Nonlinear Viscoelastic Model Research Articles

Treatment of inelastic material models within a dynamic ALE formulation for structures subjected to moving loads

AbstractThis article showcases the development of a dynamic Arbitrary Lagrangian Eulerian (ALE) formulation to account for inelastic material models within a finite element framework. Such a formulation is commonly utilized in research domains like fluid mechanics, fluid‐structure interaction, quasi static remeshing techniques, and quasi static load movement. The work at hand describes the application of the ALE formulation to efficiently analyse structures subjected to moving loads in the field of transient inelastic solid mechanics. In particular, structures such as pavements, gantry crane girders etc., which are subjected to moving loads, can be numerically simulated, and their transient response in the relevant region around the load can be obtained without relying on moving loads. The focus of this article is to facilitate the treatment of history variables stemming from inelastic material models. Of particular interest is the advection procedure required to transport the history variables through the mesh, as the material appears to flow through it. The mathematical framework necessary to treat this advection process is described in detail, considering a nonlinear viscoelastic material model on a neo‐Hookean base at finite deformations. Then, four methods for numerically achieving the advection are implemented within a transient finite element ALE formulation. These methods are compared against each other, and additionally with the conventional Lagrangian method for validation. The results demonstrate satisfactory agreement with conventional simulation methods, while offering a significant improvement in terms of computation speed. With the work at hand, the dynamic response of inelastic materials subjected to moving loads can be numerically simulated in a computationally efficient manner.

Investigating fracture mechanisms in glassy polymers using coupled particle-continuum simulations

We study the fracture behavior of glassy polymers using a multiscale simulation method that integrates a molecular dynamics (MD) system within a continuum domain. By employing a nonlinear viscoelastic constitutive model in the continuum domain, the MD system undergoes non-uniform deformation with flexible boundaries through interaction with the surrounding continuum. Systems with pre-defined double cracks are subjected to tensile stretch under various geometric constraints and bond breakage criteria. The simulation results show that geometric constraints primarily affect deformation behavior at small strains, but their influence diminishes at larger strains. In the stage of fracture, increased deformation correlates with a narrowing distribution of microscopic structures at molecular scales, such as bond length. This distribution converges across different systems just before fracture, irrespective of ultimate stress, fracture strain, bond breakage criteria, and geometric constraints. This convergence suggests that the distribution of microscopic structures is a fundamental property linked to the fracture behavior of glassy polymers.

Investigations on the finite strain behavior of standard PVB: experiment and modeling

AbstractThe numerical simulation of the residual load-bearing capacity of laminated glass is one of the most relevant topics of current research in structural glass engineering. The mechanical description of the behavior of the interlayer plays a crucial role. Several approaches and models already exist in automotive engineering and structural protection. Structural glass design assuming quasi-static loads misses a detailed model so far. The aim of this work is the mechanical description of the time-dependent behavior of Polyvinylbutyral (PVB) under large deformations considering quasi-static loading. The results of experiments on single-layer Standard PVB are presented, and a model of finite nonlinear viscoelasticity is derived. In particular, the formulation of phenomenologically motivated viscosity functions is described. Subsequently, the model parameters are calibrated on the experimental results, and the uniaxial stress state is simulated.

Nonlinear Viscoelastic Modeling of Finger Arteries: Toward Smartphone-Based Blood Pressure Monitoring via the Oscillometric Finger Pressing Method.

