures above and below the glass transition temperature.

Thermo-Viscoelastic Characterization of 3D Printing Polymers

Polymer materials used in 3D printing exhibit degradation of material mechanical properties when exposed to thermal environments and thermal expansions can induce residual stresses in products or molds, which may result in dimensional instability and subsequent structural failures. In this study, based on linear thermo-viscoelastic principles, material degradation master curves, shift functions, and glass transition temperatures for four different polymers used for 3D printing techniques such as MultiJet Printing and Digital Lighting Process were measured by using a dynamic mechanical analyzer. Based on the single frequency test, the glass transition temperature was measured. In addition, dynamic measurements were carried out over a frequency range at isothermal condition and storage modulus vs. frequency curves were obtained. Then, the storage moduli curves measured at different temperatures were superposed into master curves using the frequency–temperature superposition principle and shift factors were calculated as a function of temperature. Subsequently, the complex moduli curves that were measured in the frequency were curve-fitted onto generalized Maxwell models by using the least squares method and the master curves of relaxation moduli at reference temperature were obtained. The effects of temperature, frequency, and time on dynamic moduli and relaxation behaviors of four polymers used for 3D printing were evaluated. Experimental results showed that Polymers C and D could be suitable to use at the service temperature above 100 °C and Polymer C was highly crosslinked and showed low modulus reduction after about a year. The master relaxation curves obtained through this process can be utilized to predict the long-term performance of polymer molds made by 3D printing at a given environmental condition.

Microscopic activated dynamics theory of the shear rheology and stress overshoot in ultradense glass-forming fluids and colloidal suspensions

We formulate a particle and force level, activated dynamics-based statistical mechanical theory for the continuous startup nonlinear shear rheology of ultradense glass-forming hard sphere fluids and colloidal suspensions in the context of the elastically collective nonlinear Langevin equation approach and a generalized Maxwell model constitutive equation. Activated structural relaxation is described as a coupled local-nonlocal event involving caging and longer range collective elasticity which controls the characteristic stress relaxation time. Theoretical predictions for the deformation-induced enhancement of mobility, the onset of relaxation acceleration at remarkably low values of stress, strain, or shear rate, apparent power law thinning of the steady-state structural relaxation time and viscosity, a nonvanishing activation barrier in the shear thinning regime, an apparent Herschel–Buckley form of the shear rate dependence of the steady-state shear stress, exponential growth of different measures of a yield or flow stress with packing fraction, and reduced fragility and dynamic heterogeneity under deformation were previously shown to be in good agreement with experiments. The central new question we address here is the defining feature of the transient response—the stress overshoot. In contrast to the steady-state flow regime, understanding the transient response requires an explicit treatment of the coupled nonequilibrium evolution of structure, elastic modulus, and stress relaxation time. We formulate a new quantitative model for this aspect in a physically motivated and computationally tractable manner. Theoretical predictions for the stress overshoot are shown to be in good agreement with experimental observations in the metastable ultradense regime of hard sphere colloidal suspensions as a function of shear rate and packing fraction, and accounting for deformation-assisted activated motion appears to be crucial for both the transient and steady-state responses.

Methodologies and models for measuring viscoelastic properties of cancer cells: Towards a universal classification

Different methods and several physical models exist to study cell viscoelasticity with the atomic force microscope (AFM). In search of a robust mechanical classification of cells through AFM, in this work, viscoelastic parameters of the cancer cell lines MDA-MB-231, DU-145, and MG-63 are obtained using two methodologies; through force–distance and force–relaxation curves. Four mechanical models were applied to fit the curves. The results show that both methodologies agree qualitatively on the parameters that quantify elasticity but disagree on the parameters that account for energy dissipation. The Fractional Zener (FZ) model represents well the information given by the Solid Linear Standard and Generalized Maxwell models. The Fractional Kelvin (FK) model concentrates the viscoelastic information mainly in two parameters, which could be an advantage over the other models. Therefore, the FZ and FK models are proposed as the basis for the classification of cancer cells. However, more research using these models is needed to obtain a broader view of the meaning of each parameter and to be able to establish a relationship between the parameters and the cellular components.

