# The Effects of an Interlayer Debond on the Flexural Behavior of Three-Layer Beams

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{c}and Young’s modulus E

_{c}, enclosed by two stiffer outer faces, having equal thickness h

_{f}and Young’s modulus E

_{f}, was considered. The beam is simply supported at both ends and subjected to a point load at center, as shown in Figure 3. The lower interface between the core and the lower face (layer 3) was assumed to be perfect also in the longitudinal direction, so no slip can occur along this interface. The upper interface between the core and the upper face (layer 1) was assumed to be perfect as well, except for the presence of a debonded portion having length a at a distance d < l/2 from the left end. Then, the upper interface can exhibit one of the two configurations shown in Figure 4 depending on the value of d + a.

^{AD}denotes the deflection function in portion AD. Furthermore, the continuity conditions at D can be written as:

_{max}exhibited by the composite beam at midspan under the load point can be evaluated by considering the proper deflection function. For the configuration shown in Figure 4a, w

_{max}is given indifferently by the deflection function w

^{EC}or w

^{CB}evaluated at C. For the configuration shown in Figure 4b, w

_{max}is given indifferently by the deflection function w

^{DC}or w

^{CE}evaluated at C. It is straightforward that the value of w

_{max}depends on those of all the geometrical and physical parameters of the model, as discussed in Section 3, where the solution described above is employed to analyze the effects of an interlayer debond at the upper interface of three-layer beams on their flexural behavior.

## 3. Results

_{c}, η = h

_{f}/h

_{c}, Ω = E

_{f}/E

_{c}

_{c}= Λa/l, δ = d/h

_{c}= Λd/l

_{max}/w

_{0}, where for each set of results, w

_{max}and w

_{0}are the maximum deflections in the cases of upper interface with an initial debond and intact upper interface, respectively. It is well known that for a simply supported beam under a point load at midspan:

_{0}= 1/48Pl

^{3}/K

_{lim}/w

_{0}, which corresponds to the case of fully debonded upper interface; it results in:

_{c}–80h

_{c}= 0.2l–0.8l for face layers little stiffer than the core (Ω = 1.5 and 2.5) and a = 15h

_{c}–85h

_{c}= 0.15l–0.85l for face layers much stiffer than the core (Ω = 5 and 10).

_{c}= 0.3l for a = 40h

_{c}= 0.4l, d = 35h

_{c}= 0.35l for a = 30h

_{c}= 0.3l, and d = 40h

_{c}= 0.4l for a = 20h

_{c}= 0.2l) is minimum independently of layer materials and geometry. Such a decrease becomes more significant for the debond approaching one end of the beam and is maximum for an end debond (d = 0). However, in many cases, the stiffness reduction with respect to that of three perfectly bonded layers results in being independent of the debond position unless the debond approaches midspan. As an example, for faces moderately stiffer than the core (Ω = 1.5), significant reductions are induced by 20h

_{c}= 0.2l (circular markers in Figure 7) and 30h

_{c}= 0.3l (triangular markers in Figure 7) long debonds at a distance from the end longer than 30h

_{c}= 0.3l and 20h

_{c}= 0.2l, respectively. Analogous considerations follow from the results related to faces much stiffer than the core (Ω = 10).

## 4. Discussion and Conclusions

## Funding

## Conflicts of Interest

## Appendix A

_{tj}at the interface between layers j and j + 1 (j = 1,2). The first equation follows from the bending problems of the three layers which undergo equal deflections w, and then equal rotations φ = −w′, because of the assumption of perfect connections in the transverse direction, and can be written as:

_{tj}and slip Δs

_{tj}at the j-th interface with interfacial coefficients A

_{j}and B

_{j}(j = 1, 2):

_{zi}and q

_{yi}, are applied (i = 1, …, 3). The system of differential Equations (A1), (A3), and (A5) admits closed form solutions for all possible combinations of the regimes experienced by the two interfaces. In [7], the solution to the case of three-layer beams uniformly loaded and having two imperfect interfaces or only one imperfect interface is detailed. Closed form expressions for axial displacements, interlayer slips, and all the remaining kinematic and static unknowns (deflection, internal forces, and interfacial normal tractions) are derived as well.

_{zi}= q

_{yi}= 0 with i = 1, …, 3) and under the assumption of no interlayer slip at the lower interface (Δs

_{t2}= 0), Equation (A3b) gives:

_{t1}= 0), Equations (A3a) and (A6) give:

_{1}= B

_{1}= 0) we have:

_{h}(h = 1, …, 3) as three arbitrary constants. It is straightforward that interfacial shear tractions are obtained, substituting Equation (A10) in Equations (A6) and either (A8) or (A9).

_{4}is one more arbitrary constant and, for brevity, hereinafter, $\int}\xb7\mathrm{d}z$ indicates integration of ∙ with respect to z.

_{h}(h = 5, 6).

_{t1}= 0:

_{h}(h = 7, 8).

_{t2}= 0:

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**Figure 2.**Composite beam assumed in the present model: Free-body diagram of an infinitesimal element.

**Figure 3.**Simply supported composite sandwich beam case study: Geometry, constraining and loading conditions.

**Figure 4.**Simply supported composite sandwich beam case study: Upper interface generic configurations (

**a**) for d + a ≤ l/2 and (

**b**) for d + a > l/2.

**Figure 5.**Simply supported sandwich beam with a debond at one end of the upper interface: Midspan deflection vs. debond length for varying layer material and thickness (Ω = 1.5 and 2.5).

**Figure 6.**Simply supported sandwich beam with a debond at one end of the upper interface: Midspan deflection vs. debond length for varying layer material and thickness (Ω = 5 and 10).

**Figure 7.**Simply supported sandwich beam with a debond within the upper interface: Midspan deflection vs. debond distance from the left end for varying layer thickness and debond length (Ω = 1 and 5).

**Figure 8.**Simply supported sandwich beam with a debond within the upper interface: Midspan deflection vs. debond distance from the left end for varying layer thickness and debond length (Ω = 10).

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**MDPI and ACS Style**

Monetto, I.
The Effects of an Interlayer Debond on the Flexural Behavior of Three-Layer Beams. *Coatings* **2019**, *9*, 258.
https://doi.org/10.3390/coatings9040258

**AMA Style**

Monetto I.
The Effects of an Interlayer Debond on the Flexural Behavior of Three-Layer Beams. *Coatings*. 2019; 9(4):258.
https://doi.org/10.3390/coatings9040258

**Chicago/Turabian Style**

Monetto, Ilaria.
2019. "The Effects of an Interlayer Debond on the Flexural Behavior of Three-Layer Beams" *Coatings* 9, no. 4: 258.
https://doi.org/10.3390/coatings9040258