Viscoplastic modeling of asphalt mixes with the effects of anisotropy, damage and aggregate characteristics

This paper presents an approach for constitutive modeling of the viscoplastic behavior of asphalt mixes. This approach utilizes an anisotropic non-associated flow rule based on the Drucker–Prager yield surface. The selection of this yield surface is motivated by the field stress paths and material properties associated with permanent deformation at high temperatures. The efficacy of the model is demonstrated by analyzing data from compressive triaxial tests conducted at different confining pressures and strain rates for three different mixes. The model parameters are related to the experimental measurements of aggregate shape characteristics, aggregate surface energy, inherent anisotropic distribution of aggregates, and microstructure damage measured using X-ray computed tomography and image analysis techniques. Establishing the relationship between the model parameters and material properties is important in order to optimize the mix properties, and achieve desirable mix performance.     

Effect of waste plastic bottles on the stiffness and fatigue properties of modified asphalt mixes

Nowadays, the use of recycled waste materials as modifier additives in asphalt mixes could have several economic and environmental benefits. The main purpose of this research was to investigate the effect of waste plastic bottles (Polyethylene Terephthalate (PET)) on the stiffness and specially fatigue properties of asphalt mixes at two different temperatures of 5 and 20 °C. Likewise, the effect of PET was compared to styrene butadiene styrene (SBS) which is a conventional polymer additive which has been vastly used to modify asphalt mixes. Different PET contents (2–10% by weight of bitumen) were added directly to mixture as the method of dry process. Then the resilient modulus and fatigue tests were performed on cylindrical specimens with indirect tensile loading procedure. Overall, the mix stiffness reduced by increasing the PET content. Although stiffness of asphalt mix initially increased by adding lower amount of PET. Based on the results of resilient modulus test, the stiffness of PET modified mix was acceptable and warranted the proper deformation characteristics of these mixes at heavy loading conditions. At both temperatures, PET improved the fatigue behavior of studied mixes. PET modified mixes revealed comparable stiffness and fatigue behavior to SBS at 20 °C. However, at 5 °C the fatigue life of SBS modified mixes was to some extent higher than that of PET modified ones especially at higher strain levels of 200 microstrain.     

A micro-damage healing model that improves prediction of fatigue life in asphalt mixes

The focus of the current paper is on the development and validation of a micro-damage healing model that improves the ability of an integrated nonlinear viscoelastic, viscoplastic, and viscodamage constitutive model based on continuum damage mechanics for predicting the fatigue life of asphalt paving mixtures. The model parameters of the continuum-based healing model are related to fundamental material properties. Recursive-iterative and radial return algorithms are used for the numerical implementation of viscoelasticity and viscoplasticity models respectively, whereas the viscodamage and micro-damage healing models are implemented using the concept of the effective undamaged-healed natural configuration. Numerical algorithms are implemented into the well-known finite element code Abaqus via the user material subroutine UMAT. Finally, the model is validated by comparing its predictions with experimental data on an asphalt mix that include repeated creep-recovery tests for different loading times and rest periods in both tension and compression. The significant enhancement of the ability of the constitutive model to predict fatigue life due to inclusion of the micro-damage healing is clearly demonstrated.     

Fracture mechanics of idealised bituminous mixes

The durability of asphalt pavements is strongly impaired by cracks, caused primarily by traffic loads and environmental effects. In this work, fracture behaviour of idealised asphalt mixes is investigated. Experiments on idealised asphalt mixes under pure-tension mode (mode I cracking) were performed and fracture parameters were evaluated. In these three-point bend fracture tests, the test variables were temperature and load rate. The test data were stored in an asphalt materials database and special-purpose tools were implemented to analyse and handle the laboratory data automatically. Fracture mechanism maps were constructed, showing the conditions associated with ductile, brittle and ductile–brittle transition regimes of behaviour. The mechanism maps show the failure response of the material in terms of the stress intensity factor, strain energy release rate and J-integral as a function of the temperature-compensated crack mouth opening strain rate. Fracture behaviour of asphalt mix specimens was simulated by cohesive zone model in conjunction with a novel material constitutive model for asphalt mixes. The finite element model agrees well with the experimental results and provides insights into fracture response of the notched asphalt mix beam specimens.     

