Formation and relaxation of residual facesheet dent on foam-core sandwich structures by localized transverse loading

Abstract

A residual facesheet dent on an indented or impacted foam-core sandwich structure significantly affects the residual mechanical properties and the detectability of the damage. However, previous studies have not sufficiently addressed residual dent formation during unloading and the subsequent relaxation. This study quantitatively clarifies the mechanism underlying the formation and the relaxation of a facesheet dent using a segment-wise modeling approach that was formulated for honeycomb sandwich structures in our previous study. We utilize fully backed foam-core sandwich beams under static indentation loading, focusing on the interaction between local deformation of the facesheet and crushing/stretching of the core. This study begins by evaluating the indentation response using quasi-static tests. The indentation behavior is then predicted by extending our previous model. The underlying mechanism of facesheet dent formation and relaxation is quantitatively discussed from the viewpoint of a reaction force from the core to the facesheet.

Keywords

Sandwich foam impact indentation core crushing facesheet dent relaxation modeling

Introduction

Carbon fiber reinforced plastic (CFRP) is used in almost all modern commercial aircraft as a primary structural material. However, the potential capability of CFRP cannot be maximized under the conventional structural design concept, which consists of skins, stringers, and frames. One innovative structural concept is a foam-core sandwich structure [1 –6]. The integral construction consists of two thin facesheets and a lightweight foam core, which can considerably reduce the weight and the number of parts compared to conventional structures. However, since the composite facesheet is very thin and the lightweight foam core is weak, they can be easily damaged when an impact or indentation load is applied [7 –9]. As illustrated in Figure 1, the sandwich structure is deformed globally under localized transverse loading, and the upper facesheet, to which the load is applied, locally deflects against the lower facesheet, followed by through-thickness deformation of the core. When the local deformation exceeds the elastic limit, core crushing and facesheet damage around the loading point occur. In contrast, under unloading, the facesheet deflects upward as the localized load decreases, and the crushed core, whose height in the stress-free state is lower than its uncrushed height, pulls the facesheet down. Finally, the facesheet stops at the position where the deflection of the facesheet and the deformation of the crushed core are in equilibrium. As a result, a residual dent remains after unloading. The core crushing and the facesheet dent significantly degrade the mechanical properties of the structure, even when the dent is small and barely visible [10 –13]. Furthermore, the facesheet dent depth determines the detectability of the damage by visual inspection during structural maintenance [14,15]. Thus, residual dent formation is a key phenomenon under localized loading conditions. Many researchers have investigated the response of composite foam-core sandwich structures to localized transverse loading [16 –22]. Their studies include modeling the indentation response, offering insight into the mechanical behavior during loading, and the key parameters affecting damage growth. In contrast, research on the unloading process has been very limited [8], and the underlying mechanism of dent formation during unloading has not been sufficiently clarified. Furthermore, even though significant relaxation of the residual facesheet dent occurs after unloading [8,23], this relaxation behavior has not been modeled in any study.

This study quantitatively clarifies the mechanism of dent formation and relaxation using a segment-wise modeling approach developed for honeycomb sandwich beams in our previous studies [24,25]. We utilize fully backed foam-core sandwich beams under indentation loading, focusing on the interaction between the local facesheet deformation and the core crushing and stretching. This study begins by evaluating the indentation response using quasi-static indentation tests. The indentation behavior is then predicted by extending our previous model. The underlying mechanism of facesheet dent formation and relaxation is quantitatively discussed from the viewpoint of the reaction force from the core to the facesheet.

Quasi-static indentation test of foam-core sandwich beams

Materials and methods

Figure 2 presents a schematic of the specimens used for the quasi-static indentation test. The specimens consisted of unidirectional CFRP facesheets (T700S/2592, Toray Industries, Inc., [0₈] or [0₁₂], thickness 1.14 or 1.65 mm, Young’s modulus in fiber direction E_f = 123 GPa), an aerospace-grade foam core (PMI Rohacell WF-51, Evonik Rohm GmbH, thickness 35 mm), and thermosetting adhesive films (AF-163-2 K, 3 M Co.). This study evaluated the effect of facesheet rigidity on the indentation response by using CFRP facesheets with different thicknesses. The CFRP laminates were manufactured in advance. The laminates and the foam core were then secondarily bonded to form a sandwich panel. Beam specimens (width 25 mm) were cut out from the panel using a diamond blade saw, and each specimen was bonded to a flat steel plate with an adhesive (AF-163-2 K, 3 M Co.), eliminating overall bending. A 10-mm-diameter steel cylinder was attached to a material testing system (AG-50kNI, Shimadzu Co.), and a concentrated line-load was applied to the center of each specimen. A constant displacement rate of 5 mm/min was used. After the maximum indentation displacement of 5 or 2.5 mm was reached, the specimens were unloaded.

