Failure analysis in pure bending of single-face honeycomb core sandwich beams

Abstract

This study is focused on failure of single-face honeycomb core sandwich beams loaded in four-point flexure. The beams are configured for pure moment loading of a single-face sandwich (SFS) where the lower region of the core is under compression. Failure of the beam is assumed to occur when the maximum compressive strain in the core reaches its ultimate value in uniaxial compression. Analysis of the stiffness and failure load of the beam utilizes laminate beam theory. Constraint on the dominant bending deformation of core cell walls develop in honeycomb core sandwich structures because the cell walls are adhesively bonded to rigid face sheets. This constraint elevates the effective extensional modulus of the core. This effect is incorporated in the analysis by replacing part of the core with a gradient layer. Single-face sandwich beam specimens consisting of carbon/epoxy face sheets and Nomex honeycomb core were tested in flexure. Experiments and analysis demonstrate that the one-sided constraint on the core significantly increase the bending stiffness and bending strength of the single face sandwich beams. For a typical SFS specimen, it is shown that the constraint elevates the stiffness and strength by 13 and 16%, respectively. The gradient core model predictions of the failure load of the single-face specimens are in good agreement with experiments.

Keywords

single-face sandwich honeycomb core deformation constraints decay length end notch flexure (ENF) test mixed mode bending (MMB) test

Introduction

One serious failure mode of sandwich panels is the separation of face and core denoted as face/core debonding.^1,2 The presence of a face/core interface crack poses a serious threat to the integrity of sandwich structures because the debond allows for opening and sliding relative displacement of the crack faces, disruptions of shear flow and stiffness loss. Enabling safe design of sandwich structures thus requires characterization of face/core debonding. Several test procedures have been proposed for the characterization of face/core debonding failure.^3–5 A popular test used for mode II delamination testing of monolithic unidirectional composites⁶ called ENF test has been extended to sandwich specimen to characterize mode II face/core debonding.^7,8 A mixed mode I and II composite delamination specimen⁹ called MMB specimen has likewise been modified for face/core debond testing of sandwich specimens.¹⁰ The ENF and MMB sandwich tests consist of a beam with a face/core debond at the upper face/core interface at the end of the specimen. Loading of such a specimen in a fracture test cause opening and relative sliding displacements of the crack faces and intense strains and stresses at the crack tip which will propagate the precrack.

The region below the debond in these specimens is a single-face sandwich beam that supports the majority of the bending moment and the shear force in the end region. Determination of fracture toughness from ENF and MMB test results requires knowledge of the bending stiffness and strength of the end region. Traditional laminate approach¹¹ and finite element analysis¹² are suitable for the analysis of the mechanical response of sandwich beams with a foam core. For a sandwich with a honeycomb core, however, such analysis is inappropriate since it is assumed that the mechanical properties of the core are the same in-situ as isolated. The bonding of honeycomb core to stiff face sheets will constrain the dominant bending deformation of inclined core cell walls and prohibit the potentially large transverse deformation of the core. This constraint, to a very large extent, elevates the in-plane extensional modulus of the core in a region close to the face/core interface.^13–15 For the debonded end regions of the ENF and MMB specimens the constraint is one-sided. The significant influence of one-sided constraint on the elastic response, that is, the flexural stiffness of single-face honeycomb core sandwich was recently examined experimentally and analytically using a laminate beam gradient core model.^16,17

The objective of the current study is to examine flexural failure of single face sandwich beams and analyze the response of the specimens in terms of the uniaxial in-plane compressive behavior of isolated Nomex core studied earlier by the first two authors.^18,19 Four-point single-face sandwich specimens are prepared and loaded in pure bending until failure of the core occurs. The stiffness of the specimen is analyzed using laminate beam theory incorporating a core gradient model and the failure load is predicted from the maximum strain failure criterion. The experimentally measured stiffness and strength of the beams are compared to analytic predictions.

SFS flexure test specimen

Failure tests of single-face sandwich (SFS) specimens utilized the test specimen and loading configuration shown in Figure 1.

