Vibration behaviours of multi-debonded sandwich beams with symmetric angle-ply facesheets

Abstract

Sandwich beams, comprising two stiff facesheets bonded to a core, are light and strong and see widespread application. However, debonded regions between the facesheets and the core can significantly reduce the strength and have an impact on the vibration behaviour. In this study, the previous literature was referred to, and finite element simulations were conducted using ANSYS on debonded sandwich beams with [0°]₄, [+45°/−45°]_S and [90°]₄ carbon fibre reinforced polymer facesheets. The effects of the configurations of the three debonded regions and facesheet stacking orientations were studied. The most important finding is that if debonds cover vibration nodes, the natural frequencies drop more significantly because the stiffness decreases more. It was determined that when there were debonded regions, the natural frequencies of the bending modes of the beams using [0°]₄ facesheets decreased more compared to those of the beams using [+45°/−45°]_S and [90°]₄ facesheets; while the trends are opposite for the natural frequencies of torsion modes of the beams using [+45°/−45°]_S facesheets. Reducing the natural frequency can cause a structure to vibrate at the resonance frequency, leading to structural failure. This study contributes to furthering our understanding of how different debonds and facesheet stacking orientations affect the vibrations of sandwich beams.

Keywords

Composite sandwich beams debonding vibration finite element modelling composite laminates

Introduction

Composite sandwich structures, comprising a light core and two strong and stiff facesheets, are being widely used for various applications, such as wind turbine blades, aircraft wings and ship decks. While the core of sandwich structures is usually thick and light, the facesheets are thin and strong, rendering these structures with a low density and high flexural stiffness and, thus, the ability to withstand heavy loads. Fibre reinforced polymers (FRPs), such as carbon fibre reinforced polymers (CFRPs), are often used as facesheets. Because sandwich structures comprise at least two different materials, the mechanical behaviours of composite structures are more complicated and, thus, widely studied.^1–3 It was found that the stacking orientations of FRP laminates can significantly affect the mechanical behaviours, e.g. flexural stiffness, of sandwich structures.^4,5

To prevent the failure of sandwich structures, it is essential to perfectly bond the facesheets and the core. Face-core interfacial debonding is a critical issue that causes the failure of sandwich structures^6,7; this is because debonding at the adhered regions between the facesheets and the core may occur due to manufacturing flaws or in-service damage.^8–11 Debonded regions can lead to crack growth, strength and stiffness reduction, and failure. The flexural rigidity, stiffness and strength reductions are more significant for a longer debonded region.^12,13

Vibration is a key factor in structural applications because large displacements, damage and structural failure can be induced by resonance. Since sandwich structures comprise at least two different materials, their vibration behaviours are more complicated. Various approaches have been explored to analyse the vibration behaviours of these structures and predict the mode shapes and natural frequencies.^14–22 A spline finite strip method was developed to analyse the natural frequencies of rectangular sandwich plates; the results were in sound agreement with those of other studies.¹⁵ It was found that the fundamental frequencies increase with an increase in the skew angles of the parallelogram sandwich plates.^18,21 The modified displacement field expressions used to analyse the free vibrations of cylindrical sandwich structures contributed to rapidly converging and accurate results.²²

Among the various methods developed for analysing the vibrations of sandwich structures, finite element analysis is one of the most studied approaches. A higher-order layerwise finite element model was developed and used to predict the free vibration behaviours of multilayer sandwich plates. The designed model is simpler than previous ones and can predict accurate results.¹⁷ Additionally, an improved higher-order one-dimensional beam element was proposed to derive reasonably accurate results as three-dimensional elements while significantly reducing the computation cost.²³ Through finite element analysis, it was found that facesheet stacking orientations have a significant impact on the stiffness of the structures, thereby impacting the natural frequencies and mode shapes.^{17,18,24–26} It was also found that the geometry and curvature of the structures can significantly influence vibration behaviours.^25–27 Moreover, increasing the layers of the cores or using multiple cores can also affect the natural frequencies of sandwich structures significantly.^28,29

Since debonded regions in sandwich structures can reduce system stiffness, the vibration behaviours of sandwich structures are also affected by debonded regions. The effects of size and location of debonded regions on the vibrations of sandwich structures have been widely studied and are deemed essential factors.^27,30,31 Since a longer debonded region results in greater stiffness reduction, the structure’s natural frequencies decrease more significantly.^13,32–34 If two debonded regions are located symmetrically with respect to the middle plane of the structure, the natural frequencies can reduce further.³² The curvature of structures can also influence vibration behaviours.^25,27,31,35 Compared to flat structures, it was revealed that the mode shapes and natural frequencies of the lateral and torsion modes of curved sandwich structures are affected more significantly by debonded regions.^25–27,31

For practical applications, operating structures or devices near low natural frequencies should be avoided because resonance is more likely to be excited at these frequencies, which can cause large displacements and lead to failure. To prevent structural resonance, measuring the vibration behaviours of debonded structures can help identify damaged areas and locations, as a reduction in stiffness affects natural frequencies and mode shapes. Therefore, the measurement of vibration behaviours can be used as a non-destructive testing technique.^36–38