Results

Figure 3 presents photographs of a specimen during the test. The core near the facesheet-core interface was crushed, and the damaged area gradually expanded from the loading point to the outer area as the indentation displacement increased. Neither facesheet damage nor core cracking was observed. Finally, a shallow residual facesheet dent remained after unloading. Figure 4 presents the measured indentation load-displacement curves. The curves start to bend after the initial elastic deformation, showing softening behavior, then they increase in an approximately constant ratio as the core crushing evolves. In contrast, during the unloading process, the load rapidly decreased and the residual displacement, which was identical to the dent depth at the loading point, remained after unloading, depending on the maximum applied indentation displacement. The facesheet rigidity was found to affect the indentation load. A specimen with 12-ply facesheets had about 1.2 times greater load than another specimen with 8-ply facesheets. However, the residual deformation was almost the same when the maximum indentation displacement was identical.

After unloading, we measured the change in the dent depth at the loading point over time using a laser displacement meter (LK-030, Keyence Co., Ltd., sampling time 512 µs, spatial resolution 30 µm). Figure 5 plots the time change of the dent depth. Again, two specimens with different facesheet thicknesses exhibited similar behavior. Within the first few minutes after unloading, significant relaxation of the facesheet dent occurred. After 60 min, the relaxation almost stopped, and the dent depth became less than half compared to the depth immediately after unloading. As described above, the final facesheet dent shape not only affects the residual mechanical properties of the indented structure but also determines the detectability of the damage by visual inspection during structural maintenance. Hence, the relaxation behavior after unloading is crucially important.

Simulation of indentation response

Extension of segment-wise model

Next, the indentation loading-unloading characteristics were simulated by an extended “segment-wise model.” The segment-wise model, originally formulated for honeycomb sandwich beams, is based on a deformation theory of an Euler beam on an elastic Winkler foundation [7,24]. As depicted in Figure 6(a), an indentation load P is applied to a sandwich beam and a consequent indentation displacement α is induced. With the back facesheet supported by a rigid facing, an upper facesheet of unit width can be considered as a beam of rigidity D_f supported by a foundation that provides a reaction r(x) per unit length. The equilibrium of the beam is governed by the generalized equation

D_{f} \frac{d^{4} w (x)}{{dx}^{4}} + r (x) = 0

(1)

where w(x) is the transverse deflection of the facesheet as seen in Figure 6(b). When r(x) is positive, the core pushes up the facesheet, and vice versa. Since the transverse deflection of the facesheet induces the same amount of through-thickness deformation of the core, the functional form of r(x) is given by

r (x) = σ (w (x))

(2)

where σ(w(x)) is a relational expression between compressive stress and the through-thickness deformation of the core that can be obtained from flatwise compression-tension tests of the core. In the segment-wise model, it is assumed that the through-thickness stress field of the core is governed by the deformation of adjacent intersection lines, defined as line joints between the honeycomb cell walls (Figure 7) [24]. The honeycomb sandwich beam is divided into several segments centering around the intersection lines. Each segment has a material property determined from the through-thickness deformation of the intersection line within it. The model can calculate the indentation loading-unloading response of honeycomb sandwich beams. Two modifications were made in order to apply this modeling approach to foam-core sandwich beams.