The specimen consists of a central region, BC, with a core bonded to a single face sheet on top, and two end regions, AB and CD, with face sheet only. When a vertical load, P, is applied, it is shared in two equal parts (P/2) acting at the ends, A and D. The central, SFS region BC, will be loaded under pure bending. The bending moment acting on the central region, BC, will load the lower region of the core in compression. This situation is representative for the loading of the end regions of the ENF and MMB sandwich specimens.^7,8,10

Stiffness analysis

In this section, laminate beam theory analysis^20,21 will be used to determine displacement and strain in the SFS specimen. A general case of a laminate beam consisting of N layers, Figure 2, loaded axially and in bending is first considered.

The thickness of the beam is denoted by h, and the width is b. The x coordinate is along the beam axis while the z coordinate is through-thickness with z = 0 at the geometric mid-plane of the beam. Each layer is indicated by a number, k, starting with k = 1 at the bottom and ending with k = N at the top.

Strain and stress develop in the layers. The strain is assumed to be linear

ε_{x} = ε_{x}^{0} + {z ϰ}_{x}

(1)

where z is the thickness coordinate,

ε_{x}^{0}

is the strain at the mid-plane (z = 0), and

ϰ_{x}

is the curvature of the mid-plane. The stress in a given layer, k, is given by

{(σ_{x})}_{k} = {(E_{x})}_{k} ε_{x}

(2)

where

{(E_{x})}_{k}

is the Young’s modulus of ply k in the x-direction. Integration of the stress through the thickness^20,21 provides the force and moment resultants per unit width,

N_{x}

and

M_{x}

N_{x} = A ε_{x}^{0} + B ϰ_{x}

(3a)

M_{x} = B ε_{x}^{0} + D ϰ_{x}

(3b)

where the stiffnesses A, B, and D are

A = \sum_{K = 1}^{N} {(E_{x})}_{k} (z_{k} - z_{k - 1})

(4a)

B = \frac{1}{2} \sum_{K = 1}^{N} {(E_{x})}_{k} (z_{k}^{2} - z_{k - 1}^{2})

(4b)

D = \frac{1}{3} \sum_{K = 1}^{N} {(E_{x})}_{k} (z_{k}^{3} - z_{k - 1}^{3})

(4c)

For pure bending

(N_{x} = 0),

equation (3a) gives

ε_{x}^{0} = - B ϰ_{x} / A

(5)

Substitution of equation (5) into (3b) gives the curvature-moment relation

ϰ_{x} = \frac{M_{x}}{D - B^{2} / A} = \frac{M_{x}}{D_{x}}

(6)

where

D_{x}

is the bending stiffness of the beam

D_{x} = D - \frac{B^{2}}{A}

(7)

Bending of SFS with constrained core

As discussed earlier, bonding of the core to a rigid face sheet will constrain the potentially large transverse in-plane deformation of the core. A gradient core model for bending stiffness of SFS specimens was developed recently.¹⁷ The results from this analysis will be briefly presented here.

Figure 3 illustrates the gradient model of the constrained core. The beam divided into three layers, that is, bottom region of unconstrained core (k = 1) where the core modulus $(E_{c})$ is unaffected by the constraint. To represent the constraint effect, a gradient layer (k = 2) of size $λ$ is introduced in the upper part of the core where the modulus increases linearly from a value $E_{c}$ to $k_{c} E_{c}$ , where $k_{c}$ is the amplification of core modulus due to the above discussed core deformation constraint, and the third top layer (k = 3) being the face sheet $(modulus E_{f})$ .

The layer moduli are

{k = 1 : E}_{1} = E_{c}

(8a)

{k = 2 : E}_{2} (z) = α + m z

(8b)

{k = 3 : E}_{3} = E_{f}

(8c)

where

E_{c}

is the unconstrained core modulus. The parameters m and α are given by

m = \frac{E_{c} (k_{c} - 1)}{λ}

(9a)

α = E_{c} - m (h / 2 - (h_{f} + λ))

(9b)

Notice that unconstrained core is represented by $k_{c} = 1$ , giving m = 0, and $α = E_{c} .$ Expressions for the stiffnesses A, B, and D for a single-face sandwich with constrained core were derived in Refs. 16,17 and are provided in Appendix A.