The locations and sizes of debonded regions can significantly influence the vibration behaviours of sandwich structures. Also, according to the literature, stiffness and natural frequencies drop more significantly when the debonded region is longer than about one-fifth of the beam’s length.¹³ However, the effects of multi-debonding have been less discussed. It was revealed that the shortest length of a single debond is equal to one-fifth (40 mm) of the beam’s length (200 mm) and that the longest length is up to two-fifths (80 mm) of the beam’s length.^24,26 Furthermore, a single 80-mm debond was divided into two 40-mm debonds, and the effects of the locations of the two debonds on the vibration behaviours of sandwich structures were investigated.^24,26 To further explore the effects of debonds, this study examined the effects of three 40-mm debonds on the mode shapes, natural frequencies and vibration nodes of sandwich structures with symmetric angle-ply facesheets. The aim was to identify how these factors affect the vibrations of composite sandwich beams.

Materials and material properties

The sandwich beams comprised an epoxy core and two CFRP facesheets. The epoxy resin, Araldite LY 556 from Huntsman, UK, was crosslinked and cured using Albidur HE 600, a curing agent supplied by Evonik, Germany. The CFRP facesheets were manufactured using the unidirectional carbon fibre fabric Gurit UT-C300, supplied by Marineware, UK. The glass transition temperature of the resin system was about 150°C.^39,40

Tensile and density tests were conducted on the materials according to ASTM standards, and the densities, tensile moduli, shear moduli and Poisson’s ratios were measured. The measured results are indicated in Table 1.^24–26 The values were used in finite element simulation. The dimensions of the sandwich beams are listed in Table 2.

Table 1.

Materials and mechanical properties.⁶

	Density, ρ (kg/m³)	Modulus (GPa)	Poisson’s ratio
Bulk epoxy	1154	E = 2.8	ν = 0.35
CFRP of 0° lay-up	1557	E₁ = 122.6, E₂ = 6.7,	ν₁₂ = 0.33
CFRP of 0° lay-up	1557	G₁₂ = 4.5, G₂₃ = 2.3	ν₂₃ = 0.48

Note: E₂ = E₃, ν₁₂ = ν₁₃, G₁₂ = G₁₃.

Table 2.

Sandwich model dimensions.

	Length (mm)	Width (mm)	Thickness (mm)
Core	200	25	6
Facesheet CFRP			1.3/Piece
Sandwich model			8.6

In this study, the stacking orientations of the facesheet CFRP were [0°]₄, [+45°/−45°]_S (symmetric angle-ply) and [90°]₄. The bending stiffness matrix, D, of the sandwich beams, using the three types of facesheets, has also been calculated and reported.^24–26 The values are listed below:

D_{0 \times 4} = [\begin{array}{l} 4373.96 & 97.84 & 0 \\ 97.84 & 293.02 & 0 \\ 0 & 0 & 175.49 \end{array}] (GPa - {mm}^{3})

(1)

D_{\pm 45 S} = [\begin{array}{l} 1391.15 & 1040.18 & 8.00 \\ 1040.18 & 1391.15 & 8.00 \\ 8.00 & 8.00 & 1117.82 \end{array}] (GPa - {mm}^{3})

(2)

D_{90 \times 4} = [\begin{array}{l} 292.20 & 97.56 & 0 \\ 97.56 & 4373.14 & 0 \\ 0 & 0 & 175.22 \end{array}] (GPa - {mm}^{3})

(3)

where the subscripts “0 × 4”, “±45S” and “90 × 4” denote the sandwich beams using [0°]₄, [+45°/−45°]_S and [90°]₄ facesheets, respectively. The components in the matrices are indicated using the terms, d_ij, where i and j are 1, two and 6.

Finite element simulation

The commercial finite element simulation software ANSYS was used to simulate the natural frequencies and mode shapes of the debonded sandwich beams. The engineering data were set up based on the measured mechanical properties listed in Table 1. The dimensions of the developed three-dimensional models are listed in Table 2. While the debonded regions between the facesheets and the core were modelled using the “frictionless contact” function to prevent the facesheets from penetrating the core, the bonded regions were modelled using the “bonded contact” function. To investigate the effects of friction between the facesheets and the core on the vibration behaviours, “friction contact” (with different coefficients of friction [μ]) was also introduced into the debonded regions.

Each facesheet of the sandwich model comprised a 4-plies CFRP laminate. The stacking orientations were [0°]₄, [+45°/−45°]_S and [90°]₄ by assigning different orientation angles to every ply. Free-free boundary conditions were set, and the modal analysis module, with the Block Lanczos method, was used to simulate the vibration behaviours of the intact and debonded sandwich beams. A convergence study was also performed on the perfectly bonded model. A linear 8-node SOLID185 cuboid element was assigned for meshing. The elements of the core had dimensions of 1 mm × 1 mm × 1 mm (length × width × thickness), while the elements of the facesheets had dimensions of 1 mm × 1 mm × 0.325 mm. The mesh resulted in 70,000 elements (20,000 for each facesheet and 30,000 for the core) and was adopted for subsequent simulations.

