**1. Introduction**

Asphalt pavement is generally considered as being a multilayered structure comprising of successive material layers. The kinematics of the disorders in this type of structure are related to the nature of the materials used, to the conditions of the construction and more particularly to the layers properties as well as the bonding conditions between layers. Among these conditions, a good interface bond between the asphalt layers ensures the estimated performance of the designed pavement structure. Moreover, the majority of current works for the rehabilitation of existing road network as well as for new pavement structures use thinner and thinner overlayers, which require an effective bonding. However, conventional design methods consider that the interface between two pavement layers is perfectly bonded, or unbonded, depending on the nature of the layers involved. In situ inspections revealed that lack of bonding or damage to the bonding layer (interface) leads to rapid and considerable structural damage. The principle of dimensioning is based on the fact that the layers deformed by bending depend on their own characteristics (thickness, Young modulus and Poisson ratio), but also on the other layers on which they are glued. When there is an absence or failure of bonding at the interfaces (on the top or at the bottom of the layers), each layer works independently. Deformations and constraints on both sides of the interface are then more important than when the layers are glued.

Burmister [1] first derived the analytical solutions for a two-layered elastic system and subsequently extended them to a three-layered system [2–4]. Over the years, the theory has been extended to an arbitrary number of layers [5]. However, the interface bonding condition still has not been well considered in most of the modelling processes. Since the 1970s, many experimental methods have been applied to assess the capability of tack coats as well as the internal cohesion of the two involved pavement layers. Experimental methods can be divided into two main groups according to the situation of testing, in laboratory or in situ. In laboratory, direct shear tests with or without normal stress are most commonly used in the assessment of adhesion properties between two asphalt layers. Shear tests with normal stress allow the consideration of the presence of a wheel load on the road by not only its horizontal force but also its vertical influence [6,7]. However, the application of normal stress makes the experiment much more complicated. Therefore, the direct shear test method without normal stress is the most utilized one [8–11]. Most of these tests are inspired from the Leutner shear test [12]. With monotonic loading, they allow us to rapidly evaluate the influence of different factors on bond strength at the interfaces between pavement layers [13,14]. In parallel to these quasi-static tests, several dynamic shear tests developed recently [15,16] should lead to more reliable field performance characteristics. In field evaluation, until now there have been very few methods. Some pull-off test methods can be found in the literature, but are rare or only in development. In France, the destructive ovalization test has been developed since 1970s, aiming to evaluate bond conditions at the interface between pavement layers under moving wheel loads [17,18]. However, it is not often used due to the complex interpretation of the measurements. Recently, the non-destructive method of using a Falling Weight Deflectometer (FWD) [19] device has been applied quite commonly for pavement assessment through measured pavement surface deflections. Several researches using this method were performed with the same objective of investigating pavement layers interface bonding, but without relating the measured pavement deflections with interface bonding characteristics.

This present paper focuses on numerical and experimental investigations of asphalt pavement behaviour taking into account actual bonding condition at the interface between the asphalt layers. For that purpose, a theoretical background on the analytical solution of multilayered pavement structure is firstly presented. It is then improved by introducing a shear reaction modulus to take into account the bonding condition of the interface between the pavement layers. Next, the improved solution is implemented in a numerical program, which is used to perform a parametric study to investigate the sensitivity of pavement responses to the interface bonding conditions. Finally, the developed solution is applied through an original experimental case study where falling weight deflectometer (FWD) tests were carried out on two full-scale pavement structures to investigate field condition of the interface bond between the asphalt layers.

This paper is an expanded version of the conference paper [20] from the same authors. All parts of the work have been developed with more completed and self-supported elements, in particular, the analytical solution and the experimental case study. New elements have also been added in this expanded version to support both the model developed in the analytical solution and the result obtained in the original experimental study. They are the sensitivity analysis part and the characteristics of materials and structures of a full-scale pavement in the experimental part.

#### **2. Analytical Solution Background and Improvement**

#### *2.1. Analytical Solution Background*

Asphalt pavement is typically modelled using a multilayered structure based on the layered theory of Burmister. Each layer is considered as linear elastic isotropic (having an elastic modulus and a Poisson ratio) and infinite in the horizontal plan. The thickness of each layer is finite, except the bottom layer which is infinite. The interface bonding conditions between the layers are only bonded or unbonded. Figure 1 presents the multilayered pavement structure in cylindrical coordinates with *r* and *z* are the coordinates in the radial and vertical directions, respectively. The load applied on the surface of the pavement is a uniform vertical pressure of magnitude *q* and has a circular form of radius *a*. The analytical results to the problem described above are the stress, strain and displacement fields in the pavement structure. As discussed in the objectives of the work, for further improvement purpose in the paper and especially with the numerical implementation developed by the authors, the main steps and equations of the analytical solution to the problem described above, to which improvements will be made in the next paragraph, are presented here. Other details for this analytical solution can be found in the literature [5].