The constraint factor $k_{c}$ for a honeycomb core was determined using finite element analysis of a unit cell of constrained core.¹⁵ The size of the constrained region is quantified by a decay length, $λ$ . Sue and Whitcomb²² examined the core constraint effect in a honeycomb core sandwich element using finite element analysis. For their specific case they found that the constraint imposed by the face sheets on cell wall deformation decayed over a distance $λ = 5.3 m m$ away from the face/core interface. This value, however, is expected to depend on core cell dimensions and cell wall properties.

Analysis of beam deflection

Consider the SFS specimen shown in Figure 1. The displacement, $δ$ , of the loading points, A and D, is due to bending of the face sheet in regions AB and CD, and rotation of the ends B and C of the central, pure bending region, BC

δ = δ_{F} + δ_{R}

(10)

where

δ_{F}

and

δ_{R}

are the deflections due to bending and rotation. Treating the end regions as cantilever beams²³ yields

δ_{F} = \frac{P a^{3}}{6 {(D_{x})}_{f} b}

(11)

where a is the length of the end regions,

{(D_{x})}_{f}

is the bending stiffness of the face sheet and b is the width of the beam specimen, Figure 1. The deflection due to rotation of the cross section at B is

δ_{R} = \frac{d w}{d x} \cdot a

(12)

where the slope

, \frac{d w}{d x},

is given by¹⁶

\frac{d w}{d x} = \frac{M_{0} L}{2 {(D_{x})}_{S F S}}

(13)

M_{0}

is the moment per unit width acting on the central region (BC), L is the length of the central region, Figure 1, and

{(D_{x})}_{S F S}

is the bending stiffness of the single-face sandwich central region. The bending moment,

M_{0}

is given by

M_{0} = \frac{P a}{2 b}

(14)

Equations (10)–(14) give

δ = \frac{P a^{3}}{4 b} (\frac{2}{3 {(D_{X})}_{f}} + \frac{L}{a {(D_{X})}_{S F S}})

(15)

Failure analysis

Failure prediction of the SFS beam will utilize the maximum strain criterion.¹¹ Hence, failure of the SFS specimen in bending is assumed to occur when the maximum bending strain, $ε_{x},$ in the core reaches the failure strain $ε_{c}$ measured in uniaxial compression of the core

ε_{x} (- h / 2) = - ε_{c}

(16)

The region BC, Figure 1, is under constant moment loading. Combination of equations (1), (5), and (16) gives the curvature at failure

ϰ_{x}^{f} = \frac{ε_{c}}{B / A + h / 2}

(17)

The curvature, $ϰ_{x}$ , is related to the bending moment and bending stiffness ${(D_{x})}_{S F S}$ ²¹

ϰ_{x} = \frac{M_{x}}{{(D_{x})}_{S F S}}

(18)

Combining equations (14), (17), and (18) provides the failure load

P_{f} = \frac{2 b}{a} {(D_{x})}_{S F S} ϰ_{x}^{f}

(19)

with

ϰ_{x}^{f}

given by equation (17).

Experimental

In order to examine the failure mechanism of SFS specimens loaded in pure bending, SFS specimens were prepared from a sandwich panel obtained from Airbus, Hamburg. The sandwich panel consists of carbon/epoxy fabric face sheets of 0.7 mm thickness bonded to a 30 mm thick Nomex honeycomb core (4.8–32) with cell size of 4.8 mm and density of 32 kg/ $m^{3}$ . SFS specimens, Figure 1, were prepared by slicing the sandwich panel into 25.4 mm wide strips along the L-direction of the core and then cutting the strips to pieces of 130 mm length. To achieve the SFS flexure specimens, end sections of the core, approximately 27 mm long, were removed from each piece leaving a 76.4 mm long single-face region at the center. The desired core thicknesses ( $h_{c}$ = 21.6 and 25.4 mm) were achieved by slicing the $130 m m \times 25.4 m m$ sandwich blocks perpendicular to the thickness direction to remove part of the core and the lower face. A total of five SFS specimens were prepared, three with $h_{c} = 21.6 mm and two with h_{c} = 25.4 mm,$ all with the L-direction of the core along the beam axis.