In this study, the sandwich beams with three 40-mm debonds with different separations (Δ) were studied. The debonded regions were located between the facesheets and the core, i.e., the adhered interfaces in real beams. The centre of separation was aligned with the centre of the beam and has a length of either 20, 40 or 60 mm. Models of the sandwich beams and the locations of the debonded regions are shown in Figure 1.

Figure 1.

Models of sandwich beams and debonding locations (shown in green). (a) Intact (no debond), and (b) three 40-mm debonds.

The simulated natural frequencies of the intact sandwich beams were compared with those found by previous studies that used Abaqus, as shown in Table 3.^24,26 In previous studies, the maximum difference between the results from the hammer impact tests and the simulation was about 10% (see column 5 in Table 3. The results of this study were in sound agreement with the reported results simulated by Abaqus: the maximum difference was less than 0.5% (marked with an asterisk in column 7). Also, the differences between the results of this study and the reported results from the tests were in good agreement with the differences between the reported results from Abaqus and the tests (see columns 5 and 8 in Table 3. For the same mode, the absolute values of the difference percentage in columns 5 and 8 were close but the signs were opposite because column 5 was the test compared to Abaqus, while column 8 was ANSYS (this study) compared to the test. The reasons for the differences between the test and simulation results could be experimental errors, such as the materials and the test set-up and simulation factors such as the element type used.

Table 3.

Simulated natural frequencies compared to reported values.

mode	Vibration mode shape	Reported (Abaqus) (Hz)^24,26	Reported (test) (Hz)^24,26	Difference (test to Abaqus) (%)^24,26	This study (ANSYS) (Hz)	Difference (this study to Abaqus) (%)	Difference (this study to test) (%)
Intact, [0°]₄ facesheets
1	1st bending	1645.8	1512.9 (11.9)	−8.1	1646.9	0.1	8.9
2	1st torsion	2095.4	1880.9 (87.2)	−10.2	2103.4	0.4^a	11.8
3	1st lateral	2986.4	2910.7 (22.9)	−2.5	2988.9	0.1	2.7
4	2nd bending	3890.3	3603.1 (199.8)	−7.4	3893.2	0.1	8.1
5	2nd torsion	4273.6	3921.2 (145.3)	−8.2	4290.0	0.4^a	9.4
Intact, [+45°/−45°]_S facesheets
1	1st bending	655.8	616.8 (3.1)	−5.9	655.2	−0.1	6.2
2	1st lateral	1431.9	1394.5 (9.2)	−2.6	1430.3	−0.1	2.6
3	2nd bending	1759.9	1616.4 (81.3)	−8.2	1758.8	−0.1	8.8
4	3rd bending	3320.6	3224.4 (104.1)	−2.9	3319.7	0.0	3.0
5	1st torsion	3435.7	3212.2 (321.5)	−6.5	3448.9	0.4^a	7.4
Intact, [90°]₄ facesheets
1	1st bending	452.3	447.5 (4.4)	−1.1	453.1	0.2	1.2
2	1st lateral	1092.6	1134.4 (3.5)	3.8	1092.5	0.0	−3.7
3	2nd bending	1223.7	1227.7 (10.6)	0.3	1225.9	0.2	−0.1
4	1st torsion	2072	1818.9 (84.4)	−12.2	2079.9	0.4^a	14.3
5	3rd bending	2336.4	2460.3 (191.7)	5.3	2341.1	0.2	−4.8

^aMaximum difference between the results of this study and the reported simulation results.

Since both the simulated results showed good consistency, the simulation method used in this study was validated. The simulation methodology was then used for simulating the beams with the debonded regions.

Results and discussion

This study investigated the effects of CFRP facesheet stacking orientations ([0°]₄, [+45°/−45°]_S and [90°]₄) and the configurations of three face-core interfacial debonded regions on the vibration behaviours of sandwich beams using finite element simulations. The mode shapes, natural frequencies and locations of vibration nodes are discussed in this section. To clarify the effects on the natural frequencies, the ratios of the debonded beams’ natural frequencies to the intact beams’ natural frequencies were used and presented in this study (i.e., relative frequency, f_n/f_n,intact, where n indicates the n-th vibration mode).

Intact sandwich beams

The effects of CFRP facesheet stacking orientations on the natural frequencies of the intact beams using [0°]₄, [+45°/−45°]_S and [90°]₄ CFRP facesheets are briefly discussed in this section.

