Nonlinear Viscoelastic Theory Research Articles

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.

MODELING AND RESEARCH OF PROCESSES FOR CREEP COMPLIANCE AND RELAXATION BASED ON FADING MEMORY CONCEPT

Using of rheonomic materials in engineering structures is one of the most important issues in determining strength and durability. The main task of the mechanics of a deformable solid is to simulate the processes of deformation of viscoelastic materials. Currently, there are quite well-developed theories and methods of viscoelasticity that allow to determine and describe the viscoelastic properties of materials. They differ between linear and nonlinear viscoelasticity. In the linear and nonlinear theory of viscoelasticity, such a task is reduced to finding creep and relaxation kernel. Creep and relaxation kernel are interconnected by a known integral relationship that establishes a relationship between stress, strain and time. The work is devoted to modeling of stress relaxation and sttrain of hereditary materials. Stress relaxation is described by a nonlinear integral equation with an Abel’s kernel. New efficient method has been proposed for determining of parameters (α, δ). Bisection method is used for obtaining of parameter α. Algorithm have been developed for calculating of parameters of α and δ.

Tensile Mechanical Properties and Dynamic Constitutive Model of Polyurea Elastomer under Different Strain Rates.

In order to clearly explain the large deformation mechanical characteristics of polyurea under impact and to construct a dynamic model that can be used for finite element analysis, two kinds of polyurea materials were prepared by formula design, and their uniaxial tensile properties were tested with strain rates ranging from 10−3~103 s−1 using an electronic universal testing machine and a split Hopkinson tensile bar (SHTB). The tensile stress–strain curves of polyurea were obtained under different strain rates. The difference in tensile mechanical properties of the materials was analyzed under dynamic loading and quasi-static loading. Based on the nonlinear viscoelastic theory and the energy dissipation rate inequality, a dynamic visco-hyperelastic constitutive model of polyurea elastomer was established. The research results showed that the uniaxial tensile stress–strain curves of two kinds of polyurea at different strain rates had obvious nonlinear characteristics and strain rate sensitivity and that their tensile strength increased with increased strain rate. The polyurea gradually changed from exhibiting rubbery mechanical behavior under quasi-static loading to glassy mechanical behavior under dynamic loading. The fitting analysis of experimental data and the results of finite element simulation showed that the dynamic constitutive model can predict the nonlinear mechanical behavior of polyurea elastomers over a wide range of strain rates. The research results could contribute to a deepening of the understanding of the damage and failure behavior of polyurea under impact load and provide a theoretical basis for numerical studies on impact safety design of polyurea-coated protective structures.

Analysis of MSCR test results for asphalt binders with improved accuracy

The multiple stress creep recovery (MSCR) test has been regarded by many as the best available test method to evaluate the high-temperature performance of asphalt binders. However, recent studies have shown that the 9-s recovery time may not allow a full recovery of viscoelastic strain for some modified asphalt binders. In that case, some potential viscoelastic recovery is mistakenly reported as unrecovered strain, leading to an underestimation of the asphalt binder rutting resistance. In order to address the issue of incomplete recovery, this paper developed a mechanistic modeling approach to analyze the MSCR test results of six types of asphalt binders. Before using the mechanistic modeling approach, the steady-state strain response at each stress level was firstly determined for all asphalt binders. It was found that the three neat asphalt binders reached a steady-state condition from the beginning of the last 10 cycles, whereas the SBS-modified asphalt binders reached a relatively steady state in the last 5 cycles. Afterwards, a mechanistic modeling approach was employed to analyze the shear strain data in the last 5 cycles at each stress level based on the Boltzmann superposition principle and the Schapery’s nonlinear viscoelastic theory. The viscoplastic strain at the end of the recovery portion and the viscoelastic strain at the end of the creep portion were utilized to calculate the non-recoverable creep compliance and percent recovery. It was demonstrated that the approach adopted in this study was capable of analyzing the MSCR test results of asphalt binders with improved accuracy.

Spatial estimates for Kelvin–Voigt finite elasticity with nonlinear viscosity: Well behaved solutions in space

We provide some spatial estimates for the nonlinear partial differential equation governing anti-plane motions in a nonlinear viscoelastic theory of Kelvin–Voigt type when the viscosity is a function of the strain rate. The spatial estimates we prove are an alternative of Phragmen–Lindelöf type. These estimates are possible when a precise balance between the elastic and viscoelastic nonlinearities holds.

