1. Introduction
Many modern technologies require materials with uncommon combinations of properties that cannot be achieved by a single material. The combinations and the range of material properties are constantly evolving with the development of composite materials. The aeronautic and aerospace industries have used an enormous amount of composite materials in their products, particularly laminated composites. However, these composites pose a challenge in devising methods to predict mechanical strength, primarily when they exhibit stress concentrations.
The laminated composite category includes a wide range of materials, from the most common place to the sophisticated. Glass, carbon and aramid fibers are among the most common reinforcement materials in these composites, and the aeronautic and aerospace industries have driven a large part of their development^{1}.
Within this category of laminated composites are sandwich structures, which consist of two faces of a highstrength laminate separated by a lowdensity core with a honeycomb structure. The function of these two faces in the sandwich composite is to resist tensile and compression stresses on its opposite faces when the panel is submitted to external bending forces. They are also thick enough to withstand buckling and failures caused by localized impacts^{2}. The sandwich laminated composite under study was manufactured with Fibrelam® panels (Hexcel Composites), consisting of two thin unidirectional layers of Eglass fiber impregnated with F155 epoxy resin containing fiber volume of 34% in the laminates and separated by a honeycomb core of phenolic resin, which exhibits low stiffness.
The most widely used methods for optimal preliminary design of laminates with stress concentrations are the Point Stress Criterion (PSC) and Average Stress Criterion (ASC) proposed by Whitney and Nuismer, or their variations^{3}^{}^{5}. The PSC assumes that failure occurs when the stress at a characteristic distance from the hole or notch edge reaches the tensile strength of an unnotched material. The ASC predicts failure when the average stress over a characteristic distance is equal to the unnotched strength. Recent research has reported several methods, such as the IFM and finite fracture mechanics (FFM) model^{5}^{}^{7}. However, all of these methods have been applied to study laminate composites with stress concentrations that underwent tensile strength testing^{8}. No study was found that used these methodologies to sandwichlaminated composites design, when submitted to bending forces.
This study conducted fourpoint bending testing in sandwich laminate plate samples with a central hole to determine bending resistance, modulus of elasticity in bending and flexural strain. After load behavior as a function of strain was determined for this bending test, the PSC, ASC and the IFM were used to study the applicability of these failure criteria.
Finally, in addition to the cases studied, these failure prediction models were modified in order to apply them in sandwich laminates with a central hole. The results obtained show the applicability of the proposed formulation in the study of laminates with stress concentrations submitted to bending.
2. Methodology
The composite was studied at the behest of the ALLTEC Company, located in São Jose dos Campos, São Paulo, Brazil. Two Fibrelam® sandwich laminated plates (Hexcel Composites) were acquired, with two unidirectional Eglass fiber laminas, impregnated with F155 epoxy resin, containing fiber volume of 34% (V_{f}) in the laminas and separated by a honeycomb core of phenolic resin (355 mm wide by 1835 mm long). This material is used in the construction of aircraft components such as floor and ceiling reinforcement.
The panels manufactured using an autoclave, exhibit excellent quality, with total (h) and core (c) thickness of 10.8 and 9.8 mm, respectively.
For the present study, ASTM standard code C393 (2016) was used as reference^{9}. A fourpoint bending tests were carried out, with and without a hole, as shown in Figure 1. The specimens were tested with the fibers parallel to their length (300 mm). The width (W) of the test specimens without a hole was 75 mm and those with a hole had the following combinations of width (W) and diameter (D), in mm: W75D16, W75D22, W85D14, W90D16, W90D22, W90D32 and W90D38 (Table 1). These combinations were selected in order to increase the region of experimental data analysis when related to the failure criteria studied here and to make better use of the material.
Nomenclature  Width (W)  Diameter (D) 

W75D16  75 mm  16 mm 
W75D22  75 mm  22 mm 
W85D14  85 mm  14 mm 
W90D16  90 mm  16 mm 
W90D22  90 mm  22 mm 
W90D32  90 mm  32 mm 
W90D38  90 mm  38 mm 
During the tests, a length of 250 mm between supports (S) was used and a distance of 83 mm (S/3) from the support to the load application lines (Figure 2). A total of 43 tests were conducted, with five samples for each width/diameter combination and eight samples for the laminate without a hole. All the tests were carried out in a Shimadzu universal testing machine with maximum capacity of 300 kN.
To calculate the flexural modulus (E_{1}) and bending stress on the face (σ) of the material, Equations 1 and 2 from ASTM C393 (2016) and ASTM 7249 (2016), respectively, were used^{9}^{,}^{10}. In these expressions, P is the applied load, S is the span, t is the ply thickness, c is the core thickness, and h is the total sandwich thickness. ε_{3000}, ε_{1000}, σ_{3000} and σ_{1000} are the strains and stresses corresponding to 3,000 and 1,000 microstrain, respectively. It is noteworthy that Equation 2 was used to calculate ultimate strength on the face of the material with (σ_{N}) and without a hole (σ_{0}).
