Linear and nonlinear viscoelastic and viscoplastic analysis of asphalt binders with warm mix asphalt additives

Warm-Mix Asphalt (WMA) is a widely used product, which proved a contribution to the reduction in asphalt mixing and compaction temperatures. This reduction leads to lower fuel consumption and smoke emission in asphalt plants. Most of the characterisation of binders used in WMA has focused in the past on measuring linear viscoelastic properties and associated Superpave parameters. Several studies have shown that the average stresses and strains of the asphalt mixture remain mostly within the linear viscoelastic response. However, localised strains in the binder phase of the mixture could reach values high enough to induce nonlinear viscoelastic and viscoplastic deformations. Therefore, this study focuses on an experimental and analytical evaluation of linear, nonlinear viscoelastic and viscoplastic responses of selected binders modified for use in WMA. The first part of the paper analyses the linear viscoelastic material properties and their ability to evaluate permanent deformation resistance. Then, the non-recoverable creep compliance parameter obtained from the Multiple Stress Creep Recovery (MSCR) test is analysed to assess the nonlinear response and permanent deformation of asphalt binders. The paper utilises a nonlinear plasto-viscoelastic (NPVE) approach to assess and quantify the nonlinear plasto-viscoelastic response of binders by separating the recoverable and irrecoverable strains measured in the MSCR test. Two WMA additives were included in this study by mixing them with polymer-modified and unmodified asphalt binders. Analysis of results showed that the NPVE approach captured a higher percentage of recovery than the NLVE approach. However, binder’s performance evaluation and ranking did not change by adopting the NPVE approach. The nonlinear viscoelastic parameters provided insight on the behaviour of asphalt binders mixed with WMA additives during loading cycles. Sasobit showed higher influence than Advera on binders in resisting permanent deformation by increasing the recoverable strain during the unloading phase.     

Creep and recovery of nonlinear viscoelastic materials of the differential type

In this work, the creep and recovery properties of rubberlike viscoelastic materials in simple shear are studied by two special constitutive equations for isotropic, nonlinear incompressible viscoelastic material of the differential type. The creep and recovery processes are of significant importance to both the mechanics analysis and engineering applications. The constitutive equations introduced in this work generalize the Voigt-Kelvin solid and the 3-parameter model of classical linear viscoelasticity. They describe the uncoupled non-Newtonian viscous and nonlinear elastic response of an isotropic, incompressible material. The creep and recovery processes are treated for simple shear deformation superimposed on a longitudinal static stretch. Closed form solutions are provided and both processes are described effectively by the exponential function.     

Development of a stress-mode sensitive viscoelastic constitutive relationship for asphalt concrete: experimental and numerical modeling

Asphalt binder is responsible for the thermo-viscoelastic mechanical behavior of asphalt concrete. Upon application of pure compressive stress to an asphalt concrete specimen, the stress is transferred by mechanisms such as aggregate interlock and the adhesion/cohesion properties of asphalt mastic. In the pure tensile stress mode, aggregate interlock plays a limited role in stress transfer, and the mastic phase plays the dominant role through its adhesive/cohesive and viscoelastic properties. Under actual combined loading patterns, any coordinate direction may experience different stress modes; therefore, the mechanical behavior is not the same in the different directions and the asphalt specimen behaves as an anisotropic material. The present study developed an anisotropic nonlinear viscoelastic constitutive relationship that is sensitive to the tension/compression stress mode by extending Schapery’s nonlinear viscoelastic model. The proposed constitutive relationship was implemented in Abaqus using a user material (UMAT) subroutine in an implicit scheme. Uniaxial compression and indirect tension (IDT) testing were used to characterize the viscoelastic properties of the bituminous materials and to calibrate and validate the proposed constitutive relationship. Compressive and tensile creep compliances were calculated using uniaxial compression, as well as IDT test results, for different creep-recovery loading patterns at intermediate temperature. The results showed that both tensile creep compliance and its rate were greater than those of compression. The calculated deflections based on these IDT test simulations were compared with experimental measurements and were deemed acceptable. This suggests that the proposed viscoelastic constitutive relationship correctly demonstrates the viscoelastic response and is more accurate for analysis of asphalt concrete in the laboratory or in situ.     