First, the crushing and stretching properties of the foam core were comprehensively evaluated, and the obtained stress-displacement curves were implemented in the model through equation (2). Cylindrical foam specimens with a diameter of 35 mm were cut out from a foam bulk (PMI Rohacell WF-51, Evonik Rohm GmbH, thickness 35 mm) and were transversely loaded in flatwise compression-tension tests [24], with a displacement rate of 5 mm/min using a material testing system (AG-50kNI, Shimadzu Co.). During the tests, the foam specimens were observed by an optical microscope (VH-7000, Keyence Co., Ltd.) to clarify the deformation mechanism of foam cells (Figure 8). One of the obtained compressive stress-displacement curves and micrographs of unit closed cell of the foam are presented in Figures 9 and 10. The stress curve in Figure 9 has several distinct sections depending on the deformation of the foam cells [26]. After elastic compressive deformation (Figure 10 (a)), parts of the cell edges elastically buckled (Figure 10 (b)), resulting in a slight decrease in the stiffness. In the weakest layer of the foam specimen, the buckled cell edges then plastically collapsed (Figure 10 (c)) and the crushing area expanded under constant stress (Figure 11), which was described as a “progressive crushing mechanism” in Ref. [8]. In the subsequent tension process, the crushed layers were gradually stretched under tensile stress, and the plastically collapsed cell edges partially recovered their original shape (Figure 10(d)). This inelastic crushing-stretching behavior of the foam core was implemented in the segment-wise model. Figure 12 presents the utilized stress-displacement curves for the foam core. First, the stress curves were divided into several sections based on the deformation behavior of the foam cells (Figure 10). The curves were then approximated as a set of lines. Thus, the reaction r(x) in equation (2) was determined as

r (x) = σ (w (x)) = k_{i} w (x) + q_{i} when W_{i} \leq w (x) \leq W_{i + 1}

(3)

where W_i is the displacement at which section i starts, and foundation parameters k_i and q_i are the gradient and the intercept of the section line, which depend on w(x). In the indentation simulation, the crushing-stretching property r(x) is implemented in each segment of the sandwich beams, and parameters k_i and q_i are updated as the transverse deformation of facesheet w(x) in each segment increases or decreases. It is important to note that, in indentation unloading, the core crushed during indentation loading is stretched from the deformed state at the end of the loading process. The stretching of the crushed core differs depending on the maximum deformation applied during indentation loading, as seen in Figure 12. Hence, a stretching curve following an arbitrary maximum deformation was estimated by a “complement method” using nodes of the three stretching curves, a method that was proposed and validated in our previous studies [24,25]. Estimated stress curves were used to simulate the unloading process.

Next, the segmentation method was modified. In the segment-wise model for honeycomb sandwich beams, beams were divided into several segments based on the periodic shape of the honeycomb. However, foam cores do not have such in-plane periodicity and are macroscopically homogeneous. Thus, the foam-core sandwich beam was divided into several segments of a constant width a_seg based on the mechanical properties rather than the geometrical properties of the core and the facesheet. “Virtual” intersection lines were placed at the center of each segment. The segment width a_seg most significantly affects indentation damage initiation. Figure 13 presents a schematic of elastically deforming models just before the first segment below the loading point begins to be crushed. The left side of the figure is a model with small segments, and the right side is one with large segments, comparing the facesheet deformations at the damage initiation point. As described above, in the segment-wise model, the core failure of each segment is judged based on its displacement at the position of an intersection line within it. Thus, when the segment is too wide, damage initiation is delayed, as represented by the displacement gap at the loading point in Figure 13, and the simulation significantly overestimates the indentation load. When an indentation displacement α is applied and all of the segments behave elastically (i.e., before damage initiation), the facesheet deflection w(x) and the indentation load P are given from equations (1) and (3) by

w (x) = α e^{- \frac{x}{λ}} (\sin \frac{x}{λ} + \cos \frac{x}{λ})

(4)

P = 8 \frac{D_{f}}{λ^{3}} α

(5)

with

λ = \sqrt[4]{\frac{4 D_{f}}{k_{1}}}

(6)

where k₁ is the elastic foundation parameter of the core (equation (3)) [7,8,24]. Thus, the decay and oscillation of the facesheet deformation w(x) are governed by the ratio between the elastic stiffness of the core and the bending rigidity of the facesheet. Damage initiation load P_d, which depends on the segment width, was calculated in order to determine the appropriate segment width a_seg. From equations (4) and (5), the failure load P_d is given by