In addition to the modulus of elasticity (E_{1}), Poisson’s ratio (v_{12}) was measured using rosette extensometers at 90°. Figure 2 demonstrates the test scheme and Table 2 the results obtained for the sandwich laminate without a hole. It is important to say that the sandwich composite behavior was linear throughout the test, both for the material without notch and for the notched material. In addition to the experimental values obtained (σ_{0}, E_{1}, v_{12}), Table 2 also presents the transverse modulus of elasticity (E_{2}), shear modulus (G_{12}) and the stress concentration factor for an infinite plate (
In Equation 3, P can be substituted by E_{2} or G_{12}, P_{m} is the value of the matrix modulus (E_{m}= 3.0 GPa, G_{m}= 1.3 GPa) and P_{f} is the value of the fiber modulus (E_{f}= 71 GPa, G_{f}= 29 GPa). The bestfit factor was ξ= 2 used to calculate E_{2} and was equal one to calculate G_{12}.
It is important to highlight that, despite the fact that Equations 4 to 16 were used to analyze failure behavior in composites under tension or compression (and not bending), these equations will be used here to determine their viability. The laminate under study is symmetrical and has only one thin layer of Eglass fiber in the outer layers, which simplifies analysis (in this layer the shear stresses can be considered nil and these lamina will only be submitted to tension or compression and the honeycomb stiffness can be neglected).
Furthermore, the failure obtained with a hole always occurred in the region with the smallest width and consequently greatest stress. None of the samples exhibited displacement between the laminae and honeycomb, or localized damage only in the honeycomb.
3. Analytical Prediction of Notched Strength
Loadings in an infinite plate were considered to create a failure criterion model in the presence of a stress gradient (in this case due to the presence of hole). In this situation, there is an infinite orthotropic plate containing a circular hole with radius R, submitted to uniform stress (σ^{∞}) applied perpendicularly to the xaxis, and longitudinal stress distribution (σ_{y}) along the xaxis ahead of the hole, expressed by Equation 5 ^{12}.
However, although stress analyses are conducted considering infinite plates, they cannot be performed in practice. Thus, an expression that serves as a correction factor for an infinite plate (FWC – finite width correction factor) was used, as demonstrated in Equation 6 and 7 ^{13}^{,}^{14}. Equation 8 relates stress in a finite (σ) and infinite plate (σ^{∞}).
There are other methods for calculating FWC besides the use of the analytical equation presented in Equation 6. TaheriBehrooz and Bakhshan^{15} for example, used the finite element method, applying the Progressive failure analysis technique, for calculating FWC for laminates manufactured with fiberglass woven.
3.1 Maximum stress criterion (MSC)
This criterion defines the failure threshold for material that is totally fragile and therefore insensitive to the hole, or totally ductile, and totally sensitive to stress concentration^{5}^{,}^{16}^{,}^{17}. Equations 9 and 10 represent these two situations, respectively.
D represents the diameter of the hole and W the width of the specimen, σ_{0} and σ_{N} the ultimate bending stress on the face of the material without and with a hole, respectively.
3.2 Inherent Flaw Model (IFM)
Based on linear elastic fracture mechanics^{18}^{}^{20}, this model assumes that there is a region of intense energy with characteristic length a, in which the critical stress intensity factor of an infinite plate (K_{IC}) with a hole whose radius is R, is given by Equation 11 ^{5}^{,}^{17}^{,}^{21}^{,}^{22}.
Where
Considering the situation in which the infinite plate has no hole (R= 0) and substituting this value in Equation 12, we have f(a,0) = 1, and applying this result in Equation 11, we have the result presented in Equation 13. In this situation, the ultimate stress in an infinite plate (
Equation 14 establishes the variation in the residual strength of the material in the presence of a hole in an infinite plate, and Equation 8 can be used to convert its values to a finite plate. The characteristic length a can be obtained interactively, relating the residual strength response obtained by Equation 14 with the residual resistance value obtained experimentally, and ASTM D5766^{24} serves as the basis for this analysis. When several tests are carried out with a change in radius (R) and width (W), the mean a value can be used.
3.3 Average Stress Criterion (ASC) and Point Stress Criterion (PSC)
Whitney and Nuismer (1974) proposed two failure criteria to predict stress at the hole of composite laminates containing circular holes^{3}. As mentioned in the introduction, these are known as the point stress criterion (PSC) and average stress criterion (ASC). The PSC estimates that failure occurs when the point stress found in Equation 5 is equal to the ultimate stress of the laminate with no hole at a fixed characteristic distance d_{0}. With respect to ASC, failure occurs when the average stress value is equal to the ultimate stress of the laminate with no hole at a characteristic distance a_{0}. The ASC and PSC can be represented by Equations 15 and 16, respectively.
As with IFM, the characteristic lengths a_{0} and d_{0} can be obtained interactively for each width (W) and hole diameter (D) combination. For the material investigated here, these values are presented in Table 3. As mentioned in item 2, the behavior obtained for the material was linear throughout the test, it is important to remember that if the material had a nonlinear behavior, the methodology for the numerical analysis would be different^{25}^{,}^{26}.