Phenomenological formulation of viscoplastic constitutive equation for polyethylene by taking into account strain recovery during unloading

A viscoplastic constitutive equation for polyethylene that properly describes significant strain recovery during unloading was proposed. The constitutive equation was formulated by combining the kinematic hardening creep theory of Malinin and Khadjinsky with the nonlinear kinematic hardening rule of Armstrong and Frederick. In order to describe the strain recovery, the nonlinear kinematic hardening rule was modified. First, a loading surface was defined in a viscoplastic strain space. A loading–unloading criterion was then introduced using the loading surface. Moreover, a new parameter was defined by the relationship between the loading surface and the current state of the viscoplastic strain, and the evolution equation of back stress was modified using this parameter, which has some value only during unloading. Experimental results for polyethylene were simulated by using the modified constitutive equations, and cyclic inelastic deformation in both uniaxial and biaxial states of stress was predicted. Finally, the validity of the above-described modification was verified, and the features of the constitutive equation and the deformation were discussed.     

Influence of surface energy on the nanoindentation response of elastically-layered viscoelastic materials

In the context of integrated nonlinear viscoelastic contact mechanics, a nonlinear finite element model is developed to predict and analyze the quasistatic response of nanoindentation problems of an elastically-layered viscoelastic materials considering the surface elasticity effects. Effects of surface energy are accounted for by employing the Gurtin–Murdoch continuum model for surface elasticity. The linear viscoelastic response is modeled by the Schapery’s creep model with a Prony’s series to express the transient component in the creep compliance. The viscoelastic constitutive equations are cast into a recursive form that needs only the previous time increment rather than the entire strain history. To satisfy the contact constraints exactly, the Lagrange multiplier method is adopted to enforce the contact conditions into the system. The equilibrium indentation configuration is obtained through the Newton–Raphson iterative procedure. The developed model is verified then applied to investigate the quasistatic nanoindentation response of two different indentation problems with different geometry and loading conditions. Results show the significant effects of surface energy and viscoelasticity on the quasistatic nanoindentation response.     

Viscoplastic response and creep failure time prediction of polymers based on the transient network model

The nonlinear viscoelastic/viscoplastic response of polymeric materials is described by a new model based on previous works in terms of monotonic loading, stress–relaxation, and creep. In the proposed analysis, following a constitutive equation of viscoelasticity, based on the transient network theory, essential modifications are introduced, which account for the nonlinearity and viscoplasticity at small elastic and finite plastic strain regime. In addition, viscoplastic response is successfully analyzed by a proper kinematic formulation, which is combined with a functional form of the rate of plastic deformation. A three-dimensional constitutive equation is then derived for an isotropic incompressible medium. This analysis is capable of capturing the main aspects of inelastic response and the instability stage taking place at the tertiary creep, related to the creep failure. Model simulations described successfully the experimental data of polypropylene, which were performed elsewhere.     

Bending and creep buckling response of viscoelastic functionally graded beam-columns

The viscoelastic creep response of flexural beams and beam-columns made with functionally graded materials is numerically investigated. The paper highlights the challenges associated with the modeling and analysis of such structures, and presents a nonlinear theoretical model for their bending and creep buckling analysis. The model accounts for the viscoelasticity of the materials using differential-type constitutive relations that are based on the linear Boltzmann’s principle of superposition. The model is general in terms of its ability to deal with any material volume faction distribution through the depth of the beam, and with different linear viscoelastic laws, boundary conditions, and loading schemes. The governing equations are solved through time stepping numerical integration, which yields an exponential algorithm following the expansion of the relaxation function into a Dirichlet series. A numerical study that examines the capabilities of the model and quantifies the creep response of functionally graded beam-columns is presented, with special focus on the stresses and strains redistribution over time and on the creep buckling response. The results show that the creep response of such structures can be strongly nonlinear due to the variation of the viscoelastic properties through the depth, along with unique phenomena that are not observed in homogenous structures.     