Dynamic mechanical analysis of PA 6 under hydrothermal influences and viscoelastic material modeling

Polyamides serve as matrix material for fiber reinforced composites and are widely applied in many different engineering applications. In this context, they are exposed to various environmental influences ranging from temperature to humidity. Thus, the influence of these environmental conditions on the mechanical behavior and the associated implications on the performance of the material is of utmost importance. In this work, the thermoviscoelastic behavior of polyamide 6 (PA 6) for two equilibrium moisture contents is investigated. To this end, dynamic mechanical analysis tests with and without humidity control of the environmental chamber were performed. In terms of relaxation tests, the experimental results reveal drying effects and increased diffusion activities when the sample’s equilibrium moisture content differs from the ambient humidity level within the testing chamber. Temperature-frequency tests quantify the humidity-induced shift of the glass transition temperature. The linear generalized Maxwell model (GMM) and time-temperature superposition are used to analyze the hydrothermal effects on the linear viscoelastic material properties and the onset of mechanical nonlinearity. Based on these investigations and findings, insight is gained on the humidity influence on the material properties and the limitations of linear thermoviscoelastic modeling. Furthermore, the computational construction of master curves and the parameter identification for a generalized Maxwell model are described in detail.

A time-temperature-ageing shift model for bitumen and asphalt mixtures based on free volume theory

ABSTRACT This paper aims to develop a general form of the time-temperature-ageing shift model for bituminous materials based on free volume theory and validate its application in the Finite Element (FE) modelling for predicting aged asphalt mixtures’ mechanical responses. The frequency sweep, dynamic modulus, and Fourier Transform Infrared Spectroscopy (FTIR) tests were used to validate the proposed time-temperature-ageing shift model. A generalised Maxwell model incorporating the proposed shift model of asphalt mixtures was used to predict the mechanical responses of aged asphalt mixtures using FE modelling, verified via laboratory cyclic load tests. Results indicate that the time-temperature-ageing shift function can model the complex shear modulus and dynamic modulus master curves of bituminous materials, addressing the modulus dependency of temperature and ageing. Temperature dominates the modulus change of bituminous materials in the ageing process. The coefficients of the time-temperature-ageing shift model have clear physical meanings and can potentially quantify the effects of asphalt mixtures’ air voids on their temperature and ageing susceptibilities. The time-temperature-ageing shift model can be incorporated into the constitutive relations of the FE model to effectively couple and efficiently predict the effects of temperature and ageing on the mechanical responses of aged asphalt mixtures.

Application of the Small Punch Creep-Recovery Test (SPCRT) for the Estimation of Large-Amplitude Viscoelastic Properties of Polymers.

The Small Punch Creep-Recovery Test (SPCRT) is a novel miniature test used to estimate the viscoelastic properties of polymers and biomaterials. The current investigation related to the SPCRT is limited to Finite Element Method (FEM) simulations and experimental tests on PVC. The aim of this investigation was focused on: (i) extending the experimental tests to other polymers with dissimilar viscoelastic properties; (ii) deepening the influence of non-linear viscoelastic properties in the estimation capabilities of the SPCRT; and (iii) developing a numerical methodology to estimate and take into account the viscoelastic recovery produced during the unloading step of compressive creep-recovery tests (CCRT) and SPCRTs. The experimental tests (CCRTs and SPCRTs) were done on polyethylene PE 500, polyoxymethylene POM C, nylon PA 6, and polytetrafluoroethylene (PTFE), with a range of creep loads, in the case of CCRTs, in the whole elastic regime and the surroundings of the yield strength of each material. The experimental results confirmed that the SPCRT was an accurate and reliable testing method for linear viscoelastic polymers. For a non-linear viscoelastic behavior, SPCRT estimated the viscoelastic properties obtained from CCRTs for creep loads near the yield strength of the polymer, which corresponded with large-amplitude viscoelastic properties in dynamic creep testing. In order to consider the viscoelastic recovery generated in the unloading step of CCRTs and SPCRTs, a Maxwell-Wiechert model with two branches was used, simulating the different steps of the experimental tests, and solving numerically the differential equation of the Maxwell-Wiechert model with the Runge-Kutta-Fehlberg (RKF) numerical method. The coefficients of the elements of the Maxwell-Wiechert model were estimated approaching the straining curve of the recovery step of the simulation with the same curve registered on each experimental test. Experimental CCRTs with different unloading times demonstrated that the use of this procedure derived in no influence of the unloading step time in the viscoelastic properties estimation.

A new flexible regression model with application to recovery probability Covid-19 patients

The aim of this study is to propose a generalized odd log-logistic Maxwell mixture model to analyze the effect of gender and age groups on lifetimes and on the recovery probabilities of Chinese individuals with COVID-19. We add new properties of the generalized Maxwell model. The coefficients of the regression and the recovered fraction are estimated by maximum likelihood and Bayesian methods. Further, some simulation studies are done to compare the regressions for different scenarios. Model-checking techniques based on the quantile residuals are addressed. The estimated survival functions for the patients are reported by age range and sex. The simulation study showed that mean squared errors decay toward zero and the average estimates converge to the true parameters when sample size increases. According to the fitted model, there is a significant difference only in the age group on the lifetime of individuals with COVID-19. Women have higher probability of recovering than men and individuals aged ≥ 60 years have lower recovered probabilities than those who aged < 60 years. The findings suggest that the proposed model could be a good alternative to analyze censored lifetime of individuals with COVID-19.

