Strength prediction of woven hybrid composite laminates each with a center hole

Abstract

In this study, the notched strengths of woven hybrid composite laminates each with a center circular hole were investigated. The effective engineering moduli of the composite laminate were first predicted by the spring model method and the stiffness averaging method. Then, a finite element-based point stress criterion proposed previously by the authors was employed to predict the notched strengths of those woven hybrid composite laminates each with a center circular hole. It is shown that the predicted notched strengths by this approach agree well with the experimental results. It is also noted that the notched strength decreases with the radius of the center circular holes for both the drilling and water jet cases. However, the decreasing rate of the notched strength for water jet is obviously higher than those for drilling.

Keywords

Woven hybrid composite laminate stress criterion characteristic length hole

Introduction

Composite materials have been widely applied in many kinds of industries, such as sport, construction, transportation and aerospace etc.,¹ because they are able to withstand the influence from acids, sodas and weather in addition to their high strength and light weight. However, in using structural composites, holes or cut-outs are often made to facilitate joining of structural parts or to provide access to the interior of the structure. These holes or cut-outs will produce stress concentrations, which can significantly reduce the strength and service life of the composite structures. Therefore, many fracture models have been established for predicting the notched strengths of composites.^2-13

Among the fracture models, the stress fracture models have been found to be used widely because of simplicity. In 1974, Whitney and Nuismer (WN)³ first proposed the point stress criterion (PSC) and average stress criterion (ASC) based on the normal stress distribution adjacent to the notch edge. These two parameter models were then modified into a three-parameter criterion by Pipes et al.⁴ Besides, Tan⁵ also extended the idea of the WN fracture model into the point strength model (PSM) and minimum strength model (MSM). To consider the effect of finite width, Tan⁶ proposed finite width correction factors, and Gillespie and Carlsson⁷ presented an approximate finite width stress distribution for orthotropic plates with circular holes.

Although the failure models mentioned above are mainly developed for composite laminates,^2-22 they have also been applied to predict the notched strength of fabric composites.^23-38 Traditionally, both PSC and ASC were only applied for predicting the notched strengths of composite plates each with a center hole because analytical solutions were required. In order to predict the notched strengths of composite plates each with notches located either at the edge or in the center of the plate, Tsai et al.³⁹ and Hwan et al.⁴⁰ have recently developed a finite element-based point stress criterion (FEBPSC), which can consistently predict the notched strength of composite plates without using analytical solutions.

Nowadays, woven hybrid composite laminates have been widely used in various industries because the woven fabric on their outer layer not only can improve their aesthetic appearance but also decrease the possibility of delamination during drilling. However, the notched strength of woven hybrid composite laminates each with a center hole has not been studied so far. In this paper, FEBPSC will be applied to investigate the problem. Moreover, the effect of machining method, such as drilling and water jet cutting, on the notched strengths and fracture mechanism of woven hybrid composite laminates each with a center hole will also be studied.

Theoretical aspects

In this study, despite many available methods for calculating engineering moduli,⁴¹ the effective engineering moduli of the composite laminate are first predicted by the spring model method⁴² and the stiffness averaging method.⁴³ The PSC is then used to find the characteristic length for each composite laminate with different size of hole by the finite element analysis. The characteristic length is then expressed as an empirical function of the hole size as well as the width of the composite laminate. The notched strengths of woven hybrid composite laminates are finally predicted based on the empirical function and the finite element analysis results incorporated with the principle of superposition in elasticity. Details in each step are described as follows.

Computation of equivalent elastic moduli of the woven hybrid composite laminates

The cross section of the woven hybrid composite laminates used in this study is shown in Figure 1. The stacking sequence is [w/ ± 45°/0°/ ± 45°/w] in which w stands for woven fabric layer. In the computation of the equivalent elastic moduli of the woven hybrid composite laminates, the equivalent elastic moduli of each woven fabric layer are first predicted by using spring model method.⁴²

Figure 1.