Figure 2 shows the first five mode shapes of the intact sandwich beam using [+45°/−45°]_S facesheets. The first five modes and natural frequencies of the intact sandwich beams using the three types of CFRP facesheets are listed in Table 3. Notably, different facesheet stacking orientations resulted in different sequences and natural frequencies for the same mode. The natural frequencies of the first and second bending modes of the beam using [0°]₄ facesheets (1646.9 Hz and 3893.2 Hz) were higher than those of the beam using [+45°/−45°]_S facesheets (655.2.0 Hz and 1758.8 Hz) and [90°]₄ facesheets (453.1 Hz and 1225.9 Hz). Additionally, the natural frequency of the first torsion mode of the beam using [+45°/−45°]_S facesheets (2696.7 Hz) was higher than that of the beam using [0°]₄ facesheets (2103.4 Hz) and [90°]₄ facesheets (2079.9 Hz).

Figure 2.

Mode shapes of the intact sandwich beam using [+45°/−45°]_S facesheets. (a) Mode 1: first bending. (b) Mode 2: first lateral. (c) Mode 3: second bending. (d) Mode 4: third bending and (e) Mode 5: first torsion.

These differences stemmed from the anisotropic mechanical properties of CFRP facesheets. As shown in the bending stiffness matrices (see equations (1)–(3)), the d₁₁ term in equation (1) is greater than that in equations (2) and (3) (4373.96 GPa-mm³ vs 1391.15 GPa-mm³ and 292.20 GPa-mm³), meaning that the beam using [0°]₄ facesheets had a higher bending stiffness along the longitudinal direction and, thus, contributed to the higher natural frequencies of the bending modes. Meanwhile, the d₆₆ term in equation (2) is greater than that in equations (1) and (3) (1117.82 GPa-mm³ vs 175.49 GPa-mm³ and 175.22 GPa-mm³), meaning that the beam using [+45°/−45°]_S facesheets had a higher twisting stiffness and, thus, resulted in higher natural frequencies of the torsion modes.

Comparison with single and double debonds

Figure 3(a) compares the relative natural frequencies of the beams with three debonds (this study) to those of the beams with single or double debonds (previous studies).^24,26 The stacking orientation of the facesheets of the compared beams was [0°]₄. The models of the beams with single or double debonds are shown in Figure 3(b)–(e). The beams with two 40-mm debonds had the debonds at the same side, diagonal locations (D) and symmetric locations (S). The separation (Δ) between the two debonds was 20 mm.

Figure 3.

(a) Relative natural frequencies of the intact sandwich beam and the beams with one, two and three debonds and the models of sandwich beams with debonds: (b) single 40-mm, (c) two 40-mm on the same side, (d) two 40-mm at diagonal locations (d) and (e) two 40-mm at symmetrical locations (S). (b)–(e) are reproduced from previous studies.^24,26

According to previous studies, reductions in natural frequencies have not shown the same trend. For example, the natural frequency of the first bending mode decreased when it changed from “single 40 mm” to “double Δ20”, while the natural frequency of the second bending mode increased for the same change in debond configuration. As revealed by previous studies, in addition to the number of debonds, the vibration nodes being overlapped by debonds can also affect the stiffness of the beam and, thus, the natural frequencies.^24–26

When the number of debonds increased from two to three, except for the first lateral mode, all the modes showed reductions in the natural frequencies (see Figure 3(a)). This is because the total length of the debonded regions increased, which led to more reduction in the stiffness, thereby causing the natural frequencies to decrease more significantly. The first lateral mode showed no significant change in the natural frequencies since the lateral vibration was not sensitive to the horizontal debonds.^24–26

Effects of friction at the debonded regions

To simulate a more realistic condition that there can be friction between the facesheets and the core, different coefficients of friction (μ) were introduced into the debonded regions. The effects of friction on the relative natural frequencies of the beams with three debonds are shown in Figure 4. The facesheet stacking orientation of the beams was [0°]₄. Notably, when there was no friction, the natural frequencies decreased the most for the bending and torsion modes; however, when μ was increased slightly to 0.01, the natural frequencies of the bending and torsion modes increased significantly. When μ was further increased to 0.05, the relative natural frequencies became almost equal to 1, i.e., the frequencies of the debonded beam became the same as the intact beam. On the other hand, friction had little effect on the relative natural frequencies of the lateral modes because the lateral modes were not sensitive to the debonds.

Figure 4.

Effects of coefficient of friction on the relative natural frequencies of the triple-debonded beams using [0°]₄ facesheets.

The reason for the increase in the natural frequencies with an increase in friction is that friction limited the relative movements between the facesheets and the core at the debonded regions. Since the beams could not have relative movements at the debonded regions, their stiffness increased. Therefore, the natural frequencies also increased.

Effects of increasing the separation between the three debonds

Figure 5 shows the frequency ratios of the sandwich beams with three debonds. The beam model is shown in Figure 1(b). The beams with three debonds were left-right asymmetric, resulting in the asymmetry of the mode shapes. According to Figure 5, when Δ was increased, the relative frequencies of the debonded beams using the three types of facesheets showed the same trend for the same mode.

