2.1.3. At the Lowest Layer, *i* = *n*, *z* ≥ H

The bottom layer is semi-infinite ( *z* → ∞) and all responses (stresses, displacements) approach zero as *z* approaches ∞, so λ approaches infinity. From Equation (1) for the lowest layer with *i* = *n* and <sup>λ</sup> approaches infinity, we have (*e*−m(λ*n*−λ)→∞) and (*e*−m(λ−λ*n*−1)→0), to all responses (stresses, displacements and strains) approach zero, coefficients A*<sup>n</sup>* and C*<sup>n</sup>* will become zero.

#### *2.2. Improvement Taking into Account Actual Bonding Condition*

In a general case, the layers interface bonding condition can be considered as partially bonded. The layers interface behavior can be described according to Goodman's constitutive law [21] (Figure 2) in which the interface shear stress can be expressed as follows:

$$
\pi = \mathsf{K}\_{\mathsf{s}} \,\mathsf{\varDelta u} \tag{11}
$$

where Δu is the relative horizontal displacement of the two layers at the interface; K*<sup>s</sup>* is the horizontal shear reaction modulus at the interface.

**Figure 2.** Modeling of the bonding between two faces at the interface.

The continuity conditions for this general case are:

$$(\sigma\_{zz}^\*)\_{\dot{i}} = (\sigma\_{zz}^\*)\_{\dot{i}+1} \tag{12}$$

$$(\tau\_{rz}^\*)\_{i} = (\tau\_{rz}^\*)\_{i+1} \tag{13}$$

$$\mathbf{K}\left(\tau\_{rz}^{\*}\right)\_{i} = \mathbb{K}\_{\mathbf{s}}\left[ (\mathbf{u}^{\*})\_{i+1} - (\mathbf{u}^{\*})\_{i} \right] \tag{14}$$

$$(w^\*)\_{i} = (w^\*)\_{i+1} \tag{15}$$

Substituting Equation (1) by these above conditions, one obtains:

⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 1 F*<sup>i</sup>* −(1 − 2ν*<sup>i</sup>* − mλ*i*) (1 − 2ν*<sup>i</sup>* + mλ*i*)F*<sup>i</sup>* 1 −F*<sup>i</sup>* 2ν*<sup>i</sup>* + mλ*<sup>i</sup>* (2ν*<sup>i</sup>* − mλ*i*)F*<sup>i</sup>* mE*<sup>i</sup>* (1+ν*i*)K*<sup>s</sup>* <sup>+</sup> <sup>1</sup> <sup>1</sup> <sup>−</sup> mE*<sup>i</sup>* (1+ν*i*)K*<sup>s</sup> Fi* <sup>1</sup> <sup>+</sup> <sup>m</sup>λ*<sup>i</sup>* <sup>+</sup> (2ν*i*+mλ*i*)mE*<sup>i</sup>* (1+ν*i*)K*<sup>s</sup>* (2ν*i*−mλ*i*)m*Ei* (1+ν*i*)K*<sup>s</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> <sup>m</sup>ν*<sup>i</sup>* F*i* 1 −*Fi* −(2 − 4ν*<sup>i</sup>* − mλ*i*). −(2 − 4ν*<sup>i</sup>* + mλ*i*)F*<sup>i</sup>* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ A*i* B*i* C*i* D*i* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ F*i*+<sup>1</sup> 1 −(1 − 2ν*i*+<sup>1</sup> − mλ*i*)F*i*+<sup>1</sup> 1 − 2ν*i*+<sup>1</sup> + mλ*<sup>i</sup>* F*i*+<sup>1</sup> −1 (2ν*i*+<sup>1</sup> + mλ*i*)F*i*+<sup>1</sup> 2ν*i*+<sup>1</sup> − mλ*<sup>i</sup>* R*i*F*i*+<sup>1</sup> R*<sup>i</sup>* (1 + mλ*i*)R*i*F*i*+<sup>1</sup> −(1 − mλ*i*)R*<sup>i</sup>* R*i*F*i*+<sup>1</sup> −R*<sup>i</sup>* −(2 − 4ν*i*+<sup>1</sup> − mλ*i*)R*i*F*i*+<sup>1</sup> −(2 − 4ν*i*+<sup>1</sup> + mλ*i*)R*<sup>i</sup>* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ A*i*+<sup>1</sup> B*i*+<sup>1</sup> C*i*+<sup>1</sup> D*i*+<sup>1</sup> ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)

with F*<sup>i</sup>* = *<sup>e</sup>*−m(λ*i*−λ*i*−1); R*<sup>i</sup>* = <sup>E</sup>*<sup>i</sup>* E*i*+<sup>1</sup> 1+ν*i*+<sup>1</sup> <sup>1</sup>+ν*<sup>i</sup>* .

In Equation (2), the stress or displacement function for each layer has four coefficients of integration: A*i*, B*i*, C*<sup>i</sup>* and D*i*. All responses (stresses, displacements) can be calculated by these coefficients and integrations.

For n-layers system, the total number of unknown coefficients is 4n, which must be evaluated by the boundary and continuity conditions. With the lowest layer A*<sup>n</sup>* = C*<sup>n</sup>* = 0, there are only (4n-2) unknown coefficients.

