Nonlinear Viscoelastic Constitutive Model Research Articles

Dynamic compressive properties and constitutive model of waste crumb rubber (CR) modified ultra-high performance engineered cementitious composites (UHP-ECC)

Ultra-high-performance engineered cementitious composites (UHP-ECC) are well-known for their outstanding mechanical properties under static and dynamic loads. However, the effect of modifying UHP-ECC with crumb rubber (CR) on its performance, especially under high strain rates, remains poorly understood. This study systematically examined the dynamic compressive behavior and characteristic failure patterns of CR-modified UHP-ECC under high strain rates using a split Hopkinson pressure bar (SHPB) system. The study focused on evaluating the dynamic compressive strength, stress-strain response, and strain energy density (SED). A dynamic increase factor (DIF) formula was developed based on experimental data, and a nonlinear viscoelastic constitutive model was proposed to predict compressive behavior across various strain rates. Results indicate that while the addition of 10% CR slightly reduces the quasi-static compressive and flexural strength, it significantly enhances tensile ductility. Under high strain rates, CR-modified UHP-ECC shows no substantial effect on peak compressive stress but a marked increase in peak compressive strain. The SED also increased by 20.2%, outperforming other high-performance materials such as UHPC and normal ECC. These findings demonstrate that incorporating 10% CR into UHP-ECC offers considerable advantages, particularly in enhancing structural resilience under dynamic loading conditions. The proposed nonlinear viscoelastic model accurately predicts the compressive behavior of CR-modified UHP-ECC, presenting significant implications for practical engineering applications where high strain rate performance is crucial.

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.

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.

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.

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.

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.

Failure mechanical properties of lumbar intervertebral disc under high loading rate

BackgroundLumbar disc herniation (LDH) is the main clinical cause of low back pain. The pathogenesis of lumbar disc herniation is still uncertain, while it is often accompanied by disc rupture. In order to explore relationship between loading rate and failure mechanics that may lead to lumbar disc herniation, the failure mechanical properties of the intervertebral disc under high rates of loading were analyzed.MethodBend the lumbar motion segment of a healthy sheep by 5° and compress it to the ultimate strength point at a strain rate of 0.008/s, making a damaged sample. Within the normal strain range, the sample is subjected to quasi-static loading and high loading rate at different strain rates.ResultsFor healthy samples, the stress–strain curve appears collapsed only at high rates of compression; for damaged samples, the stress–strain curves collapse both at quasi-static and high-rate compression. For damaged samples, the strengthening stage becomes significantly shorter as the strain rate increases, indicating that its ability to prevent the destruction is significantly reduced. For damaged intervertebral disc, when subjected to quasi-static or high rates loading until failure, the phenomenon of nucleus pulposus (NP) prolapse occurs, indicating the occurrence of herniation. When subjected to quasi-static loading, the AF moves away from the NP, and inner AF has the greatest displacement; when subjected to high rates loading, the AF moves closer to the NP, and outer AF has the greatest displacement. The Zhu–Wang–Tang (ZWT) nonlinear viscoelastic constitutive model was used to describe the mechanical behavior of the intervertebral disc, and the fitting results were in good agreement with the experimental curve.ConclusionExperimental results show that, both damage and strain rate have a significant effect on the mechanical behavior of the disc fracture. The research work in this article has important theoretical guiding significance for preventing LDH in daily life.

Dynamic compressive behavior of polyethylene fiber reinforced high strength and high ductility concrete over wide ranges of strain rate

High strength and high ductility concrete (HSHDC) possesses extraordinary mechanical properties, including high compressive strength (123.3 MPa), high tensile strength (9.1 MPa), and ideal tensile strain capacity (6.6%), and exhibits promising application prospects in structures against dynamic loadings. This study systematically explored the dynamic compressive behavior of Polyethylene (PE) fiber reinforced HSHDC over wide ranges of strain rates (10−5–102 s−1) using the hydraulic servo-controlled testing machine and split Hopkinson pressure bar device. The dynamic stress-strain relations and failure patterns of HSHDC were obtained, based on which the dynamic compressive strength, peak compressive strain, and specific energy absorption (SEA) were analyzed in detail. It indicated that the compressive strength increased moderately at low strain rates (2.4 × 10−5–2.4 × 10−2 s−1) and obviously at high strain rates (105.8–258.8 s−1) with the increase of strain rate. To describe the dynamic compressive strength, several dynamic increase factor (DIF) formulae in the literature were evaluated, and an experimentally fitted DIF formula was proposed. The peak compressive strain remained almost constant at low strain rates, while it decreased significantly with the increase of strain rate at high strain rates. Besides, the dynamic SEA shows a bilinear growth law with the increasing strain rate and has a more obvious strain rate effect than the dynamic compressive strength. Finally, a nonlinear viscoelastic constitutive model was established, which satisfactorily predicts the compressive stress-strain relationship of HSHDC at different strain rates.