The failure analysis is based on assumption of linear-elastic behavior and perfect bonding between the layers. Initial SFS tests revealed that the thin carbon/epoxy face sheet ( $h_{f}$ = 0.7 mm) could not support the loads required to cause core failure. Large deflections occurred followed by nonlinear response. Different methods to reinforce the face sheet, such as bonding of end tabs were tried. The end tabs, however, debonded before the SFS specimen failed in flexure. Adhesive bonding of a 0.2 mm thick stainless-steel layer was tried but did not provide sufficient reinforcement. It was found that bonding of a 0.4 mm stainless steel layer with epoxy adhesive worked satisfactory. Figure 4 shows a schematic of the layered structure of the central SFS region of the stainless-steel reinforced specimen.

The reinforced face sheet consists of the base 0.7 mm thick carbon/epoxy layer, the adhesive layer and the 0.4 mm stainless steel layer on the top. The face sheet thickness was measured by a digital caliper at the ends and the edges of the central SFS region. The accuracy of the measurements was hampered by core residues and adhesive on the face sheet. The results are listed in Table 1.

Table 1.

Face and core thicknesses of SFS specimen.

Specimen #	$h_{f},$ mm	$h_{c}$ , mm
1	1.5–1.9	21.6
2	1.7–2.5	21.6
3	1.5–2.6	21.6
4	2.1–3.4	25.4
5	2.4–3.0	25.4

Face thickness varies quite much for any given specimen and between specimens. The major factor contributing to the variation in face thickness is non-uniform application of adhesive.

SFS test results

The beam specimens (total of 5) considered were 130 mm long and 25.4 mm wide. The test geometry dimensions a and L in Figure 1 are a = 19.05 mm and L = 76.4 mm. The face and core thicknesses are listed in Table 1. The determination of the actual face thickness will be discussed below. Load-displacement curves for the SFS specimens are shown in Figure 5.

After an initial low stiffness region, the response curves are reasonably linear. The initial low stiffness is attributed to twisting of the face sheets, that is, deviation from flatness of the face sheet. After the low stiffness region, the response is linear, interrupted by a discontinuity, as a result of core failure. The failure load is indicated by a horizontal dashed line.

After core failure initiation, all specimens, except for specimen #3 (Figure 5), continued to support increasing load, but the tests were interrupted shortly after core failure was indicated. Figure 6 shows a photograph of specimen #3 loaded to the maximum load.

Inspection of the lower region of the core, Figure 6, where failure is expected to occur, reveals no obvious indication of broken cells. Observation of the core from below would probably reveal more detail on the deformation and failure of the core, but it was not possible for this set-up. To illustrate the failure mechanism of honeycomb core, a rectangular sample of Nomex 4.8–32 core was loaded in uniaxial in-plane compression. The in-plane compression failure mechanism of Nomex 4.8–32 core (L direction) is shown in Figure 7. For a slightly loaded core, Figure 7(a), it is observed that the inclined cell walls bend slightly. After loading the core close to compression failure, Figure 7(b), substantial bending of inclined cell walls is observed which transforms the core cells to assume a more rectangular shape. Further compressive loading revealed that the specimens continued to support load at a level close to the maximum load, see also.^18,19

For loading of the core in bending, failure in bending occurs when the stress in the outer layers of the core reaches the failure stress of the core ( $σ_{c})$ . As further bending deformation is imposed, the region of constant stress ( $σ_{c})$ will increase which corresponds to increased load after the failure point as seen for most specimens in Figure 5. On the other hand, if the core compressive failure was followed by collapse of the structure, a drop in the load versus displacement curves in Figure 5 would be expected.

Test results, that is, stiffness (slope of the linear region) and failure load for each specimen determined from Figure 5 are listed in Table 2.

Table 2.

Measured stiffness and failure load for SFS test specimens.

Specimen #	$h_{c}$ mm	Stiffness, N/mm	Failure load, N
1	21.6	39	250
2	21.6	100	500
3	21.6	66	410
4	25.4	89	490
5	25.4	87	510

The large variations in stiffness and failure load for specimens # 1–3 are attributed to the variation in the face sheet thickness, Table 1. For specimens with a 21.6 mm core, specimen# 2 has the highest stiffness and failure load which is due to the high thickness of the face sheet due to the adhesive layer.

