Appendix A
Review of Literature for Constitutive Models in Asphalt Mixes
by: Thomas Weaver
1 Numerical Modeling of Asphalt Concrete
Numerical models are being used increasingly in research and practice to predict the performance of asphalt concrete pavement performance. Numerical models able to account for many parameters that impact asphalt concrete pavement behavior are: discrete element models and constitutive models.
The discrete element methodology models each individual particle within a material and uses contact laws to govern the interaction of the particles. This procedure was initially developed by Cundall (1971). Application of the discrete element method may lead to better understanding of the physics governing asphalt concrete behavior and is a valuable research tool. However, use of this method is not likely in the near future by engineering practitioners to model asphalt concrete pavement performance.
Constitutive modeling is another method useful for modeling asphalt concrete behavior and models material response on a more global level. This methodology is more developed than the discrete element methodology and with more research, may be useful for engineering practitioners to assess long term asphalt concrete pavement performance.
The following section presents background information on constitutive models, in particular, models that have been implemented in finite element programs (Note, more work is required to complete this literature review).
1.1 Constitutive Models
Constitutive models are relationships correlating stress with strain. The simplest constitutive model is Hooke’s law which states:
σ=Eϵ
where s is stress, E is the modulus of elasticity, and e is strain. More complex models are required to capture non-linear response of materials due to yielding, fracture, and other variables impacting material behavior. Two constitutive models that have been implemented in finite element programs are the disturbed state concept (DSC) constitutive model and an elastoviscoplastic model. These models are described below.
1.1.1 Disturbed State Concept
The disturbed State concept (DSC) constitutive model was developed by Desai (2001) and has been used to model asphalt concrete pavement (Desai 2007). This model is capable of including the effect of elasticity, plasticity, creep, microcracking, and fracture under mechanical and environmental loading.
The DSC considers a material to exist in two states, a relatively intact state (RI) or a microcracked state/fully adjusted (FA) state. As the material is loaded, the material transforms from RI to FA. The DSC relates these two states and the transformation from RI to the FA state through a disturbance value. The disturbance value ranges from 0 to 1 corresponding to relatively intact to significantly damaged material, respectively. The disturbance value controls the softening that occurs in a material after yielding has occurred.
The relatively intact state is modeled as an elastic-plastic material using a yield function and elastic material properties. The parameters used to define material response can be defined as functions of temperature and loading rate. Creep can also be included.
The fully adjusted state may be defined as a critical state where no volume changes occur during yielding under a constant shear.
The disturbance value is obtained using the equation:
D=Du1-e-AεDZ
Where D is the disturbance, eD is the acculuative deviatoric platic strain, and the values Du, A, and Z are disturbance parameters.
1.1.1.1 Determination of DSC Paramters
The DSC model requires elastic parameters, plasticity parameters, creep parameters, disturbance parameters, and thermal effects. Not all parameters are needed if a certain aspect of the model is neglected. The model parameters and methods for obtaining these parameters as provided by Desai (2007) and are summarized below.
The elastic parameters are Young’s modulus, E, and Poisson’s ratio, n. The modulus value can be obtained from a stress-strain curve and is a function of mean pressure.
The hierarchical single yield surface developed by (Desai et al. 1986) is used for the plasticity model. Parameters needed to define this yield surface are g, b, n, a1, h1, and R. The value of g is determined from the plot of the ultimate envelope based on the asymptotic stress in the ultimate region. The value n is determine from the stress condition at which transition from compaction to dilation occurs. Hardening parameters a1 and h1 are determined by computing the accumulated plastic strain from the incremental stress-strain curve, and the cohesive intercept, c, is used to find the value R.
Other parameters associated with creep, disturbance, and thermal effects are presented below. Creep parameters may be obtained from a plot of strain versus time. The disturbance parameters, A and Z, are found from a stress vs. strain curve where softening occurs. The value Du is the residual response and can be assumed to be 0.9. Thermal effects can be determined by plots of the parameter of interest with temperature.
The number of parameters and effort required to obtain these parameters will likely be cumbersome to most engineers.
1.1.2 Elastoviscoplastic Model
A number of researchers have developed elastoviscoplastic constitutive models for application to asphalt concrete (Abdulshafi and Majidsadeh 1985, Scarpas et al. 1997, Lu and Wright 1998, Seibi et al. 2001, Collop et al. 2003, and Masad et al. 2007). Differences in these models include the yield surface employed (e.g. Drucker-Prager, HISS, Mohr-Coulomb) and whether the model accounts for microstructure behavior that influences anisotropy and damage. In general, these models decompose the total strain rate into a viscoelastic strain rate and a viscoplastic strain rate. The constitutive model proposed by Masad et al. (2007) is discussed in more detail below since this model accounts or material anisotropy associated with the aggregate and damage associated with material cracking.