Dynamic Mechanical Performance of Self-compacting Concrete Containing Crumb Rubber under High Strain Rates

The parameters of dynamic mechanical performance at high strain rates are of significant importance in the structural designs and numerical simulations. The aim of this paper is to study the strain rate sensitivity of self-compacting rubberized concrete (SCRC) in the dynamic compressive, splitting tensile and bending tests by using the split Hopkinson pressure bar (SHPB) technique. The failure modes, stress-strain curves and dynamic strength changes of SCRC under various loading conditions were also observed. The specific energy absorption, dynamic tensile-compressive strength ratios, and dynamic constitutive model of SCRC with different rubber content were also analyzed. The results indicate that SCRC can exhibit stronger strain rate sensitivity compared to SCC when strain rate is enough high, while the strain rate sensitivity of SCRC is lower than that of SCC at when strain rate is less than a critical value. To understand the strain rate sensitivity of SCRC, dynamic constitutive model (ZWT model) based on nonlinear viscoelastic theory was proposed. For impact toughness, although the specific energy absorption decreases with increasing rubber content, the ratios of tensile strength to compressive strength of SCRC have an increasing tendency when rubber content increases.

Thermo-mechanical coupling analysis of transient temperature and rolling resistance for solid rubber tire: Numerical simulation and experimental verification

The achievement of low rolling resistance and long-term durability of tires on various vehicles is of great challenge. Tire performances heavily depend on rubber properties; however, the thermo-mechanical coupling characteristics of rubber composites are complicated rendering the design of high-performance tires time-consuming and costly. In our research, the transient temperature and rolling resistance of a solid rubber tire were performed based on the thermo-mechanical coupling approach and nonlinear viscoelastic theory by using finite element method. Particular attention was paid to the strain cycles as the tire rolling on the road presents non-sinusoidal deformation. First, a static three dimensional tire-road contact analysis was conducted to obtain the principal strain cycles. Second, the 100th-order Fourier sine series was used to approximate the strain amplitude. Third, the heat generation rate proportional to the product of the loss modulus and the square of strain amplitude was calculated. The loss modulus was updated as a function of strain amplitude, temperature and frequency. Loss modulus softening effect was also considered. A practical method was proposed to compute the rolling resistance and transient temperature distributions by establishing a 2-D axisymmetric model. A rubber rolling tester was used to verify the numerical results. The comparison between numerical data and test data reveals that the proposed analytical method is a reliable approach to predict rolling resistance and transient temperature distribution for rubber tires. At last, the dependence of rolling resistance and heat build-up on thermal conductivity and loss factor were investigated by the parametric numerical experiments.

Relationship Between Strain and Tension of the Base Thread and Weft on a Loom

Simplified formulas for use in establishing the relationship between the tension and strain of base threads and weft on a loom, taking into account their viscoelastic properties, are obtained. It is established that the relationship between the tension and strain of carbon and quartz threads may be described by equations from the nonlinear theory of viscoelasticity.

Numerical simulation and experimental verification of heat build-up for rubber compounds

Abstract Based on nonlinear viscoelastic theory and coupled thermo-mechanical approach, heat build-up analysis of a rubber specimen under cyclic loading was performed through finite element analysis. The determination of the rubber material properties attracted particular attention. The thermo-mechanical coupling approach can be divided into three major parts: deformation, dissipation, and thermal modules. In the deformation module, uniaxial, planar, and equibiaxial tensile tests were used to determine the hyperelastic constitutive equation. In the dissipation module, an analytical method for calculating the energy dissipation rate was established. The dynamic properties were updated as a function of the strain and temperature based on the modified Kraus model. Creep effect and dynamic property softening effect on the viscoelastic properties were firstly considered. In the thermal module, the dependence of thermal parameters on temperature was established. A highly sophisticated equipment which can measure the temperature rise curves both on the surface and at the heart of the rubber specimen was used to verify the calculated results. The comparison between numerical results and experimental data shows that the proposed analysis method provides a satisfactory way to predict heat build-up for rubber compounds.

Creep and recovery behavior analysis of space mesh structures

The Schapery's nonlinear viscoelastic theory and nonlinear force-density method have been investigated to analyze the creep and recovery behaviors of space deployable mesh reflectors in this paper. Based on Schapery's nonlinear viscoelastic theory, we establish the creep and recovery constitutive model for cables whose pretensions were applied stepwise in time. This constitutive model has been further used for adjustment of cables’ elongation rigidity. In addition, the time-dependent tangent stiffness matrix is calculated by the partial differentiation of the corresponding load vector with respect to the nodal coordinate vector obtained by the nonlinear force-density method. An incremental-iterative solution based on the Newton-Raphson method is adopted for solving the time-dependent nonlinear statics equations. Finally, a hoop truss reflector antenna is presented as a numerical example to illustrate the efficiency of the proposed method for the creep and recovery behavior analysis of space deployable mesh structures.