Figure 5.

Relative natural frequencies of the sandwich beams with three debonds. (a) [0°]₄, (b) [+45°/−45°]_S and (c) [90°]₄ facesheets.

Notably, for the same bending mode and same Δ, the frequency ratios of the beam using [0°]₄ facesheets reduced more compared to those of the beams using [+45°/−45°]_S and [90°]₄ facesheets. Moreover, for the torsion modes, the frequency ratios of the beam using [+45°/−45°]_S facesheets decreased more compared to those of the beams using [0°]₄ and [90°]₄ facesheets. Although [0°]₄ facesheets provided the highest bending stiffness (i.e., d₁₁ in equation (1), 4373.96 GPa-mm³), when there were debonded regions, the reduction in the bending stiffness was more significant, resulting in greater reductions in the natural frequencies of the bending modes. For the same reason, [+45°/−45°]_S facesheets provided the highest twisting stiffness (d₆₆ in equation (2), 1117.82 GPa-mm³). When there were debonded regions, the reduction in the twisting stiffness was more significant, resulting in greater reductions in the natural frequencies of the torsion modes.

In this study, if a vibration node was not covered by any debond and was situated more than 5 mm away from the edges of the debond, it was defined as not covered (NC); if a node was not covered but was located at a distance of within 5 mm from the edges of the debond, it was defined as near debond (ND); if a node was covered by a debond and was located within 5 mm from the edges of the debond, it was defined as just covered (JC); and if a node was covered by a debond and was located 5 mm away from the edges of the debond, it was defined as fully covered (FC).

When Δ increased from 20 to 60, the frequencies of the first bending mode of the beams using all types of the facesheets showed decreasing trends (see Figure 5). This can be attributed to the overlaps between the nodes and the debonds. For example, for the beam using [0°]₄ facesheets, when Δ increased from 20 mm to 40 mm, the right node changed from JC (Figure 6(a)) to FC (Figure 6(b)) by the right debond, while the left node remained FC by the left debonds. For the beam using [+45°/−45°]_S facesheets, when Δ increased from 20 mm to 60 mm, the two nodes changed from ND (Figure 6(c)) to FC (Figure 6(d)) by the debonds. Also, for the beam using [90°]₄ facesheets, the two nodes also became FC by the debonds with an increase in Δ from 20 mm to 60 mm, see Figure 6(e) and (f).

Figure 6.

Mode shapes and nodes of the first bending mode of the sandwich beams with three debonds. (a) [0°]₄, Δ20. (b) [0°]₄, Δ40. (c) [+45°/−45°]_S, Δ20. (d) [+45°/−45°]_S, Δ60. (e) [90°]₄, Δ20, and (f) [90°]₄, Δ60.

For vibration, the nodes were located where the structures had no displacements. However, when the debonds covered the vibration nodes, the facesheets and the core near these regions could exhibit relative movement, which led to a significant reduction in the bending stiffness and, thus, resulted in greater reductions in the natural frequencies.

For the second bending mode of the beams using all types of facesheets, the frequencies first increased (from Δ20 to Δ40) and then decreased (Δ40 to Δ60), see Figure 5. When Δ increased from 20 mm to 40 mm, the overlapping conditions of the left and right nodes remained uncovered by any debond. However, the overlapping condition of the centre node of the beam using [0°]₄ facesheets changed from FC (Figure 7(a)) to JC (Figure 7(b)) by the two debonds on the left. The overlapping condition of the centre node of the beam using [+45°/−45°]_S facesheets changed from JC (Figure 7(d)) to NC (Figure 7(e))by the two debonds on the left, and that of the centre node of the beam using [90°]₄ facesheets remained uncovered by any debond (see Figure 7(g) and (h)). Therefore, the relative movements between the facesheets and core around the centre nodes became smaller or restricted. Therefore, the natural frequencies of the second bending mode of the beams increased.

Figure 7.

Mode shapes and nodes of the second bending mode of the sandwich beams with three debonds. (a) [0°]₄, Δ20. (b) [0°]₄, Δ40. (c) [0°]₄, Δ60. (d) [+45°/−45°]_S, Δ20. (e) [+45°/−45°]_S, Δ40. (f) [+45°/−45°]_S, Δ60. (g) [90°]₄, Δ20. (h) [90°]₄, Δ40, and (i) [90°]₄, Δ60.

However, for the second bending mode, when Δ was increased further from 40 mm to 60 mm, the centre nodes of all the beam types did not get covered by any debond. However, the left and right nodes of the beam using [0°]₄ facesheets were fully covered by the debonds (see Figure 7(c). Meanwhile, the left and right nodes of the beam using [+45°/−45°]_S facesheets were covered by the two debonds on the left and the debond near the right, respectively (see Figure 7(f)), and the left and right nodes of the beam using [90°]₄ facesheets approached closer to the debonds (see Figure 7(i). This allowed for larger relative movements between the facesheets and the core around the left and right nodes, resulting in reductions in the frequencies.