Analysis of SFS test results

The experimental test results were analyzed using laminate beam theory outlined earlier. Specifically, the following equation for the beam compliance, obtained from equation (15), is utilized

\frac{δ}{P} = \frac{a^{3}}{4 b} (\frac{2}{3 {(D_{x})}_{f}} + \frac{L}{a {(D_{x})}_{S F S}})

(20)

Most parameters required for equation (20) are well defined, Table 3.

Table 3.

Layer thickness and modulus.^16,23,24

Layer	Thickness, mm	Modulus, MPa
HC-core	21.6 and 25.4	0.088
Carbon/epoxy	0.7	$8.89 \times 10^{3}$
Adhesive	Variable	$1 \times 10^{3}$
Stainless steel	0.4	$193 \times 10^{3}$

As indicated earlier, however, the thickness of the adhesive layer is uncertain, Table 1. The dimensions a, b, and L are easily determined. The bending stiffness of the face sheet ${(D_{x})}_{f}$ , however, depends strongly on the adhesive layer thickness, which is uncertain. The bending stiffness of the single-face region, ${(D_{x})}_{S F S}$ , similarly depends on the adhesive layer thickness, but also on the elevated core modulus due to the core deformation constraint, active over the region of size $λ$ , Figure 3. The core modulus amplification factor for the 4.8-32 core loaded in the L direction is $k_{c} = 350 [15] .$

As discussed earlier, the deformation of the honeycomb core bonded to a rigid face sheet is severely constrained which causes stiffening of the core, especially for in-plane extension.^13–15 The constraint effect decays over a certain distance from the face sheet, characterized by a decay length ( $λ$ ). Sue and Whitcomb²² conducted 3D finite element analysis of a honeycomb core sandwich element loaded in-plane. The constraint was examined by monitoring bending deflection of inclined cell walls in the core.²² Experimentally, however, such deformation is difficult to observe.

The decay length $λ$ is here determined indirectly. In a previous study of elastic behavior of SFS specimens, $λ$ was determined by matching the experimentally measured bending stiffness of the SFS specimen, ${(D_{x})}_{S F S}$ , with analytically calculated values.¹⁷ This procedure will be described shortly. For a SFS specimen with a 21.6 mm core, $λ = 7.5 m m$ ¹⁷.

A procedure similar to the one used in Ref. 17 was adopted in this study for the specimens with h_c = 25.4 mm (#4 and #5 in Table 2). Prior to bonding stainless steel reinforcement to the face sheet, the specimens #4 and #5 were tested in flexure using the same fixture and same dimensions (a, b, and L) as for the flexure tests described earlier in this paper (Figure 1). By using unreinforced specimens with a carbon/epoxy face sheet of known stiffness, ${(D_{x})}_{f} = 254 N m m$ , the determination of $λ$ should be accurate.²¹ P/δ, for these specimens were 3.65 and 3.81 N/mm. The average stiffness ( $3.73 \frac{N}{m m}$ ) and the face bending stiffness ${(D_{x})}_{f}$ were used to determine the bending stiffness of the single face region using equation (20), providing ${(D_{x})}_{S F S}$ = 3.05 $\times 10^{3}$ Nmm.

To determine $λ$ , a MATLAB²⁵ code was developed to calculate ${(D_{x})}_{S F S}$ for the unreinforced specimen based on the equations in Appendix A. MATLAB runs were conducted for $k_{c} = 350$ and a range of $λ$ from 0 to 10 mm in increments of 0.5 mm. For an unconstrained core $(λ = 0)$ , the analytical value of ${(D_{x})}_{S F S} = 755 Nmm$ , provides a stiffness (P/δ) much less than the measured stiffness $(3.05 \times 10^{3} N m m)$ . Agreement with the experimental stiffness for specimen #4 and #5 was achieved at a $λ$ value at 9.2 mm, slightly larger than that for the 21.6 mm thick core $(λ = 7.5$ mm).¹⁷

After establishing $λ$ values for the 21.6 and 25.4-mm thick cores, attention was turned to the test results from the specimens with stainless steel reinforced face sheets, Figure 4. Another MATLAB code²⁵ was developed for the reinforced SFS specimens, based on ply coordinates, $z_{k}$ , for the 5-layer configuration shown in Figure 8, and the stiffness expressions listed in Appendix B. In addition, a MATLAB code²⁴ was developed to calculate ${(D_{x})}_{f}$ for the three-layer stainless-steel reinforced face sheet, Figure 4.