P_{d} = 8 \frac{D_{f}}{λ^{3}} α_{d} = 8 \frac{D_{f}}{λ^{3}} \frac{W_{2}}{e^{- \frac{0.5 a_{seg}}{λ}} (\sin \frac{0.5 a_{seg}}{λ} + \cos \frac{0.5 a_{seg}}{λ})}

(7)

where α_d is the indentation displacement at the damage initiation point and W₂ is the transverse displacement at which the core crushing begins (equation (3) and Figure 13). Figure 14 plots the failure load, which depends on the segment width. The segment width is normalized by the governing parameter λ, and the failure load is normalized by the load obtained from a model with segment width a_seg = λ. The normalized failure load decreased as the segment became narrower and ultimately converged to about 0.8. The decrease in the failure load almost stopped when the normalized segment width was smaller than 1/3. This convergence led us to decide that λ/3 is the upper limit of the appropriate segment width for accurately simulating indentation responses, as expressed in the following.

a_{seg} \leq \frac{λ}{3} = \frac{1}{3} \sqrt[4]{\frac{4 D_{f}}{k_{1}}}

(8)

The upper limit value for a sandwich specimen with [0₈] ([0₁₂]) facesheets was calculated to be 4.5 (5.9) mm, and this value was utilized for the indentation simulation discussed in the next section. As the authors' tests have indicated, the new segmentation method, based on the mechanical properties, worked well for several material combinations of the facesheet and the core because the method is derived from the governing equation of the facesheet deformation and thus possesses general versatility.

The indentation response for a given indentation displacement α can be calculated by determining P and all of the integral constants included in the general solutions obtained from equation (1), using boundary conditions between the segments. We can simulate indentation loading-unloading by renewing the foundation parameters as α increases or decreases. Full details of the simulation procedure are given in Ref. [24].

Results and discussion

Figure 15 presents the simulated indentation load-displacement curves. The load curves are smooth, confirming that the new segmentation method is feasible. The segment-wise model, which takes into consideration the detailed crushing/stretching characteristics of the foam core, reasonably reproduced the nonlinear load-displacement curves obtained in the experiment. The slight difference between the simulation and the experiment may be attributed to the shear stress distribution in the core and the viscoelastic behavior of the foam, which were not modeled in the segment-wise model. Next, the mechanism of facesheet dent formation is interpreted from the viewpoint of the reaction stress r(x) from the core to the facesheet. Figure 16 presents the calculated residual transverse stress distributions after loading-unloading of a 5-mm indentation. The severely damaged part of the core near the loading line was stretched from the crushed state, generating a tensile “pull-in” reaction stress of 0.2 MPa. This is the force that caused the residual facesheet dent after unloading. Furthermore, the damage boundary area, including the elastically-deformed undamaged part, supplied high compressive “push-up” reaction stress, since the pull-in force around the center of the specimen prevented the boundary area from being fully unloaded. This high compressive stress may serve as a source of additional damage growth under subsequent bending or in-plane compressive loading. The facesheet thickness affected the size of the damaged area. This is because a thicker facesheet with higher bending rigidity deforms more widely under the same indentation displacement, and core crushing occurs over a larger area. In contrast, the magnitude of the residual pull-in stress at the center of the specimen was almost the same regardless of the facesheet thickness. This implies that the residual facesheet dent depth of a specimen with thick and thus stiff facesheets (i.e., 12 plies) would be less than the dent depth of a thin-facesheet specimen. However, since the pull-in area was wider in the thick-facesheet specimen, almost the same dent depth remained after unloading at the loading point in both specimens (Figure 15). Furthermore, the thick-facesheet specimen had a wider push-up area, which may serve as a source of additional damage. This indicates that strength reduction due to localized damage may become more significant as the facesheet thickness increases.