Oscillometric finger pressing is a smartphone-based blood pressure (BP) monitoring method. Finger photoplethysmography (PPG) oscillations and pressure are measured during a steady increase in finger pressure, and an algorithm computes systolic BP (SP) and diastolic BP (DP) from the measurements. The objective was to assess the impact of finger artery viscoelasticity on the BP computation. Nonlinear viscoelastic models relating transmural pressure (finger BP - applied pressure) to PPG oscillations during finger pressing were developed. The output of each model to a measured transmural pressure input was fitted to measured PPG oscillations from 15 participants. A parametric sensitivity analysis was performed via model simulations to elucidate the viscoelastic effect on the derivative-based BP computation algorithm. A Wiener viscoelastic model comprising a first-order transfer function followed by a static sigmoidal function fitted the measured PPG oscillations better than an elastic model containing only the static function (median (IQR) error of 30.5% (25.6%-34.0%) vs 50.9% (46.7%-53.7%); p<0.01). In Wiener model simulations, the derivative algorithm underestimated SP, especially with high pulse pressure and low transfer function cutoff frequency (i.e., greater viscoelasticity). The mean of the normalized PPG waveform at the maximum oscillation beat was found to correlate with the cutoff frequency (r = -0.8) and could thus possibly be used to compensate for viscoelasticity. Finger artery viscoelasticity negatively impacts oscillometric BP computation algorithms but can potentially be compensated for using available measurements. These findings may help in converting smartphones into truly cuffless BP monitors for improving hypertension awareness and control.

Study on fragmentation characteristics of carbon nanotube-enhanced concrete and a comparative evaluation of the ZWT and ottosen constitutive model

In order to study the difference between the crushing characteristics of carbon nanotubes (CNTs) concrete and ordinary concrete, this paper conducts a test between ordinary concrete and CNTs concrete with a volume content of 0.3 % at different impact speeds through the Hopkinson test, using fragmentation and particle size distribution as evaluation indexes. Crushed fragments are analyzed from the perspective of concrete pores. Finally, the nonlinear viscoelastic constitutive model and the nonlinear elastic constitutive model of CNTs concrete are then developed and compared. Based on the results, the average pores of the CNTs concrete decreased from 42.0 nm to 38.0 nm, but the pore area increased from 5.743 m2/g to 6.305 m2/g, indicating that CNTs can fill the internal pores. Meanwhile, the average particle size of broken CNTs concrete is significantly larger than that of ordinary concrete. However, the relationship between the crushing particle size and impact velocity is approximately y=A-Bln(x+C) for both concretes. According to the study of concrete constitutive models, compared to the Loand damage model, the combination of Loand model and Mazars model is more accurate in describing concrete damage characteristics, and the average correlation coefficient increases from 0.84 to 0.92. Meanwhile, compared to the Zhu-Wang-Tang (ZWT) model, Ottosen model can better capture the dynamic mechanical properties of CNT concrete, with a correlation coefficient over 0.98.

Numerical approximation for algorithmic tangent moduli for nonlinear viscoelastic model with CSDA method

This work revisits the notion of complex step derivative approximation (CSDA) and presents its use in constitutive model of a class of nonlinear viscoelastic materials. The effectiveness of a CSDA is evaluated by putting it through a series of straightforward examples. After that, the idea of the CSDA is put to use in order to carry out a numerical evaluation of the algorithmic tangent moduli of a viscoelastic constitutive model. The performance of the constitutive models is evaluated through the use of three different numerical tests, and the results are compared to those that were achieved by the application of an analytical method. In comparison to other numerical differentiation techniques, It has been found that the CSDA scheme is the most computationally efficient and robust method of numerical differentiation, regardless of the size of the finite difference interval.

Rate-Dependent Tensile Properties of Aluminum-Hydroxide-Enhanced Ethylene Propylene Diene Monomer Coatings for Solid Rocket Motors.

Quasi-static and dynamic tensile tests on aluminum-hydroxide-enhanced ethylene propylene diene monomer (EPDM) coatings were conducted using a universal testing machine and a Split Hopkinson Tension Bar (SHTB) over a strain rate range of 10-3 to 103 s-1. This comprehensive study explored the tensile performance of enhanced EPDM coatings in solid rocket motors. The results demonstrated a significant impact of strain rate on the mechanical properties of EPDM coatings. To capture the hyperelastic and viscoelastic characteristics of EPDM coatings at large strains, the Ogden hyperelastic model was used to replace the standard elastic component to develop an enhanced Zhu-Wang-Tang (ZWT) nonlinear viscoelastic constitutive model. The model parameters were fitted using a particle swarm optimization (PSO) algorithm. The improved constitutive model's predictions closely matched the experimental data, accurately capturing stress-strain responses and inflection points. It effectively predicts the tensile behavior of aluminum-hydroxide-enhanced EPDM coatings within a 20% strain range and a wide strain rate range.