Nonlinear viscoelastic analysis of orthotropic beams using a general third-order theory

The displacement based finite element model of a general third-order beam theory is developed to study the quasi-static behavior of viscoelastic rectangular orthotropic beams. The mechanical properties are considered to be linear viscoelastic in nature with a scope to undergo von Kármán nonlinear geometric deformations. A differential constitutive law is developed for an orthotropic linear viscoelastic beam under the assumptions of plane-stress. The fully discretized finite element equations are obtained by approximating the convolution integrals using a trapezoidal rule. A two-point recurrence scheme is developed that necessitates storage of data from the previous time step only, and not from the entire deformation history. Full integration is used to evaluate all the stiffness terms using spectral/hp lagrange polynomials. The Newton iterative scheme is employed to enhance the rate of convergence of the nonlinear finite element equations. Numerical examples are presented to study the viscoelastic phenomena like creep, cyclic creep and recovery for thick and thin beams using classical mechanical analogues like generalized n-parameter Kelvin-Voigt solids and Maxwell solids.     

Analysis of nanocontact problems of layered viscoelastic solids with surface energy effects under different loading patterns

Surface stresses have a remarkable effect on nanocontact response of layered viscoelastic solids, especially under specific loading patterns. In the framework of nonlinear viscoelastic contact mechanics, a numerical model is developed to investigate the quasistatic nanocontact response of elastically layered viscoelastic solids under different loading patterns. The developed model accounts for surface energy effects by adopting the complete Gurtin–Murdoch surface elasticity model. The Schapery’s constitutive viscoelastic creep model is used for the stress, strain, and time relationships. The transient term in the creep compliance is expressed by Prony’s series. Frictionless contact condition is assumed throughout the contact interface. The equilibrium contact configuration, in which the contact constraints are exactly satisfied without any need for an appropriate value for the penalty parameter, is obtained by using the Lagrange multiplier method in the framework of the Newton–Raphson procedure. The developed model is applied to study and analyze the quasistatic nanocontact response of two different problems under different loading patterns. Results show the significant effect of the type of loading pattern and its rate on the nanocontact response of elastically layered viscoelastic solids.     

Influence of confinement pressure and air voids on the repeated creep and recovery of asphalt concrete mixtures

A laboratory investigation was conducted to capture the influence of confinement pressure and specimen air voids on the creep and recovery response of asphalt concrete (AC) mixtures. AC specimens were fabricated at 2% and 7% air voids and tested at three temperatures (20, 40 and 55°C) and at unconfined and confined conditions (100 and 200 kPa). A total of 20,000 repetitions of a repeated trapezoidal loading and recovery cycle were applied. The resulting creep curves showed four distinct patterns of the three-stage creep curve depending on the loading condition and specimen density. To quantify the mechanical response during the secondary stage where the response was found to be linear, linear viscoelastic modelling was carried out. Using creep time, energy stored and energy dissipated, which were determined from model parameters; the influence of air voids and confinement pressure was quantified.     

Thermal-stress concentration near inclusions in viscoelastic random composites

The determination of thermal-stress concentrations near inclusions in viscoelastic random composites is concerned with the prediction of the overall response of random nonlinear viscoelastic multi-component media. The continuum considered here is assumed to be subjected to a finite deformation. First Piola’s stress tensor and deformation gradient are used as conjugate field variables in a fixed reference state. A nonlinear problem is investigated in a second-order approximation theory when the gradient deformation terms higher than second order are neglected. A convex potential function in a thermo-elastic problem and time functionals in a viscoelastic one are used to construct overall constitutive relations. The technique of surface operators developed by R. Hill and others is used to determine stress concentrations near inclusions for nonlinear matrix creep.     