All of these above conditions result in four equations for each of (n-1) interfaces and two equations at the surface, there are so (4n-2) independent equations. Thus, the (4n-2) unknown constants can be solved.

#### *2.3. Numerical Implementation and Backcalculation Principle*

The analytical solution including its improvement was implemented in a numerical program using Matlab [22]. This implementation is very important for research studies of the authors because it, with regard of specific or new features of pavement materials and structures, allows evaluating pavement responses under different loading configurations without depending on existing commercial software.

The developed numerical program can be used to determine pavement responses by forward calculation or to evaluate pavement properties by backcalculation. In forward calculation, based on given properties of pavement materials and structures, pavement responses in terms of stress, strain or deflection can be calculated directly. In backcalculation, which is frequently applied for FWD measurements, pavement properties can be evaluated by adjusting their seed values until getting the least squares differences between the calculated and measured pavement deflections. These investigations where the bonding condition at the interface of the asphalt layers were taken into account are presented in the following paragraphs 3 and 4, respectively.

#### **3. Sensitivity Analysis**

Sensitivity analysis using the developed numerical program is presented in this paragraph. The variation of some most important pavement responses under the loading of an FWD (with a circular plate of 0.3 m in diameter and a vertical static pressure of 0.92 MPa) in function of the interface bonding condition were evaluated. The main characteristics (with nominal values of the asphalt layers thickness) of the pavement structure used for this analysis are presented in Table 1.


**Table 1.** Pavement structure characteristics.

<sup>1</sup> Values used for sensitivity analysis; <sup>2</sup> Values measured in actual pavement structures in paragraph 4.

#### *3.1. Strain Sensitivity to the Interface Bonding Conditions*

In an asphalt pavement, the horizontal strain at the bottom of the asphalt layer is among the most important parameters because its magnitude will directly affect the pavement performance. Generally, the higher this magnitude is, the lower the pavement performance is. Figure 3 presents the horizontal strain at the bottom of each of the two asphalt layers of the investigated pavement structure in function of the bonding condition at the interface between the asphalt layers. As can be seen in this figure, when the bond modulus Ks decreases from infinite to nil, the horizontal strain at the bottom of the asphalt surface layer (EpsilonT\_bottom\_AC1) increases from 47 to 360 microstrains. The horizontal strain at the bottom of the asphalt base layer (EpsilonT\_bottom\_AC2) increases from 243 to a maximum value of 251 before decreasing down to 233 microstrains when Ks decreases from infinite to about 10 MPa/mm then continues to decrease to nil, respectively. Compared to the first strain, the shape of the second strain is different. This can be explained by the fact that in this case, the interface between the two asphalt layers is below their neutral axis. The position of the last one is a result from a combination of the pavement layers thicknesses and moduli. For the considered pavement structure, while the first strain is smaller than the second one when Ks > 2 MPa/mm, opposite result is obtained when Ks < 2 MPa/mm. The first strain is even much higher than the second one when Ks is close to nil, i.e., close to the unbonded condition of the interface. Based on these evaluations, it is possible to classify the interface bonding condition as follows:


**Figure 3.** Impact of the interface bonding conditions between the asphalt layers on the horizontal strains at the bottom of the asphalt layers.

Moreover, the pavement responses are more sensible for Ks between 0.1 and 100 MPa/mm than when Ks ≥ 100 MPa/mm or Ks ≤ 0.1 MPa/mm. Among the two horizontal strains, the one at the bottom of the surface layer is more sensible with variation of Ks than the other one of the base layer. That means that the influence of the interface bonding condition is higher on the bottom of the surface layer than in the bottom of the base layer. This result can be explained by the fact that the interface is much closer to the bottom of the surface layer than the base layer.

#### *3.2. Deflection Sensitivity to the Interface Bonding Conditions*

In this parametrical study, five different deflection bowls of the pavement surface were calculated for five different bonding levels at the interface between the asphalt layers. The results are presented in Figure 4. It can be observed that when Ks = 100 MPa/mm, the pavement response is very close to the one where the interface is fully bonded. Similarly, when Ks = 0.1 MPa/mm, the pavement response is very close to that where the interface is fully debonded. For Ks = 5 MPa/mm, this bonding level gives a deflection bowl near to the middle position between the two previous cases. These observations affirm once more the classification in the previous paragraph.

#### **4. Evaluation of Pavement Interface Bonding Condition in an Experimental Case Study**

The developed solution is applied in this part to evaluate field conditions of the interface bond between the asphalt layers of full-scale pavement structures in an experimental case study.