For the first torsion mode, increasing Δ from 20 mm to 60 mm increased the frequencies (see Figure 5(a) and (b)). For the beam using [0°]₄ facesheets, the overlapping condition of the only node changed from JC by the two debonds on the left (Δ20, Figure 8(a)) to NC (Δ40 and Δ60, Figure 8(b) and (c)) by any debond. For the beam using [+45°/−45°]_S facesheets, the overlapping condition of the node changed from FC by the two debonds on the left (Δ20, Figure 8(d)) to JC (Δ40 and Δ60, Figure 8(e) and (f)) by the two debonds. Additionally, for the beam using [90°]₄ facesheets, the overlapping condition of the node changed from FC by the two debonds on the left (Δ20, Figure 8(g)) to near the two debonds; however, they were NC (Δ60, Figure 8(i)). Therefore, the relative movements between the facesheets and the core around the node became smaller or restricted, resulting in increases in the natural frequencies of the first torsion mode.

Figure 8.

Mode shapes and nodes of the first torsion mode of the sandwich beams with three debonds. (a) [0°]₄, Δ20. (b) [0°]₄, Δ40. (c) [0°]₄, Δ60. (d) [+45°/−45°]_S, Δ20. (e) [+45°/−45°]_S, Δ40. (f) [+45°/−45°]_S, Δ60. (g) [90°]₄, Δ20. (h) [90°]₄, Δ40, and (i) [90°]₄, Δ60.

From the discussion thus far, it is clear that facesheet stacking orientation can impact the locations of vibration nodes. However, by investigating the coordinates of the vibration nodes for each mode and debonding configuration, it was found that compared to the shift amount of the debonds, the changes in locations of the nodes were not significant; this can be observed from Figures 6–8. The effects of facesheet stacking orientation will be further discussed in the next section.

Notably, for the debond configurations used in this study, no significant local vibration mode of the debonded regions was observed, based on the simulated mode shapes, as shown in Figures 6–8. The local modes appeared when the debonded regions became longer or were at a higher mode. Since the length of the debonded regions simulated in this study was 40 mm (one-fifth of the beam length) and since the modes discussed were the lowest five ones (which were at lower frequencies), the facesheets and the core at the debonded regions did not exhibit local vibration modes.

Effects of facesheet stacking orientation on the node position

Facesheet stacking orientation affects the positions of the vibration nodes, which further affects the reductions in the natural frequencies when there are debonded regions in the sandwich beams. Table 4 summarises the overlapping conditions of the vibration nodes of the first and second bending and the first torsion modes of the debonded beams explored by this study. For the modes with more than one node, the node sequence was from left to right, as shown in Figures 6–8. Also, as previously defined, the abbreviations of the overlapping conditions from not covered to fully covered are NC (not covered), ND (near debond), JC (just covered) and FC (fully covered).

Table 4.

Overlapping conditions of the nodes (from left to right) of the sandwich beams with three debonds.

		Δ20	Δ40	Δ60
1st bending (2 nodes)	[0°]₄	FC/JC	FC/FC	FC/FC
	[+45°/−45°]_S	ND/ND	FC/FC	FC/FC
	[90°]₄	ND/ND	FC/FC	FC/FC
2nd bending (3 nodes)	[0°]₄	NC/FC/NC	NC/JC/NC	FC/NC/FC
	[+45°/−45°]_S	NC/JC/NC	NC/NC/NC	JC/NC/ND
	[90°]₄	NC/NC/NC	NC/NC/NC	ND/NC/ND
1st torsion (1 node)	[0°]₄	JC	NC	NC
	[+45°/−45°]_S	FC	JC	JC
	[90°]₄	FC	JC	ND

NC: not covered, ND: near debond, JC: just covered, FC: fully covered.

By investigating the overlapping conditions in Table 4, it was found that for the bending modes of the beams using [0°]₄ facesheets, these nodes showed a tendency to be located closer to the debonds compared to the nodes of the beams using [+45°/−45°]_S and [90°]₄ facesheets. For the first bending mode of the beams with Δ20, the overlapping conditions of the two nodes of the beam using [0°]₄ facesheets were FC and JC, while all of the overlapping conditions of the nodes of the beams using [+45°/−45°]_S and [90°]₄ facesheets were ND. For the second bending mode with Δ20, the centre node of the beam using [0°]₄ facesheets was FC, that of the beam using [+45°/−45°]_S facesheets was JC and that of the beam using [90°]₄ facesheets was NC. Also, for the 2nd bending mode with Δ60, the left and right nodes of the beam using [0°]₄ facesheets were FC, and those of the beams using [+45°/−45°]_S and [90°]₄ facesheets were JC or ND.

On the other hand, for the torsion modes, the nodes of the beam using [0°]₄ facesheets tended to be located farther away from the debonds compared to those of the beams using [+45°/−45°]_S facesheets. For example, for the first torsion mode of the beams with Δ20, the node of the beam using [0°]₄ facesheets was JC, while the node of the beam using [+45°/-45°]_S facesheets was FC (see Table 4).