Non-linear viscoelasticity and viscoplasticity of isotactic polypropylene

Observations are reported in tensile tests with constant cross-head speeds (ranging from 5 to 200 mm/min), relaxation tests (at strains from 0.02 to 0.08), creep tests (at stresses from 15.0 to 25.0 MPa) and recovery tests (after straining up to the maximal strains ranging from 0.04 to 0.12 and subsequent retraction) on isotactic polypropylene at room temperature. A constitutive model is derived for the time- and rate-dependent responses of a semicrystalline polymer at isothermal deformation with small strains. A polymer is treated as an equivalent heterogeneous network of chains bridged by temporary junctions (entanglements, physical cross-links and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. The viscoelastic behavior of the ensemble reflects thermally-induced rearrangement of strands (separation of active strands from temporary junctions and merging of dangling strands with the network). To describe the viscoplastic response, the entire plastic deformation is split into the sum of two components: one of them is associated with sliding of junctions in the non-affine network of chains, while the other accounts for coarse slip and fragmentation of lamellar blocks. Stress–strain relations and kinetic equations for the plastic strains are developed by using the laws of thermodynamics. The constitutive equations involve five material constants that are found by fitting the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation.     

Single and double-step stress relaxation and constitutive modeling of viscoelastic behavior of swelled and un-swelled natural rubber loaded with carbon black

A phenomenological one-dimensional constitutive model, characterizing the mechanical behavior of viscoelastic natural rubber filled with the percolation concentration of HAF carbon black is developed in this investigation. This simple differential form model is based on a combination of linear and non-linear springs with dashpots, incorporating typical polymeric behavior such as shear thinning, thermal softening and non-linear dependence on deformation. The material parameters for this model are determined for the investigated vulcanizates. The model was also developed on same samples after immersion in kerosene for different intervals of times. One step mechanism of relaxation was appeared for straining the samples to different strain levels with constant strain rate. On the other hand, two step mechanisms of relaxation were appeared on straining specimens to same strain level but with different strain rates.     

Viscoelasticity and viscoplasticity of semicrystalline polymers: Structure–property relations for high-density polyethylene

Observations are reported on two commercial grades of high-density polyethylene (HDPE) in uniaxial tensile tests, relaxation tests, creep tests and cyclic tests with a strain-controlled deformation program. Constitutive equations are derived for the viscoelastic and viscoplastic responses of semicrystalline polymers at three-dimensional deformation with small strains. A polymer is modeled as a two-phase continuum consisting of a crystalline skeleton and an amorphous phase treated as a transient network of chains. Its viscoelastic response is associated with thermally activated rearrangement of strands in the temporary network. The viscoplastic behavior reflects fine and coarse slip of lamellar stacks and sliding of junctions between chains in the network. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. The study focuses on the effect of molecular weight of HDPE on its mechanical properties.     

Creep rupture and viscoelastoplasticity of polypropylene

Observations are reported on isotactic polypropylene in tensile tests with various strain rates, relaxation tests at various strains, and creep tests with various stresses at room temperature. Constitutive equations are derived for the viscoelastic and viscoplastic responses of semicrystalline polymers at three-dimensional deformations with small strains. The stress-strain relations involve eight material constants that are found by fitting the experimental data. The model is applied to the numerical analysis of creep failure of polypropylene under various deformation modes (uniaxial tension, equi-biaxial tension, shear, multiple-step creep tests).     