Cross section of the woven hybrid composite laminates.

In the spring model method,⁴² it is first hypothesized that solid element and spring element are used to simulate the matrix and the fiber tow, respectively, and the unit cell representing the composite material is selected. Next, the basic material properties of the matrix and the fiber are adjusted according to the mixture rule to develop a finite element model. Proper boundary conditions of six independent deformations are then assigned respectively to obtain the mean stresses and mean strains of the unit cell under various independent deformation conditions. Finally, using the generalized stress–strain relationship, an equivalent stiffness matrix or a compliance matrix of the unit cell is obtained. Various equivalent engineering constants are then acquired.

The equivalent elastic moduli of each unidirectional layer are then calculated by the rules of mixture. Finally, the overall equivalent elastic moduli of a woven hybrid composite laminate are evaluated by the stiffness averaging method, which is carried out by averaging the stiffness matrices of all layers with the weighting of their corresponding volume fraction.

Finite element based PSC

Among the two-stress criteria proposed by Whitney and Nuismer,³ PSC has been used most often because of its simplicity. As shown in Figure 2, consider a laminate containing a center hole of radius R is subjected to a uniform stress ${\bar{σ}}_{y}$ , in the PSC it was assumed that failure occurs when the stress $σ_{y}$ at some fixed distance d₀ ahead of the notch reaches the unnotched strength $σ_{0}$ of the laminate. Mathematically, it can be expressed as

PSC : σ_{y} (x, 0) |_{x = R + d_{0}} = σ_{0}

(1)

Figure 2.

A schematic illustration of Whitney-Nuismer point stress criterion.

The dependence of characteristic length d₀ on the specimen geometry has been proposed by Kim²⁹ as:

d_{0} = k - 1 (\frac{2 R}{W}) m

(2)

where k is the notch sensitivity factor concerning 2R and W, and m is an exponential parameter.

Traditionally, PSC was only applied for predicting the notched strengths of composite plates each with a center hole because analytical solutions were required. Recently, Tsai et al.³⁹ and Hwan et al.⁴⁰ have developed an FEBPSC for predicting the notched strength of composite plates each with notches located either at the edge or in the center of the plate.

In this paper, FEBPSC was employed for predicting the notched strength of woven hybrid composite laminates each with a center circular hole. In this approach, we first obtain the stress distribution of a woven hybrid composite laminate with a center circular hole by a finite element analysis, in which the experimental notched strength is applied as the loading at the boundary of the finite element model. We then apply the PSC to find the characteristic length for each plate with different size of hole by an interpolation of the finite element analysis results. Using Kim’s model,²⁹ the characteristic lengths of composite laminates are then expressed as an empirical function of hole size as well as width of the laminate. Finally, the characteristic lengths of composite laminates are predicted based on the empirical function, and the notched strengths of composite laminates are acquired by the predicted characteristic lengths and the finite element analysis results incorporated with the principle of superposition in elasticity.

Experiment

Woven fabrics and unidirectional composites were used to make woven hybrid composite laminates for investigation in this paper. As shown in Figure 1, the woven fabric layers in the woven hybrid composite laminates were fabricated by plain weave with T300 carbon fibers (3000 filaments), while the fiber types for UD(0°) and UD( ± 45°) are CN60 and HTA, respectively. The elastic moduli of the three fibers are depicted in Table 1. The longitudinal elastic moduli of three different fibers, T300, CN60 and HTA are provided by Toray industries Inc., Nippon Graphite Fiber Corp. and Toho Rayon Co., Ltd, respectively.

Table 1.

The elastic moduli of three fibers in the woven hybrid composite laminates.