Because of the uncertainty of the adhesive layer thickness, Table 1, the appropriate adhesive layer thickness was determined by matching experimental (Table 2) and analytical stiffness values (P/

δ

), equation (20). Parametric calculations of the bending stiffnesses for the face sheet and single-face sandwich,

{(D_{x})}_{f}

and

{(D_{x})}_{S F S}

, were done for a range of adhesive thickness,

h_{a d h}

, from 0 to 1.5 mm in increments of 0.1 mm. The values of

{(D_{x})}_{f}

and

{(D_{x})}_{S F S}

for each adhesive layer thickness were substituted in equation (20) to obtain the stiffness (P/

δ

h_{a d h}

was increased until a stiffness match was achieved. The results are listed in Table 4.

Table 4.

Calculated adhesive layer thickness, $h_{a d h}$ obtained from stiffness matching.

Specimen #	$h_{c}$ mm	$h_{a d h}$ mm	$h_{f}^{*}$ $m m$
1	21.6	0.7	1.8
2	21.6	1.4	2.5
3	21.6	1.1	2.2
4	25.4	1.4	2.6
5	25.4	1.4	2.6

{*) h}_{f} = h_{s s} + h_{a d h} + h_{c a r} = 1.1 + h_{a d h} (m m)

The corresponding face thicknesses values are also listed in Table 4. These values are within the range of measured face thicknesses listed in Table 1.

Experimental and predicted failure loads

With all key parameters for the SFS specimens established, it is now possible to predict the failure load of the SFS specimens. First, the curvature,

ϰ_{x}^{f}

, is calculated from equation (17) based on the calculated stiffness ratio,

\frac{B}{A}

, measured core failure strain and specimen thickness. The failure load,

P_{f}

, is then calculated from equation (19) using the calculated bending stiffness, (D_x)_SFS, and measured dimension, a and b. The measured compressive failure strain in the L direction of the 4.8-32 core is

ε_{c} = 0.19

¹⁸. Table 5 lists bending stiffness of the single face region, the failure curvature of the core, and predicted and measured failure loads.

Table 5.

Bending stiffness ${(D_{x})}_{S F S}$ , failure curvature ${(ϰ_{x}^{f})}_{f}$ , and measured and predicted failure loads ( $P_{f})$ .

Specimen #	$h_{f}$ $m m$	${(D_{x})}_{S F S} \times 10^{3}$ Nmm	$ϰ_{x}^{f} \times 10^{- 3} m m^{- 1}$	$P_{f} (N)$
Specimen #	$h_{f}$ $m m$	${(D_{x})}_{S F S} \times 10^{3}$ Nmm	$ϰ_{x}^{f} \times 10^{- 3} m m^{- 1}$	Meas.	Pred.
1	1.8	13.0	8.22	250	286
2	2.5	27.4	8.0	500	586
3	2.2	20.4	8.1	410	442
4	2.6	29.3	6.9	490	540
5	2.6	29.3	6.9	510	540

Comparison of measured and predicted failure loads reveal that the maximum strain criterion tends to overpredict the failure load, but the percentage difference is in all cases quite small providing confidence to the laminate beam model. Furthermore, bending strength tends in general to be higher than uniaxial strength due to a lower stressed volume in bending.^26,27 Notice also that the predicted curvature at failure is smaller for the specimens with 25.4 mm thick core than for the 21.6 mm thick core, as would be expected from equation (17).

The indirect determination of the adhesive layer thickness may lead to uncertainty in the prediction of the failure load from equation (19). The uncertainty of failure load prediction stems mostly from the uncertainty in the bending stiffness, (D_x)_SFS. For specimen # 2, a change of 0.1 mm of the adhesive thickness resulted in 0.6% change of the ratio $\frac{B}{A}$ , and 9.5% change of (D_x)_SFS and a 9.5% change of the failure load. Thus, the stiffness and failure load predictions are sensitive to changes (uncertainty) in the adhesive layer thickness. But the method for determination of adhesive layer thickness based on stiffness matching seems robust. The closeness between predicted and measured failure loads, Table 5, is further support for this approach.

