Nonlinear Viscoelastic Model Research Articles

Mapping deformation and dissipation during fracture of soft viscoelastic solid

Energy dissipation around a propagating crack is the primary mechanism for the enhanced fracture toughness in viscoelastic solids. Such dissipation is spatially non-uniform and is highly coupled to the crack propagation process due to the history-dependent nature of viscoelasticity. We present an experimental approach to map the dissipation field during crack propagation in soft viscoelastic solid. Specifically, we track randomly distributed tracer particles to measure the evolving deformation field. The measured deformation field is then put into a nonlinear constitutive model to determine the dissipation field. Our methodology was used to investigate the deformation and dissipation fields around a propagating crack in a Polyampholyte (PA) hydrogel. The deformation field measurements allowed us to assess whether the commonly assumed translational invariance in viscoelastic fracture theories holds true in practical experiments. Furthermore, by combining the obtained deformation fields with a nonlinear viscoelastic model, we captured the complete history of the dissipation field during crack propagation. We found that dissipation occurred even at material points that are a few millimeters away from the crack tip. The mapped dissipation field also enabled the separate determination of the intrinsic and dissipative components of fracture toughness for the viscoelastic hydrogel.

Energy of a non-linear viscoelastic model compatible with fractional relaxation

Recently, a non-linear model of viscoelasticity based on Rational Extended Thermodynamics was proposed in Ruggeri (2024). This theory extends the evolution of the viscous stress beyond the linear framework of the Maxwell model to the non-linear realm, provided that the viscous energy function is given. This work aims at establishing a possible constitutive law for the viscous energy such that the relaxation modulus of the fractional Maxwell model with order α∈(1/2,1] is contained within the solutions of the (non-linear) relaxation experiment. Necessary and sufficient conditions for the existence of this coincident solution are discussed, together with a numerical evaluation of the viscous energy associated with the nonlinear model.

Time-Temperature-Stress Equivalent Characteristics and Nonlinear Viscoelastic Model of Asphalt Mixture under Triaxial Compressive Stress State

Time-Temperature-Stress Equivalent Characteristics and Nonlinear Viscoelastic Model of Asphalt Mixture under Triaxial Compressive Stress State

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.

General analysis of the dissipation of strain energy in circular columns

This theoretical paper is concerned with general analysis of the strain energy dissipation in columns of circular cross-section. The columns are under longitudinal displacements that vary continuously with time. The columns exhibit non-linear viscoelastic behavior that is studied by mechanical models constructed by using non-linear springs and dashpots. Besides, the columns are functionally graded along the radius of the cross-sections. A simple expression for the dissipated strain energy in the columns under consideration is derived. The expression holds for columns having portions with different radius of the cross-section. Also, the expression derived is applicable for columns built-up by concentric layers. Results indicating how the dissipated strain energy is influenced by various factors (distribution of material properties, geometry of the columns and loading) are presented. These results are found by using a non-linear viscoelastic model with one spring and one dashpot. A check-up is performed by determining the dissipated strain energy through subtracting the strain energy in the spring of viscoelastic model from the whole strain energy in the column.

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

A Nonlinear Viscoelastic Model for Silicon Rubber Foam Cushion Considering Time-Varying Evolution Characteristics

A Nonlinear Viscoelastic Model for Silicon Rubber Foam Cushion Considering Time-Varying Evolution Characteristics

Evaluation of the mechanical properties of porcine kidney.

With the development of medical diagnosis and treatment, knowing the mechanical properties of living tissues becomes critical. The aim of this study was to investigation material properties of the fresh porcine kidney and the parametric characterization of its viscoelastic material behavior. The material investigation included uniaxial tension tests in different strain rates, relaxation tests, as well as hydrostatic compression tests on the samples extracted from the fresh porcine kidney cortex. Tension tests and relaxation tests were performed by a planar dog-bone specimen with a micron loading testing machine. Hydrostatic compression tests were performed on the kidney cylinder sample which was placed in a compression chamber. Furthermore, a nonlinear viscoelastic model recently proposed by us was employed to characterize the tension data at different strain rates and relaxation test data. The the experimental and numerical results show that the stress-strain relations of the porcine kidney cortex at different strain rates in tension are presented for the first time and a higher strain rate results in higher ultimate strength and initial Young modulus but a lower rupture strain. A damage-dependent visco-elastic model is employed to model the tension data at different strain rates and relaxation data and exhibits a good agreement with the experimental data, which also demonstrates that the damage has an obvious influence on the stress-strain relation. Through comparison with the existing reference covering the uniaxial compression data, it seems that the mechanical behavior of the porcine kidney cortex manifests a stress state-dependent mechanical behavior. The ultimate strength and rupture strain are larger in compression than that in tension.