T300	CN60	HTA
Axial modulus ( $E_{11}$ in GPa)	235	620	235
Transverse modulus ( $E_{22}$ in GPa)	16.7	16.7	16.7
Axial shear modulus ( $G_{12}$ in GPa)	24	24	24
Transverse shear modulus ( $G_{23}$ in GPa)	24	24	24
Axial Poisson’s ratio ( $ν_{12}$ )	0.2	0.2	0.2
Transverse Poisson’s ratio ( $ν_{23}$ )	0.2	0.2	0.2

After stacking prepregs in the mold, hot pressing was then employed to make the laminates which were cured at 130℃ for 40 min under pressure of 5 MPa. Resin content (RC), fiber volume fraction (V_f) and thickness of the corresponding layers in the woven hybrid composite laminates are listed in Table 2. The same epoxy resin was used in all layers with material properties shown in Table 3. Using both the spring model and the rule of mixture, the elastic moduli of woven layers and unidirectional layers are calculated and shown in Table 4. Then, employing the stiffness averaging method, the equivalent elastic moduli of woven hybrid composite laminates are shown in Table 5. Axial young’s modulus (E_y) and in-plane Poisson’s ratio (

ν_{yx}

) from experiment are also given in Table 5 for comparison.

Table 2.

Resin content, fiber volume fraction and thickness of each layer.

Woven fabric	UD (0°)	UD ( ± 45°)
RC (%)	45	33	42
V_f (%)	45.16	54.64	48.35
Thickness (mm)	0.48	2.08	0.53

Table 3.

Material properties of epoxy.

Young’s Modulus	3.50 GPa
Poisson Ratio	0.38

Table 4.

Predicted elastic moduli of woven layers and unidirectional layers.

E_y (GPa)	E_x (GPa)	$G_{xy}$ (GPa)	$ν_{yx}$
0° layer	271.06	6.10	2.61	0.34
$\pm 45 o$ layer	8.66	8.66	29.37	0.86
Woven layer	54.29	54.29	2.88	0.025

Table 5.

Predicted elastic moduli of woven hybrid composite laminates.

E_y(GPa)	E_x(GPa)	$G_{xy}$ (GPa)	$ν_{yx}$
Predicted value	227.53	18.17	7.30	0.344
Experiment	232	0.354
Error (%)	−1.93%	−2.91%

All specimens tailored from the laminates are 35-mm wide, 3-mm thick and 250-mm long with 160-mm gage length. A schematic diagram of the fabricated specimens is depicted in Figure 3. A circular hole with radius R is formed at the center of each specimen. In this study, specimens were cut with 3-, 4-, 5- and 6-mm radius circular holes by both drilling (spindle speed: 900 r/min) and water jet (water pressure: 310 MPa; abrasive mass flow rate: 6.45 kg/min). According to ASTM-D3039, all the tensile tests are conducted on the AG-100KNX machine using serrated wedge grips. The specimens are loaded in tension to failure at a crosshead speed of 2 $mm / min$ , and at room temperature. For each case, five identical specimens are tested and the results are averaged with standard deviation indicated. The unnotched strength of the specimens is 671.48 ± 16.93 MPa. The notched strengths of woven hybrid composite laminate specimens by experiment are listed in Table 6.

Figure 3.

Geometry and dimensions of specimens.

Table 6.

Notched strengths of woven hybrid composite laminate specimens.

Type of machining	R (mm)	2R/W	$σ_{N}$ (MPa)
Drilling	3	0.1714	363.18 ± 33.14
4	0.2286	313.93 ± 17.17
5	0.2857	295.73 ± 23.97
6	0.3429	282.18 ± 32.49
Water jet	3	0.1714	491.91 ± 26.54
4	0.2286	396.98 ± 10.18
5	0.2857	280.59 ± 11.07
6	0.3429	220.12 ± 14.27

Results and discussions

Prediction of notched strength

The corresponding characteristic lengths, d₀, to plates with different size of holes are acquired by interpolation of the finite element analysis results as mentioned in references 39 and 40. The results for those woven hybrid composite laminates with four different sizes of holes are listed in Table 7 in which d₀ is not constant and varies with the hole size for both the drilling and water jet cases. In order to evaluate the notched strength of the woven hybrid composite laminates more accurately, d₀ is expressed as a function of specimen geometry by the same exponential relationship as shown in equation (2). The values of k and m for those woven hybrid composite laminates are calculated and shown in Table 8.

