For this investigation, four trapezoidal corrugated core sandwich panels were designed with fixed thickness of 4.0 mm and corrugation pitch of 12 mm as the test specimens. They were modeled as 3D elastic solid as shown in Fig. 1. The thickness of the face sheet and core sheet was set at 1 mm. Therefore, the skins have the thickness of 1 mm each and the core has the thickness of 2 mm. Engineering-grade thermoplastic ABS was chosen for the whole specimens. The type of specimens is tabulated in Table 1. The depth and length of the model were derived based on the ASTM D790 specification.²³

Fig. 1 
Specimen model LW1-45

Model’s core unit	Type of specimen	θ (°)	h (mm)	m (g)
	LW1-45	45	4.0	5.178
	LW1-60	60	4.0	5.224
	LW1-75	75	4.0	5.277
	LW1-90	90	4.0	5.342
	LW2-45	45	4.0	5.178
	LW2-60	60	4.0	5.224
	LW2-75	75	4.0	5.277
	LW2-90	90	4.0	5.342
	S4.0	-	4.0	6.748
	S3.1	-	3.1	5.230

Due to the corrugation construction as in Fig. 1, there are alternating thin and thick sides for each corrugation pitch. Notation LW1 and LW2 are assigned to the specimen where the loading nose is on the thin side and thick side, respectively.

Fig. 2 shows the three-point bending test setup for the study of the flexural characteristic of the panel. The loading nose and support radius parameters for ASTM D790 standardized normal, maximum and minimum test setup are listed in Table 2.

Fig. 2 
Test setup for trapezoidal corrugated core sandwich panel

A three-dimensional non-linear FEA was carried out using ABAQUS Explicit V6.11. The test setup was modeled as a quarter model with symmetrical planes normal to X- and Z-axis at the midpoint of the specimen. The specimens were meshed with solid element type of 8-node linear brick reduced integration (C3D8R), which had been applied in studies by Raj et al. and Magnucki et al..^7,24 Five layers of elements were chosen for the face sheets and core sheets to simulate the bending behavior appropriately. Therefore, there were at least 68,900 solid elements and 84,600 nodes to represent the sandwich panel specimens. No core debonding was considered.

Table 3 shows the material properties of the ABS that had being assigned as a homogenous isotropic material for the specimens. The loading nose and supports were modeled as 3D discrete rigid shell with quadrilateral elements. The loading nose was set to displace 5 mm against the Y-axis direction for the simulation. The supports were constrained for translation and rotation in all directions. The friction coefficient between the specimen, loading nose and supports is 0.38.

The maximum stress always occurs at the bottom midpoint element for the bending of a homogeneous solid specimen. However, this is not the case for the corrugated sandwich panel due to the existence of stress concentration area at the corrugation core. Therefore, the bending pattern of the core unit at mid span of the corrugated sandwich panels was investigated using the results of the FEM analysis, as shown in Figs. 3-6.

Fig. 3 
Maximum principal stress plot for specimens LW1 under normal test setup at d = 5 mm

Fig. 4 
Maximum principal stress plot for specimens LW1-45 under different test setups at d = 5 mm

Fig. 5 
Maximum principal stress plot for specimens LW2 under normal test setup at d = 5 mm

Fig. 6 
Maximum principal stress plot for specimens LW2-45 under different test setups at d = 5 mm

The stress plot in Fig. 3 shows various stress zones in the mid span core unit. The largest tensile stress zone (zone 1) occurs at the bottom surface of the specimen and it indicates the overall bending stress acting on the specimen. The bottom surface of the thin side of LW1 specimens is also in tension (zone 2) and this implies the occurrence of that local bending. High tension near the pin contact area (zone 3) supports the local bending. As the corrugation angle increases, tension at zones 2 and 3 decreases. Based on these observations, it can be concluded that the local bending effect is greater on smaller corrugation angle. This local bending pattern is a result of the bending load not being transferred directly to the lower thick side due to the lack of direct connection at the mid span. This is evidenced by the existence of two neutral axes at the mid span plane, as shown in Fig. 7.

Fig. 7 
Neutral axis for half specimens LW1 under normal setup

In addition, the result shows that maximum tensile stress changes location as the pin deflection increases. The highest tensile stress point is typically located at Point A. This critical stress point has been identified as the parting point between the thick side and thin side on the bottom face, which has significant difference in strength. This maximum stress point changes position to zone 2 when the effect of local bending becomes severe at higher load. When the displacement is approximately 5 mm, except for LW1-90, stress concentration at corrugation web edge (zone 4) is observed to be dominant.

Fig. 4 shows that zones 2 and 3 become smaller as the pin radius increases. Hence, it is deduced that the effect of local bending on the upper face is crucial for LW1 arrangement, particularly when the radius of the loading nose is relatively smaller than the groove width of corrugation. This happens because load from the nose contact is away from the corrugation web, resulting in the increase of local bending moment.

In Fig. 5, the tensile stress zones are uniformly located at the bottom of both thin (zone 5) and thick (zone 6) side faces. It can be observed that the thin side at the mid span is straightened instead of bended, because only tensile stress is acting at zone 5, resembling a tensile case. This is due to the neutral axis location at thick side of LW2, as shown in Fig. 8, and the direction of the corrugation web has diverted the compressive load away from the mid span thin face of LW2.

