Advances in Materials Science and Engineering

Volume 2015, Article ID 937126, 8 pages

http://dx.doi.org/10.1155/2015/937126

## Multiscale Validation of the Applicability of Micromechanical Models for Asphalt Mixture

^{1}School of Highway, Chang’an University, Shaanxi, Xi’an 710064, China^{2}School of Materials Science and Engineering, Chang’an University, Shaanxi, Xi’an 710064, China

Received 22 April 2015; Revised 22 July 2015; Accepted 28 July 2015

Academic Editor: Antônio G. B. de Lima

Copyright © 2015 Jiupeng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Asphalt mixture is more complicated than other composite materials in terms of the higher volume fraction of aggregate particles and the viscoelastic property of asphalt matrix, which obviously affect the applicabilities of the micromechanical models. The applicabilities of five micromechanical models were validated based on the shear modulus of the multiscale asphalt materials in this paper, including the asphalt mastic, mortar, and mixture scales. It is found that all of the five models are applicable for the mastic scale, but the prediction accuracies for mortar and mixture scales are poorer. For the mixture scale, all models tend to overestimate at the intermediate frequencies but show good agreement at low and high frequencies except for the Self-Consistent (SC) model. The Three-Phase Sphere (TPS) model is relatively better than others for the mortar scale. The applicability of all the existing micromechanical models is challenged due to the high particle volume fraction in the multiscale asphalt materials as well as the modulus mismatch between particles and matrix, especially at the lower frequencies (or higher temperatures). The particle interaction contributes more to the stiffening effect within higher fraction than 30%, and the prediction accuracy is then deteriorated. The higher the frequency (or the lower the temperature) is, the better the model applicability will be.

#### 1. Introduction

Asphalt mixture is a kind of heterogeneous composite material, consisting of asphalt binder and mineral aggregates with different sizes. The traditional researches on the mechanical properties of asphalt mixture and its failure mechanism are mostly based on the continuum mechanics theory and the experimental method. However, the mechanical properties of asphalt mixture are closely related to the complicate internal structure, which is dependent on the various raw materials’ properties, shapes, sizes, and proportions. The traditional analyzing methods fail to reveal the microstructure related failure mechanism of asphalt mixture, such as the formation and propagation of microcracks, the micro damage caused due to the heterogeneous material, and the local failures caused due to the stress concentration. Recently, researchers have realized the importance of the internal microstructure to the macromechanics properties, and the micromechanics theory of composites has been introduced to research the macromechanical properties from the meso- and microscale [1–4].

At the microscale, the asphalt mixture could be considered as a kind of heterogeneous multiphase composite which is composed of asphalt binder, aggregates, asphalt-aggregate interphase, microcrack, and void. The homogenization approach of the heterogeneous material is a fundamental problem in the area of composite, which mainly involves the prediction of the effective modulus of the composite. Micromechanical modeling techniques have long been successfully used to predict the effective modulus from mechanical properties and volume fractions of individual constituents for composite materials such as metal and polymer matrix composites [5–9]. For asphalt mixture, the homogenization method could be applied to acquire the effective properties, like the effective modulus. However, compared with other composites, asphalt mixture exhibits some special characteristics both due to the complicated microstructure and mechanical properties of raw materials. Firstly, the aggregates play the role of reinforcement phase in this composite, and its volume fraction is usually up to 80%, which is far higher than the reinforcement phase in other composite materials. Secondly, the aggregates in common asphalt mixtures are with various sizes from 0.075 mm up to 37.5 mm while the reinforcement phase is with a relatively uniform size in other composite materials. Well-graded aggregates obviously exhibit the size effect, which means aggregate with different size plays different role in this composite [10, 11]. Moreover, asphalt binder is a viscoelastic matrix of this composite material, which also affects the asphalt-aggregate interphase, and causes more difficulties to the applications of these micromechanical models [12, 13].

Though considerable achievements have been made by previous researchers, few have involved the applicability of the existing micromechanical models on asphalt mixture. So, in this paper, the micromechanical models for composite materials were firstly introduced and evaluated. Then, these models were applied to predict the mechanical properties of asphalt materials from the multiscales and compared with the experiment results to validate the applicability of micromechanical models. In addition, the effects of particle volume fraction and modulus mismatch between reinforcement phase and matrix phase were also analyzed from the mastic scale.

#### 2. Literature Review on the Micromechanical Models

Since Eshelby’s pioneering work [14] on elastic solutions for an infinite medium including a single inclusion, a lot of micromechanical models have been developed in the past decades, including the Self-Consistent (SC) model, the Mori-Tanaka (M-T) model, the Generalized Self-Consistent (GSC) model, the Differential Scheme Effective Medium (DSEM) model, and the Three-Phase Sphere (TPS) model.

In the SC model, any material point on the particles is isolated as an infinitesimal volume element. Then the rest of the material is homogenized as the uniform material, whose mechanical property is identical to the composite itself [5, 6, 12]. The effective bulk modulus and shear modulus of the SC model can be written aswhere denotes the volume fraction of particles and , ; , ; and , represent the bulk modulus and shear modulus of effective medium, matrix, and particle, respectively.

The M-T model involves complicated manipulation of the field variables in Mori and Tanaka [7]. In this method, a particle is embedded into the matrix with a uniform strain same as the matrix’s averaged strain, and the particle’s averaged strain is derived from the solution for single particle embedded in the infinite matrix [12]. The effective bulk modulus and shear modulus can be expressed as

In the GSC model, the spherical particle is embedded in a concentric spherical shell of the matrix material. Shell and particle dimensions are chosen to correspond to the prescribed volume fraction, and the particle-shell assembly in turn is embedded in an infinite medium with unknown effective properties [15]. The solution for effective shear modulus is the solution of a quadratic equationThe expressions for , , and can be found in Christensen and Lo [8]. Shashidhar and Shenoy [16] provided a simplification form of the GSC model that is suitable for mastic. The effective bulk modulus is the same as the M-T model.

By introducing the Differential Scheme Effective Medium theory into the micromechanics field, Mclaughlin [9] and Norris [17] successfully developed the differential method. In this method, the composite material is viewed as a sequence of dilute composites to which an increment of inclusions is added, as shown in Figure 1. This method shows great potential in predicting the effective properties of composites with high particle volume fraction. Shu and Huang [18] and Kim and Buttlar [19] further derived new models suit for asphalt mixture, respectively, and only the latter is given in this research. The effective bulk modulus and shear modulus can be written as