Characterization of nonlinear viscoelastic behaviors of CF/EP laminates with consideration to physical aging effect under thermo-mechanical loading

The characterization of the time-dependent deformation of fiber-reinforced polymer laminates is critical when evaluating the reliability of composite structures over long-term service lifetime. Several isothermal tensile creep tests under different temperatures and mechanical load levels for two types of carbon fiber-reinforced epoxy composite laminates are conducted in this work. The results indicate that the composites deformed in a time-dependent way and their axial strains are non-monotonic, which implies that both the creep temperature effect and physical aging effect come into play and the creep deformation was affected by the physical aging within the range of loads and temperatures involved in the tests. To formulate this kind of time-dependent behavior, a phenomenological nonlinear viscoelastic constitutive model of single-integral form was proposed and the effective time theory was introduced to describe the effects of temperature, stress, and physical aging. The measured strain was departed into two parts, one is the creep strain and the other is caused by the physical aging effect. The model was implemented within a finite element software by employing a recursive-iterative algorithm and was verified by comparing the simulation results with the experimental ones.

A nonlinear viscoelastic constitutive model with damage and experimental validation for composite solid propellant

The development of a nonlinear viscoelastic constitutive model of composite solid propellant (CSP) coupled with effects of strain rate and confining pressure is essential to assess the reliability of solid propellant grains during ignition operation process. In the present work, a nonlinear viscoelastic constitutive model with novel energy-based damage initiation criterion and evolution model was firstly proposed to describe the coupled effects of confining pressure and strain rate on mechanical responses of CSP. In the developed damage initiation criterion and evolution model, the linear viscoelastic strain energy density was introduced as the damage driving force, and the coupled effects of strain rate, damage history and confining pressure on damage growth were taken into account. Then, uniaxial tensile tests from low strain rates to medium strain rates and various confining pressures, and stress relaxation tests were conducted using a self-made active confining pressure device. Finally, the identification procedures of model parameters and validation results of the constitutive model were presented. Moreover, the master curve of damage initiation parameter was constructed through the time-pressure superposition principle (TPSP). The results show that the developed nonlinear constitutive model is capable of predicting the stress–strain responses of CSP under different strain rates and confining pressures.

Characterization of the nonlinear viscoelastic constitutive model of asphalt mixture

Asphalt mixtures are generally considered viscoelastic materials that exhibit time and frequency dependency and have been the subject of numerous studies. However, some studies have shown that the performance of asphalt mixtures changes with the magnitude of stress/strain, which is considered nonlinear viscoelasticity. Concurrently, the accurate establishment of an asphalt mixture constitutive model is the basis of the accurate prediction of the asphalt pavement mechanical response. Hence, the nonlinear viscoelastic constitutive in asphalt mixtures is analyzed and established. First, the uniaxial compressive creep experiment was conducted for three types of asphalt mixtures subjected to 0.1–1.1 MPa. The Schapery, Findley and multiple creep integral models were adopted to characterize the nonlinear viscoelastic performance of asphalt mixtures. The results showed that it was difficult to establish the multiple creep integral model due to its complex kernel function. The processes of establishing the Schapery and Findley models were similar, and the fitting degree of the Schapery model was superior to that of the Findley model. Finally, a Schapery nonlinear viscoelastic model of asphalt mixtures was proposed at 20 and 35 ℃.

Multiscale viscoelastic constitutive modeling of solid propellants subjected to large deformation

This paper develops a multiscale nonlinear viscoelastic constitutive model for the Nitrate Ester Plasticized Polyether (NEPE) propellant with high elongation. The proposed model considers viscoelasticity, strain rate, confining pressure, and softening effects. The macroscopic damage due to interface debonding or failure in the matrix itself is investigated by capturing the microstructural changes in a representative volume element (RVE). To account for the evolution of interface debonding, a physics-based normalized debonding ratio is introduced and modeled using a cohesive zone model (CZM). As the damage functions are obtained from the RVE, decoupled calibration of the material parameters is possible. The model is then programmed into ABAQUS via a subroutine for numerical analysis. The effects of strain rate, multi-step relaxation, creep, and confining pressure on the stress response are presented. Overall, the predicted results are in good agreement with the experiments.

Experimental and constitutive model study on the mechanical properties of a structural plane of a rock mass under dynamic disturbance.