Form Finding of Cable Net Reflector Antennas Considering Creep and Recovery Behaviors

This paper investigates a whole process for the form-finding analysis of cable net reflector antennas with creep and recovery behaviors in consideration. Both the force-density and nonlinear finite element methods are extended together to establish and solve the nonlinear optimization model for the form-finding analysis. First, the equal force-density method is applied to the form-finding analysis of the initial cable net configuration, in which the creep and recovery behaviors are not taken into account. Second, the nonlinear force-density method and Schapery’s nonlinear viscoelastic theory are used to build the mathematical model of the creep and recovery behaviors of cable net reflector antennas, and the incremental-iterative method is then adopted to solve such a mathematical model. Based on aforementioned two steps, the nonlinear optimization model is established, and it is solved by the inverse iteration algorithm. Eventually, the hoop truss reflector antenna is presented as a numerical example to illustrate the efficiency of the proposed method for the form-finding design of the creep and recovery behaviors of cable net reflector antennas.

Finite element modelling of field compaction of hot mix asphalt. Part II: Applications

A constitutive model is developed and implemented in the finite element system three-dimensional computer-aided pavement analysis for the simulation of hot mix asphalt field compaction. The details of this model are presented in a companion paper (Masad et al., Finite element modelling of field compaction of hot mix asphalt. Part I: Theory, International Journal of Pavement Engineering, Accepted, 2014). This model is based on nonlinear viscoelasticity theory and can accommodate large deformations that occur during the compaction process. The model was used to study the influence of frequency and amplitude of vibratory compaction rollers on the level of compaction. In addition, it was used to analyse the influence of various methods for compacting longitudinal joints on the percent air voids near these joints. The model was used to simulate the compaction of asphalt pavements with different structures and compacted using various equipment and patterns. The finite element results of the level of compaction and percent air voids were in reasonable agreement with the measurements. The model offers the opportunity to simulate and predict the compaction of asphalt mixtures under various rolling patterns and for different pavement structures.

Nonlinear Viscoelastic Model for Describing the Response of Asphalt Binders within the Context of a Gibbs-Potential–Based Thermodynamic Framework

Many researchers have asserted the importance of considering the nonlinearity of the mechanical behavior of asphalt binders for accurately estimating their performance under field conditions, and for comparing and ranking them accordingly. To do so, it is necessary to have a robust and reliable nonlinear viscoelastic theory and a model derived under its purview that can describe the mechanical response of asphalt binders reasonably well. The objective of this study is to develop such a model. A new Gibbs-potential-based thermodynamic framework is used for this purpose. The model developed in this paper is a generalization of the Oldroyd-B model. It is used successfully to describe results of several experiments from the literature concerning the nonlinear response of asphalt binders, including nonlinear creep-recovery and stress-relaxation behavior, thinning behavior, and the appearance of normal forces perpendicular to the plane of shear in simple shear tests.

Determining the Parameters of the Hereditary Kernels of Nonlinear Viscoelastic Materials in Tension

A method of determining the parameters of the hereditary kernels in a viscoelastic model with time-independent nonlinearity is tested. The parameters are determined by fitting the discrete values of the kernels that are obtained considering the similarity of isochronous creep curves and instantaneous deformation curve. The discrete values of the kernels in the zone of singularity are found using weight functions. The Abel function, a combination of power and exponential functions, and fractional-exponential functions are used as hereditary kernels. The method is tested by analyzing the creep, creep recovery, and stress relaxation in laminated composites, polymeric binders, and fiber reinforcements under uniaxial tension Introduction. The main difficulties faced in determining the parameters of the hereditary kernels of nonlinear viscoelastic materials are associated with the selection of the most adequate nonlinear viscoelastic model and with the inadequate accuracy of measurements in short-term tests in which dynamic effects are manifested. The most effective approaches to and methods of resolving such challenges are reviewed in (6, 9, 10, 15, 22, 25). In (2), a method was proposed for determining the parameters of the hereditary kernels in a Rabotnov-type nonlinear theory of viscoelasticity with time-independent nonlinearity. The method is based on the unified isochronous deformation curve that reflects the similarity of isochronous creep curves to the instantaneous deformation curve. The nonlinearity of the constitutive equations is determined by the nonlinearity of the instantaneous deformation curve. We will use the creep kernel rather than the relaxation kernel because creep tests are easier to conduct than relaxation tests. The parameters of creep kernels are determined by fitting their discrete values obtained by differentiating the average similarity function. The discrete values of the kernels in the zone of singularity occurring in short-term tests are obtained using weight functions. Here the method of determining the parameters of the hereditary kernels of nonlinear viscoelastic materials outlined in (2) is tested by calculating creep strains at constant stresses, creep recovery under complete unloading, and stress relaxation. 1. Problem Formulation. Subject of Analysis. We will test the method of determining the parameters of the hereditary kernels of nonlinear viscoelastic materials in the hereditary theory of viscoelasticity with time-independent nonlinearity. We start with the following one-dimensional constitutive equations (3, 5, 15, 23):