基于不可逆内变量热力学的岩石材料粘弹-粘塑性本构方程

岩石材料的粘弹性和粘塑性变形是与时间相关的能量耗散行为。在Rice不可逆内变量热力学框架下,引入两组内变量分别用来描述在粘弹性和粘塑性变形过程中材料的内部结构调整。通过给定比余能的具体形式和内变量的演化方程,推导出内变量粘弹-粘塑性本构方程。粘弹性本构方程具有普遍性,能涵盖Kelvin-Voigt和Poynting-Thomson在内的经典粘弹性模型的本构方程。并指出热力学力与应力呈线性关系是组合元件模型为线性模型的根本原因。粘塑性本构方程能较好地刻画岩石材料在粘塑性变形过程中的硬化现象。对模拟岩石的模型相似材料进行单轴加卸载蠕变试验,将蠕变过程中的粘弹性和粘塑性变形分离并根据试验数据对本构方程的材料参数进行辨识。试验数据和理论曲线对比结果表明该文提出的本构方程能很好地模拟材料的蠕变行为。该类型的本构方程能为岩石工程的长期稳定性的预测、评价以及加固分析提供基础。     

Strain accumulation in bituminous binders under repeated creep-recovery loading predicted from small-amplitude oscillatory shear (SAOS) experiments

The creep-recovery (CR) test starts out with a period of shearing at constant stress (creep) and is followed by a period of zero-shear stress where some of the accumulated shear strain gets reversed. Linear viscoelasticity (LVE) allows one to predict the strain response to repeated creep-recovery (RCR) loading from measured small-amplitude oscillatory shear (SAOS) data. Only the relaxation and retardation time spectra of a material need to be known and these can be determined from SAOS data. In an application of the Boltzmann superposition principle (BSP), the strain response to RCR loading can be obtained as a linear superposition of the strain response to many single creep-recovery tests. SAOS and RCR data were collected for several unmodified and modified bituminous binders, and the measured and predicted RCR responses were compared. Generally good agreement was found between the measured and predicted strain accumulation under RCR loading. However, in the case of modified binders, the strain accumulation was slightly overestimated (≤20% relative error) due to the insufficient SAOS information at long relaxation times. Our analysis also demonstrates that the evolution in the strain response under RCR loading, caused by incomplete recovery, can be reasonably well predicted by the presented methodology. It was also shown that the outlined modeling framework can be used, as a first approximation, to estimate the rutting resistance of bituminous binders by predicting the values of the Multiple Stress Creep Recovery (MSCR) test parameters.     

Viscoplastic response and modeling of asphalt-aggregate mixes

The strain response of asphalt-aggregate mixes to applied stresses is decomposed additively into a viscoelastic part and a viscoplastic part. The paper focuses on the response and modeling of the viscoplastic component; it includes the development of a multiaxial constitutive formulation that is capable of generating: (i) strain hardening when the loading is applied in one direction; (ii) strain softening immediately after stress reversals; (iii) volumetric changes under uniaxial conditions or isotropic conditions, or both; and (iv) directional non-symmetry. In order to investigate the model’s capabilities, four tests were performed sequentially on one asphalt sample. The tests were limited to pre-peak conditions and one temperature and consisted of creep and recovery sequences in uniaxial tension, uniaxial compression, isotropic compression and uniaxial tension-compression. Analysis of the results showed that the new theory, once calibrated, was able to adequately reproduce the viscoplastic strain component; its forecastability, however, was found limited.     

Viscoelastic incremental formulation using creep and relaxation differential approaches

A new incremental formulation in the time domain for linear, non-ageing viscoelastic materials undergoing mechanical deformation is presented in this work. The formulation is derived from linear differential equations based on a discrete spectrum representation for the creep and relaxation tensors. The incremental constitutive equations are then obtained by finite difference integration. Thus the difficulty of retaining the stress and strain history in computer solutions is avoided. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of strains and stresses in order to establish the constitutive stress–strain relationship. The presented method is validated using numerical simulations and reliable results are obtained.     