Simulating a facesheet dent after relaxation

Relaxation characteristics of crushed foam core

Next, the segment-wise modeling approach was further extended to simulate facesheet-dent relaxation. As discussed in the previous section, the reaction force from the core to the facesheet controls the residual dent formation phenomenon, and the pull-in stress around the loading point induces a residual facesheet dent. Thus, the time change of the facesheet dent can be attributed to pull-in stress relaxation of the crushed foam core (Figure 17(a)). Figure 17(b) illustrates the reaction stress change during loading-unloading and subsequent relaxation at position A in Figure 17(a). It is important to note that, during facesheet dent relaxation, stress relaxation and deformation of the crushed core occur simultaneously because the stress relaxation induces upward deformation of the facesheet, stretching the crushed core. The reaction stress r(x) during relaxation depends on both the elapsed time after unloading and the deformation history. Hence, the facesheet deformation “during” relaxation is a highly complex phenomenon and will not be addressed in this study. However, we can calculate the facesheet dent profile w(x) and the reaction stress distribution r(x) “after” relaxation by modeling the stress-displacement relationships “after” relaxation (gray dashed line in Figure 17(b)) in the indentation unloading simulation.

In order to obtain the relaxed stress-displacement curves, we began by determining the test method. First, the deformation path dependency of the stress after relaxation was comprehensively investigated using flatwise compression-tension tests as described above. Figure 18 presents one of the obtained results, showing three different displacement time changes during relaxation following a series of induced stress changes. This test simulated a core being crushed to 5 mm displacement, stretched to 1 mm, and then relaxed for 1 h to 0 mm. Relaxation path A was the simplest, where the displacement rate was the same as in the crushing-stretching process (5 mm/mm), and the displacement was then stopped at 0 mm for 1 h. It is interesting to note that the stress values after relaxation were almost the same, even though the paths during relaxation totally differed. This was probably because, during all of the relaxation processes, additional damage was not induced in the crushed layers of the core except for the stretching of the collapsed cell edges, and so the consequent stress of the relaxed core was determined only by the final displacement. Thus, the path dependency during relaxation was negligible, and the relaxed stress-displacement curves could be obtained by using the simplest method, following relaxation path A. The implemented curves are presented in Figure 19. The stress-displacement curves without relaxation are also presented for comparison, confirming that the absolute value of the stress in the indentation simulation was significantly decreased by the relaxation.

Results and discussion

Figure 20 presents the calculated facesheet-dent profiles w(x). The experiment data were obtained by measuring the shape of the facesheets after the 1-h relaxation using a laser displacement meter (LK-030, Keyence Co., Ltd., sampling time 512 μs, spatial resolution 30 µm). The simulated dent shape after relaxation agreed well with the experiment in all of the specimens, confirming that the relaxation of the facesheet dent can be predicted by using stress-displacement curves obtained by a simplified relaxation test of the crushed foam core, neglecting the path dependency. Figure 21 compares the reaction stress distribution r(x) just after indentation unloading (i.e., before relaxation) with the distribution after relaxation (facesheet: [0₈], maximum indentation displacement: 5 mm). The length of the tensile pull-in stress area around the loading point was not changed by the relaxation. However, the pull-in stress significantly decreased to less than half of that just after unloading, and this induced significant relaxation of the facesheet dent (Figures 5 and 20). Furthermore, the compressive push-up stress in the damage boundary area, which may serve as a source of additional damage growth during subsequent loading, also decreased. This is because the pull-in stress, which prevented the boundary area from being fully unloaded, was relaxed, implying that the strength of the indented specimen was partly recovered by relaxation. Facesheet-dent relaxation makes it difficult to detect damage by visual inspection during structural maintenance, while it has a positive effect on the damage tolerance of foam-core sandwich structures.

Conclusions

This study clarified the underlying mechanism of facesheet-dent formation and relaxation by focusing on the interaction between local facesheet deformation and core crushing-stretching. First, the indentation response was evaluated using quasi-static indentation tests. A segment-wise modeling approach was then extended to simulate the indentation characteristics. It was confirmed that tensile “pull-in” reaction stress around the loading point induced a residual facesheet dent, and that the compressive “push-up” stress in the damage boundary area may serve as a source of additional damage growth during subsequent loading. Furthermore, the relaxation characteristics of the facesheet dent were predicted by considering the stress-relaxation effect of the foam core. Both the pull-in and push-up stresses were found to be significantly reduced after relaxation, indicating that facesheet-dent relaxation positively affects the damage tolerance of foam-core sandwich structures. In future work, it will be necessary to investigate the hygrothermal influence. In this study, all of the tests were conducted at room temperature. However, the mechanical properties and the relaxation of the foam exhibit significant temperature and humidity dependency, and thus future work needs to address these environmental effects on the indentation response.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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