**Figure 1.** Multi-layered pavement structure.

Equation (1) presents the axisymmetric layered elastic responses (stresses and displacements) under a concentrated load.

$$
\begin{bmatrix}
\left(\boldsymbol{\sigma}\_{zz}^{\star}\right)\_{i} \\
\left(\boldsymbol{\tau}\_{rz}^{\star}\right)\_{i} \\
\left(\mathbf{u}^{\star}\right)\_{i} \\
\left(\mathbf{w}^{\star}\right)\_{i}
\end{bmatrix} = \begin{bmatrix}
\mathbf{m}\mathbf{J}\_{1}(\mathbf{m}\boldsymbol{\rho})\{1 & -1 & (2\nu\_{i}+\mathbf{m}\lambda) & (2\nu\_{i}-\mathbf{m}\lambda)\} \\
\frac{1+\nu\_{i}}{\mathbf{E}}\mathbf{J}\_{1}(\mathbf{m}\boldsymbol{\rho})\mathbf{H}\{1 & -1 & (1+\mathbf{m}\lambda) & (1-\mathbf{m}\lambda)\} \\
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\varepsilon}^{-\mathbf{m}(\boldsymbol{\lambda}\_{i}-\boldsymbol{\lambda})}\mathbf{A}\_{i} \\
\boldsymbol{\varepsilon}^{-\mathbf{m}(\boldsymbol{\lambda}-\boldsymbol{\lambda}\_{i-1})}\mathbf{B}\_{i} \\
\boldsymbol{\varepsilon}^{-\mathbf{m}(\boldsymbol{\lambda}\_{i}-\boldsymbol{\lambda})}\mathbf{C}\_{i} \\
\boldsymbol{\varepsilon}^{-\mathbf{m}(\boldsymbol{\lambda}-\boldsymbol{\lambda}\_{i-1})}\mathbf{D}\_{i}
\end{bmatrix} \tag{1}
$$

where (σ∗ *zz*)*<sup>i</sup>* and (τ<sup>∗</sup> *rz*)*<sup>i</sup>* are the vertical and shear stresses, (u<sup>∗</sup> )*<sup>i</sup>* and (*w*<sup>∗</sup> )*<sup>i</sup>* are the horizontal and vertical displacements of layer *i*; H is the distance from the pavement surface to the upper boundary of the bottom layer ρ = *r*/H and λ = *z*/H; J0 and J1 are Bessel functions of the first kind and order 0 and 1 respectively; A*i*, B*i*, C*<sup>i</sup>* and D*<sup>i</sup>* are constants of integration to be determined from boundary and continuity conditions; m is a parameter. The superscript *i* varies from 1 to *n* and refers to the quantities corresponding to the *i th* layer. A star super is placed on these stresses and displacement due to a concentrated vertical load −mJ0(mρ), not the actual stresses and displacements due to a uniform pressure *q* distributed over a circular are of radius a.

The stresses and displacements as a result of the uniform pressure *q* distributed over the circular load of radius *a* are obtained by using the Hankel transform (Equation (2)):

$$\mathbf{R} = q\alpha \int\_0^\infty \frac{\mathbf{R}^\*}{\mathbf{m}} \mathbf{J}\_1(\mathbf{m}\alpha) \mathbf{d}\mathbf{m} \tag{2}$$

where α = *a*/H; R<sup>∗</sup> is the stress or displacement as a result of concentrated load −mJ0(mρ); R is the stress or displacement as a result of load uniform *q*. So, the boundary and continuity of the multilayered pavement structure by the load −mJ0(mρ) and uniform *q* distributed are the same.

## 2.1.1. At the Surface, *z* = 0

At this position, *i* = 1 and λ = *z*/H = 0, the surface stresses conditions are:

$$(\sigma\_{zz}^\*)\_1 = -\mathbf{m} l\_0(\mathbf{m}\rho) \text{ with } 0 \le \mathbf{r} \le a \tag{3}$$

$$(\sigma\_{zz}^\*)\_1 = 0 \text{ with } \mathbf{r} \succ a \tag{4}$$

$$
\pi\_{rz}^\* = 0 \tag{5}
$$

2.1.2. Between the Layers *i* and *i* + 1, 0 < *z* < H

#### (a) Fully bonded

The layers are fully bonded with the same vertical stress, shear stress, vertical displacement and radial displacement at every point along the interface. Therefore λ = λ*i*. The continuity conditions are:

$$(\sigma\_{zz}^\*)\_{\dot{i}} = (\sigma\_{zz}^\*)\_{\dot{i}+1} \tag{6}$$

$$(\pi\_{rz}^\*)\_{\dot{i}} = (\pi\_{rz}^\*)\_{\dot{i}+1} \tag{7}$$

$$(\mathbf{u}^\*)\_i = (\mathbf{u}^\*)\_{i+1} \tag{8}$$

$$(w^\*)\_i = (w^\*)\_{i+1} \tag{9}$$

(b) Unbonded

At the interface, the vertical stress and vertical displacement remain the same, but the shear stresses are equal to zero on both sides of the interface. Equation (7) is replaced by:

$$(\tau\_{rz}^\*)\_{i+1} = (\tau\_{rz}^\*)\_i = 0 \tag{10}$$