Mathematical modeling of hydrodynamic resistance in an oscillatory flow of a viscoelastic fluid

The problems of the oscillatory flow of a viscoelastic fluid in a flat channel for a given harmonic oscillation of the fluid flow rate are solved based on the generalized Maxwell model. The transfer function of the amplitude-phase frequency characteristics is determined. Using this function, the dependence of the hydrodynamic resistance on the dimensionless oscillation frequency is studied for various values of the elastic Deborah number and the concentration of the Newtonian fluid. It is shown that in an oscillatory flow of a viscoelastic fluid, the hydrodynamic resistance decreases depending on the Deborah number. With an increase in this number, the decrease becomes more pronounced than before. This effect allows us to evaluate the hydrodynamic resistance for a given law; the change in the longitudinal velocity averaged over the channel section and for the motion of a viscoelastic fluid in an unsteady flow allows us to determine the dissipation of the mechanical energy of the medium, which is important in the regulation of hydraulic and pneumatic systems.

Efficient Analytical Model for Long‐Term Performance of Laminated Tube with Viscoelastic Interlayer by Quasi‐Elastic Approximation

An efficient analytical method is proposed so as to study the long‐term performance of laminated tubes with neighboring layers glued by viscoelastic interlayers bearing inhomogeneous long‐duration loads. In the analysis, the formulations for describing the mechanical performance of each tube layer are based on the elasticity theory, while the viscoelastic behavior of the glue interlayer is modeled by the generalized Maxwell model. Making use of the quasielastic approximation and the recursive matrix method, the analytical solution for the long‐term performance of the laminated tube is solved efficiently. The present analytical solution is verified by comparing it with other solutions. The influences of the interlayer thickness, the material property, as well as the load distribution on the long‐term performance of the laminated tube are discussed.

Development of a fast and stable dynamics algorithm for soft robots including viscoelastic body

In recent years, research and development of robots that exist in the same space as humans and can collaborate with humans have been actively carried out. If the body of a robot is made of a hard material, it may cause injury. Therefore, attempts have been made to make a robot with a soft body using rubber or resin. In order to accelerate such research on soft robotics, it is necessary to establish fast and stable simulation algorithm for robots containing viscoelastic bodies such as rubber and resin. Therefore, in this study, we consider to approximate viscoelastic bodies with finite rigid body segments and connect them with joints and linear viscoelastic elements such as Voigt model, Maxwell model and generalized Maxwell model to approximate viscoelastic properties. The recursive dynamics algorithm is used to speed up the calculation, and the generalized-α method is used to stabilize the numerical integration. In particular, we propose a new method on how to incorporate the Maxwell model and generalized Maxwell model into recursive dynamics algorithm and generalized-α method. The effectiveness of the proposed method is confirmed by some numerical examples. In addition, the effectiveness of the proposed method is verified on a real system by applying Particle Swarm Optimization (PSO) to identify the dynamic parameters in linear viscoelastic elements.

EXPERIMENTAL STUDY OF THE RHEONOMIC PROPERTIES OF BREAKER RUBBER AND RUBBER-CORD COMPOSITE UNDER QUASI-STATIC LOADING

Samples of filled breaker rubber and rubber-cord composite were tested for quasi-static cycling loadings with constant strain rates. The dependence of the rubber mix stress-strain diagram on the strain rate was revealed even for low strain rates. A difference was observed between stress-strain diagrams for the first loading and for subsequent ones. The second and subsequent diagrams are close to each other. This effect is referred to in the literature as the Mullins effect. Eight experiments were conducted with different loading programs to determine the manifestation of this effect for breaker rubber. The stress-strain curves depend differently on the strain rate for the first and subsequent loadings. The dependence on strain rate can be described as using the generalized Maxwell model for the first loading, but not for subsequent ones. Moreover, a similar study was done for a bi-ply rubber-cord sample with a 45&amp;deg; cord angle. The same conclusion regarding the Maxwell model applicability was also proven. Finally, numerical simulation of the bi-ply rubber-cord composite was carried out to determine the dimensions of the edge effect zones and, thereby, to estimate the allowable dimensions of the sample. Two types of boundary effects were revealed. Their dimensions were calculated for the 45&amp;deg; rubber-cord sample with various lengths. It was shown that short samples can also be used, if the longitudinal strain is properly recalculated.