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The increasing development of smart materials, such as piezoelectric and shape memory alloys, has opened new opportunities for improving repair techniques. Particularly, active repairs, based on the converse piezoelectric effect, can increase the life of a structure by reducing the crack opening. A deep characterization of the electromechanical behavior of delaminated composite structures, actively repaired by piezoelectric patches, can be achieved by considering the adhesive layer between the host structure and the repair and by taking into account the frictional contact between the crack surfaces. In this paper, Boundary Element (BE) analyses performed on delaminated composite structures repaired by active piezoelectric patches are presented. A two-dimensional boundary integral formulation for piezoelectric solids based on the multi-domain technique to model the composite host damaged structures and the bonded piezoelectric patches is employed. An interface spring model is also implemented to take into account the finite stiffness of the bonding layers and to model the frictional contact between the delamination surfaces, by means of an iterative procedure. The effect of the adhesive between the plies of piezoelectric bimorph devices on the electromechanical response is first pointed out for both sensing and actuating behavior. Then, the effect of the frictional contact condition on the fracture mechanics behavior of actively repaired delaminated composite structures is investigated.

The increasing development of smart materials, such as piezoelectric and smart memory alloys, has opened new opportunities for practical and useful engineering applications. Focusing on piezoelectric materials, they allow the design of effective sensors and actuator devices, by means of the direct and converse piezoelectric effect [

Recently, the use of piezoelectric actuators has been also proposed and studied, by both numerical and experimental analyses, for applications in so-called active repair technology [

In spite of its attractiveness, active repair is a very difficult problem involving both design and technological aspects. From the design point of view, active repairs could be, in fact, arranged by bonding piezoelectric actuators to the host damaged structures and, in order to gain full advantage of the active repair technology, a deep characterization of both the electromechanical and fracture mechanics behavior of the actively repaired damaged structures is needed. In fact, the complex active repair mechanism, which stems from the strain induced by the piezoelectric actuator to the damaged structure by means of the bonding layer, can be fully understood through efficient numerical tools, taking into account both the adhesive between the bonded layers and the frictional contact condition between the crack or delamination surfaces. With this aim, Liu [

As already mentioned, a more accurate characterization of the repairing mechanism of piezoelectric active patches can be achieved through the modeling of the frictional contact between the crack surfaces. The interaction between two crack surfaces in contact, in fact, gives rise to a sliding resistance, depending on the normal and tangential tractions at the interface. It is evident that the modeling of such a situation can lead to a more accurate prediction of the electromechanical response of the actively repaired damaged structure [

Thus, the present paper deals with the analysis of the electromechanical response of piezoelectric active patches applied on delaminated composite structures, accomplished through a boundary element code [

The content of the paper is arranged as follows: the Multi-domain Boundary Element formulation and the spring interface iterative procedure for the modeling of the Coulomb's frictional contact are discussed in Section 2; The Modified Crack Closure Integral (MCCI) [_{T}

The Boundary Integral formulation is developed for a two-dimensional piezoelectric domain, Ω, with boundary, ∂Ω, lying in the _{1}x_{2}_{j}_{0}, F_{j} = _{j}δ_{0}_{j}_{j}_{j}_{0}_{0}

The modeling of the laminated structure, as well as the assembling between the host structure and the piezoelectric active patch is then obtained by means of the multi-domain approach [

The global system of equations pertaining to the overall assembled structure is then obtained by applying the compatibility and equilibrium conditions along all the sub-region interfaces:
_{ij}^{ij}. Conversely, the elastic interface conditions are addressed by means of an interface Spring Model [_{N}_{T}

More particularly, by considering the local reference system of

The spring interface, expressed by _{N}_{T}_{N}_{T}

The aim at issue is achieved by varying the _{N}_{T}_{N}_{T}_{T}_{N}_{T}

In this paper, the total ERR, _{T}_{i}, i_{i}_{T}

Thus, the mode-mixed phase angle, Ψ, is computed, under the assumption of negligible oscillatory behavior of the crack front stress and displacement fields [

The modified crack closure integral MCCI technique is thus used for the computation of the total ERR, _{T}

the energy released is identical to the work required to close the crack;

Δ being small enough, the crack extends in a self-similar manner.

The second assumption, in particular, implies that the stress field does not change as the crack extends from

The traction variation along the boundary element, including the crack tip, identified by node

In the present section, the numerical results obtained are discussed. More particularly, Coulomb's friction modeling strategy as proposed is first validated through the analysis of a flat punch over an elastic foundation previously studied by Man [

Coulomb's friction modeling strategy as proposed in the present paper is validated through the analysis of a flat punch over an elastic foundation previously studied by Man [_{o}, while the tangential one, with respect to _{0}_{1}_{2} = 0.8, the transition point from stick to slip condition can be found, and good agreement with the results found in the literature is evidenced.

In order to point out the effect of the adhesive between the piezoelectric plies of bimorph devices to be used as sensors and actuators, two different analyses of a piezoelectric series bimorph are performed for both sensing and actuating functions. The geometry of the piezoelectric bimorph devices has been chosen, since 3-D electromechanical data for the perfect bonding condition were available in the literature [

The first configuration deals with the piezoelectric device used as a sensor in a closed circuit; see

_{1} =

To point out the effect of the adhesive layer on the actuating performances of a piezoelectric bimorph with a series arrangement, in the second configuration analyzed, an electric potential is applied on the top and bottom faces of the device, _{B}_{2} = 0 and _{U}_{2} =

The analysis discussed in the present section deals with a drop-ply delaminated composite structure repaired through a multi-layered PZT-4 active patch, as shown in

A multi-layered piezoelectric patch having length, _{P}_{a}_{N}^{5}_{T} = 9.33 × 10-^{5}_{T}

_{r}_{r}_{T}_{r}_{r}

In this paper, BE analyses performed on delaminated composite structures repaired by active piezoelectric patches have been presented. A two-dimensional boundary integral formulation for piezoelectric solids based on the multi-domain technique to model the composite host damaged structures and the bonded piezoelectric patches has been employed. An interface spring model has been also implemented to take into account the finite stiffness of the bonding layers and to model the frictional contact between the delamination surfaces, by means of an iterative procedure. The analyses have shown that the finite stiffness of the bonding layer deeply affects the electromechanical response of piezoelectric devices. On the other hand, it has been pointed out that the frictional contact condition has a negligible effect on the repairing mechanism for the analyzed delaminated structure.

Multi-domain assembling.

Interface local reference system and Spring Model.

Flow chart of the frictional contact iterative procedure.

Smoothing scheme linear element.

(

Series bimorph configuration as sensor (

Through-the-thickness vertical displacement _{2}

Vertical Displacement Boundary Element Method (BEM), Electric Potential BEM (perfect/imperfect bonding).

Through-the-thickness vertical displacement and longitudinal displacement.

(

Effect of the friction coefficient on the fracture parameters.