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The mechanical behavior of the adhesive interface between the fiber-reinforced polymer (FRP) strip and the concrete substrate often controls the response of FRP-strengthened reinforced concrete (RC) members. Plenty of studies devoted to understanding the mechanical behavior of FRP strips glued to concrete are currently available in the scientific literature. However, they are mainly focused on the response under monotonic actions, which is certainly relevant in a wide class of practical applications. Conversely, few contributions are currently available to better understand the response of FRP-to-concrete interfaces under cyclic actions, such as those deriving from either seismic excitations or traffic loads. This paper presents a unified numerical approach to simulate both monotonic and cyclic behavior of FRP plates glued on quasi-brittle substrates like those made of concrete. Particularly, a damage-based approach is proposed to simulate the fracture behavior of FRP-to-concrete joints under loading/unloading cycling tests. The model is formulated within the general framework of Fracture Mechanics and is based on assuming that fracture at the FRP-to-concrete interface develops in (pure shear) mode II, as widely accepted in similar problems. Two alternative expressions of the bond-slip behavior are herein considered and their preliminary validation is finally proposed. The proposed results highlight the difference between the monotonic and the cyclic response; particularly, they show that the latter is characterized by a significantly lower force and displacement capacity.

Fiber-reinforced polymer (FRP) materials recently gained popularity in a variety of retrofitting solutions aimed at upgrading structural members in existing civil engineering structures, such as concrete columns [

Nevertheless, FRP strips are widely used in practical applications with the aim of enhancing the structural performance of RC beams under cyclic actions possibly induced by either traffic loads or earthquakes. Despite the significant differences between the two aforementioned load cases, the state of knowledge about the actual behavior of both the adhesive FRP-to-concrete interface and the performance of EB-FRP strengthened RC members under cyclic actions is still in need for dedicated investigations under both the experimental and theoretical standpoints. In fact, few studies are available nowadays on this topic. Particularly, some authors [

This paper is intended as a further contribution to the modeling of FRP-to-concrete adhesive interface under cyclic actions: it presents a theoretical model formulated within the general framework of Fracture Mechanics (FM) to describe the post-elastic behavior of the aforementioned adhesive interface. In fact, the formulation is mainly intended at simulating the force-displacement response of FRP strips glued to a concrete substrate, as those generally adopted in single pull-out tests [

Although it is well known that normal stresses arising through the FRP-to-concrete interface significantly affect the behavior of FRP strips glued to concrete substrates and, then, the debonding process generally developing throughout such an interface is characterized by a mixed I/II mode of fracture [

After a short literature review,

Finally, in the authors’ best knowledge, although several studies, both theoretical and experimental in nature, are already available to investigate the mechanical behavior of FRP-to-concrete adhesive joints under monotonic actions, no well-established formulation is available yet for simulating the cyclic response of such joints, being this topic only approached in few experimental studies, one of which is considered herein as a reference [

A simplified theoretical model is proposed to model the cyclic response of FRP strips glued to brittle substrates, made of materials such as concrete or masonry. Particularly, the present proposal is based upon the following key assumptions:

the crack develops at the FRP-to-concrete interface in (pure shear) “mode II”;

the analytical expression of the monotonic softening branch of the bond-slip relationship is described “

stiffness degradation in the unloading stages depends upon the actual value of the “fracture work” developed in each interface point;

“small” displacements are assumed at the interface and axial strains possibly developing in the concrete substrate are neglected.

The four assumptions listed above lead to define the general governing equations for the mechanical behavior of FRP strips glued to a brittle substrate. They are derived by writing the classical “equilibrium”, “compatibility” and “(generalized) stress–strain” relationships, in both monotonic and cyclic response.

The proposed model is intended at simulating the FRP strip glued to a brittle support and schematically depicted in

Single-lap shear test of a fiber-reinforced polymer (FRP)-to-concrete bonded joint.

The assumptions of uniform width and thickness, _{p}_{p}_{p}

In this study, two alternative bond-slip laws are considered for the adhesive behavior, as discussed in

The linear elastic behavior of the FRP strip can be easily represented by the following relationship:
_{p}

Finally, the differential equation which relates the two field

The unloading/reloading stiffness is modeled within the framework of fracture mechanics (FM) theory by considering, for each point of the adhesive interface, the fracture work _{sl}_{sl}

The fracture work, developed during the sliding fracture process, controls the evolution of damage. Particularly, the variable _{sl}

Since a unique bond-slip law, possibly defined by Equations (1) and (2), is assumed through the bond length, the value of

Furthermore, a damage parameter _{d}

The two following subsections propose the explicit analytical expressions of Equations (5–8) corresponding to two alternative assumptions of the bond-slip law τ[

The general relationships introduced in _{E}_{e} = _{0}/_{E}_{0} is the shear strength, while β is the exponential parameter of the post-peak τ-

Bond-slip relationship and fracture work: exponential softening law.