聚合物基复合材料有效蠕变响应与单轴拉伸行为的细观力学模拟

基于变分渐近均匀化理论框架建立表征线性黏弹性聚合物基复合材料有效蠕变响应和宏观应力-应变行为的细观力学模型。从线性黏弹性聚合物基复合材料本构方程中构建能量泛函变分表达式出发,采用变分渐近法求解线性黏弹性聚合物基复合材料的有效蠕变柔度系数,并以此为基础计算聚合物基复合材料的时变和单轴拉伸行为。通过算例验证了构建模型的适用性和准确性。由于所有计算均在时间域内完成,不再需要传统线黏弹性复合材料使用的Laplace转换和反演,计算效率大为提高。      

复合材料蠕变本构关系及实验测定Ⅱ实验测定

在前一部分,本文得到复合材料蠕变的本构关系,在此基础上,本文进一步分析了复合材料蠕变本构关系的具体形式,实验测得了长纤维增强复合材料在蠕变、恢复两个阶段的应变,以用来确定本构关系中的待定参数,考虑到本构关系为复杂的非线性方程,本文提出了用离散变量和最小二乘法联合的方法确定参数,进而拟合蠕变本构关系的理论公式,分离出了蠕变过程中的弹性变形、粘弹性变形和粘塑性变形,对本构关系中的几个参函数,本文根据有限的实验数据拟合了其函数。      

沥青砂混合料粘弹塑力学特性研究

在0.1MPa、0.15MPa、0.2MPa、0.25MPa和0.3MPa下进行了沥青砂试样单轴压缩和蠕变实验,分析了其压缩和蠕变性质,根据变形机理提出了粘弹塑本构模型可由粘弹性和粘塑性的两个子模型串联构成,通过对粘塑性子模型中粘性系数进行改进,理论推导了模型蠕变本构方程,确定了模型参数,并求得模型参数与加载应力函数关系。进行模型预测与实验结果对比,结果表明:该模型能够描述沥青砂试样在不同应力下蠕变变形的3个阶段,反映了沥青砂混合料粘弹塑变形特点。     

复合材料蠕变本构关系及实验测定 Ⅱ实验测定

在前一部分,本文得到复合材料蠕变的本构关系,在此基础上,本文进一步分析了复合材料蠕变本构关系的具体形式,实验测得了长纤维增强复合材料在蠕变、恢复两个阶段的应变,以用来确定本构关系中的待定参数,考虑到本构关系为复杂的非线性方程,本文提出了用离散变量和最小二乘法联合的方法确定参数,进而拟合蠕变本构关系的理论公式,分离出了蠕变过程中的弹性变形、粘弹性变形和粘塑性变形,对本构关系中的几个参函数,本文根据有限的实验数据拟合了其函数.     

Creep and strain recovery in hot-pressed silicon nitride

It is observed that creep response in hot-pressed silicon is characterized by two parallel phenomena; one accounts for a persistent non-recoverable plastic deformation and the other for a transient viscoelastic recoverable deformation. The persistent creep component is time-dependent, and apparently follows parabolic time kinetics. It is further observed that creep is characterized by a power law stress exponent of about 4 and an activation energy of 848 kJ mol^–1. The viscoelastic recoverable component of strain is found to be independent of the total plastic strain in the material. The recovery rate at any given time is directly proportional to the preceding creep stress and therefore can be considered linear viscoelastic. The creep compliance of the viscoelastic transient is temperature-dependent with an activation energy of about 722 kJ mol^–1. It is further observed that the viscoelastic recovery is characterized by a spectrum of retardation times and can be modelled by a series of Kelvin analogue models. Finally, the viscoelastic recovery and the viscoelastic component of subsequent creep appear to be inversely related and apparently obey Boltzman superposition. A model is developed for the creep and recovery behaviour of hot-pressed silicon nitride consistent with all experimental observations and based in relative grain motion accommodated by the fluid grain-boundary glass liquid flow, cavitation and wedge opening.