Nonlinear time-dependent behavior of rheodictic polymers: A theoretical and experimental investigation

The present study examined the nonlinear time-dependent behavior of rheodictic polymers, a class of noncrosslinked materials that exhibit flow. Such behavior was addressed with extended Schapery's nonlinear viscoelastic model by introducing new physical quantities, i.e., the flow term Φflow and corresponding nonlinear shift parameter g2,flow. While Φflow portrays irrecoverable deformation, g2,flow depicts a nonlinear contribution to flow acceleration. This theory was accompanied by analytical and experimental methodologies for identifying all the parameters in the linear and nonlinear viscoelastic domains. Predictions of long-term time-dependent behavior (in shear) at various stress states show excellent agreement with the experimental data, i.e., within 5% error, obtained for polycarbonate at 130°C. Surprisingly, the newly introduced g2,flow indicates that flow retardation occurs with increasing stress, implying that a highly deformed entangled system hinders molecular reptation/disentanglement. Nevertheless, the proposed extension of Schapery's nonlinear viscoelastic model not only allows accurate predictions of the nonlinear time-dependent behavior of rheodictic polymers but also enables a detailed outlook on the underlying molecular mechanisms under severe environmental and loading conditions.

Mechanistic Pavement Modeling of Typical Brazilian Pavements: Cohesive Zone Fracture Modeling and Continuum Damage Modeling

Pavement performance evaluation with respect to fatigue cracking has seen a major shift toward more rigorous mechanistic models that can capture the complex damage response of the mixtures more accurately. This study utilized two mechanistic methods (cohesive zone [CZ]-based fracture modeling and continuum damage [CD] modeling) to study the cracking behavior of asphalt mixtures and subsequently predict the damage-associated performance of pavements. A Superpave mixture designed for Brazilian highway conditions was selected to evaluate the two modeling approaches for the performance of the mixture when utilized as a surface course within typical Brazilian highway pavements. The mixture was first evaluated for its linear viscoelastic properties, and then the damage characteristics of the mixture were obtained through two different experiments: the semi-circular bending (SCB) beam test and the cyclic fatigue (CF) test. The SCB tests performed at different loading rates were used to obtain the model parameters of the nonlinear viscoelastic CZ model with a Gaussian damage evolution criterion, while the CF tests were performed at different strain amplitudes, and the mixture-specific damage characteristic curve and the fatigue failure criterion were obtained using the simplified-viscoelastic continuum damage model. From a design perspective, three pavement structures that varied in geometry (pavement layer thickness) and underlying layer properties were selected while retaining the same asphalt mixture. The three pavement scenarios were evaluated for the fatigue cracking performance from each of the mechanistic modeling methods. The results indicate that both methods rank the performance of the pavement structures in a similar manner, while the damage initiation and progression were seen to be different because of the different mechanics in modeling cracks.

Finite element analysis of a rubber bearing base isolator under vertical and horizontal loads using a nonlinear hyper-viscoelastic material model

This research is devoted to developing a finite element model using Abaqus code for the computer simulation of an in-house developed and manufactured rubber bearing subjected to static vertical and cyclic horizontal loads. A high-damping rubber compound was designed. The material behavior of the rubber was assumed to be described by the hyper-viscoelastic model. Both linear (Prony series) and nonlinear (strain hardening power law) viscoelastic relationships were used in conjunction with the Ogden-Roxburgh equation to take the addition of the stress softening phenomenon or Mullins effect into consideration. The parameters of the material model were determined using MCalibration code in which an optimization technique was used, and data obtained in experiments carried out on test specimens were fitted into the selected model. The results of the simulations were compared with their corresponding experimental data carried out on the rubber bearing. The force-displacement behavior, stress and strain fields, and computed energy were presented and discussed. It is shown that the nonlinear viscoelastic model accompanied by the Mullins effect gives the best results. Moreover, the model could accurately predict the energy variations during the earthquake loading.