Viscoelastic effects on electromechanical instabilities in dielectric elastomers

We present a computational study of the effects of viscoelasticity on the electromechanical behavior of dielectric elastomers. A dynamic, finite deformation finite element formulation for dielectric elastomers is developed that incorporates the effects of viscoelasticity using the nonlinear viscoelasticity theory previously proposed by Reese and Govindjee. The finite element model features a three-field Q1P0 formulation to alleviate volumetric locking effects caused by material incompressibility. We apply the formulation to first perform a fundamental examination of the effects of the viscoelastic deviatoric and volumetric response on dielectric elastomers undergoing homogeneous deformation. Specifically, we evaluate the effects of the shear and bulk relaxation times on the electromechanical instability, and demonstrate that while the bulk relaxation time has a negligible impact, the shear relaxation time substantially increases the critical electric field needed to induce electromechanical instability. We also demonstrate a significant increase in the critical voltage needed to induce electromechanical instability in the presence of a distribution of relaxation times, compared to a single relaxation time, where the former is more representative of viscoelastic behavior of polymers. We then study the effects of viscoelasticity on crack-like electromechanical instabilities that have recently been observed in constrained dielectric films with a small hole containing a conductive liquid. Viscoelasticity is shown again to not only significantly increase the critical electric field to induce the electromechanical instability, but also to substantially reduce the crack propagation speeds in the elastomer.

Adequacy of a nonlinear theory of viscoelasticity

Some features of the behavior of viscoelastic materials whose existence leads to the choice of nonlinear constitutive relations are discussed. A classification of such constitutive relations is given and a number of requirements imposed by practice on their adequacy are formulated. A nonlinear theory of viscoelasticity is proposed; this theory offers the advantages over the theory in which stresses are expressed in terms of strains by integral operators of increasing multiplicity. By a one-dimensional example, it is shown that the constitutive operator relations are reciprocal.

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

It is proved that fractional exponential hereditary kernels of nonlinear viscoelasticity can be used to evaluate creep strains and stress relaxation. A nonlinear theory of viscoelasticity with time-independent nonlinearity described as a nonlinear curve of instantaneous elastoplastic deformation is used. The calculated results are validated against experimental data on the viscoelastic deformation of laminated and unidirectional fibrous composites and their components under the conditions of constant stresses, complete unloading, incremental loading, pure torsion, and constant strains

On the nonlinear theory of viscoelasticity of differential type

We consider nonlinear viscoelastic materials of differential type and for some special models we derive exact solutions of initial boundary value problems. These exact solutions are used to investigate the reasons for the non-existence of global solutions for such equations.

Two-scale mechanism-based theory of nonlinear viscoelasticity

The paper presents a mechanism-based two-scale theory for a generalized nonlinear viscoelastic continuum. The continuum is labeled as generalized since it contains extra degrees of freedom typical of past high-order continuum theories, though a new formulation is presented here tailored to meet the needs of the physical description of the viscoelastic solid. The microstress that appears in the equations, often criticized for a lack of physical meaning, is assigned in this work to viscous free chains superimposed on a nonlinear elastic backbone composed of crosslinks and reinforcement. Mathematically, hyperelasticity is used to describe the equilibrium backbone (macroscale), and an improvement of tube models for reptation dynamics describes the free chain motion at the microscale. Inhomogeneous deformation is described by inclusion of a microstrain gradient into the formulation. Thus, the theory is nicely suited for materials with microstructure where localization of strains and inhomogeneous deformation occur in addition to viscoelastic damping mechanisms due to free chains. Besides the microstress, physical meaning of the additional boundary conditions arising in the general theory is also presented. Since the proposed material model is mechanism-based, macroscopic performances are functions of microstructural variables describing the polymer chemistry so that parametric material design concepts may be gleaned from the model. Several physical phenomena are captured through numerical simulation of the class of materials of interest: size effects, strain localization, and the fracture process. Results agree qualitatively with both experimental data and direct numerical simulation for filled elastomeric solids.

A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity

The analytical and experimental procedures of the method for determining the parameters of the hereditary kernels in the modified Rabotnov’s nonlinear viscoelastic model are outlined. The method is based on the similarity of the isochronous creep curves to the instantaneous-deformation curve. The parameters of the kernels are determined by fitting the discrete values of the kernels that are found by differentiating the average similarity function. The discrete values of the kernels in the domain of singularity are calculated using weight functions