Since overlapping between the debonds and vibration nodes results in greater reductions in natural frequencies, facesheet stacking orientations that affect the positions of vibration nodes can also have an impact on the reductions in natural frequencies. The nodes of bending modes of the beams using [0°]₄ facesheets tended to be closer to or covered by the debonds, which permitted more significant relative movement between the facesheets and the core around the nodes, thus resulting in a more significant reduction in the bending stiffness. Therefore, the natural frequencies of bending modes of the beams using [0°]₄ facesheets decreased more when there were debonds; in contrast, the nodes of the torsion modes of the beams using [+45°/−45°]_S facesheets tended to be closer to or covered by the debonds. Therefore, the natural frequencies of the torsion modes of the beams using [+45°/−45°]_S facesheets decreased more when there were debonds.

Conclusions

Sandwich structures comprising two stiff facesheets and a lightweight core are widely used. However, delamination or debonds in sandwich structures can reduce stiffness and strength, affecting the vibration behaviours of such structures. This study conducted finite element simulations to investigate the effects of three face-core interfacial debonds and facesheet stacking orientations. It was found that when the separation between the debonds was increased, the three facesheet stacking orientations ([0°]₄, [+45°/−45°]_S and [90°]₄) show the same trend.

The most significant effect of maintaining a separation between the debonds was the location of the vibration nodes. Increasing the separation resulted in the movement of the vibration nodes. When the debonds moved and covered the nodes, there was a larger relative movement between the facesheets and the core at the regions around the nodes. A larger relative movement reduced the stiffness, resulting in greater reductions in the natural frequencies. On the other hand, if the debonds moved but did not cover the nodes, the natural frequencies increased because the relative movement was restricted.

The facesheet stacking orientation also affected the locations of the vibration nodes. For the bending modes, the nodes of the beams using [0°]₄ facesheets tended to be closer to and covered by the debonds. Therefore the natural frequencies of the bending modes of the beams using [0°]₄ facesheets dropped more significantly. Meanwhile, for the torsion modes, the nodes of the beams using [+45°/−45°]_S facesheets tended to be closer to and covered by the debonds. Therefore the natural frequencies of the torsion modes of the beams using [+45°/−45°]_S facesheets showed more significant reductions.

This study investigated how the vibration nodes and the natural frequencies of composite sandwich beams are affected by different debonding configurations and facesheet stacking orientations. In conclusion, overlapping between the debonded regions and the vibration nodes results in greater reductions in the natural frequencies.

Footnotes

Acknowledgements

The author would like to thank the National Science and Technology Council, Taiwan, for supporting this research under contract No. 112-2221-E-110-058.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by National Sun Yat-sen University, Taiwan, under projects No. 12DS02 and 12C030105, National Science and Technology Council, for supporting this research under contract No. 112-2221-E-110-058.

ORCID iD

Shang-Nan Tsai

Data availability statement

The raw/processed required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

References

Jin

Yao

. Hygrothermal analysis of laminated composite plates in terms of an improved c0-type global–local model. Aero Sci Technol 2017; 63: 328–343.

Gibson

. A review of recent research on mechanics of multifunctional composite materials and structures. Compos Struct 2010; 92(12): 2793–2810.

Cameron

Wennhage

Göransson

, et al. Structural-acoustic design of a multi-functional sandwich panel in an automotive context. J Sandw Struct Mater 2010; 12(6): 684–708.

Steeves

Fleck

. Material selection in sandwich beam construction. Scripta Mater 2004; 50(10): 1335–1339.

Mudassir

Shankar

Hussain

. Flexural behavior of sandwich composite panels under 4-point loading. Int J Mater Sci 2016; 11(1): 47–55.

Hohe

Becker

. Assessment of the delamination hazard of the core face sheet bond in structural sandwich panels. Int J Fract 2001; 109(4): 413–432.

Xue

Wei

, et al. Face-core interfacial debonding characterization model of an all-composite sandwich beam with a hexagonal honeycomb core. Eng Fract Mech 2022; 269: 108554.

Beaumont

. The structural integrity of composite materials and long-life implementation of composite structures. Appl Compos Mater 2020; 27(5): 449–478.

Singher

Segal

Shamir

. Interaction of a guided wave with a nonuniform adhesion bond. Ultrasonics 1997; 35(5): 385–391.

10.

Brighenti

Carpinteri

Scorza

. Mechanics of interface debonding in fibre-reinforced materials. J Compos Mater 2016; 50(19): 2699–2718.

11.

Safri

SNA

Sultan

MTH

Jawaid

, et al. Impact behaviour of hybrid composites for structural applications: a review. Compos B Eng 2018; 133: 112–121.

12.

Mouritz

Thomson

. Compression, flexure and shear properties of a sandwich composite containing defects. Compos Struct 1999; 44(4): 263–278.