1.1.2.1 Anisotropy and Damage
Material anisotropy is a result of the anisotropy associated with granular material in the asphalt concrete mix. Masad et al. (2007) have proposed using image analysis of an asphalt concrete cross section to determine the orientation of aggregates within the material. Based on the orientation of the aggregates, a vector magnitude D is computed. The vector magnitude D is used to modify the stresses in the material computed using a viscoplastic model.
The concept of modifying the effective stress in the material as a result of crack growth as proposed by Kachanov (1958) has been implemented within the model proposed by Masad et al. (2007). When computing the stress as implemented in the viscoplastic model, the effective stress is multiplied by a factor that increases the effective stress in the material. In this model, the variable controlling damage ranges from 0 for undamaged to 1 for a completely damaged material.
1.1.2.2 Material Parameters
The material parameters required for the constitutive model proposed by Masad et al. (2007) are listed in Table 1 below. Similar to the DSC model, many parameters are needed to define the material behavior with an elastoviscoplastic model that can account for anisotropy and damage. However, based on results presented by Masad et al. (2007), some parameters do not change significantly from one material to another. For example, the anisotropy vector magnitude for granite and general gravel aggregates were 28.6 and 26.1, respectively. As long as typical aggregates are consistently utilized, image analysis tests may be eliminated from the required testing sequence for determining this parameter. Correlations with other simplistic tests may also be possible for estimating appropriate material parameters for this model.
Table 1 Material Parameters for Elastoviscoplastic Model (Masad 2007)
Parameter / Test for Determining ParameterAnisotropy, D / Image analysis
Viscoelastic stiffness, E1 / Uniaxial compression
Viscoelastic stiffness, E2 / Uniaxial compression
Poisson’s ratio, n / Triaxial shear
Drucker-Prager friction angle, a / Triaxial shear
Drucker-Prager cohesion, k0 / Triaxial shear
Perzyna’s viscoplastic parameter, G / Triaxial shear
Perzyna’s viscoplastic parameter, N / Triaxial shear
Dilation Parameter, b
Damage Parameter, x1 / Triaxial shear
Damage Parameter, x1 / Triaxial shear
Damage Parameter,x1 / Triaxial shear
Hardening Parameter, k1 / Triaxial shear
Hardening Parameter, k2 / Triaxial shear
1.2 Summary
Numerical models have been developed for modeling asphalt concrete behavior. Two types of constitutive models were described above: the disturbed state concept and the elastoviscoplastic models. The challenge associated with using constitutive models is the number of parameters that must be defined for using the model. Results of some laboratory testing by Masad et al. (2007) indicate that it may be possible to reduce the amount of testing required to determine all of the material parameters for the elastoviscoplastic model. If the amount of laboratory testing can be reduced or more simple tests can be substituted for some of the triaxial shear tests, this model may become useful for engineers in assessing long-term asphalt concrete pavement performance.
2 References
Abdulshafi, A., and Majidzadeh, K. (1985). “Combo viscoelastic-plastic modeling and rutting of asphaltic mixtures.” Transportation Research Record 968, Transportation Research Board, Washington, D.C., 19-31.
Collop, C., Scarpas, A.T., Kasbergen, C., and de Bondt, A. (2003). “Development and finite element implementation of a stress dependent elasto-visco-plastic constitutive model with damage for asphalt.” Transportation Research Record 1832, Transportation Research Board, Washington, D.C., 96-104.
Desai, C.S., (2001). Mechanics of materials and interfaces: The disturbed state concept, CRC, Boca Raton, FL.
Desai, C.S. (2007). “Unified DSC constitutive model for pavement materials with numerical implementation,” International Journal of Geomechanics, ASCE, Vol. 7, No. 2, 83 – 101.
Desai, C.S., Somasundaram, S., and Frantiziskonis, G. (1986). “A hierarchical approach for constitutive modeling of geologic materials.” Internation Journal for Numerical and Analytical Methods in Geomechanics, 10(3), 225-257.
Lu, Y., and Wright, P.J. (1988). “Numerical approach of viscoelastoplastic analysis for asphalt mixtures.” Computers & Structures, 69, 139-157.
Masad, E., Dessouky, S., Little, D. (2007). “Development of an elastoviscoplastic microstructural-based continuum model to predict permanent deformation in hot mix asphalt,” International Journal of Geomechanics, ASCE, Vol. 7, No. 2, 119 – 130.
Scarpas, A., Al-Khoury, R., Van Gurp, C., and Erkens, S.M. (1997). “Finite element simulation of damage development in asphalt concrete pavements.” Proc., 8th Int. Conf. on Asphalt Pavements, Univ. of Washington, Seattle, 673-692.
Seibi, A.C., Sharma, M.G., Ali, G.A., and Kenis, W.J. (2001). “Constitutive relations for asphalt concrete under high rates of loading.” Transportation Research Record 1767, Transportation Research Board, Washington, D.C., 111-119.