Fig. 8 
Neutral axis for half specimens LW2 under normal setup

There is no significant difference observed in stress distribution when the corrugation angle changes, except for the stress area at web edge in zone 6, which diminishes as the corrugation angle increases. This happens because the bending load is transferred from the top face to the bottom face through the corrugation web. As the corrugation angle increases from 45° to 90°, the downward shear component acting on the corrugation web edge reduces. Besides, the stress distribution does not vary at different test setup, as shown in Fig. 6. This indicates that the corrugation angle and the test setup have minimal influence on bending behavior of LW2.

Similar to the LW1, the highest tensile stress point is observed at Point B for most of the time throughout the bending. This critical stress point shifts from the bottom face towards the nearest corrugation web edge as the bending continues, forming a locus that represents the parting line between the thick side and thin side of the bottom face. When the local tension becomes greater, the highest tensile stress point is found in the middle of the thin side face of zone 5, where necking is expected.

For the corrugated sandwich panels, the actual neutral axes (as shown in Figs. 7 and 8, which are at the boundary between tension and compression obtained from the FEA analysis, reveal that they cannot be predicted using the theoretical neutral axis based on cross section, as shown in Fig. 9. The compressive shear on the top face is found to be higher for both LW1 and LW2. The bending load at the empty gap in the core is distributed horizontally through the upper face and reaches the bottom face via the corrugation web. Thus, at different corrugation angle and test position (LW1 & LW2), the neutral axes and shear load distribution are different too.

Fig. 9 
Theoretical neutral axis for LW1 and LW2

The force-displacement curves for the designed specimens under different test setups were plotted, as shown in Fig. 10. Fig. 10(a) shows that the specimen of different corrugation angle has different maximum load F capability, especially for LW1 specimens. The obvious difference in LW1 is due to the local bending effect, which is greater on smaller corrugation angle. Therefore, the maximum load decreases as the corrugation angle decreases in LW1 specimens. The local bending effect in LW2 is minimal as the corrugation angle changes. This explains the similarities of the bending characteristics between LW2 specimens.

Fig. 10 
Force-Displacement curves

Comparing LW1 specimens in Figs. 10(a)-10(c), it is noticed that the specimen is able to sustain higher bending load at maximum test setup compared to the normal and minimum test setup. The occurrence of local bending causes the bending load distribution from the loading nose to be ineffectively reflected on the overall bending of the specimen, which explains the lower bending load characteristics of LW1 shown in Figs. 10(a) and 10(c). In contrast, the specimens within the LW2 and solid groups show similar force-displacement relationship regardless of the test setup. This is consistent with the observation of similar stress distribution for LW2, as discussed previously.

When comparing the slope of the force-displacement curve in Figs. 10(a)-10(c), it is noticed that the flexural stiffness of the sandwich panels is approximately 13 - 18% lower than solid S4.0, but 74-84% better than solid S3.1. The LW2 group of specimens is 2-4% stiffer than the LW1 group of specimens. Due to the corrugated core, the load is not distributed equally across the cross section of the specimen. The thick side of the load face in LW2 is observed to accommodate more compressive load directly from the bending load, compared to the thin side of the load face of LW1, as shown in Figs. 7 and 8. Hence, LW2 specimens show better flexural stiffness compared to LW1 specimens.

The specific stiffness of the sandwich panels are 7-12% and 74-84% better than the solid S4.0 and S3.1, respectively, as shown in Fig. 11. This advantage is due to the lower apparent density. Specific stiffness for LW1 specimens is near constant. However, LW2 specimens have their specific stiffness reduced as the corrugation angle increases.

Fig. 11 
Specific stiffness for various specimens at normal setup

Using Eq. (2), the specific modulus was compared, as shown in Fig. 12. These sandwich panels have higher specific modulus compared to the solid panel S4.0 by 26-30%. The corrugation angle does not have significant influence on the flexural modulus. Hence, this specific modulus improvement is mainly due to their weight reduction of 20-24%. The solid S3.1 specimen, which has apparent density similar to LW1-60 and LW2-60 specimens, has similar specific modulus, but with 76% and 82% lower in stiffness, respectively.

Fig. 12 
Specific modulus for various specimens at normal setup

To investigate the maximum stroke that the specimen can sustain before breakage occurs, ABS tensile strength of 40 MPa and 6% elongation at break limit were applied. The maximum stroke is limited by either tensile strength limit or elongation at break limit, whichever lower, as shown in Fig. 13.

Fig. 13 
Maximum limit stroke

From Fig. 13, the maximum stroke increases as the corrugation angle increases, except for the case of LW1 under maximum test setup. When comparing between the LW1 and LW2 specimens, it is observed that the LW2 specimens have lower maximum stroke. This is due to the tension is acting locally on the thin side of LW2 during the bending test, as illustrated in Fig. 8. The tensile stress on the thin side of LW2 becomes dominant at maximum test setup due to the removal of the notch stress concentration effect of the loading nose. Therefore, for LW2 and solid specimens, the maximum strokes are observed to have a decreasing trend when the radius of the loading nose and support increases from minimum to maximum test setup.

Besides, it is also observed that the maximum stroke is limited by the tensile strength of the material, except for all LW1 specimens under minimum test setup and LW1 specimens at 45 to 75° corrugation angles under normal test setup. The local bending at the thin side causes the elongation at break limit to occur first, which explains why the limit stroke due to elongation at break limit happens at all LW1 specimens under minimum test setup and some LW1 specimens under normal test setup.