Continuum Mechanics Models for Describing the Creep and Physical Aging Behaviors of Laminated Composites

Two types of carbon fiber-reinforced epoxy composite laminates are chosen for long-term tensile creep tests under different temperatures and load levels. Their time-dependent and non-monotonic deformations indicate clearly both temperature effect and physical aging effect. To characterize these viscoelastic behavior, two phenomenological constitutive models and one physical model are developed. The linear viscoelastic model based on the Boltzmann superposition principle is able to describe reasonably the deformations at relatively lower stress levels and temperatures. The nonlinear viscoelastic model of Schapery’s single-integral form, together with a usage of effective time theory, could describe nicely all the effects of temperature, stress, and physical aging. The physical model based on Ngai’s coupling mechanism concept is further combined with the framework of Schapery’s nonlinear viscoelastic theory, which may provide certain physical understanding about the effect of aging behavior on long-term creep deformation of the laminated composites. Numerical modelling by finite element method are implemented, and comparisons between the experimental and simulation results are demonstrated.

A macroscopic viscoelastic viscoplastic constitutive model for porous polymers under multiaxial loading conditions

A macroscopic constitutive model, the Porous Eindhoven Glass Polymer (Porous EGP) model, is presented to describe the deformation behavior of cavitated rubber toughened polymers under multiaxial loading conditions. It is shown that the proposed macroscopic constitutive model is able to describe the non-linear pre-yield regime, strain rate dependence, post-yield behavior (strain softening and hardening) and void evolution for loading conditions ranging from shear to equi-triaxial (pure triaxial) tension and compression. The Porous EGP model is a combination of a well established non-linear viscoelastic viscoplastic model, the Eindhoven Glassy Polymer (EGP) model, and the modified Gurson model. The Gurson model is adopted to determine the equivalent stress and plastic rate of deformation tensor making it depending on the void volume fraction, deviatoric and hydrostatic stress. The macroscopic constitutive model is developed based on the response of realistic 3D representative volume elements (RVEs) containing randomly positioned mono-disperse inclusions. The constitutive behavior of the matrix phase in this full-field model is described by the EGP model, and the cavitated inclusions are idealized as voids. Their response is studied for a range of void volume fractions, multiaxial loading conditions, strain rates and thermodynamic states. The yield behavior of the heterogeneous material depends non-linearly on the macroscopic hydrostatic stress. This response is well captured with the proposed macroscopic constitutive model.

Generalization of nonlinear Murnaghan elastic model for viscoelastic materials

This paper presents a generalization of Murnaghan elastic material to viscoelastic behavior using the Green–Rivlin multiple-integral approach. In the linear limit, the model coincides with the generalized Maxwell model. To create a nonlinear generalization, all possible second-order corrections were included in the constitutive equations written in the internal strains representation. Using this approach, we obtained expressions for the time- and frequency-dependent nonlinear dynamic moduli. We applied the developed nonlinear viscoelastic model to the description of infinitesimal strain waves superposed on a finite prestrain. Furthermore, we considered the generation of higher harmonics by the nonlinear interaction of two strain waves, which we showed can provide a method for measuring all viscoelastic constants of the developed model.

Implementation and analysis of viscoelastic damping in a 2D + 1D model of railway track vibrations

One of the most important technological challenges in the railway context nowadays is reducing the noise produced by the interaction between the train wheels and the track surface, that is the rolling noise. Introducing viscoelastic damping in the railway structure is one method to mitigate this rolling noise. In the paper, we investigate the impact of damping in the vibrational response of railway tracks using different viscoelastic models and parameters available in the literature. For this purpose, the main focus here is on developing an advanced numerical model in which viscoelastic models representing the rail foundation and/or pad are implemented in the framework of the Semi-Analytical Finite Element (SAFE) method. Linear or nonlinear viscoelastic models are considered, compared and effectively implemented for accelerance computation. The obtained numerical strategy leads to affordable and accurate predictions on dynamic features of the railway structure compared to classical methods.