#### *4.1. Pavement Structures and Materials Characteristics*

In order to evaluate the field interface bonding conditions, two specific full-scale pavement structures at the accelerated pavement testing (APT) facility of IFSTTAR were chosen. They have the same design, which is composed of two asphalt concrete layers built on a homogenous and well-controlled subgrade of 2.9-m-thick unbound granular material and sand. The subgrade has a mean value of stiffness modulus of 184 MPa. All pavement layers were built above a concrete raft inside a watertight concrete lining. The same asphalt concrete material was used for both asphalt layers in both structures. The asphalt material is a hot mix whose formulation is a standard semi-coarse asphalt concrete of class 3 (according to the standard EN 13108-1). The unique difference between the two structures is the bonding condition at the interface between the asphalt layers. In the first structure, noted S-I, the asphalt surface layer was laid directly above the asphalt base layer. In the second one, noted S-II, there is a geogrid at the interface between the asphalt layers. One can notice that the surface layer is thicker than the base layer. The reason is that in order to get advantage of geogrid-based reinforcement in new pavement, the geogrid must be installed below the apparent neutral axis of the asphalt layers. For rehabilitated pavement, the overlay above the geogrid is often thinner than the existing base layer. A same tack coat material made of a classical cationic rapid setting bitumen emulsion (classified as C69B3 according to EN 13808) was applied at the interface between the asphalt layers with an application rate of 350 g/m<sup>2</sup> and 700 g/m2 in the case without and with geogrid, respectively.

Asphalt concrete material was extracted during the construction of the full-scale pavement. The loose mix was then used for fabrication in the laboratory by a roller compacter of slab with the same air voids content as targeted in the field. The complex modulus of the obtained asphalt material was measured using two points bending test (according to EN 12697-26). The results obtained at five different frequencies (3, 6, 10, 25 and 40 Hz) and six different temperatures (−10, 0, 10, 15, 20 and 30 ◦C) are plotted in Figure 5 in isotherm curves.

**Figure 5.** Isotherms of complex modulus of the tested asphalt concrete material.

#### *4.2. Evaluation of Bonding Condition at the Interface of the Asphalt Layers*

For this evaluation, a dedicated FWD tests campaign was carried out. Measurements were performed at three different locations on each pavement structure with the same load level of 65 kN. The circular load plate of the FWD used for these measurements has 0.3 m in diameter. The distances of the geophone sensors are 0, 0.3, 0.45, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1 m from the load plate, respectively. The temperature measured by thermocouple sensors in the middle depth of the asphalt surface and base layers during these FWD measurements were close to 23 ◦C and 21.5 ◦C, respectively.

The actual thicknesses (Table 1) of the pavement layers were obtained from levelling measurement during the construction. The stiffness modulus of each asphalt layer (the same as in Table 1) was taken from the complex modulus measured in the laboratory. They were determined taking into account the temperature and frequency variations in function of the asphalt layer depth according to [23]. The Poisson's ratio of each pavement layer material was assumed to be equal to 0.35 for asphalt and unbound granular materials and 0.25 for concrete raft.

The backcalculation process was applied here to determine the shear reaction modulus Ks at the interface between the asphalt layers. In this case, all the pavement layers moduli were known, only the interface bonding condition was the unknown parameter.

Figure 6 presents the measured and calculated deflections associated with a value of shear reaction modulus for each point of FWD measurement. Good results of calculated deflections can be observed. They fit well with the measured values. These obtained values of Ks are in accordance with the initial assumption of the interface bonding condition between the asphalt layers of the two investigated pavement structures: structure S-I has good interface bond condition at points 1, 2, 3 with Ks equal to 531, 109 and 131 MPa/mm (>100 MPa/mm), respectively; intermediate interface bond conditions were obtained in structure S-II at points 4, 5, 6 with Ks equal to 74, 76 and 69 MPa/mm (0.01 MPa/mm < Ks < 100 MPa/mm), respectively.

**Figure 6.** Measured and calculated deflections in structures S-I (points 1, 2, 3) and S-II (points 4, 5, 6) and the associated interface shear reaction moduli.

One can note some differences in the Ks values obtained for structure S-I, which vary between 109 and 531 MPa/mm. However, as analyzed in paragraph 3, when Ks is higher than 100 MPa/mm (good bond), pavement responses (strains and deflections) are much closer to the case with fully bonded condition. In that case, even though the difference in terms of Ks value is high, the difference in terms of pavement deflection is little. This experimental result confirms those observed in paragraph 3.2 of the sensitivity analysis. For structure S-II, the three Ks values are very similar, which means that the interface bonding condition is quite homogeneous, at least within the investigated pavement section, and is at the same intermediate bonding level. Moreover, Ks values in structure S-II with geogrid at the interface between the asphalt layers are smaller than the ones in structure S-I without geogrid. It confirms the literature review made in [24] that the use of a geogrid reduces the interlayer bond and hence reduces the instantaneous structural response of the pavement. However, as the geogrid could delay the reflective cracking, if properly installed, it can contribute to the long-term performance of the pavement. Furthermore, one can note that the experimental Ks values obtained for both pavement structures in this case study are at the same order of magnitude as those from dynamic shear tests [15,16] than from quasi-static shear tests [6,14]. This result confirms the position, as stated in [25] that dynamic tests represent better the field condition of interface bonding than static tests and hence are more suitable for characterization, modelling and design studies of the structural behaviors of pavements. It joints also the point of view of the Task Group 3 of the actual RILEM Technical Committee 272-PIM [26] working on dynamic interlayer shear testing.