13.

Kim

Hwang

. Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams. Compos Struct 2002; 55(1): 51–62.

14.

Wang

Ang

Yang

, et al. Free vibration of skew sandwich plates with laminated facings. J Sound Vib 2000; 235(2): 317–340.

15.

Yuan

Dawe

. Free vibration of sandwich plates with laminated faces. Int J Numer Methods Eng 2002; 54(2): 195–217.

16.

Liu

Zhong

Shi

, et al. Free and forced vibration analysis of pyramid lattice sandwich plate based on the dimensional reduction model. Thin-Walled Struct 2022; 174: 109155.

17.

Belarbi

Tati

Ounis

, et al. On the free vibration analysis of laminated composite and sandwich plates: a layerwise finite element formulation. Lat Am J Solid Struct 2017; 14: 2265–2290.

18.

Dhotre

Srinivasa

. On free vibration of laminated skew sandwich plates: a finite element analysis. Nonlinear Eng 2021; 10(1): 66–76.

19.

Raville

Ueng

. Determination of natural frequencies of vibration of a sandwich plate. Exp Mech 1967; 7(11): 490–493.

20.

Zhai

Liang

. Damping properties analysis of composite sandwich doubly-curved shells. Compos B Eng 2019; 161: 252–262.

21.

Srinivasa

Suresh

Prema Kumar

. Experimental and finite element studies on free vibration of skew plates. Int J Adv Struct Eng 2014; 19(1): 365–377.

22.

Yang

Jin

Liu

, et al. Vibration and damping analysis of thick sandwich cylindrical shells with a viscoelastic core under arbitrary boundary conditions. Int J Mech Sci 2015; 92: 162–177.

23.

Hui

Giunta

Belouettar

, et al. A free vibration analysis of three-dimensional sandwich beams using hierarchical one-dimensional finite elements. Compos B Eng 2017; 110: 7–19.

24.

Tsai

Taylor

. Vibration behaviours of single/multi-debonded composite sandwich structures with nanoparticle-modified matrices. Compos Struct 2019; 210: 590–598.

25.

Tsai

Taylor

. Vibration behaviours of single/multi-debonded curved composite sandwich structures. Compos Struct 2019; 226: 111291.

26.

Tsai

. The effect of debonding on the vibration behaviour of CFRP/epoxy sandwich panels. PhD thesis. London, UK: Imperial College London, 2019.

27.

Baba

. Free vibration analysis of curved sandwich beams with face/core debond using theory and experiment. Mech Adv Mater Struct 2012; 19(5): 350–359.

28.

, et al. Vibration characteristics of novel multilayer sandwich beams: modelling, analysis and experimental validations. Mech Syst Signal Process 2020; 142: 106799.

29.

Selvaraj

Maneengam

Sathiyamoorthy

. Characterization of mechanical and dynamic properties of natural fiber reinforced laminated composite multiple-core sandwich plates. Compos Struct 2022; 284: 115141.

30.

Mujumdar

Suryanarayan

. Flexural vibrations of beams with delaminations. J Sound Vib 1988; 125(3): 441–461.

31.

Baba

Thoppul

. Experimental evaluation of the vibration behavior of flat and curved sandwich composite beams with face/core debond. Compos Struct 2009; 91(1): 110–119.

32.

Della

Shu

. Free vibration analysis of composite beams with overlapping delaminations. Eur J Mech A Solids 2005; 24(3): 491–503.

33.

Burlayenko

Sadowski

. Nonlinear dynamic analysis of harmonically excited debonded sandwich plates using finite element modelling. Compos Struct 2014; 108: 354–366.

34.

Burlayenko

Sadowski

. Finite element nonlinear dynamic analysis of sandwich plates with partially detached facesheet and core. Finite Elem Anal Des 2012; 62: 49–64.

35.

Zhang

Qing

. Free vibration analysis of Euler-Bernoulli curved beams using two-phase nonlocal integral models. J Vib Control 2021; 28: 10775463211022483.

36.

Zimmerman

Simmermacher

Kaouk

. Structural damage detection using frequency response functions. SPIE 2014; 13: 179–184. ISBN 0361-0748.

37.

Pan

Shankar

Singh

, et al. Vibration-based detection of skin-stiffener debonding on composite stiffened panels using surrogate-assisted algorithms. Compos Struct 2021; 270: 114090.

38.

Yang

Gibson

. Nondestructive detection of weak joints in adhesively bonded composite structures. Compos Struct 2001; 51(1): 63–71.

39.

Awang Ngah

Taylor

. Toughening performance of glass fibre composites with core–shell rubber and silica nanoparticle modified matrices. Composites Part A Appl Sci Manuf 2016; 80: 292–303.

40.

Lim

Carolan

Taylor

. Simultaneously tough and conductive rubber-graphene-epoxy nanocomposites. J Mater Sci 2016; 51(18): 8631–8644.