Mathematical Modeling of Collisional Heat Generation and Convective Heat Transfer Problem for Single Spherical Body in Oscillating Boundaries

The application of high-energy ball milling in the field of advanced materials processing, such as mechanochemical alloying and ammonia synthesis, has been gaining increasing attention beyond its traditional use in material crushing. It is important to recognize the role of thermodynamics in high-energy processes, including heat generation from collisions, as well as ongoing investigations into grinding ball behavior. This study aims to develop a mathematical model for the numerical analysis of a spherical ball in a shaker mill, taking into account its dynamics, contact mechanics, thermodynamics, and heat transfer. The complexity of the problem for mathematical modeling is reduced by limiting the motion to one-dimensional translation and representing the vibration of the vial wall in a shaker mill as rigid boundaries that move in a linear fashion. A nonlinear viscoelastic contact model is employed to construct a heat generation model. An equation of internal energy evolution is derived that incorporates a velocity-dependent heat convection model. In coupled field modeling, equations of motion for high-energy impact phenomena are derived from energy-based Hamiltonian mechanics rather than vector-based Newtonian mechanics. The numerical integration of the governing equations is performed at the system level to analyze the general heating characteristics during collisions and the effect of various operational parameters, such as the oscillation frequency and amplitude of the vial. The results of the numerical analysis provide essential performance metrics, including steady-state temperature and time constant for the characteristics of temperature evolution for a high-energy shaker milling process with a computation accuracy of 0.1%. The novelty of this modeling study is that it is the first to obtain such a high accuracy numerical solution for the temperature evolution associated with a shaker mill process.

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.

Nonlinear viscoelastic model for time-dependent mechanical characterization of PMMA acrylic bone cements

Acrylic bone cement materials are widely used in prosthetic implants, orthopaedics and others trauma surgery. From the mechanical constitutive behaviour viewpoint, experimental analyses have provided ample evidence that such materials exhibit time-dependent properties. In this context, this work addresses the formulation of a nonlinear viscoelastic model for the behaviour of PMMA bone cements under compressive creep loading. Relying upon experimental data available for four PMMA bone cement types, a nonlinear Burgers-like rheological model is formulated and related parameters calibrated for the mechanical description of the time-dependent behaviour of these materials under isothermal conditions and one dimensional setting. The proposed model reveals relevant in reproducing both instantaneous and delayed properties of studied PMMA bone cements.

Approach for accurately characterizing nonlinear viscoelastic and nonrecoverable properties of asphalt binders based on MSCR test

The multiple stress creep and recovery (MSCR) test has been widely accepted as a more practical method for accessing the rutting performance of asphalt binders than the conventional Superpave rutting factor test. Nevertheless, there still exists a major concern that 9-s recovery time in each cycle may not guarantee a sufficient recovery of the recoverable viscoelastic strain, especially for modified asphalts, and this could result in a potential inaccuracy in characterizing rutting resistance of binders. To effectively separate the recoverable nonlinear viscoelastic (NLVE) and nonrecoverable responses in the MSCR test, this paper presents a new approach that adopts an additional one-cycle low-stress creep and recovery (LSCR) test and the Schapery’s NLVE model in analyzing the MSCR test data. The LSCR test, covering the entire duration of the MSCR test (200 s), can well determine the linear viscoelastic (LVE) recoverable properties that the Schapery’s model requires for NLVE analysis. The results show that with the recoverable and nonrecoverable responses accurately isolated, the proposed approach rigorously redefines the indicators of the percent recovery R and nonrecoverable creep compliance Jnr in the MSCR test, and therefore effectively eliminates the inaccuracy of the original indicators caused by incomplete recovery of the recoverable strain component in each cycle. Also, the developed method exactly characterizes stress dependence and permanent deformation resistance of asphalt binder, thus offering an effective and accurate approach for high-temperature performance evaluation of asphalt binders.

Ultra-slow transverse waves during continental breakup

Continental rifting is one of the four fundamental geological processes of the Wilson cycle. Rifting results from the continuous stretching of continental lithosphere and involves mechanical, thermodynamic, and rheological processes. It may be followed by a catastrophic breakup stage, which determines cessation of extensional deformation and the final separation of a continent into two distinct tectonic plates that grow by accretion of oceanic lithosphere. To date, the transition to sea−floor spreading and the conditions for the development of a new ocean have not been fully understood. We propose that a consistent description of this process must consider the existence of long−term retarded elasticity in the mantle lithosphere, the superadiabatic conditions of this layer, and the combined action of such elastic forces with the localized buoyancy arising from thermal anomalies. We present a solution of the rheological equation for a nonlinear viscoelastic model of the lithosphere mantle and numerical experiments showing that transient thermal anomalies are generated during the extension, which lead to the formation of transverse waves having wavelengths of the order of hundreds to thousands km and periods of several tens kyrs. These waves induce oscillating topography and influence the relief. Therefore, they could be responsible for eustatic cycles both in the axial rift lacustrine system and in off−axis (dendritic) lakes placed in areas of reversed drainage. At sufficiently high extension rates, deformation localizes and these ultra-slow waves determine cyclic shear failure, with formation of X−shaped cross structures through the lithosphere that prelude to the final rupture.