Impact properties of an end of life tires’ rubber. Numerical validation considering large strain and strain rate conditions

The objective of the authors is to demonstrate that the inclusion of renewed rubber from recycled end of life tires (ELT) can improve the performance of any structural system designed to dissipate impact energy. An interesting application would be its use in road safety barriers. This research starts with the rigorous characterization of the recycled material in order to include it in a viable numerical model. The authors presented, in a previous work (*), the experimental viscoelastic properties of recycled rubber, obtained under impact conditions. Bergström–Boyce (BB) nonlinear viscoelastic model was selected as the most suitable to fit the material behavior. This model is defined by nine material constants that are impossible to obtain, uniquely and directly, considering that only compression test results are available as input data. To overcome this challenge, optimization methods were applied resulting in as many sets of parameters as used optimization methods were considered and, moreover, remarkable differences between constants were observed. Solving this problem is the motivation of the present research: the validation of the optimization method by mean of the numerical evaluation of the obtained sets. The software considered for the numerical evaluation of impact tests (LS-DYNA®) had two slightly different implementations for the BB model: the original, based on the work published in 2009 by Dal & Kaliske and a more recent one implemented by Bergström, based on his works published in 1998 and 2000. This leads to a new question: which is the most appropriate implementation-optimization method for calibrating this material? This paper provides the answer to this question carrying out a complete comparative numerical analysis of all sets by means of explicit dynamics simulations. All calibrations, using the original LS-DYNA® implementation, were in perfect agreement with the experimental results. However, only one optimization method produced acceptable results for the most recent approach. A robust method to characterize recycled rubber from recycled tires using the Bergström–Boyce nonlinear viscoelastic model is presented, despite the experimental limitations (González-Vega et al. 2022).

A Constitutive Framework for Modeling of the Visco-Hyperelastic Materials: Application to the Mechanical Behavior of Cylinders Made of the Rate-Dependent Elastomers

In this paper, the formulation of the mechanical behavior of visco-hyperelastic materials is developed in a thermo-dynamically compatible framework. In this viewpoint, the stress power is assumed to be equal to the sum of the reversible (elastic) energy rate stored and the irreversible energy rate dissipated. Based on this assumption and using the second law of thermodynamics, the constitutive modeling framework is obtained for the rate-dependent materials. Inspired by the constitutive law of a Newtonian fluid and also, models of density strain energy of an elastic material, a framework for the dissipative energy rate related to the irreversible part is proposed for viscoelastic materials. In this framework, the reversible energy part is considered a function of the right Cauchy–Green deformation tensor, and the irreversible energy part is considered a function of the same tensor’s rate. To evaluate the proposed model for nonlinear viscoelastic behavior modeling, the results of tests performed on non-compressible materials at different rates are used, and the proposed model provided acceptable results. Finally, the proposed model is implemented to find a closed-form analytical solution for mechanical behavior modeling of the thick-walled cylindrical shells made of visco-hyperelastic materials, also, stress and stability analyses are assessed for these types of cylinders. To achieve this goal, the effect of pre-stretch on the stability of cylinders has been substantially investigated. Furthermore, based on the proposed model, the effect of the parameters such as the wall thickness and behavior of the material used in the cylinder are examined on the stability of open-ended and closed-ended cylinders.

On the Schapery nonlinear viscoelastic model: A review

Among the various models presented so far to describe the nonlinear viscoelastic behavior of materials, the model proposed by Schapery in the 1960s stands out for its wide acceptance and significant contributions to the field. Through a detailed inspection of the main articles presented by Schapery and other related studies, the present paper reviews the Schapery nonlinear viscoelastic model. One-dimensional and three-dimensional formulations of the model for both isotropic and anisotropic materials, along with model accuracy, are discussed. The present article highlights the challenges of material characterization methods, such as creep-recovery testing, stress relaxation testing, and dynamic mechanical analysis. The article also addresses the issues of characterizing the material's behavior under non-recoverable strain and varying temperatures. Finally, the future research needs in this field are outlined, emphasizing the potential for further advancements in our understanding of nonlinear viscoelasticity.