An accurate description of the mechanical properties and deformation characteristics of a structural plane of a rock mass with a large chamber or slope under the ultimate stress with periodic stress disturbances is of great significance to ensure the stability and safety of underground rock engineering. By theoretically analysing the strength effect of a structural plane of a rock mass under dynamic disturbance, a criterion for the occurrence of shear damage on a structural plane of a compressed rock mass under dynamic disturbance is proposed. The results of the cyclic disturbance kinetic test show that there is a disturbance threshold for the shear failure of the structural plane under different disturbance stresses. When the disturbance stress is lower than the disturbance threshold, the cumulative plastic strain stabilizes with an increasing number of cycles; when the disturbance stress is higher than the disturbance threshold, an S-shaped curve of cumulative plastic strain versus the number of cycles is observed, revealing the progressive damage process and mechanism of such a rock structure plane under periodic dynamic disturbance. Based on perturbation concept theory, the relationship between the accumulated plastic strain and the number of cyclic loadings is similar to the relationship between strain and time, the creep curve. A new nonlinear viscous element is proposed, and the nonlinear element and the deformation element considering structural plane closure and sliding are combined with the Burgers model to form an 8-element nonlinear viscoelastic‒plastic creep constitutive model. Using the global optimization algorithm of 1stOpt, model validation and parameter identification are performed on the experimental data, and the results show that the model curve has a very good agreement with the experimental data. The model can accurately reflect the deformation characteristics of a structural plane of a rock mass under periodic dynamic disturbance. These research results provide a new idea for analysing disturbance-induced geohazards.

A Study on the Interaction Behavior between an Earth-Rock Dam and a New-Typed Polymer Anti-Seepage Wall

Polymer anti-seepage wall has been gradually applied in earth-rock dam reinforcement projects as a new seepage control technique. However, due to all-pervasive properties of the new materials and root-like connection between the materials and soils, the interface characteristics between the polymer wall and the earth-rock dam, as well as the interaction behavior of both, are complex and still not clear, which obstruct studying coordination mechanism of dam and wall under earthquake. Therefore, the interface characteristics and interaction behavior of dam and wall were studied in the article. Firstly, the dynamic shear stress-displacement, shear stiffness and damping ratio of the interface between polymer and soil were investigated by ring shear test. In addition, the viscoelastic constitutive model of polymer materials were researched by dynamic mechanical analysis (DMA) test. Based on tests results, a finite element model of earth-rock dam with polymer wall was established, including a non-linear simulation interface element and viscoelastic polymer constitutive model. Next, the validity of the simulation model was verified based on dynamic centrifuge test results. Then, the interaction behavior and seismic response of the dam with polymer wall were explored by using the verified model. The research results provide a scientific basis for the development and application of new-typed polymer anti-seepage wall in reinforcement engineering.

A Rate‐Dependent Constitutive Model of HTPB Propellant Based on Parallel Rheological Framework

AbstractNumerous studies have shown that highly strain‐rate dependent behavior is present in the mechanical properties of Hydroxyl‐terminated polybutadiene(HTPB)propellants, and many experts and scholars have attempted to develop constitutive models to describe such rate‐dependent properties of propellants. From the linear viscoelastic model to the Zhu‐Wang‐Tang model, the strain‐rate dependent properties of viscoelastic materials have been described to some extent. But due to the highly nonlinear mechanical behavior exhibited in propellant materials, the nonlinear behavior of propellants at different strain rates is difficult to be captured in the above constitutive models which are all based on linear elasticity. Therefore, in this paper, a nonlinear viscoelastic constitutive model was proposed based on the parallel rheological framework, in which the hyperelastic and viscoelastic flow laws were embedded. Because of the excellent ability to capture the nonlinear behavior of the material, the parallel rheological framework was used to predict the mechanical response at six sets of experiments with different rates. Surprisingly, this constitutive structure model performed good prediction of the mechanical properties of the material by comparison between theoretical calculations and experimental data. The constitutive model was verified by finite element method with numerical simulation software, and the robustness of the constitutive model was checked by a new set of experimental data. A good foundation was established for the accurate description of the propellant constitutive model strain‐rate dependent characteristics by the results of this paper, which could provide a new method for the rapid prediction and evaluation of the propellant mechanical response.

A Nonlinear Viscoelastic Constitutive Model for Solid Propellant with Rate-Dependent Cumulative Damage.