Quantification of viscous and damage dissipation of bituminous binder and mastic using White-Metzner model

ABSTRACT Bituminous pavement layers are composite in nature, consisting of binder, filler, and aggregates. Understanding the mechanical behaviour of bituminous layers demands investigation of their different material scales, such as the binder, mastic, and mixture scales. While substantial investigation has been carried out on binder and mixture related to fatigue damage, not much research is carried out on mastic. Of the methods used to characterise fatigue damage, energy dissipation is widely used; however, it is complicated due to the presence of other modes of dissipation, such as viscous dissipation. In this study, an attempt is made to separate viscous dissipation and dissipation due to damage for both binder and mastic using a nonlinear viscoelastic fluid model. For this purpose, six types of materials were tested at temperatures of 20 ∘ C and 25 ∘ C, 10 Hz frequency, and five different strain levels spanning the linear and non-linear regimes, for 20,000 cycles. Material parameters for the non-linear White-Metzner model incorporated with the Williamson model were determined for an initial cycle and using these parameters, the viscous component of energy dissipation was determined. Dissipation due to damage was estimated by separating viscous dissipation from total dissipation at every 2000th cycle and based on the evolution of viscous and damage dissipation across cycles, three trends were observed. Based on the trends observed, the influence of strain amplitude, temperature and modifiers on the fatigue resistance of the materials was ascertained.

Rubber Comprehensive Constitutive Equation and the Prediction of Tire Temperature and Rolling Resistance

ABSTRACT Cooler running tires with a reduced rolling resistance is a consideration for both tire makers and automotive original equipment manufacturers. To investigate the mechanics of energy loss and temperature rise, a nonlinear viscoelastic model suitable for numerical analysis of rolling tires is developed and demonstrated. The rubber compounds are represented as nonlinear viscoelastic materials with temperature and frequency, strain, and strain history–dependent response. Several applications to different tire designs are provided to demonstrate the impact of this new material constitutive law in improving the quality of the numerical predictions over other methods. One application is a rolling tire model that was used to delineate the tire temperature distribution and the resultant rolling resistance for different three-dimensional tread patterns.

A proposed approach for accurate fatigue quantification of the base asphalt binders coupling nonlinear viscoelastic effects in S-VECD theoretical framework

The nonlinear viscoelasticity (NLVE) effects at high loading amplitudes are neglected in the stiffness evolution and fatigue quantification for asphalt binders. This study is to determine the applicability of the simplified viscoelastic continuum damage (S-VECD) model considering NLVE effects to the fatigue test protocols of asphalt binders at high strain levels, and to identify the real damage degradation and fatigue life quantification. An equivalent non-linear complex modulus (|G*|NLVE) of the NLVE-based S-VECD model was proposed and quantified by incremental stress sweep testing. A refined fatigue failure criterion GRNL was proposed in the NLVE model. The effectiveness of NLVE on the released pseudo-strain energy (PSE) and failure criteria was also verified. |G*|NLVE values in NLVE region are lower than |G*|LVE values in LVE region. The linear viscoelasticity (LVE)-based S-VECD model overestimates the damage effectiveness on the stiffness degradation. A significant difference exists between the NLVE and LVE models in the damage characteristic curves of the asphalt binders. The actual damage quantified by the NLVE model is lower than that of the LVE model. For both types of binder, the GRNL parameter remains well linear with the fatigue life (Nf). The linear relationship between the released PSE shrinkage values (ΔΣWrR) and the reduced damage (ΔS) is clear when comparing the LVE and NLVE models. The NLVE model predicted a higher fatigue life for asphalt binders than that predicted by the LVE model. The fatigue life determined by the NLVE model exhibits a linear functionality to that quantified by the LVE model, which further validates the applicability and validity of the NLVE model for the diagnosis of fatigue measurement. In contrast, the NLVE-based S-VECD model predicts the fatigue life of asphalt binders under practical service conditions more closely to the mechanical response under actual traffic loads.