Insight into compression-rebound behavior of lignocellulose-derived aerogel through finite element modeling based dynamic impact simulation

In this study, lignocellulose-derived aerogel (LA) with compression-rebound was synthesized from liquefied wood through sol-gel polymerization, aging and freeze-drying process. The results presented a gradual increase in the compressive strength, flexural strength, and maximum decomposition temperature of the LA sample with prolonged aging time from 3 to 10 h, demonstrating excellent elasticity and thermal stability. After more than 10 h, the bending strength began to decrease, indicating that the material’s rigidity was greater than its elastic toughness at this time. A nonlinear viscoelastic constitutive model based on the Ogden function and parallel rheological framework showed very high correlation with experimental measurement results, and also demonstrated admirable mechanical properties. These predictions can provide an effective approach for the application of LA in buffer packaging.

Viscoelastic Model of Fiber Concrete under Dynamic Loading Considering the Effect of Initial Stresses

Statement of the problem. The objective of the study was to identify the peculiarities of the stress-strain state of fiber concrete under two-stage static-dynamic loading and to develop a variant of a nonlinear viscoelastic model of concrete reinforced with steel fiber. Results. The paper presents the results of experimental tests of fiber-concrete prisms under quasi-static, dynamic and static-dynamic loading conditions. On the basis of the analysis of experimental data it is established that at dynamic loading there is an upward shift of the boundary of macrocrack formation. A nonlinear viscoelastic model of fiber concrete is proposed and compared with experimental data, taking into account the effect of the initial stress level caused by the service load. The parametric study performed using this model shows that as the loading rate increases, a more intensive dynamic hardening of concrete is observed. Conclusions. The proposed viscoelastic model of fiber concrete allows to estimate the ultimate strength and corresponding strain under static-dynamic mode of loading due to an accidental action caused by the collapse of one of the load-bearing members of the structural system of the building.

Analysis of the mechanical response of solid rocket engine propellant under acceleration shock

The increased-range solid rocket engine of new ammunition often needs to withstand extremely high acceleration overload, leading to damage to the propellant’s structural integrity. To address this issue, this paper constructs a nonlinear viscoelastic constitutive model for the CMDB propellant by using low- and high-strain rate mechanical property tests. Combined with the secondary development of explicit dynamics and numerical simulation technology, the mechanical response of a certain type of solid rocket propellant under different acceleration impacts is analyzed. The results show that under acceleration impact loads, the propellant will compress, rebound, and recover over time, continuously cycling. Its axial displacement, maximum equivalent stress, and maximum equivalent strain exhibit irregular sinusoidal wave-like periodic cycles. Looking at the time when the peaks appear, the time when the maximum equivalent stress appears always lags behind the time when the maximum axial displacement peak appears. Due to the viscous effect of the viscoelastic material of the propellant, the time when the equivalent strain peak appears will lag behind the equivalent stress. Because of the material’s damping effect, both the peak values of the maximum equivalent stress and equivalent strain decrease over time. Under continuous high acceleration impact loads, this viscous damping phenomenon continuously diminishes, and the peak value of the propellant’s axial displacement gradually increases.

Damage creep model of viscoelastic rock based on fractional derivative and experimental verification

Exploring the creep law of sandstone provides a theoretical basis for evaluating the long-term stability of geotechnical engineering projects in red beds. Based on a conventional triaxial test of sandstone, a progressive loading triaxial creep test is conducted. The deformation characteristics and laws of each sample in different deformation stages are summarized, and the laws relating steady creep rate, stress and time are analyzed. On this basis, a nonlinear viscoelastic‒plastic creep model based on fractional derivative theory and damage theory is established. According to the nonlinear fitting results, the parameter sensitivities are analyzed. The results verify the rationality of the model; this model has a good fitting effect for each creep deformation stage, especially for the accelerated creep stage. The constitutive relationship of the model is simple, clear and easily applicable. The research results provide a reference for studying the long-term stability of geotechnical engineering projects.