Solid propellant is a composite material exhibiting classic nonlinear viscoelastic mechanical characteristic, which is due in a large part to a cumulative damage process caused by the formation and growth of microflaws inside. The standard relaxation tests and uniaxial tension tests under different velocities of hydroxyl-terminated polybutadiene (HTPB) propellant are carried out in this paper, where Digital Image Correlation (DIC) technique is applied to record deformation. The experimental results show that the material mechanical behavior is rate-dependent. It is also observed that the yield stress and failure stress are significantly rate-dependent on the tensile velocity. Based on these experimental results, it can be inferred that the stiffness degradation and damage evolution of HTPB propellant are a rate-dependent processes. Therefore, the damage accumulation of HTPB propellant is considered rate-dependent in this research. In order to describe the mechanical characteristic precisely, a nonlinear viscoelastic constitutive model with rate-dependent cumulative damage is developed. The damage model is developed based on the concept of pseudo strain, in which a Prony series representation of viscoelastic material functions is applied. Besides, a rate-dependent damage variable is introduced into the model through considering the rate-dependent characteristics of cumulative damage process. In addition, a new normalized failure criterion is derived on the basis of the proposed damage model, which is independent of strain-rate after normalization. Finally, it is implemented in commercial finite element software for stress analysis to verify the predictive capacities of the damage model. The accuracy of the constitutive model and failure criterion is validated under uniaxial tensile tests of various strain rates.

Study on the relaxation behavior of bamboo fiber as a function of temperature and moisture content

The tensile relaxation behavior of bamboo fiber was evaluated. Bamboo fibers were extracted via a mechanical method and treated in alkaline solutions of sodium hydroxide. The relaxation tests of bamboo fibers were conducted by applying a constant strain at one end of a single bamboo fiber while the other end was fixed. The reaction force in the bamboo fiber was measured at the same time. The relaxation tests of the bamboo fibers were conducted under different temperature and moisture contents. The results showed that in the wet and/or high temperature conditions, the relaxation rate was faster than that in the dry and/or room temperature conditions. For a low strain, the bamboo fiber can be totally relaxed within 10 min under a high temperature. A nonlinear viscoelastic constitutive model was proposed to fit the relaxation curves, which could model the stress relaxation behavior of a bamboo fiber under different temperature or moisture content conditions. Compared with the experimental data, the model gave reasonable values for the relaxation behavior of the bamboo fiber.

Profile extrusion die design: A comparative study between elastic and inelastic fluids

AbstractComputational modeling is widely used to support the design of profile extrusion dies. However, despite the viscoelastic nature of the polymer melts, the majority of the computational approaches resort to inelastic models. With the aim of assessing the accuracy of the approach usually employed on the modeling of profile extrusion dies, this work aims at comparing the behavior of profile extrusion dies when interrelated viscoelastic (elastic) and generalized Newtonian (inelastic) constitutive models are used. For this purpose, the polymer melt employed in the study was experimentally characterized, being the data collected used to fit a nonlinear multimode viscoelastic (elastic) Giesekus constitutive model. Subsequently, the fitted model was used to generate the material shear flow curve, which was then used to fit a Bird‐Carreau (inelastic) constitutive model. The numerical studies undertaken were focused on the extrusion die of two profiles: a simple one, with a rectangular cross‐section, and a complex swimming pool cover profile. The results obtained showed that in realistic case studies, the effect of elasticity might be relevant, even when modeling just the flow in the extrusion die flow channel and, thus, should be considered when designing profile extrusion dies.

The Application of Micromechanical Modeling to Study the Non-Linear Behavior of Flexible Pavement under Truck Loading in Pakistan

Abstract: This report presents the work done to create a 3D model of a pavement structure, which incorporates in a most accurately way non damaging effects of asphalt layer considering the different layers that compose the pavement structure. A series of approaches have been considered from linear viscoelastic modelling using Maxwell model via UMAT subroutine to integrated viscoelastic response using Prony series available in the built in ABAQUS material library. During this work also other effects have been considered including the viscoelastic behaviour caused by creep, which is included via a separate CREEP subroutine. In order to predict also the non-linear effect of viscous elastic response the well-known Schapery's nonlinear viscoelastic constitutive model was considered. Implementing 3D Schapery’s model can be done by using again user material subroutine UMAT. It was found that the use of a nonlinear viscoelastic model substantially transformed the pavement response. A series of python scripts have been developed to create a multi-step analysis to simulate a cyclic loading application. Keywords: Viscoelasticity, Abaqus. Asphalt, Damage, Simulation, Modelling, Pavement design

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 ∼7.68%), showing improvements compared to its hyperelastic (mean error ∼24%) 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. Statement of SignificanceThe viscoelastic response of human heart tissues has yet to be integrated into common constitutive models describing cardiac mechanics. In this work, a fractional viscoelastic modeling approach is introduced based on the hierarchical structure of heart tissue. From these foundations, the current state-of-the-art biomechanical models of the heart muscle are transformed using fractional viscoelasticity, replicating passive muscle function across multiple experimental tests. Comparisons are drawn with current models to highlight the improvements of this approach and predictive responses show strong qualitative agreement with experimental data. The proposed model presents the first constitutive model aimed at capturing viscoelastic nonlinear response across multiple testing regimes, providing a platform for better understanding the biomechanics of myocardial tissue in health and disease.