Table 7.

Characteristic lengths of woven hybrid composite laminate specimens.

Type of machining	R (mm)	2R/W	d₀ (mm)
Drilling	3	0.1714	0.7487
4	0.2286	0.7221
5	0.2857	0.8219
6	0.3429	0.9393
Water jet	3	0.1714	1.7679
4	0.2286	1.314
5	0.2857	0.724
6	0.3429	0.5150

Table 8.

Values of k and m.

Type of machining	k	M
Drilling	0.7854	0.3308
Water jet	13.387	−1.8397

Using the abovementioned exponential relationship the predicted values of d₀ can be calculated. Then by the predicted values of d₀ the predicted notched strengths of those woven hybrid composite laminates are acquired by FEBPSC and shown in Table 9. It is observed that the maximum error between the experimental and predicted values of notched strength for all the woven hybrid composite laminates is about 5.57%. Therefore, we may say that the predicted values of notched strength by FEBPSC are very accurate compared with those experimental values.

Table 9.

Prediction of notched strengths by FEBPSC.

Type of machining	R (mm)	2R/W	Measured $σ_{N}$ (MPa)	FEBPSC $σ_{N}$ (MPa)	Relative error (%)
3	0.1714	363.2	355.8	−2.04
Drilling	4	0.2286	313.9	324.2	3.26
5	0.2857	295.7	298.6	0.96
6	0.3429	282.2	276.4	−2.05
3	0.1714	491.9	501.1	1.88
Water jet	4	0.2286	397.0	374.9	−5.57
5	0.2857	280.6	284.6	1.41
6	0.3429	220.1	223.6	1.58

FEBPSC: finite element-based point stress criterion.

Further, in order to depict the trend of notched strength versus hole radius, the predicted values of notched strength are acquired by FEBPSC for various hole radii from 2 mm to 8 mm for those woven hybrid composite laminates and the results are shown as diagrams in Figure 4. It is obvious that the notched strength decreases with the radius of the center circular holes for both the drilling and water jet cases. However, the decreasing rate of the notched strength for water jet is obviously higher than those for drilling.

Figure 4.

Diagram of notched strength versus hole radius for three composite plates.

Fracture mechanism

As shown in Figure 5, after the specimen is loaded and fractured, delamination between +45° and −45° layers was observed. In addition, fracture occurs in the lateral direction for the woven and 0° layers and along the fiber direction for the ±45° layers. To investigate the micro-damage of the notched specimen after the tensile test, one of the fractured specimens was sliced near the hole edge along two directions as shown in Figure 6 and then observed by a metalloscope. As shown in Figure 7, delamination occurs among all neighboring layers and matrix crack do exist clearly in the ±45° layers.

Figure 5.

A fractured specimen after tensile loading.

Figure 6.

A schematic illustration of the slice lines on the fractured specimen.

Figure 7.

Micro-damage in a fractured specimen (a) along slice line 1 (b) along slice line 2.

Conclusions

The notched strengths of those woven hybrid composite laminates each with a center circular hole are predicted by an FEBPSC. It is shown that the predicted notched strengths of woven hybrid composite laminates each with a center circular hole by this approach agree well with the experimental results. It is also noted that the notched strength decreases with the radius of the center circular holes for both the drilling and water jet cases. However, the decreasing rate of the notched strength for water jet is obviously higher than those for drilling.

Footnotes

Acknowledgements

This study was supported by the National Science Council of the Republic of China under the Contract NSC100-2622-E-035-010-CC3. The financial support is gratefully acknowledged.

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