Influence of core constraint on stiffness and failure load

To illustrate the influence of core constraint on stiffness and failure load, a SFS specimen with face sheet consisting of a 0.4 mm stainless steel layer, 1 mm thick adhesive layer, and a 0.7 mm carbon layer over a 21.6 mm honeycomb core was considered. This configuration is very similar to specimen #3 except for the adhesive layer thickness.

Calculations of stiffness, P/

δ

, curvature at failure, and failure load were done for both unconstrained core (

λ = 0)

and constrained core

(k_{c} = 350, λ = 7.5 m m)

as explained earlier. The results are listed in Table 6.

Table 6.

Influence of core constraint on stiffness curvature at failure and failure load of a SFS specimen ( $h_{s s}$ = 0.4 mm, $h_{a d h}$ = 1.0 mm, $h_{c}$ = 21.6 mm).

	$P / δ$ $N / m m$	$ϰ_{x}^{f}$ $m m^{- 1} \times 10^{- 3}$	P_f N
Unconstrained core	50.0	8.18	345
Constrained core	56.5	8.13	400

The results in Table 6 show that core constraints will elevate stiffness and strength, by 13 and 16%, respectively. The influence of constraint on failure curvature is only about 1%, but it should be recognized that the neutral axis locations $z_{N A} = \frac{B}{A}, are similar$ for the unconstrained and constrained cores, $z_{N A}$ = $11.5 and 11.4$ mm. The increased failure load for the constrained case is beneficial for ENF and MMB fracture testing, but the increased stiffness of the end region should be accounted for in the analysis of these tests.

Conclusion

Failure analysis of single-face honeycomb core sandwich beams loaded in four-point flexure has been presented. Single-face sandwich beams were tested under pure moment loading where the lower region of the core is under compression. Failure of the beam is assumed to occur when the maximum compressive strain in the core reaches its ultimate value in uniaxial compression. Analysis of the stiffness and failure load of the beam utilizes laminate beam theory. Constraints on deformation of core cell walls due to bonding of the cell walls to a rigid face sheet are incorporated by introducing a gradient region in the core. Single-face honeycomb core sandwich beam specimens with a range of face and core thicknesses were prepared. To find the size of the gradient region $(λ)$ unreinforced beams were first tested in flexure. The value of $λ$ was determined by matching calculations of bending stiffness with the gradient model to measure stiffness values. To allow failure testing, the face sheets of the specimens were reinforced with a stainless-steel layer. This introduced an adhesive layer with undetermined thickness. The adhesive layer thickness was determined indirectly by matching calculated and measured stiffness values of the SFS failure specimens.

Experiments and analysis demonstrate that the one-sided constraint on the core significantly increases the bending stiffness and bending strength of the single face sandwich beams. For a typical SFS specimen, it is shown that the constraint elevates the stiffness and strength by 13 and 16%, respectively. The gradient model predictions of the failure load of the single-face specimens were in good agreement with experiments.

Footnotes

Acknowledgments

Thanks are due to Mr Barry Milward of Eurocomposites for supplying Nomex honeycomb core panels and Ralf Hilgers of Airbus, Germany, for supply of honeycomb core sandwich panels. Thanks go to the Ocean and Mechanical Engineering Department of Florida Atlantic University for financial support. Typing by Mr Juan Pena is much appreciated.

ORCID iDs

Mustafa O. Ayanoglu

Leif A. Carlsson

Md Aasef Azhar Khan

Ethical considerations

This article does not contain any studies with human or animal participants.

Author contributions

Mustafa O. Ayanoglu—Conducted specimen preparation, experimental testing and data collection, wrote MATLAB codes. Leif A. Carlsson—Responsible for manuscript preparation and writing. Md Aasef Azhar Khan—Performed MATLAB-based data analysis and interpretation.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data Availability Statement

All relevant data are contained within the manuscript.*

Appendix

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