• Concrete reinforcement
• Fiber Reinforced Polymer (FRP)
• Cohesive fracture
• Elastic contact
• Fracture mechanics
• Adhesion
• Debonding
• Concrete fracture
• Distributed dislocation technique
• Chebyshev polynomials

Fiber Reinforced Polymers (FRP) are commonly used for strengthening and rehabilitation of concrete or masonry structures, by gluing strips or plates made of this material on the surface of the weak material. Experimental studies have provided evidence that the main failure mode is the debonding of the FRP stiffener from the support, triggered by high stress concentrations at the extremities of the stiffener. Fracture propagates firstly parallel to the interface and then in the substrate, until complete separation between the two adherents occurs. Final failure is often characterized by the detachment of a wedge-shaped portion of the substrate, which remains bonded to the FRP strip. In order to describe the whole process, the model problem considered here is that of a finite thin elastic stiffener, bonded to an elastic half-space in generalized plane stress, pulled at one end by an axial force. The thickness of the stiffener is supposed very small, so that its bending stiffness can be considered negligible and only shear stresses act at the interface. On the contrary to the common assumptions of current models, the elastic deformations of the substrate are not neglected here: this is the main novelty of the proposed approach.
Compatibility equations between the stiffener and the substrate allow to write a singular integral equation for the contact problem, whose solution can then be obtained through an expansion in Chebyshev's series. The debonding process in pure mode II is supposed to be activated by an energetic balance, i.e., when the release of elastic strain energy equals the surface energy associated with material separation.
If the bond is perfect, the theory of elasticity predicts stress singularities at both ends of the stiffener. The shear stresses in a neighborhood of the singularity at the loaded end of the FRP strip is sufficient to counterbalance, in practice, the whole load applied, while the experimental evidence shows instead an effective bond length (EBL), over which the load transfer occurs gradually. To solve this inconsistency, in a second model a cohesive zone has been introduced at the loaded end of the stiffener, where slippage can occur according to an interface constitutive law, until a limit slip value is reached. Following an approach “à là” Barenblatt, the length of this zone is found by imposing that the stress intensity factor is null at the transition zone between the completely bonded part and the cohesive part, so to annihilate the stress singularity. There is a maximal reachable length of this cohesive zone, in which cohesive forces counterbalance, in practice, all the applied load, and which, therefore, can be referred to as the EBL. It can be also demonstrated that the second singularity at the free end of the stiffener plays a minor role, being negligible the load associated with it.
In order to describe the phenomenon of the wedge-shaped fracturing of concrete, a fracture mechanics problem “à là” Griffith has been considered for the substrate, assuming the crack propagation occurs in steps of finite length (quanta), of the same order of the intrinsic material length scale. From the energetic and tensional competition between the failure of the adhesive joint and the fracturing of the substrate, it has been possible to determine a critic propagation angle which coincides with the characteristic angle of the detached wedge-shaped bulb.
Results obtained from the analytical models are in very good agreement with the experimental results.