FWC  σ_{N}(MPa) 


d_{0}(mm)  a_{0}(mm)  a(mm)  

W75D16  1.02  152.65  156.36  0.644  3.01  12.12  6.90 
W75D22  1.05  127.87  134.08  0.552  3.06  10.76  6.50 
W85D14  1.01  173.95  176.41  0.726  3.62  16.53  8.75 
W90D16  1.02  178.27  181.22  0.746  4.60  21.65  11.22 
W90D22  1.03  167.27  172.71  0.711  5.68  25.85  13.71 
W90D32  1.07  133.95  144.04  0.593  5.52  21.54  12.48 
W90D38  1.11  118.99  132.52  0.546  5.75  21.01  12.54 
Average  4.46  18.50  10.3 
Based on these results, the graph in Figure 3 was created to compare failure methods MSC, IFM, ASC and PSC. The sandwich laminate submitted to bend testing showed a result within the failure region (assessed by the MSC method) and exhibited a behavior closer to the notchinsensitive curve (Equation 9) than that of the notchsensitive curve. Furthermore, the three methods are satisfactorily close to experimental data, where the ASC method displays the best correlation coefficient (85.22%), followed by the IFM and PSC methods, with 84.92% and 84.44%, respectively.
The average values obtained in Table 3 for each characteristic distance are very close to that observed for the W85D14 combination. This is important since ASTM D5766^{24} uses a D/W ratio of 0.166, which is similar to the ratio of the combination tested (for W85D14, D/W = 0.165). Thus, the result used in the ASTM standard can also be applied to the fourpoint bending test with a hole.
3.4 IFM, ASC and PSC models as threeparameter linear models
In order to improve the correlation between the aforementioned models and the experimental data, the threeparameter model was used to relate the characteristic length (d_{0}, a_{0} and a), represented by l, to an equation with K_{F} and m as constants to be obtained (Equation 17)^{27}^{}^{30}. These constants can be obtained using the NewtonRaphson method^{31}, and m values must lie between 0 and 1 for the result to be valid.
For the sandwich laminate studied here, this model was not adequate, since the value of m obtained by the three methods exhibited negative values. Thus, a model was created to improve the correlation between the failure criteria and experimental data, as demonstrated in the next item.
3.5 Modifying PSC, ASC and IFM methods
The results presented in Table 3 show that an increase in width always occurred with a rise in characteristic length, and that the ratio between thickness (h) and width (W) also influenced the residual bending strength of the material^{32}. As such, a linear equation that relates the characteristic length (d_{0}, a_{0} and a) represented by l with an h/W ratio, was proposed in Equation 18.
Where l is the characteristic length (d_{0}, a_{0} and a) that depends on the method used, and A and B are the fit constants between the method and experimental data. Table 4 shows the values of A and B for each case.
Based on these results, new σ_{N}/σ_{0} values can be calculated as a function of h/W for each criterion and the determined correlation coefficients can be verified, as shown in Table 5.




Experimental  PSC  ASC  IFM  
W75D16  0.628  0.618  0.611  0.612 
W75D22  0.526  0.520  0.542  0.535 
W85D14  0.716  0.748  0.721  0.733 
W90D16  0.734  0.745  0.721  0.732 
W90D22  0.689  0.665  0.657  0.658 
W90D32  0.551  0.546  0.568  0.558 
W90D38  0.490  0.485  0.520  0.507 
Correlation coefficient (%)  98.98  98.78  98.73 
The three methods showed more accurate results than those exhibited in the previous items, increasing the correlation coefficient from 83% to 98%. Furthermore, the PSC method produced the best result for this case.
It is important to emphasize that this method is not described in any of the articles researched^{3}^{,}^{6}^{,}^{33}^{}^{37}, and represents a new proposal in which the width and thickness of the analyzed specimens were considered.
Figures 4 6 were created to better analyze the curves formed by these three methods. In this case, rather than having a single curve for all the experimental values, we have a curve for each h/W ratio, thereby obtaining more accurate results.
4. Conclusions
Based on the results presented in previous topics, the following conclusions can be drawn:
The maximum stress criterion (MSC) showed that the material is within the region of notchsensitivity, but very close to the notchinsensitive curve.
The PSC, ASC and IFM methods were successfully applied in the present study to analyze the sandwich composite under bending. The three methods satisfactorily represent the material under study, but the ASC method exhibited better correlation between the equation and experimental data.
The average values of PSC, ASC and IFM characteristic length (d_{0}, a_{0} and a) was close to that of the W85D14 configuration, corresponding to the same D/W ratio in ASTM D5766, suggesting that this ratio can also be applied to the fourpoint bending test.
the m values obtained in the PSC, ASC and IFM models with three parameters were all negative, so these model cannot be applied to this material.
The relationship between width and characteristic length enabled the creation of an equation that better approximates the results of the three models (PSC, ASC and IFM) with the experimental data and increases the correlation coefficient from 83% to 98%.
In the present study, there was no delamination failure between the outer layers and their core during the bend test. The damage obtained in the samples with a hole always occurred in the region of greatest stress and smallest width.