Parametric simulations of fractal-fractional non-linear viscoelastic fluid model with finite difference scheme

Fractal-fractional derivatives are more general than the fractional derivative and classical derivative in terms of order. Fractal-fractional derivative is used in those models where the classical continuum hypothesis theory fails. More precisely, these derivative operators are used where the surface or space is discontinuous, e.g., porous medium. Fractal-fractional derivative is considered advance tool to analyze the fluid dynamic model more than fractional and classical model. Given the extensive applicability of fractal-fractional derivatives, the current analysis focuses on investigating the behavior of a non-linear Walter’s-B fluid model under the influence of time-varying temperature and concentration During the simulation process, we have also taken into account the effects of first-order chemical reactions, Soret numbers, thermal radiation, Joule heating, and viscous dissipation of energy. A magnetic field with a strength of B0 was applied to the left plate in the transverse direction. The classical mathematical model was first developed using relative constitutive equations and later generalized with the fractal-fractional derivative operator. Numerical solutions to the generalized model have been obtained using the finite difference method. Various graphs are drawn from the obtained numerical solutions to study the influence of physical parameters on the rheology of Walter’s-B fluid. It has been observed that by varying the fractional and fractal order of the generalized model, one can easily derive fractal, fractional, and classical models.

Performance Evaluation of Nonlinear Viscoelastic Materials using Finite Element Method

This research paper applies the finite element method as a methodology to evaluate the structural performance of nonlinear viscoelastic solids. A finite element algorithm was built and developed to simulate the mathematical nonlinear viscoelastic material behavior based on incremental constitutive equations. The derived Equation of the incremental constitutive included the complete strain and stress histories. The Schapery’s nonlinear viscoelastic material model was integrated within the displacement-based finite element environment to perform the analysis. A modified Newton-Raphson technique was used to solve the nonlinear part in the resultant equations. In this work, the deviatoric and volumetric strain–stress relations were decoupled, and the hereditary strains were updated at the end of each time increment. It is worth mentioning that the developed algorithm can be effectively employed for all the permissible values of Poisson’s ratio by using a selective integration procedure. The algorithm was tested for a number of applications, and the results were compared with some previously published experimental results. A small percentage error of (1%) was observed comparing the published experimental results. The developed algorithm can be considered a promising numerical tool that overcomes convergence issues, enhancing equilibrium with high-accuracy results.

Establishment of strain rate‐dependent intrinsic models for short‐cut carbon fiber/nitrile composites in different pyrolysis states

AbstractFlexible nozzle extension technology has been extensively researched throughout the world, but traditional materials cannot meet the growing demands in harsh environments. Short‐cut carbon fiber‐reinforced nitrile rubber (CF/NBR) composites, which are similar to the phenolic impregnated carbon ablator, are currently a hot research topic in this field. The mechanical properties of CF/NBR composites were experimentally tested to study the change in the mechanical behavior of a rocket engine's flexible nozzle extension section under thermal coupling. As indicated by the results, existing models cannot accurately predict changes in materials' mechanical behavior. The theoretical foundation of the nonlinear viscoelastic intrinsic model of rubber is utilized. The uniaxial tensile intrinsic model of CF/NBR composites has been improved by introducing physical parameters, such as fiber reinforcement, strain rate‐dependent correction and temperature‐dependent correction coefficients, and fiber volume fraction. Results show that the intrinsic model developed in this article provides a guide to the uniaxial tensile mechanical behavior of randomly distributed short‐cut CF/NBR composites with known mass‐part formulations and fiber volume fractions at room temperature and under different pyrolysis conditions.