The assumption of a bond-slip law described by Equation (9) leads to the following expression for the work _{sl}_{sl} =_{e}

Moreover, the corresponding value of fracture energy in mode II _{E}_{e}

_{E}_{sl}

An alternative and more common assumption for the bond-slip law τ[_{S}_{u}_{0}/_{E}_{0}/_{S}_{sl} =_{e}

Bond-slip relationship and fracture work: linear softening law.

Moreover, the fracture energy can be easily expressed through the following relationship, as a function of the key parameters of the bond-slip law:
_{sl}_{u}

Finally, the unloading/reloading stiffness _{sl}

A Finite Difference (FD) procedure is developed for integrating Equation (4) under monotonic and cyclic actions. Particularly, a central-difference (CD) expression is assumed to express the second derivative of Equation (4) in the internal nodes of the FD mesh represented in _{T,i}_{i}^{j−1}

Finite difference discretization of the FRP-to-concrete interface.

The set of (_{j}^{j−1}^{j−1}_{T}_{T,i}_{cr,i}_{cr,i} = s_{i} − s_{e}^{j}_{c}^{j}

If this is not the case, the slip increment _{i}^{j}|_{k}_{i}^{j}|_{k,el}_{i}^{j}|_{k,cr} = Δs_{i}^{j}|_{k}_{i}^{j}|_{k,el}_{cr}_{cr,i}_{i}^{j}|_{k,cr}

Then, in the following incremental analysis steps, the same node

If this is the case for all the nodes, the corresponding force can be determined through Equation (21) and the status variable updated via Equation (22). Otherwise, an unloading stage starts in the nodes where the inequality (23) is not satisfied and an iterative predictor-corrector search leads to the new system status.

Finally, the incremental analysis proceeds up to the achievement of a given failure condition which could be practically formulated in terms of maximum slip occurring at the unloaded end.

The formulation presented in

Experimental data characterizing both of the above mentioned experimental situations, on three types of FRP sheets are available in the scientific literature [

Two groups of three nominally equal specimens were tested under monotonic and cyclic actions, respectively. They are characterized by an A-FRP strip with relative axial stiffness _{p}_{p} =_{p} =_{E} =_{0} =_{S}_{d}

Finally,

Load-slip response under monotonic and cyclic actions of FRP strips glued on concrete [

Load-slip response under monotonic and cyclic actions of FRP strips glued on concrete [

It is apparent that such an assumption, generally accepted to simulate the monotonic response of FRP strips glued to concrete, is less fit for simulating the cyclic behavior of their adhesive interface, as it generally results in an overestimation of damage which leads to debonding failure after a lower number of cycles and a smaller ultimate value of slip. Moreover, after the computational standpoint, the exponential softening law is more convenient because it simulate cracking propagation as an asymptotical process: in fact, _{sl}

This paper presented a mechanically based theory and a simple numerical procedure for analyzing the debonding phenomenon which generally develops throughout the interface of FRP plate glued on concrete substrates. The following comments can be finally highlighted:

the proposed model has been formulated within the framework of Fracture Mechanics and assumed two alternative expressions for the softening branch of the bond-slip relationship;

the closed-form expressions obtained for determining the key fracture-related quantities are among the novel and most attractive features of the present formulation;

the comparison between some experimental results available in the literature and the numerical simulations performed by means of the present model highlighted its high predictive potential;

the proposed experimental comparisons pointed out the higher accuracy obtained by assuming an exponential softening branch, with respect to the linear one, generally accepted for simulating the response under monotonic actions;

the extension of such comparisons to further experimental observation is among the future steps of this research, that finally aims at characterizing the force and displacement capacity of FRP-to-concrete joints subjected to cyclic actions, which are quite common in seismic retrofitting of existing buildings.

This work was realized during the visit of the second co-author at the University of Salerno (Italy), as part of the “EnCoRe” Project (FP7-PEOPLE-2011-IRSES no. 295283;

The authors declare no conflict of interest.