Kinematic hardening rules for modeling uniaxial and multiaxial ratcheting

Ratcheting, which is the strain accumulation observed under the unsymmetrical stress controlled loading and non-proportional loadings, is modeled using the simplified viscoplasticity theory based on overstress (VBO). The influences of kinematic hardening laws on the uniaxial and multiaxial non-proportional ratcheting behavior of CS 1026 carbon steel have been investigated. The following kinematic hardening rules have been considered: the classical kinematic hardening rule, the kinematic hardening rules introduced by Armstrong–Frederick, Burlet–Cailletaud and the modified Burlet–Cailletaud. The investigated loading conditions include uniaxial stress controlled test with non-zero mean stress, and axial strain controlled cyclic test of thin-walled tubular specimen in the presence of constant pressure. Numerical results are compared with the experimental data obtained by Hassan and Kyriakides [Hassan T, Kyriakides S. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int J Plast 1992;8:91–116] and Hassan et al. [Hassan T, Corona E, Kyriakides S. Ratcheting in cyclic plasticity, part I: multiaxial behavior. Int J Plast 1992;8:117–146]. It is observed that all investigated kinematic hardening rules do not improve ratcheting behavior under multiaxial loading, but over-prediction still exists.     

Evaluation of rutting potential for asphalt concrete mixes containing copper slag

Copper slag (CS) is a by-product of the copper extraction process, which can be used as coarse and/or fine aggregate in hot mix asphalt (HMA) pavements. This study used CS as a replacement of the fine aggregate with a percentage of up to 40% by total aggregate weight. The objective of this study was to evaluate the effect of CS on the rutting potential of the asphalt concrete mix using two methods. One method is based on the Dynamic modulus |E*| testing result. Actual pavement temperature data from a test section were used with the developed |E*| master curves. EverStressFE finite element program was used to perform a linear elastic load-deformation analysis for a pavement section and to determine the vertical resilient strain in a 40-mm HMA surface layer. The M-E PDG permanent deformation model was used with and Excel Visual Basic for Applications code to predict the accumulated rutting for different CS mixes for 10 million ESALs. The other method used the data from the flow number (FN) test. Based on the |E*| approach, the results indicated that adding 5% CS in the mix increased the predicted rutting from 0.59 to 0.98 mm at 10 million ESALs (increase by 68%). When 40% CS was used, rutting increased by more than 700% compared with the control mix. After analysing the FN results with the Francken model, the results indicated a decrease in FN as CS content is increased, indicating higher rutting potential. The decrease in FN ranged from 9% for 5% CS to 95% for 40% CS. The mixes containing up to 10% CS satisfied the minimum FN criteria for rutting. A calibration process for the M-E PDG distress prediction models that allows the use of waste and by-product materials such as CS should be considered in the future.     

Characterization of petroleum pitch using steady shear experiments

Asphalt for highway and runway construction is processed by either air blowing or blending with different petroleum streams. In the blending process, petroleum pitch, a by-product of solvent deasphalting of the vacuum residue is mixed with heavy extract to produce asphalt of the desired specifications. The rheological response of blended asphalt hence depends to a large extent on the constitutive property of petroleum pitch. In an aim to develop robust models for blended asphalt, modeling the mechanical behavior of petroleum pitch hence becomes necessary.In this work reported here, petroleum pitch from crude sources such as Basrah Light, Arab Mix and Arab Light are subjected to steady shear for 99 min at temperatures ranging from 70 to 120 °C for different shear rates. Each of these material exhibited different stress overshoot and decay during steady shear depending on the temperature and shear rate. A viscoelastic fluid model of the rate type is selected to model the response of the material. Using the recent thermodynamic framework based on Gibbs potential proposed by Rajagopal and Srinivasa [27], restrictions on the proposed model are obtained. The rotational flow problem is solved and the material parameters are estimated. The model predictions are corroborated with the experimental observations and they are found to be reasonably good.     

Applicability of the time–temperature superposition principle in modeling dynamic response of a polyurea

This paper addresses the applicability of the Time–Temperature Superposition Principle in the dynamic response of a polyurea polymer at high strain rates and different temperatures. Careful and extensive measurements in the time domain of the relaxation behavior and subsequent deduction of a master-relaxation curve establish the mechanical behavior for quasistatic deformations over a time range of 16 decades. To examine its validity in a highly dynamic environment, experiments with the aid of a split Hopkinson (Kolsky) pressure bar are carried out. The use of a two-material pulse shaper allows for stress equilibrium across the specimen during the compression process, to concentrate on the initial, small deformation part that characterizes linearly viscoelastic behavior. This behavior of polyurea at high strain rates and different temperatures is then investigated by comparing results from a physically fully three-dimensional (axisymmetric) numerical model, employing the quasistatically obtained properties, with corresponding Hopkinson bar measurements. The experimentally determined wave history entering the specimen is used as input to the model. Experimental and simulation results are compared with each other to demonstrate that the Time–Temperature Superposition Principle can indeed provide the requisite data for high strain rate loading of viscoelastic solids, at least to the extent that linear viscoelasticity applies with respect to the polyurea material.     

A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete

An existing viscoelastic constitutive model which accounts for the effects of rate-dependent damage growth is described and applied successfully to characterize the uniaxial stress, constant strain rate behavior of asphalt concrete. The special case of an elastic continuum damage model with multiaxial loading, which is based upon thermodynamics of irreversible processes with internal state variables, is first reviewed and then it is shown how this model has been extended to a corresponding viscoelastic damage model through the use of an elastic-viscoelastic correspondence principle. The general mathematical model is next specialized to uniaxial loading. A rate-type evolution law, similar in form to a crack growth law for a viscoelastic medium, is adopted for describing the damage growth within the body. Results from laboratory tests of uniaxial specimens under axial tension at different strain rates are then shown to be consistent with the theory. The discussion of data analysis describes the specific procedure used here to obtain the material parameters in the constitutive model for uniaxial loading and how the method may be generalized for multiaxial loading.     

Predicting the tensile relaxation modulus of asphalt mixes based on the mix design and environmental factors

The main objective of this study was to predict the tensile relaxation modulus of asphalt mixes, without having to perform the common relaxation modulus tests, by developing a predictive model based on the mix characteristics, ageing condition, temperature and loading time. To this end, cylindrical asphalt mixture specimens containing crushed stone aggregates with 60/70 penetration asphalt binder were fabricated using two aggregate gradations, two binder contents, two air void levels and three ageing conditions with four replicates. Uniaxial tensile relaxation modulus tests were conducted on the specimens at four temperatures using the trapezoidal loading pattern at a low level of strain. Tensile relaxation modulus master curves of all the experimental combinations were constructed by the sigmoidal model. Statistical analysis of variance and regression analysis was performed on the test data and a predictive model was developed. Finally, the predictive model was verified using a group of measured values other than those used for the development of the model, and it was found that the predicted values correlated well with the measured ones.     

On the strengthening effect of increasing cycling frequency on fatigue behavior of some polymers and their composites: Experiments and modeling

Effect of cycling frequency on fatigue behavior of neat, talc filled, and short glass fiber reinforced injection molded polymer composites was investigated by conducting load-controlled fatigue tests at several stress ratios (R = −1, 0.1, and 0.3) and at several temperatures (T = 23, 85 and 120 °C). A beneficial or strengthening effect of increasing frequency was observed for some of the studied materials, before self-heating became dominant at higher frequencies. A reduction in loss tangent (viscoelastic damping factor), width of hysteresis loop, and displacement amplitude, measured in load-controlled fatigue tests, was observed by increasing frequency for frequency sensitive materials. Reduction in loss tangent was also observed for frequency sensitive materials in DMA tests. It was concluded that the fatigue behavior is also time-dependent for frequency sensitive materials. A Larson–Miller type parameter was used to correlate experimental fatigue data and relate stress amplitude, frequency, cycles to failure, and temperature together. An analytical fatigue life estimation model was also used to consider the strengthening effect of frequency in addition to mean stress, fiber orientation, and temperature effects on fatigue life.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

Essential work of fracture and viscoplastic response of a carbon black-filled thermoplastic elastomer

Observations are reported on a carbon black-filled thermoplastic elastomer in uniaxial cyclic tensile tests with various maximum strains and double-edge-notched-tensile (DENT) tests with various ligament widths at ambient temperature. It is shown that the stress-strain diagrams in DENT tests measured relatively far away from the ligament coincide with those in tensile cyclic tests on un-notched samples. To describe the viscoplastic response of un-notched specimens, constitutive equations are derived, and adjustable parameters are found by fitting the experimental data. It is demonstrated how the energy stored in a DENT sample under tension can be accounted for in calculations of the specific essential work of fracture.